How do we determine a redshift error?

Question

I am struggling to calculate the redshift error of a galaxy. I was able to determine the redshifts of the absorption and emission lines from the galaxy's spectrum and then took the average of the redshift of the lines to determine the (overall) redshift of the galaxy. I know "error" is associated with the true and measured values but I don't know the true value of the redshift of the galaxy. Is there an explicit formula or something that I can use to calculate the redshift error of the galaxy without reference to the true redshift of the galaxy? All I know are the emitted and observed wavelengths of the emission and absorption lines of the galaxy spectrum.

Answer 1

You're confusing error and uncertainty.

In school labs or other situations where the true value of a quantity is known, the difference between a measured value and that true value is sometimes referred to as the "error" in the measurement. A lower error is taken as an indication that the results are successful.

When making observations, the true value of a quantity is seldom known. Indeed, that's usually the purpose of making observations: not as a pedagogical tool but as a way to determine something about the world. In this case, we're concerned with the uncertainty in the measurement - that is, essentially how confident we are in the value we determine. More quantitatively, we might give something like a $3\sigma$ uncertainty that explains how likely it is that the true value falls within 3 standard deviations of the measurement. That's a way of showing how robust the results are.

It's also helpful for comparing sets of measurements - if one group says that (and I'm making up numbers here) that their measurements have determined the Hubble constant to be $H_0=67.0\pm1.2\text{ km}\text{ s}^{-1}\text{ Mpc}^{-1}$ and another group says that their methods have found it to instead be $H_0=74.00\pm0.75\text{ km}\text{ s}^{-1}\text{ Mpc}^{-1}$, then we can see from the uncertainties that the two results are in fairly strong tension with one another.

If you're determining something like redshift, you presumably have some raw data measuring other quantities. Perhaps you've identified H$\alpha$ emission from a source and found it to be at $\lambda=8540\pm30$ angstroms, while the rest wavelength is $\lambda_0=6562.8$ angstroms. Then you can use standard uncertainty propagation methods to determine the uncertainty in the redshift of the source. In this case, the redshift is $$z=\frac{\lambda}{\lambda_0}-1$$ The uncertainty in $z$ is then $$\sigma_z=\sqrt{\left(\frac{\partial z}{\partial\lambda}\right)^2\sigma_{\lambda}^2+\left(\frac{\partial z}{\partial\lambda_0}\right)^2\sigma_{\lambda_0}^2}$$ where I've included $\sigma_{\lambda_0}$, though realistically it should be much smaller than $\sigma_{\lambda}$. Keep in mind also that $\sigma_{\lambda}$ has itself presumably been determined through error propagation from the raw data collected.

(Side note: As pela pointed out, this assumes that the redshift is entirely due to the motion of the object you're looking at, and that's not always the case. I originally picked Lyman $\alpha$ has my line, which turns out to not be particularly useful. Additionally, the toy redshift ended up being quite low and therefore even more sensitive to other not-recessional contributions, like stellar winds. H$\alpha$ is, I think, a better example, especially with this higher redshift.)