Proper distance for co-moving bodies

[TheCanadian]

I was told in my cosmology class that there are two planets separated by a distance L at time t = 0. L is known as the co-moving distance between the planets. I was told the equation for proper distance, $d_p$, is given by:

$d_p = a(t) L$

where $a(t)$ is the spatial expansion factor of the universe, and it is a function of time. This implies the proper distance is also a function of time. But I was told that L is constant and is the distance of separation as measured by someone on either planet to the other planet and it would be constant and equal to L for ALL times--not just t = 0. I guess I'm wondering 2 things:

1) What exactly does the proper distance correspond to physically in this case? Who could observe the proper distance?

2) Why doesn't L change with time? If the universe is expanding, wouldn't the distance of separation a person from either planets measures relative to the other planet be changing with time?

I tried to clarify this with my instructor, but I could have very well misinterpreted what he was saying. So any help would be greatly appreciated!

 

[Bystander]

Google "Lagrange points."

 

[TheCanadian]

Google "Lagrange points."

Okay, I've looked into it a bit and see that the distance of a body relative to others is kept constant because of the forces of attraction holding it in that configuration. But for the example I mentioned, to my knowledge, it is simply two bodies and I was never told we were to assume the force of attraction between the two planets allows them to maintain a constant L. That does make sense and I should have realized it is necessary. But that still leaves me to my other question: what exactly is the proper distance in this case? What does it represent physically and is it possible to observe?

 

[PeterDonis]

what exactly is the proper distance in this case? What does it represent physically and is it possible to observe?

"Proper distance" means the actual, physical distance between two objects. You seem to be confusing two different scenarios: one in which the planets are gravitationally bound to each other, and another in which the planets are not bound but are "comoving", i.e., are moving along with the Hubble flow, the general expansion of the universe. In the first scenario (gravitationally bound planets), the proper distance between the planets should be, on average, the same over time, and will not be affected by the expansion of the universe. In the second scenario (planets moving along with the expansion of the universe), the proper distance between the planets will increase with time; that is what "expansion of the universe" means, that the proper distance between comoving objects increases with time.

 

[Chalnoth]

Google "Lagrange points."

Lagrange points are a completely different concept that's unrelated to TheCanadian's question.

1) What exactly does the proper distance correspond to physically in this case? Who could observe the proper distance?

The proper distance is the distance that would be measured if we could lay down a ruler between galaxies.

It isn't measured directly. It's inferred from other distance measures. For example, the angular diameter distance (how big an object appears vs. how big it actually is) or the luminosity distance (how bright an object appears vs. how bright it actually is). We measure one of these other distances and then use that to calculate the proper distance. If you're curious about all of these various distance measures, this is a good overview (though it gets rather complicated):
http://arxiv.org/abs/astro-ph/9905116

2) Why doesn't L change with time? If the universe is expanding, wouldn't the distance of separation a person from either planets measures relative to the other planet be changing with time?

L isn't a distance. It's a sort of label. The distance is L a(t). That does change with time. We separate out the effect of the expansion (a(t)) to make the math simpler and easier to grasp: now instead of describing the individual distances of millions of galaxies, we can simply describe the expansion rate a(t), which is a single function of time.