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Black Hole Temperature Calculator

Calculate Hawking temperature, Schwarzschild radius, evaporation time, and luminosity of black holes from mass input.

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Enter a positive number. Scientific notation supported (e.g., 1e30)
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Enter black hole mass and click Calculate to see Hawking temperature and other properties.

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About

Black holes emit thermal radiation due to quantum effects near the event horizon, a phenomenon predicted by Stephen Hawking in 1974. The temperature T is inversely proportional to mass: a stellar black hole of 10 M☉ radiates at approximately 6.17 × 10−9 K, colder than the cosmic microwave background. Microscopic black holes, however, reach extreme temperatures exceeding 1012 K and evaporate in fractions of a second. Miscalculating mass-temperature relationships leads to orders-of-magnitude errors in evaporation timescales, critical for primordial black hole cosmology and Hawking radiation detection experiments.

This calculator derives temperature from the Hawking formula using fundamental constants G, ℏ, c, and kB. It computes Schwarzschild radius, total evaporation time (Page approximation), and instantaneous luminosity. Results assume non-rotating (Schwarzschild) black holes in vacuum. For rotating (Kerr) black holes, temperature depends on angular momentum and differs significantly near extremal spin.

black hole hawking radiation astrophysics thermodynamics event horizon schwarzschild

Formulas

The Hawking temperature for a non-rotating (Schwarzschild) black hole derives from quantum field theory in curved spacetime. The formula connects mass to thermal radiation:

T = ℏc38πGMkB

where T = Hawking temperature (K), ℏ = reduced Planck constant, c = speed of light, G = gravitational constant, M = black hole mass, kB = Boltzmann constant.

The Schwarzschild radius defines the event horizon boundary:

rs = 2GMc2

Evaporation time uses the Page approximation for a black hole radiating into vacuum:

tevap = 5120πG2M3ℏc4

Instantaneous luminosity (radiated power):

L = ℏc615360πG2M2

Note: These formulas assume zero angular momentum and charge. Real astrophysical black holes likely rotate (Kerr metric), which modifies temperature by factors dependent on spin parameter a.

Reference Data

Black Hole TypeMassTemperature (K)Schwarzschild RadiusEvaporation Time
Primordial (asteroid-mass)1012 kg1.23 × 10111.49 × 10−15 m2.67 × 1010 years
Primordial (mountain-mass)1015 kg1.23 × 1081.49 × 10−12 m2.67 × 1019 years
Micro black hole (LHC theoretical)10−23 kg1.23 × 10461.49 × 10−50 m10−95 s
Lunar mass7.35 × 1022 kg1.670.11 mm1.06 × 1043 years
Earth mass5.97 × 1024 kg0.0218.87 mm5.69 × 1050 years
Stellar (3 M☉, minimum)5.97 × 1030 kg2.06 × 10−88.87 km5.69 × 1068 years
Stellar (10 M☉)1.99 × 1031 kg6.17 × 10−929.5 km2.10 × 1071 years
Stellar (20 M☉)3.98 × 1031 kg3.09 × 10−959.1 km1.68 × 1072 years
Intermediate (1000 M☉)1.99 × 1033 kg6.17 × 10−112954 km2.10 × 1077 years
Intermediate (10,000 M☉)1.99 × 1034 kg6.17 × 10−1229,540 km2.10 × 1080 years
Sagittarius A* (Milky Way)4.15 × 106 M☉1.49 × 10−141.23 × 1010 m1.90 × 1087 years
M87* (Virgo A)6.5 × 109 M☉9.49 × 10−181.92 × 1013 m7.30 × 1096 years
TON 618 (ultramassive)6.6 × 1010 M☉9.35 × 10−191.95 × 1014 m7.65 × 1099 years
Phoenix A (largest known)1011 M☉6.17 × 10−192.95 × 1014 m2.10 × 10101 years
Physical Constants Used
Gravitational constant G6.67430 × 10−11 m3 kg−1 s−2
Reduced Planck constant ℏ1.054571817 × 10−34 J·s
Speed of light c299,792,458 m/s (exact)
Boltzmann constant kB1.380649 × 10−23 J/K (exact)
Solar mass M☉1.98892 × 1030 kg
CMB Temperature2.725 K

Frequently Asked Questions

Hawking temperature scales inversely with mass: T ∝ 1/M. A 10 M☉ black hole has temperature ~6×10⁻⁹ K, while the CMB is 2.725 K. The black hole absorbs more CMB radiation than it emits, causing net mass gain rather than evaporation. Only black holes below ~10²² kg are hotter than the CMB and actively evaporating today.
A stellar-mass black hole (10 M☉) has evaporation time ~10⁷¹ years, vastly exceeding the current universe age (~1.38×10¹⁰ years). Even primordial black holes formed at the Big Bang with initial mass below ~5×10¹¹ kg would have fully evaporated by now. This constrains searches for primordial black holes to masses above this threshold.
No. This calculator assumes Schwarzschild (non-rotating) black holes. For Kerr black holes, temperature depends on both mass M and dimensionless spin parameter a* = Jc/(GM²). Near-extremal spin (a* → 1) significantly reduces temperature. The generalized formula involves surface gravity κ at the outer horizon.
As mass decreases, temperature rises exponentially. In the final second, a black hole releases ~10²² J of energy - equivalent to millions of nuclear weapons. The final 10⁻²³ seconds involve Planck-scale physics where general relativity breaks down. Whether a Planck-mass remnant persists or complete evaporation occurs remains an open question in quantum gravity.
Stefan-Boltzmann law states luminosity L = σAT⁴, where A is surface area. Event horizon area scales as M² (since radius ∝ M), so A ∝ M². Temperature T ∝ 1/M gives T⁴ ∝ 1/M⁴. Thus L ∝ M² × (1/M⁴) = 1/M². Smaller black holes radiate far more intensely per unit mass.
In standard 4D spacetime, the LHC cannot create black holes - collision energy (~14 TeV) is ~10¹⁵ times below Planck energy. In hypothetical large extra dimension scenarios, micro black holes might form at TeV scales. Such objects would have temperatures exceeding 10⁴⁶ K and evaporate in ~10⁻²⁶ seconds, producing characteristic particle signatures.