Hong Life CalculatorsUnderstanding the black hole merger energy calculator — concept and principle
When two black holes spiral together and merge, part of their mass is converted into gravitational waves. This calculator estimates how much energy is radiated, the final black hole's mass and its spin, using fits to numerical-relativity simulations of non-spinning black holes.
How it is calculated
- Total mass M = m₁ + m₂; symmetric mass ratio η = m₁m₂ ÷ M² (0.25 for equal masses).
- Energy radiated: Erad ÷ M ≈ 0.032661 (4η)² + 0.004458 (4η)⁴.
- Final mass = M − Erad; final spin a ÷ M ≈ 3.272η − 2.075η².
- Energy in joules uses M☉c² = 1.787 × 10⁴⁷ J.
- Mass ratio q = larger mass ÷ smaller mass.
The principle
Einstein's general relativity predicts that accelerating masses radiate gravitational waves. In the last orbits, merger and ringdown of a binary black hole, the system loses energy so fast that a few per cent of its total mass disappears as waves (E = mc²). The equations cannot be solved by hand, so physicists run supercomputer simulations and fit simple formulas to the results. Equal masses radiate the most; very unequal pairs radiate much less because the small black hole barely disturbs the large one.
Worked example
Default inputs: 36 and 29 solar masses (close to the first detected event, GW150914).
- M = 65 M☉; η = 36 × 29 ÷ 65² ≈ 0.2471; q = 36 ÷ 29 ≈ 1.24.
- 4η ≈ 0.9884; Erad ÷ M ≈ 0.032661 × 0.9769 + 0.004458 × 0.9544 ≈ 3.62%.
- Energy radiated ≈ 0.0362 × 65 ≈ 2.35 M☉c², about 4.201 × 10⁴⁷ J.
- Final mass ≈ 65 − 2.35 ≈ 62.6 M☉.
- Final spin ≈ 3.272 × 0.2471 − 2.075 × 0.2471² ≈ 0.682.
| Mass of the first black hole (M☉) | Energy radiated (M☉c²) |
|---|---|
| 9 | 0.7 |
| 18 | 1.54 |
| 27 | 2.07 |
| 36 | 2.35 |
| 45 | 2.47 |
| 54 | 2.49 |
| 72 | 2.41 |
How to read the result
For GW150914, LIGO reported about 3 solar masses radiated, a final mass of about 62 M☉ and a spin of about 0.67, close to this estimate. The fit ignores the black holes' own spins and the long early inspiral, so treat the energy as a lower estimate. Even so, the peak power of such mergers briefly exceeds the light output of all stars in the observable universe.
Common mistakes
- Using the fit outside its range (mass ratios beyond about 4:1, or spinning black holes).
- Expecting the final mass to equal the sum of the masses.
- Confusing source-frame and detector-frame (redshifted) masses.
- Reading the spin as a rotation rate; a/M is a dimensionless spin parameter between 0 and 1.
Frequently asked questions
How much energy did the first black hole merger release?
About 3 solar masses of energy as gravitational waves, according to the LIGO and Virgo analysis of GW150914.
Why does the final black hole weigh less than the two originals?
Energy carried away by gravitational waves has mass equivalent E ÷ c².
Why does the final black hole spin?
The orbital angular momentum of the pair is largely transferred to the remnant.
Sources: LIGO Scientific Collaboration and Virgo Collaboration (2016). Observation of Gravitational Waves from a Binary Black Hole Merger. Physical Review Letters, 116, 061102. · Pretorius, F. (2005). Evolution of Binary Black-Hole Spacetimes. Physical Review Letters, 95, 121101.