# Measuring more accurately the distance of remote galaxies

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?

• A very good question about a subtle issue. I think this touches on the oft-neglected aspect of cosmology (which would certainly be an appropriate tag here) that homogeneity/isotropy kicks in only at a certain scale, but I'm not confident enough to turn such into an answer. By the way, there's no fixed definition of "group" - just whatever algorithm you choose to employ.
– user10851
Sep 8, 2014 at 12:29
• Downvotes are fascinating. It is obvious that I am asking because my knowledge is not sufficient. If my question is silly enough to justify a dawnvote, why not tell me why? I am interested. Sep 9, 2014 at 20:10

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.

• Amazing, simply amazing. I mean, I was thinking of going with explaining how there are no gravitationally bound structures on the largest scales and that the velocity of objects in smaller scales is due to orbiting or being gravitationally bound. But you did exactly that only in a decisive and irrefutable way. No messy explanation required. I'm jealous. +1
– Jim
Sep 8, 2014 at 14:14
• Thanks a lot for this very interesting answer (I suspected that the virial theorem was involved, but I do not master it enough). Still, I am bothered by some questions. First, of course, assuming what you said, am I correct in syaing that the Wikipedia assertion is significantly too strong? (that was my question). Another point that bothers me is why should larger structures have larger potential energy? Mass matters, but so does size in the energy formula. And it is strange that potential energy should increase up to some size (galactic cluster?), and then decrease. Sep 8, 2014 at 22:19
• @babou: you hve two competing effects - gravity tends to pull matter together and the expansion of space tends to pull it apart. At the small scale the expansion is negligable and gravity wins, but make the scale large enough and the expansion wins. That's why there is a maximum gravitational potential energy somewhere in the middle. Sep 9, 2014 at 4:54
• @babou: re your first point - if you sample all galaxies at some distance $r$ over the whole sky (all $4\pi$ of it) then you're sampling galaxies from many different clusters. You would expect the velocities of galaxies from different clusters to be uncorrelated and therefore average out to the Hubble velocity. Sep 9, 2014 at 4:57
• Gravity against expansion is a nice explanation for a maximum in speed. However, I do not see how we can average velocities for a bunch of galaxies without ascertaining that they are indeed more or less at the same distance, which is what we are trying to determine. So we are in a circular argument. Having them belong to a unique structure is the only way that seems to help. Another point is that it is not obvious why the potential energy should be greater on large structures. The virial theorem gives apparently an average speed on the order of $v = \sqrt{\frac{GM}{size}}$, but then what? Sep 9, 2014 at 8:07