# Flux vs. Velocity representation of spectra?

I'm looking for a basic explanation of the flux vs. velocity representation of a spectrum and how it's obtained from the regular flux vs. wavelength representation. A good example of this is in Arav et al 2001 (see here, Fig. 2 and 3).

More specific questions I can ask are:

• How do you convert between wavelength and velocity in these figures (i.e. what phenomenon are we using to make the conversion & what's the equation)?
• What thing has the velocity in question/what does the velocity represent? For example, is it the astrophysical object moving in this direction?

... But any further elaboration is much appreciated. As much as I've seen these (I'm trying to brush up, it's been a while since I've done astro stuff), I can't find any explanation for this; perhaps I don't know the right jargon, but yeah it's difficult to find a simple answer. Thanks in advance for the help.

• First part is a duplicate of astronomy.stackexchange.com/questions/35231/… Commented May 11, 2021 at 6:57
• I'm not sure I understand why it's a duplicate. I see that my post was edited to include Fig. 2 from my reference, but not Fig. 3 -- the question you reference here shows the conversion for a single spectral line, which as I understand you're converting to velocity based on deviation from $\lambda_0$, which is ok. In Fig 3 of my reference you have the exact same spectrum as in Fig 2 (the picture above), except the x-axis is on a scale of velocity; it's not dealing with a single peak.
– Erik
Commented May 11, 2021 at 15:38
• You may be right. Commented May 11, 2021 at 16:05

I think the velocity scale is calculated something like this: $$v_r = c\left( \frac{\lambda - \lambda_0}{\lambda_0}\right),$$ where $$\lambda$$ is the observed wavelength and $$\lambda_0$$ is wavelength of the Mg II line, corrected for the redshift of the quasar ($$z=0.868)$$ it is observed towards. The confusing thing is, for Fig.~3 in the cited paper, that the Mg II feature is actually doublet with lines at rest wavelengths of $$\lambda_r =2796.35$$ and $$2803.53$$ Angstroms. Therefore $$\lambda_0 = \lambda_r (1+z) = 5223.58\ \ {\rm or} \ \ 5236.99\ \ {\rm Angstroms}$$ corresponding to the thick and thin solid lines on Fig.3 respectively.

In other words, zero on the velocity scale would correspond to a Mg II absorber at the redshift of the quasar (an intrinsic absorber), whereas negative velocities correspond to absorbers at lower redshift, by the velocities indicated.

For a check on this - calculate the velocity of feature "b", which appears to be at about 5221 Angstroms in the observed spectrum for the redder doublet component (the thin solid line in Fig.3). Using the above definition, this would be a velocity of -915 km/s.

Looking at Fig.3, there seems to be a major problem. Feature "b" is at -680 km/s. However, if we look further on at Figs. 4 and 5, we see feature "b" plotted at -850 km/s in the Mg II spectrum, which is much closer, and would perhaps be in exact agreement if feature "b" was actually at 5222 Angstroms (which looks unlikely) or that a 4th, unspecified decimal place had been used in the redshift. e.g. if $$z=0.8676$$ then $$\lambda_0 = 5235.87$$ Angstroms, and the velocity of a feature measured at 5221 Angstroms would then be -852 km/s in perfect agreement with Figs. 4 and 5.

Either way, there is major disagreement between Fig. 3 (which I suspect is in error) and Figs. 4 and 5 (which could be reconciled with the above analysis if the redshift was 0.8676).

I'm sorry this is not very conclusive and I certainly wouldn't accept it without someone else verifying that the x-axis of Fig.3 appears to be problematic.

• Thanks for pointing out this discrepancy in the paper. I'll add here for anyone else that might read, that the opening paragraph in section 2.5 (He I) along with the info in Table 1 adds additional clarity to your answer: the transition wavelength for MgII is given as 2803 A, which I guess gives us the zero point for the velocity scale. So it seems to me (and I'm guessing I'm partially if not totally wrong in this conclusion) that the velocity scale doesn't represent anything physical, so much as it provides a common scale for various absorption features that incorporates the redshift.
– Erik
Commented May 11, 2021 at 23:23