Logarithms and units

One of the things that every intro calculus student learns is: $${d\ln x\over dx} = {1\over x}$$ This property of the logarithm leads to something else, which turns out to be useful to physicists and astronomers, but is never explicitly taught. If we rearrange this equation to read $${d\ln x} = {dx\over x}$$ we see that a given change in the logarithm (\(d\ln x\)) corresponds to a given fractional change in x. This equation also implies that the logarithm of anything is unitless, as follows:

• the right side of this equation, \({dx\over x}\), is unitless regardless of the units of x;
• therefore the left side, \(d\ln x\), must also be unitless; 
• \(d\ln x\) must have the same units as \(\ln x\);
• therefore \(\ln x\) must also be unitless, regardless of the units of x. 

Physics students keeping track of their units can be stumped: what units does the log of a current or a voltage have? This tiny bit of math helps us see that the answer is "none."

The fact that \(d \ln x\) specifies a fractional change in x has further repercussions in astronomy, because it is traditional to quote the measurement of a flux \(f\) in the magnitude system: $$m = -2.5 \log_{10} {f\over f_0}$$ where \(f_0\) is some reference flux. This means that a quoted uncertainty in the magnitude of a star or galaxy, \(dm\), specifies a fractional uncertainty in the flux. Let's work out the details: \(\log_{10} x\) is the same as \({\ln x \over \ln 10}\) so $$dm = -{2.5\over \ln 10}  d\ln{f\over f_0} $$ $$dm = -{2.5\over \ln 10} {df\over f} $$ Because \(\ln 10\approx 2.30\), we get \(dm \approx -1.086 {df\over f}\).  For quick estimation purposes, the magnitude uncertainty is about the same as the fractional uncertainty in flux.

This explains why a 0.1 mag uncertainty is about a 10% flux uncertainty, regardless of the magnitude. One should not say that a 0.1 mag uncertainty is a 1% uncertainty in an \(m=10\) star, nor a 0.5% uncertainty in an \(m=20\) galaxy.  For the quantity that matters---the flux of the object---a 0.1 mag uncertainty implies about a 10% uncertainty regardless of the flux.

One Plus z

This marks the launch of a new series of posts, aimed at astronomy and physics majors. In the course of my teaching I've noticed a few topics---such as propagation of errors and reduced mass---which seem to fall through the cracks between classes.  Students hear a bit about reduced mass in more than one class, but never seem to get a satisfying explanation in any one class.  Their lab instructor taught them how to propagate errors but never made them think about why.  And so on.  This first post is much more specific---how to think about redshifts and velocity dispersions in cosmology---but fits the bill because it seems to fall through the cracks between textbooks.  Practitioners know that "you need to divide by 1+z" but documentation of this is hard to come by.  So here we go.

In cosmology, we often want to measure the rest-frame velocity dispersion of a galaxy cluster, but what we actually measure is the redshift dispersion. How are they related? Redshift z is defined in terms of emitted and observed wavelengths as

This means that 1+z is a stretching factor; it is the ratio of observed to emitted wavelengths.  So you will see the combination 1+z over and over, rather than z by itself.  Get used to thinking in terms of 1+z!

The Doppler shift formula tells us the wavelength stretching factor in terms of velocity:

You will often see this called the relativistic Doppler formula, as opposed to the simpler low-velocity approximation used in many situations. But I suggest thinking of this as the Doppler formula because  high velocities are common in astrophysics, and this correct version is simple enough to memorize. Habitually using the low-velocity approximation can get you in trouble.

The Doppler formula can be inverted to obtain

Now imagine two galaxies, one at rest¹ in the cluster frame (with velocity v₁ in our frame) and a second moving with some velocity v₂₁ relative to the cluster which implies some velocity v₂ in our frame.  According to the Einstein velocity addition law,

Substituting the inverted Doppler formula into this, we obtain a complicated-looking expression for v₂₁/c:

which we can simplify in a few steps:

Because of my poor equation formatting, I have to remind you here that this is an expression for v₂₁/c, where v₂₁ represents a velocity in the cluster frame rather than in our frame. This gets us close to our goal because we want to know the velocity dispersion in the cluster frame. But this is as far as we can go without an approximation. A useful approximation in this context is that so define and eliminate z₂ using :

Taylor expanding this about we obtain

This is true for any small redshift difference, so it must be true if delta represents the redshift dispersion of the cluster (thus making v₂₁ represent the velocity dispersion of the cluster). Therefore

However, there is a much more elegant way to derive the same result. Imagine a hypothetical observer on the first galaxy. Because of the definition of 1+z as the ratio of wavelengths, it must be true that 1+z₂ = (1+z₁)(1+z₂₁) where z₂₁ is the redshift of galaxy 2 as seen by galaxy 1 (z₁ and z₂ are, as before, redshifts seen by us). Therefore

Again we use an approximation: so that we can use the low-velocity approximation for the Doppler shift, . Therefore

which is the same result as before.  We don't actually need special-relativistic reasoning if we simply use the definition of redshift to isolate the one nonrelativistic velocity in the problem.

We can better expose the equivalence of these two approaches by taking the idea of daisy-chaining wavelength ratios and applying it directly to the Doppler law:

This just says that galaxy 2's wavelength ratio ("ratio'' here is relative to a laboratory standard) observed by us is its wavelength ratio observed by galaxy 1, times galaxy 1's wavelength ratio observed by us.   In a few lines of algebra, you can show that the above expression leads directly to the Einstein velocity addition law.  The addition law can be derived in more than one way, but to me this is the most intuitive way.  Thus, daisy-chaining Doppler factors and using the velocity addition law are not contrasting approaches; they are actually the same thing.

Exercise for the reader: show that the expression above does indeed lead to the Einstein velocity addition law.

Footnotes:
¹ I specify "at rest" here only so that later it will be easy to think of this galaxy’s redshift as the mean cluster redshift.