Change in Moon's Orbit: Impact of Earth's Mass Doubling on Circular 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 ...

 

[premagg]

If initially you are assuming it to be circular then after that also it will be circular bcoz by Newton rule you can replace spherical symmetry by a point.

 

[superlux1234]

Hmm... but wouldn't the sudden change create an elipse? It wouldn't just magically pull it in closer, and continue on its circular orbit, would it? The algular speed is going to be too fast for this to be possible?

 

[D H]

Just one magic trick at a time, please. The Earth suddenly doubles its mass. This mass doubling presumably does not instantaneously alter either the relative position or velocity of the Moon.

A circular orbit at the Moon's orbital radius would have a larger velocity than the Moon's velocity. The Moon is therefore in an elliptical orbit. Since the velocity vector is normal to the position vector and since the velocity is to slow for a circular orbit, the Moon is at apogee.

Could I ask what grade level you are in? This knowledge will help me direct you to the solution.

What do you think an inertial observer who was at rest with respect to the old Earth-Moon center of mass will see?

 

[superlux1234]

2nd year univ physics.

Okay, this line "A circular orbit at the Moon's orbital radius would have a larger velocity than the Moon's velocity." ... are you saying that the velocity at the origional orbit (before mass of Earth doubles) is higher then the NEW orbit calls for, and thus cannot hold the moon in a circular oibit at its new position, so it allows the moon to escape into an elliptical orbit?

I think that the inertial observer would see the moon move towards to the earth, (1/4 closer then its origional position) ... do to d^2 = GMm/F ... as the Earth would now be pulling twice as hard on the moon. The moon would appear to move quicker then in its origional position, becasue it is closer to the earth. This greater velocity would make the moon accelerate outwards, into an elliptical path.

 

[D H]

2nd year physics ... My first opinion is that your answer is too qualitative for that level of work.

Remember that the Earth is also moving because he Earth and Moon orbit each other. Presumably the mass doubling did not instantaneously change the Earth's velocity. What happens to the center of mass after this magical doubling? It was stationary, is it still stationary? I suggest you define a new center of mass frame. What is the total angular momentum in this new frame? What does that mean the Moon's new orbit will look like?

 

[superlux1234]

Yes... well this is the first 2nd year course ... I still have a lot of learning to do. But I am going to spend a few solid days working on this stuff.

Okay, so the center of mass of the system is now closer to the earth... So it has moved, but the moon is going to be pulled closer, so the new CM is not stationary, as it depends on the position of both the Earth and the moon. Since the angular momentum has increased, the CM will be continually changing, causing an eliptical path?

 

[D H]

You are in a second year physics class, use some math. You don't have to make it purely numberic, in fact, symbols and mathematical relationships are better. But do use some math. Stop the pure qualitative stuff.

Prior to the mass doubling, the total linear momentum is tautologically zero in the original center of mass frame. The velocities of the Earth and the Moon are not zero. What are they?

The Earth and Moon will have the the same positions and same velocities the instant before and after the mass doubling. Use these values to compute a new center of mass frame. The position and velocity will be different in this frame, but that is because of the change in reference frames. The center of mass frame is the preferrable frame for doing these kinds of problems. Compute the angular momentum in this frame and determine what that means as far as the Moon's orbit is concerned.

 

[D H]

Since the angular momentum has increased, the CM will be continually changing, causing an eliptical path?

Whoa! You are only worried about the Earth and Moon, not the Earth, Moon, Sun, whatever. The Earth and Moon form a closed system. The only forces acting on the Earth/<oom system are internal forces. Newton's First Law of motion! The center of mass had doggone well better follow a straight-line trajectory or all of physics is in deep doodoo.

The only exception to a straight-line path is a stationary center of mass (think of this as a stright line path with zero velocity.) This is by far the easiest system in which to work.

 

[superlux1234]

Haha, okay, so you are saying that the CM is instantly different (as the Earth instantly becomes 2M), but then it stays constant from this point on, ie, it is stationary in its new position. The center of mass is indeed moved closer to the earth. So the moon is going to be pulled in, and as it is pulled in, it will increase in speed like a figure skater does when the arms are pulled in... thus having more angular momentum. Because of this, the gravitational force will not be able to maintain the circular orbit, which would now be closer to the earth, and thus the moon must escape the circular orbit to make up for this?

 

[superlux1234]

but the Earth is still pulling on the moon at all times, so it brings the moon back towards the Earth as the moon slows down... thus having a smaller angular momentum... as the moon approaches the earth, it will gain speed again (increase in ang momentum) ... and then protrudes into the elliptical path once again?

 

[D H]

Haha, okay, so you are saying that the CM is instantly different (as the Earth instantly becomes 2M), but then it stays constant from this point on, ie, it is stationary in its new position.

No. Do the math. Please, please, please do the math. Stop the qualitative reasoning. It isn't working. If you do the math, you should see that the CM moves at a constant but non-zero velocity with respect to the old CM after the mass doubling. I am strongly suggesting that you come up with a new CM frame in which the new CM is stationary.

but the Earth is still pulling on the moon at all times, so it brings the moon back towards the Earth as the moon slows down... thus having a smaller angular momentum... as the moon approaches the earth, it will gain speed again (increase in ang momentum) ... and then protrudes into the elliptical path once again?

Wrong, and wrong again. Angular momentum is a conserved quantity in the two-body problem. It doesn't matter whether the orbit is circular, elliptical, parabolic, or hyperbolic. In fact, the motion bodies undergo due to the inverse-r-squared gravitational force is but one kind of a very general class of problems called central force motion problems. Angular momentum must be conserved in any central force situation. Think about it for a bit: Torque is what changes angular momentum, and torque is the cross product of the displacement vector and the force vector. If the force is always parallel to the displacement the torque has to be zero. The angular momentum cannot change.

Try using the vis-viva equation. Make sure you use the form with a reduced mass.

 

[D H]

Hmm. You need the reduced mass form of the vis-viva equation only if you want more than two digit accuracy. Most on-line references (wikipedia and mathworld, for example) assume a satellite of negligible mass. The Moon's mass is 0.0123 that of the Earth, which is getting close to negligible.

 

[superlux1234]

We have not used this equation you speak of in class ... nor have we talked about it. I think this was meant to be treated as more of a conceptual question... but heck, maybe it would be good practice for me to plug some numbers in, to see what will actually happen. I am just working on some Lorentz at the moment... Lorentz is cool once you figure it out actually. After that, I will crunch some numbers, see what actually happens.

 

[D H]

The vis-viva equation is just a fancy name for conservation of energy, which you certainly should know about. Using symbols is much better than using numbers; you don't completely plug numbers until the near the end. I suggest you start with [itex]M_e, M_m, \vec r_{e\to m}, \vec v_{e\to m}[/itex] The only thing that changes is that the Earth mass doubles. The Moon mass and the Earth-to-Moon displacement and velocity vectors don't change. Moreover, the Earth-to-Moon velocity vector is known (magnitude is that for a circular orbit, direction is normal to the displacement vector). Double the mass. Its in an elliptical orbit (and stop saying escape; that means a parabolic or hyperbolic orbit). What's the relationship between central mass, distance and velocity at apofucus, and semimajor axis? Can you derive one?