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I was taught that if I wanted to find the attraction force between two objects I would use the formula: $F=(G*m_1*m_2)/d^2$. Where $d$ is the distance from the center of object $m_1$ to $m_2$. I find this counter intuitive because in orbits the orbiting object orbits the center of mass not the center of the object. For instance if we pretended that we did not know the mass of the earth and attempted to find it using the period of the moon's orbit and the distance from the center of the earth to the center of the moon, we get an answer which is too large. So what is the purpose of the formula above?

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    $\begingroup$ The force depends only on the distance between objects, not on what orbits what, in fact the solution to the two body problem, where both masses orbit the center of mass, is a consequence of this formula. Its "purpose" is to express a law of nature we discovered. Just because it is inconvenient for making the calculation you describe, or "counterintuitive", does not make it less so. $\endgroup$ – Conifold Feb 25 '17 at 22:27
  • $\begingroup$ No. The $d$ means here the distance of the bodies. The force depends from the distance of the objects. It is because the objects are attracting each other, not the common center of mass (where in many cases is only vacuum). But they orbit around their common center of mass. It has a different reason, which is totally independent from the formula. If the gravitational force would depend on $log(d)$ and not on $d^2$, they would still orbit aroung their common center of mass. If is because of the preservation of the momentum. $\endgroup$ – user259412 Feb 25 '17 at 22:28
  • $\begingroup$ The purpose of the formula is to give you the force $F$ between two massive bodies, obviously; a purpose which it achieves beautifully, one might add. Clearly the formula says nothing about orbits, nor does it claim to make such statements. $\endgroup$ – Pirx Feb 25 '17 at 23:44
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The $d$ in the formula is considered the distance between center of masses of the two objects - earth, and the satellite. If someone mentions the distance between center of the objects, they assume spherical bodies and uniform density, which is generally not true.

The distance $d$ is actually distance between centers of masses of two bodies. Center of mass and center of object are assumed to be same if someone uses "center of object" in place of "center of mass".

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Gravity is produced by mass, not by volume. Gravitational force is directly proportional to the product of each body's mass with the other, and inversely proportional to the square of the distance between their masses. The m1 and m2 in the universal gravitation formula represent each body's center of mass. In reality, each moves around the other, and both revolve around their common center of mass, which is one focus of their elliptical orbits. The focus may lie outside both bodies, or (in the case of the Earth and the Sun) within the Sun's volume, but not at the center of the Sun.

Over vast interplanetary or interstellar distances, each gravitating body may be treated as a point. When lesser distances are involved, one must take account of the distribution of mass within each body, and locate its center of mass for the purpose of computing distance in the gravitation force formula. This also applies for computing gravitational force between star clusters, and/or nebulae.

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The force depends only on the distance between the objects. The force on one object is always directed towards the other. Both objects orbit the common centre of mass and remain on opposite sides of it, each following a different ellipe. If one object is very much larger than the other, the orbit of the lighter body is the familiar circle.

The formula is the starting point to derive the equations of the orbits of 2 bodies about their common CM.

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There are two different kinds of "center of mass" here.

First, the expression $F=\frac{Gm_1m_2}{d^2}$ is correct if $m_1,m_2$ are the masses and $d$ the distance between their respective centers of mass.

However, the relative motion of one mass about the other depends on the location of the center of mass of the two-body system, not the centers of mass of the body themselves. Now, because the Earth is so much massive than the Moon, the center of mass of the two-body system nearly coincides with the center of mass of the Earth, so we very nearly see the Moon orbiting about the Earth rather than the center of mass of the Earth-Moon problem, but the location of the two-body center of mass does NOT affect Newton's law of gravitation.

To put it differently, imagine two bodies of equal mass separated by a distance $d$. Then $F=\frac{Gm_1m_2}{d^2}$ still holds, but the motions of both bodies would now take place about the midpoint of the two-body system, which is where the center of mass of this two-body system is located.

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