Question regarding maximum accretion of black holes

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The discussion revolves around calculating the maximum accretion rate of black holes using the Eddington luminosity formula. The user is trying to relate the accretion efficiency equation L=εM(dot)c² to Ledd=4πcGMmp/σT. They have substituted κ=mp/σT but are uncertain about how to derive M(dot)=κM from this. There is a mention of the accretion disk potentially being optically thin, which could be relevant to the calculations. The user seeks clarification on these concepts without assuming a maximum accretion rate.
can't.do.phys
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So I've got a question that I can't seem to figure out.

If the accretion efficiency of a black hole is L=εM(dot)c2 with ε=0.1
I have to set this to the Eddington luminosity which is Ledd = 4πcGMmp/σT to find the maximum rate at which a black hole can accrete by. I found that κ=mp/σT so I substituted that but then I got stuck. Has it got something to do with the accretion disk being optically thin? the answer should be M(dot)=κM but i can't figure out how to get it. help?
 
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Try not assuming there is a maximum accretion rate.
 

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