How Would Doubling Earth's Mass Affect the Moon's Orbit?

[superlux1234]

What would happen to the orbit of the moon if the Earth's mass were suddenly (magically!) to double? **Assume the orbit is initially circular

This is everything given in the question.



So, basically what I have come up with is using F=GMM/d^2 ... I have found that when the Earth's mass doubles, that the gravitational force also doubles. So at this point the moon will want to move closer to the earth, as the orbit will become... 3/4 of what it origionally was? When it does this, I was thinking that the moon will still have a large angular velocity (larger then is called for by its new orbit), so it will basically be sling-shot into an elliptical orbit maybe?
So... basically what I have decided so far is that the gravitational force doubles, and that brings the moon 1/4 closer to Earth ...

 

[andrevdh]

The maths indicates otherwise. That is that it will orbit at a larger distance from the "new" earth.

You have to work through the maths. Assume that the speed remains as previously (use the centripetal acceleration) and get a formula for the radius (distance between the Earth and the moon - which will change when the mass of the Earth changes) of a (circular) orbit as a function of the mass of the Earth and the speed of the moon.

The same result can be obtained by assuming the moon will enter an orbit where the gravitational potential energy remains unchanged (and the kinetic energy remains the same - that is the orbiting speed is unaltered).

 

[ogb p]

I would like to clarify a few points about your response. Firstly, the equation you have used, F=GMM/d^2, is correct in determining the gravitational force between two objects. However, it is important to note that the mass of both objects plays a role in this equation, not just the Earth's mass. Therefore, if the Earth's mass were to double, the gravitational force between the Earth and the moon would also increase, but it would not necessarily be double.

Secondly, the orbit of the moon is determined by a delicate balance between gravitational force and the moon's own velocity. If the Earth's mass suddenly doubled, the moon would indeed be pulled closer to the Earth, but its velocity would also change. This change in velocity would ultimately determine the shape of the orbit.

It is difficult to predict exactly what would happen to the moon's orbit without knowing its initial velocity and other factors such as the Earth's rotation and the moon's own gravitational pull. However, it is likely that the moon's orbit would become more elliptical as its velocity changes to accommodate the increased gravitational force from the Earth.

Overall, it is important to note that any change in the Earth's mass, whether it be a doubling or halving, would have a significant impact on the moon's orbit. This is because the gravitational force is directly proportional to the mass of the objects involved. As a scientist, it would be important to conduct further research and calculations to accurately predict the effects of such a change on the moon's orbit.