Black Holes Part 2

The mass of an isolated black hole where J = 0 \textbf{J} = 0 and Q = 0 Q = 0 is 1 0 5 M 10^{5} M_{\odot} .The black hole evaporates through Hawking radiation over a long period of time and reaches a mass of 1 0 3 M 10^{3} M_{\odot} . The change in temperature of the black hole can be expressed to three significant figures as a b c × 1 0 13 K \overline{abc} \times 10^{-13} \text{K} . What is the value of a b c \overline{abc} ?


Due to approximation of constants and potential rounding errors, we accept a 2% range of answers for this problem.


The answer is 609.0.

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2 solutions

Δ T = h c 3 16 π 2 G k B ( 1 M 1 M 0 ) 1.227 × 1 0 23 × ( 1 M 1 M 0 ) \Delta T = \frac{hc^{3}}{16\pi^{2}Gk_{B}} \left ( \frac{1}{M} - \frac{1}{M_{0}} \right ) \approx 1.227 \times 10^{23} \times \left ( \frac{1}{M} - \frac{1}{M_{0}} \right )

where M M stands for the final mass of the BH and M 0 M_{0} stands for the initial mass of the BH.

Temperature T h { T }_{ h } of black hole is given by:

T h = c 3 8 π G M k b { T }_{ h }=\frac { \hslash { c }^{ 3 } }{ 8\pi GM{ k }_{ b } }

Where:

M M is mass of black hole.

c c is speed of light.

k b { k }_{ b } is the Boltzmann constant.

\hslash is Planck's constant divided by 2 π 2\pi

G G is the universal gravitational constant.

In the question, M M_{\odot} represents mass of our sun.

Just substitute and subtract to arrive at answer.

Source: Wikipedia

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