experimental gravity formula

13 Gravitation

13.7 einstein’s theory of gravity, learning objectives.

By the end of this section, you will be able to:

Describe how the theory of general relativity approaches gravitation
Explain the principle of equivalence
Calculate the Schwarzschild radius of an object
Summarize the evidence for black holes

Newton’s law of universal gravitation accurately predicts much of what we see within our solar system. Indeed, only Newton’s laws have been needed to accurately send every space vehicle on its journey. The paths of Earth-crossing asteroids, and most other celestial objects, can be accurately determined solely with Newton’s laws. Nevertheless, many phenomena have shown a discrepancy from what Newton’s laws predict, including the orbit of Mercury and the effect that gravity has on light. In this section, we examine a different way of envisioning gravitation.

A Revolution in Perspective

In 1905, Albert Einstein published his theory of special relativity. This theory is discussed in great detail in Relativity in the third volume of this text, so we say only a few words here. In this theory, no motion can exceed the speed of light—it is the speed limit of the Universe. This simple fact has been verified in countless experiments. However, it has incredible consequences—space and time are no longer absolute. Two people moving relative to one another do not agree on the length of objects or the passage of time. Almost all of the mechanics you learned in previous chapters, while remarkably accurate even for speeds of many thousands of miles per second, begin to fail when approaching the speed of light.

This speed limit on the Universe was also a challenge to the inherent assumption in Newton’s law of gravitation that gravity is an action-at-a-distance force . That is, without physical contact, any change in the position of one mass is instantly communicated to all other masses. This assumption does not come from any first principle, as Newton’s theory simply does not address the question. (The same was believed of electromagnetic forces, as well. It is fair to say that most scientists were not completely comfortable with the action-at-a-distance concept.)

A second assumption also appears in Newton’s law of gravitation (Equation) . The masses are assumed to be exactly the same as those used in Newton’s second law, [latex] \overset{\to }{F}=m\overset{\to }{a} [/latex]. We made that assumption in many of our derivations in this chapter. Again, there is no underlying principle that this must be, but experimental results are consistent with this assumption. In Einstein’s subsequent theory of general relativity (1916), both of these issues were addressed. His theory was a theory of space-time geometry and how mass (and acceleration) distort and interact with that space-time . It was not a theory of gravitational forces. The mathematics of the general theory is beyond the scope of this text, but we can look at some underlying principles and their consequences.

The Principle of Equivalence

Einstein came to his general theory in part by wondering why someone who was free falling did not feel his or her weight. Indeed, it is common to speak of astronauts orbiting Earth as being weightless, despite the fact that Earth’s gravity is still quite strong there. In Einstein’s general theory, there is no difference between free fall and being weightless. This is called the principle of equivalence . The equally surprising corollary to this is that there is no difference between a uniform gravitational field and a uniform acceleration in the absence of gravity. Let’s focus on this last statement. Although a perfectly uniform gravitational field is not feasible, we can approximate it very well.

Within a reasonably sized laboratory on Earth, the gravitational field [latex] \overset{\to }{g} [/latex] is essentially uniform. The corollary states that any physical experiments performed there have the identical results as those done in a laboratory accelerating at [latex] \overset{\to }{a}=\overset{\to }{g} [/latex] in deep space, well away from all other masses. (Figure) illustrates the concept.

On the left is a drawing of a rocket moving upward. An arrow pointing up is labeled a (=g). A view into the rocket shows a chemistry experiment and a clock indicating an interval of 10 minutes. On the right is a drawing of the earth with the same chemistry experiment and clock indicating an interval of 10 minutes at the surface of the earth. A downward arrow is labeled g.

Figure 13.28 According to the principle of equivalence, the results of all experiments performed in a laboratory in a uniform gravitational field are identical to the results of the same experiments performed in a uniformly accelerating laboratory.

How can these two apparently fundamentally different situations be the same? The answer is that gravitation is not a force between two objects but is the result of each object responding to the effect that the other has on the space-time surrounding it. A uniform gravitational field and a uniform acceleration have exactly the same effect on space-time.

A Geometric Theory of Gravity

Euclidian geometry assumes a “flat” space in which, among the most commonly known attributes, a straight line is the shortest distance between two points, the sum of the angles of all triangles must be 180 degrees, and parallel lines never intersect. Non-Euclidean geometry was not seriously investigated until the nineteenth century, so it is not surprising that Euclidean space is inherently assumed in all of Newton’s laws.

The general theory of relativity challenges this long-held assumption. Only empty space is flat. The presence of mass—or energy, since relativity does not distinguish between the two—distorts or curves space and time, or space-time, around it. The motion of any other mass is simply a response to this curved space-time. (Figure) is a two-dimensional representation of a smaller mass orbiting in response to the distorted space created by the presence of a larger mass. In a more precise but confusing picture, we would also see space distorted by the orbiting mass, and both masses would be in motion in response to the total distortion of space. Note that the figure is a representation to help visualize the concept. These are distortions in our three-dimensional space and time. We do not see them as we would a dimple on a ball. We see the distortion only by careful measurements of the motion of objects and light as they move through space.

An illustration of space time, shown as a grid. A large mass at the center of the grid distorts space time, forming a dimple and bending the grid lines. A small mass is shown orbiting the large mass at the rim of the dimple.

Figure 13.29 A smaller mass orbiting in the distorted space-time of a larger mass. In fact, all mass or energy distorts space-time.

For weak gravitational fields, the results of general relativity do not differ significantly from Newton’s law of gravitation. But for intense gravitational fields, the results diverge, and general relativity has been shown to predict the correct results. Even in our Sun’s relatively weak gravitational field at the distance of Mercury’s orbit, we can observe the effect. Starting in the mid-1800s, Mercury’s elliptical orbit has been carefully measured. However, although it is elliptical, its motion is complicated by the fact that the perihelion position of the ellipse slowly advances. Most of the advance is due to the gravitational pull of other planets, but a small portion of that advancement could not be accounted for by Newton’s law. At one time, there was even a search for a “companion” planet that would explain the discrepancy. But general relativity correctly predicts the measurements. Since then, many measurements, such as the deflection of light of distant objects by the Sun, have verified that general relativity correctly predicts the observations.

We close this discussion with one final comment. We have often referred to distortions of space-time or distortions in both space and time. In both special and general relativity, the dimension of time has equal footing with each spatial dimension (differing in its place in both theories only by an ultimately unimportant scaling factor). Near a very large mass, not only is the nearby space “stretched out,” but time is dilated or “slowed.” We discuss these effects more in the next section.

Black Holes

Einstein’s theory of gravitation is expressed in one deceptively simple-looking tensor equation (tensors are a generalization of scalars and vectors), which expresses how a mass determines the curvature of space-time around it. The solutions to that equation yield one of the most fascinating predictions: the black hole . The prediction is that if an object is sufficiently dense, it will collapse in upon itself and be surrounded by an event horizon from which nothing can escape. The name “black hole,” which was coined by astronomer John Wheeler in 1969, refers to the fact that light cannot escape such an object. Karl Schwarzschild was the first person to note this phenomenon in 1916, but at that time, it was considered mostly to be a mathematical curiosity.

Surprisingly, the idea of a massive body from which light cannot escape dates back to the late 1700s. Independently, John Michell and Pierre Simon Laplace used Newton’s law of gravitation to show that light leaving the surface of a star with enough mass could not escape. Their work was based on the fact that the speed of light had been measured by Ole Roemer in 1676. He noted discrepancies in the data for the orbital period of the moon Io about Jupiter. Roemer realized that the difference arose from the relative positions of Earth and Jupiter at different times and that he could find the speed of light from that difference. Michell and Laplace both realized that since light had a finite speed, there could be a star massive enough that the escape speed from its surface could exceed that speed. Hence, light always would fall back to the star. Oddly, observers far enough away from the very largest stars would not be able see them, yet they could see a smaller star from the same distance.

Recall that in Gravitational Potential Energy and Total Energy , we found that the escape speed, given by (Figure) , is independent of the mass of the object escaping. Even though the nature of light was not fully understood at the time, the mass of light, if it had any, was not relevant. Hence, (Figure) should be valid for light. Substituting c , the speed of light, for the escape velocity, we have

Thus, we only need values for R and M such that the escape velocity exceeds c , and then light will not be able to escape. Michell posited that if a star had the density of our Sun and a radius that extended just beyond the orbit of Mars, then light would not be able to escape from its surface. He also conjectured that we would still be able to detect such a star from the gravitational effect it would have on objects around it. This was an insightful conclusion, as this is precisely how we infer the existence of such objects today. While we have yet to, and may never, visit a black hole, the circumstantial evidence for them has become so compelling that few astronomers doubt their existence.

Before we examine some of that evidence, we turn our attention back to Schwarzschild’s solution to the tensor equation from general relativity. In that solution arises a critical radius, now called the Schwarzschild radius [latex] ({R}_{\text{S}}) [/latex]. For any mass M , if that mass were compressed to the extent that its radius becomes less than the Schwarzschild radius, then the mass will collapse to a singularity, and anything that passes inside that radius cannot escape. Once inside [latex] {R}_{\text{S}} [/latex], the arrow of time takes all things to the singularity. (In a broad mathematical sense, a singularity is where the value of a function goes to infinity. In this case, it is a point in space of zero volume with a finite mass. Hence, the mass density and gravitational energy become infinite.) The Schwarzschild radius is given by

If you look at our escape velocity equation with [latex] {v}_{\text{esc}}^{}=c [/latex], you will notice that it gives precisely this result. But that is merely a fortuitous accident caused by several incorrect assumptions. One of these assumptions is the use of the incorrect classical expression for the kinetic energy for light. Just how dense does an object have to be in order to turn into a black hole?

Calculating the Schwarzschild Radius

Calculate the Schwarzschild radius for both the Sun and Earth. Compare the density of the nucleus of an atom to the density required to compress Earth’s mass uniformly to its Schwarzschild radius. The density of a nucleus is about [latex] 2.3\,×\,{10}^{17}\,{\text{kg/m}}^{3} [/latex].

We use (Figure) for this calculation. We need only the masses of Earth and the Sun, which we obtain from the astronomical data given in Appendix D .

Substituting the mass of the Sun, we have

This is a diameter of only about 6 km. If we use the mass of Earth, we get [latex] {R}_{\text{S}}=8.85\,×\,{10}^{-3}\,\text{m} [/latex]. This is a diameter of less than 2 cm! If we pack Earth’s mass into a sphere with the radius [latex] {R}_{\text{S}}=8.85\,×\,{10}^{-3}\,\text{m} [/latex], we get a density of

Significance

A neutron star is the most compact object known—outside of a black hole itself. The neutron star is composed of neutrons, with the density of an atomic nucleus, and, like many black holes, is believed to be the remnant of a supernova—a star that explodes at the end of its lifetime. To create a black hole from Earth, we would have to compress it to a density thirteen orders of magnitude greater than that of a neutron star. This process would require unimaginable force. There is no known mechanism that could cause an Earth-sized object to become a black hole. For the Sun, you should be able to show that it would have to be compressed to a density only about 80 times that of a nucleus. (Note: Once the mass is compressed within its Schwarzschild radius, general relativity dictates that it will collapse to a singularity. These calculations merely show the density we must achieve to initiate that collapse.)

Check Your Understanding

Consider the density required to make Earth a black hole compared to that required for the Sun. What conclusion can you draw from this comparison about what would be required to create a black hole? Would you expect the Universe to have many black holes with small mass?

The event horizon

The Schwarzschild radius is also called the event horizon of a black hole. We noted that both space and time are stretched near massive objects, such as black holes. (Figure) illustrates that effect on space. The distortion caused by our Sun is actually quite small, and the diagram is exaggerated for clarity. Consider the neutron star, described in (Figure) . Although the distortion of space-time at the surface of a neutron star is very high, the radius is still larger than its Schwarzschild radius. Objects could still escape from its surface.

However, if a neutron star gains additional mass, it would eventually collapse, shrinking beyond the Schwarzschild radius. Once that happens, the entire mass would be pulled, inevitably, to a singularity. In the diagram, space is stretched to infinity. Time is also stretched to infinity. As objects fall toward the event horizon, we see them approaching ever more slowly, but never reaching the event horizon. As outside observers, we never see objects pass through the event horizon—effectively, time is stretched to a stop.

Visit this site to view an animated example of these spatial distortions.

On the left are three illustrations of space time as a grid with increasingly deep dimples with an object at the bottom of the dimple. The top drawing is labeled sun, and has a shallow dimple. The middle figure is labeled white dwarf and has a deeper dimple and more distorted grid lines. The third figure is labeled neutron star. The dimple is very deep and its sides are nearly vertical. The region above the star is labeled distorted space time. On the right is a larger illustration of the effects of a black hole. The dimple is now a bend that becomes a flared tube that becomes vertical and is open at the bottom. The bottom of the tube is labeled singularity. The grid lines in the tube form vertical lines and a spiral. A circular cross section of the tube is labeled event horizon. A circle where the space time grid bends to form the top of the tube is labeled last stable orbit.

Figure 13.30 The space distortion becomes more noticeable around increasingly larger masses. Once the mass density reaches a critical level, a black hole forms and the fabric of space-time is torn. The curvature of space is greatest at the surface of each of the first three objects shown and is finite. The curvature then decreases (not shown) to zero as you move to the center of the object. But the black hole is different. The curvature becomes infinite: The surface has collapsed to a singularity, and the cone extends to infinity. (Note: These diagrams are not to any scale.)

The evidence for black holes

Not until the 1960s, when the first neutron star was discovered, did interest in the existence of black holes become renewed. Evidence for black holes is based upon several types of observations, such as radiation analysis of X-ray binaries, gravitational lensing of the light from distant galaxies, and the motion of visible objects around invisible partners. We will focus on these later observations as they relate to what we have learned in this chapter. Although light cannot escape from a black hole for us to see, we can nevertheless see the gravitational effect of the black hole on surrounding masses.

An infrared image of stars near the center of the Milky way. Eight orbits are shown with several data points on each. The orbits differ in eccentricity, orientation, and size, but all overlap near the center of the image.

Figure 13.31 Paths of stars orbiting about a mass at the center of our Milky Way galaxy. From their motion, it is estimated that a black hole of about 4 million solar masses resides at the center. (credit: UCLA Galactic Center Group – W.M. Keck Observatory Laser Team)

The physics of stellar creation and evolution is well established. The ultimate source of energy that makes stars shine is the self-gravitational energy that triggers fusion. The general behavior is that the more massive a star, the brighter it shines and the shorter it lives. The logical inference is that a mass that is 4 million times the mass of our Sun, confined to a very small region, and that cannot be seen, has no viable interpretation other than a black hole. Extragalactic observations strongly suggest that black holes are common at the center of galaxies.

Visit the UCLA Galactic Center Group main page for information on X-ray binaries and gravitational lensing. Visit this page to view a three-dimensional visualization of the stars orbiting near the center of our galaxy, where the animation is near the bottom of the page.

Dark matter

Stars orbiting near the very heart of our galaxy provide strong evidence for a black hole there, but the orbits of stars far from the center suggest another intriguing phenomenon that is observed indirectly as well. Recall from Gravitation Near Earth’s Surface that we can consider the mass for spherical objects to be located at a point at the center for calculating their gravitational effects on other masses. Similarly, we can treat the total mass that lies within the orbit of any star in our galaxy as being located at the center of the Milky Way disc. We can estimate that mass from counting the visible stars and include in our estimate the mass of the black hole at the center as well.

But when we do that, we find the orbital speed of the stars is far too fast to be caused by that amount of matter. (Figure) shows the orbital velocities of stars as a function of their distance from the center of the Milky Way. The blue line represents the velocities we would expect from our estimates of the mass, whereas the green curve is what we get from direct measurements. Apparently, there is a lot of matter we don’t see, estimated to be about five times as much as what we do see, so it has been dubbed dark matter . Furthermore, the velocity profile does not follow what we expect from the observed distribution of visible stars. Not only is the estimate of the total mass inconsistent with the data, but the expected distribution is inconsistent as well. And this phenomenon is not restricted to our galaxy, but seems to be a feature of all galaxies. In fact, the issue was first noted in the 1930s when galaxies within clusters were measured to be orbiting about the center of mass of those clusters faster than they should based upon visible mass estimates.

Graph of Galaxy rotation curve plotting orbital velocity in arbitrary units as a function of radius, r, in kiloparsecs. The horizontal axis scale is 0 to 14 kiloparsecs, in increments of 2. The vertical axis scale is 0 to 1.6 in increments of 0.2. A green curve is labeled Observed. The curve starts at r=0, v=0.9, rises to almost v=1.4 at r a little less than 2, then decreases to about v = 1.3 at about r = 4, then more slowly to about v = 1.2 at r = 14. A blue curve is labeled Expected. The curve starts at r=0, v=1.0 ad rises to a maximum value that is smaller than the green curve’s and at a smaller value of r. The curve then decreases smoothly with steadily decreasing slope to v approximately 0.5 at r = 14.Three additional gray curves are also shown. A dotted curve labeled dark matter starts at r=0, v=0 and rises smoothly with steadily decreasing slope to v approximately 0.9 at r = 14. A dot-dashed curve labeled Bulge (light) also starts at r=0, v=0 and rises to a maximum value of about v = 0.5 at an r between 1 and 2, then decreases smoothly with steadily decreasing slope to v approximately 0.2 at r = 14. A dashed curve labeled Disk (light) starts at r=0, v=1 and decreases smoothly with steadily decreasing slope to v approximately 0.3 at r = 14.

Figure 13.32 The blue curve shows the expected orbital velocity of stars in the Milky Way based upon the visible stars we can see. The green curve shows that the actually velocities are higher, suggesting additional matter that cannot be seen. (credit: modification of work by Matthew Newby)

There are two prevailing ideas of what this matter could be—WIMPs and MACHOs. WIMPs stands for weakly interacting massive particles. These particles (neutrinos are one example) interact very weakly with ordinary matter and, hence, are very difficult to detect directly. MACHOs stands for massive compact halo objects, which are composed of ordinary baryonic matter, such as neutrons and protons. There are unresolved issues with both of these ideas, and far more research will be needed to solve the mystery.

According to the theory of general relativity, gravity is the result of distortions in space-time created by mass and energy.
The principle of equivalence states that that both mass and acceleration distort space-time and are indistinguishable in comparable circumstances.
Black holes, the result of gravitational collapse, are singularities with an event horizon that is proportional to their mass.
Evidence for the existence of black holes is still circumstantial, but the amount of that evidence is overwhelming.

Key Equations

Newton’s law of gravitation	[latex] {\overset{\to }{F}}_{12}=G\frac{{m}_{1}{m}_{2}}{{r}^{2}}{\hat{r}}_{12} [/latex]
Acceleration due to gravity at the surface of Earth	[latex] g=G\frac{{M}_{\text{E}}}{{r}_{}^{2}} [/latex]
Gravitational potential energy beyond Earth	[latex] U=-\frac{G{M}_{\text{E}}m}{r} [/latex]
Conservation of energy	[latex] \frac{1}{2}m{v}_{1}^{2}-\frac{GMm}{{r}_{1}}=\frac{1}{2}m{v}_{2}^{2}-\frac{GMm}{{r}_{2}} [/latex]
Escape velocity	[latex] {v}_{\text{esc}}=\sqrt{\frac{2GM}{R}} [/latex]
Orbital speed	[latex] {v}_{\text{orbit}}=\sqrt{\frac{{\text{GM}}_{\text{E}}}{r}} [/latex]
Orbital period	[latex] Τ=2\pi \sqrt{\frac{{r}^{3}}{{\text{GM}}_{\text{E}}}} [/latex]
Energy in circular orbit	[latex] E=K+U=-\frac{Gm{\text{M}}_{\text{E}}}{2r} [/latex]
Conic sections	[latex] \frac{\alpha }{r}=1+e\text{cos}\theta [/latex]
Kepler’s third law	[latex] {Τ}^{2}=\frac{4{\pi }^{2}}{GM}{a}^{3} [/latex]
Schwarzschild radius	[latex] {R}_{\text{S}}=\frac{2GM}{{c}^{2}} [/latex]

Conceptual Questions

The principle of equivalence states that all experiments done in a lab in a uniform gravitational field cannot be distinguished from those done in a lab that is not in a gravitational field but is uniformly accelerating. For the latter case, consider what happens to a laser beam at some height shot perfectly horizontally to the floor, across the accelerating lab. (View this from a nonaccelerating frame outside the lab.) Relative to the height of the laser, where will the laser beam hit the far wall? What does this say about the effect of a gravitational field on light? Does the fact that light has no mass make any difference to the argument?

The laser beam will hit the far wall at a lower elevation than it left, as the floor is accelerating upward. Relative to the lab, the laser beam “falls.” So we would expect this to happen in a gravitational field. The mass of light, or even an object with mass, is not relevant.

As a person approaches the Schwarzschild radius of a black hole, outside observers see all the processes of that person (their clocks, their heart rate, etc.) slowing down, and coming to a halt as they reach the Schwarzschild radius. (The person falling into the black hole sees their own processes unaffected.) But the speed of light is the same everywhere for all observers. What does this say about space as you approach the black hole?

What is the Schwarzschild radius for the black hole at the center of our galaxy if it has the mass of 4 million solar masses?

[latex] 1.19\,×\,{10}^{7}\text{km} [/latex]

What would be the Schwarzschild radius, in light years, if our Milky Way galaxy of 100 billion stars collapsed into a black hole? Compare this to our distance from the center, about 13,000 light years.

Additional Problems

A neutron star is a cold, collapsed star with nuclear density. A particular neutron star has a mass twice that of our Sun with a radius of 12.0 km. (a) What would be the weight of a 100-kg astronaut on standing on its surface? (b) What does this tell us about landing on a neutron star?

(a) How far from the center of Earth would the net gravitational force of Earth and the Moon on an object be zero? (b) Setting the magnitudes of the forces equal should result in two answers from the quadratic. Do you understand why there are two positions, but only one where the net force is zero?

How far from the center of the Sun would the net gravitational force of Earth and the Sun on a spaceship be zero?

[latex] 1.49\,×\,{10}^{8}\text{km} [/latex]

Calculate the values of g at Earth’s surface for the following changes in Earth’s properties: (a) its mass is doubled and its radius is halved; (b) its mass density is doubled and its radius is unchanged; (c) its mass density is halved and its mass is unchanged.

Suppose you can communicate with the inhabitants of a planet in another solar system. They tell you that on their planet, whose diameter and mass are [latex] 5.0\,×\,{10}^{3}\,\text{km} [/latex] and [latex] 3.6\,×\,{10}^{23}\,\text{kg} [/latex], respectively, the record for the high jump is 2.0 m. Given that this record is close to 2.4 m on Earth, what would you conclude about your extraterrestrial friends’ jumping ability?

The value of g for this planet is 2.4 m/s 2 , which is about one-fourth that of Earth. So they are weak high jumpers.

(a) Suppose that your measured weight at the equator is one-half your measured weight at the pole on a planet whose mass and diameter are equal to those of Earth. What is the rotational period of the planet? (b) Would you need to take the shape of this planet into account?

A body of mass 100 kg is weighed at the North Pole and at the equator with a spring scale. What is the scale reading at these two points? Assume that [latex] g=9.83\,{\text{m/s}}^{2} [/latex] at the pole.

At the North Pole, 983 N; at the equator, 980 N

Find the speed needed to escape from the solar system starting from the surface of Earth. Assume there are no other bodies involved and do not account for the fact that Earth is moving in its orbit. [ Hint: (Figure) does not apply. Use (Figure) and include the potential energy of both Earth and the Sun.

Consider the previous problem and include the fact that Earth has an orbital speed about the Sun of 29.8 km/s. (a) What speed relative to Earth would be needed and in what direction should you leave Earth? (b) What will be the shape of the trajectory?

a. The escape velocity is still 43.6 km/s. By launching from Earth in the direction of Earth’s tangential velocity, you need [latex] 43.4-29.8=13.8\,\text{km/s} [/latex] relative to Earth. b. The total energy is zero and the trajectory is a parabola.

A comet is observed 1.50 AU from the Sun with a speed of 24.3 km/s. Is this comet in a bound or unbound orbit?

An asteroid has speed 15.5 km/s when it is located 2.00 AU from the sun. At its closest approach, it is 0.400 AU from the Sun. What is its speed at that point?

Space debris left from old satellites and their launchers is becoming a hazard to other satellites. (a) Calculate the speed of a satellite in an orbit 900 km above Earth’s surface. (b) Suppose a loose rivet is in an orbit of the same radius that intersects the satellite’s orbit at an angle of [latex] 90\text{°} [/latex]. What is the velocity of the rivet relative to the satellite just before striking it? (c) If its mass is 0.500 g, and it comes to rest inside the satellite, how much energy in joules is generated by the collision? (Assume the satellite’s velocity does not change appreciably, because its mass is much greater than the rivet’s.)

A satellite of mass 1000 kg is in circular orbit about Earth. The radius of the orbit of the satellite is equal to two times the radius of Earth. (a) How far away is the satellite? (b) Find the kinetic, potential, and total energies of the satellite.

a. [latex] 1.3\,×\,{10}^{7}\,\text{m} [/latex]; b. [latex] 1.56\,×\,{10}^{10}\,\text{J} [/latex]; [latex] \text{−}3.12\,×\,{10}^{10}\,\text{J} [/latex]; [latex] -1.56\,×\,{10}^{10}\,\text{J} [/latex]

After Ceres was promoted to a dwarf planet, we now recognize the largest known asteroid to be Vesta, with a mass of [latex] 2.67\,×\,{10}^{20}\,\text{kg} [/latex] and a diameter ranging from 578 km to 458 km. Assuming that Vesta is spherical with radius 520 km, find the approximate escape velocity from its surface.

(a) Using the data in the previous problem for the asteroid Vesta, what would be the orbital period for a space probe in a circular orbit of 10.0 km from its surface? (b) Why is this calculation marginally useful at best?

a. [latex] 6.24\,×\,{10}^{3}\,\text{s} [/latex] or about 1.7 hours. This was using the 520 km average diameter. b. Vesta is clearly not very spherical, so you would need to be above the largest dimension, nearly 580 km. More importantly, the nonspherical nature would disturb the orbit very quickly, so this calculation would not be very accurate even for one orbit.

What is the orbital velocity of our solar system about the center of the Milky Way? Assume that the mass within a sphere of radius equal to our distance away from the center is about a 100 billion solar masses. Our distance from the center is 27,000 light years.

(a) Using the information in the previous problem, what velocity do you need to escape the Milky Way galaxy from our present position? (b) Would you need to accelerate a spaceship to this speed relative to Earth?

a. 323 km/s; b. No, you need only the difference between the solar system’s orbital speed and escape speed, so about [latex] 323-228=95\,\text{km/s} [/latex].

Circular orbits in (Figure) for conic sections must have eccentricity zero. From this, and using Newton’s second law applied to centripetal acceleration, show that the value of [latex] \alpha [/latex] in (Figure) is given by [latex] \alpha =\frac{{L}^{2}}{\text{G}M{m}^{2}} [/latex] where L is the angular momentum of the orbiting body. The value of [latex] \alpha [/latex] is constant and given by this expression regardless of the type of orbit.

Show that for eccentricity equal to one in (Figure) for conic sections, the path is a parabola. Do this by substituting Cartesian coordinates, x and y , for the polar coordinates, r and [latex] \theta [/latex], and showing that it has the general form for a parabola, [latex] x=a{y}^{2}+by+c [/latex].

Setting [latex] e=1 [/latex], we have [latex] \frac{\alpha }{r}=1+\text{cos}\theta \to \alpha =r+r\text{cos}\theta =r+x [/latex]; hence, [latex] {r}^{2}={x}^{2}+{y}^{2}={(\alpha -x)}^{2} [/latex]. Expand and collect to show [latex] x=\frac{1}{-2\alpha }\,{y}^{2}+\frac{\alpha }{2} [/latex].

Using the technique shown in Satellite Orbits and Energy , show that two masses [latex] {m}_{1} [/latex] and [latex] {m}_{2} [/latex] in circular orbits about their common center of mass, will have total energy [latex] E=K+E={K}_{1}+{K}_{2}-\frac{G{m}_{1}{m}_{2}}{{r}^{}}=-\frac{G{m}_{1}{m}_{2}}{2{r}^{}} [/latex]. We have shown the kinetic energy of both masses explicitly. ( Hint: The masses orbit at radii [latex] {r}_{1} [/latex] and [latex] {r}_{2} [/latex], respectively, where [latex] r={r}_{1}+{r}_{2} [/latex]. Be sure not to confuse the radius needed for centripetal acceleration with that for the gravitational force.)

Given the perihelion distance, p , and aphelion distance, q , for an elliptical orbit, show that the velocity at perihelion, [latex] {v}_{p} [/latex], is given by [latex] {v}_{p}=\sqrt{\frac{2G{M}_{\text{Sun}}}{(q+p)}\,\frac{q}{p}} [/latex]. ( Hint: Use conservation of angular momentum to relate [latex] {v}_{p} [/latex] and [latex] {v}_{q} [/latex], and then substitute into the conservation of energy equation.)

Substitute directly into the energy equation using [latex] p{v}_{p}=q{v}_{q} [/latex] from conservation of angular momentum, and solve for [latex] {v}_{p} [/latex].

Comet P/1999 R1 has a perihelion of 0.0570 AU and aphelion of 4.99 AU. Using the results of the previous problem, find its speed at aphelion. ( Hint: The expression is for the perihelion. Use symmetry to rewrite the expression for aphelion.)

Challenge Problems

A tunnel is dug through the center of a perfectly spherical and airless planet of radius R . Using the expression for g derived in Gravitation Near Earth’s Surface for a uniform density, show that a particle of mass m dropped in the tunnel will execute simple harmonic motion. Deduce the period of oscillation of m and show that it has the same period as an orbit at the surface.

[latex] g=\frac{4}{3}\,G\rho \pi r\to F=mg=[\frac{4}{3}\,Gm\rho \pi ]\,r [/latex], and from [latex] F=m\,\frac{{d}^{2}r}{d{t}^{2}} [/latex], we get [latex] \frac{{d}^{2}r}{d{t}^{2}}=[\frac{4}{3}\,G\rho \pi ]\,r [/latex] where the first term is [latex] {\omega }^{2} [/latex]. Then [latex] T=\frac{2\pi }{\omega }=2\pi \sqrt{\frac{3}{4G\rho \pi }} [/latex] and if we substitute [latex] \rho =\frac{M}{4\text{/}3\pi {R}^{3}} [/latex], we get the same expression as for the period of orbit R .

Following the technique used in Gravitation Near Earth’s Surface , find the value of g as a function of the radius r from the center of a spherical shell planet of constant density [latex] \rho [/latex] with inner and outer radii [latex] {R}_{\text{in}} [/latex] and [latex] {R}_{\text{out}} [/latex] . Find g for both [latex] {R}_{\text{in}}<r<{R}_{\text{out}} [/latex] and for [latex] r<{R}_{\text{in}} [/latex]. Assuming the inside of the shell is kept airless, describe travel inside the spherical shell planet.

Show that the areal velocity for a circular orbit of radius r about a mass M is [latex] \frac{\text{Δ}A}{\text{Δ}t}=\frac{1}{2}\sqrt{GMr} [/latex]. Does your expression give the correct value for Earth’s areal velocity about the Sun?

Using the mass of the Sun and Earth’s orbital radius, the equation gives [latex] 2.24\,×\,{10}^{15}{\text{m}}^{2}\text{/s} [/latex]. The value of [latex] \pi {R}_{\text{ES}}^{2}\text{/}(1\,\text{year}) [/latex] gives the same value.

Show that the period of orbit for two masses, [latex] {m}_{1} [/latex] and [latex] {m}_{2} [/latex], in circular orbits of radii [latex] {r}_{1} [/latex] and [latex] {r}_{2} [/latex], respectively, about their common center-of-mass, is given by [latex] T=2\pi \sqrt{\frac{{r}^{3}}{G({m}_{1}+{m}_{2})}}\,\text{where}\,r={r}_{1}+{r}_{2} [/latex]. ( Hint: The masses orbit at radii [latex] {r}_{1} [/latex] and [latex] {r}_{2} [/latex], respectively where [latex] r={r}_{1}+{r}_{2} [/latex]. Use the expression for the center-of-mass to relate the two radii and note that the two masses must have equal but opposite momenta. Start with the relationship of the period to the circumference and speed of orbit for one of the masses. Use the result of the previous problem using momenta in the expressions for the kinetic energy.)

Show that for small changes in height h , such that [latex] h\text{<}\,\text{<}{\text{R}}_{\text{E}} [/latex], (Figure) reduces to the expression [latex] \text{Δ}U=m\text{g}h [/latex].

[latex] \text{Δ}U={U}_{f}-{U}_{i}=-\frac{G{M}_{\text{E}}m}{{r}_{f}}+\frac{G{M}_{\text{E}}m}{{r}_{i}}=G{M}_{\text{E}}m(\frac{{r}_{f}-{r}_{i}}{{r}_{f}{r}_{i}}) [/latex] where [latex] h={r}_{f}-{r}_{i} [/latex]. If [latex] h\text{<}\,\text{<}{\text{R}}_{\text{E}} [/latex], then [latex] {r}_{f}{r}_{i}\approx {R}_{\text{E}}^{2} [/latex], and upon substitution, we have

[latex] \text{Δ}U=G{M}_{\text{E}}m(\frac{h}{{R}_{\text{E}}^{2}})=m(\frac{G{M}_{\text{E}}}{{R}_{\text{E}}^{2}})h [/latex] where we recognize the expression with the parenthesis as the definition of g .

Using (Figure) , carefully sketch a free body diagram for the case of a simple pendulum hanging at latitude lambda, labeling all forces acting on the point mass, m . Set up the equations of motion for equilibrium, setting one coordinate in the direction of the centripetal acceleration (toward P in the diagram), the other perpendicular to that. Show that the deflection angle [latex] \epsilon [/latex], defined as the angle between the pendulum string and the radial direction toward the center of Earth, is given by the expression below. What is the deflection angle at latitude 45 degrees? Assume that Earth is a perfect sphere. [latex] \text{tan}(\lambda +\epsilon )=\frac{g}{(g-{\omega }^{2}{R}_{\text{E}})}\text{tan}\lambda [/latex], where [latex] \omega [/latex] is the angular velocity of Earth.

(a) Show that tidal force on a small object of mass m , defined as the difference in the gravitational force that would be exerted on m at a distance at the near and the far side of the object, due to the gravitation at a distance R from M , is given by [latex] {F}_{\text{tidal}}=\frac{2GMm}{{R}^{3}}\text{Δ}r [/latex] where [latex] \text{Δ}r [/latex] is the distance between the near and far side and [latex] \text{Δ}r\text{<}\,\text{<}R [/latex]. (b) Assume you are falling feet first into the black hole at the center of our galaxy. It has mass of 4 million solar masses. What would be the difference between the force at your head and your feet at the Schwarzschild radius (event horizon)? Assume your feet and head each have mass 5.0 kg and are 2.0 m apart. Would you survive passing through the event horizon?

a. Find the difference in force,

[latex] {F}_{\text{tidal}}==\frac{2GMm}{{R}^{3}}\text{Δ}r [/latex];

b. For the case given, using the Schwarzschild radius from a previous problem, we have a tidal force of [latex] 9.5\,×\,{10}^{-3}\,\text{N} [/latex]. This won’t even be noticed!

Find the Hohmann transfer velocities, [latex] \text{Δ}{v}_{\text{EllipseEarth}}^{} [/latex] and [latex] \text{Δ}{v}_{\text{EllipseMars}}^{} [/latex], needed for a trip to Mars. Use (Figure) to find the circular orbital velocities for Earth and Mars. Using (Figure) and the total energy of the ellipse (with semi-major axis a ), given by [latex] E=-\frac{Gm{M}_{\text{s}}}{2{a}^{}} [/latex], find the velocities at Earth (perihelion) and at Mars (aphelion) required to be on the transfer ellipse. The difference, [latex] \text{Δ}v [/latex], at each point is the velocity boost or transfer velocity needed.

OpenStax University Physics. Authored by : OpenStax CNX. Located at : https://cnx.org/contents/[email protected]:Gofkr9Oy@15 . License : CC BY: Attribution . License Terms : Download for free at http://cnx.org/contents/[email protected]

Physics Formulas

Gravity Formula

We know gravity is the fundamental force and is the elemental concept of Physics. Gravity or gravitational force is the contribution of the great scientist Sir Isaac Newton. We know that dropping an apple from a tree was the reason for the inception of the Law of Universal Gravitation. In this session, let us learn about the gravity formula and its application.

Gravity or Gravitation Formula

Gravity also termed as gravitation, is a force that occurs among all material objects in the universe. For any two objects or units having non-zero mass, the force of gravity has a tendency to attract them toward each other. Newton’s Law of Universal Gravitation states that :

“Every particle attracts every other particle in the universe with force directly proportional to the product of the masses and inversely proportional to the square of the distance between them”.

If the distance between two masses m 1 and m 2 is d, then the gravity formula is articulated as:

$Gravity formula$

G is a constant equal to 6.67 × 10 -11 N-m 2 /kg 2
m 1 is the mass of the body 1
m 2 is the mass of body 2
r is the radius or distance between the two bodies

The gravitational force formula is very useful in computing gravity values, larger mass, larger radius, etc.

Gravity Problems Solved Examples

Underneath are given some questions on gravity which helps one to comprehend the use of this formula.

Problem 1: Calculate the force due to gravitation being applied on two objects of mass 2 Kg and 5 Kg divided by the distance 5cm? Answer:

Given: Mass m 1 = 2 Kg, Mass m 2 = 5 Kg, Radius r = 5 cm. Gravitational Constant G = 6.67 ×× 10 -11 Nm 2 /Kg 2

$Gravity formula1$

From Newton’s law of gravitation, we know that the force of attraction between the bodies is given by $\begin{array}{l}F=\frac{Gm_{1}m_{2}}{r^{2}}\end{array} $ Where,

m1= $\begin{array}{l}6*10^{24}\end{array} $ kg
r = $\begin{array}{l}6.4*10^{6}\end{array} $

Substituting in the above equation, $\begin{array}{l}F=\frac{6.67*10^{11}(6*10^{24})*1}{(6.4*10^{6})^{2}}\end{array} $ =9.8 N This shows that the earth exerts force 9.8 N on a body of mass of 1 Kg.

See the video below, to understand what is gravitation and gravity formula explanation.

Hope you learned the gravity formula along with the law of universal gravitation. For more such valuable equations and formulas stay tuned with BYJU’S!!

Frequently Asked Questions – FAQs

Who put forth the law of universal gravitation.

The Law of Universal Gravitation was put forth by Sir Isaac Newton.

State law of universal gravitation.

The law of universal gravitation states that:

Write gravity formula.

Where, G is a constant equal to 6.67 × 10-11 N-m2/kg2 m1 is the mass of the body 1 m2 is the mass of body 2 r is the radius or distance between the two bodies

What is the value of the gravitational constant?

State true or false: gravity is a force that occurs among all material objects in the universe..

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Newton’s law of gravity

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West Texas A and M University - Science Questions with Surprising Answers - Why is there no gravity in space?
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Table Of Contents

Recent News

The Moon’s orbit has a radius of about 384,000 km (239,000 miles; approximately 60 Earth radii), and its period is 27.3 days (its synodic period , or period measured in terms of lunar phases, is about 29.5 days). Newton found the Moon’s inward acceleration in its orbit to be 0.0027 metre per second per second, the same as (1/60) 2 of the acceleration of a falling object at the surface of Earth.

In Newton’s theory every least particle of matter attracts every other particle gravitationally, and on that basis he showed that the attraction of a finite body with spherical symmetry is the same as that of the whole mass at the centre of the body. More generally, the attraction of any body at a sufficiently great distance is equal to that of the whole mass at the centre of mass. He could thus relate the two accelerations, that of the Moon and that of a body falling freely on Earth, to a common interaction, a gravitational force between bodies that diminishes as the inverse square of the distance between them. Thus, if the distance between the bodies is doubled, the force on them is reduced to a fourth of the original.

The constant G is a quantity with the physical dimensions (length) 3 /(mass)(time) 2 ; its numerical value depends on the physical units of length, mass, and time used. ( G is discussed more fully in subsequent sections.)

The weight W of a body can be measured by the equal and opposite force necessary to prevent the downward acceleration; that is M g . The same body placed on the surface of the Moon has the same mass, but, as the Moon has a mass of about 1 / 81 times that of Earth and a radius of just 0.27 that of Earth, the body on the lunar surface has a weight of only 1 / 6 its Earth weight, as the Apollo program astronauts demonstrated. Passengers and instruments in orbiting satellites are in free fall. They experience weightless conditions even though their masses remain the same as on Earth.

Kepler’s very important second law depends only on the fact that the force between two bodies is along the line joining them.

The motions of the moons of Jupiter (discovered by Galileo) around Jupiter obey Kepler’s laws just as the planets do around the Sun. Thus, Newton calculated that Jupiter, with a radius 11 times larger than Earth’s, was 318 times more massive than Earth but only 1 / 4 as dense .

13.7 Einstein's Theory of Gravity

Learning objectives.

By the end of this section, you will be able to:

Describe how the theory of general relativity approaches gravitation
Explain the principle of equivalence
Calculate the Schwarzschild radius of an object
Summarize the evidence for black holes

A Revolution in Perspective

In 1905, Albert Einstein published his theory of special relativity. This theory is discussed in great detail in Relativity , so we say only a few words here. In this theory, no motion can exceed the speed of light—it is the speed limit of the Universe. This simple fact has been verified in countless experiments. However, it has incredible consequences—space and time are no longer absolute. Two people moving relative to one another do not agree on the length of objects or the passage of time. Almost all of the mechanics you learned in previous chapters, while remarkably accurate even for speeds of many thousands of miles per second, begin to fail when approaching the speed of light.

A second assumption also appears in Newton’s law of gravitation Equation 13.1 . The masses are assumed to be exactly the same as those used in Newton’s second law, F → = m a → F → = m a → . We made that assumption in many of our derivations in this chapter. Again, there is no underlying principle that this must be, but experimental results are consistent with this assumption. In Einstein’s subsequent theory of general relativity (1916), both of these issues were addressed. His theory was a theory of space-time geometry and how mass (and acceleration) distort and interact with that space-time. It was not a theory of gravitational forces. The mathematics of the general theory is beyond the scope of this text, but we can look at some underlying principles and their consequences.

The Principle of Equivalence

Einstein came to his general theory in part by wondering why someone who was free falling did not feel their weight. Indeed, it is common to speak of astronauts orbiting Earth as being weightless, despite the fact that Earth’s gravity is still quite strong there. In Einstein’s general theory, there is no difference between free fall and being weightless. This is called the principle of equivalence . The equally surprising corollary to this is that there is no difference between a uniform gravitational field and a uniform acceleration in the absence of gravity. Let’s focus on this last statement. Although a perfectly uniform gravitational field is not feasible, we can approximate it very well.

Within a reasonably sized laboratory on Earth, the gravitational field g → g → is essentially uniform. The corollary states that any physical experiments performed there have the identical results as those done in a laboratory accelerating at a → = g → a → = g → in deep space, well away from all other masses. Figure 13.28 illustrates the concept.

A Geometric Theory of Gravity

The general theory of relativity challenges this long-held assumption. Only empty space is flat. The presence of mass—or energy, since relativity does not distinguish between the two—distorts or curves space and time, or space-time, around it. The motion of any other mass is simply a response to this curved space-time. Figure 13.29 is a two-dimensional representation of a smaller mass orbiting in response to the distorted space created by the presence of a larger mass. In a more precise but confusing picture, we would also see space distorted by the orbiting mass, and both masses would be in motion in response to the total distortion of space. Note that the figure is a representation to help visualize the concept. These are distortions in our three-dimensional space and time. We do not see them as we would a dimple on a ball. We see the distortion only by careful measurements of the motion of objects and light as they move through space.

Black Holes

Surprisingly, the idea of a massive body from which light cannot escape dates back to the late 1700s. Independently, John Michell and Pierre-Simon Laplace used Newton’s law of gravitation to show that light leaving the surface of a star with enough mass could not escape. Their work was based on the fact that the speed of light had been measured by Ole Rømer in 1676. He noted discrepancies in the data for the orbital period of the moon Io about Jupiter. Rømer realized that the difference arose from the relative positions of Earth and Jupiter at different times and that he could find the speed of light from that difference. Michell and Laplace both realized that since light had a finite speed, there could be a star massive enough that the escape speed from its surface could exceed that speed. Hence, light always would fall back to the star. Oddly, observers far enough away from the very largest stars would not be able to see them, yet they could see a smaller star from the same distance.

Recall that in Gravitational Potential Energy and Total Energy , we found that the escape speed, given by Equation 13.6 , is independent of the mass of the object escaping. Even though the nature of light was not fully understood at the time, the mass of light, if it had any, was not relevant. Hence, Equation 13.6 should be valid for light. Substituting c , the speed of light, for the escape velocity, we have

Before we examine some of that evidence, we turn our attention back to Schwarzschild’s solution to the tensor equation from general relativity. In that solution arises a critical radius, now called the Schwarzschild radius ( R S ) ( R S ) . For any mass M , if that mass were compressed to the extent that its radius becomes less than the Schwarzschild radius, then the mass will collapse to a singularity, and anything that passes inside that radius cannot escape. Once inside R S R S , the arrow of time takes all things to the singularity. (In a broad mathematical sense, a singularity is where the value of a function goes to infinity. In this case, it is a point in space of zero volume with a finite mass. Hence, the mass density and gravitational energy become infinite.) The Schwarzschild radius is given by

If you look at our escape velocity equation with v esc = c v esc = c , you will notice that it gives precisely this result. But that is merely a fortuitous accident caused by several incorrect assumptions. One of these assumptions is the use of the incorrect classical expression for the kinetic energy for light. Just how dense does an object have to be in order to turn into a black hole?

Example 13.15

Calculating the schwarzschild radius.

This is a diameter of only about 6 km. If we use the mass of Earth, we get R S = 8.85 × 10 −3 m R S = 8.85 × 10 −3 m . This is a diameter of less than 2 cm! If we pack Earth’s mass into a sphere with the radius R S = 8.85 × 10 −3 m R S = 8.85 × 10 −3 m , we get a density of

Significance

Check your understanding 13.11.

The event horizon

The Schwarzschild radius is also called the event horizon of a black hole. We noted that both space and time are stretched near massive objects, such as black holes. Figure 13.30 illustrates that effect on space. The distortion caused by our Sun is actually quite small, and the diagram is exaggerated for clarity. Consider the neutron star, described in Example 13.15 . Although the distortion of space-time at the surface of a neutron star is very high, the radius is still larger than its Schwarzschild radius. Objects could still escape from its surface.

Interactive

Visit this site to view an animated example of these spatial distortions.

Observational evidence of neutron stars and black holes

The first validated neutron star was discovered by Jocelyn Bell Burnell in 1967. Burnell observed a regular radio pulse from an distant object, a phenomenon not previously observed. After further analysis and the discovery of a second pulsar (as the objects would later be called), the researchers and scientific community recognized that the pulses were caused by rapidly rotating neutron stars. At about the same time, a number of other astronomers, including Iosof Shklovsky, observed radio sources that would be proven to be neutron stars. These discoveries renewed interest in black holes. Evidence for black holes is based upon several types of observations, such as radiation analysis of X-ray binaries, gravitational lensing of the light from distant galaxies, and the motion of visible objects around invisible partners. We will focus on these later observations as they relate to what we have learned in this chapter. Although light cannot escape from a black hole for us to see, we can nevertheless see the gravitational effect of the black hole on surrounding masses.

The closest, and perhaps most dramatic, evidence for a black hole is at the center of our Milky Way galaxy. The UCLA Galactic Group, using data obtained by the W. M. Keck telescopes, has determined the orbits of several stars near the center of our galaxy. Some of that data is shown in Figure 13.31 . The orbits of two stars are highlighted. From measurements of the periods and sizes of their orbits, it is estimated that they are orbiting a mass of approximately 4 million solar masses. Note that the mass must reside in the region created by the intersection of the ellipses of the stars. The region in which that mass must reside would fit inside the orbit of Mercury—yet nothing is seen there in the visible spectrum. In 2020, UCLA astrophysicist Andrea Ghez and Plank Institute's (Germany) Reinhardt Genzel shared the Nobel prize for the discovery of the supermassive object. They shared the prize with Richard Penrose, whose mathematical models proved that black holes can actually form and align with Einstein's theory. Since their work, the world has actually seen a black hole, thanks to the Event Horizon Telescope. The global network of observatories combined massive amounts of data in order to produce remarkable images of two black holes: first at the center of galaxy Messier 87, and second at the center of the Milky Way, Saggitarious A.

Visit the UCLA Galactic Center Group main page for information on X-ray binaries and gravitational lensing. Visit this page to view the background and the image of the black hole at the center of the Milky Way.

Dark matter

But when we do that, we find the orbital speed of the stars is far too fast to be caused by that amount of matter. Astronomer Vera Rubin disovered this phenomenon in the 1970s while researching the movement of spiral galaxies. She observed that their outermost reaches were rotating as quickly as their centers, which was not the predicted outcome. Figure 13.32 shows the orbital velocities of stars as a function of their distance from the center of the Milky Way. The blue line represents the velocities we would expect from our estimates of the mass, whereas the green curve is what we get from direct measurements. Rubin calculated that the rotational velocity of galaxies should have been enough to cause them to fly apart, unless there was a significant discrepancy between their observable matter and their actual matter. This became known as the galaxy rotation problem. Rubin proposed that massive amounts of unseen matter must be present in and around the galaxies for the rotation speeds to occur. Furthermore, the velocity profile does not follow what we expect from the observed distribution of visible stars. Not only is the estimate of the total mass inconsistent with the data, but the expected distribution is inconsistent as well. Rubin’s 1978 discovery provided much more concrete and widespread evidence of dark matter , a term Fritz Zwicky coined in 1933 when he observed a similar anomaly while examining a galaxy cluster.

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Publication date: Sep 19, 2016
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Newton's Laws
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About Concept Checkers
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Newton's Laws of Motion
Newton's First Law
Newton's Third Law
The Value of g
Gravity is More Than a Name
The Apple, the Moon, and the Inverse Square Law
Newton's Law of Universal Gravitation
Cavendish and the Value of G

In Unit 2 of The Physics Classroom , an equation was given for determining the force of gravity ( F grav ) with which an object of mass m was attracted to the earth

Now in this unit, a second equation has been introduced for calculating the force of gravity with which an object is attracted to the earth.

where d represents the distance from the center of the object to the center of the earth.

The Value of g Depends on Location

To understand why the value of g is so location dependent, we will use the two equations above to derive an equation for the value of g. First, both expressions for the force of gravity are set equal to each other.

Now observe that the mass of the object - m - is present on both sides of the equal sign. Thus, m can be canceled from the equation. This leaves us with an equation for the acceleration of gravity.

The above equation demonstrates that the acceleration of gravity is dependent upon the mass of the earth (approx. 5.98x10 24 kg) and the distance ( d ) that an object is from the center of the earth. If the value 6.38x10 6 m (a typical earth radius value) is used for the distance from Earth's center, then g will be calculated to be 9.8 m/s 2 . And of course, the value of g will change as an object is moved further from Earth's center. For instance, if an object were moved to a location that is two earth-radii from the center of the earth - that is, two times 6.38x10 6 m - then a significantly different value of g will be found. As shown below, at twice the distance from the center of the earth, the value of g becomes 2.45 m/s 2 .

The table below shows the value of g at various locations from Earth's center.

		)
Earth's surface	6.38 x 10 m	9.8
1000 km above surface	7.38 x 10 m	7.33
2000 km above surface	8.38 x 10 m	5.68
3000 km above surface	9.38 x 10 m	4.53
4000 km above surface	1.04 x 10 m	3.70
5000 km above surface	1.14 x 10 m	3.08
6000 km above surface	1.24 x 10 m	2.60
7000 km above surface	1.34 x 10 m	2.23
8000 km above surface	1.44 x 10 m	1.93
9000 km above surface	1.54 x 10 m	1.69
10000 km above surface	1.64 x 10 m	1.49
50000 km above surface	5.64 x 10 m	0.13

As is evident from both the equation and the table above, the value of g varies inversely with the distance from the center of the earth. In fact, the variation in g with distance follows an inverse square law where g is inversely proportional to the distance from earth's center. This inverse square relationship means that as the distance is doubled, the value of g decreases by a factor of 4. As the distance is tripled, the value of g decreases by a factor of 9. And so on. This inverse square relationship is depicted in the graphic at the right.

Calculating g on Other Planets

The same equation used to determine the value of g on Earth' surface can also be used to determine the acceleration of gravity on the surface of other planets. The value of g on any other planet can be calculated from the mass of the planet and the radius of the planet. The equation takes the following form:

Using this equation, the following acceleration of gravity values can be calculated for the various planets.

			)
Mercury	2.43 x 10	3.2 x 10	3.61
Venus	6.073 x 10	4.88 x10	8.83
Mars	3.38 x 10	6.42 x 10	3.75
Jupiter	6.98 x 10	1.901 x 10	26.0
Saturn	5.82 x 10	5.68 x 10	11.2
Uranus	2.35 x 10	8.68 x 10	10.5
Neptune	2.27 x 10	1.03 x 10	13.3
Pluto	1.15 x 10	1.2 x 10	0.61

The acceleration of gravity of an object is a measurable quantity. Yet emerging from Newton's universal law of gravitation is a prediction that states that its value is dependent upon the mass of the Earth and the distance the object is from the Earth's center. The value of g is independent of the mass of the object and only dependent upon location - the planet the object is on and the distance from the center of that planet.

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Kepler's Three Laws

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Gravity: The law of universal gravitation

by Nathaniel Page Stites, M.A./M.S.

Did you know that the same force that makes an apple fall to the ground holds vast galaxies together? Gravity affects our activities every day, and yet it is not well understood by scientists. However, whether it is a small marble dropping from someone's hand or the motion of planets around the sun, the behavior of objects under the influence of gravity can be described mathematically.

Though the mechanisms of gravitational force are still a mystery, physicists have been able to effectively describe the influence of gravity on objects.

Newton’s mathematical model describing gravitational attraction paved the way for other scientists to build toward an understanding of the relationships between mass, acceleration, and the force of attraction.

Using the Law of Universal Gravitation, it is possible to predict the behavior of objects under the influence of gravitational force.

According to the Inverse Square Law, as the distance between two objects doubles, the force of gravity between those two objects decreases by a factor of four.

What causes objects to fall toward Earth? Why do the planets orbit the sun? What holds galaxies together? If you traveled to another planet, why would your weight change?

All of these questions relate to one aspect of physics: gravity . For all of its influence on our daily lives, for all of its control over the cosmos, and for all of our ability to describe and model its effects, we do not understand the actual mechanisms of gravitational force . Of the four fundamental forces identified by physicists – strong nuclear, electroweak, electrostatic, and gravitational – the gravitational force is the least understood. Physicists today strive toward a "Grand Unified Theory ," wherein all four of these forces are united into one physical model that describes the behavior of everything in the universe . At this point in time, the gravitational force is the troublesome one, the force that resists unification.

In spite of the mystery behind the mechanisms of gravity , physicists have been able to describe the behavior of objects under the influence of gravity quite thoroughly. Isaac Newton , a seventeenth into eighteenth century English scientist and mathematician (among other things), was the first person to propose a mathematical model to describe the gravitational attraction between objects. Albert Einstein built upon this model in the twentieth century and devised a more complete description of gravity in his theory of general relativity . In this module, we will explore Newton 's description of gravity and some of the experimental confirmations of his theory that came many years after he proposed his original idea.

Whether or not Isaac Newton actually sat under an apple tree while pondering the nature of gravity , the fact that objects fall toward the surface of Earth was well understood long before Newton 's time. Everyone has experience with gravity and its effects near the surface of Earth, and our intuitive view of the world includes an understanding that what goes up must come down.

Galileo Galilei (1564–1642) demonstrated that all objects fall to the surface of Earth with the same acceleration , and that this acceleration was independent of the mass of the falling object (see the Concept Simulation Leaning Tower of Pisa Experiment below). Isaac Newton was no doubt familiar with this concept, and he would eventually formulate a broad and far-reaching theory of gravitation. Newton 's theory would encompass not only the behavior of an apple near the surface of Earth, but also the motions of much larger bodies quite far away from Earth.

Interactive Animation: Acceleration during Free Fall

The Planets

Early conceptions of the universe were "geocentric" – they placed Earth at the center of the universe and had the planets and stars move around Earth. This Ptolemaic Model of the universe dominated scientific thought for many centuries, until the work of such careful astronomers as Tycho Brahe , Nicolaus Copernicus , Galileo Galilei , and Johannes Kepler supplanted this view of the cosmos. The "Copernican Revolution" placed the sun at the center of the solar system and the planets, including Earth, in orbit around the sun. This major shift in perception laid the foundation for Isaac Newton to begin thinking about gravitation as it related to the motions of the planets.

Figure 1: The Solar System

An early unification theory

Just as physicists today are searching for ways to unify the fundamental forces , Isaac Newton also sought to unify two seemingly disparate phenomena: the motion of objects falling toward Earth and the motion of the planets orbiting the sun. Isaac Newton 's breakthrough was not that apples fall to Earth because of gravity ; it was that the planets are constantly falling toward the sun for exactly the same reason: gravity!

Newton built upon the work of early astronomers, in particular Johannes Kepler , who in 1596 and 1619 published his laws of planetary motion. One of Kepler's central observations was that the planets move in elliptical orbits around the sun. Newton expanded Kepler's description of planetary motion into a theory of gravitation.

Comprehension Checkpoint

Newton's Law of Universal Gravitation

The essential feature of Newton's Law of Universal Gravitation is that the force of gravity between two objects is inversely proportional to the square of the distance between them. Such a connection is known as an "inverse square" relationship. Newton derived this relationship from Kepler's assertion that the planets follow elliptical orbits. To understand this, consider the light radiating from the surface of the sun. The light has some intensity at the surface of the sun. As the light travels away from the sun, its intensity diminishes. The intensity of the light at any distance away from the sun equals the strength of the source divided by the surface area of a sphere surrounding the sun at that radius.

As the distance away from the sun ( r ) doubles, the area of the sphere surrounding the sun quadruples. Thus, the intensity of the sun's light depends inversely on the square of the distance away from the sun. Newton envisioned the gravitational force as radiating equally in all directions from a central body, just as sunlight in the previous example. Newton recognized that his gravitational model must take the form of an inverse square relationship. Such a model predicts that the orbits of objects around a central body will be conic sections , and years of astronomical observations have borne this out. Although this idea is most commonly attributed to Isaac Newton , the English mathematician Robert Hooke claimed that he originated the idea of the inverse square relationship. Nonetheless, Newton eventually published his theory of gravitation and became famous as a result.

The relationship that Newton came up with looks like this:

F = G m 1 m 2 r 2

where F is the force of gravity (in units now referred to as newtons), m 1 and m 2 are the masses of the two objects in kilograms (for example, the sun and Earth), r is the distance separating the centers of mass of the objects and G is the "gravitational constant ." The equation shows that the force of gravity is directly proportional to the product of the two masses, but inversely proportional to the square of the distance between the centers of those two masses. To understand the formula , keep in mind that the force of gravity decreases as distance increases (an inverse relationship). The distance ( r ) is squared due to the relationship between the increasing distance and the growth of the area over which the force is exerted (just as rays of light spread out as they get farther from the sun). Finally, since both masses exert some force due to gravity, it is the product of their masses – not just a single mass – that makes a difference.

This relationship has come to be known as Newton's Law of Universal Gravitation. It is "universal" because all objects in the universe are attracted to all other objects in the universe according to this relationship. Two people sitting across a room from each other are actually attracted gravitationally. As we know from everyday experience, human-sized objects don't crash into each other as a result of this force , but it does exist even if it is very small. Although Newton correctly identified this relationship between force, mass , and distance, he was able only to estimate the value of the gravitational constant between these quantities. The world would have to wait more than a century for an experimental measurement of the constant of proportionality: G .

Measuring the mass of Earth: The Cavendish experiment

In 1797 and 1798 Henry Cavendish set out to confirm Newton 's theory and to determine the constant of proportionality in Newton 's Law of Universal Gravitation. His ingenious experiment , based on the work of John Michell , was successful on both fronts. To accomplish this, Cavendish created a "torsion balance," which consisted of two masses at either end of a bar that was suspended from the ceiling by a thin wire (see Figure 2).

Figure 2 : The Torsion Balance, devised by Michell and Cavendish to determine the constant of proportionality in Newton's Law of Universal Gravitation.

Attached to the wire was a mirror, off of which a beam of light was reflected. Cavendish brought a third mass close to one of the masses on the torsion balance. As the third mass attracted one of the ends of the torsion balance, the entire apparatus, including the mirror, rotated slightly and the beam of light was deflected. Through careful measurement of the angular deflection of the beam of light, Cavendish was able to determine the extent to which the known mass was attracted to the introduced mass. Not only did Cavendish confirm Newton 's theory , but also he determined the value of the gravitational constant to an accuracy of about 1 percent.

G = 6.674 × 10 − 11 N m 2 k g 2

Cavendish cleverly referred to his research as "Measuring the Mass of Earth." Since he had determined the value of G , he could do some simple calculations to determine the mass of Earth. By Newton's Second Law, the force between an object and Earth equals the product of the acceleration ( a ) and the mass of the object ( m ):

Galileo had determined the acceleration due to gravity ( g ) of all objects near the surface of Earth in the early 1600s as g = 9.8 m s 2 .

Therefore, setting this equation equal to Newton's Law of Universal Gravitation described above, Cavendish found:

F = m g = G m m E r E 2

where m is the mass of the object, m E is the mass of Earth, and r E is the radius of Earth. Solving for the mass of Earth yields the following result:

m E = g r E 2 G = ( 9.8 m s 2 ) ( 6.38 × 10 6 m ) 2 6.67 × 10 − 11 N m 2 k g 2

m E = 5.98 × 10 24 k g

Cavendish had determined the mass of Earth with great accuracy . We can also use this relationship to calculate the force of attraction between two people across a room. To do this, we simply need to use Newton's Law of Universal Gravitation with Cavendish's gravitational constant . Assume the two people have masses of 75 and 100 kilograms, respectively, and that they are 5 meters apart. The force of gravitation between them is:

F = ( 6.67 × 10 − 11 N m 2 k g 2 ) ⋅ ( 75 k g ) ⋅ ( 100 k g ) ( 5 m ) 2

F = 2.00 × 10 − 8 N

Although it is small, there is still a force!

Newton's Law of Universal Gravitation grew in importance as scientists realized its utility in predicting the orbits of the planets and other bodies in space. In 1705, Sir Edmund Halley , after studying comets in great detail, predicted correctly that the famous comet of 1682 would return 76 years later, in December of 1758. Halley had used Newton's Law to predict the behavior of the comet orbiting the sun. With the advent of Cavendish's accurate value for the gravitational constant , scientists were able to use Newton 's law for even more purposes. In 1845, John Couch Adams and Urbain Le Verrier predicted the existence of a new, yet unseen, planet based on small discrepancies between predictions for and observations of the position of Uranus. In 1846, the German astronomer Johann Galle confirmed their predictions and officially discovered the new planet, Neptune.

While Newton's Law of Universal Gravitation remains very useful today, Albert Einstein demonstrated in 1915 that the law was only approximately correct, and that it fails to work when gravitation becomes extremely strong. Nonetheless, Newton 's gravitational constant plays an important role in Einstein's alternative to Newton's Law, the Theory of General Relativity. The value of G has been the subject of great debate even in recent years, and scientists are still struggling to determine a very accurate value for this most elusive of fundamental physical constants.

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Finding gravity through experimental data

Thread starter BayernBlues
Start date Oct 13, 2007
Tags Data Experimental Experimental data Gravity
Oct 13, 2007

Homework Statement

Homework equations, the attempt at a solution.

New computational microscopy technique provides more direct route to crisp images
New method for generating monochromatic light in storage rings
Updating the textbook on polarization in gallium nitride to optimize wide bandgap semiconductors

It gives you the values of x and t, you have the equation that relates them. Solve for g.

Oct 20, 2007

Related to Finding gravity through experimental data

In an experiment, gravity can be measured by using a device called a gravimeter, which measures the acceleration due to gravity (g). This device uses a mass and spring system to measure the effects of gravity on an object.

The relationship between mass and gravity is described by Newton's Law of Universal Gravitation, which states that the force of gravity between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

The acceleration due to gravity can be calculated by using the equation g = F/m, where g is the acceleration due to gravity, F is the force of gravity, and m is the mass of the object experiencing the force of gravity. This can be done by plotting the data points and finding the slope of the best-fit line.

Some common sources of error in gravity experiments include air resistance, friction, and human error in taking measurements. Other factors such as the curvature of the Earth and variations in local gravitational fields can also affect the results.

Gravity affects the motion of objects by exerting a force on them that causes them to accelerate towards the center of the Earth. This acceleration is constant and can cause objects to fall towards the ground or orbit around larger objects, such as the Moon orbiting the Earth.

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Determine acceleration from experiment (Newton 2nd Law)

I have done a physics experiment (setup below). And was asked to determine the experimental and theoretical acceleration.

The data I've got

Ok, am I right to say

Experimental acceleration = $2(s_f - s_i) / t^2$

Theoratical acceleration = $m_2 \times 0.98 / m_1$

Percentage discrepancy = $\frac{|(Experimental - Theoretical)|}{Theoretical} \times 100$%

homework-and-exercises
newtonian-mechanics

1 $\begingroup$ Hi Jiew - your question seems kind of unfocused. What concept is it specifically that is confusing you? Are you confused about why the mass of the cart affects its horizontal acceleration? Or are you confused about some step in the calculation of the theoretical acceleration? It would help a lot if you edit your question to focus on the one thing you want to ask. If you have multiple concepts to ask about, you can post more than one question about the same lab setup. $\endgroup$ – David Z Commented Sep 27, 2012 at 4:30
$\begingroup$ Ok, I edited my post and posted another question physics.stackexchange.com/questions/38448/… $\endgroup$ – Jiew Meng Commented Sep 27, 2012 at 6:25

If by $s_f$ and $s_i$ you mean the final and initial position, respectively --- so that $s_f-s_i$ is just $d$ in your table --- then yes, your experimental acceleration is right. As for your theoretical acceleration, it should be $\frac{9.8m_2}{m_1}$, not $0.98$ --- the acceleration due to gravity is $g=9.8$. I'm assuming you just made a typo. Your definition of percentage discrepancy is right.

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Falling for Gravity

Calculate the acceleration of gravity using simple materials, a cell phone, and a computer to record, watch, and analyze the motion of a dropped object.

Two-meter measuring tape or two meter sticks
Masking tape
Small, cheap, rugged flashlight
Towel, carpeting, or other soft material for the dropped flashlight to land on
Digital camera with video capability (the HD camera on a phone should work fine)
Computer with a program that lets you play videos frame by frame (not shown)
Pencil and paper to record data (not shown)

Acceleration due to Gravity Formula

We have no doubt seen gravity work in our life. After all, it is the force that is helping us to keep our feet on the ground. If we throw a ball in the upward direction in the air. Then it will come down on its own. Why? When the ball is going in upwards direction, its speed will be less as compared to when it comes down. This is because of the acceleration, which is produced due to the force of gravity. In this topic, we will discuss acceleration due to Gravity formula. Let us learn about acceleration due to gravity in detail.

Source: simple.wikipedia.org

What is Acceleration due to Gravity?

Acceleration due to gravity is the acceleration that is gained by an object due to the gravitational force. Its SI unit is ms². It has a magnitude as well as direction. Thus it is a vector quantity.

We represent acceleration due to gravity by the symbol g. Its standard value on the surface of the earth at sea level is 9.8 ms². Its computation formula is based on Newton’s Second Law of Motion and Newton’s Law of Universal Gravitation .

Acceleration Due to Gravity Formula

Near the surface of Earth, the acceleration due to gravity is approximately constant. But, at large distances from the Earth, or around other planets or moons, it is varying. The acceleration due to gravity depends on the terms as the following:

Mass of the body,

Distance from the center of mass,

Constant G i.e. Universal gravitational constant.

g = $ \frac {G M}{r^2} $

g	Acceleration due to gravity $(units ms^ {-1} ) $
G	The universal gravitational constant,$ = 6.673 \times 10^{-11} N m^2 Kg^2 $
m	Mass of a very large body like Earth.
r	The distance from the center of mass of the large body

Variation of g with Height:

Acceleration due to gravity varies with the height from the surface of the earth. Its computation can be done as follows:

$ g_h = g \; (1+\frac {h}{R})^{-2} $

g	Acceleration due to gravity at the surface.
$g_h$	Acceleration due to gravity at the height h.
R	The radius of the earth.
h	Height from the earth’s surface.

It is clear that the value of g decreases with an increase in height of an object. Hence the value of g becomes zero at infinite distance from the earth.

Solved Examples

Example-1: The radius of the moon is $ 1.74 \times 10^6 m$. The mass of the moon is taken as $7.35 \times 10^{22}$ kg. Find out the acceleration due to gravity on the surface of the moon.

Solution: On the surface of the moon, the distance to the center of mass will be the same as the radius.

Thus, r = $1.74 \times 10^6 m$.

Mass of the object i.e. moon,

m = $7.35 \times 10^{22}$ kg

As we know that, universal gravitational constant G = $ 6.673 \times 10^{-11} $

The acceleration due to gravity on the surface of the moon can be computed by using the formula as below:

Substituting the values,

g = $ \frac {6.673 \times 10^{-11} \times 7.35 \times 10^{22}}{(1.74 \times 10^6)^2}$

g = $ 1.620 \; ms^{-2} $

Hence, value of the acceleration due to gravity is $ 1.620 \; ms^{-2} $

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5 responses to “Spring Potential Energy Formula”

Typo Error> Speed of Light, C = 299,792,458 m/s in vacuum So U s/b C = 3 x 10^8 m/s Not that C = 3 x 108 m/s to imply C = 324 m/s A bullet is faster than 324m/s

I have realy intrested to to this topic

m=f/a correct this

Interesting studies

It is already correct f= ma by second newton formula…

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Calculate Percent Error 5

Percent Error Definition

Percent error, sometimes referred to as percentage error, is an expression of the difference between a measured value and the known or accepted value . It is often used in science to report the difference between experimental values and expected values.

$Percent Error Formula$

The formula for calculating percent error is:

Note: occasionally, it is useful to know if the error is positive or negative. If you need to know the positive or negative error, this is done by dropping the absolute value brackets in the formula. In most cases, absolute error is fine. For example, in experiments involving yields in chemical reactions, it is unlikely you will obtain more product than theoretically possible.

Steps to Calculate the Percent Error

Subtract the accepted value from the experimental value.
Take the absolute value of step 1
Divide that answer by the accepted value.
Multiply that answer by 100 and add the % symbol to express the answer as a percentage .

Example Calculation

Now let’s try an example problem.

You are given a cube of pure copper. You measure the sides of the cube to find the volume and weigh it to find its mass. When you calculate the density using your measurements, you get 8.78 grams/cm 3 . Copper’s accepted density is 8.96 g/cm 3 . What is your percent error?

Solution: experimental value = 8.78 g/cm 3 accepted value = 8.96 g/cm 3

Step 1: Subtract the accepted value from the experimental value.

8.78 g/cm 3 – 8.96 g/cm 3 = -0.18 g/cm 3

Step 2: Take the absolute value of step 1

|-0.18 g/cm 3 | = 0.18 g/cm 3

$Percent Error Math 3$

Step 3: Divide that answer by the accepted value.

Step 4: Multiply that answer by 100 and add the % symbol to express the answer as a percentage.

0.02 x 100 = 2 2%

The percent error of your density calculation is 2%.

5 thoughts on “ calculate percent error ”.

Percent error is always represented as a positive value. The difference between the actual and experimental value is always the absolute value of the difference. |Experimental-Actual|/Actualx100 so it doesn’t matter how you subtract. The result of the difference is positive and therefore the percent error is positive.

Percent error is always positive, but step one still contains the error initially flagged by Mark. The answer in that step should be negative:

experimental-accepted=error 8.78 – 8.96 = -0.18

In the article, the answer was edited to be correct (negative), but the values on the left are still not in the right order and don’t yield a negative answer as presented.

Mark is not correct. Percent error is always positive regardless of the values of the experimental and actual values. Please see my post to him.

Say if you wanted to find acceleration caused by gravity, the accepted value would be the acceleration caused by gravity on earth (9.81…), and the experimental value would be what you calculated gravity as 🙂

If you don’t have an accepted value, the way you express error depends on how you are making the measurement. If it’s a calculated value, like, based on a known about of carbon dioxide dissolved in water, then you have a theoretical value to use instead of the accepted value. If you are performing a chemical reaction to quantify the amount of carbonic acid, the accepted value is the theoretical value if the reaction goes to completion. If you are measuring the value using an instrument, you have uncertainty of the instrument (e.g., a pH meter that measures to the nearest 0.1 units). But, if you are taking measurements, most of the time, measure the concentration more than once, take the average value of your measurements, and use the average (mean) as your accepted value. Error gets complicated, since it also depends on instrument calibration and other factors.

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NEWS FEATURE
27 June 2024

Five new ways to catch gravitational waves — and the secrets they’ll reveal

Davide Castelvecchi

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Two black holes orbiting around each other generate gravitational waves of increasing frequency. New instruments and techniques could watch this merger for weeks or even years. Credit: NASA's Goddard Space Flight Center Conceptual Image Lab

In September 2015, a vibration lasting just one-fifth of a second changed the history of physics. It was the first direct detection of gravitational waves — perturbations in the geometry of space-time that move across the Universe at the speed of light.

Astronomers say it was like gaining a new sense — as if, until 2015, they had only been able to ‘see’ cosmic events, and now could ‘hear’ them, too. Since then, it has become almost a matter of daily routine to record the passage of gravitational waves at the two massive facilities of the Laser Interferometer Gravitational-wave Observatory (LIGO) in Louisiana and Washington state, along with their sibling Virgo observatory near Pisa, Italy.

The detection of gravitational waves has provided new ways to explore the laws of nature and the history of the Universe , including clues about the life story of black holes and the large stars they originated from. For many physicists, the birth of gravitational-wave science was a rare bright spot in the past decade, says Chiara Caprini, a theoretical physicist at the University of Geneva in Switzerland. Other promising fields of exploration have disappointed: dark-matter searches have kept coming up empty handed; the Large Hadron Collider near Geneva has found nothing beyond the Higgs boson; and even some promising hints of new physics seem to be fading. “In this rather flat landscape, the arrival of gravitational waves was a breath of fresh air,” says Caprini.

That rare bright spot looks set to become brighter.

All of the more than 100 gravitational-wave events spotted so far have been just a tiny sample of what physicists think is out there. The window opened by LIGO and Virgo was rather narrow, limited mostly to frequencies in the range 100–1,000 hertz. As pairs of heavy stars or black holes slowly spiral towards each other, over millions of years, they produce gravitational waves of slowly increasing frequency, until, in the final moments before the objects collide, the waves ripple into this detectable range. But this is only one of many kinds of phenomenon that are expected to produce gravitational waves.

LIGO and Virgo are laser interferometers: they work by detecting small differences in travel time for lasers fired along perpendicular arms, each a few kilometres long. The arms expand and contract by minuscule amounts as gravitational waves wash over them. Researchers are now working on several next-generation LIGO-type observatories, both on Earth and, in space, the Laser Interferometer Space Antenna; some have even proposed building one on the Moon 1 . Some of these could be sensitive to gravitational waves at frequencies as low as 1 Hz.

But physicists are also exploring entirely different techniques to detect gravitational waves. These strategies, which range from watching pulsars to measuring quantum fluctuations, hope to catch a much wider variety of gravitational waves, with frequencies in the megahertz to nanohertz range (see ‘Opening the window on gravitational waves’).

Opening the window on gravitational waves: graphic that shows a range of new detectors, and the range of frequencies from different sources that they will be able to detect.

By broadening their observational window, astronomers should be able to watch black holes circling each other for days, weeks or even years, rather than just catching the last few seconds before collision. And they’ll be able to spot waves made by totally different cosmic phenomena — including mega black holes and even the start of the Universe itself. All this, they say, will crack open many remaining secrets of the cosmos.

Pulsar timing array: catching waves that last a decade

Last year, one viable alternative to interferometers entered the game.

Since the early 2000s, radio astronomers have been attempting to use the entire Galaxy as a gravitational-wave detector. The trick is to monitor dozens of neutron stars called pulsars. These spin on their axis hundreds of times per second while emitting a radio-frequency beam, producing what looks like a pulse of light on each turn.

Gravitational waves sweeping the Galaxy would change the distance between Earth and each pulsar, creating anomalies in detected pulsar frequencies from one year to the next. Observations of a collection or array of pulsars — called a pulsar timing array (PTA) — should be able to detect changes induced by gravitational waves with frequencies of just nanohertz, as might be produced by pairs of supermassive black holes, for example. It takes tens of years for successive crests of such waves to pass a given vantage point, meaning that tens of years of observations are needed to spot them.

Giant gravitational waves: why scientists are so excited

In 2023, the PTA technique began to bear fruit . Four separate collaborations, in North America, Europe, Australia and China, unveiled tantalizing hints of a pattern expected from a random ‘stochastic background’ of gravitational waves that make Earth slosh around, probably caused by a cacophony of supermassive black-hole binaries, says astrophysicist Chiara Mingarelli at Yale University in New Haven, Connecticut.

The teams have not yet used the word ‘discovery’, because the evidence that each collaboration unveiled is not yet rock solid. But three of the groups — all but the Chinese one — are now pooling their data and conducting a joint analysis in the hope of getting to the ‘D’ word. This requires painstaking work, because each group processed its raw data in slightly different ways, and so it could take at least another year to get to publication, says Scott Ransom, an astrophysicist at the US National Radio Astronomy Observatory in Charlottesville, Virginia, and a senior member of the North American collaboration.

“In our current data, we almost certainly have the hints of individual supermassive black-hole binaries out there,” says Ransom. With each extra year of observation, they should get closer to resolving single black-hole pairs out of the cacophony, he adds. “Things are just going to get better and better.”

Microwave telescopes: spotting waves from the Big Bang

A year before LIGO’s 2015 detection, a team of cosmologists using a South Pole telescope called BICEP2 claimed to have spotted gravitational waves — not directly, but in the pattern of light called the cosmic microwave background (CMB), sometimes described as the afterglow of the Big Bang.

The BICEP2 claim turned out to be premature , but cosmologists are now doubling down on this idea. An array of telescopes much more powerful than BICEP2, called the Simons Observatory , is being set up on a mountaintop in northern Chile’s Atacama Desert. Some researchers are holding out hope for an even more powerful array called CMB-S4 (originally proposed to include 12 telescopes in Chile and at the South Pole) — although in May, plans for that project were put on hold because of the disrepair of the US South Pole base.

Gravitational waves: 6 cosmic questions they can tackle

What cosmologists are looking for in the CMB is a specific ‘B mode’ pattern in the swirls of its polarization — the preferential directions in which the microwaves wiggle — that would have been imprinted by the passage of gravitational waves. The theory is that such waves should have been produced by inflation, a quick burst of exponential cosmic expansion thought to have happened around the time of the Big Bang 2 . Inflation would explain many of the Universe’s most striking properties, such as its flatness and how mass is distributed. The gravitational waves that inflation produced would have started at high frequencies, but would by now be at incredibly low frequencies of around 10 −14 Hz.

Although inflation is a cornerstone of accepted cosmological theory, there’s no proof of it yet. The B-mode pattern would be the smoking gun and, moreover, would reveal the energy scales involved, which would be a first step towards understanding what powered inflation.

The problem is, no one knows whether that energy scale was large enough to have left a noticeable mark. “Inflation predicts the B modes, but we don’t know if it’s big enough to be detected,” says Marc Kamionkowski, a theoretical astrophysicist at Johns Hopkins University in Baltimore, Maryland. But if the leading models are right, either the Simons Observatory or CMB-S4 should eventually find it, he says.

Atom interferometry: closing the gap

Although many of these projects push gravitational-wave science towards lower frequencies, they leave a crucial gap just below 1 Hz.

Detecting such frequencies could reveal mergers of black holes much more massive than those seen by LIGO (which spots waves from collapsing stars that weigh at most a few tens of solar masses). “This is an unexplored region, but it could be populated with lots of black holes,” says Caprini.

Physicists Jason Hogan and Mark Kasevich pictured next to equipment they are developing for measuring gravitational waves.

Jason Hogan (left) and Mark Kasevich work on an atom interferometer — a device that could reveal mergers of black holes much more massive than those seen by current laser interferometers. Credit: L.A. Cicero and Stanford University

A nascent technique could come to the rescue, according to physicist Oliver Buchmüller at Imperial College London. “Atom interferometry sits in that gap which we currently cannot explore with any other technology,” he says. An atom interferometer is a vertical high-vacuum pipe in which atoms can be released and allowed to fall under gravity. As they do so, physicists tickle the atoms with laser light to toggle them between an excited and a relaxed state — the same principle used by atomic clocks. “We’re trying to push this atomic-clock technique to what’s ultimately possible,” says Jason Hogan, a physicist at Stanford University in California.

To detect gravitational waves, physicists plan to drop two or more sets of atoms at different heights inside the same vertical pipe, and to measure the time it takes for a laser pulse to travel from one set of atoms to the next, says Hogan. The passage of gravitational waves would result in light spending either slightly less or slightly more time travelling between them — a variation smaller than one part in 100 billion billion.

Pioneering experiments at Stanford University have developed atom interferometers with 10-metre drops, but detecting gravitational waves would require devices at least 1 kilometre in height, which could be installed in a mine shaft, say, or even in space. As a first step, several groups around the world are planning to build 100-m atom interferometers as test beds. One such facility, called MAGIS-100, is already under construction in an existing shaft at the Fermi National Accelerator Laboratory outside Chicago, Illinois, and is scheduled for completion in 2027.

Desktop detectors: pushing the frequency up

Other researchers are exploring ways of detecting gravitational waves with much, much smaller (and cheaper) detectors — including some that could fit on a desktop. These are designed to watch for extremely high-frequency gravity waves. Known phenomena probably don’t produce such waves, but some speculative theories do predict them.

The Levitated Sensor Detector (LSD) at Northwestern University in Evanston, Illinois looks like a toy LIGO: it bounces lasers between pairs of mirrors just 1 metre apart. The LSD is a prototype for a new type of instrument designed to sense gravitational waves using resonance: the same principle by which even little pushes can make a child on a swing go higher and higher if they are timed just right 3 .

Will the Einstein Telescope be split in two?

In a vacuum inside each of the LSD’s arms, laser light suspends a particle just micrometres wide. As with an interferometer, the passage of gravitational waves will alternately elongate and compress the length of each arm. If the frequency of the gravitational waves resonates with that of the device, the lasers will then give many tiny kicks to the particle. The LSD can track the particle’s motion with a precision of femtometres, says Northwestern physicist Andrew Geraci, who is leading the project.

The LSD is designed to be sensitive to gravitational waves with frequencies of around 100 kHz. This prototype might already have a shot at detecting some, if the team can keep experimental noise in check — and provided that such waves exist. “Depending how optimistic you are, we may be able to measure a real signal in that band even with a 1-m instrument,” Geraci says. Future versions could be scaled up to 100-m-long arms, he adds, which would increase their sensitivity.

Theoretical physicist Ivette Fuentes at the University of Southampton, UK, has an idea for making an even smaller resonant detector. She aims to exploit sound waves in an exotic state of matter called a Bose–Einstein condensate (BEC) — a cloud of atoms kept at temperatures as low as a few millionths of a degree above absolute zero. If a gravitational wave passes through at a frequency that resonates with the sound wave, it can be detected. Because the act of looking for this signal destroys the BEC, a new flood of atoms needs to be released every second. The process might need to be repeated for months for a successful detection, Fuentes says.

In principle, a BEC-based detector could expand the search for gravitational waves to extremely high frequencies of 1 MHz or more — again, provided they exist. Fuentes says her scheme would require pushing BEC techniques just a little beyond the current state of the art. “I think the idea is very bold,” she says. Physicists have posited that high-frequency gravitational waves could reveal exotic physics that went on in the first second or so after the Big Bang. “We could use it to study the state of the Universe at very high energies,” says Caprini.

Quantum crystal: only takes a second

A final, more radical proposal for detecting gravitational waves involves putting objects in two places at once.

‘Best view ever’: observatory will map Big Bang’s afterglow in new detail

Sougato Bose, a physicist at University College London, has proposed a device in which a micrometre-sized diamond crystal is put in a superposition of two quantum states. In his scheme, the crystal’s two ‘personas’ would be pushed apart by as much as 1 metre and then brought together again — an extremely delicate procedure that has been compared to putting the nursery-rhyme character Humpty Dumpty back together after a fall. The passage of gravitational waves would make one persona travel further than the other when apart, putting them out of sync — in a measurable way — when reunited. The whole process would take around one second to complete, which would make the device sensitive to gravitational waves of around 1 Hz.

This idea is extremely ambitious: such quantum tricks have so far been shown to work only for objects the size of molecules, and no one has ever tested whether quantum weirdness can be pushed to such extremes. “Putting Humpty Dumpty back together has never been demonstrated for crystals,” says Bose.

But if the technique can be perfected, then table-top experiments such as this one could take gravitational-wave detection out of the hands of just a few large-scale labs. Together, these techniques could blow open the window on what can be seen. “The outlook is very positive,” says Caprini.

doi: https://doi.org/10.1038/d41586-024-02003-6

Branchesi, M. et al. Space Sci. Rev. 219 , 67 (2023).

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Kamionkowski, M., Kosowsky, A. & Stebbins, A. Phys. Rev. Lett. 78 , 2058 (1997).

Arvanitaki, A. & Geraci, A. A. Phys. Rev. Lett. 110 , 071105 (2013).

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June 26, 2024

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Experiment captures atoms in free fall to look for gravitational anomalies caused by dark energy

by University of California - Berkeley

Precision instrument bolsters efforts to find elusive dark energy

Dark energy—a mysterious force pushing the universe apart at an ever-increasing rate—was discovered 26 years ago, and ever since, scientists have been searching for a new and exotic particle causing the expansion.

Pushing the boundaries of this search, University of California, Berkeley physicists have now built the most precise experiment yet to look for minor deviations from the accepted theory of gravity that could be evidence for such a particle, which theorists have dubbed a chameleon or symmetron. The results are published in the June 11, 2024, issue of Nature Physics .

The experiment, which combines an atom interferometer for precise gravity measurements with an optical lattice to hold the atoms in place, allowed the researchers to immobilize free-falling atoms for seconds instead of milliseconds to look for gravitational effects, besting the current most precise measurement by a factor of five.

Though the researchers found no deviation from what is predicted by the theory spelled out by Isaac Newton 400 years ago, expected improvements in the precision of the experiment could eventually turn up evidence that supports or disproves theories of a hypothetical fifth force mediated by chameleons or symmetrons.

The ability of the lattice atom interferometer to hold atoms for up to 70 seconds—and potentially 10 times longer—also opens up the possibility of probing gravity at the quantum level , said Holger Müller, UC Berkeley professor of physics. While physicists have well-tested theories describing the quantum nature of three of the four forces of nature—electromagnetism and the strong and weak forces—the quantum nature of gravity has never been demonstrated.

"Most theorists probably agree that gravity is quantum. But nobody has ever seen an experimental signature of that," Müller said.

"It's very hard to even know whether gravity is quantum, but if we could hold our atoms 20- or 30-times longer than anyone else, because our sensitivity increases exponentially, we could have a 400 to 800,000 times better chance of finding experimental proof that gravity is indeed quantum mechanical."

Aside from precision measurements of gravity, other applications of the lattice atom interferometer include quantum sensing.

"Atom interferometry is particularly sensitive to gravity or inertial effects. You can build gyroscopes and accelerometers," said UC Berkeley postdoctoral fellow Cristian Panda, who is first author of the paper. "But this gives a new direction in atom interferometry, where quantum sensing of gravity, acceleration and rotation could be done with atoms held in optical lattices in a compact package that is resilient to environmental imperfections or noise."

Because the optical lattice holds atoms rigidly in place, the lattice atom interferometer could even operate at sea, where sensitive gravity measurements are employed to map the geology of the ocean floor.

Screened forces can hide in plain sight

Dark energy was discovered in 1998 by two teams of scientists: a group of physicists based at Lawrence Berkeley National Laboratory, led by Saul Perlmutter, now a UC Berkeley professor of physics, and a group of astronomers that included UC Berkeley postdoctoral fellow Adam Riess. The two shared the 2011 Nobel Prize in Physics for the discovery.

The realization that the universe was expanding more rapidly than it should came from tracking distant supernovas and using them to measure cosmic distances. Despite much speculation by theorists about what's actually pushing space apart, dark energy remains an enigma—a large enigma, since about 70% of the entire matter and energy of the universe is in the form of dark energy.

One theory is that dark energy is merely the vacuum energy of space. Another is that it is an energy field called quintessence, which varies over time and space.

Another proposal is that dark energy is a fifth force much weaker than gravity and mediated by a particle that exerts a repulsive force that varies with the density of surrounding matter. In the emptiness of space, it would exert a repulsive force over long distances, able to push space apart. In a laboratory on Earth, with matter all around to shield it, the particle would have an extremely small reach.

This particle has been dubbed a chameleon, as if it's hiding in plain sight.

In 2015, Müller adapted an atom interferometer to search for evidence of chameleons using cesium atoms launched into a vacuum chamber , which mimics the emptiness of space.

During the 10 to 20 milliseconds it took the atoms to rise and fall above a heavy aluminum sphere, he and his team detected no deviation from what would be expected from the normal gravitational attraction of the sphere and Earth.

The key to using free-falling atoms to test gravity is the ability to excite each atom into a quantum superposition of two states, each with a slightly different momentum that carries them different distances from a heavy tungsten weight hanging overhead. The higher momentum, higher elevation state experiences more gravitational attraction to the tungsten, changing its phase.

When the atom's wave function collapses, the phase difference between the two parts of the matter wave reveals the difference in gravitational attraction between them.

"Atom interferometry is the art and science of using the quantum properties of a particle, that is, the fact that it's both a particle and a wave. We split the wave up so that the particle is taking two paths at the same time and then interfere them at the end," Müller said.

"The waves can either be in phase and add up, or the waves can be out of phase and cancel each other out. The trick is that whether they are in phase or out of phase depends very sensitively on some quantities that you might want to measure, such as acceleration, gravity, rotation or fundamental constants."

In 2019, Müller and his colleagues added an optical lattice to keep the atoms close to the tungsten weight for a much longer time—an astounding 20 seconds—to increase the effect of gravity on the phase. The optical lattice employs two crossed laser beams that create a lattice-like array of stable places for atoms to congregate, levitating in the vacuum. But was 20 seconds the limit, he wondered?

During the height of the COVID-19 pandemic, Panda worked tirelessly to extend the hold time, systematically fixing a list of 40 possible roadblocks until establishing that the wiggling tilt of the laser beam, caused by vibrations, was a major limitation.

By stabilizing the beam within a resonant chamber and tweaking the temperature to be a bit colder—in this case less than a millionth of a Kelvin above absolute zero, or a billion times colder than room temperature—he was able to extend the hold time to 70 seconds.

Gravitational entanglement

In the newly reported gravity experiment, Panda and Müller traded a shorter time, 2 seconds, for a greater separation of the wave packets to several microns, or several thousandths of a millimeter. There are about 10,000 cesium atoms in the vacuum chamber for each experiment—too sparsely distributed to interact with one another—dispersed by the optical lattice into clouds of about 10 atoms each.

"Gravity is trying to push them down with a force a billion times stronger than their attraction to the tungsten mass, but you have the restoring force from the optical lattice that's holding them, kind of like a shelf," Panda said.

"We then take each atom and split it into two wave packets, so now it's in a superposition of two heights. And then we take each one of those two wave packets and load them in a separate lattice site, a separate shelf, so it looks like a cupboard. When we turn off the lattice, the wave packets recombine, and all the quantum information that was acquired during the hold can be read out."

Panda plans to build his own lattice atom interferometer at the University of Arizona, where he was just appointed an assistant professor of physics. He hopes to use it to, among other things, more precisely measure the gravitational constant that links the force of gravity with mass.

Meanwhile, Müller and his team are building from scratch a new lattice atom interferometer with better vibration control and a lower temperature. The new device could produce results that are 100 times better than the current experiment, sensitive enough to detect the quantum properties of gravity .

The planned experiment to detect gravitational entanglement, if successful, would be akin to the first demonstration of quantum entanglement of photons performed at UC Berkeley in 1972 by the late Stuart Freedman and former postdoctoral fellow John Clauser. Clauser shared the 2022 Nobel Prize in Physics for that work.

Other co-authors of the gravity paper are graduate student Matthew Tao and former undergraduate student Miguel Ceja of UC Berkeley, Justin Khoury of the University of Pennsylvania in Philadelphia and Guglielmo Tino of the University of Florence in Italy.

Journal information: Nature , Nature Physics

Provided by University of California - Berkeley

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Article Contents

1. introduction, 2. definitions and notations.

3. HQET factorization and form factors for W → B ℓ + ℓ −

4. Numerical analysis

< Previous

Semileptonic W Decay to the B Meson with Lepton Pairs in Heavy Quark Effective Theory Factorization up to |$\mathcal {O}(\alpha _s)$|

Article contents
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Saadi Ishaq, Sajawal Zafar, Abdur Rehman, Ishtiaq Ahmed, Semileptonic W Decay to the B Meson with Lepton Pairs in Heavy Quark Effective Theory Factorization up to |$\mathcal {O}(\alpha _s)$| , Progress of Theoretical and Experimental Physics , Volume 2024, Issue 6, June 2024, 063B05, https://doi.org/10.1093/ptep/ptae080

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Motivated by the study of heavy–light meson production within the framework of heavy quark effective theory (HQET) factorization, we extend the factorization formalism for a rather complicated process W + → B + ℓ + ℓ − in the limit of a nonzero invariant squared mass of the dilepton, q 2 , at the lowest order in 1/ m b up to |$\mathcal {O}(\alpha _s)$|⁠ . The purpose of the current study is to extend the HQET factorization formula for the W + → B + ℓ + ℓ − process and subsequently compute the form factors for this channel up to next-to-leading-order corrections in α s . We explicitly show that the amplitude of the W + → B + ℓ + ℓ − process can also be factorized into a convolution between the perturbatively calculable hard-scattering kernel and the nonperturbative yet universal light-cone distribution amplitude (LCDA) defined in HQET. The validity of the HQET factorization depends on the assumed scale hierarchy m W ∼ m b ≫ Λ QCD . Within the HQET framework, we evaluate the form factors associated with the W + → B + ℓ + ℓ − process, providing insights into its phenomenology. In addition, we also perform an exploratory phenomenological study on W + → B + ℓ + ℓ − by employing an exponential model for the LCDAs for the B + meson. Our findings reveal that the branching ratio for W + → B + ℓ + ℓ − is below 10 −10 . Although the branching ratios are small, this channel in high-luminosity LHC experiments may serve to further constrain the value of λ B .

Since the discovery of heavy–light mesons, the exclusive B -decays in particular not only offer an excellent laboratory to extract the Standard Model (SM) parameters or to look for yet-unknown particles and interactions but also help to pin down the strong interaction dynamics at different scales from the quantum chromodynamics (QCD) point of view. Over the past few decades, a vast amount of literature has been devoted to heavy–light mesons whose underlying weak decays are understandable, but complications appear for their theoretical elucidation in the context of perturbative and nonperturbative QCD effects.

For their theoretical description, numerous techniques have been introduced to disentangle perturbative and nonperturbative effects of QCD that rely on the relatively large mass of the b quark as compared to the strong interaction scale Λ QCD . The mass of the bottom quark m b provides a scale at which the strong coupling α s is smaller such that the short-distance effects can be calculated in a perturbative manner. Aiming to deal with nonperturbative effects, various theoretical approaches are developed. Among these, QCD factorization has emerged as the predominant theoretical framework, which derives from first principles [ 1–3 ].

The most straightforward application of QCD factorization is for the exclusive heavy-to-light radiative B meson transitions [ 4–6 ]. Where the amplitude can be factorized, in the heavy quark limit, as a convolution between a hard-scattering kernel perturbative in nature and nonperturbative light-cone distribution amplitudes (LCDAs) of B mesons. Furthermore, the Lorentz invariant form factors of the exclusive B -decays can be calculated in terms of process-independent B meson LCDAs. As a result, the radiative B -decays serve as an excellent testing ground for exploring the characteristics of B meson LCDAs and verifying the validity of the factorization approach. This factorization formula has also been extended for somewhat more complicated channels, e.g. B → M 1 M 2 [ 2 , 3 ] and B → γγ [ 7 ].

The significant generation of weak gauge bosons at the Large Hadron Collider (LHC) is a source of inspiration to validate the predictions of the SM, search for new physics (NP), improve our understanding of QCD dynamics at different regimes, and also offer opportunities to investigate the exclusive W gauge boson decays. Among these decay modes, W → D s γ stands out with the highest branching fraction. The first detailed analysis of the radiative decay process W → D s γ was studied decades ago [ 8 , 9 ]. The upper limit was set by the CDF Collaboration with the value Br Exp ( W → D s γ) < 1.3 × 10 −3 [ 10 ]. This high-yield production of W ± and Z has been a pivotal driver in unraveling their decay characteristics with increased precision. In this context, several exclusive radiative decays of W and Z bosons into heavy–light mesons have been investigated in the standard collinear factorization (or light-cone factorization) [ 11 ]. The heavy meson LCDAs that appear in this factorization formula are not completely nonperturbative, as they still entail the hard scale because the b quark field is defined in full QCD.

Nonetheless, the LCDA of a heavy–light meson entering the heavy quark effective theory (HQET) factorization formula is entirely nonperturbative because the b quark field is defined in HQET rather than in full QCD. It is worth highlighting that both types of LCDA associated with heavy–light mesons are connected through a perturbatively calculable matching coefficient [ 12 ]. The HQET factorization formula crucially depends on the mass hierarchy: m W ∼ m b ≫ Λ QCD . This hierarchy ensures that the LCDA’s dependence is confined to the soft scale; consequently, the LCDA’s behavior is not entangled with perturbative effects. This separation of scales, facilitated by the mass hierarchy, allows for a more tractable and precise description of the heavy meson HQET LCDA. Notably, the production of heavy–light mesons within the HQET factorization formalism has been comprehensively addressed in Ref. [ 13 ] up to next-to-leading order (NLO) in α s , shedding light on the application of this factorization formula in the study of these processes. Furthermore, the exclusive production of flavored quarkonia, such as |$W^+ \rightarrow B_c^+ \gamma$|⁠ , has also been investigated through |$\mathcal {O}(\alpha _s)$| within the nonrelativistic QCD (NRQCD) factorization framework [ 14 ].

The HQET factorization formula for the decay process W + → B + γ has previously been established in Ref. [ 13 ] for the scenario where the photon is energetic, q − ≫ Λ QCD . This factorization formula for the production of B mesons contains only |${\phi }^{+}_{B}(\omega )$|⁠ , the leading-twist B meson LCDA. Consequently, the relevant transition form factors are expressed in terms of poorly constrained first-inverse moment |$1/ \lambda _B\equiv \int _0^\infty d\omega ~\phi _B^+(\omega )/ \omega$|⁠ , in the same fashion as appeared in QCD calculation of exclusive B meson decays. To constrain λ B , the exclusive production of B mesons through W radiative decay and the radiative B -decays would serve as clean channels. One can find the most recent constraint on λ B from Belle [ 15 ]; this result is expected to be updated at Belle II. However, implementing this strategy at LHCb is challenging due to the difficulty in reconstructing the photon involved in the radiative B → γℓν decay. The analysis would be feasible once the photon further decays into dileptons. Following the measurement reported by the LHCb experiment [ 16 ] on branching ratio, |$\text{Br}(B^+\rightarrow \mu ^+\bar{\nu }_{\mu }\mu ^-\mu ^+)\lt 1.6\times 10^{-8}$|⁠ , recent theoretical studies have been focused on B -decays to four leptons [ 17 , 18 ] within the QCD factorization framework to constrain the phenomenological parameter λ B .

In light of these developments, there is a compelling interest in investigating the potential of HQET factorization for a complicated process W + → B + ℓ + ℓ − . In this paper, our study extends the HQET factorization formula for the process W + → B + ℓ + ℓ − in which q 2 is nonzero, where q 2 is the invariant mass squared of the ℓ + ℓ − pair originating from the virtual photon. Our primary objective is to perform a comprehensive calculation of the form factors associated with the W + → B + ℓ + ℓ − process for both intermediate |$q^2\sim \mathcal {O}(m_b\, \Lambda _{\mathrm{QCD}})$| and small scale |$q^2 \lt \lt m_B^2$|⁠ , within HQET factorization framework, up to NLO in α s at the lowest order in 1/ m b . We also explore the capability of W + → B + ℓ + ℓ − to constrain λ B and hence provide an alternative measurement.

The article is organized as follows: In the next section, we provide the necessary definitions and notations to specify the setup for the calculation of the hard-scattering kernel in HQET. In Section 3 , we present the details of the perturbative QCD calculation on the hard-scattering kernel of the W + → B + ℓ + ℓ − decay process. In Section 4 , we study the phenomenology of this decay process and report numerical predictions on the branching ratio of W + → B + ℓ + ℓ − by invoking an exponential LCDA model. We summarize the work in Section 5 .

In this section, we introduce the notations and definitions used in this work. We start with the kinematics for the process W + → B + ℓ + ℓ − where the momentum of W + is represented by Q with Q = P + q . Here P is the momentum of B + and |$q\, (=q_1+q_2)$| is the momentum of the dilepton (ℓ + ℓ − ) satisfying q 2 ≠ 0. The polarization vector for W + is denoted by ε W .

For convenience, we work in light-cone coordinates by introducing light-like reference vectors, |$n_{\pm }^{\mu }\equiv \frac{1}{\sqrt{2}}(1,0,0,\mp 1)$| that satisfy the conditions |$n_\pm ^2=0$| and n + · n − = 1. This allows us to write any four vector a μ = ( a 0 , a 1 , a 2 , a 3 ) as

where |${a}^{\mu }_{\perp }=(0,a^{1},a^{2},0)$| represents the transverse component of the four vector. It would be convenient to stick with the W boson rest frame to investigate this process as long as we follow the standard collinear factorization approach. In the frame where the W boson is at rest, we assume that the B meson moves along the positive |$\hat{z}$| axis, while the virtual photon moves in the opposite direction to the B meson.

Since the HQET is formulated in the rest frame of the B meson, it is inevitable to boost this process to the B meson rest frame. To achieve this, a dimensionless four velocity v μ is introduced via P μ = m B v μ , satisfying v 2 = 1. The momentum of the virtual photon in the light-cone basis can be decomposed as

The large component of q μ reads

where |$\lambda \equiv m_{W}^{4}+(m_{B}^2-q^{2})^2-2m_{W}^{2}(q^{2}+m_{B}^2)$|⁠ . The physical range of the invariant squared mass of a dilepton is |$4 m_{\ell }^{2} \le q^2\le (m_W-m_B)^2$|⁠ . It is crucial to identify that the amplitude for the exclusive production of a heavy–light meson is highly suppressed for very large |$q^2\sim \mathcal {O}(m_B^2)$| at the heavy quark limit. However, an interesting scenario emerges when |$q^2 \lt \lt m_B^2$|⁠ , while one component of q μ still remains large, |$q^-\sim \mathcal {O}(m_B)$|⁠ , and q + is an order of Λ QCD or even smaller. Hence, with the aid of q 2 = 2 q + q − ≡ s , one can deduce that the “+”-component of q μ is suppressed relative to q − .

For our convenience, the transition amplitude of exclusive W + → B + ℓ + ℓ − decay can be written as

where e and e u are the electric charges of the leptons and the u quark, respectively, θ W is the weak mixing angle, and V ub denotes the CKM matrix element. The scalar quantities F V and F A are the vector and axial-vector form factors, respectively, for the process W + → B + ℓ + ℓ − . These form factors depend on the kinematical variables m W , m B , which encode the nontrivial QCD dynamics and have to be computed for the predictions of exclusive heavy–light meson production. The decay rate for the processes W + → B + ℓ + ℓ − in the W rest frame reads

with α being the QED fine structure constant and

In the following sections, we compute the analytical expression of the form factors F V / A up to the NLO QCD correction at leading order in 1/ m b . Note that any frame of reference can be used to compute these form factors because they are Lorentz scalar. In this study, we continue to use the B rest frame for their computation to make the picture of HQET factorization more transparent.

2.1. B meson LCDA in HQET

In this subsection, we summarize the pivotal aspects of B meson LCDA. Within the framework of factorization, B meson LCDA emerges as the main nonperturbative input for describing the numerous exclusive decays and production processes involving the B meson. Since the LCDA of heavy–light mesons can be defined either in standard QCD or HQET, however, the convolution of the hard-scattering kernel with the LCDA remains identical. In the HQET factorization approach, the LCDA of the B meson can be expressed as an independent pair of nonperturbative functions |$\widetilde{\phi }_{B}^{\pm }$| [ 19 , 20 ]:

In Eq. ( 9 ), the B meson has been deliberately placed in the bra rather than the ket. This choice is made due to our focus on the production of the B meson instead of its decay processes, where z 2 = 0, t = v · z; u is the standard QCD light quark field; and h v refers to the |$\bar{b}$| quark field with the velocity label v defined in HQET. In addition, |$\hat{f}_{B}$| signifies the B meson decay constant defined in HQET. α, β are spinor indices, and [ z , 0] is a light-like gauge link to ensure the gauge invariance of the LCDA,

where |$\mathcal {P}$| indicates the path ordering, g s is a strong coupling constant, and t a ( a = 1, ⋅⋅⋅, 8) refers to SU (3) generators in the fundamental representation. |$A_\mu ^a$| is the gauge field and ξ is the momentum distribution along the Wilson line. The QCD decay constant f B can be found through perturbative matching [ 21 , 22 ],

where μ F is the factorization scale and C F is the color factor. We can obtain the momentum space representation of B meson LCDA by Fourier transforming the coordinate-space correlators provided in Eq. ( 9 ):

Here ω indicates the “+”-momentum carried by the spectator quark in the B rest frame, whose typical value is ∼Λ QCD . A pair of independent, nonperturbative functions is defined through

We will see explicitly that only |$\phi ^{+}_{B}(\omega )$| survives in the HQET factorization approach, which contributes to the form factors F V, A . Similar to B meson LCDA defined in standard QCD, |$\phi ^{+}_{B}(\omega )$| is also scale-dependent, with a scale dependence governed by Lange and Neubert [ 23 ]:

where μ is the renormalization scale. |$\phi ^{+}_{B}(\omega )$| can be deduced at some initial scale |$\mu _0=1\, \mathrm{GeV}$|⁠ ; one can then determine its form at any other scale (typically, |$1\, \mathrm{GeV}\le \mu \le m_b$|⁠ ) by solving the evolution equation ( 14 ), whereas the scale dependence of the meson LCDA defined in full QCD is governed by the Efremov–Radyushkin–Brodsky–Lepage (ERBL) equation [ 24–31 ].

3. HQET factorization and form factors for W → B ℓ + ℓ −

In this section we present the calculation of the hard-scattering kernel at |$\mathcal {O}(\alpha _s)$| to the leading order in 1/ m b . The computation of the form factors F V, A is conducted in the B meson’s rest frame. For this, we follow the NLO calculation of W + → B + γ [ 13 ]; therefore, it is instructive to modify the HQET factorization formula reported in Ref. [ 13 ] for W + → B + ℓ + ℓ − as

where T μ (ω, m b , q 2 , μ F ) is the hard-scattering kernel, which can be computed in perturbation theory by employing the perturbative matching technique. The hard-scattering kernel is also a function of the invariant squared mass of the dilepton, q 2 . In the following section, it is explicitly shown that the hard-scattering kernel for the W → B + γ process can be recovered by the substitution of q 2 → 0 in the expression of T μ (ω, m b , q 2 , μ F ). In perturbative calculation, the hard kernel is independent of the external state, and we safely choose a convenient partonic state as B + meson consists of a |$\bar{b}$| antiquark and a u quark. As a result, the LCDA in Eq. ( 12 ) takes the form

To extract the amplitude of the W + → B + ℓ + ℓ − process in the factorization approach, one can use the B meson momentum projector. For this, the substitution reported in Ref. [ 20 ] may be incorporated at the quark-level amplitude:

where i, j = 1, 2, ⋅⋅⋅, N c are color indices and N c = 3. The first Kronecker symbol serves as the color-singlet projector. The momentum of the spectator quark u is k μ , which is soft and scales as k μ ∼ Λ QCD .

The HQET factorization approach for the exclusive production of heavy–light mesons arises from the heavy quark recombination (HQR) mechanism [ 32 ], specifically for the color-singlet channel. This mechanism achieved noteworthy success in explaining the observed charm/anticharm hadron production asymmetry [ 33–35 ]. The HQR mechanism offers a shortcut to efficiently replicate the heavy meson production amplitude with less computational effort, in contrast to Ref. [ 20 ]. For this, one can invoke the following projector on the quark-level amplitude:

which ensures that the fictitious B + meson is the color- and spin-singlet. It is noteworthy that this simplification has been utilized to evaluate the NLO correction to form factors for the W → B + γ process [ 13 ] in HQET factorization and for W → B c + γ within the NRQCD factorization framework [ 14 , 36 ].

3.1. Factorization at tree level

At the tree level, only three diagrams contribute to the quark-level process, |$W\rightarrow [\bar{b}(P-k) u(k)]+\ell ^+\ell ^-$|⁠ , as shown in Fig. 1 . The momentum of the u quark scales as |$\left(k^+,k^-,|\boldsymbol{k}_\perp |\right) \sim \mathcal {O}(\Lambda _{\mathrm{QCD}},\Lambda _{\mathrm{QCD}},\Lambda _{\mathrm{QCD}})$| while the momentum of the dilepton scales as |$\left(q^+,q^-,|\boldsymbol{q}_\perp |\right) \sim \mathcal {O}(\Lambda _{\mathrm{QCD}},m_{b},0)$|⁠ . Therefore, it is easy to identify that the u quark propagator in Fig. 1(a) is at |$\mathcal {O}(1/\Lambda _{\mathrm{QCD}})$|⁠ , which is the leading order, while in Fig. 1(b) the internal propagator is |$\mathcal {O}(1/m_{b})$|⁠ . Thus, Fig. 1(b) is suppressed by one power of m b compared to Fig. 1(a) ; consequently, the contribution of this diagram could be neglected at the leading twist. Figure 1(c) is also dropped because it is at |$\mathcal {O}\left(1/m_W^{2}\right)$| due to the intermediate W propagator. Hence the leading order contribution comes from the diagram in which the virtual photon is radiated from the spectator quark. In the light of Eq. ( 4 ) the tree-level QCD amplitude in the heavy quark limit reads:

where q 2 represents the propagator of the virtual photon and |$\mathcal {M}^{(0)}$| takes the form

On the other hand, the LCDAs for the fictitious B + meson given in Eq. ( 16 ) come out in the following simple form:

found to be

while the hard kernel at tree level will be

An analytical expression for the form factors can be obtained by comparing the Lorentz decomposition specified in Eq. ( 5 ) with the factorization formula proposed in Eq. ( 15 ). At tree level, it reads

By invoking Eq. ( 17 ), one can prove that the |$\phi _B^-\big (\omega \big )$| terms vanish and, as a result, |$\Phi _{[\bar{b}u]}^-$| does not enter the HQET factorization formula. Here |$\lambda ^{-1}_B(q^{+})$| is the inverse moment of the B meson LCDA, which depends on the invariant squared mass of the dilepton through q + = q 2 /2 q − and is defined as

Note that |$\lambda ^{-1}_B$| scales as |$\lambda ^{-1}_B\sim \Lambda _{\mathrm{QCD}}$| and is also scale-dependent. For q 2 → 0, the expressions of form factors and inverse moment reduce to those [ 13 ] for the real photon.

$Feynman diagrams for $W^{+}\rightarrow [\bar{b} u]+\ell ^+\ell ^-$ at tree level. The bold line represents the $\bar{b}$ quark.$

Feynman diagrams for |$W^{+}\rightarrow [\bar{b} u]+\ell ^+\ell ^-$| at tree level. The bold line represents the |$\bar{b}$| quark.

3.2. Factorization at one-loop level

In this subsection, we present the analytical expression of the hard-scattering kernel at next-to-leading order (NLO) in α s . We can expand the matrix element on the left-hand side of Eq. ( 15 ) in perturbation theory. Thus, up to |${\cal O}(\alpha _s)$|⁠ , it takes the following schematic form:

where ⊗ encodes the convolution integral in ω and superscripts represent the power of α s . We calculate |$T_{\mu }^{(1)}$| by taking |$B^+ =[\bar{b}(P-k) u(k)]$| as

At lower order in 1/ m b , the dominant contribution is an order of |$\Lambda _{\rm QCD}^{-1}$| that arises only through those diagrams for which the virtual photon is emitted from the spectator quark. To have the hard-scattering kernel at NLO, one needs to evaluate |$\mathcal {M}^{(1)}$| from Eq. ( 29 ) in standard QCD and Φ (1) in HQET. The one-loop diagrams for Φ (1) and |$\mathcal {M}^{(1)}$| are represented in Figs. 2 and 3 , respectively. The general principle of effective field theory dictates that the infrared (IR) finite hard kernel at NLO precision can be extracted by evaluating the difference of |$\mathcal {M}^{(1)}$| and |$\Phi ^{(1)}\otimes T_{\mu }^{(0)}$| on a diagram-by-diagram basis, as these quantities contain the same IR singularities. Therefore, it appears instructive to regulate mass (collinear) singularity in the same way for both |$\mathcal {M}^{(1)}$| and Φ (1) ⊗ T (0) ; this can be achieved by taking a nonzero mass m u for the spectator u quark. However, dimensional regularization (with space-time dimensions d = 4 − 2ϵ) is used to regularize UV divergences, and we used the |$\overline{\rm MS}$| renormalization scheme by redefining the ’t Hooft unit mass through |$\mu ^2\rightarrow \mu ^2 {e^{-\gamma _E}\over 4\pi }$|⁠ . The ’t Hooft unit mass μ R is designated for the QCD amplitude |$\mathcal {M}^{(1)}$| calculation, while a different ’t Hooft unit mass μ F is used in computing Φ (1) . We perform the calculation in the Feynman gauge for our convenience.

$One-loop QCD correction to LCDA for a fictitious B meson. The double line represents the $\bar{b}$ field in HQET, the dashed line represents the gauge link.$

One-loop QCD correction to LCDA for a fictitious B meson. The double line represents the |$\bar{b}$| field in HQET, the dashed line represents the gauge link.

$One-loop QCD correction to the amplitude for $W^{+}\rightarrow [\bar{b}u]+\ell ^+\ell ^-$. We consider only those diagrams in which the virtual photon is emitted from the spectator u quark.$

One-loop QCD correction to the amplitude for |$W^{+}\rightarrow [\bar{b}u]+\ell ^+\ell ^-$|⁠ . We consider only those diagrams in which the virtual photon is emitted from the spectator u quark.

To find out |$T_{\mu }^{(1)}(\omega )$| in the light of Eq. ( 29 ), we must calculate |$\mathcal {M}^{(1)}$|⁠ . This can be done by evaluating the one-loop QCD diagrams in Fig. 3 . It is easy to evaluate the electromagnetic vertex correction, weak vertex correction, and internal quark self-energy QCD diagrams as represented in Figs. 3(a) , (b), and (d), respectively. We have also tacitly included those quark mass counterterm diagrams to obtain UV finite results. On the other hand, the NLO perturbative contributions to Φ (1) ⊗ T (0) can be obtained from the soft loop region of the electromagnetic vertex correction, weak vertex correction, and light quark propagator correction in their respective QCD counterparts, as depicted in Fig. 3 . For the calculation of |$\Phi _{+}^{(1)}$|⁠ , one needs the Feynman rules for the Wilson line. If p is the momentum flowing in the gauge link [ 37 ], then the Feynman rules for the propagator are 1/ p + , and |$-ig_sT^a n_+^\mu$| for the eikonal vertex. Applying these Feynman rules to evaluate Fig. 2 diagram by diagram, we found that the contribution to |$\Phi _{+}^{(1)}\otimes T^{(0)}$| is the same as reported in Ref. [ 13 ]. We also noticed that there is no contribution to |$\Phi _{+}^{(1)}\otimes T_{\mu }^{(0)}$| from the gauge link self-energy diagram as shown in Fig. 2(d) because its contribution to |$\Phi _{+}^{(1)}$| is proportional to |$n^2_+=0$|⁠ . Hence, on subtracting the contributions of |$\Phi _{+}^{(1)}\otimes T_{\mu }^{(0)}$| from their respective QCD counterparts as depicted in Figs. 3(a) , (b), and (d), we have the corresponding IR finite hard kernel:

where z = 2 q · k . Now, we look at the wave function correction to the external quark fields. First, we start by considering the wave function correction to the u quark, as shown in Figs. 3(e) and 2(e) . By employing the Lehmann–Symanzik–Zimmermann (LSZ) reduction formula, the corresponding hard kernel reads

For external heavy quark field correction, one needs to consider the external |$\bar{b}$| correction in QCD amplitude, Fig. 3(f) , and HQET LCDA, as illustrated in Fig. 2(f) . One then finds the contribution to |$T_{\mu }^{(1)}$| due to the external heavy quark field correction,

where δ Z q represents the standard q quark wave function renormalization constant defined in full QCD. Now we consider the final one-loop graph, which is the box diagram shown in Fig. 3(c) . At the lowest power of 1/ m b , one can easily find that only the soft region of loop momenta ( l μ ∼ Λ QCD ) yields the leading-order contribution to |$\mathcal {M}^{(1)}$|⁠ . This contribution is exactly equal to |$\Phi ^{(1)}_{\mathrm{box}}\otimes T_{\mu }^{(0)}$| and thus a vanishing contribution to |$T_{\mu }^{(1)}$| from the box diagram has been found, identical to the case observed in Refs. [ 6 , 13 ]. Hence, the hard-scattering kernel at NLO in perturbation theory can be obtained by summing up the nonvanishing hard kernel corresponding to each one-loop diagram:

The hard-scattering kernel ( 33 ) at NLO precision is IR safe, which guarantees that the HQET factorization equation ( 15 ) is applicable for the exclusive production of heavy–light mesons through a semileptonic W gauge boson. It is natural to compare Eq. ( 33 ) with the corresponding result of the rather simpler process, the exclusive production of B mesons via radiative W decay [ 13 ], and find that the expression in Eq. ( 33 ) at q 2 → 0 reduces to the corresponding expression for W + → B + γ [ 13 ]. Now we are in a position to present the central result of this study, the form factors. At leading-order expansion in 1/ m b , the form factor reads

where |$r\equiv m_b^2/m_W^2$| is a dimensionless constant. At the lowest order in 1/ m b , the NLO expression of form factors for W + → B + ℓ + ℓ − also depends on the “+”-momentum of the dilepton pair. This explicitly depends on accounting for the difference when compared with the corresponding expression of form factors reported in Ref. [ 13 ]. However, in the limit q 2 → 0, the expression in Eq. ( 34 ) coincides with the result presented in Ref. [ 13 ]. Here it is important to note that Eq. ( 34 ) exhibits a symmetry relationship among form factors associated with different currents, thereby guaranteeing the preservation of heavy quark spin symmetry at the leading order in the 1/ m b expansion.

In this section, we perform numerical computations to predict the vector/axial-vector form factors associated with the W + → B + ℓ + ℓ − process as well as the corresponding decay width and branching fractions. For this purpose, we use the numerical values of input parameters from PDG [ 38 ] unless stated otherwise. However, for the evaluation of QCD running coupling α s at one-loop accuracy, we invoke the atuomated package HOPPET [ 39 ]:

It is worth mentioning here that our theoretical predictions are based on the Grozin–Neubert exponential model [ 19 ], where heavy–light meson LCDA at the initial scale μ 0 = 1 GeV is defined as

with |$\lambda _B\equiv \lambda ^{+}_B(q^{+}=0)=0.350\pm 0.15$| GeV [ 40 ].

In the current study, we have calculated the leading-order (LO) and next-to-leading-order (NLO) QCD corrections to the form factors for the decay W + → B + ℓ + ℓ − , which are defined as |$F_{V/A}^{\text{LO}}\equiv F_{V/A}^{(0)}$| and |$F_{V/A}^{\text{NLO}}\equiv F_{V/A}^{(0)}+ F_{V/A}^{(1)}$|⁠ . The form factors |$F_{V/A}^{(0)}$| and |$F_{V/A}^{(1)}$| are given in Eqs. ( 25 ) and ( 34 ), respectively. One can see from these equations that the form factors rely on B meson LCDA, which demonstrates scale dependence μ F . Therefore, to understand the variation in the vector/axial-vector form factors arising due to the factorization scale μ F , it is essential to first know the μ F dependence of LCDA. Consequently, one can calculate the sensitivity of physical observables such as decay rates and branching fractions to the scale μ F . To achieve this goal, we use the analytical solutions of the Lange–Neubert evolution equation ( 14 ) as reported in Refs. [ 41 , 42 ], to obtain the form factors at desired scales as a function of the invariant squared mass of the dilepton, q 2 . In Fig. 4 , we plot the form factors at LO and NLO in α s as a function of q 2 . The band indicates the uncertainty arising from the factorization scale, μ F . We divide the variation in μ F into two intervals: |$1\, \mathrm{GeV}\le \mu _1\le m_B$| (left) |$\&$| |$m_B\le \mu _2\le 10\, \mathrm{GeV}$| (right); one can see that the q 2 dependence of the form factors is very mild for both intervals while the color bands depict the scale dependence. The blue, red, and gray bands correspond to the LO form factors ( ⁠|$F_{V/A}^{\text{NLO}}$|⁠ ), the real part of the NLO form factors ( ⁠|$\mathcal {R}e[F_{V/A}^{\text{NLO}}]$|⁠ ), and the imaginary part of the NLO form factors ( ⁠|$\mathcal {I}m[F_{V/A}^{\text{NLO}}]$|⁠ ), respectively. It is observed that these uncertainty bands of |$F_{V/A}^{\text{LO}}$| and |$F_{V/A}^{\text{NLO}}$| (both computed at precision in α s ) turn out to be well separated as μ varies within the μ 1 interval while, for the μ 2 interval, the uncertainty bands reduce and overlap for both |$F_{V/A}^{\text{LO}}$| and |$\mathcal {R}e[F_{V/A}^{\text{NLO}}]$|⁠ ; however, |$\mathcal {I}m[F_{V/A}^{\text{NLO}}]$| is not significantly changed. As one can also see from Fig. 4(b) , the reduction in |$F_{V/A}^{\text{NLO}}$| is greater than that for |$F_{V/A}^{\text{LO}}$| for |$m_B\le \mu _2\le 10\, \mathrm{GeV}$|⁠ , which is attributed to the inclusion of the α s correction in the form factors. This ensures that the decay rates at NLO for W + → B + ℓ + ℓ − processes in the μ 2 interval are almost insensitive as shown in Fig. 5 . However, for relatively small scales, the NLO form factors, |$F_{V/A}^{\text{NLO}}$|⁠ , still reflect a notable dependence on μ F . This residual scale dependence could be eliminated by incorporating higher-order QCD corrections.

The q 2 dependence of the vector/axial-vector form factors at LO and NLO in α s . The band represents the uncertainty from μ F = 1 to m B (left) and μ F = m B to 10 GeV (right).

Decay rates of W + → B + ℓ + ℓ − as a function of μ F , which varies from 1–10 GeV.

Moreover, in our NLO predictions for decay rates, we also include the imaginary part of the one-loop corrections to the form factors, |$\mathcal {I}m[F_{V/A}^{(1)}]$|⁠ , without strictly truncating the decay width at |$\mathcal {O}(\alpha _s)$|⁠ . To show the μ F dependence, for the decay rates W + → B + ℓ + ℓ − (ℓ = e , μ, τ) against μ F , we integrated over |$q^2\epsilon [4m_\mu ^2,6]$| for the muon; to avoid the photon pole, we use the same q 2 bin for the electron as well, while for the tauon we take q 2 ϵ[14, 20], plotted in Fig. 5 . From this figure, one can see that for the case of the tauon as a final state lepton, the decay rate is suppressed by around three orders in comparison with the electron and muon cases; this is quantified in Table 1 . However, for W + → B + ℓ + ℓ − (ℓ = e , μ), the LO decay rates (Γ LO ) strongly depend upon μ F . Similarly, the NLO decay rates (Γ NLO ) at a low μ F scale, from 1–4 GeV, are also highly sensitive. On the other hand, the sensitivity is very mild at a relatively large scale, above 4 GeV. This feature emerges because one-loop QCD corrections generate significant precision in the form factors. Consequently, the scale dependence in the NLO decay rates (Γ NLO ) is greatly reduced, particularly for relatively large μ F . The profiles of the decay rates against the factorization scale show similar behavior as seen in W + → B + γ [ 13 ]. We have also calculated the numerical values of these decay rates (Γ LO , Γ NLO ) and the branching fractions (Br NLO ) at NLO by varying μ F from 1–10 GeV; these are listed in Table 1 . We found that the NLO corrections turn out to be substantial and may vary from −76% to +58% of the LO decay rates for the case of the electron and −79% to +52% for the muon, whereas for the tauon it varies from −63% to +72%.

Numerical predictions of the decay rates and branching ratios for the processes W + → B + ℓ + ℓ − with ℓ = e , μ, τ. The uncertainty is estimated by varying μ F from 1–10 GeV at λ B = 0.35 GeV after integrating over |$q^2\epsilon [4m_\mu ^2,6]$| for the electron and the muon, and over q 2 ϵ[14, 20] for the tauon.

Decay channel .	\|$\vphantom{\frac{L^L}{L^L}}\Gamma ^{\mathrm{LO}}$\| .	Γ .	Br .
\|$\vphantom{\frac{L^L}{L^L}}W^{+}\rightarrow B^{+}e^+e^-$\|	(6.37–2.71) × 10 GeV	(1.51–4.10) × 10 GeV	(0.72–1.97) × 10
\|$\vphantom{\frac{L^L}{L^L}}W^{+}\rightarrow B^{+}\mu ^+\mu ^-$\|	(4.23–1.80) × 10 GeV	(0.87–2.73) × 10 GeV	(0.42–1.31) × 10
\|$\vphantom{\frac{L^L}{L^L}}W^{+}\rightarrow B^{+}\tau ^+\tau ^-$\|	(2.39–1.06) × 10 GeV	(0.87–1.82) × 10 GeV	(0.42–0.87) × 10

Decay channel .	\|$\vphantom{\frac{L^L}{L^L}}\Gamma ^{\mathrm{LO}}$\| .	Γ .	Br .
\|$\vphantom{\frac{L^L}{L^L}}W^{+}\rightarrow B^{+}e^+e^-$\|	(6.37–2.71) × 10 GeV	(1.51–4.10) × 10 GeV	(0.72–1.97) × 10
\|$\vphantom{\frac{L^L}{L^L}}W^{+}\rightarrow B^{+}\mu ^+\mu ^-$\|	(4.23–1.80) × 10 GeV	(0.87–2.73) × 10 GeV	(0.42–1.31) × 10
\|$\vphantom{\frac{L^L}{L^L}}W^{+}\rightarrow B^{+}\tau ^+\tau ^-$\|	(2.39–1.06) × 10 GeV	(0.87–1.82) × 10 GeV	(0.42–0.87) × 10

In addition, the theoretical predictions for the branching ratios are also influenced by the parameter λ B (μ 0 ), because it affects the HQET factorization through |$\phi _{B}^{+}$|⁠ . Therefore, the analysis of branching fractions as a function of λ B is a handy tool to precisely constrain this parameter. For this purpose, to see the sensitivity of the branching fraction to λ B , we plotted it against λ B by using the range |$\lambda _B(\mu _0)=0.35\, \mathrm{GeV}\pm 0.15 \, \mathrm{GeV}$| [ 40 ], shown in Fig. 6 by the green and red bands. Figure 6(a) depicts when the leptons in the final state are muons, while Fig. 6(b) corresponds to the case of tauons as final state leptons. The widths of the green and red bands represent the variation by μ F in the intervals |$1\, \mathrm{GeV}\le \mu _1\le m_B$| and |$m_B\le \mu _2\le 10\, \mathrm{GeV}$|⁠ , respectively.

$Illustration of the inverse moment λB(μ0) dependence of the branching fraction for the decay modes W+ → B+ℓ+ℓ−. The uncertain band shows the variation in μF from 1 GeV to the meson mass.$

Illustration of the inverse moment λ B (μ 0 ) dependence of the branching fraction for the decay modes W + → B + ℓ + ℓ − . The uncertain band shows the variation in μ F from 1 GeV to the meson mass.

The green band indicates a strong dependence on λ B compared to uncertainty arising from the scale μ F . Similarly, the branching fraction exhibits higher sensitivity to smaller values of λ B compared to larger values. Therefore, the branching fraction in the interval |$m_B\le \mu _2\le 10\, \mathrm{GeV}$| is more suitable to extract the precise value of λ B , particularly around the lower value of λ B ≃ 0.24 GeV, which is measured by Belle with |$90\%$| C.L.

To further explore how the parametric dependence of the decay rates varies in the different q 2 bins, numerical values of the decay rates are calculated for the processes W → B + ℓ + ℓ − where ℓ = e , μ, τ and listed in Table 2 . To get the numerical values, we have integrated over three q 2 bins, |$[4m_\mu ^2,0.96]$|⁠ , |$[4m_{\mu }^2,6]$|⁠ , and [2,6], for the case of the electron and the muon, while for the tauon we have selected the q 2 bin above the |$c\bar{c}$| resonance region, i.e. [14,20]. In the first and second columns, we have listed the numerical values for the hard (μ h ) and hard-collinear (μ hc ) scales by setting μ hc = 1.5, μ h = 5 GeV and λ B = 0.35 GeV for LO and NLO, respectively. In the remaining columns, we have given the uncertainties in the decay rates by using the ranges of parameters: μ hc = 1.5 ± 0.5 GeV, |$\mu _h=5^{+5}_{-2.5}$| GeV, and λ B = 0.35 ± 0.15 GeV. The total uncertainty is calculated by adding the uncertainties due to μ h, hc and λ B in quadrature.

The numerical values of decay rates are integrated over the different q 2 bins for the processes W → B + ℓ + ℓ − where ℓ = e , μ, τ. In the first and second columns, we have listed the numerical values by setting |${\mu _{hc}=1.5\, \mathrm{GeV},\mu _h=5}$| GeV, and λ B = 0.35 GeV for LO and NLO, respectively. In the remaining columns, we have given the uncertainties in the decay rates by using the ranges of parameters: μ hc = 1.5 ± 0.5 GeV, |$\mu _h=5^{+5}_{-2.5}$| GeV, and λ B = 0.35 ± 0.15 GeV. The total uncertainty is calculated by adding the uncertainties due to μ h, hc and λ B in quadrature.

Decay .	bin .	LO .	NLO .	Uncertainty .
.	GeV .	λ = 0.35 GeV .	λ = 0.35 GeV .	LO .	NLO .	LO (λ ) .	NLO (λ ) .	LO (tot) .	NLO (tot) .
.	.	(μ , μ ) = (1.5, 5) GeV .	(μ , μ ) = (1.5, 5)GeV .	(μ , μ ) .	(μ , μ ) .	(1.5, 5) GeV .	(1.5, 5) GeV .	(1.5, 5) GeV .	(1.5, 5) GeV .
	\|$[4m_\mu ^2,0.96]$\|	(5.25, 3.35) × 10	(2.14, 3.74) × 10	\|$(^{+0.88}_{-0.52},^{+1.00}_{-0.75})$\|	\|$(^{+0.58}_{-0.90},^{+0.20}_{-0.65})$\|	\|$(^{+9.72}_{-2.56},^{+0.49}_{-1.47})$\|	\|$(^{+4.27}_{-0.93},^{+5.56}_{-1.60})$\|	(⁠\|$^{+9.76}_{-2.61},^{+1.11}_{-1.65}$\|⁠)	\|$(^{+4.31}_{-1.29},^{+5.56}_{-1.73})$\|
	\|$[4m_\mu ^2,6]$\|	(5.46, 3.48) × 10	(2.24, 3.90) × 10	\|$(^{+0.91}_{-0.54},^{+1.04}_{-0.77})$\|	\|$(^{+0.61}_{-0.93},^{+0.21}_{-0.68})$\|	\|$(^{+10.09}_{-2.66},^{+5.14}_{-1.53})$\|	\|$(^{+4.45}_{-0.97},^{+5.79}_{-1.67})$\|	\|$(^{+10.13}_{-2.71},^{+5.24}_{-1.71})$\|	\|$(^{+4.49}_{-1.34},^{+5.79}_{-1.80})$\|
	[2,6]	(7.78, 5.00) × 10	(3.81, 6.01) × 10	\|$(^{+1.27}_{-0.76},^{+1.47}_{-1.10})$\|	\|$(^{+0.86}_{-1.52},^{+0.19}_{-0.84})$\|	\|$(^{+13.71}_{-3.74},^{+7.04}_{-2.16})$\|	\|$(^{+6.77}_{-1.65},^{+8.72}_{-2.57})$\|	(⁠\|$^{+13.77}_{-3.82},^{+7.19}_{-2.42}$\|⁠)	\|$(^{+6.82}_{-2.24},^{+8.72}_{-2.70})$\|
μ μ	\|$[4m_\mu ^2,0.96]$\|	(3.41, 2.18) × 10	(1.40, 2.44) × 10	(⁠\|$^{+0.57}_{-0.34},^{+0.65}_{-0.48}$\|⁠)	\|$(^{+0.38}_{-0.58},^{+1.29}_{-0.42})$\|	(⁠\|$^{+6.31}_{-1.66},^{+3.22}_{-0.96}$\|⁠)	\|$(^{+2.79}_{-0.61},^{+3.62}_{-1.05})$\|	(⁠\|$^{+6.34}_{-1.69},^{+3.29}_{-1.07}$\|⁠)	\|$(^{+2.82}_{-0.84},^{+3.84}_{-1.13})$\|
	\|$[4m_\mu ^2,6]$\|	(3.62, 2.31) × 10	(1.49, 2.59) × 10	\|$(^{+0.61}_{-0.36},^{+0.69}_{-0.51})$\|	\|$(^{+0.40}_{-0.62},^{+0.14}_{-0.45})$\|	(⁠\|$^{+6.68}_{-1.76},^{+3.40}_{-1.01}$\|⁠)	\|$(^{+2.97}_{-0.65},^{+3.85}_{-1.11})$\|	(⁠\|$^{+6.71}_{-1.80},^{+3.47}_{-1.13}$\|⁠)	\|$(^{+3.00}_{-0.90},^{+3.85}_{-1.20})$\|
	[2,6]	(7.72, 4.97) × 10	(3.78, 5.96) × 10	\|$(^{+1.26}_{-0.76},^{+1.45}_{-1.09})$\|	\|$(^{+6.72}_{-1.64},^{+8.66}_{-2.55})$\|	(⁠\|$^{+13.60}_{-3.71},^{+6.99}_{-2.15}$\|⁠)	\|$(^{+2.42}_{-0.60},^{+3.13}_{-0.93})$\|	(⁠\|$^{+13.66}_{-3.79},^{+7.14}_{-2.41}$\|⁠)	\|$(^{+7.14}_{-1.75},^{+9.21}_{-2.71})$\|
τ τ	[14,20]	(0.21, 0.14) × 10	(0.13, 0.18) × 10	\|$(^{+0.32}_{-0.19},^{+0.38}_{-0.29})$\|	\|$(^{+0.45}_{-0.22},^{+0.00}_{-0.16})$\|	(⁠\|$^{+3.23}_{-0.95},^{+1.70}_{-0.56}$\|⁠)	(⁠\|$^{+1.89}_{-0.55},^{+2.40}_{-0.75}$\|⁠)	(⁠\|$^{+3.25}_{-0.97},^{+1.74}_{-0.63}$\|⁠)	(⁠\|$^{+1.94}_{-0.59},^{+2.4}_{-0.77}$\|⁠)

Decay .	bin .	LO .	NLO .	Uncertainty .
.	GeV .	λ = 0.35 GeV .	λ = 0.35 GeV .	LO .	NLO .	LO (λ ) .	NLO (λ ) .	LO (tot) .	NLO (tot) .
.	.	(μ , μ ) = (1.5, 5) GeV .	(μ , μ ) = (1.5, 5)GeV .	(μ , μ ) .	(μ , μ ) .	(1.5, 5) GeV .	(1.5, 5) GeV .	(1.5, 5) GeV .	(1.5, 5) GeV .
	\|$[4m_\mu ^2,0.96]$\|	(5.25, 3.35) × 10	(2.14, 3.74) × 10	\|$(^{+0.88}_{-0.52},^{+1.00}_{-0.75})$\|	\|$(^{+0.58}_{-0.90},^{+0.20}_{-0.65})$\|	\|$(^{+9.72}_{-2.56},^{+0.49}_{-1.47})$\|	\|$(^{+4.27}_{-0.93},^{+5.56}_{-1.60})$\|	(⁠\|$^{+9.76}_{-2.61},^{+1.11}_{-1.65}$\|⁠)	\|$(^{+4.31}_{-1.29},^{+5.56}_{-1.73})$\|
	\|$[4m_\mu ^2,6]$\|	(5.46, 3.48) × 10	(2.24, 3.90) × 10	\|$(^{+0.91}_{-0.54},^{+1.04}_{-0.77})$\|	\|$(^{+0.61}_{-0.93},^{+0.21}_{-0.68})$\|	\|$(^{+10.09}_{-2.66},^{+5.14}_{-1.53})$\|	\|$(^{+4.45}_{-0.97},^{+5.79}_{-1.67})$\|	\|$(^{+10.13}_{-2.71},^{+5.24}_{-1.71})$\|	\|$(^{+4.49}_{-1.34},^{+5.79}_{-1.80})$\|
	[2,6]	(7.78, 5.00) × 10	(3.81, 6.01) × 10	\|$(^{+1.27}_{-0.76},^{+1.47}_{-1.10})$\|	\|$(^{+0.86}_{-1.52},^{+0.19}_{-0.84})$\|	\|$(^{+13.71}_{-3.74},^{+7.04}_{-2.16})$\|	\|$(^{+6.77}_{-1.65},^{+8.72}_{-2.57})$\|	(⁠\|$^{+13.77}_{-3.82},^{+7.19}_{-2.42}$\|⁠)	\|$(^{+6.82}_{-2.24},^{+8.72}_{-2.70})$\|
μ μ	\|$[4m_\mu ^2,0.96]$\|	(3.41, 2.18) × 10	(1.40, 2.44) × 10	(⁠\|$^{+0.57}_{-0.34},^{+0.65}_{-0.48}$\|⁠)	\|$(^{+0.38}_{-0.58},^{+1.29}_{-0.42})$\|	(⁠\|$^{+6.31}_{-1.66},^{+3.22}_{-0.96}$\|⁠)	\|$(^{+2.79}_{-0.61},^{+3.62}_{-1.05})$\|	(⁠\|$^{+6.34}_{-1.69},^{+3.29}_{-1.07}$\|⁠)	\|$(^{+2.82}_{-0.84},^{+3.84}_{-1.13})$\|
	\|$[4m_\mu ^2,6]$\|	(3.62, 2.31) × 10	(1.49, 2.59) × 10	\|$(^{+0.61}_{-0.36},^{+0.69}_{-0.51})$\|	\|$(^{+0.40}_{-0.62},^{+0.14}_{-0.45})$\|	(⁠\|$^{+6.68}_{-1.76},^{+3.40}_{-1.01}$\|⁠)	\|$(^{+2.97}_{-0.65},^{+3.85}_{-1.11})$\|	(⁠\|$^{+6.71}_{-1.80},^{+3.47}_{-1.13}$\|⁠)	\|$(^{+3.00}_{-0.90},^{+3.85}_{-1.20})$\|
	[2,6]	(7.72, 4.97) × 10	(3.78, 5.96) × 10	\|$(^{+1.26}_{-0.76},^{+1.45}_{-1.09})$\|	\|$(^{+6.72}_{-1.64},^{+8.66}_{-2.55})$\|	(⁠\|$^{+13.60}_{-3.71},^{+6.99}_{-2.15}$\|⁠)	\|$(^{+2.42}_{-0.60},^{+3.13}_{-0.93})$\|	(⁠\|$^{+13.66}_{-3.79},^{+7.14}_{-2.41}$\|⁠)	\|$(^{+7.14}_{-1.75},^{+9.21}_{-2.71})$\|
τ τ	[14,20]	(0.21, 0.14) × 10	(0.13, 0.18) × 10	\|$(^{+0.32}_{-0.19},^{+0.38}_{-0.29})$\|	\|$(^{+0.45}_{-0.22},^{+0.00}_{-0.16})$\|	(⁠\|$^{+3.23}_{-0.95},^{+1.70}_{-0.56}$\|⁠)	(⁠\|$^{+1.89}_{-0.55},^{+2.40}_{-0.75}$\|⁠)	(⁠\|$^{+3.25}_{-0.97},^{+1.74}_{-0.63}$\|⁠)	(⁠\|$^{+1.94}_{-0.59},^{+2.4}_{-0.77}$\|⁠)

In this paper, we have studied the production of heavy–light mesons at NLO in strong coupling α s at leading order in 1/ m b through semileptonic W boson decay in the HQET factorization approach. Here, we have extended the HQET factorization formula for a rather sophisticated decay for W → B ℓ + ℓ − validating the scale hierarchy, m W ∼ m b ≫ Λ QCD , in a similar fashion to that reported in Ref. [ 13 ]. However, the amplitude of the decay process under consideration is highly suppressed when |$q^2\sim \mathcal {O}(m_B^2)$|⁠ . However, we have shown that the amplitude for the exclusive production of a heavy–light meson can be factorized at both the soft scale, |$q^2 \lt \lt m_b^2$|⁠ , and the intermediate scale, |$q^2\sim \mathcal {O}(m_b\, \Lambda _{\mathrm{QCD}})$|⁠ . To achieve this goal, we explicitly calculated the hard-scattering kernel at NLO, which is IR finite.

In addition, we have calculated the form factors associated with the process W → B ℓ + ℓ − up to α s at the lowest order in 1/ m b . It has been determined that both of these form factors are identical up to α s at the lowest order in 1/ m b . This confirms the heavy quark spin symmetry and supports the accuracy of our results. Furthermore, it is important to emphasize that our results for form factors and hard kernels are consistent with the corresponding findings of Ref. [ 13 ] when the value of q 2 approaches zero. We have also investigated the impact of NLO perturbative corrections for the numerical prediction of vector and axial-vector form factors appearing in W + → B + ℓ + ℓ − and presented the q 2 as well as the factorization scale μ F dependence of these form factors. The results of our study suggest that the form factors show very little variation for q 2 . Additionally, within the range of |$m_B\le \mu _F\le 10\, \mathrm{GeV}$|⁠ , the form factors also become unaffected by variations in the factorization scale μ F .

Moreover, by using these form factors, we have calculated the decay rates and branching ratios for the processes W → B + ℓ + ℓ − where ℓ = e , μ, τ at λ B = 0.35 GeV. We have observed significant NLO corrections, ranging from a decrease of |$-76\%$| to an increase of |$+58\%$| in the decay rates for electrons; for muons the range is |$-79\%$| to |$+52\%$|⁠ . Although for the case of tauons the decay rate is suppressed by around three orders in comparison with the electron and muon cases, the NLO corrections range from a decrease of |$-63\%$| to an increase of |$+72\%$|⁠ , depending on the variation of the scale parameter μ F from 1–10 GeV. The factorization formula in the current study is valid for both low- and intermediate- q 2 regions. Therefore, we have also computed the numerical values of the decay rates in different q 2 bins. It is found that the branching ratios are sensitive to the first inverse moment λ B , particularly for relatively large values of the factorization scale. Therefore, the branching ratios of W → B + ℓ + ℓ − within the range of |$m_B\le \mu _F\le 10\, \mathrm{GeV}$| are more suited to constrain the value of λ B . Hence, this decay channel could be used as an additional source to pin down the precise value of λ B , which explicitly appears in both the production and decay processes of B mesons.

Open Access funding: SCOAP 3 .

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Research progress of advanced design method, numerical simulation, and experimental technology of pumps in deep-sea resource exploitation.

1. Introduction

2. design and optimization methods, 2.1. performance of pump, 2.2. performance of deep sea lift pump.

References	Existing Correlation for Head Reduction Factor “K ”
Fairbank [ ]
Cave [ ]
Wesserroth [ ]
Gahlot [ ]
Kazim [ ]
Vocadlo [ ]
Engin [ ]
Engin [ ]

2.3. Design Method of Deep Sea Lift Pump

2.3.1. enlarged flow design method, 2.3.2. inverse design method, 2.4. optimization methods, 2.4.1. parameter modeling optimization, 2.4.2. alternative optimization methods, 3. numerical simulation methods, 3.1. control equations of fluids, 3.2. euler–euler model, 3.2.1. two-fluid model, 3.2.2. mixture model, 3.3. discrete element method, 3.3.1. soft-sphere models, 3.3.2. particle–fluid interaction forces, 3.3.3. coupling scheme between cfd and dem, 3.3.4. applications of cfd–dem in deep-sea lift pump, 3.4. discrete phase model (dpm), 3.5. model modification, modified drag model, 4. experimental research, 4.1. external characteristic experiment, 4.2. visualization experiment, 4.3. wear experiment, 5. conclusions and outlook, 5.1. conclusions, 5.2. outlook.

Presently, no robust design methodologies exist for deep-sea lifting pumps; however, employing numerical simulations in conjunction with optimization algorithms and parametric modeling proves significantly efficacious for the design of these pumps under complex operational conditions. However, the optimization of deep-sea lifting pumps in solid–liquid two-phase flow conditions remain inadequately explored.
Optimization studies of deep-sea lifting pumps predominantly consider efficiency in steady-state conditions designed for constant speed. Considering the complexity of deep-sea mining applications, it is pertinent to evaluate efficiency across variable speeds and different operating states. Moreover, attention must be given to the impacts of wear on the pump’s efficiency and reliability, suggesting a shift towards optimization goals that include multi-speed, multi-state efficiency, and wear characteristics.
In solid–liquid two-phase flows, the presence of solid particles can induce changes in the flow pattern, particle clogging and deposition, wear, and erosion, as well as uneven particle concentration and distribution. These factors may lead to flow instability and separation, thereby triggering rotational stalls. Rotational stalls can significantly affect the efficiency of pumps, increase vibration and noise, exacerbate component wear, and lead to operational instability, negatively impacting the performance and lifespan of the pump. Therefore, to effectively prevent and mitigate the occurrence of rotational stalls, it is necessary to optimize the design of impellers and flow passages and to adjust and maintain operating conditions. These strategies are crucial for enhancing system reliability and efficiency, and for extending the service life of the equipment.
Considering the particle systems within deep-sea lifting pumps, which often comprise a large number of particles, the associated (DEM) simulations require substantial computational effort. The computational efficiency is further compromised when DEM is coupled with computational fluid dynamics (CFD). Thus, enhancing the computational efficiency of the integrated CFD-DEM approach is crucial for practical deployment. Integration of the Coarse-Grained Methodology (CGM) into the CFD-DEM framework could substantially lower computational demands. Modern commercial software now often incorporates GPU-accelerated parallel computation techniques to facilitate CFD-DEM solvers, as evidenced by platforms such as EDEM and Rocky DEM.
The CFD-DEM approach is predominantly employed for investigating solid–liquid two-phase flows in rotating machinery. However, the complex structural demands of these machines make mesh generation particularly challenging. The Smoothed Particle Hydrodynamics (SPH) method, representing a novel mesh-free approach, provides extensive potential for addressing complex challenges such as free surface flows, multiphase flows, and interactions between fluids and particles.
Modifications to computational models for two-phase flow pumps have traditionally focused on the effects of turbulence on drag forces. However, the influence of additional forces warrants consideration. It is imperative that these models are refined to enhance the simulation accuracy of slurry pumps.
Particles in deep-sea lifting pumps are typically heterogeneous and non-spherical. Much of the extant research is based on spherical particles, with a notable deficiency in studies addressing non-spherical particles and their fragmentation.
For visualization experiments concerning deep-sea lifting pumps, the identification and velocimetry of particles present intricate challenges. A promising resolution could involve deploying advanced image processing technologies anchored in machine learning methodologies. Such enhancements are expected to enrich the database for impellers in two-phase flow pumps significantly.

Data Availability Statement

Conflicts of interest.

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Click here to enlarge figure

Parameter	Equation
Impeller inlet diameter
Impeller blade outlet width
Impeller outlet diameter
Number of impeller blades
The width of diffuser blades
Number of diffuser blades

Authors	Optimization Algorithm	Pump Type	Efficiency Optimization	Head Optimization
Pei et al. [ ]	Improved particle swarm optimization	Centrifugal pump	-	-
Gong et al. [ ]	Improved bat algorithm	Seawater desalination high pressure multistage pump	+3.98%	-
Lu et al. [ ]	Radial basis function (RBF) neural network multi-islands genetic algorithm (MIGA)	Mixed-flow pump	+4.3%	-
Derakhshan et al. [ ]	Artificial colony algorithm	Centrifugal pump	+3.59%	+6.89 m
Wu et al. [ , ]	Artificial neural network Genetic algorithm	Multistage centrifugal pump	+2.8%, (CMEI decreases by 1.34%)	+8.8%

Normal Contact Force	Equation
Linear spring–dashpot model [ ]
Hysteretic linear spring model [ ]
Hertzian spring–dashpot model [ ]

References	Correlations
Haider and Levenspiel [ ]
Ganser [ ]
Leith [ ]
Tran-Cong et al. [ ]
Hölzer and Sommerfeld [ ]

References	Correlations	Notes
Sanjeevi et al. [ ]		For spherocylinder and ellipsoid Valid when 0.1 ≤ Rep ≤ 2000
Zastawny et al. [ ]		For spherocylinder and ellipsoid Valid when 0.1 ≤ Rep ≤ 300
Cao and Tafti [ ]		For cylinder with w = 0.25 Valid when 10 ≤ Rep ≤ 300
Richter and Nikrityuk [ ]		For ellipsoid with w = 2 Valid when 10 ≤ Rep ≤ 200

Consortium	KCON (Kennecott Consortium)	OMA (Ocean Mining Association)	OMI (Ocean Management Incorporated)	OMCO (Ocean Mineral Company)
Established	1974	1974	1975	1977
Mining	1975–1976; collector model test at depth of 5000 m (Shaw [ ])	1970; mining system test at the depth of 800 m	1976; collector test in deep sea	1978; mining system test in shallower water
Test	1978; lift up test on-land	1978; mining pilot test in the DOMES site C (4400 m) and lift up 500 t nodules	1978; pilot miner test in the DOMES A(5000–5200 m) and lift up 500 t nodules (Chung [ ])	1979; mining system test in deep sea (Xiao [ ])
Target	Ni, Co, Cu	Ni, Cu, Co, Mn	Ni, Cu, Co	-

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Ji, L.; He, X.; Li, W.; Tian, F.; Shi, W.; Zhou, L.; Liu, Z.; Yang, Y.; Xiao, C.; Agarwal, R. Research Progress of Advanced Design Method, Numerical Simulation, and Experimental Technology of Pumps in Deep-Sea Resource Exploitation. Water 2024 , 16 , 1881. https://doi.org/10.3390/w16131881

Ji L, He X, Li W, Tian F, Shi W, Zhou L, Liu Z, Yang Y, Xiao C, Agarwal R. Research Progress of Advanced Design Method, Numerical Simulation, and Experimental Technology of Pumps in Deep-Sea Resource Exploitation. Water . 2024; 16(13):1881. https://doi.org/10.3390/w16131881

Ji, Leilei, Xinrui He, Wei Li, Fei Tian, Weidong Shi, Ling Zhou, Zhenbo Liu, Yang Yang, Cui Xiao, and Ramesh Agarwal. 2024. "Research Progress of Advanced Design Method, Numerical Simulation, and Experimental Technology of Pumps in Deep-Sea Resource Exploitation" Water 16, no. 13: 1881. https://doi.org/10.3390/w16131881

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As we know that, universal gravitational constant G = 6.673 ×10−11. The acceleration due to gravity on the surface of the moon can be computed by using the formula as below: g = GM r2. Substituting the values, g = 6.673×10−11×7.35×1022 (1.74×106)2. g = 1.620 ms−2. Hence, value of the acceleration due to gravity is 1.620 ms−2.
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mytrialbuisness October 10, 2017 at 12:41 pm. Say if you wanted to find acceleration caused by gravity, the accepted value would be the acceleration caused by gravity on earth (9.81…), and the experimental value would be what you calculated gravity as 🙂
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Semileptonic W Decay to the B Meson with Lepton Pairs in Heavy Quark
where |${a}^{\mu }_{\perp }=(0,a^{1},a^{2},0)$| represents the transverse component of the four vector. It would be convenient to stick with the W boson rest frame to investigate this process as long as we follow the standard collinear factorization approach. In the frame where the W boson is at rest, we assume that the B meson moves along the positive |$\hat{z}$| axis, while the virtual ...
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Gahlot et al. investigated the performance of centrifugal slurry pumps under different particle size distributions, by using existing empirical correlations to predict performance with large deviations compared to experimental data, due to improper calculations of the specific gravity of solids or the use of d 50 as representative particle size.

Decay .	bin .	LO .	NLO .	Uncertainty .
.	GeV .	λ = 0.35 GeV .	λ = 0.35 GeV .	LO .	NLO .	LO (λ ) .	NLO (λ ) .	LO (tot) .	NLO (tot) .
.	.	(μ , μ ) = (1.5, 5) GeV .	(μ , μ ) = (1.5, 5)GeV .	(μ , μ ) .	(μ , μ ) .	(1.5, 5) GeV .	(1.5, 5) GeV .	(1.5, 5) GeV .	(1.5, 5) GeV .
	\|$[4m_\mu ^2,0.96]$\|	(5.25, 3.35) × 10	(2.14, 3.74) × 10	\|$(^{+0.88}_{-0.52},^{+1.00}_{-0.75})$\|	\|$(^{+0.58}_{-0.90},^{+0.20}_{-0.65})$\|	\|$(^{+9.72}_{-2.56},^{+0.49}_{-1.47})$\|	\|$(^{+4.27}_{-0.93},^{+5.56}_{-1.60})$\|	(⁠\|$^{+9.76}_{-2.61},^{+1.11}_{-1.65}$\|⁠)	\|$(^{+4.31}_{-1.29},^{+5.56}_{-1.73})$\|
	\|$[4m_\mu ^2,6]$\|	(5.46, 3.48) × 10	(2.24, 3.90) × 10	\|$(^{+0.91}_{-0.54},^{+1.04}_{-0.77})$\|	\|$(^{+0.61}_{-0.93},^{+0.21}_{-0.68})$\|	\|$(^{+10.09}_{-2.66},^{+5.14}_{-1.53})$\|	\|$(^{+4.45}_{-0.97},^{+5.79}_{-1.67})$\|	\|$(^{+10.13}_{-2.71},^{+5.24}_{-1.71})$\|	\|$(^{+4.49}_{-1.34},^{+5.79}_{-1.80})$\|
	[2,6]	(7.78, 5.00) × 10	(3.81, 6.01) × 10	\|$(^{+1.27}_{-0.76},^{+1.47}_{-1.10})$\|	\|$(^{+0.86}_{-1.52},^{+0.19}_{-0.84})$\|	\|$(^{+13.71}_{-3.74},^{+7.04}_{-2.16})$\|	\|$(^{+6.77}_{-1.65},^{+8.72}_{-2.57})$\|	(⁠\|$^{+13.77}_{-3.82},^{+7.19}_{-2.42}$\|⁠)	\|$(^{+6.82}_{-2.24},^{+8.72}_{-2.70})$\|
μ μ	\|$[4m_\mu ^2,0.96]$\|	(3.41, 2.18) × 10	(1.40, 2.44) × 10	(⁠\|$^{+0.57}_{-0.34},^{+0.65}_{-0.48}$\|⁠)	\|$(^{+0.38}_{-0.58},^{+1.29}_{-0.42})$\|	(⁠\|$^{+6.31}_{-1.66},^{+3.22}_{-0.96}$\|⁠)	\|$(^{+2.79}_{-0.61},^{+3.62}_{-1.05})$\|	(⁠\|$^{+6.34}_{-1.69},^{+3.29}_{-1.07}$\|⁠)	\|$(^{+2.82}_{-0.84},^{+3.84}_{-1.13})$\|
	\|$[4m_\mu ^2,6]$\|	(3.62, 2.31) × 10	(1.49, 2.59) × 10	\|$(^{+0.61}_{-0.36},^{+0.69}_{-0.51})$\|	\|$(^{+0.40}_{-0.62},^{+0.14}_{-0.45})$\|	(⁠\|$^{+6.68}_{-1.76},^{+3.40}_{-1.01}$\|⁠)	\|$(^{+2.97}_{-0.65},^{+3.85}_{-1.11})$\|	(⁠\|$^{+6.71}_{-1.80},^{+3.47}_{-1.13}$\|⁠)	\|$(^{+3.00}_{-0.90},^{+3.85}_{-1.20})$\|
	[2,6]	(7.72, 4.97) × 10	(3.78, 5.96) × 10	\|$(^{+1.26}_{-0.76},^{+1.45}_{-1.09})$\|	\|$(^{+6.72}_{-1.64},^{+8.66}_{-2.55})$\|	(⁠\|$^{+13.60}_{-3.71},^{+6.99}_{-2.15}$\|⁠)	\|$(^{+2.42}_{-0.60},^{+3.13}_{-0.93})$\|	(⁠\|$^{+13.66}_{-3.79},^{+7.14}_{-2.41}$\|⁠)	\|$(^{+7.14}_{-1.75},^{+9.21}_{-2.71})$\|
τ τ	[14,20]	(0.21, 0.14) × 10	(0.13, 0.18) × 10	\|$(^{+0.32}_{-0.19},^{+0.38}_{-0.29})$\|	\|$(^{+0.45}_{-0.22},^{+0.00}_{-0.16})$\|	(⁠\|$^{+3.23}_{-0.95},^{+1.70}_{-0.56}$\|⁠)	(⁠\|$^{+1.89}_{-0.55},^{+2.40}_{-0.75}$\|⁠)	(⁠\|$^{+3.25}_{-0.97},^{+1.74}_{-0.63}$\|⁠)	(⁠\|$^{+1.94}_{-0.59},^{+2.4}_{-0.77}$\|⁠)

Decay .	bin .	LO .	NLO .	Uncertainty .
.	GeV .	λ = 0.35 GeV .	λ = 0.35 GeV .	LO .	NLO .	LO (λ ) .	NLO (λ ) .	LO (tot) .	NLO (tot) .
.	.	(μ , μ ) = (1.5, 5) GeV .	(μ , μ ) = (1.5, 5)GeV .	(μ , μ ) .	(μ , μ ) .	(1.5, 5) GeV .	(1.5, 5) GeV .	(1.5, 5) GeV .	(1.5, 5) GeV .
	\|$[4m_\mu ^2,0.96]$\|	(5.25, 3.35) × 10	(2.14, 3.74) × 10	\|$(^{+0.88}_{-0.52},^{+1.00}_{-0.75})$\|	\|$(^{+0.58}_{-0.90},^{+0.20}_{-0.65})$\|	\|$(^{+9.72}_{-2.56},^{+0.49}_{-1.47})$\|	\|$(^{+4.27}_{-0.93},^{+5.56}_{-1.60})$\|	(⁠\|$^{+9.76}_{-2.61},^{+1.11}_{-1.65}$\|⁠)	\|$(^{+4.31}_{-1.29},^{+5.56}_{-1.73})$\|
	\|$[4m_\mu ^2,6]$\|	(5.46, 3.48) × 10	(2.24, 3.90) × 10	\|$(^{+0.91}_{-0.54},^{+1.04}_{-0.77})$\|	\|$(^{+0.61}_{-0.93},^{+0.21}_{-0.68})$\|	\|$(^{+10.09}_{-2.66},^{+5.14}_{-1.53})$\|	\|$(^{+4.45}_{-0.97},^{+5.79}_{-1.67})$\|	\|$(^{+10.13}_{-2.71},^{+5.24}_{-1.71})$\|	\|$(^{+4.49}_{-1.34},^{+5.79}_{-1.80})$\|
	[2,6]	(7.78, 5.00) × 10	(3.81, 6.01) × 10	\|$(^{+1.27}_{-0.76},^{+1.47}_{-1.10})$\|	\|$(^{+0.86}_{-1.52},^{+0.19}_{-0.84})$\|	\|$(^{+13.71}_{-3.74},^{+7.04}_{-2.16})$\|	\|$(^{+6.77}_{-1.65},^{+8.72}_{-2.57})$\|	(⁠\|$^{+13.77}_{-3.82},^{+7.19}_{-2.42}$\|⁠)	\|$(^{+6.82}_{-2.24},^{+8.72}_{-2.70})$\|
μ μ	\|$[4m_\mu ^2,0.96]$\|	(3.41, 2.18) × 10	(1.40, 2.44) × 10	(⁠\|$^{+0.57}_{-0.34},^{+0.65}_{-0.48}$\|⁠)	\|$(^{+0.38}_{-0.58},^{+1.29}_{-0.42})$\|	(⁠\|$^{+6.31}_{-1.66},^{+3.22}_{-0.96}$\|⁠)	\|$(^{+2.79}_{-0.61},^{+3.62}_{-1.05})$\|	(⁠\|$^{+6.34}_{-1.69},^{+3.29}_{-1.07}$\|⁠)	\|$(^{+2.82}_{-0.84},^{+3.84}_{-1.13})$\|
	\|$[4m_\mu ^2,6]$\|	(3.62, 2.31) × 10	(1.49, 2.59) × 10	\|$(^{+0.61}_{-0.36},^{+0.69}_{-0.51})$\|	\|$(^{+0.40}_{-0.62},^{+0.14}_{-0.45})$\|	(⁠\|$^{+6.68}_{-1.76},^{+3.40}_{-1.01}$\|⁠)	\|$(^{+2.97}_{-0.65},^{+3.85}_{-1.11})$\|	(⁠\|$^{+6.71}_{-1.80},^{+3.47}_{-1.13}$\|⁠)	\|$(^{+3.00}_{-0.90},^{+3.85}_{-1.20})$\|
	[2,6]	(7.72, 4.97) × 10	(3.78, 5.96) × 10	\|$(^{+1.26}_{-0.76},^{+1.45}_{-1.09})$\|	\|$(^{+6.72}_{-1.64},^{+8.66}_{-2.55})$\|	(⁠\|$^{+13.60}_{-3.71},^{+6.99}_{-2.15}$\|⁠)	\|$(^{+2.42}_{-0.60},^{+3.13}_{-0.93})$\|	(⁠\|$^{+13.66}_{-3.79},^{+7.14}_{-2.41}$\|⁠)	\|$(^{+7.14}_{-1.75},^{+9.21}_{-2.71})$\|
τ τ	[14,20]	(0.21, 0.14) × 10	(0.13, 0.18) × 10	\|$(^{+0.32}_{-0.19},^{+0.38}_{-0.29})$\|	\|$(^{+0.45}_{-0.22},^{+0.00}_{-0.16})$\|	(⁠\|$^{+3.23}_{-0.95},^{+1.70}_{-0.56}$\|⁠)	(⁠\|$^{+1.89}_{-0.55},^{+2.40}_{-0.75}$\|⁠)	(⁠\|$^{+3.25}_{-0.97},^{+1.74}_{-0.63}$\|⁠)	(⁠\|$^{+1.94}_{-0.59},^{+2.4}_{-0.77}$\|⁠)

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