<p>Epsilon is a factor to describe how much of the mass that is falling towards the black hole is converted to photons. The general equation for radiation pressure is (<a href="http://www-astro.physics.ox.ac.uk/%7Egarret/teaching/lecture7-2012.pdf" rel="nofollow noreferrer">source</a>)</p>
<p><span class="math-container">$$P_{\rm rad}=\frac{L}{4\pi R^2c}$$</span></p>
<p>In which <span class="math-container">$L$</span> is the luminosity of the star or central body. But for a black hole, the luminosity comes from the rate of accretion into the central part of the black hole, and this is <span class="math-container">$dM/dt$</span> or <span class="math-container">$\dot M$</span>. If all the mass were converted to energy then the luminosity would be <span class="math-container">$\dot Mc^2$</span>, but in fact some of the matter does actually fall into the black hole. For a non-rotating black hole, 6% of the mass can be converted to energy.  For a rotating black hole as much as 42% can be converted (<a href="https://www.popularmechanics.com/space/a14524087/black-holes-mass-energy/" rel="nofollow noreferrer">source</a>) and so <span class="math-container">$\varepsilon$</span> has a value between 0.06 and 0.42.  A value of <span class="math-container">$\varepsilon=0.1$</span> is a reasonable empirical value.</p>
<p>So the formula above becomes</p>
<p><span class="math-container">$$P_{\rm rad}=\frac{\varepsilon \dot M c^2}{4\pi R^2c}$$</span></p>
<p>Then kappa is the opacity of the material that is being accreted. If the material is perfectly transparent, then it won't experience any radiation pressure.</p>
<p>If we assume the body is a plasma of fully ionised hydrogen, then its opacity is due to the <a href="https://en.wikipedia.org/wiki/Thomson_scattering" rel="nofollow noreferrer">Thompson scattering</a> by the electrons in the plasma, and has a value of <span class="math-container">$\kappa = 0.4 \,\mathrm{cm}^2\mathrm{g}^{-1}$</span> (<a href="https://www.astro.princeton.edu/%7Egk/A403/opac.pdf" rel="nofollow noreferrer">source1</a>, <a href="https://www.sciencedirect.com/topics/earth-and-planetary-sciences/thomson-scattering#:%7E:text=If%20electron%20scattering%2C%20also%20called,that%20is%2C%20independent%20of%20frequency." rel="nofollow noreferrer">source2</a>)</p>
<p>To find the force on a body, and noting that radiation pressure is force per unit area, whereas opacity is the
cross-sectional area per unit mass for radiation scattering. So</p>
<p><span class="math-container">$$L_{\rm rad} = P_{\rm rad}\kappa m$$</span></p>
<p>Combining these gives the formula in the video.</p>
<p>Garret Cotter's <a href="http://www-astro.physics.ox.ac.uk/%7Egarret/teaching/lecture7-2012.pdf" rel="nofollow noreferrer">slides on accretion</a> gives a lot more details.</p>

CC BY-SA 4.0
2023-04-07T19:04:07.297Z
2023-04-09T08:08:59.467Z
2023-04-09T08:08:59.467Z
James K
<p><a href="https://i.sstatic.net/sX7A1.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/sX7A1.png" alt="Rebecca Smethurst radiation pressure force" /></a></p>
<p>In her most recent video, &quot;An ULTRAMASSIVE black hole has been discovered in a BRAND NEW way!&quot;, as well as a previous video, &quot;How massive can black holes get?&quot;, Dr. Becky Smethurst shows an equation relating to the radiation pressure of black holes:</p>
<p><span class="math-container">$$F_{\rm rad} = \epsilon \dot{M}c \frac{\kappa m}{4 \pi R^2}$$</span>.</p>
<p>With Newton's derivative dot over the capital <span class="math-container">$M$</span>. What do the lowercase epsilon, M-dot and lowercase kappa stand for?</p>

CC BY-SA 4.0
2023-04-07T18:26:23.200Z
Note the site standard for displaying math is [Mathjax](https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference) and it's much preferred to images of math.
StephenG - Help Ukraine
CC BY-SA 4.0
2023-04-09T08:13:46.267Z
Does "CREDIT: R SMETHURST" appear in the actual video, or has this been added by the person who took the screenshot?
kaya3
<black-hole><astrophysics><supermassive-black-hole><radiation><education>
What do the symbols signify in Dr. Becky Smethurst's radiation pressure equation for black holes?
Kurt Hikes
# What do the symbols signify in Dr. Becky Smethurst's radiation pressure equation for black holes?

(Asked by: username_0 on 2023-04-07)

[![Rebecca Smethurst radiation pressure force](https://i.sstatic.net/sX7A1.png)](https://i.sstatic.net/sX7A1.png)

In her most recent video, "An ULTRAMASSIVE black hole has been discovered in a BRAND NEW way!", as well as a previous video, "How massive can black holes get?", Dr. Becky Smethurst shows an equation relating to the radiation pressure of black holes:

$$F\_{\rm rad} = \epsilon \dot{M}c \frac{\kappa m}{4 \pi R^2}$$.

With Newton's derivative dot over the capital $M$. What do the lowercase epsilon, M-dot and lowercase kappa stand for?

**Tags:** black-hole, astrophysics, supermassive-black-hole, radiation, education

> **Comments:**
> **username_1**: Note the site standard for displaying math is [Mathjax](https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference) and it's much preferred to images of math.  
> **username_2**: Does "CREDIT: R SMETHURST" appear in the actual video, or has this been added by the person who took the screenshot?  

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## 1 answers

### Answer by username_3 on 2023-04-07. Score: 13

Epsilon is a factor to describe how much of the mass that is falling towards the black hole is converted to photons. The general equation for radiation pressure is ([source](http://www-astro.physics.ox.ac.uk/%7Egarret/teaching/lecture7-2012.pdf))

$$P\_{\rm rad}=\frac{L}{4\pi R^2c}$$

In which $L$ is the luminosity of the star or central body. But for a black hole, the luminosity comes from the rate of accretion into the central part of the black hole, and this is $dM/dt$ or $\dot M$. If all the mass were converted to energy then the luminosity would be $\dot Mc^2$, but in fact some of the matter does actually fall into the black hole. For a non-rotating black hole, 6% of the mass can be converted to energy. For a rotating black hole as much as 42% can be converted ([source](https://www.popularmechanics.com/space/a14524087/black-holes-mass-energy/)) and so $\varepsilon$ has a value between 0.06 and 0.42. A value of $\varepsilon=0.1$ is a reasonable empirical value.

So the formula above becomes

$$P\_{\rm rad}=\frac{\varepsilon \dot M c^2}{4\pi R^2c}$$

Then kappa is the opacity of the material that is being accreted. If the material is perfectly transparent, then it won't experience any radiation pressure.

If we assume the body is a plasma of fully ionised hydrogen, then its opacity is due to the [Thompson scattering](https://en.wikipedia.org/wiki/Thomson_scattering) by the electrons in the plasma, and has a value of $\kappa = 0.4 \,\mathrm{cm}^2\mathrm{g}^{-1}$ ([source1](https://www.astro.princeton.edu/%7Egk/A403/opac.pdf), [source2](https://www.sciencedirect.com/topics/earth-and-planetary-sciences/thomson-scattering#:%7E:text=If%20electron%20scattering%2C%20also%20called,that%20is%2C%20independent%20of%20frequency.))

To find the force on a body, and noting that radiation pressure is force per unit area, whereas opacity is the
cross-sectional area per unit mass for radiation scattering. So

$$L\_{\rm rad} = P\_{\rm rad}\kappa m$$

Combining these gives the formula in the video.

Garret Cotter's [slides on accretion](http://www-astro.physics.ox.ac.uk/%7Egarret/teaching/lecture7-2012.pdf) gives a lot more details.


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