# How to disentangle a very distant star's relative velocity vs. redshift distance

We measure a star's relative velocity towards or away from us via its Doppler-shifted spectrum. This is also how we measure the distance of very distant stars: measuring the shifts in the spectrum tells us its relative velocity, which we interpret as a distance via Hubble's Law.

Suppose we knew the star's relative velocity and distance exactly. It seems to me that if the star were moving away from us slightly less than would be expected from expansion alone, it would be because it was moving towards us more than the matter surrounding it: it would have some additional velocity component towards us. If we were to try to infer that star's distance from expansion alone, we would measure its relative velocity via the Doppler shift and would conclude that it was closer than it actually is. Conversely, if it were moving away from us faster than it should based on expansion alone, we would estimate its distance further than it actually is.

Is there any way to disentangle this degeneracy?

• There is a mis-perception that we know astronomical distances with high precision. Its not the case. The proper motion is just one of the uncertainties that we have to deal with when estimating the distance of a galaxy, and in practice, not the greatest one. – James K Oct 29 '15 at 11:30

To put some numbers on this. Galaxy peculiar motions tend to be a few 100 km/s, as do the individual velocities of stars with respect to their galaxies. Taking a Hubble constant of 70 km/s per Mpc, we see that we need to be at distances of 15 Mpc before Hubble recession velocities ($v = H_0 d$) become large compared with peculiar motions. At these distances we cannot observe individual stars - they are too faint and unresolved from the bulk of the Galactic light.
But does it matter? Even if we look at a "low redshift" supernova at $z=0.1$, its Hubble recession velocity is 30,000 km/s and far in excess of any peculiar velocity contribution at the level of $\sim 1$%.