Black Hole Temperature Calculator

Black Hole Temperature Calculator — fast, accurate results online. Enter your values and get instant answers.

Results

Calculated
Hawking temperature
—
In K
Schwarzschild radius
—
In km
Evaporation time
—
In years
Mass
—
In kg

What black hole temperature means

Classically a black hole emits nothing, but in 1974 Stephen Hawking showed that quantum effects near the event horizon give it a thermal spectrum, as if it were a hot body. The temperature depends on the mass in a surprising way: the more massive the black hole, the colder it is. This is a theoretical prediction that has not been observed directly for astrophysical black holes.

Enter a mass in solar masses, Earth masses or kilograms and the calculator returns the Hawking temperature, the Schwarzschild radius, the estimated evaporation time and the mass converted to kilograms. Use it for astronomy coursework, science writing and intuition-building, for example to see why stellar-mass black holes are far colder than the cosmic microwave background and therefore cannot evaporate today.

Formula and variables

T = ℏc³ / (8π G M kB)

  • ℏ is the reduced Planck constant, 1.054572 × 10-34 J·s.
  • c is the speed of light, 299,792,458 m/s.
  • G is the gravitational constant, 6.6743 × 10-11 m³/(kg·s²).
  • kB is the Boltzmann constant, 1.380649 × 10-23 J/K.
  • M is the black hole mass in kilograms.

The Schwarzschild radius is rs = 2GM/c², and the evaporation time is estimated as 5120π G² M³ / (ℏ c&sup4;), a simplified figure that ignores particle-species corrections and any radiation the hole absorbs.

Worked example

For one solar mass, M = 1.98847 × 1030 kg.

  • Numerator: ℏc³ ≈ 2.8414 × 10-9 in SI units.
  • Denominator: 8π G M kB ≈ 4.605 × 10-2.
  • T ≈ 6.170 × 10-8 K, about 62 nanokelvin.
  • rs = 2GM/c² ≈ 2.953 km.
  • Evaporation time ≈ 2.10 × 1067 years.

Entering 1 with solar masses displays these values. Ten solar masses gives one tenth the temperature, 6.170 × 10-9 K, but a thousand times the lifetime. A black hole at the temperature of the cosmic microwave background, 2.725 K, would need a mass of only about 4.5 × 1022 kg, roughly 60% of the Moon.

Common mistakes and how to interpret the result

  • Expecting bigger holes to be hotter. Temperature is inversely proportional to mass, while the lifetime grows with the cube of mass.
  • Entering a temperature. The input is a mass; temperature is the output, in kelvin.
  • Confusing the number of digits with accuracy. The formulas are idealised for an isolated, non-rotating, uncharged hole.
  • Reading evaporation time as a real forecast. A hole sitting in the 2.725 K background absorbs more than it emits, so real astrophysical black holes are currently growing, not shrinking.

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Frequently Asked Questions

Why do heavier black holes have lower temperatures?
Hawking temperature is set by the surface gravity at the horizon, which is weaker for larger holes. The result is T proportional to 1/M.
Has Hawking radiation been observed?
No. The radiation from stellar or supermassive black holes is far too faint to detect. It remains a well-motivated theoretical prediction, and laboratory analogues are studied in other systems.
Why is the evaporation time so large?
Power radiated falls as 1/M squared while the energy to be radiated is proportional to M, so lifetime scales as M cubed. A solar-mass hole would outlast the present age of the universe, about 1.4 x 10^10 years, by a factor near 10^57.
Does the calculator cover rotating or charged black holes?
No. It uses the Schwarzschild (non-rotating, uncharged) case. Spin and charge change the temperature and radius.