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Rebecca Smethurst radiation pressure force

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?

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    $\begingroup$ Note the site standard for displaying math is Mathjax and it's much preferred to images of math. $\endgroup$
    – StephenG - Help Ukraine
    Commented Apr 7, 2023 at 18:26
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    $\begingroup$ Does "CREDIT: R SMETHURST" appear in the actual video, or has this been added by the person who took the screenshot? $\endgroup$
    – kaya3
    Commented Apr 9, 2023 at 8:13

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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)

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

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) 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 by the electrons in the plasma, and has a value of $\kappa = 0.4 \,\mathrm{cm}^2\mathrm{g}^{-1}$ (source1, source2)

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 gives a lot more details.

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