Accretion Disk Luminosity Calculator
Accretion Disk Luminosity Calculator computes luminosity of a thin disk around a black hole, with a temperature plot.
Formulas Used in Accretion Disk Luminosity Calculator
The calculator computes the luminosity of a thin accretion disk around a black hole:
Total Luminosity:
\\[ L = \eta \dot{M} c^2 \\]Temperature Profile:
\\[ T(r) = \left( \frac{3 G M \dot{M}}{8 \pi r^3 \sigma} \left(1 – \sqrt{\frac{r_{\text{in}}}{r}}\right) \right)^{1/4} \\]Where:
- \\(L\\): Total luminosity (W)
- \\(\eta\\): Radiative efficiency
- \\(\dot{M}\\): Accretion rate (kg/s)
- \\(c\\): Speed of light (\\(2.99792458 \times 10^8 \, \text{m/s}\\))
- \\(M\\): Central object mass (kg)
- \\(G\\): Gravitational constant (\\(6.67430 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2}\\))
- \\(r_{\text{in}}\\): Inner disk radius (\\(r_{\text{in}} = 6 R_s\\) for Schwarzschild BH)
- \\(R_s = \frac{2 G M}{c^2}\\): Schwarzschild radius
- \\(\sigma\\): Stefan-Boltzmann constant (\\(5.670367 \times 10^{-8} \, \text{W} \text{m}^{-2} \text{K}^{-4}\\))
- \\(T(r)\\): Temperature at radius \\(r\\) (K)
Example Calculation
Example: Stellar-Mass BH (\\(M = 10 M_\odot, \dot{M} = 0.01 M_\odot/\text{yr}, \eta = 0.1\\))
\\[
L = 0.1 \times (0.01 \times 1.989 \times 10^{30} / 3.15576 \times 10^7) \times (2.99792458 \times 10^8)^2 \approx 5.67 \times 10^{36} \, \text{W}
\\]
\\[
L \approx 1.48 \times 10^{10} \, L_\odot
\\]