# How are galaxy distances inferred from their recessional velocities?

*I have a Ph.D in physics from Caltech (1948).I am living in a retirement community in Redlands, CA. By request, I am giving lectures
on astronomy every two weeks. A question that I am asked but cannot
answer satisfactorily is "How do they determine the age of the
universe and the distance in light-years (or kilometers) to distant
galaxies?"
*

*
I understand that the redshift determines the recession velocity, but
I do not understand the relation of this to the distance.
*

There is a simple answer and a not-so-simple complication to it.

The simple answer is that it has been known for some time that the distance to a galaxy is proportional to its recessional velocity: this is called Hubble's Law, and it was demonstrated observationally by Hubble in the late 1920s. It turns out that if you assume that the Universe is homogeneous and isotropic (which we think is the case), then Hubble's Law can also be predicted by theory. The constant of proportionality between the recessional velocity of galaxies and their distance is called Hubble's constant. Astronomers have tried to measure Hubble's constant ever since the term was coined: the simplest way is to look at the recessional velocities of galaxies for which the distances are known by other means (like looking at the period of variable stars). The current best estimate of Hubble's constant today is about 20 km/s per Mly, so that a galaxy with a recessional velocity of 2000 kilometres per second is 100 mega-light-years away, and so on.

We can therefore use Hubble's law to tell us distances to galaxies by simply measuring their redshifts, which is a relatively easy thing to do even for very distant objects. We can also estimate the age of the Universe: you'll notice that the units of Hubble's constant are actually 1/time, so one over Hubble's constant must be the characteristic age of the Universe. From the Hubble constant above, we therefore calculate that the Universe is about 14 billion years old.

Now for the complications: it turns out that for *very* distant objects (at redshifts of, say, 2 and greater) Hubble's Law doesn't quite hold, because at very great distances we have to start taking the (4-dimensional) curvature of the Universe into account. A more important effect is that in most cosmologies, Hubble's constant isn't really a constant: it actually increases as you look back in time, so that its value was different for galaxies at redshifts of 5 than it is today. So, to get estimates of the distances to galaxies and the age of the Universe, astronomers must assume a set of cosmological parameters for the Universe (for instance, the total amount of "normal" matter it contains) and model its age and the distances of galaxies as a function of redshift by integrating the equations of motion for cosmological evolution. The answers they get are not that different from what we estimated above, but can be crucial to learning about galaxies at high redshift (for instance, the extraction of their sizes depends linearly on their distance, and their masses on the distance squared).

For all intents and purposes, however, Hubble's law is a very powerful tool for getting distances from velocities (and you may want to stop at that in your lectures).

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