Q. 7.53 - An Introduction to Thermal Physics

Question

A black hole is a blackbody if ever there was one, so it should emit blackbody radiation, called Hawking radiation. A black hole of mass M has a total energy of $M{c}^2$ , a surface area of $16\pi G^2{M}^2/c^4$, and a temperature of$h{c}^3/16{\pi }^{2}kGM$ (as shown in Problem 3.7).

(a) Estimate the typical wavelength of the Hawking radiation emitted by a one-solar-mass (2 x 1030 kg) black hole. Compare your answer to the size of the black hole. 

(b) Calculate the total power radiated by a one-solar-mass black hole.

(c) Imagine a black hole in empty space, where it emits radiation but absorbs nothing. As it loses energy, its mass must decrease; one could say it "evaporates." Derive a differential equation for the mass as a function of time, and solve this equation to obtain an expression for the lifetime of a black hole in terms of its initial mass.

(d) Calculate the lifetime of a one-solar-mass black hole, and compare to the estimated age of the known universe (1010 years).

(e) Suppose that a black hole that was created early in the history of the universe finishes evaporating today. What was its initial mass? In what part of the electromagnetic spectrum would most of its radiation have been emitted? 

Step-by-Step Solution

Answer

(a) The typical wavelength of the Hawking radiation is $28r$.

(b)  The total power is $9.18\times {10}^{31}W$.

(c) The differential equation for the mass as a function of time is$\tau =\frac{M^{3}i}{3C}$.

(d) The lifetime of a one-solar-mass black hole is $\tau =2.072\times {10}^{69}y$.

(e) The initial mass is $3.94\times {10}^{10}kg$ and the part of the electromagnetic spectrum would most of its radiation has been emitted is $1.636\times {10}^{-15}m$.

Part(a) Step 1: Given Information

We need to find the typical wavelength of the Hawking radiation emitted by a one-solar-mass.

Part (a) Step 2: Simplify

Consider we have a black hole, the energy, temperature radius and surface area are given by respectively:

$u=m{c}^2 T=\frac{h{c}^3}{16\pi kGM} r=\frac{2GM}{c^2} A=4\pi r^2$

The spectrum is given by:

$u\left(T\right)=\frac{8\pi }{{\left(hc\right)}^3}\frac{{\epsilon }^3}{e^{\epsilon /kT}-1}$

let $x=\epsilon /kT$, then the spectrum will become

$u\left(T\right)=\frac{8\pi {\left(kT\right)}^3}{{\left(hc\right)}^3}\frac{x^3}{e^x-1}$

to find peak of the spectrum, we set the partial derivative with respect to energy (or x since the constant doesn't equal to zero) of the specturm equals to zero, that is

$\frac{a}{ax}\left(\frac{x^3}{e^x-1}\right))=0$

$\frac{3x^2(e^x-1)-x^3{e}^x}{{(e^x-1)}^2}=0$

$3x^2(e^x-1)-x^3{e}^x=0$

$3e^x-3-x{e}^x=0$

the solution of this equation is $x=2.8214$, which obtained using Matlab, and the code is shown in the following picture

$$\begin{gathered} >>fun = @ \left(x\right) 3* (exp (x) -1) -x, *exp\left(x\right); \\ >>x0 = (1 3); \\ >>x=fzero(fun,x0) \\ x= \\ 2.8214 \end{gathered}$$

but,

$x=\frac{\epsilon }{kT}=\frac{hc}{kT\lambda }$where $\epsilon =\frac{hc}{\lambda }$

so the peak wavelength is:

$\lambda =\frac{hc}{(2.8214)kT}$

substitute form (1) to get:

$\lambda =\frac{hc}{(2.8214)k}\left(\frac{16{\pi }^{2}kGM}{h{c}^3}\right)$

$\lambda =\frac{16{\pi }^{2}GM}{(2.8214)c^2}=\left(28\right)\frac{2GM}{c^2}$

$\lambda =28r$

thus the peak wavelength equals 28 times the radius of the black hole or 14 times the diameter of the black hole

Part (b) Step 1: given Information

We need to calculate the total power radiated by a one-solar-mass black hole.

Part (b) Step 2: Simplify

The irradiated power is given by:

$P=Ae\sigma T^4$

where A is the area and e is the emissivity of the material. Substitute form (1) to get:

$P=4\pi e\sigma {\left(\frac{2GM}{c^2}\right)}^2{\left(\frac{h{c}^3}{16\pi 2kGM}\right)}^4$

$p=\frac{h{c}^6}{\left(30720\right){\pi }^6{G}^2{M}^2}$

substitute with the gives to get(note that l set e=1 since it is black body):

$$\begin{gathered} p=\frac{6.626\times {10}^{-34}J\times s){(3.00\times {10}^{8}m/s)}^6}{\left(30720\right){\pi }^6{(6.674\times {10}^{-11}{m}^3/kg\times s^2)}^2(2.0\times {10}^{30}kg{)}^2} \\ =9.18\times {10}^{-31}W \\ P=9.18\times {10}^{31}W \end{gathered}$$

Part (c) Step 1: Given Information

We need to obtain an expression for the lifetime of a black hole in terms of its initial mass.

d U d t = - A σ T 4 Part (c) Step 2: Simplify

The power equals the time derivative of the energy that is$\frac{dU}{dt}$, so we can write:

$\frac{dU}{dt}=-A\sigma T^4$

where the negative sign came from the fact that the mass of the black hole decreases with time. Substitute from get:

$\frac{d\left(M{c}^2\right)}{dt}=-\frac{h{c}^6}{\left(30720\right){\pi }^6{G}^2{M}^2}$

$\frac{dM}{dt}=-\frac{h{c}^4}{\left(30720\right){\pi }^6{G}^2{M}^2}$

$\frac{dM}{dt}=-\frac{C}{M^2}$

where,

$C=\frac{h{c}^4}{\left(30720\right){\pi }^6{G}^2}$

the value of this constant is:

$$\begin{gathered} C=\frac{(6.626\times {10}^{-34}J\times s){(3.00\times {10}^{8}m/s)}^4}{\left(30720\right){\pi }^6{(6.674\times {10}^{-11}{m}^3/kg\times s^2)}^2} \\ =4.08\times {10}^{13}k{g}^3/s \end{gathered}$$

now we need to solve the differential equation as:

$M^{2}dM=-Cdt$

Integrate from the initial mass $M_i$ to zero, and from  o to the lifetime $\tau$ to get:

$$\begin{gathered} {\int }_{M_1}^0{M}^{2}dM=-C{\int }_0^{T}dt \\ -\frac{{M_i}^3}{3}=-C_T \\ \tau =\frac{{M^3}_i}{3C} \end{gathered}$$

Part (d) Step 1: Given Information

We need to calculate the lifetime of a one-solar-mass black hole.

Part (d) Step 2: Simplify

For one solar mass black hole (the initial mass is $M_i=2.0\times {10}^{30}kg)$the life time is:

$$\begin{gathered} \tau =\frac{{(2.0\times {10}^{30}kg)}^3}{3(4.08\times {10}^{13}k{g}^3/s} \\ =6.536\times {10}^{76}s \\ =6.536\times {10}^{76}s\left(\frac{1y}{365\times 24\times 60\times {60}_s}\right) \\ =2.072\times {10}^{69y} \\ \tau =2.072\times {10}^{69}y \end{gathered}$$

which is more than ${10}^{59}$times the age of the known universe

Part (e) Step 1: Given Information

We need to find the initial mass of the black hole and the electromagnetic spectrum would most of its radiation have been emitted.

Part (e) Step 2: Simplify

The age of the known universe is $1.5\times {10}^{10}y, or 5\times {10}^{17}s,$ we need to find the mass of a black hole, such that it started with mass of $M_i$ at the beginning of the universe and completely vaporized after the age of universe elapsed, so:

$M_i={\left(3C_{\tau }\right)}^{\frac{1}{3}}$

substitute with the givens to get:

$$\begin{gathered} M_i=\left(3\right(4.08\times {10}^{13}k{g}^3/s){(5\times {10}^{17}s))}^{\frac{1}{3}}=3.94\times {10}^{10}kg \\ {M}_i=3.94\times {10}^{10}kg \end{gathered}$$

to find what part of the spectrum that the most radiation have been emitted, we use the expression of part (a), so:

$\lambda =\frac{56(6.674\times {10}^{-11}{m}^3/kg\times s^2)(3.94\times {10}^{10}kg)}{{(3.0\times {10}^{s}m/s)}^2}$

$$\begin{gathered} =1.636\times {10}^{-15}m \\ \lambda =1.636\times {10}^{-15}m \end{gathered}$$