Gravitational attraction of a spherically symmetrical mass

[Bernie G]

Many sources on basic gravity, like this quote from Wikipedia, say:

“In the case of a spherically symmetric mass distribution we can conclude (by using a spherical Gaussian surface) that the field strength at a distance r from the center is inward with a magnitude of G/ r^2 times only the total mass within a smaller distance than r. All the mass at a greater distance than r from the center can be ignored.”

Proofs or explanations of this make sense, but I have a problem with the following scenario:

Suppose an object P orbits directly above a spherical mass M of constant density and radius R, and the object P only reacts gravitationaly with this constant density material. Then suppose the constant density material is extended to infinity. Should the object P still continue to orbit at radius R, since it is supposedly only affected by the mass within R? It seems under the new circumstances that P should move in a straight line.

 

[pallidin]

Well, a radius extended to infinity would indeed cause the object to appear flat for VERY great distances. Thus, an orbiting object would seem to travel in a straight line.

 

[Bernie G]

Well, if that's true then an object is affected by symmetrical mass outside the radius R... the gravitational force from mass from outside the radius R apparently subtracting from the gravitational force from the mass with within radius R. The object can't be both affected by mass outside the radius R and not affected by mass outside the radius R.

 

[Bernie G]

Correct typos:

Well, if that's true then an object is affected by symmetrical mass outside the radius R... the gravitational force from mass outside the radius R apparently subtracting from the gravitational force from the mass within radius R. The object can't be both affected by mass outside the radius R and not affected by mass outside the radius R.

 

[kcdodd]

I am not sure you can construct such a scenario. If you extend the radius of the object to infinity keeping the total mass constant, then the gravitational force drops to zero and the object would thus travel in a straight line. You could allow the mass to go to infinity too to keep the force nonzero. However, there would be no way "orbit" such an object. The centripetal force is mv^2/r, and if r->inf then that goes to zero unless you allow the velocity to go to infinity too. No matter how fast it goes it would eventually crash into the surface in a parabolic path.

 

[Bernie G]

I see there's potential confusion about the scenario. The mass isn't constant and increases greatly beyond the original R. The density is constant. Rephrased scenario:

Suppose an object P orbits directly above a spherical mass M of constant density and radius R, and the object P only reacts gravitationaly with this constant density material. Then suppose the constant density material is extended to infinity. Should the object P still continue to orbit at the original radius R, since it is supposedly only affected by the mass within the original radius R? It seems under the new circumstances that P should move in a straight line.

 

[Bernie G]

pallidin: I not so sure very great distances are required for a significant effect, since gravity goes as 1/r^2.

 

[granpa]

Suppose an object P orbits directly above a spherical mass M of constant density and radius R,
and the object P only reacts gravitationally with this constant density material.
Then suppose the constant density material is extended to infinity.
Should the object P still continue to orbit at radius R, since it is supposedly only affected by the mass within R?
It seems under the new circumstances that P should move in a straight line.

For any finite R it will make no difference to the orbit of object P.

I'm not going to get involved in an infinity argument.
You all will have to hash that out for yourselves

 

[Bernie G]

"For any finite R it will make no difference to the orbit of object P."

Yes, that makes a lot of sense.

The infinity case is puzzling.

 

[Cleonis]

I see there's potential confusion about the scenario. The mass isn't constant and increases greatly beyond the original R. The density is constant. Rephrased scenario:

Suppose an object P orbits directly above a spherical mass M of constant density and radius R, and the object P only reacts gravitationaly with this constant density material. Then suppose the constant density material is extended to infinity. Should the object P still continue to orbit at the original radius R, since it is supposedly only affected by the mass within the original radius R? It seems under the new circumstances that P should move in a straight line.

Let me rephrase your scenario once more.

Suppose an object P orbits directly above a spherical mass M of constant density and radius R, and the object P only reacts gravitationaly with this constant density material.

Then suppose a hollow sphere pops into existence, same density as M, enveloping the original system concentrically, the cavity in it leaves just enough room for object P to continue its orbit.
Then object P will continue its orbit at radius R.


It seems to me that if you suggest that the gravitating mass M expands with constant density, then most people will automatically move M's center of mass away from test mass P, so that P can continue to orbit outside of M.

 

[Bernie G]

"Then suppose a hollow sphere pops into existence, same density as M, enveloping the original system concentrically, the cavity in it leaves just enough room for object P to continue its orbit. Then object P will continue its orbit at radius R."

Yes, that makes total sense.

"It seems to me that if you suggest that the gravitating mass M expands with constant density, then most people will automatically move M's center of mass away from test mass P, so that P can continue to orbit outside of M."

I'm probably misinterpreting what you are saying... wouldn't the center of mass stay at the same location... the exact center?

 

[Cleonis]

It seems to me that if you suggest that the gravitating mass M expands with constant density, then most people will automatically move M's center of mass away from test mass P, so that P can continue to orbit outside of M.

I'm probably misinterpreting what you are saying... wouldn't the center of mass stay at the same location... the exact center?

And I am problaby misinterpreting what you are saying.

If mass M expands with constant density, with the center of mass remaining in the same location, and test mass P remains at the same distance to M's center of mass, then P would suddenly be embedded inside M.

That's another thought experiment entirely: what would happen if we make two solids suddenly occupy the same space? You would suddenly have twice as much nuclei and electrons in the same space. Kinda crowded.

 

[Bernie G]

"If mass M expands with constant density, with the center of mass remaining in the same location, and test mass P remains at the same distance to M's center of mass, then P would suddenly be embedded inside M."

Yes, P would be embedded at the same original radius R, if R is finite. But the qualification of the scenario was the object P only reacts gravitationaly with the constant density material... so it could pass through it.

Maybe grandpa was right... that an infinity argument has inherent problems.

"if we make two solids suddenly occupy the same space..."

Yes, but the qualification was

 

[kcdodd]

I think I see what your question is now. You are wondering about the case where all space is filled with uniform mass density. It does not seem like the object should be gravitationally attracted to any particular point. Is that a correct characterization of your question?

 

[Bernie G]

Yes, that is the characterization of the question.

 

[kcdodd]

Ah, ok. well, usually to solve the field it is assumed the field goes to zero at infinity. Can't assume that in this case because if you did you would get undefined (infinite) field strengths at your test point. I do not believe the field can be defined in this situation. In the 2D (infinite line) and 1D (infinite plane) situations there are ways around this problem (not very satisfactory too I might add). What you have set up is basically a zero-D problem.

 

[Bernie G]

"I do not believe the field can be defined in this situation."

Thanks. I think that expresses what grandpa wanted to convey. Thread answered to my satisfaction.