Black Hole Temperature Calculator (Hawking Radiation)

Background.

Compress the Sun into a sphere 5.9 kilometres across and it becomes a black hole with a Hawking temperature of 6.17 × 10⁻⁸ kelvin, sixty-two billionths of a degree above absolute zero. Shrink the mass to that of the Moon and the temperature climbs only to 1.67 K. Both are colder than the 2.72548 K microwave background that fills the sky, so both would take in more energy from their surroundings than they give out.

That inverse relationship is the core of the calculation. Stephen Hawking's 1975 paper showed that a horizon radiates like a warm body at a temperature fixed by its surface gravity, and for a hole with no spin and no charge this works out to T_H = ħc³/(8πGMk_B). Double the mass and the temperature halves; nothing about what fell in, or when, enters at all. Start from a mass to get the temperature, or from a temperature to get the mass that would produce it. Either way you also get the horizon radius, the Bekenstein–Hawking entropy, the radiated power and an evaporation time.

Those outputs are not equally solid. Temperature and entropy are exact results of the semiclassical theory for a non-rotating hole. Power and lifetime rest on a textbook simplification: the horizon treated as a perfect blackbody that emits only photons. Don Page's 1976 calculation of the real emission found about 9.7 times more power once neutrinos and gravitons are counted and the filtering effect of the hole's own gravitational field is included, so an evaporating hole really dies roughly ten times sooner than the printed lifetime. Neither figure allows for the microwave background feeding the hole, and a spinning hole of the same mass is colder than the temperature shown.

What is black hole temperature calculator?

Hawking temperature is the temperature of the thermal radiation that a distant observer would see coming from a black hole's horizon. It is set by the horizon's surface gravity, κ = c⁴/(4GM), through T_H = ħκ/(2πck_B), so for a non-rotating, uncharged hole the mass alone determines it. The companion quantity is the Bekenstein–Hawking entropy, which counts the horizon's area in units of four Planck areas and grows with the square of the mass.

How to use this calculator.

1. 01Choose whether you are starting from a black hole's mass or from a Hawking temperature.
2. 02For mass, enter kilograms. The field hint gives the Sun (1.98841 × 10³⁰ kg), Sagittarius A* and M87*; to convert solar masses, multiply by 1.98841 × 10³⁰.
3. 03For temperature, enter kelvin on the absolute scale. Entering 2.72548 K finds the hole that is exactly as warm as the microwave background.
4. 04Check T_H ÷ T_CMB before reading the lifetime. Below 1, the hole is gaining mass from the background today and the lifetime describes an empty, cold universe instead.
5. 05Read the power and evaporation time alongside the note beneath them, which states the photon-only simplification and flags any hole short-lived enough to have evaporated already.

The formula.

Hawking's result is cleanest in natural units, where Wald's review states it as T = κ/2π and the entropy as S = A/4. Restoring SI units and substituting κ = c⁴/(4GM) gives T_H = ħc³/(8πGMk_B); with the horizon area A = 16πG²M²/c⁴ the entropy becomes S = 4πk_B GM²/(ħc), or S/k_B = A/(4ℓ_P²) with ℓ_P² = ħG/c³. The radiated power applies the Stefan–Boltzmann law to the horizon, P = σAT_H⁴ with σ = π²k_B⁴/(60ħ³c²), which collapses to P = ħc⁶/(15360πG²M²). Setting the rate of mass loss to −P/c² and integrating gives the lifetime t = 5120πG²M³/(ħc⁴). A quick consistency check: P × t = Mc²/3 exactly. The constants ħ, k_B and c are exact under CODATA 2022; G = 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻² carries a relative uncertainty of 2.2 × 10⁻⁵, the only real uncertainty in the temperature. Lifetimes are given in Julian years of 31,557,600 s. A mass or temperature of zero is rejected, since the temperature diverges as the mass goes to zero and a zero temperature would need infinite mass.

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 calculator's T_H and S exactly.
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 this calculator 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.