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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?

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  • $\begingroup$ 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. $\endgroup$ – James K Oct 29 '15 at 11:30
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Conrad is almost right. It is true generally that if a Galaxy is close enough to take spectra of individual stars (e.g. luminous supergiants) then it is not far enough away to be regarded as part of the "Hubble flow" and so applying Hubble's law to this star, or its host galaxy, would not yield a reliable distance in any case, but would reflect the "peculiar motion" of that galaxy.

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.

The exceptions are supernovae. The redshifts of individual supernovae, that briefly outshine their galaxies, can be measured right across the universe. Here you are correct that the measured redshift is a combination of cosmological redshift due to the expansion of the universe and a velocity of the star relative to the Hubble flow at that distance. There is no way to distinguish between these two unless velocity measurements could be obtained for other objects in the same galaxy. Given the rarity of supernovae, we might wait a long time for this.

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 $\sim 1$%.

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We measure the aggregated red-shift of galaxies not single stars, which averages out the motion of the stars within the galaxy. This leaves us with the proper motion of the galaxy to deal with, which can be estimated from the red-shifts of other galaxies in the same group/cluster/super-cluster and their distribution. But this is all moot at high red-shift since proper motions become small compared to the Hubble flow.

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