<p>You are specifically talking about absolute <em>visual</em> magnitude; how bright an object would appear to our eyes if the object were 10 parsecs away from us. However, you must remember that stars emit a wide spectrum of light — much of it isn't visible to the naked eye.</p>

<p>Luminosity is a measure of the total amount of energy given off by a star (usually as light) in a certain amount of time. Thus, luminosity includes both visible light and invisible light emitted by a star. So there isn't a precise conversion between luminosity and absolute visual magnitude, although there is an approximation we can do.</p>

<p>Absolute <em>bolometric</em> magnitude is an analogous magnitude system considering all the energy emitted by the star. The formula relating absolute bolometric magnitude with luminosity is as follows:
<span class="math-container">$$L_\text{star} = L_0 10^{-0.4 M_\text{Bol}}$$</span></p>

<p>where <span class="math-container">$L_\text{star}$</span> is the star's luminosity, <span class="math-container">$M_{\text{Bol}}$</span> is the bolometric magnitude of the star, and <span class="math-container">$L_{0}$</span> is the zero-point luminosity (the luminosity of a star with <span class="math-container">$M_{\text{Bol}} = 0$</span>) arbitrarily defined as <span class="math-container">$L_{0}=3.0128×10^{28}$</span> watts by the <a href="http://www.iau.org/news/announcements/detail/ann15023/" rel="nofollow noreferrer">IAU</a>. We can approximately convert between the two magnitudes via a <a href="https://en.wikipedia.org/wiki/Bolometric_correction" rel="nofollow noreferrer">bolometric correction</a>:
<span class="math-container">$$M_{\text{bol}}\approx M_V + BC$$</span>
The bolometric correction term <span class="math-container">$BC$</span> is empirically determined for the spectral class and evolutionary stage (although it can be modeled; see <a href="https://arxiv.org/abs/1008.3913v1" rel="nofollow noreferrer">Torres (2010)</a>). A table of values is given on the Wikipedia page.</p>

<p>The idea is that stars of the same spectral class and evolutionary stage have a similar distribution of wavelengths, so the discrepancy between all emitted wavelengths (<span class="math-container">$M_{\text{bol}}$</span>) and visible wavelengths (<span class="math-container">$M_V$</span>) should be similar.</p>

<p>Thus, the conversion between luminosity and absolute visual magnitude is approximately
<span class="math-container">$$L_\text{star} \approx L_0 10^{-0.4 (M_V + BC)}$$</span>
Of course, this is only an approximation, but it can give us a pretty decent idea of what to expect. Trying it on <a href="https://en.wikipedia.org/wiki/9_Pegasi" rel="nofollow noreferrer">9 Pegasi</a>, I predicted it would be around <span class="math-container">$1000 L_\odot$</span>, which is still in the ballpark of the actual value <span class="math-container">$1950 L_\odot$</span>. Mind you, the approximation would be better suited for less extreme stars than 9 Pegasi.</p>

CC BY-SA 4.0
2016-10-04T15:04:09.740Z
2019-04-28T21:29:27.850Z
2019-04-28T21:29:27.850Z
Sir Cumference
<ul>
<li>Why does this relationship involves the absolute magnitude of the sun and its luminosity?</li>
<li>How to derive an expression relating the absolute magnitude of a star to its luminosity?</li>
</ul>

<p>So according to the magnitude equation, $$m - M = 2.5\log\left(\frac{d^2}{d_0^2}\right)$$ 
$$\Rightarrow M = m - 2.5\log\left(\frac{d^2}{d_0^2}\right)\ \ (eq1)$$</p>

<p>and Luminosity is $$L = 4\pi(d^2) \times f$$<br>
 $$\Rightarrow d^2 = \frac{L}{4\pi \times f}\ \ (eq2)$$</p>

<p>Plugging eq2 to eq1 would have seem reasonable, but how is absolute magnitude of the sun and its luminosity used?</p>

CC BY-SA 3.0
2016-02-13T21:37:30.537Z
This site has TeX enabled. You can format your equations using TeX
James K
<astrophysics>
Relationship between absolute magnitude of a star and its luminosity?
J.S. M
# Relationship between absolute magnitude of a star and its luminosity?

(Asked by: username_0 on 2016-02-12)

* Why does this relationship involves the absolute magnitude of the sun and its luminosity?
* How to derive an expression relating the absolute magnitude of a star to its luminosity?

So according to the magnitude equation, $$m - M = 2.5\log\left(\frac{d^2}{d\_0^2}\right)$$
$$\Rightarrow M = m - 2.5\log\left(\frac{d^2}{d\_0^2}\right)\ \ (eq1)$$

and Luminosity is $$L = 4\pi(d^2) \times f$$  
$$\Rightarrow d^2 = \frac{L}{4\pi \times f}\ \ (eq2)$$

Plugging eq2 to eq1 would have seem reasonable, but how is absolute magnitude of the sun and its luminosity used?

**Tags:** astrophysics
> **Comments:**
> **username_1**: This site has TeX enabled. You can format your equations using TeX  

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## 1 answers

### Answer by username_2 on 2016-10-04. Score: 6 | Accepted answer

You are specifically talking about absolute *visual* magnitude; how bright an object would appear to our eyes if the object were 10 parsecs away from us. However, you must remember that stars emit a wide spectrum of light — much of it isn't visible to the naked eye.

Luminosity is a measure of the total amount of energy given off by a star (usually as light) in a certain amount of time. Thus, luminosity includes both visible light and invisible light emitted by a star. So there isn't a precise conversion between luminosity and absolute visual magnitude, although there is an approximation we can do.

Absolute *bolometric* magnitude is an analogous magnitude system considering all the energy emitted by the star. The formula relating absolute bolometric magnitude with luminosity is as follows:
$$L\_\text{star} = L\_0 10^{-0.4 M\_\text{Bol}}$$

where $L\_\text{star}$ is the star's luminosity, $M\_{\text{Bol}}$ is the bolometric magnitude of the star, and $L\_{0}$ is the zero-point luminosity (the luminosity of a star with $M\_{\text{Bol}} = 0$) arbitrarily defined as $L\_{0}=3.0128×10^{28}$ watts by the [IAU](http://www.iau.org/news/announcements/detail/ann15023/). We can approximately convert between the two magnitudes via a [bolometric correction](https://en.wikipedia.org/wiki/Bolometric_correction):
$$M\_{\text{bol}}\approx M\_V + BC$$
The bolometric correction term $BC$ is empirically determined for the spectral class and evolutionary stage (although it can be modeled; see [Torres (2010)](https://arxiv.org/abs/1008.3913v1)). A table of values is given on the Wikipedia page.

The idea is that stars of the same spectral class and evolutionary stage have a similar distribution of wavelengths, so the discrepancy between all emitted wavelengths ($M\_{\text{bol}}$) and visible wavelengths ($M\_V$) should be similar.

Thus, the conversion between luminosity and absolute visual magnitude is approximately
$$L\_\text{star} \approx L\_0 10^{-0.4 (M\_V + BC)}$$
Of course, this is only an approximation, but it can give us a pretty decent idea of what to expect. Trying it on [9 Pegasi](https://en.wikipedia.org/wiki/9_Pegasi), I predicted it would be around $1000 L\_\odot$, which is still in the ballpark of the actual value $1950 L\_\odot$. Mind you, the approximation would be better suited for less extreme stars than 9 Pegasi.


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