When gas falls onto a compact object, it speeds up enormously, and when it is stopped or swirls into a disk, that energy turns into heat and light. In X-ray binaries a neutron star or a black hole pulls gas off a companion star; around supermassive black holes, accretion powers quasars. How much energy can falling release?
Why it works
A kilogram falling from far away to the surface of a body of mass $M$ and radius $R$ loses potential energy $GM/R$ (with $U = 0$ at infinity). If that energy is radiated, a stream of $\dot m$ kilograms per second gives $$L_{\text{acc}} = \frac{GM\dot m}{R}.$$ Compare with rest energy: falling releases a fraction $GM/(Rc^2)$ of $mc^2$. For fusion of hydrogen into helium the fraction is 0.007. For a neutron star it is about 0.15 to 0.2.
Key idea
Accretion releases $GM/R$ per kilogram, a fraction $\eta \approx GM/(Rc^2)$ of the rest energy. The more compact the object, the larger $\eta$: about $10^{-4}$ for a white dwarf, about 0.15 to 0.2 for a neutron star, and about 0.06 to 0.4 for gas spiraling into a black hole (depending on its spin), compared with 0.007 for hydrogen fusion.
| Process | Energy per kg | Fraction of $mc^2$ |
|---|---|---|
| Burning coal | about $3 \times 10^{7}$ J | $3 \times 10^{-10}$ |
| Falling onto a white dwarf (0.6 $M_\odot$, 8,700 km) | $9 \times 10^{12}$ J | $1 \times 10^{-4}$ |
| Fusing hydrogen to helium | $6.3 \times 10^{14}$ J | 0.007 |
| Falling onto a neutron star (1.4 $M_\odot$, 12 km) | $1.5 \times 10^{16}$ J | 0.17 |
Worked examples
Worked example: Falling onto a neutron star
How much energy does each kilogram release falling onto a $1.4\,M_\odot$ neutron star of radius 12 km? Compare with fusion.
- $$\frac{GM}{R} = \frac{(6.67 \times 10^{-11})(2.79 \times 10^{30})}{1.2 \times 10^{4}\text{ m}}$$
- $= 1.5 \times 10^{16}$ J/kg; fusion gives $0.007c^2 = 6.3 \times 10^{14}$ J/kg.
Answer: About 25 times more than fusing the same kilogram of hydrogen.
Worked example: An X-ray binary
Gas lands on that neutron star at $10^{14}$ kg/s. How luminous is it?
- $L = (GM/R)\dot m = 1.5 \times 10^{16} \times 10^{14}$ W.
- $= 1.5 \times 10^{30}$ W.
Answer: About 4,000 times the Sun's luminosity, mostly as X-rays, since the gas near the surface is heated to millions of kelvins.
Worked example: Feeding a quasar
A quasar shines with $10^{40}$ W at an efficiency $\eta = 0.1$. How much mass does its black hole swallow per year?
- $\dot m = L/(\eta c^2) = 10^{40}/(0.1 \times 9.00 \times 10^{16}) = 1.1 \times 10^{24}$ kg/s.
- Per year: $3.5 \times 10^{31}$ kg.
Answer: About 18 solar masses per year.
Common mistake
Thinking fusion always releases more energy per kilogram than falling. Fusion beats falling onto ordinary stars and white dwarfs, but near a neutron star or black hole, gravity wins by a wide margin: a kilogram dropped onto a neutron star releases more energy than fusing it would, about 25 times more. That is why the brightest steady X-ray sources in the sky, such as Scorpius X-1, are accreting neutron stars and black holes. Check: compare $GM/(Rc^2)$ with 0.007.