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

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 $$[v] =1 km/s$$ ?

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 $$\Delta v$$.
• Makes sense, but what do I need to change in the formula to make it consistent. Because looking at the formula that I gave for the cross-correlation function the function argument of the galaxy spectrum$(x+z)$ inside the integral needs to have the same units. May 26 at 13:47
• @trynerror I don;t understand your comment. The x-axes of both spectrum and template need to be the same. The cross correlation is a function of $z$, which is the lag in pixels. May 26 at 14:22
• When I write $K_{S, G}(z)= \int dx G(x+z)S(x)$, then the units of x and z should be the same formally ? I understand your answer and it makes sense, I guess the remaining doubt is more about this formula. May 26 at 14:37