Black Hole Temperature Calculator

In 1974, Stephen Hawking proved that black holes are not perfectly black — quantum mechanical effects near the event horizon cause them to radiate thermally, with a temperature inversely proportional to their mass. The greater the mass, the colder the black hole. This counterintuitive result unifies quantum mechanics, general relativity, and thermodynamics in one formula, and it remains one of the most profound theoretical results in all of physics — though no black hole Hawking radiation has yet been directly observed. The black hole temperature calculator applies the exact Hawking formula to any mass.

The Hawking Temperature Formula

T_H = ℏc³ / (8πGMk_B)

Constants: ℏ = 1.055 × 10⁻³⁴ J·s, c = 2.998 × 10⁸ m/s, G = 6.674 × 10⁻¹¹ N·m²/kg², k_B = 1.381 × 10⁻²³ J/K. Simplified in solar masses: T_H ≈ 6.17 × 10⁻⁸ K / (M/M_☉). Notable temperatures:

• 1 solar mass black hole: T_H = 6.17 × 10⁻⁸ K ≈ 0.0000000617 K
• Stellar black hole (10 M_☉): T_H ≈ 6 × 10⁻⁹ K
• Sagittarius A* (4.15 × 10⁶ M_☉): T_H ≈ 1.5 × 10⁻¹⁴ K — coldest objects in the theoretical universe
• Micro-black hole (10¹² kg): T_H ≈ 1.2 × 10¹¹ K — hotter than the Big Bang's first second
• Planck mass black hole (2.18 × 10⁻⁸ kg): T_H ≈ Planck temperature = 1.42 × 10³² K

Use this online calculator for any black hole mass. The black hole calculator provides a complete set of black hole properties including Schwarzschild radius and entropy.

Why Hawking Radiation Makes Black Holes Unstable

Because T_H ∝ 1/M, as a black hole loses mass via Hawking radiation, its temperature increases — which increases the radiation rate, which reduces mass faster, increasing temperature further. This positive feedback produces an accelerating evaporation that culminates in an explosive burst as the mass approaches zero. The luminosity of Hawking radiation: L = (ℏc⁶)/(15360πG²M²). For a stellar-mass black hole, this luminosity is approximately 10⁻²⁸ watts — less than any conceivable measurement. For a Planck-mass black hole, the luminosity approaches 3.56 × 10⁵² watts — a burst comparable to 10²⁶ supernovae in a single moment.

Why Hawking Radiation Has Never Been Detected

The cosmic microwave background (CMB) has a temperature of 2.73 K. Any black hole with temperature below 2.73 K will absorb more CMB radiation than it emits via Hawking radiation — net mass gain. Since T_H = 2.73 K corresponds to M ≈ 2.26 × 10²² kg ≈ 0.01 times the Moon's mass, all known astrophysical black holes (minimum 1–3 solar masses) have Hawking temperatures far below 2.73 K and are currently gaining mass from the CMB. Detection of Hawking radiation would require black holes lighter than about 10²² kg — objects for which no confirmed observational evidence exists. Analog gravity experiments in condensed matter physics have observed acoustic analogs of Hawking radiation, providing indirect experimental support for the theoretical framework. The astrophysics calculators cover the complete astronomical calculation toolkit.

Theoretical Significance: The Four Laws of Black Hole Thermodynamics

Hawking's temperature result is embedded in a complete thermodynamic framework for black holes: Zeroth law — surface gravity κ is uniform over the event horizon (analogy: temperature in equilibrium). First law — dM = (κ/8π)dA + ΩdJ + ΦdQ (energy conservation with angular momentum J, charge Q). Second law (Bekenstein) — the total black hole area A never decreases (analogy: entropy never decreases). Third law — it is impossible to reduce surface gravity κ to zero (analogy: absolute zero unreachable). The second law generalization (GSL): total entropy = black hole entropy + matter entropy outside never decreases. These four laws, derived purely from classical general relativity by Bardeen, Carter, and Hawking in 1973, were initially considered analogies — until Hawking's radiation calculation gave them quantitative physical reality.