Accretion onto Compact Objects

Accretion onto Compact Objects

When matter falls onto a compact object — a white dwarf, neutron star, or black hole — gravitational potential energy is released as radiation. Accretion is one of the most powerful energy sources in the universe.

Accretion Luminosity

The luminosity generated by mass falling from infinity onto a compact object of mass $M$ and radius $R$ at a mass accretion rate $\dot{M}$ (kg/s) is:

$L_{\text{acc}} = \frac{G M \dot{M}}{R}$

Radiative Efficiency

The efficiency $\eta$ compares the radiated energy to the rest-mass energy of the accreted material:

$\eta = \frac{G M}{R c^2}$

so that $L_{\text{acc}} = \eta \dot{M} c^2$.

For a neutron star with $M = 1.4\,M_\odot$ and $R = 10$ km:

$\eta_{\text{NS}} = \frac{G \times 1.4 M_\odot}{10^4 \text{ m} \times c^2} \approx 0.207$

For a Schwarzschild black hole, the innermost stable circular orbit (ISCO) is at $R_{\text{ISCO}} = 6 G M / c^2 = 3 r_s$, giving $\eta = 1/6 \approx 16.7\%$.

Your Task

Implement three functions. All constants must be defined inside each function.

• accretion_luminosity_W(M_kg, Mdot_kg_s, R_m) — returns $L_{\text{acc}} = G M \dot{M} / R$ in watts
• accretion_efficiency(M_kg, R_m) — returns $\eta = G M / (R c^2)$ (dimensionless)
• mdot_from_luminosity(L_W, M_kg, R_m) — returns $\dot{M} = L R / (G M)$ in kg/s

Use $G = 6.674 \times 10^{-11}$ N m² kg⁻² and $c = 2.998 \times 10^8$ m/s.