ALL 11 companions are currently exhibiting significant proper motion
We can't measure the proper motions of galaxies. He's using the
radial velocities, which is what we're claiming the redshift measures.
Anyway, he's just treating this like a coin flip. That is, if there are N galaxies orbiting a more massive galaxy, then the probability of them all moving in the same direction relative to the massive galaxy is just:
P(N)=\frac{1}{2^N}
Same as the probability of getting N successive heads on a coin flip. There's more to the story, however.
turbo-1 said:
If the smaller objects are physically associated with (and gravitationally effected by) the more massive object, we would reasonably expect about half the smaller objects moving around that massive galaxy would be moving toward us relative to the host, and would therefore be blueshifted in relation to the host.
Look at the phrase I highlighted; it's the important one. If the galaxies were orbiting the most massive galaxy in the cluster and it was feeling a negligible influence from them, then this analysis would be correct.
However, if this isn't the dynamical situation, then his analysis collapses. Can you see why?
To elucidate, let's try exploring the other limit; that the galaxies are all equivalent and moving randomly through space. In this case, all of the galaxies are equally likely to be the most or least redshifted one, so the probability of the "dominant" galaxy having the lowest redshift is:
P(N)=\frac{1}{N}
Surely we must be agreed up to this point, as this is just basic statistics.
Now, here's where the astrophysics comes in. The galaxies in a cluster
cannot be said to be orbiting around the most massive member, with it being stationary relative to them. There are galaxies that can be thought of approximately in this way (called satellite galaxies), but Arp wasn't just counting satellite galaxies. He included larger galaxies, like the Milky Way and M33, which will themselves have a non-negligible effect on the "dominant" galaxy (M31, according to Arp). Furthermore, in the standard model (the one he's trying to show to be inconsistent), there is a lot of unseen intracluster matter (and even a lot that is seen in the intracluster medium) which will be effecting the motion of the galaxies in the cluster. All in all, it's not consistent with the picture of one galaxy being orbited by the rest.
Does this mean that the galaxy velocities should be treated as a uniform random distribution, as assumed by Newman and Terzian? Probably not, but given what we know, it's a
much better approximation than Arp's. To get a better approximation of the relevant probability, one should do Monte Carlo simulations, evolving the motion of a set of galaxies of equivalent masses to the Local Group galaxies, each iteration having different initial conditions. If you're so set on verifying Arp's arguments you could do this yourself and determine more accurate probabilities.
I wouldn't count on it being particularly shocking, however. There are two further reasons why I think Arp's result is uninteresting (in the way that he intends, that is). The first is that galaxies in clusters tend to have very little angular momentum relative to on another, so the motion of two very massive galaxies relative to one another ought to be nearly radial. This means that, in the Local Group at least, it should not be so surprising if M31 has an unusually small or large redshift relative to the Milky Way. Since they're the two most massive members of the cluster, their gravitational interaction will be the strongest, so it's not a bad approximation to treat them as a two-body system with low eccentricity motion. If the motions of the other galaxies are randomly distributed, then there is a larger than 1/6 chance that M31 will lie at one of the extremes (that is, either the smallest or largest redshift).
The other reason it's uninteresting is that these probabilities were calculated
after the observation. That is, Arp was looking for something unusual in the redshift distributions and reported the first one he found. This is sort of like looking for license plates that say funny things and then reporting the probability of seeing the first funny one you saw (OMG, it's 1 in a million!). The appropriate probability to calculate is rather the probability that the distributions would be such that Arp would write a paper about them. This is obviously not possible to do, but it's at least worth considering that if the "dominant" galaxies had had the
highest redshifts, he also would have pointed it out as unusual.