What would the mass of a supermassive black hole at the center of a quasar need to be, in terms of the solar mass, in order to produce a luminosity of 10¹² times that of our Sun, assuming that this luminosity is at the Eddington limit? A)10⁹ B)3.3 × 10⁷ C)10¹² D)3.3 × 10⁸

Question

Quizplus · Accepted Answer

According to the Eddington luminosity limit, the luminosity of a black hole with mass $M$ is given by:

$L_{	ext{Edd}} = \frac{4\pi GMc}{\kappa}$

where $G$ is the gravitational constant, $c$ is the speed of light, and $\kappa$ is the opacity of the material surrounding the black hole. Assuming a solar metallicity, $\kappa = 0.34	ext{ cm}^2/	ext{g}$.

We need to solve for the mass $M$ of the black hole that produces a luminosity of $10^{12}$ times that of our Sun, so:

$10^{12} L_{\odot} = 10^{12} 	imes 3.828 	imes 10^{33}	ext{ erg/s}$

$= 3.828 	imes 10^{45}	ext{ erg/s}$

Using this in the Eddington luminosity equation, we get:

$3.828 	imes 10^{45}	ext{ erg/s} = \frac{4\pi GMc}{0.34	ext{ cm}^2/	ext{g}}$

Solving for $M$, we get:

$M = \frac{L_{	ext{Edd}}\kappa}{4\pi Gc} = \frac{(3.828 	imes 10^{45}	ext{ erg/s})(0.34	ext{ cm}^2/	ext{g})}{4\pi (6.674 	imes 10^{-8}	ext{ cm}^3/	ext{g s}^2)(2.998 	imes 10^{10}	ext{ cm/s})}$

$= 3.3 	imes 10^{17}	ext{ g} = 3.3 	imes 10^{10} M_{\odot}$

Therefore, the mass of the supermassive black hole at the center of a quasar that produces a luminosity of $10^{12}$ times that of our Sun is approximately $3.3 	imes 10^{10}$ solar masses, which corresponds to answer choice B.

What Would the Mass of a Supermassive Black Hole at the Center