# How does one measure velocities of far-off, bright objects

As the title already says, I want to know how one measures velocities of far-off, bright objects, e.g. when the mean parallax drift isn't measurable with current apparatus (this means when there is only a "non-moving" picture of that object available)

I know that one can measure the redshift of spectral lines and correct them for gravitational redshifting if the distance is already known. But then the relation between source-frequency $f_S$ and observed frequency $f_O$ is $$f_O = f_S \frac{\sqrt{1-\frac{v^2}{c^2}}}{1+\frac{v}{c}\cos(\alpha)}$$, where $\alpha$ is the angle between line of sight and the velocity vector.

If one only has a "non-moving" picture of the object, there is no way to determine this angle $\alpha$ and therefore no way to get the velocity by means of redshift.

As an addition: At which distance does the mean parallax drift become unmeasurble with current apparatus?

The "redshift" of a distant galaxy is defined in terms of its line of sight velocity. In our model of the expanding universe, once we move away from the local group of galaxies (which have their own peculiar motions), distant galaxies follow the Hubble flow and to first order have a line of sight velocity tht is proportional to their distance way (it gets more complicated for very distant galaxies).

Distant galaxies may well have a "tangential" velocity too, but for galaxies outside the local group these velocities will be be negligible compared with the redshift. i.e. The line of sight velocity due to the expansion of the universe is dominant.

I guess by "parallax drift" you actually mean proper motion - which is the rate at which a star's position changes with respect to the celestial coordinate system. This proper motion depends on how far away the star is and how fast it is moving tangentially with respect to the solar system.

Thus to estimate a tangential velocity you need both the proper motion and the distance to the star.

I think the most distant object for which a proper motion has been determined with any accuracy is the Andromeda galaxy, which is a couple of million light years away. This was achieved by studying the position of many stars in Andromeda over a 7 year period using the Hubble Space Telescope. The details can be found in Sohn et al. (2012); but the headline numbers are that the proper motion is a mere $\sim 0.05$ milli-arcseconds per year(!) , implying a tangential velocity (with respect to the solar system) of about 150 km/s.

Another candidate is measuring the velocities of material in the jet of the active galaxy M87 by Meyer et al. (2013). This galaxy is at 50 million light years, but the motion of the jet is only detectable here because it is moving relativistically.

These are quite special cases. In general, the tangential velocities of stars in our Galaxy are small and large-scale susrveys of proper motions are generally inaccurate beyond a few thousand light years. The upcoming Gaia results will improve this dramatically meaning we have good proper motions for objects out to tens of thousands of light years.

In general the total velocity of a faraway object is not obtainable, only the component along the line of sight. However, if a star is not too far away, and its position on the sky is measured very accurately with an interval of some years, the velocity component perpendicular to the line of sight can be determined, yielding together with the redshift the total velocity. The recently launched Gaia spacecraft is expected to be measure the proper motion of millions of stars in this way.

In distant galaxies, this is not possible. If one is interested in, say, the 3D velocity dispersion of the gas, the 1D dispersion is measured, and assuming (fairly) an isotropic dispersion, this number is then multiplied by $\sqrt{3}$ to get the total velocity.

The distance to which parallax is measurable depends on the angular resolution of your instrument. For milliarcsec resolution, parallax measurements are good out to roughly 1 kpc. If you wait long enough, the motion around the galactic center increases your baseline such that larger distance can be probed.