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## Homework Statement

This is the original problem for part 1

A)

Suppose that a black hole of mass M accretes mass at a rate ##\dot{M}##. Further suppose that accretion of mass Δm leads to the radiation of energy ##\Delta E= \eta Δmc^2##, for some effeciency of energy conversion, ##\eta##. What is the luminosity of emitted radiation in terms of ##\dot{M}## and ##\eta##?

B)For what accretion rate ##\dot{M}## does this luminosity equal the Eddington luminosity

for the black hole? Leave your answer as an expression, without plugging in

numerical values.

C) In terms of ##\eta## what is the shortest amount of time that a black hole could increase

its mass by a factor of e≈2.71828, assuming that all of its mass growth occurs

through accretion?

D) Supermassive black holes of mass M≈10

^{9}M

_{sun}have been detected as ultra luminous quasars in the very early universe, roughly 780 million years after the

Big Bang. Assuming that these black holes have been growing at the maximal

rate possible that you computed in part (c), since the beginning of the universe,

what is the smallest initial mass they have begun with to reach 10

^{9}M

_{sun}after 780 million years? Assume that ##\eta## =0.1.

## Homework Equations

L=η##\dot M##c^2 For A

## The Attempt at a Solution

For B I set L=L

_{e}. Where L

_{e}= 10

^{31}watts ##\frac{M}{M_{sun}}##

[tex]\eta \dot M c^2=10^{31} \frac {M}{M_{sun}}[/tex]

[tex] \dot M=\frac{10^{31} \frac {M}{M_{sun}}}{\eta c^2}[/tex]

Now C) I took ##\dot M## = ##\frac{dm}{dt}## and moved that dt to the right hand side and integrated from M→2.71828M and 0→t respectively.

This leaves an M on both sides of the equation that cancels and only the ##\eta## variable remains (M

_{sun}is known of course). The units work out to seconds as I assumed and I'm left with:

t=1.03x10

^{16}##\eta##

Part D) For this I just took the 2.71828M and divided by the time and got 2.64x10

^{-16}.

So, the initial mass plus the mass rate times initial mass times 780 million years equals 10

^{9}M

_{sun}.

[tex]M+2.64(10^{-16})M (7.8(10^8))=10^9(M_{sun})[/tex]

Pulling out M and solving I get what amounts to M=10

^{9}M

_{sun}because I'm basically dividing by 1. This does not make sense. SO for 780 million years the mass didnt change. I must be doing something wrong.

Anyone have a good idea?? My rate must be wrong but I don't know where my mistake is. Thanks in advance.