What is the unit of the correlation function between logarithmically rebinned galaxy spectrum and stellar template spectrum?

Question

Assume I have a logarithmically rebinned galactic spectrum $G(x)$ and a stellar template spectrum $S(x)$, where $x = \ln{\lambda}$. When $[\lambda] = 1 Å$, then $[x] = 1 Np/Å$. The fourier transformed galaxy spectrum is $\tilde{G}(\kappa)$, then $[\kappa]= 1/(Np/Å)$.

Now the correlation function between galaxy and spectra is defined as: $K_{S, G}(z)= \int dx G(x+z)S(x)$, so the unit of $z$ should be $[z] = [x]= 1 Np/Å $? Since the rebinning on a logarithmic wavelength scale introduces an easy conversion from measured wavelength shifts to velocity shifts: $\Delta x = \Delta \lambda/\lambda=\Delta v/c$. So the way $K_{S,G}(z)$ is defined above it measures shifts in wavelengths between the spectra. For my analysis, I needed a velocity axis, so I plotted my correlation function on a velocity axis (which would be wrong to label with Np/Å), at the same time I just wonder how to do this consistently in the mathematical formalism. One would have to do something like $\bar{K}_{S,G}(v) \equiv K_{S,G}(v/c)$ ?

And then one would need a different variable in fourier space: $\tilde{\bar{K}}_{S, G}(\bar{\kappa})$, where $\bar{\kappa} = 1 s/km$, when $[v] =1 km/s$ ?

Answer 1

The unit is that of a velocity. If you have bins of equal $\ln \lambda$ (which equals $\Delta \lambda/\lambda =\Delta v/c$), then each of your pixels (in your spectrum and in the template) represents a constant velocity increment $\Delta v = c |\Delta \ln \lambda|$, which could have units of km/s.

The shift in the peak of a cross-correlation function is known as "the lag" and the velocity shift is given by the lag (in pixels) multiplied by $\Delta v$.