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This page contains answers to questions Mr Sunspot received about Gravity. The questions answered on this page are:
[127] Why don't we feel the acceleration that the Earth suffers in its motion around the Sun, because its speed is different (second Kepler's Law)? asked by Maria Eugenia of Madrid, Spain. 30 January 1998
[123] If there is no gravity in outer space, how does the landing vehicle stay on the moon's surface? asked by Cal. 28 November 1997
[83] Does the Sun have a gravitational field? asked by Lauren McGovern (10). 6 May 1997.
[49] What is the gravity of the Sun? Asked by Chad Parker. 25 November 1996.
[32] At what distance from the centre of the planet must an artificial satellite orbit Jupiter so that it is always over the same place on the surface? Asked by Patricia (17) from Australia.. 22 August 1996.
[31] As a spacecraft travels from Earth to the Moon, the gravitational pull of the Earth gets weaker and that of the moon increases. How far from the centre of Earth are these two gravitational forces balanced? Asked by Patricia (17) from Australia.. 22 August 1996.
[30] How far above the Moon's surface will the gravitational pull be half that on the surface? Asked by Patricia (17) from Australia.. 22 August 1996.
[7] At about what height above the Earth's surface does weightlessness start? Asked by Igor Votava of Vienna, Austria.
More questions about gravity are answered on Page 2. You may also want to read about Black Holes.
[Mathematical Notation]
Gravity
Everything that has mass (including black holes, stars, planets, dust specks, and even you) has gravity as well. The amount of gravity is proportional to the amount of mass, so if you make the mass twice as large, then the gravity gets twice as strong, too. Gravity is the weakest fundamental force that is known. It only amounts to something if the mass that generates the gravitation is quite large.Since you have very much less mass than either the Earth or the Sun, your gravitational field is much weaker than those of the Earth and the Sun. The Sun is about 330,000 times more massive than the Earth is, so the gravitational field of the Sun at any distance from the Sun is about that many times stronger than the gravitational field of the Earth at the same distance. Gravity decreases with distance, so because the Sun is also much larger than the Earth is the gravity at the surface of the Sun is not 330,000 times stronger than the gravity at the surface of the Earth, but only 28 times stronger. If you weigh 100 pounds then the gravity at your surface due to your mass is about 300 million times weaker than the gravity at the surface of the Earth - that's why you don't notice your own gravitational field.
[LS 7 May 1997]
Planetocentric Orbits
The surface of Jupiter is not solid but consists of clouds, and because these clouds move with respect to each other Jupiter does not have a single rotation period. To hover above a particular place on a planet, an artificial satellite must have an orbit that follows the equator of the planet. The orbital period P of a satellite at distance r (in km) from the center of a planet which is M times as heavy as the Earth is equal toP = 0.0100 sqrt(r^3/M) secondswhere "sqrt" is the square-root function. It is assumed here that there are no other objects nearby that influence the orbit of the satellite. One mile equals 1.60934 km. (See Equation 11 on explanation page 2.) We can rewrite this equation to
r = 21.6157 (P^2 M)^(1/3) km.
The clouds near the equator of Jupiter have a rotation period ("Jupiter day") of about 9 hours, 50 minutes, and 30 seconds, which is equal to 35430 seconds (P = 35430) and Jupiter is 318 times as heavy as the Earth (M = 318). These values of P and M yield r = 159147 km. Jupiter's equatorial radius is 71492 km, so a satellite in a jovistationary orbit (an orbit in which it hovers above a "fixed" point on Jupiter) must orbit at about 87700 km above Jupiter's surface.
The following table lists the radius of the planetocentric orbit for all the planets and the Sun and Moon. The radii are determined with the formula given above, and are given both in kilometers and relative to the radius of the planet itself (R/R_P). The planetary sidereal rotation period is also listed. Saturn has the smallest ratio between planetocentric orbital radius relative and the planetary radius, of 1.86. Pluto has the smallest planetocentric orbital radius, of 18,900 km.
| Object | R_eq | P_eq | R_Orbit | R_O/R_eq |
|---|---|---|---|---|
| km | sec | km | ||
| Sun | 25,240,000 | 2,214,000 | 25,449,500 | 36.6 |
| Mercury | 242,100 | 5,067,020 | 242,832 | 99.6 |
| Venus | 1,534,500 | 21,009,000 | 1,537,150 | 254.0 |
| Earth | 42,200 | 86,164 | 42,168 | 6.61 |
| Moon | 88,300 | 2,360,592 | 88,570 | 51 |
| Mars | 20,400 | 88,643 | 20,431 | 6.01 |
| Jupiter | 158,800 | 35,430 | 159,147 | 2.23 |
| Saturn | 112,000 | 38,360 | 112,263 | 1.86 |
| Uranus | 79,600 | 58,700 | 79,659 | 3.12 |
| Neptune | 90,500 | 65,520 | 90,590 | 3.66 |
| Pluto | 18,300 | 551,819 | 18,939 | 16.2 |
[LS 22 August 1996 - 11 February 1998]
Balanced Gravity
If two bodies A and B are a distance d apart and body A (the heaviest one) is m times as heavy as body B, then there are two points at which the gravitational attraction of both bodies is equally strong. These two points lie on the straight line that passes through the centers of A and B, at distancesr1 = d * (m - sqrt(m/(m - 1)))and
r2 = d * (m + sqrt(m/(m - 1)))from body A in the direction of body B. The first point lies between A and B, and there both bodies pull in opposite directions. The second point lies on the other side of B (as seen from A), and there both bodies pull in the same direction.
The mass of the Earth is 81 times the mass of the Moon (m = 81) and the mean distance between the Moon and the Earth is 384400 km. The formulas then indicate that the gravity of the Earth and the Moon are equal at distances of 9/10 and 9/8 times the mean distance, or 345960 and 432450 km from the center of the Earth, and 38440 and 48050 km from the center of the Moon.
Because the Moon orbits around the Earth (or more accurately: both the Earth and the Moon orbit around their common "center of mass", which on average lies 4688 km from the center of the Earth and 379712 km from the center of the Moon), the r1 point also rotates around the center of mass, so if you put an object (such as a spacecraft) in the r1 point it will not stay there because the r1 point moves in a circle while the spacecraft wants to move in a straight line (because the gravitational forces on it are exactly balanced).
Lagrange Points
![[Lagrange Point Image]](../gifs/lagrange.gif)
If two massive bodies orbit their common center of mass in strictly circular orbits, then there are five points at fixed positions relative to the two bodies where a spacecraft is in balance with the gravities of the two objects and their orbital motion. These points are called the "Lagrange points". If the bodies move in orbits which are mostly circular but not quite (like the Earth and the Moon) then the Lagrange points move around a little bit.
The first three Lagrange points (L1 through L3) lie on the straight line that passes through the centers of the two objects. Usually the Lagrange point between the two bodies is called L1, the one on the far side of the secondary (= lightest) body is called L2, and the one on the far side of the primary (= heaviest) body is called L3. The remaining two Lagrange points (L4 and L5) form equilateral triangles with both bodies, i.e., they are as far away from both bodies as the bodies are from each other. L4 precedes the primary in its orbital rotation, and L5 follows. As seen from either one of the bodies, L4 and L5 lie 60 degrees away in the sky from the other body.
In case of the Earth-Moon system, L2 lies 64,500 km from the center of the Moon on the side facing away from the Earth, L1 lies between the Earth and the Moon, at 58,000 km from the center of the Moon and 326,400 km from the center of the Earth, and L3 lies 381,700 km from the center of the Earth on the side facing away from the Moon. The plot shows the Earth-Moon system with its five Lagrange points, its common center of mass, and the orbits of the Earth and Moon around that center. The size of the Earth and Moon has been exaggerated in the plot.
The first three Lagrange points provide an unstable balance: if you move the spacecraft a little bit away from such a point, then it will continue to move away from it, just like a ball on top of a hill keeps rolling away from the top if you give it a little push. L4 and L5 provide a stable balance, as long as the mass ratio between the two objects is larger than 24.960: if you move the spacecraft away slowly enough from such a point, then it will orbit around that point.
The Lagrange points are important in astronomy. In 1906, Max Wolf discovered asteroid 588 Achilles, which orbits the Sun in one of the stable Lagrange points of the Jupiter-Sun system. Since then, more asteroids have been discovered in L4 and L5 of the Jupiter-Sun system. Some of the smaller moons of Saturn orbit in the Lagrange points of Saturn and one of its major moons. Also, the SOHO satellite is currently observing the Sun from the L1 point of the Sun-Earth system, at about 1.5 million km from the Earth in the direction of the Sun. This vantage point offers unlimited vision of the Sun, since the satellite is always between the Earth and the Sun. Because the L1 point is unstable, the satellite has to use its propulsion system once in a while to remain close to it. And thirdly, when one star in a double star system turns into a red giant, it may grow so big that it reaches to the L1 point between it and its companion star. When that happens, material from the giant star may flow through the L1 point to the other star. This added mass speeds up the evolution of the companion star.
To calculate the positions of the first three Lagrange points, you can use the following method that Mr Sunspot derived: First calculate
n = m/(m + 1)and
z = (n/3)^(1/3)where ^ means power-taking (x^(1/3) is the cube root of x) and m is the ratio of the masses of the secondary and primary objects. This number is always between 0 and 1. Then, the distance between L1 and the secondary, divided by the distance between the two objects, is approximately
x1 = z - z*z/3.Likewise, the distance between L2 and the secondary is approximately
x2 = z + z*z/3and the distance between L3 and the primary is approximately
x3 = 1 - (7/12)*n.These approximations are quite good: the error in x1 is always less than 0.00006 for m smaller than 1/100 (i.e., when the primary is more than 100 times as massive as the secondary), and less than 0.004 for m smaller than 1/10. Likewise, the error in x2 is always less than 0.00003 for m smaller than 1/100, and less than 0.004 for m smaller than 1/10, and the error in x3 is less than 0.00006 for m smaller than 1/10. The errors decrease when m gets smaller. For approximations up to fourth order in n, study the derivation of these formulas.
The L1 and L2 Lagrange points are furthest away from the secondary object (relative to the distance between the two objects) when the two objects have equal mass. In that case, x2 and x3 are equal to 0.6984, and x1 is equal to 0.5.
The following table lists the distance of L1 to the planet for all Sun-planet systems, both in AU and in millions of kilometers (also called Gm). Satellites in orbit around the planet at a distance near that of L1 have very strange orbits. The satellite systems of all the planets remain well within the distance of the L1 point of that planet: the Moon reaches a quarter of the Earth's distance to its L1, and the moons of Jupiter reach up to a third of Jupiter's distance to its L1; The moons of the other planets remain relatively closer to their planet.
| Planet | AU | Gm |
|---|---|---|
| Mercury | 0.0015 | 0.22 |
| Venus | 0.0067 | 1.01 |
| Earth-Moon | 0.0100 | 1.50 |
| Mars | 0.0072 | 1.08 |
| Jupiter | 0.3470 | 51.91 |
| Saturn | 0.4297 | 64.29 |
| Uranus | 0.4661 | 69.72 |
| Neptune | 0.7708 | 115.32 |
| Pluto-Charon | 0.0576 | 8.62 |
The L2 points have slightly larger distances to the planets. For Jupiter, the L2 point is almost 5 percent further away than the L1 point, and for the lighter planets the relative difference is less. The L3 point of each planet is on the other side of the Sun from the planet, and almost as far away from the Sun as the planet is. [LS 22-29 August 1996]
Limits on Rings and Moons
Moons cannot form or orbit at every distance from a planet. Moons cannot orbit too close to the planet because of tidal stress: stress that occur because the near side of the moon (relative to the planet) is slightly more strongly attracted by the planet than the far side. If the moon is too close to the planet, then it will be torn apart into many small pieces which will form a ring. The smallest distance to the planet that the moon can be at without being torn apart is called the Roche limit. The value of the Roche limit depends on what the moon and planet are made of and on how fast they rotate. For moons that always show the same face to the planet and that are held together only by their own gravity (for instance a moon made completely out of sand grains) the Roche limit is equal toR_l = 2.46 (d_p/d_s)^(1/3) R_pwhere R_p is the radius of the planet, d_p is the density of the planet, and d_s is the density of the moon. If such a moon and planet have about the same density, then the Roche limit is about 2.5 times the radius of the planet. In practice, moons are not made out of sand grains, but out of materials, like stone, that hold together strongly, so moons can actually get a bit closer to the planet than the formula suggests. In very small objects, such as artificial satellites, the forces that bind the object are always more important than the tidal forces, so such objects can get as close as they want to a planet without being torn apart by the tidal forces. For Jupiter and Saturn, the radius of the planetocentric orbit is smaller than the Roche limit for grainy moons.
[LS 22 August 1996]
Gravity of an Object
The gravitational pull of a spherical object decreases with the square of the distance to the object, so the pull is only a quarter as much if you go twice as far away from the center of the object. If you go sqrt(2) = 1.414 times as far away, then the pull is only half as much.The radius of the Moon is 1738 km, so if you increase your distance to the center of the Moon to 1738*1.414 = 2458 km (or 720 km above the surface of the Moon) then the gravitational pull is only half as much as it is at the surface of the Moon. Likewise, the radius of the Earth is 6378 km and there you have to go 2642 km above the surface to feel gravity only half as much as at the surface.
The formula is
g/g_Earth = k * (M/M_Earth)/(r * r)where g is the gravity, g_Earth is the gravity at the surface of the Earth (which is about 32 ft/s^2 or 9.8 m/s^2), M is the mass, M_Earth is the mass of the Earth (which is about 1.3*10^25 lb or 6.0*10^24 kg), r is the distance to the center of the object, and k is a number that depends on the units you use to measure r. If you measure r in units of the radius of the Earth (which is about 3963 mi or 6378 km), then k = 1. If you measure r in AU, then k = 0.000,000,0018. For instance, the Sun has a mass 332,946 times greater than that of the Earth, and a radius 109.12 times greater than that of the Earth, so the gravity at the surface of the Sun is 332,946/(109.12*109.12) = 28.0 times as strong as at the surface of the Earth. You'd weigh 28.0 times as much on the Sun as you do on the Earth.
If you measure the distance in miles rather than Earth radii, then k = 15,707,000, and if you measure r in kilometers, then k = 40,682,000. For instance, at 200 miles above the surface of the Earth (a typical Space Shuttle orbit) the gravity is equal to g = 15,707,000/(4163*4163) = 0.91, where the 4163 is equal to the radius of the Earth (3963 miles) plus the 200 miles, so if you were 200 miles above the Earth without moving (so that the centrifugal force cannot help keep you weightless) then you'd weigh about 9 percent less than on the Earth.
Only the gravity of spherical objects follows the formula exactly. If the object is not quite spherical, then the gravity is a little bit different. However, if the object is almost spherical (like the Earth) then the difference between its gravity and that of the formula is small, or if you are much farther away from the object than its size (even if the object is potato-shaped), then the difference is again small.
The gravitational acceleration on Earth due to the Sun is about 0.0006 times as large as the Earth's gravity at the surface: on a 50 kg (150 lb) person that corresponds to a weight of 30 grams (1 oz). You feel a force as large as that if you accelerate from 0 to 5 km/h (3 mph, walking speed) in 4 minutes. I'm not surprised we don't feel the Sun's gravity.
The gravitational acceleration of the Earth due to the Sun, Moon, all planets (at their closest), and some things on Earth is listed in the following table.
| Object | Mass | Dist | Acceleration |
|---|---|---|---|
| Sun | 332,946 E | 1 AU | 0.0006 |
| Moon | 1/81 E | 60 R | 0.000,003 |
| Mercury | 0.055 E | 0.60 AU | 0.000,000,0003 |
| Venus | 0.815 E | 0.28 AU | 0.000,000,02 |
| Earth | 1 E | 1 R | 1 |
| Mars | 0.107 E | 0.52 AU | 0.000,000,0007 |
| Jupiter | 318 E | 4.2 AU | 0.000,000,03 |
| Saturn | 95 E | 8.5 AU | 0.000,000,002 |
| Uranus | 15 E | 17 AU | 0.000,000,000,09 |
| Neptune | 17 E | 29 AU | 0.000,000,000,04 |
| Pluto | 0.0022 E | 29 AU | 0.000,000,000,000,05 |
| man | 82 kg | 0.6 m | 0,000,000,002 |
| full 747 | 363,000 kg | 20 m | 0,000,000,006 |
| ESB | 365,000,000 kg | 200 m | 0,000,000,06 |
| FKG | 4,400,000 kg | 10 m | 0,000,0003 |
[LS 22 August 1996 - 29 January 1998]
Weightlessness
You are weightless when gravity can act on you without interference from other forces, i.e., when you are in an orbit, for example around a planet or the Sun. When you fall down on a planet without an atmosphere, then you follow a tiny part of an orbit around the center of the planet, and then you are weightless. If you fall down on Earth, then the air around the Earth pushes against you, and this slows you down and changes your orbit from what it would be without the atmosphere, and then you are no longer weightless. However, when you're falling and the wind past you is going rather slow (less than, say, 30 mph or 50 km/h), then you are mostly weightless. So, you can be weightless anywhere, even inside a mine shaft far below the surface.Of course, its no fun being weightless if your orbit goes through the Earth: then you'd hit the ground at some point. To be able to stay weightless for a long time, you need to be in an orbit that goes around the Earth and not through it. To get to and stay in such an orbit, you need to be able to go at least 5 miles per second (8 km/s, that's 18,000 mph or 29,000 km/h). If your orbit is too close to the surface (say, less than 100 miles or 150 km high), then you'd be slowed down by the air and you'd still fall to the surface after a while. If you go higher up, then there is so little air that you can stay in that orbit and be weightless for many years. For instance, the Hubble Space Telescope is in an orbit 370 miles (600 km) above the surface of the Earth and is expected to stay up there for at least 15 years.
If the air around the Earth did not slow down falling objects, then we'd be in big trouble. The biggest raindrops (with a diameter of about 1/4 of an inch or 6 mm) hit the surface at a speed of about 22 mph (36 km/h), and smaller raindrops go slower. If these drops were not slowed down by the air, then they might reach speeds of more than 100 mph (160 km/h) and hurt us.
[LS 11-25 April 1996]
Weightlessness and Gravity
When you're standing on the surface of the Earth there is a force of gravity acting on you. That you yet stay at the surface is because the surface pushes up against you, exactly balancing the force of gravity which pulls you down. It is this force opposing gravity that causes you to feel weight. If there are no forces opposing gravity (for instance, in space), then you are weightless. So, weightlessness is not the same as absence of gravity, and on other planets and moons you have weight, just like you have weight on Earth.[LS 5 January 1998]
For more answers to questions about gravity, see Page 2.