<p>The UV luminosity of a galaxy can be calculated, given a stellar population. This population, in turn, can be calculated given an <em>initial mass function</em> (IMF), i.e. the distribution function of stellar masses. In this case, the UV luminosity should be linearly proportional to the star formation rate (SFR), sometimes written <span class="math-container">$\Psi$</span>.</p>
<p>The UV is primarily emitted by young stars with lifetimes of <span class="math-container">$\lesssim10^8\,\mathrm{yr}$</span>. Assuming that the SFR is constant over longer time scales, <a href="https://ui.adsabs.harvard.edu/abs/1998ARA%26A..36..189K" rel="nofollow noreferrer">Kennicutt (1998)</a> calculates in his seminal paper (with ~5000 citations) for a <a href="https://ui.adsabs.harvard.edu/abs/1955ApJ...121..161S" rel="nofollow noreferrer">Salpeter (1995)</a> IMF (equally seminal; ~6700 citations) with mass limits 0.1 and 100 <span class="math-container">$M_\odot\,\mathrm{yr}^{-1}$</span> that
<span class="math-container">$$
\frac{\Psi_\mathrm{Sal}}{M_\odot\,\mathrm{yr}^{-1}} = 1.4\times10^{-28}\frac{L_\nu}{\mathrm{erg}\,\mathrm{s}^{-1}\,\mathrm{Hz}^{-1}}.
$$</span>
were <span class="math-container">$L_\nu$</span> is the emitted UV luminosity <em>density</em>, i.e. luminosity per frequency bin. At a (luminosity) distance <span class="math-container">$d_L$</span> from the galaxy, the observed flux density <span class="math-container">$f_\nu$</span> is then
<span class="math-container">$$
f_\nu = \frac{L_\nu}{4\pi d_L^2}.
$$</span></p>
<p>The flux density is related to the apparent (observed) UV magnitude <span class="math-container">$m_\mathrm{UV}$</span> as (<a href="https://ui.adsabs.harvard.edu/abs/1983ApJ...266..713O" rel="nofollow noreferrer">Oke &amp; Gunn 1983</a>; ~1900 citations despite their +/- error in their most important equation)
<span class="math-container">$$
m_\mathrm{UV} = -2.5\log \frac{f_\nu}{\mathrm{erg}\,\mathrm{s}^{-1}\,\mathrm{cm}^{-1}\,\mathrm{Hz}^{-1}} - 48.6.
$$</span></p>
<h3>Distance modulus and dust extinction</h3>
<p>The apparent magnitude is related to the absolute magnitude <span class="math-container">$M$</span> through the distance modulus
<span class="math-container">$$
\mu \equiv m - M.
$$</span></p>
<p>Furthermore, the above relations don't take into account the fact that some of the emitted light is absorbed by dust. Dust extinction <em>increases</em> the observed magnitude (because magnitudes are backwards) by a number <span class="math-container">$A_\mathrm{UV}$</span>.</p>
<p>That is, the apparent UV magnitude is
<span class="math-container">$$
m_\mathrm{UV} = M_\mathrm{UV} + \mu + A_\mathrm{UV}.
$$</span></p>
<p>Typically, the extinction <span class="math-container">$A_\mathrm{UV}$</span> isn't measured explicitly (because UV measurements are difficult); rather the extinction is measured in two optical bands, e.g. <span class="math-container">$B$</span> and <span class="math-container">$V$</span>. The extinction <span class="math-container">$A_\lambda$</span> at a given wavelength <span class="math-container">$\lambda$</span> can then be estimated as
<span class="math-container">$$
A_\lambda = k_\lambda E(B-V),
$$</span>
where <span class="math-container">$E(B-V)\equiv A_B - A_V$</span> is the color excess, and <span class="math-container">$k_\lambda$</span> is given by the assumed extinction law, e.g. a <a href="https://ui.adsabs.harvard.edu/abs/1989ApJ...345..245C" rel="nofollow noreferrer">Cardelli (1989)</a> law or a <a href="https://ui.adsabs.harvard.edu/abs/2000ApJ...533..682C" rel="nofollow noreferrer">Calzetti (2000)</a> law.</p>
<h3>The role of the initial mass function</h3>
<p>The Salpeter IMF is old, and at that time we didn't have too good data for the low-mass stars (which are faint), so he just fitted a single power-law. With better data, it was eventually found that there were less low-mass stars than previously thought, i.e. it takes less stellar mass to produce a given UV luminosity. A popular choice today is the <a href="https://ui.adsabs.harvard.edu/abs/2003PASP..115..763C" rel="nofollow noreferrer">Chabrier (2003)</a> IMF (~5100 citations), which gives <em>roughly</em> a factor of 1.8 more UV than the Salpeter IMF per star formed, on average. That is, the conversion factor is 1.8 times smaller.</p>
<p>With a Chabrier IMF, you hence get that the relation between absolute magnitude of UV and star formation rate (which is what you're asking) is
<span class="math-container">$$
\frac{\Psi_\mathrm{Cha}}{M_\odot\,\mathrm{yr}^{-1}} = 7.8\times10^{-29}\,\times\,4\pi d_L^2 \times10^{-(M_\mathrm{UV} + \mu + A_\mathrm{UV}+48.6)/2.5},
$$</span>
where <span class="math-container">$d_L$</span> is measured in cm.</p>
<h3>Simpler relation</h3>
<p>The above formula resembles the one you give, but is a bit strange in that it both contains <span class="math-container">$\mu$</span> <em>and</em> <span class="math-container">$d_L$</span>, which are just two different measures of the same thing. Using the definition
<span class="math-container">$$
\mu = 5\log\left(\frac{d_L}{10\,\mathrm{pc}}\right)
$$</span>
we can — after <a href="https://www.evernote.com/shard/s283/sh/ff7310fa-e363-4486-80e1-0b38c2489588/3103e43bd0ad772c/res/1b8b415d-2107-4bcd-9a16-0a49b2609a61" rel="nofollow noreferrer">a bit of algebra</a> — express the definition of the Oke &amp; Gunn (AB) magnitude as
<span class="math-container">$$
M_\mathrm{UV} = -2.5\log L_\nu + 51.6.
$$</span>
With this, we can can write the relation as
<span class="math-container">$$
\frac{\Psi_\mathrm{Cha}}{M_\odot\,\mathrm{yr}^{-1}} = 7.8\times10^{-29}
\,\times
10^{-(M_\mathrm{UV}+A_\mathrm{UV}-51.6)/2.5},
$$</span>
or even
<span class="math-container">$$
\boxed{
\frac{\Psi_\mathrm{Cha}}{M_\odot\,\mathrm{yr}^{-1}} = 3.4\times10^{-8}
\,\times
10^{-(M_\mathrm{UV}+A_\mathrm{UV})/2.5}.}
$$</span></p>

CC BY-SA 4.0
2020-11-13T21:45:44.330Z
@RobJeffries Hmm… yeah, maybe… But the title asks for the relation between _absolute_, not apparent, magnitude and SFR, so the way I interpreted this is that they found that formula, but didn't know what the terms meant. But I agree it's a somewhat peculiar formula, so I'll edit to give a simpler form.
pela
CC BY-SA 4.0
2020-11-13T12:30:07.690Z
2020-11-14T12:15:04.793Z
2020-11-14T12:15:04.793Z
pela
<p>I surfed the internet and searched a lot of sites,
so far I got this</p>
<p><a href="https://i.sstatic.net/Au5Ci.gif" rel="nofollow noreferrer"><img src="https://i.sstatic.net/Au5Ci.gif" alt="enter image description here" /></a></p>
<p>but unable to find the formula which relates only M<sub>uv</sub> and the Ψ as I don't know the value of parameters A<sub>uv</sub> and μ(z).</p>
<p>If there exists any such formula, then do help!</p>

CC BY-SA 4.0
2020-11-12T20:27:08.147Z
Explain all the terms please.
ProfRob
CC BY-SA 4.0
2020-11-14T23:52:04.500Z
Hi space_nkc, I don’t know if you saw it, but I edited my answer after you accepted it, to give a simpler relation :)
pela
<star><galaxy><mass><star-formation><formation>
Relation between absolute magnitude of UV and star formation rate
Nitin Kumar Chauhan
# Relation between absolute magnitude of UV and star formation rate

(Asked by: username_0 on 2020-11-12)

I surfed the internet and searched a lot of sites,
so far I got this

[![enter image description here](https://i.sstatic.net/Au5Ci.gif)](https://i.sstatic.net/Au5Ci.gif)

but unable to find the formula which relates only Muv and the Ψ as I don't know the value of parameters Auv and μ(z).

If there exists any such formula, then do help!

**Tags:** star, galaxy, mass, star-formation, formation

> **Comments:**
> **username_1**: Explain all the terms please.  
> **username_2**: Hi space_nkc, I don’t know if you saw it, but I edited my answer after you accepted it, to give a simpler relation :)  

---

## 1 answers

### Answer by username_2 on 2020-11-13. Score: 4 | Accepted answer

The UV luminosity of a galaxy can be calculated, given a stellar population. This population, in turn, can be calculated given an *initial mass function* (IMF), i.e. the distribution function of stellar masses. In this case, the UV luminosity should be linearly proportional to the star formation rate (SFR), sometimes written $\Psi$.

The UV is primarily emitted by young stars with lifetimes of $\lesssim10^8\,\mathrm{yr}$. Assuming that the SFR is constant over longer time scales, [Kennicutt (1998)](https://ui.adsabs.harvard.edu/abs/1998ARA%26A..36..189K) calculates in his seminal paper (with ~5000 citations) for a [Salpeter (1995)](https://ui.adsabs.harvard.edu/abs/1955ApJ...121..161S) IMF (equally seminal; ~6700 citations) with mass limits 0.1 and 100 $M\_\odot\,\mathrm{yr}^{-1}$ that
$$
\frac{\Psi\_\mathrm{Sal}}{M\_\odot\,\mathrm{yr}^{-1}} = 1.4\times10^{-28}\frac{L\_\nu}{\mathrm{erg}\,\mathrm{s}^{-1}\,\mathrm{Hz}^{-1}}.
$$
were $L\_\nu$ is the emitted UV luminosity *density*, i.e. luminosity per frequency bin. At a (luminosity) distance $d\_L$ from the galaxy, the observed flux density $f\_\nu$ is then
$$
f\_\nu = \frac{L\_\nu}{4\pi d\_L^2}.
$$

The flux density is related to the apparent (observed) UV magnitude $m\_\mathrm{UV}$ as ([Oke & Gunn 1983](https://ui.adsabs.harvard.edu/abs/1983ApJ...266..713O); ~1900 citations despite their +/- error in their most important equation)
$$
m\_\mathrm{UV} = -2.5\log \frac{f\_\nu}{\mathrm{erg}\,\mathrm{s}^{-1}\,\mathrm{cm}^{-1}\,\mathrm{Hz}^{-1}} - 48.6.
$$

### Distance modulus and dust extinction

The apparent magnitude is related to the absolute magnitude $M$ through the distance modulus
$$
\mu \equiv m - M.
$$

Furthermore, the above relations don't take into account the fact that some of the emitted light is absorbed by dust. Dust extinction *increases* the observed magnitude (because magnitudes are backwards) by a number $A\_\mathrm{UV}$.

That is, the apparent UV magnitude is
$$
m\_\mathrm{UV} = M\_\mathrm{UV} + \mu + A\_\mathrm{UV}.
$$

Typically, the extinction $A\_\mathrm{UV}$ isn't measured explicitly (because UV measurements are difficult); rather the extinction is measured in two optical bands, e.g. $B$ and $V$. The extinction $A\_\lambda$ at a given wavelength $\lambda$ can then be estimated as
$$
A\_\lambda = k\_\lambda E(B-V),
$$
where $E(B-V)\equiv A\_B - A\_V$ is the color excess, and $k\_\lambda$ is given by the assumed extinction law, e.g. a [Cardelli (1989)](https://ui.adsabs.harvard.edu/abs/1989ApJ...345..245C) law or a [Calzetti (2000)](https://ui.adsabs.harvard.edu/abs/2000ApJ...533..682C) law.

### The role of the initial mass function

The Salpeter IMF is old, and at that time we didn't have too good data for the low-mass stars (which are faint), so he just fitted a single power-law. With better data, it was eventually found that there were less low-mass stars than previously thought, i.e. it takes less stellar mass to produce a given UV luminosity. A popular choice today is the [Chabrier (2003)](https://ui.adsabs.harvard.edu/abs/2003PASP..115..763C) IMF (~5100 citations), which gives *roughly* a factor of 1.8 more UV than the Salpeter IMF per star formed, on average. That is, the conversion factor is 1.8 times smaller.

With a Chabrier IMF, you hence get that the relation between absolute magnitude of UV and star formation rate (which is what you're asking) is
$$
\frac{\Psi\_\mathrm{Cha}}{M\_\odot\,\mathrm{yr}^{-1}} = 7.8\times10^{-29}\,\times\,4\pi d\_L^2 \times10^{-(M\_\mathrm{UV} + \mu + A\_\mathrm{UV}+48.6)/2.5},
$$
where $d\_L$ is measured in cm.

### Simpler relation

The above formula resembles the one you give, but is a bit strange in that it both contains $\mu$ *and* $d\_L$, which are just two different measures of the same thing. Using the definition
$$
\mu = 5\log\left(\frac{d\_L}{10\,\mathrm{pc}}\right)
$$
we can — after [a bit of algebra](https://www.evernote.com/shard/s283/sh/ff7310fa-e363-4486-80e1-0b38c2489588/3103e43bd0ad772c/res/1b8b415d-2107-4bcd-9a16-0a49b2609a61) — express the definition of the Oke & Gunn (AB) magnitude as
$$
M\_\mathrm{UV} = -2.5\log L\_\nu + 51.6.
$$
With this, we can can write the relation as
$$
\frac{\Psi\_\mathrm{Cha}}{M\_\odot\,\mathrm{yr}^{-1}} = 7.8\times10^{-29}
\,\times
10^{-(M\_\mathrm{UV}+A\_\mathrm{UV}-51.6)/2.5},
$$
or even
$$
\boxed{
\frac{\Psi\_\mathrm{Cha}}{M\_\odot\,\mathrm{yr}^{-1}} = 3.4\times10^{-8}
\,\times
10^{-(M\_\mathrm{UV}+A\_\mathrm{UV})/2.5}.}
$$

> **Comments:**
> **username_2**: @RobJeffries Hmm… yeah, maybe… But the title asks for the relation between _absolute_, not apparent, magnitude and SFR, so the way I interpreted this is that they found that formula, but didn't know what the terms meant. But I agree it's a somewhat peculiar formula, so I'll edit to give a simpler form.  

---

