I've read that Uranus has a lower surface gravity due to its lower density than Earth. However, does this mean Uranus has a weaker pull on its moons than Earth does for the same reason?
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5$\begingroup$ $GM/r^2$..............? $\endgroup$– ProfRobFeb 2, 2018 at 0:04
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2$\begingroup$ Earth’s influence on its moon, or Earth’s influence on Uranus’s moons? As it is stated, the question is about the latter. $\endgroup$– chirluFeb 2, 2018 at 2:43
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$\begingroup$ You can tell Uranus has a larger gravitational pull on its moons, because the moons are orbiting Uranus. They aren’t more attracted to the Earth. $\endgroup$– Jack MoodyFeb 7, 2018 at 14:06
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$\begingroup$ I know, but if it is relative... $\endgroup$– StellarExileFeb 8, 2018 at 14:37
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2 Answers
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The "strength of gravity" (ie gravitational acceleration) is determined by the mass of the planet and the distance between the planet and the moon. The formula is $$GM/r^2.$$
For the Earth (mass= 1 Earth, distance to moon=390000km) the acceleration is $0.003 ms^{-2}$
For Uranus and Titania (mass =14.5 Earth, distance=435000km) the acceleration is $0.03 ms^{-2}$
In summary, the gravitational pull of Uranus on its moon Titania is 10 times greater than the gravitational pull of Earth on her moon.
The same formula applies to surface gravity:
For the Earth, (mass = 1, radius = 6370km) the acceleration due to gravity on the surface is $9.8 ms^{-2}$
For Uranus (mass =14.5 Earths, radius = 25400km) the acceleration due to gravity is $9.0ms^{-2}$.
The surface gravity on Uranus is less that that of Earth, due mostly to the relatively low density of Uranus meaning that you are much further from the centre of the planet when you are at the surface.
(These value vary due to the neither the Earth nor Uranus being perfectly spherical, and the effective gravity is also lower due to centrifugal effects)
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$\begingroup$ Interesting. Does this apply to surface gravity? I'm a little confused. $\endgroup$ Feb 3, 2018 at 11:07
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$\begingroup$ Edit made to explain why surface gravity is less on Uranus. $\endgroup$– James KFeb 3, 2018 at 11:56
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Provided that you are the same distance from the centre and you are on or above the surface then the heavier (more mass) planet will have a greater gravitational attraction. It is still possible that the heavier planet has a lower attraction at the surface since it may be bigger. If you can find the mass and radius then you can use the formula that James gives to calculate the attraction at the surface.
