<p>You are familiar with the relationship between flux and luminosity, right? This is usually taken as the definition of luminosity distance in extragalactic astronomy,
<span class="math-container">$$ F = \frac{L}{4\pi D_L^2}.$$</span>
The quickest way to convert this into a relationship with spectral variables is to use <A HREF="https://en.wikipedia.org/wiki/Differential_form" rel="nofollow noreferrer">differential 1-forms</A>. So, <span class="math-container">$L \rightarrow L_\lambda \,\mathrm{d}\lambda_e$</span> and <span class="math-container">$F \rightarrow F_\lambda \,\mathrm{d}\lambda_o$</span>, with <span class="math-container">$\lambda_e$</span> the photon wavelength at emission (also called the <em>rest frame</em> wavelength) and <span class="math-container">$\lambda_o$</span> the observed wavelength at <span class="math-container">$z=0$</span> (also called the <em>comoving</em> wavelength). Thus
<span class="math-container">\begin{align}
  F_\lambda \,\mathrm{d}\lambda_o &amp; = \frac{L_\lambda \,\mathrm{d}\lambda_e}{4\pi D_L^2} \Rightarrow\\
  F_\lambda &amp; = \frac{L_\lambda}{4\pi D_L^2} \times \frac{\mathrm{d}\lambda_e}{\mathrm{d}\lambda_o}. 
\end{align}</span>
Now you just need to take your derivative using the definition of redshift, <span class="math-container">$\lambda_o = (1+z)\lambda_e$</span>, and use some way of calculating the luminosity distance in your cosmology.</p>

<p>Formally, a lot of people like to use <span class="math-container">$\nu F_\nu = \lambda F_\lambda$</span> as though it were a total flux, but it's not. The derivation will work, but only because redshifting scales all frequencies/wavelengths by a single constant. <span class="math-container">$\nu F_\nu$</span> isn't a measure of the flux under any part of the spectrum, it's a spectral flux using a different coordinate system. Consider the following
<span class="math-container">\begin{align}
   F &amp; = F_\nu \,\mathrm{d}\nu \\
    &amp;= \nu F_\nu \frac{\mathrm{d}\nu}{\nu} \\
    &amp;= F_{\ln \nu} \,\mathrm{d}\ln \nu.
\end{align}</span>
In other words, <span class="math-container">$\nu F_\nu$</span> is still a spectral flux. If <span class="math-container">$F_\nu$</span> is in <span class="math-container">$\mathrm{ergs}\,\mathrm{cm}^{-2}\,\mathrm{Hz}^{-1}$</span> then <span class="math-container">$\nu F_\nu$</span> will be in <span class="math-container">$\mathrm{ergs}\,\mathrm{cm}^{-2}\,(e\text{-fold})^{-1}$</span>. The <span class="math-container">$e\text{-fold}$</span> is a pseudo-unit, like radians, dex, octave, decibel, or magnitudes. In principle it can be omitted, just like how angular frequencies can be in radians per second, or just <span class="math-container">$\mathrm{s}^{-1}$</span>.</p>

<p>Fun math exercise: show that <span class="math-container">$\ln(10) \nu F_\nu$</span> has units of <span class="math-container">$\mathrm{ergs}\,\mathrm{cm}^{-2}\,\mathrm{dex}^{-1}$</span>.</p>

CC BY-SA 4.0
2019-07-19T11:07:22.163Z
2019-07-19T11:07:22.163Z
Sean Lake
<p>Say I have a spectrum of a galaxy at a redshift <span class="math-container">$z$</span>, in flux density units of erg/s/cm^2/Angstrom. I'd like to recover the spectrum (in the same flux density units) at z=0, i.e. at its rest wavelengths. How do I do this? </p>

<p>I saw this paper (<a href="https://arxiv.org/pdf/astro-ph/9905116.pdf" rel="nofollow noreferrer">Distance measures in cosmology, Hogg (2000)</a>) but am still confused, because of their use of luminosity and luminosity densities, and in my class I only have fluxes.</p>

CC BY-SA 4.0
2019-04-16T01:26:53.187Z
Since these are differential quantities, a careful application of the [chain rule](https://en.wikipedia.org/wiki/Chain_rule) should be all that's necessary for a basic answer. Since I am not careful, I'll defer to someone who is ;-)
uhoh
<galaxy><cosmology><redshift><spectra><luminosity>
How to correct observed flux densities for redshift
curious_cosmo
# How to correct observed flux densities for redshift

(Asked by: username_0 on 2019-04-15)

Say I have a spectrum of a galaxy at a redshift $z$, in flux density units of erg/s/cm^2/Angstrom. I'd like to recover the spectrum (in the same flux density units) at z=0, i.e. at its rest wavelengths. How do I do this?

I saw this paper ([Distance measures in cosmology, Hogg (2000)](https://arxiv.org/pdf/astro-ph/9905116.pdf)) but am still confused, because of their use of luminosity and luminosity densities, and in my class I only have fluxes.

**Tags:** galaxy, cosmology, redshift, spectra, luminosity

> **Comments:**
> **username_1**: Since these are differential quantities, a careful application of the [chain rule](https://en.wikipedia.org/wiki/Chain_rule) should be all that's necessary for a basic answer. Since I am not careful, I'll defer to someone who is ;-)  

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## 1 answers

### Answer by username_2 on 2019-07-19. Score: 2

You are familiar with the relationship between flux and luminosity, right? This is usually taken as the definition of luminosity distance in extragalactic astronomy,
$$ F = \frac{L}{4\pi D\_L^2}.$$
The quickest way to convert this into a relationship with spectral variables is to use [differential 1-forms](https://en.wikipedia.org/wiki/Differential_form). So, $L \rightarrow L\_\lambda \,\mathrm{d}\lambda\_e$ and $F \rightarrow F\_\lambda \,\mathrm{d}\lambda\_o$, with $\lambda\_e$ the photon wavelength at emission (also called the *rest frame* wavelength) and $\lambda\_o$ the observed wavelength at $z=0$ (also called the *comoving* wavelength). Thus
\begin{align}
F\_\lambda \,\mathrm{d}\lambda\_o & = \frac{L\_\lambda \,\mathrm{d}\lambda\_e}{4\pi D\_L^2} \Rightarrow\\
F\_\lambda & = \frac{L\_\lambda}{4\pi D\_L^2} \times \frac{\mathrm{d}\lambda\_e}{\mathrm{d}\lambda\_o}.
\end{align}
Now you just need to take your derivative using the definition of redshift, $\lambda\_o = (1+z)\lambda\_e$, and use some way of calculating the luminosity distance in your cosmology.

Formally, a lot of people like to use $\nu F\_\nu = \lambda F\_\lambda$ as though it were a total flux, but it's not. The derivation will work, but only because redshifting scales all frequencies/wavelengths by a single constant. $\nu F\_\nu$ isn't a measure of the flux under any part of the spectrum, it's a spectral flux using a different coordinate system. Consider the following
\begin{align}
F & = F\_\nu \,\mathrm{d}\nu \\
&= \nu F\_\nu \frac{\mathrm{d}\nu}{\nu} \\
&= F\_{\ln \nu} \,\mathrm{d}\ln \nu.
\end{align}
In other words, $\nu F\_\nu$ is still a spectral flux. If $F\_\nu$ is in $\mathrm{ergs}\,\mathrm{cm}^{-2}\,\mathrm{Hz}^{-1}$ then $\nu F\_\nu$ will be in $\mathrm{ergs}\,\mathrm{cm}^{-2}\,(e\text{-fold})^{-1}$. The $e\text{-fold}$ is a pseudo-unit, like radians, dex, octave, decibel, or magnitudes. In principle it can be omitted, just like how angular frequencies can be in radians per second, or just $\mathrm{s}^{-1}$.

Fun math exercise: show that $\ln(10) \nu F\_\nu$ has units of $\mathrm{ergs}\,\mathrm{cm}^{-2}\,\mathrm{dex}^{-1}$.


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