How do gravity and cosmological expansion strike an equilibrium?

[cephron]

I was just reading on wikipedia about the expansion of space. ( http://en.wikipedia.org/wiki/Metric_expansion_of_space )

It explains that the expansion of space can be treated as if it were a repulsive force between particles proportional to the distance between them, ie the further apart they are, the more strongly they are pushed apart.

But then it says: "However this does not cause the objects [talking mainly about galaxies here] to grow steadily or to disintegrate; unless they are very weakly bound, they will simply settle into an equilibrium state which is slightly (undetectably) larger than it would otherwise have been."

This doesn't make sense to me - if the "equilibrium state" is slightly larger, then the force of gravity is reduced and the "force" of expansion is increased. This would tend to push the objects even further apart. I don't see any kind of negative feedback here to keep the cosmological expansion from pushing any orbiting bodies away from each other.

Another way of looking at it -
Imagine an empty universe with two equal-mass objects orbiting each other in a perfectly circular path. If there were no expansion of space, they would orbit each other infinitely without getting any closer (Right?). Now add small repulsive force between them, proportional to their distance from each other. What can happen except that they get pushed slightly further apart by it? And now that they're slightly further apart, the attraction is weaker and the repulsion is stronger. What can happen except they again get pushed slightly further apart? I must be missing something...

 

[pervect]

If it weren't for the fact that most bound systems involve orbits, your argument would probably be right.

But the bound systems do have orbits - galaxies orbit their center of mass, planets orbit the sun, etc.

And when you have an orbit, adding the disturbing force just changes the size of the orbit. And when you work out the numbers,the cosmological influences are small on the scale of galaxies - you need to go to a cosmological scale before they become dominant.

 

[cephron]

And when you have an orbit, adding the disturbing force just changes the size of the orbit.

Ok, but I'd like to ask for clarification on this. Given a continuous disturbing force, would the size change of the orbit also be continuous?

Pretend we of the Earth aimed a rocket engine at the sun and fired it continuously, pushing the Earth away from the sun. Would we measure the change in Earth's radius from the sun in meters, or meters per second? My intuition tells me meters per second - our radius would be continuously, ever-so-slowly increasing, not just settling at a new altitude.

And when you work out the numbers,the cosmological influences are small on the scale of galaxies - you need to go to a cosmological scale before they become dominant.

By this, do you mean that over a cosmological time scale, galaxies will eventually drift apart? If so, that's different from an equilibrium, which is what the article seems to be claiming.

 

[Chronos]

Dark energy upsets the balance of the apple cart.

 

[cephron]

Dark energy upsets the balance of the apple cart.

Sounds insightful and mysterious!

...but if that's a serious answer, please elaborate?

 

[pervect]

Ok, but I'd like to ask for clarification on this. Given a continuous disturbing force, would the size change of the orbit also be continuous?

Pretend we of the Earth aimed a rocket engine at the sun and fired it continuously, pushing the Earth away from the sun. Would we measure the change in Earth's radius from the sun in meters, or meters per second? My intuition tells me meters per second - our radius would be continuously, ever-so-slowly increasing, not just settling at a new altitude.

The Earth's would initially accelerate, but would reach a steady state. The exact details on the evolution of the orbit in time would be complicated, the orbit would no doubt be elliptical.

If we ignore the eccentricity of the ellipse, though, we can estimate what the new radius would be if the orbit was circular, with the idea that the average of the major and minor axes of the ellipse would be in the same ballpark as the new circular equilbrium orbit.

The tangential velocity v of the Earth would not change if the force was always directed radially.

initially v^2 / r = GM/r^2, or v^2 = GM/r
setting the centripetal acceleration v^2/r to the gravitational acceleration

v = tangential orbital velocity
r = orbital radius
G = gravitational constant
M = sun's mass

The new equilibrium point would be

v^2 / r' = GM/r'^2 + delta

so v62 = GM/r' + delta

We can then write GM/r' + delta = GM/r

or

GM + delta r' = GM r' / r

r' (GM/r - delta) = GM

r' = GM / (GM/r - delta)

By this, do you mean that over a cosmological time scale, galaxies will eventually drift apart? If so, that's different from an equilibrium, which is what the article seems to be claiming.

As the universe expands, the distances between galaxies increases.

 

[cephron]

Thank you for the explanation concerning the orbits. This seems to be telling me what I wanted to know, that two orbiting bodies would not be eventually pushed to infinity from each other by cosmological expansion.

About that last question, I'm sorry, it was terribly ambiguous. What I meant was drift apart from themselves.
I'll rephrase the question, and humbly ask that you answer the revised question:

And when you work out the numbers,the cosmological influences are small on the scale of galaxies - you need to go to a cosmological scale before they become dominant.

By this, do you mean that over a cosmological time scale, a galaxy will eventually drift apart from itself, eventually ceasing to be a galaxy? If so, that's different from an equilibrium, which is what the article seems to be claiming, but if not, that would be consistent with your claim about orbits and the article's claim of an equilibrium.