Black Hole Temperature Calculator (Hawking Radiation)

Background.

Black holes are not black. In 1974 Stephen Hawking showed that quantum field theory in the curved spacetime around a horizon forces the hole to radiate with an exactly thermal spectrum, at a temperature fixed entirely by its mass: T_H = ħc³/(8πGMk_B). This calculator returns that temperature, along with the Bekenstein–Hawking entropy, the radiated power and an evaporation-time estimate — and it will run the relation backwards, giving the mass a hole must have to sit at a temperature you specify.

The first thing the numbers tell you is that Hawking radiation is astrophysically irrelevant for every black hole anyone has ever observed. A one-solar-mass hole has a Hawking temperature of 6.17026 × 10⁻⁸ kelvin — sixty-two nanokelvin, about 44 million times colder than the cosmic microwave background. Sagittarius A* comes in at 1.4 × 10⁻¹⁴ K and M87* at 9.5 × 10⁻¹⁸ K. Because temperature falls as 1/M, bigger holes are colder, and because the radiated power falls as 1/M², bigger holes are dimmer. The only holes that radiate appreciably are the ones far too light to form from a collapsing star.

That leads directly to the caveat that matters most, and it belongs here rather than buried in an FAQ. **Any black hole colder than the 2.72548 K microwave background absorbs more energy from that background than it emits, so it is growing, not evaporating.** The crossover sits at 4.5016 × 10²² kg, roughly six-tenths of a percent of the Moon's mass. Every black hole ever detected is enormously heavier than that, which means every evaporation time this page prints for a real object is not a prediction — it is the lifetime the hole would have in an empty, cold universe. The calculator says so on the result itself whenever the condition applies.

The second caveat concerns which numbers are exact and which are idealisations, because they are not all the same kind of quantity. The temperature and the entropy are exact semiclassical results: they follow from the horizon's surface gravity and area with no free parameters, and Wald's review states them in natural units as T = κ/2π and S = A/4, which restore to precisely the SI expressions used here. The radiated power and the evaporation time are different. They come from the textbook idealisation of a perfect blackbody of area 4πr_s² emitting photons and nothing else. Don Page computed the real emission in 1976 and found roughly 2 × 10⁻⁴ ħc⁶G⁻²M⁻² for a hole hot enough to matter — about 9.7 times the naive coefficient — of which 81% is neutrinos, 17% photons and 2% gravitons. So the true lifetime of an evaporating hole is roughly an order of magnitude shorter than the figure shown. This page prints the standard textbook number, because that is what the formula in every textbook gives, and states the correction rather than silently applying one particular refinement.

Everything here assumes the Schwarzschild solution: zero spin, zero charge. A rotating Kerr hole of the same mass is colder, because its surface gravity is lower, and its emission is anisotropic and spin-dependent in ways no closed-form single-line formula captures. Treat the answer as exact for a non-spinning hole and as an upper bound on temperature for a spinning one.

The constants carry named sources and revisions. ħ = 1.054571817 × 10⁻³⁴ J s and k_B = 1.380649 × 10⁻²³ J/K are exact under CODATA 2022, as is c. G = 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻² carries a relative uncertainty of 2.2 × 10⁻⁵ and is the only real source of error in the temperature. The CMB temperature of 2.72548 ± 0.00057 K is Fixsen (2009); the 13.787-billion-year age of the universe used for the "already evaporated" flag is Planck 2018.

What is black hole temperature calculator?

Hawking temperature is the thermodynamic temperature of the radiation a distant observer sees emerging from a black hole horizon. For a Schwarzschild (non-rotating, uncharged) hole it is T_H = ħc³/(8πGMk_B), inversely proportional to mass. Nothing about the hole's composition or history enters; the mass alone fixes it.

The result comes from quantum field theory on a classical curved background — the semiclassical regime. Hawking's 1975 calculation showed that the vacuum state defined before collapse does not look like the vacuum afterwards, and the mismatch is exactly a Planck spectrum at κ/2π, where κ is the horizon's surface gravity. It is the same κ = c⁴/(4GM) that the Schwarzschild radius calculator reports.

Bekenstein–Hawking entropy is the companion result: S = k_B A c³/(4ħG), or equivalently S/k_B = A/(4ℓ_P²) — the horizon area measured in units of four Planck areas. It is a genuine thermodynamic entropy, satisfying a first law dM c² = T dS, and it is enormous. A single solar-mass black hole has S/k_B ≈ 1.05 × 10⁷⁷, which exceeds the entropy of all ordinary matter in the observable universe by a wide margin.

The evaporation time is not on the same footing. It comes from integrating a model of the radiated power, and the standard formula t = 5120πG²M³/(ħc⁴) assumes photon-only blackbody emission from area 4πr_s². It is the number in the textbooks and the number this page prints, but it is a model output, not a theorem, and the real answer depends on which particle species the hole is hot enough to emit.

How to use this calculator.

1. 01Choose whether you are starting from a mass or from a temperature.
2. 02Enter the mass in kilograms — the field hint lists the Sun, Sagittarius A*, M87*, the CMB-balance mass and the mass that evaporates in one age of the universe.
3. 03Or enter a Hawking temperature in kelvin to find the mass that produces it.
4. 04Read the Hawking temperature and the entropy first. Those two are exact semiclassical results.
5. 05Read the power and evaporation time second, and read them as idealisations — the note beneath states the Page (1976) correction factor and whether the hole is a net CMB absorber.
6. 06Check T_H ÷ T_CMB. If it is below 1, the hole is currently growing and the evaporation figure describes a hypothetical empty universe rather than ours.

The formula.

Start from the horizon's surface gravity, κ = c⁴/(4GM). Hawking's result, stated in the form Wald's review uses (units with G = c = ħ = k_B = 1), is simply T = κ/2π. Restoring SI units gives T = ħκ/(2πck_B), and substituting κ yields T_H = ħc³/(8πGMk_B). The same review states the entropy as S = A/4 in Planck units; restoring units gives S = k_B Ac³/(4ħG), and substituting A = 16πG²M²/c⁴ gives S = 4πk_B GM²/(ħc). Both restorations are asserted numerically in this page's test suite, which is how the formulas were verified against a source that states them in a completely different notation.

The power follows from treating the horizon as a blackbody: P = σAT⁴ with σ = π²k_B⁴/(60ħ³c²), A = 16πG²M²/c⁴ and T = T_H. The algebra collapses to P = ħc⁶/(15360πG²M²). Setting dM/dt = −P/c² and integrating M² dM gives t = 5120πG²M³/(ħc⁴). A useful check on those two together: P × t = Mc²/3 exactly, which the tests assert as a dimensional guard.

Units and dimensions. Mass in kilograms, temperature in kelvin (thermodynamic, never Celsius), entropy in J/K and also as the dimensionless S/k_B, power in watts, lifetime in seconds and Julian years of exactly 31 557 600 s. The dimensional guard the test suite runs is that S/k_B computed as 4πGM²/(ħc) equals the horizon area divided by four Planck areas, ℓ_P² = ħG/c³ — and that √(ℓ_P²) reproduces the PDG's published Planck length of 1.616255 × 10⁻³⁵ m.

Rounding stage. No intermediate rounding. Forty significant digits throughout, rounded once at the return boundary to twelve significant digits. Decimal-place rounding was rejected deliberately: these outputs span about 180 orders of magnitude — a supermassive hole radiates around 10⁻⁴⁸ W — and ten-decimal-place rounding would report that as exactly zero, a wrong answer disguised as a rounded one.

Significant figures. Quote at most five. The temperature, entropy, power and lifetime all inherit G's 2.2 × 10⁻⁵ relative uncertainty; the lifetime inherits it cubed through M³ if you supply the mass in kilograms rather than as a GM product. Beyond that, the power and lifetime carry a modelling uncertainty of roughly an order of magnitude, which dwarfs anything the constants contribute.

Invalid-domain behaviour. Zero or negative mass raises a field error rather than returning an infinite temperature: T_H diverges as M → 0, and the entropy, power and lifetime all become 0/0 or division by zero. Zero or negative temperature raises the same kind of error, since T = 0 would require infinite mass. Two thresholds attach notes rather than errors: T_H below 2.72548 K flags the hole as a net CMB absorber, and an idealised lifetime below 13.787 billion years flags it as something that would already have evaporated.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Hawking, S. W. (1975), "Particle Creation by Black Holes", Communications in Mathematical Physics 43, 199–220. The original derivation of the thermal spectrum at temperature ħκ/(2πk_B). Peer-reviewed; the journal copy is paywalled at Project Euclid/Springer — cited here as the origin of the result, with the formulas themselves verified against the open-access Wald review below. Retrieved (metadata) 2026-07-29.
2. [2]Wald, R. M. (2001), "The Thermodynamics of Black Holes", Living Reviews in Relativity 4, 6; open-access full text retrieved 2026-07-29 via PubMed Central. States, in units with G = c = ħ = k = 1, that a black hole "radiates to infinity all species of particles with a perfect black body spectrum, at temperature" T = κ/(2π), and that the entropy is S_bh = A/4 in Planck units. This is the second, independent authority: restoring SI units from those two statements alone reproduces this engine's T_H and S exactly, and the tests assert it numerically.
3. [3]Page, D. N. (1976), "Particle emission rates from a black hole: Massless particles from an uncharged, nonrotating hole", Physical Review D 13, 198. Finds that a hole of mass M ≫ 10¹⁷ g emits a total power of 2 × 10⁻⁴ ħc⁶G⁻²M⁻², of which 81% is neutrinos, 17% photons and 2% gravitons, and that primordial black holes would have decayed within the present age of the universe if and only if their initial mass was below (5 ± 1) × 10¹⁴ g. This is the source of the ~9.7× correction stated beside the power and lifetime outputs. Peer-reviewed; paywalled at APS, abstract freely visible.
4. [4]Bekenstein, J. D. (1973), "Black Holes and Entropy", Physical Review D 7, 2333. The argument that a black hole must carry entropy proportional to horizon area. Peer-reviewed; paywalled at APS. Bibliographic reference — no number is taken from it that is not independently in Wald.
5. [5]Fixsen, D. J. (2009), "The Temperature of the Cosmic Microwave Background", Astrophysical Journal 707, 916; arXiv:0911.1955, retrieved 2026-07-29. Gives T_CMB = 2.72548 ± 0.00057 K, the value this page compares every Hawking temperature against. The Particle Data Group independently prints 2.7255(6) K. Peer-reviewed; open-access preprint.
6. [6]NIST/CODATA 2022 fundamental constants, retrieved 2026-07-29: reduced Planck constant ħ = 1.054 571 817 × 10⁻³⁴ J s (exact), Boltzmann constant k_B = 1.380 649 × 10⁻²³ J K⁻¹ (exact), speed of light c = 299 792 458 m s⁻¹ (exact), Newtonian constant of gravitation G = 6.674 30(15) × 10⁻¹¹ m³ kg⁻¹ s⁻² (u_r = 2.2 × 10⁻⁵). Standards body; open access.
7. [7]Particle Data Group (Workman et al.), Review of Particle Physics, "Astrophysical Constants and Parameters", Table 2.1, revised August 2023; 2024 PDF retrieved 2026-07-29. Source of M☉ = 1.98841(4) × 10³⁰ kg and of the Planck length ℓ_P = 1.616255(18) × 10⁻³⁵ m, which the entropy area-law test reproduces from ħ, G and c. Also prints T_CMB = 2.7255(6) K, agreeing with Fixsen. Open access.
8. [8]Planck Collaboration (2020), "Planck 2018 results. VI. Cosmological parameters", Astronomy & Astrophysics 641, A6. Source of the 13.787 ± 0.020 Gyr age of the universe used only to decide when to flag a hole as one that would already have evaporated. Peer-reviewed; open access.