How do mass and temperature change in black holes due to Hawking Radiation?

[James Way]

Hello, I am rather new to Physics and for a class project on exponential growth and decay in nature and I chose the effects Hawking Radiation on black holes. If anyone could help explain how the mass and temperature change over time and how to calculate them(this one especially) that would be wonderful

 

[James Way]

Simplified, why are these decreases(in mass) and increases(in heat) expoenential.

 

[James Way]

*exponential

 

[GW150914]

how the mass and temperature change over time and how to calculate them

According to the Bekenstein–Hawking formula in black hole thermodynamics, $$S=\frac{kAc^3}{4G\hbar}$$ Consider a Schwarzschild black hole, we know that $$r_{\rm g}=\frac{2GM}{c^2},\ A=4\pi r_{\rm g}^2$$ Therefore, we can get $$M^2=\frac{\hbar c^3}{4\pi k_{\rm B} G}S$$ From thermodynamics, we learn that ${\rm d}M=T{\rm d}S$, so we have $$T=\frac{{\rm d} M}{{\rm d} S}=\frac{\hbar c^3}{8\pi k_{\rm B}G}\frac{1}{M}$$I believe this is what you want.

By the way, according to the Stefan-Boltzmann law, $$\frac{{\rm d}M}{{\rm d}t}=-\sigma AT^4$$ and thus you can know how the mass and temperature change over time.