"P" == Painius writes:

P With stars, the measurements become a little trickier because they
P are so far away from us. It seems that the radial velocity is
P still easy to compute, while the transverse component of the
P velocity gets more and more difficult to measure the farther away
P the celestial object is from us.

In general, yes, though I'd use the term "measure" not "compute" in
reference to velocities.

P The farther away an object is, the more *time* we must wait to be
P able to sense how far to the left or right the object has moved.

P We now know that both of the other large galaxies in our Local
P Group have a radial velocity that is in our general direction, that
P is, they both show a blue shift. This could mean many things, but
P i think it means that all three galaxies, Andromeda, Triangulum and
P our own Milky Way are gravitationally bound to a common center of
P gravity. All three are "falling toward" each other as they revolve
P around... something. (What this "something" is can be anybody's
P guess. Since scientists are finding huge masses of dark matter in
P the center of other galaxy clusters, it may follow that our Local
P Group revolves around a huge clump of this dark matter?)

Yes and no. It is probably the case that the three major galaxies are
gravitational bound together. This does not mean that there is
something "at" the common center of mass. Remember that saying one
object "orbits" another is an approximation. The Moon does not orbit
the Earth. The Earth and Moon orbit a common center of mass, which is
somewhere inside the Earth but not inside the center of the Earth.
Jupiter and the Sun orbit a common center of mass that happens to be
just outside the Sun's surface. The Milky Way, Andromeda, and
Triangulum galaxies orbit a common center of mass.

P So my question is this... Is there a formula to compute how long we
P must wait before we can gather fairly accurate measurements of
P transverse velocity? In other words...

Sure. The amount of time you have to wait is

t = (D/v)*theta

where v is the velocity of the object, D is its distance, and theta is
the size of the angle that it needs to move for you to determine that
it has moved. For instance, suppose you could observe water masers in
the Andromeda galaxy using VLBI techniques. You might hope to obtain
a resolution of about 0.3 milliarcseconds, so that you could determine
they had moved after they shifted about 1 mas (= 5 nanoradians = 5E-9
radians). Suppose that the Andromeda galaxy has a transverse velocity
of 200 km/s, and it is 750 kpc distant (= 2.3E19 km). Then t =
561777408 seconds ~ 17 years.

P How long would we have to wait to notice a sideways movement of the
P Andromeda and/or Triangulum Galaxies? a movement of, say, 1 mm?
P Would we already have to know the transverse velocity to compute
P this? or is there another way to measure the amount of time needed?
P (so we can use it to compute the transverse velocity?)

P Such a formula and measurement would shed an amazing amount of
P light on the dynamics of our Local Group!

Yes!

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