Hawking Radiation Temperature Calculator

Compute the Hawking radiation temperature, evaporation lifetime, and peak emission wavelength for a black hole of any mass.

🌡️ Hawking Radiation Temperature Calculator
Black hole mass
Hawking temperature
K
Hawking Temperature
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Evaporation Time
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Hawking Luminosity
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Peak Wavelength (Wien)
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Implied Black Hole Mass
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Evaporation Time
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🌡️ What is Hawking Radiation?

Hawking radiation is the theoretical thermal radiation emitted by black holes as a consequence of quantum mechanical effects near the event horizon. Stephen Hawking predicted it in 1974 by applying quantum field theory to a curved spacetime background. The result was profound: black holes are not perfectly black. They slowly radiate energy, lose mass, and eventually evaporate completely.

The physical mechanism involves the constant creation and annihilation of virtual particle-antiparticle pairs near the event horizon. Occasionally, one particle of the pair falls inward (carrying negative energy from the black hole's perspective) while the other escapes to infinity as real radiation. The black hole therefore loses energy at a rate that depends on its temperature. Smaller black holes are hotter and evaporate faster. Larger black holes are extremely cold and evaporate on timescales many orders of magnitude beyond the current age of the universe.

For stellar-mass and supermassive black holes, the Hawking temperature is far below the cosmic microwave background temperature of 2.725 K. This means they absorb more radiation than they emit and effectively grow rather than shrink. Only hypothetical primordial black holes with masses below about 10^12 kg would be warm enough to emit detectable Hawking radiation and evaporate on cosmological timescales. The evaporation time of a 1.73 × 10^11 kg black hole is roughly equal to the current age of the universe, making such objects prime targets for gamma-ray observatories.

Hawking radiation has never been directly observed from an astrophysical black hole. However, analogue experiments in acoustic black holes (sonic horizons in fluid flow) and in optical fibres have produced Hawking-like correlations, providing indirect support for the underlying physics. Detecting real Hawking radiation remains one of the grand open challenges of experimental physics.

📐 Formula

TH  =  ℏc3 ÷ (8πGMkB)  ≈  1.227 × 1023 / Mkg K
ℏ = reduced Planck constant = 1.0546 × 10−34 J⋅s
c = speed of light = 2.998 × 108 m/s
G = gravitational constant = 6.674 × 10−11 m3 kg−1 s−2
M = black hole mass in kg
kB = Boltzmann constant = 1.381 × 10−23 J/K
Evaporation time: tevap = 5120πG2M3 / (ℏc4) ≈ 8.41 × 10−17 × Mkg3 seconds
Hawking luminosity: LH = ℏc6 / (15360πG2M2) ≈ 3.56 × 1032 / Mkg2 W
Peak wavelength (Wien): λpeak = 2.898 × 10−3 / TH m

📖 How to Use This Calculator

Steps

1
Select a mode — choose Find Temperature to compute the Hawking temperature, evaporation time, and luminosity from a black hole mass, or Find Mass to find the implied mass from a known temperature.
2
Enter the mass or temperature — in Find Temperature mode, type a mass and select the unit (kg, grams, metric tonnes, or solar masses). Scientific notation like 1e12 is accepted. In Find Mass mode, enter a Hawking temperature in kelvin.
3
Read the results — the calculator shows Hawking temperature, evaporation time (in seconds, years, Gyr, or Tyr as appropriate), Hawking luminosity in watts, and the peak emission wavelength from Wien's displacement law.

💡 Example Calculations

Example 1 — Primordial Black Hole (1012 kg)

A hypothetical primordial black hole with mass 1012 kg (1 billion metric tons)

1
TH = 1.227 × 1023 / 1012 = 1.227 × 1011 K = 122.677 billion K.
2
Evaporation time: t = 8.41 × 10−17 × (1012)3 = 8.41 × 1019 s = 2.665 Tyr (2.665 × 1012 years).
3
Hawking luminosity: L = 3.56 × 1032 / (1012)2 = 3.563 × 108 W (comparable to a large nuclear power plant).
4
Peak wavelength: λ = 2.898 × 10−3 / 1.227 × 1011 = 2.36 × 10−14 m = 23.623 fm (gamma-ray range).
TH = 122.677 billion K, evaporation = 2.665 Tyr, L = 3.563 × 108 W
Try this example →

Example 2 — Evaporating Now (1.73 × 1011 kg)

A black hole whose evaporation time equals the current age of the universe

1
TH = 1.227 × 1023 / 1.73 × 1011 = 7.091 × 1011 K = 709.115 billion K.
2
Evaporation time: t = 8.41 × 10−17 × (1.73 × 1011)3 = 4.355 × 1017 s = 13.798 Gyr (about the age of the universe).
3
Hawking luminosity: L = 3.56 × 1032 / (1.73 × 1011)2 = 1.190 × 1010 W. This is detectable in principle as a hard gamma-ray source.
TH = 709.115 billion K, evaporation = 13.798 Gyr, L = 1.190 × 1010 W
Try this example →

Example 3 — Find Mass from Temperature (T = 1012 K)

Reverse mode: what mass produces a Hawking temperature of 1012 K?

1
Switch to Find Mass mode. Enter temperature = 1e12 K (1 trillion kelvin).
2
M = 1.227 × 1023 / 1012 = 1.227 × 1011 kg.
3
Evaporation time: t = 8.41 × 10−17 × (1.227 × 1011)3 = 1.555 × 1017 s = 4.920 Gyr (comparable to Earth's age).
Mass = 1.227 × 1011 kg, evaporation = 4.920 Gyr
Try this example →

❓ Frequently Asked Questions

What is the Hawking radiation temperature formula?+
T_H = hbar c^3 / (8 pi G M k_B). Evaluating the constants gives T_H = 1.227 x 10^23 / M kelvin, where M is the black hole mass in kilograms. A smaller black hole has a higher temperature and evaporates faster.
Why is Hawking radiation temperature inversely proportional to mass?+
Hawking radiation arises from quantum fluctuations near the event horizon. The event horizon size (Schwarzschild radius) grows linearly with mass. A larger horizon means longer de Broglie wavelengths for the radiated quanta and therefore lower temperature, following the inverse relationship T proportional to 1/M.
How long does it take a black hole to evaporate completely?+
The evaporation time is t_evap = 5120 pi G^2 M^3 / (hbar c^4) = 8.41 x 10^-17 x M^3 seconds, where M is in kilograms. Because it scales as M^3, doubling the mass increases the lifetime by a factor of 8. A solar mass black hole would take about 2 x 10^67 years to evaporate, compared to the universe's age of 1.38 x 10^10 years.
Which primordial black holes are evaporating today?+
Black holes with initial mass around 10^11 to 10^12 kg have evaporation times comparable to the current age of the universe. A 1.73 x 10^11 kg black hole has a lifetime of exactly 13.8 Gyr. These objects, if they exist, would be observable as point sources of hard gamma rays and are targets of searches with the Fermi Gamma-ray Space Telescope.
What type of radiation does Hawking radiation produce?+
For typical primordial black holes near the end of their life, the Hawking temperature is 10^11 to 10^13 K, corresponding to peak wavelengths of femtometres. The peak emission is in the gamma-ray regime or harder. The spectrum is approximately thermal (blackbody), but Hawking's calculation gives greybody factors that modify the exact spectrum.
Is Hawking radiation purely thermal?+
Hawking showed the radiation is exactly thermal in the semiclassical approximation. However, this leads to the information paradox: a black hole formed from a pure quantum state would seem to evaporate into a mixed thermal state, violating unitarity. The resolution, possibly involving quantum gravity corrections, remains one of the deepest unsolved problems in theoretical physics.
What is the Hawking luminosity of a black hole?+
The Hawking luminosity is L = hbar c^6 / (15360 pi G^2 M^2) = 3.56 x 10^32 / M^2 watts, where M is in kilograms. For a 10^12 kg primordial black hole, L is about 3.6 x 10^8 W. For a 1 solar mass black hole, L is about 9 x 10^-29 W, effectively zero.
Why can't we detect Hawking radiation from stellar black holes?+
A stellar black hole of 10 solar masses has a Hawking temperature of about 6 x 10^-9 K, compared to the cosmic microwave background at 2.725 K. The black hole absorbs CMB photons at a rate vastly exceeding its tiny Hawking luminosity of 10^-29 W. The signal is completely buried under background radiation for any realistically sized black hole.
Do rotating or charged black holes have different Hawking temperatures?+
Yes. The surface gravity of a Kerr (rotating) black hole is lower at the outer event horizon than for the Schwarzschild case with the same mass, so T_H is lower. For a maximally rotating black hole, T_H approaches zero even for small masses. Charged (Reissner-Nordstrom) black holes also have a modified temperature formula. This calculator assumes the non-rotating, uncharged Schwarzschild case.
What happens to the information inside a black hole when it evaporates?+
This is the black hole information paradox. Classical Hawking radiation is perfectly thermal and carries no information about what fell in. But quantum mechanics requires information to be conserved. Current research suggests information may be encoded in subtle correlations in the Hawking radiation, possibly escaping as the Page time (when half the black hole mass has evaporated). The Page curve calculation in recent years supports this view, though a complete proof is still lacking.
Tags: Hawking Radiation Hawking Temperature Black Hole Evaporation Calculator Formula Free Online Quantum Gravity

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What temperature is Hawking radiation for a 1 solar mass black hole?

A 1 solar mass black hole has a Hawking temperature of about 6.17 x 10^-8 K. This is billions of times colder than the cosmic microwave background temperature of 2.725 K, so the black hole absorbs CMB photons far faster than it emits Hawking radiation.

How long would it take a black hole to evaporate via Hawking radiation?

The evaporation time is t = 5120 pi G^2 M^3 / (hbar c^4). For a solar mass black hole, this gives about 2 x 10^67 years, vastly longer than the age of the universe. Only primordial black holes with mass below about 10^11 kg could have evaporated by now.

What is the Hawking radiation temperature formula?

T_H = hbar c^3 / (8 pi G M k_B), where hbar is the reduced Planck constant, c is the speed of light, G is Newton's constant, M is black hole mass in kg, and k_B is the Boltzmann constant. The constant evaluates to approximately 1.227 x 10^23 K per kilogram.

Can Hawking radiation be observed?

Not directly from astrophysical black holes, because their temperature is far below the CMB. Laboratory analogue systems (acoustic black holes, optical fibres) have demonstrated Hawking-like radiation in controlled settings. Detection from a real astrophysical black hole would require a tiny primordial black hole near the end of its evaporation.

What is the Hawking luminosity of a black hole?

The Hawking luminosity is L = hbar c^6 / (15360 pi G^2 M^2). For a solar mass black hole, L is about 10^-28 watts. For a 10^12 kg black hole, L is about 3.56 x 10^8 watts (roughly a large power plant output).

What are primordial black holes and how does Hawking radiation affect them?

Primordial black holes (PBHs) are hypothetical black holes formed in the early universe from density fluctuations. Those with initial mass below about 5 x 10^11 kg would have completely evaporated by now, producing a burst of gamma rays. PBHs of mass around 10^11 kg are evaporating today and are targets of observational searches.

Does Hawking radiation violate conservation of energy?

No. Near the event horizon, virtual particle pairs constantly form. When one falls inward with negative energy (from the black hole frame) and one escapes, the black hole loses mass equivalent to the emitted energy. The total energy of the system is conserved; the black hole simply converts rest mass into radiation.

What happens at the end of black hole evaporation?

As mass decreases, Hawking temperature increases and evaporation accelerates. The final stages are thought to produce an intense burst of radiation. Whether the process ends in complete evaporation, a Planck-mass remnant, or something else remains an open question in quantum gravity.

📌 Quick Tips

💡Stellar-mass black holes have Hawking temperatures around 10^-8 K, far colder than the cosmic microwave background (2.7 K), so they absorb more radiation than they emit.
💡A black hole with mass around 1.73 x 10^11 kg would be evaporating right now with a lifetime equal to the current age of the universe.
💡Hawking radiation is a theoretical prediction combining quantum mechanics and general relativity. It has never been directly observed.
💡The evaporation time scales as M^3, so halving the mass reduces the lifetime by a factor of 8.
💡The peak emission wavelength follows Wien's law but reaches femtometre scales for primordial black holes, placing the emission in the gamma-ray and beyond regime.