Black Hole Calculator (Formulas), Fabio Pacucci

This page contains the formulas used in the Black Hole Calculator, along with a brief explanation.
The symbols used are: $M_{\bullet}$ (black hole mass), $a$ (dimensionless spin parameter), $G$ (gravitational constant), $c$ (speed of light), $J$ (angular momentum), $c_s$ (speed of sound), $m_p$ (proton mass), $\sigma_T$ (Thomson scattering cross section), $\epsilon_{\mathrm{rad}$ (radiative efficiency), $\hbar$ (reduced Planck’s constant), $k_B$ (Boltzmann’s constant).

EDDINGTON LUMINOSITY

The Eddington luminosity is the maximum luminosity that a black hole can achieve when there is balance between the radiation force in the outward direction and the gravitational force in the inward direction.

$L_{\mathrm{Edd}} = \frac{4\pi G M_{\bullet} m_p c}{\sigma_T}$

EDDINGTON ACCRETION RATE

The Eddington accretion rate is the accretion rate for which the black hole radiates at the Eddington luminosity.

$\dot{M}_{\mathrm{Edd}} = \frac{L_{\mathrm{Edd}}}{\epsilon_{\mathrm{rad}} c^2}$

ANGULAR MOMENTUM

The space-time around a rotating (and uncharged) black hole is correctly described by the Kerr metric. The angular momentum of a black hole will cause an inertial reference frame to be entrained by the rotating mass to participate in the rotation. This effect is known as frame dragging. The dimensionless spin parameter $a$ can assume values within 0 (non-rotating, or Schwarzschild black hole) and 1 (maximally-rotating Kerr black hole).

$J= \frac{a G M_{\bullet}^2}{c}$

EVENT HORIZON RADIUS

The event horizon radius for a Schwarzschild (i.e., uncharged and non-rotating) black hole is calculated as follows.

$R_S= \frac{2 G M_{\bullet}}{c^2}$

In the more general case of a Kerr (i.e., uncharged and rotating) black hole with dimensionless spin parameter $a$ , the appropriate equation is the following.

$R_S= \frac{G M_{\bullet}}{c^2}(1+\sqrt{1-a^2})$

INNERMOST STABLE CIRCULAR ORBIT RADIUS

The radius of the innermost stable circular orbit (often abbreviated as ISCO) is the smallest circular orbit in which a massive particle can stably orbit a black hole. For a Schwarzschild black hole (i.e., uncharged and non-rotating) the innermost stable circular orbit is:

$R_{\mathrm{ISCO}}= \frac{6 G M_{\bullet}}{c^2} = 3 R_S$

For a Kerr black hole (i.e., uncharged and rotating) the radius of the ISCO is different and depends on the spin parameter $a$ . We first need to distinguish between prograde and retrograde orbits. A prograde orbit occurs in the same sense of the spin of the black hole, while a retrograde orbit occurs in the opposite sense. The following image (credit: NASA/JPL-Caltech) can help in understanding this concept.

For a Kerr black hole the radius of the ISCO is computed as follows.

$R_{\mathrm{ISCO}}= \frac{G M_{\bullet}}{c^2}[3 + Z_2 + \sqrt{(3-z_1)(3+Z_1+2Z_2)}]$

with:

$Z_1= 1 + (1-a^2)^{1/3}[(1+a)^{1/3} + (1-a)^{1/3}]$

$Z_2= (3a^2+Z_1^2)^{1/2}$

PHOTON SPHERE RADIUS

The radius of the photon sphere is the dimension of the spherical region around a black hole where gravity is strong enough that photons are forced to travel in orbits. For a Schwarzschild black hole (i.e., uncharged and non-rotating) the photon sphere radius is:

$R_{\mathrm{ps}}= \frac{3 G M_{\bullet}}{c^2} = \frac{3 R_S}{2}$

For a Kerr black hole (i.e., uncharged and rotating) the radius of the photon sphere is different and depends on the spin parameter $a$ .

$R_{\mathrm{ps}}= \frac{2 G M_{\bullet}}{c^2} \left[1 + \cos{\left(\frac{2}{3}\arccos{(\pm a)}\right)} \right]$

where the positive sign is for retrograde orbits and the negative sign is for prograde orbits.

BONDI RADIUS

The Bondi radius (Bondi, 1952) is the radius of the sphere of gravitational influence of the black hole. A test mass inside this sphere feels the gravitational presence of the black hole. As it is derived from setting the escape speed equal to the sound speed, it also represents the boundary between subsonic and supersonic infall.

$R_S= \frac{2 G M_{\bullet}}{c_s^2}$

EFFECTIVE LUMINOSITY

The effective luminosity takes into account the Eddington ratio (ratio between actual accretion rate and Eddington accretion rate)

$\lambda_{\mathrm{Edd}} = \frac{\dot{M}}{\dot{M}_{\mathrm{Edd}}}$

and the radiative efficiency $\epsilon_{\mathrm{rad}}$ , whose standard value is $0.1$ . The resulting effective luminosity is:

$L_{\mathrm{eff}} = \epsilon_{\mathrm{rad}} \lambda_{\mathrm{Edd}} \dot{M}_{\mathrm{Edd}} c^2$

GROWTH TIME

Assuming a continuous accretion with Eddington ratio $\lambda_{\mathrm{Edd}}$ , radiative efficiency $\epsilon_{\mathrm{rad}}$ , the growth time from the initial mass $M_{\mathrm{seed}}$ is:

$\tau_{\mathrm{growth}} = \frac{\sigma_T c}{4\pi G M_{\bullet} m_p} \frac{\epsilon_{\mathrm{rad}}}{(1-\epsilon_{\mathrm{rad}})\lambda_{\mathrm{Edd}}} \ln{\frac{M_{\bullet}}{M_{\mathrm{seed}}}}$

HAWKING TEMPERATURE

The Hawking temperature (Hawking, 1974) is the black body temperature at which a black hole emits radiation due to quantum effects close to the event horizon.

$T_{\mathrm{H}} = \frac{1}{M_{\bullet}} \frac{\hbar c^3}{8 \pi k_B G}$

BLACK HOLE ENTROPY

The second law of classical thermodynamics requires that black holes have entropy. If black holes carried no entropy, it would be possible to violate the second law by throwing mass into the black hole and thus reducing the entropy of the Universe. The entropy of a black hole is proportional to its surface area.

$S = M_{\bullet}^2 \frac{4 \pi G}{\hbar c}$

BLACK HOLE EVAPORATION TIME

Due to Hawking’s radiation, a black hole is predicted to slowly lose energy. If the black hole has no influx of mass, then it slowly evaporates. Following the analytical estimate by Don Page in his 1976 paper “Particle emission rates from a black hole: Massless particles from an uncharged, non-rotating hole”, the evaporation time for a black hole of mass $M_{\bullet}$ with no influx of mass is:

$\tau_{\mathrm{ev}} \approx 8.66 \times 10^{-27} \left( \frac{M_{\bullet}}{1 \, \mathrm{g}} \right)^3 \, \, \mathrm{s}$

For a rotating (i.e., Kerr) black hole the evaporation time can be shorter, due to the phenomenon of the super-radiance. Particularly in the near-maximal spin case ( $a \approx 0.998$ see Thorne 1974) the mass-loss is enhanced. See e.g. Arbey, Auffinger & Silk (2019) for a thorough description of this effect.

SURFACE AREA

Following the definition of the event horizon radius $R_S$ , the surface area of a black hole is simply $A = 4 \pi R_S^2$ , or:

$A= \frac{16 \pi G^2 M^2_{\bullet}}{c^4}$

SURFACE GRAVITY

In general, the surface gravity of a body is defined as the gravitational acceleration experienced at its surface. This value is infinite at the event horizon of a black hole. For this reason, in the case of a black hole, the surface gravity is usually defined as the product of the local proper acceleration (which diverges at the event horizon) and the gravitational time dilation factor (which goes to zero at the event horizon).

$k = \frac{1}{M_{\bullet}} \frac{c^4}{4 G}$

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