About

Binary black hole mergers radiate a fraction of their total rest-mass energy as gravitational waves. The exact fraction depends on the symmetric mass ratio η = m₁ ⋅ m₂(m₁ + m₂)² and the individual spin vectors. For equal-mass non-spinning mergers, numerical relativity simulations show approximately 5.5% of the total mass is radiated. GW150914, the first detected event, released roughly 3 M_⊙c² in under 0.2 s, briefly outshining all stars in the observable universe combined. Getting the energy budget wrong by even a factor of two changes post-merger mass estimates and disrupts population synthesis models for stellar evolution.

This calculator uses peer-reviewed numerical relativity fitting formulas from Rezzolla et al. (2008) for final spin, Tichy & Marronetti (2008) for radiated energy, and Peters (1964) for inspiral timescales. It approximates gravitational wave peak frequency from quasi-normal mode fits. Limitations: spins are assumed aligned or anti-aligned with the orbital angular momentum. Precessing spin configurations require full 3D numerical relativity and are beyond the scope of analytic fits. Results carry systematic uncertainties of order 1 - 5% relative to full NR simulations.

Formulas

The total mass and symmetric mass ratio define the merger dynamics:

M = m₁ + m₂

η = m₁ ⋅ m₂M²

The Schwarzschild radius for each component:

r_s = 2GMc²

Radiated energy fraction using the non-spinning NR fit (Tichy & Marronetti 2008):

E_radMc² ≈ 0.2η² (1 − 0.63(1 − 4η))

Final spin of the remnant (Rezzolla et al. 2008, aligned spins):

a_f ≈ √12η − 3.87η² + 4.028η³ + η(a₁ ⋅ m₁²M² + a₂ ⋅ m₂²M²)

Quasi-normal mode (ringdown) frequency:

f_QNM ≈ c³2πGM_f ⋅ F(a_f)

Peters (1964) inspiral timescale for circular orbits:

T_inspiral ≈ 5256 ⋅ c⁵ a₀⁴G³ m₁ m₂ M

where m₁, m₂ are component masses, M is total mass, η is symmetric mass ratio (maximum 0.25 for equal masses), a₁, a₂ are dimensionless spin parameters (0 ≤ a < 1), G is Newton’s gravitational constant (6.674×10⁻¹¹ m³ kg⁻¹ s⁻²), c is speed of light (2.998×10⁸ m/s), a₀ is initial orbital separation, M_f is the final remnant mass, and F(a_f) is the dimensionless QNM frequency factor.

Reference Data

Event	m₁ (M_⊙)	m₂ (M_⊙)	M_final (M_⊙)	E_rad (M_⊙c²)	a_final	f_peak (Hz)
GW150914	35.6	30.6	63.1	3.1	0.69	150
GW151226	14.2	7.5	20.8	1.0	0.74	450
GW170104	31.2	19.4	48.7	2.0	0.64	180
GW170814	30.5	25.3	53.2	2.7	0.70	160
GW170817 (BNS)	1.46	1.27	2.73*	-	-	~1500
GW190521	85	66	142	8.0	0.72	60
GW190814	23.2	2.6	25.0	0.8	0.28	300
GW200115	5.7	1.5	7.0	0.2	0.43	800
GW200225	19.3	16.3	33.5	2.1	0.70	220
Sgr A* + Stellar	4.0×10⁶	30	~4.0×10⁶	~0.01	~0.0	~10⁻³
Equal 10 M_⊙	10	10	18.9	1.1	0.69	320
Equal 50 M_⊙	50	50	94.5	5.5	0.69	64
Equal 100 M_⊙	100	100	189	11	0.69	32
10:1 Ratio (100+10)	100	10	109	1.0	0.26	88
SMBH Merger (10⁹)	5×10⁸	5×10⁸	~9.5×10⁸	~5×10⁷	0.69	~10⁻⁷

Frequently Asked Questions

The radiated energy scales with the symmetric mass ratio η squared. Equal-mass mergers (η = 0.25) radiate the maximum fraction (~5.5% of Mc² for non-spinning cases). A 10:1 mass ratio gives η ≈ 0.083, radiating roughly 1%. Extreme mass ratios (>100:1) enter the EMRI regime where the lighter object spirals in slowly, radiating energy over millions of orbits but with very small total fractional loss.

Even if both progenitors are non-spinning (a₁ = a₂ = 0), the final remnant acquires spin from the orbital angular momentum. For equal-mass non-spinning mergers, the final spin is a_f ≈ 0.69. Adding aligned spins increases this (up to ~0.95 for maximally spinning progenitors). Anti-aligned spins can reduce the final spin below 0.4. The Kerr limit a < 1 is never violated; cosmic censorship holds in all NR simulations to date.

Peters' formula shows T ∝ a₀⁴. This strong dependence arises because gravitational wave luminosity scales as v¹⁰/c⁵ in the quadrupole approximation, and orbital velocity relates to separation via Kepler's law. Doubling the initial separation increases inspiral time by a factor of 16. A stellar-mass binary at 1 AU takes longer than the age of the universe to merge; the same pair at 10 Schwarzschild radii merges in seconds.

Yes. Asymmetric gravitational wave emission produces a net linear momentum. For non-spinning unequal masses, kicks reach ~175 km/s. For spinning BHs with specific orientations ("superkick" configurations), NR simulations predict kicks up to ~5000 km/s. Typical galaxy escape velocities are 500 - 2000 km/s, so superkicks can eject supermassive BH remnants from their host galaxies entirely. This calculator uses the Gonzalez et al. fitting formula for aligned-spin recoil estimates.

The same fitting formulas apply regardless of mass scale because general relativity is scale-invariant in vacuum. The physics is identical; only the characteristic frequencies and timescales change. A 10⁹ M_⊙ merger produces gravitational waves at ~10⁻⁷ Hz (nanohertz, detectable by pulsar timing arrays like NANOGrav), while a 30 M_⊙ merger peaks at ~150 Hz (LIGO band). Enter any mass in solar masses and the calculator scales automatically.

The Rezzolla final spin formula has residuals of ±0.01 against NR simulations for aligned spins but degrades for precessing configurations (not modeled here). The radiated energy fit carries ~3 - 5% relative error. Peters' timescale is exact for circular orbits in the adiabatic limit but overestimates merger time when the binary enters the strong-field regime (last ~10 orbits). The quasi-normal mode frequency fit is accurate to <1% for the dominant (l = 2, m = 2) mode.