Measuring more accurately the distance of remote galaxies

Question

From what I read in Wikipedia, the velocity of a Galaxy has two components: one is due to Hubble's law for cosmic expansion, and the other is the peculiar velocity of the galaxy.

Since the peculiar velocity of galaxies can be over 1.ooo km/s in random direction, this causes an error in evaluating their distance using Hubble's law (I am summarizing from Wikipedia).

A more accurate estimate can be made by taking the average velocity of a group of galaxies: the peculiar velocities, assumed to be essentially random, will cancel each other, leaving a much more accurate measurement.

I assume that "group of galaxies" actually means the cosmic structure by that name, rather than just any collection of galaxies that seem to be in "the same neighborhood", though the Wikipedia text does not explicitly reference that, as it usually would. But I will ignore that issue.

The major problem that I see is that the speed of celestial structures with respect to their "surroundings" seems to be in proportion to their size: 30 km/s for Earth, 200 km/s for the Sun, 600 km/s for the Milky Way, and generally up to 1000 km/s and more for galaxies.

So I would expect this to go up again for even larger structures, such as groups or clusters of galaxies.

Hence, while averaging velocities may give some correction in the measurement, the major source of error should come from the group velocity itself, and would not be corrected by that procedure.

This would weaken significantly the Wikipedia assertion that it produces a "much more accurate" measurement.

Am I right, or do I make an error in my reasonning?

Answer 1

The behaviour you are describing is a consequence of the virial theorem. Without going into the gory details this tells us that if some interacting system of many objects has an average total potential energy of $<U>$ then its average total kinetic energy $<T>$ is related to $<U>$ by:

$$ <T> = \tfrac{1}{2} <U> $$

The proof of this is somewhat daunting, but let's take a simple example. The potential energy of the Earth in the Sun's gravitational well is:

$$ U = \frac{GMm}{r} $$

where $M$ is the Sun's mass, $m$ is the Earth's mass and $r$ is the orbital radius. The kinetic energy of the Earth is:

$$ T = \tfrac{1}{2}mv^2 $$

where $v$ is the orbital velocity. If we set $T = \tfrac{1}{2}U$ we get:

$$ \tfrac{1}{2}mv^2 = \tfrac{1}{2}\frac{GMm}{r} $$

and with a bit of rearrangement we get an expression for the orbital velocity:

$$ v = \sqrt{\frac{GM}{r}} $$

and bingo, this is exactly the expression for the orbital velocity.

The point of all this is that the increasing velocities that you describe are a result of the increasing potential energies of the interactions. The potential energy of the Solar System relative to the Milky Way is greater than that of the Earth relative to the Sun, so we expect the resulting velocities to be higher. Likewise the potential energy of the Milky Way relative to the Local Group is higher again, and so on.

So you are quite correct that larger structures are associated with larger velocities, but it's really a consequence of the larger potential energies involved rather than directly due to the size. This matters because on the very largest scales there are no gravitationally bound structures i.e. the potential energies are zero (relative to the universe as a whole).

So as we increase in size we expect kinetic energies to increase, then reach a maximum and start decreasing again. Exactly where the maximum is I'm not sure - presumably somewhere between the galaxy cluster and supercluster scale. Anyhow the point is that if we average galaxy velocities on a large enough scale we will average out the peculiar velocities and get an accurate value for the Hubble constant.