Why do smaller black holes have a higher temperature?

Since the mass is in the denominator of the Hawking temperature formula, the temperature is inversely proportional to the mass. Halving the mass doubles the temperature, meaning that smaller black holes are hotter and supermassive black holes are extremely cold.

How long does it take for a black hole to evaporate?

The time for evaporation is proportional to the third power of mass. A black hole with a solar mass can live about 10 to the 67 years, which is far longer than the age of the universe. In contrast, tiny black holes can evaporate almost instantly.

Can you determine mass from temperature?

Yes it is possible. If you change the calculation type to "Calculation from Temperature" and enter a value in Kelvin, then the same formula will be adapted to give the mass that can radiate at this temperature as well as the radius, entropy and lifetime.

Has Hawking radiation ever been observed?

No, there has been no direct observation. The temperature of actual black holes in space is much lower than the cosmic microwave background and cannot be detected with current measurement equipment. Still, this formula is a central prediction that connects gravity, quantum mechanics, and thermodynamics together.

Black Hole Temperature Calculator

Black hole

What do you know?

From the mass

From the temperature

Black hole mass

Enter a mass and read off the temperature and every related property below.

Key results

Hawking temperature (K)

Schwarzschild radius

Mass in solar masses

Colder than the cosmic microwave background (2.725 K), so today it absorbs more radiation than it emits and is still growing.

Hawking temperature (K)

Temperature relative to the CMB
Entropy (multiples of k_B)
Event horizon area (square metres)

Hawking luminosity (W)
Evaporation time (years)
Peak emission wavelength (m)

Black holes to scale

Compare famous black holes

A table of real black holes with your own placed alongside them.

Black holes have an unusual property: the smaller and lighter they are, the hotter they run. A mountain-mass black hole would glow far hotter than any star, while a black hole heavier than the Sun is colder than deep space.

Hawking temperature falls as mass rises: bigger black holes are colder.
Black hole	Mass (suns)	Temperature	Schwarzschild radius	Evaporation time
Primordial (1e12 kg)	5.03 x 10^-19	1.23 x 10^11 K	1.49 x 10^-15 m	2.67 x 10^12 yr
Stellar (10 suns)	1 x 10^1	6.17 x 10^-9 K	2.95 x 10^1 km	2.1 x 10^70 yr
Your black hole	1 x 10^0	6.17 x 10^-8 K	2.95 x 10^0 km	2.1 x 10^67 yr
Sagittarius A* (4.3e6 suns)	4.3 x 10^6	1.43 x 10^-14 K	1.27 x 10^7 km	1.67 x 10^87 yr
M87* (6.5e9 suns)	6.5 x 10^9	9.49 x 10^-18 K	1.92 x 10^10 km	5.76 x 10^96 yr

This black hole temperature calculator allows you to convert the mass of a black hole into the tiny temperature it radiates through Hawking radiation or vice versa. Enter the mass in solar masses, kilograms or Earth masses and get the Hawking temperature in Kelvin along with information about the size of its event horizon, entropy, luminosity and time for the black hole to evaporate. Conversely you can also calculate the mass that produces a given temperature.

What is the temperature of a black hole?

For decades it was thought that black holes were completely black. In 1974 Stephen Hawking proved that due to quantum effects near the event horizon, black holes emit a weak thermal radiation. This radiation is now called Hawking radiation. The radiation has a certain temperature and for simple non-rotating and uncharged (Schwarzschild) black holes it is determined only by mass.

For objects of normal astronomical size this temperature is incredibly low; the temperature of a black hole with the same mass as the Sun would be about 62 billionths of a degree above absolute zero, which is much colder than even the emptiest parts of outer space. Only very small black holes are hot.

The formula for the Hawking temperature:

The Hawking temperature of a Schwarzschild black hole is given by:

T = \frac{\hbar\, c^{3}}{8\pi G M k_{B}}

where T is the temperature in kelvins, ħ is the reduced Planck constant, c is the speed of light, G is the gravitational constant and M is the mass in kilograms while k_B is the Boltzmann constant. As all terms except for the mass are constants, the whole formula reduces to a simple inverse proportionality.

T \approx \frac{1.227 \times 10^{23}}{M\ (\text{kg})}\ \text{K}

Since the mass is in the denominator, temperature and mass change inversely. If the mass doubles, then the temperature halves. This explains why lighter black holes are hotter.

A calculation example can make this more intuitive. For a black hole with the mass of the Sun, M is about 1.989 x 10^30 kilograms.

T = \frac{1.227 \times 10^{23}}{1.989 \times 10^{30}} \approx 6.17 \times 10^{-8}\ \text{K}

The smaller the black hole is, the higher its temperature.

This inverse proportionality holds over a very wide range. The radiation from black holes of enormous mass is brighter than that from stars, while supermassive black holes at the centers of galaxies are among the coldest objects in the universe. The following table shows this trend across the entire range.

Black hole	Mass	Hawking temperature
Primordial (1e12 kg)	1 x 10^12 kg	1.2 x 10^11 K
Stellar mass	10 solar masses	6.2 x 10^-9 K
Sun mass	1 solar mass	6.2 x 10^-8 K
Sagittarius A*	4.3 x 10^6 solar masses	1.4 x 10^-14 K
M87*	6.5 x 10^9 solar masses	9.5 x 10^-18 K

The following dimensions are calculated by this calculation tool.

Since all properties of a Schwarzschild black hole are determined by its mass, other important values can be derived from the same inputs. The radius of the event horizon can be calculated using the Schwarzschild formula.

r_{s} = \frac{2 G M}{c^{2}}

Bekenstein and Hawking discovered that the entropy is proportional to the area of the event horizon, not the volume, which hints at a deep relationship between gravity and information.

\frac{S}{k_{B}} = \frac{4\pi G M^{2}}{\hbar c}

The power of the radiation (i.e. luminosity) and the time it takes to completely evaporate can both be derived from the temperature and event horizon.

L = \frac{\hbar c^{6}}{15360\,\pi G^{2} M^{2}} \qquad t \approx \frac{5120\,\pi G^{2} M^{3}}{\hbar c^{4}}

Since the evaporation time is proportional to the cube of the mass, for black holes with large masses this time is unimaginably long. A black hole with a mass equal to that of the Sun would take about 10^67 years to evaporate, which is much longer than the current age of the universe.

Black holes and cosmic microwave background radiation

Space is neither a truly empty nor an extremely cold place. Instead it is pervaded by the cosmic microwave background radiation, a pervasive glow with a temperature of 2.725 K that has existed since the beginning of the universe. Black holes only get smaller if they are hotter than their surroundings.

If the Hawking temperature is set to 2.725 K, this gives a critical mass of about 4.5 x 10^22 kg. This corresponds roughly to the mass of a small satellite. Black holes with a mass above this value have a temperature below that of the background and therefore absorb more energy than they radiate, causing them to grow larger. Only black holes with less mass are actually evaporating at present.

Do black holes really evaporate?

Hawking radiation extracts energy from the black hole. By the mass-energy equivalence principle this means that the black hole slowly loses mass. As it loses mass, the black hole heats up which causes it to radiate faster. This runaway behavior is an expression of the fact that black holes have negative heat capacity, eventually leading to a black hole explosion.

For any black hole we can actually observe, this effect is far too weak to be detectable at all. Hawking radiation has never been directly measured. Still, the concept remains a cornerstone of theoretical physics because it unites gravity, quantum mechanics and thermodynamics in one single equation.

This calculator models an idealized Schwarzschild black hole (non-rotating and electrically neutral) in a vacuum, and is intended for educational purposes. Since real black holes rotate and interact with their environment, the exact values will differ.

Frequently asked questions

What is the temperature of a black hole?: This is the Hawking temperature, which is the temperature of the weak radiation emitted by a black hole. For a Schwarzschild black hole it depends only on the mass and is inversely proportional to it. The temperature of a black hole with the same mass as the Sun would be about 62 billionths of a Kelvin.
Why do smaller black holes have a higher temperature?: Since the mass is in the denominator of the Hawking temperature formula, the temperature is inversely proportional to the mass. Halving the mass doubles the temperature, meaning that smaller black holes are hotter and supermassive black holes are extremely cold.
How long does it take for a black hole to evaporate?: The time for evaporation is proportional to the third power of mass. A black hole with a solar mass can live about 10 to the 67 years, which is far longer than the age of the universe. In contrast, tiny black holes can evaporate almost instantly.
Can you determine mass from temperature?: Yes it is possible. If you change the calculation type to "Calculation from Temperature" and enter a value in Kelvin, then the same formula will be adapted to give the mass that can radiate at this temperature as well as the radius, entropy and lifetime.
Has Hawking radiation ever been observed?: No, there has been no direct observation. The temperature of actual black holes in space is much lower than the cosmic microwave background and cannot be detected with current measurement equipment. Still, this formula is a central prediction that connects gravity, quantum mechanics, and thermodynamics together.

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Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

Hawking, S. W. (1975). Particle creation by black holes
The original paper deriving black hole radiation and its temperature.
Wikipedia: Hawking radiation
Overview of the temperature, luminosity, and evaporation formulas.
HyperPhysics: Hawking radiation and black hole temperature
Georgia State University reference on black hole thermodynamics.