Hawking Temperature Calculator: How Cold, and How Long to Evaporate
Hawking Temperature Calculator: the coldest things in the universe
A black hole glows, very faintly, at a temperature set entirely by its mass — and the bigger it is, the colder it gets. Put in a mass and this gives you the Hawking temperature, the power it radiates, and the evaporation lifetime. For a black hole the mass of the Sun that lifetime is 2×1067 years. The absurdity of that number is the lesson.
Hawking temperature, radiated power and evaporation time
Find the Hawking temperature of a black hole of one solar mass, M = 1.988 41×1030 kg, and then its evaporation lifetime. Use ℏ = 1.054 571 817×10−34 J s, k = 1.380 649×10−23 J K−1, G = 6.674 30×10−11 and c = 2.997 924 58×108 m s−1.
Four constants, one mass, and an inverse
- T
- Hawking temperature, in kelvin. Proportional to 1/M: double the mass and you halve the temperature.
- P
- power radiated, in watts, treating the hole as a black body at temperature T with area 4πrs². Proportional to 1/M².
- t
- evaporation lifetime, in seconds, for an isolated hole in empty space. Proportional to M³, which is why the numbers get out of hand so fast.
- ℏ
- reduced Planck constant, h/2π = 1.054 571 817×10−34 J s. Its presence is the whole point: this is a quantum effect, and if ℏ were zero the temperature would be zero too.
- k
- Boltzmann constant, exactly 1.380 649×10−23 J K−1 since the 2019 SI revision. It is only here to turn an energy into a temperature.
- G, c
- the gravitational constant and the speed of light, as on the Schwarzschild radius page. Four constants from four different parts of physics appear in one formula, which is why this result mattered so much.
Worked example
Find the Hawking temperature of a black hole of one solar mass, M = 1.988 41×1030 kg, and then its evaporation lifetime. Use ℏ = 1.054 571 817×10−34 J s, k = 1.380 649×10−23 J K−1, G = 6.674 30×10−11 and c = 2.997 924 58×108 m s−1.
Do the constants once and keep the answer. ℏc³ = 1.0545718×10−34 × 2.6944002×1025 = 2.84146×10−9. And 8πGk = 25.13274 × 6.67430×10−11 × 1.380649×10−23 = 2.31622×10−32. Dividing those two gives T = 1.2269×1023 / M kelvin, with M in kilograms. That one line is worth memorising; everything else on this page follows from it.
Put the Sun in. T = 1.2269×1023 / 1.98841×1030 = 6.1703×10−8 K, which is 61.70 nanokelvin. The coldest temperature ever reached in a laboratory is in the picokelvin range, so this is cold but not record-breakingly cold; what is remarkable is that it is a property of the mass alone.
Compare it with the sky. The microwave background is at 2.725 K. This hole is 4.4×107 times colder than the emptiest place in the present universe, so heat flows into it. A solar-mass black hole today is not evaporating at all; it is growing.
Now the lifetime. Collect the constants again: 5120πG²/(ℏc⁴) = 8.4115×10−17, so t = 8.4115×10−17 M³ seconds. Cube the solar mass: (1.98841×1030)³ = 7.8614×1090 kg³. Multiplying gives t = 6.613×1074 s.
Convert to years. Divide by 3.15576×107 s per Julian year: t = 2.095×1067 years. The present age of the universe is 1.38×1010 years, so this lifetime is about 1.5×1057 times the age of the universe.
Sit with that number. 1057 is roughly the number of atoms in the Sun. If you ran one age-of-the-universe for every atom in the Sun, you would be about done. This is not a long time in any sense a human word covers, and that is the honest reason evaporation is a statement about physics rather than about anything that will be observed.
The same formula across sixty orders of magnitude of mass
| Mass | Hawking temperature | Evaporation lifetime | What it is |
|---|---|---|---|
| 108 kg | 1.2×1015 K | 2.7 s | gone in the time it takes to read this row |
| 1.7×1011 kg | 7.2×1011 K | 1.3×1010 years | one formed in the Big Bang would be finishing now |
| 1012 kg | 1.2×1011 K | 2.7×1012 years | a small mountain |
| 4.5×1022 kg | 2.725 K | 2.4×1044 years | exactly as warm as the sky; the break-even mass |
| 1 Msun | 61.70 nK | 2.1×1067 years | a typical stellar-mass black hole |
| 30 Msun | 2.06 nK | 5.7×1071 years | the sort LIGO and Virgo detect |
| 4.3×106 Msun | 1.4×10−14 K | 1.7×1087 years | Sagittarius A*, at the Galactic centre |
Why nobody has seen it
| Obstacle | The number |
|---|---|
| A stellar-mass hole is colder than empty space | 61.70 nK against 2.725 K: a factor of 4.4×107 |
| so it is absorbing, not emitting, today | net heat flow is inward until the sky cools below 61.70 nK |
| The power radiated by a solar-mass hole | 9×10−29 W — about 10−10 of the power in one visible photon per second |
| Mass lost per second | 10−45 kg, which is 10−18 of the mass of an electron |
| The one detectable case | a primordial hole of about 1011 kg finishing now, which is why gamma-ray searches target that mass |
Why a black hole has a temperature at all
A black hole is supposed to be the one object nothing escapes from, so a temperature is the last thing it should have. In 1974 Stephen Hawking showed that it does. The argument runs through quantum field theory in curved spacetime, but the result is startlingly simple: a non-rotating black hole radiates exactly like a black body at a temperature inversely proportional to its mass.
The inverse is what makes it strange. Everything else you heat up gets hotter when you add energy. A black hole gets colder. Feed it mass and the temperature drops; let it radiate and the temperature rises. That means it has a negative heat capacity, which is why the evaporation runs away at the end: the smaller it gets, the hotter it gets, the faster it radiates. The last tonne of a black hole goes in a fraction of a second.
Four constants in one formula. T = ℏc³/(8πGMk) contains Planck’s constant from quantum mechanics, c from relativity, G from gravity and k from thermodynamics. No other formula in physics brings those four together, and that is why this result is treated as a genuine clue about what a theory of quantum gravity will have to look like, rather than as a curiosity about an exotic object.
The numbers are the problem, and they are also the lesson. For anything astrophysical the temperature is a fraction of a nanokelvin and the lifetime is 1067 years or more. Those are not large numbers in the way a national debt is a large number; they are outside the range in which the word “large” does any work. If you want a feel for them, use the orders of magnitude page and count: the age of the universe is 1010 years, so a solar-mass hole lives about 1057 ages of the universe. There are roughly 1057 atoms in the Sun.
Where this goes wrong in homework. The commonest error is a mass in solar masses used as though it were kilograms, which puts the temperature out by thirty orders of magnitude. The second is forgetting that the lifetime goes as M³ and not as M, so a ten-times-heavier hole lives a thousand times longer rather than ten. The third is quoting an evaporation time for a stellar-mass hole as though it were something that is happening: it is not happening, because the hole is colder than the microwave background and is gaining energy, and it cannot begin to shrink until the expanding universe has cooled the sky below its own temperature.
What is actually in the formula and what is not. This page uses the simplest version: a non-rotating, uncharged hole radiating only photons, with the standard 5120π coefficient in the lifetime. A real calculation counts every particle species light enough to be emitted at that temperature, and the number of species rises as the hole gets hotter, which shortens the lifetime by a factor of a few. Spin changes it as well. So treat the lifetime as an order-of-magnitude figure with the right scaling, not as a precise prediction — the uncertainty is in the coefficient, not in the M³.
Frequently asked questions
Has Hawking radiation ever been detected?
No. There is no direct observation of Hawking radiation from any black hole, and on the numbers above there is no prospect of one from a stellar-mass hole: the power is around 10−28 W and the hole is colder than the sky around it. Analogue systems — sound waves in flowing fluids, light in optical fibres, Bose–Einstein condensates — have produced thermal spectra matching the prediction, which supports the kinematic argument, but they are not tests of gravity. The honest position is that this is well-established theory with no direct experimental confirmation.
Why does adding mass make a black hole colder?
Because T goes as 1/M. The physical picture is that the temperature is set by how sharply spacetime is curved at the horizon, and a bigger hole has a gentler horizon: the surface gravity falls as 1/M even though the total mass rises. The consequence is a negative heat capacity, which is thermodynamically peculiar — it means a black hole can never reach equilibrium with an infinite heat bath, and it is the reason evaporation accelerates rather than levelling off.
What is the break-even mass, and why does it matter?
4.50×1022 kg, about three fifths of the Moon’s mass. At that mass the Hawking temperature is exactly 2.725 K, the temperature of the microwave background. Lighter holes are hotter than the sky and lose mass on balance; heavier ones are colder and gain. It matters because almost every statement you read about black holes evaporating quietly assumes an isolated hole in empty space, and every astrophysical black hole is on the wrong side of this line. The calculator gives this mass as a power of ten, and it does not depend on what you type.
Does the lifetime formula really go as the cube of the mass?
Yes, and that is where the absurd numbers come from. The power radiated goes as 1/M² (temperature to the fourth power times area, and the area goes as M²), so dM/dt goes as 1/M², and integrating gives t proportional to M³. Ten times the mass is a thousand times the lifetime. It also means the end is sudden: a hole spends almost all of its life at its original mass and disposes of the last tiny fraction in a burst.
Is a black hole the coldest thing in the universe?
A large one is colder than anything else known, yes: a supermassive hole sits around 10−14 K, far below the 2.725 K background and far below the picokelvin temperatures produced in laboratories. But a small one is the hottest thing in the universe — a 108 kg hole is at 1015 K, hotter than the core of any star. Both statements come out of the same one-line formula, which is a good reason to look at the whole curve rather than one point on it.
Why are there so many powers of ten on this page instead of numbers?
Because the numbers do not fit. An evaporation lifetime of 1067 years written out is a sixty-eight digit integer, and the page’s formatter prints ordinary decimals. A row reading 67.32 means 1067.32 years. Reading quantities as powers of ten is the normal working habit in astrophysics for exactly this reason, and it is worth getting comfortable with before you need it — the scientific notation page covers the mechanics.
Related calculators
References
- NIST, Fundamental Physical Constants (SP 961, 2022 wallet card), and E. Tiesinga et al., CODATA Recommended Values of the Fundamental Physical Constants: 2022. Every constant on this page is taken from here: G = 6.674 30(15)×10−11 m3 kg−1 s−2, ℏ = 1.054 571 817×10−34 J s, k = 1.380 649×10−23 J K−1 and c = 299 792 458 m s−1. The 2022 report states that no new determination of G was available, so its 2022 and 2018 values are identical; c, h and k are exact by the 2019 SI definitions, and ℏ and the Stefan–Boltzmann constant follow from them exactly.
- D. J. Fixsen, A. Kogut, S. Levin and others, ARCADE 2 measurement of the extra-galactic sky temperature at 3–90 GHz, arXiv:0901.0555. Gives the microwave background temperature as 2.725 ± 0.001 K from ARCADE 2 combined with earlier data, and 2.730 ± 0.004 K from ARCADE 2 alone.
- S. W. Hawking, Black hole explosions?, Nature 248, 30–31 (1974), doi:10.1038/248030a0, and Particle creation by black holes, Communications in Mathematical Physics 43, 199–220 (1975). The original result. Cited for provenance: the abstract and citation details were confirmed, but the full text was not read for this page, and the numerical work here comes from the standard formulae with the CODATA constants above.
- J. D. Bekenstein, Black holes and entropy, Physical Review D 7, 2333–2346 (1973). The entropy argument that preceded Hawking’s temperature and that the S/k row on this page computes. Cited for provenance on the same terms.
- Planck Collaboration, Planck 2018 results. VI. Cosmological parameters, Astronomy & Astrophysics 641, A6 (2020). The abstract of the version readable here gives H0 = 67.4 ± 0.5 km s−1 Mpc−1 for base ΛCDM, from which the Hubble time is 14.51 billion years. The age row of its parameter table could not be read from the copy available, so the age of the universe is quoted on this page as 13.8 billion years to three figures and no further.