Black Hole Temperature Calculator - Hawking Temperature & Entropy

A black hole temperature calculator turns a single mass into the Hawking temperature of a Schwarzschild black hole together with the surface gravity at the horizon, the Bekenstein-Hawking entropy, and the peak wavelength of the Hawking spectrum. It is built for physics coursework on quantum field theory in curved spacetime, where the same closed-form formulas are needed again and again.

• • Stellar-mass homework: Compute the Hawking temperature of a 5 to 50 solar mass remnant and compare it to the cosmic microwave background.
• • Supermassive reference: Check M87* or Sgr A* in solar masses and read the Wien peak wavelength to see where the Hawking spectrum sits in the electromagnetic band.
• • Primordial limits: Estimate when a primordial black hole of a given mass would have evaporated and what Hawking temperature it reached.
• • Relativity review: Show how Hawking temperature scales as 1/M, surface gravity as 1/M, and Bekenstein-Hawking entropy as M^2.

Every output in this black hole temperature calculator comes from a single closed-form expression evaluated with NIST CODATA 2018 constants. Once the mass is set, the surface gravity, entropy, and Wien peak follow without further input.

The page covers the textbook non-rotating uncharged Schwarzschild solution; spin dependence (Kerr) and charge dependence (Reissner-Nordstrom) are out of scope.

For the geometric and lifetime side of the same physics unit, the Black Hole Calculator takes the same mass and adds the Schwarzschild radius, evaporation lifetime, and average density that this temperature page does not duplicate.

The calculator takes the mass, converts it to kilograms, then evaluates four closed-form expressions that share the same constants. The Hawking temperature is the headline output; the rest follow from the same NIST CODATA constant set.

• M: Black hole mass in kilograms, converted from solar masses via M_sun = 1.98892e30 kg.
• G: Gravitational constant, 6.67430e-11 m^3 kg^-1 s^-2 (NIST CODATA 2018).
• c: Speed of light, 299792458 m/s exactly (NIST CODATA 2018).
• hbar: Reduced Planck constant, 1.054571817e-34 J s.
• k_B: Boltzmann constant, 1.380649e-23 J/K.
• b: Wien displacement constant, 2.897771955e-3 m K.

All four expressions live in one pure JavaScript function, so the same input flows through every formula. The unit selector only affects the mass-to-kilogram conversion; the rest are evaluated in SI base units and converted to the user-friendly display units.

The Hawking temperature and surface gravity both scale as 1/M, so heavier black holes are colder and have weaker surface gravity. The Bekenstein-Hawking entropy scales as M^2, so a 10-solar-mass hole has 100 times the entropy of a 1-solar-mass hole even though its temperature is 10 times lower. The Wien peak scales linearly with M, so heavier holes radiate at longer wavelengths.

According to NIST CODATA 2018 - Fundamental Physical Constants, G, c, hbar, k_B, M_sun, and the Wien displacement constant take the values used in the formulas above, which is the same fixed set evaluated by the calculator.

The same NIST CODATA constants that govern the Hawking temperature also drive atomic radiation, and the Bohr Model Calculator uses that shared hbar and c to compute the photon wavelength and energy of a hydrogen-like transition between any two quantum levels.

Four ideas hold the calculator together: quantum pair creation near the horizon, the surface gravity that sets the spectrum, entropy proportional to horizon area, and the Wien peak of the resulting black-body curve.

These four concepts share a common thread: every quantity depends only on the mass. Once M is set, the Hawking temperature, surface gravity, entropy, and Wien peak are fixed by the same input.

If you want to see how the surface gravity translates into observable redshift, the same mass drives a gravitational time dilation curve.

The surface gravity at the horizon fixes the Hawking temperature, and the Gravitational Time Dilation Calculator uses the same Schwarzschild mass for the gravitational redshift and time dilation at any chosen radius outside the horizon.

Enter a mass in either kilograms or solar masses, then read the seven outputs. The calculation runs on every change, so you can scan a range of masses without reloading the page.

1. 1 Choose a mass unit: Solar masses for stellar or supermassive black holes; kilograms for primordial or Earth-mass candidates.
2. 2 Enter the mass: Defaults to 10 solar masses, a textbook size for a stellar black hole from core collapse.
3. 3 Read the Hawking temperature: Headline output in kelvin, usually in scientific notation because astrophysical black holes are far colder than the cosmic microwave background.
4. 4 Check the Schwarzschild radius: Geometric context for the size of the horizon that produces the temperature.
5. 5 Compare the entropy: Bekenstein-Hawking entropy in k_B is far larger than any ordinary thermodynamic system of the same mass.
6. 6 Scan the Wien peak wavelength: Heavier holes peak at longer wavelengths, lighter holes at gamma-ray wavelengths.

For a textbook stellar-mass check, leave the unit on solar masses and enter 1; the Hawking temperature reads about 6.17e-8 K, the Schwarzschild radius 2.95 km, and the entropy 1.05e77 k_B. For the supermassive M87* limit, stay on solar masses and enter 4.3e9; the Hawking temperature drops to roughly 1.4e-17 K with a Wien peak around 2e14 m. For the primordial case, switch to kilograms and enter 1e12; the Hawking temperature rises to about 1.2e11 K.

When the question shifts from the horizon to a stable orbit just outside it, the Orbital Period Calculator takes the same central mass and returns the orbital period and energy for a particle in a circular Schwarzschild orbit.

The black hole temperature calculator packages four Hawking-derived formulas behind one mass input, so homework, exam prep, and research-style checks share the same numbers.

• • Seven outputs from one mass: Hawking temperature, Schwarzschild radius, surface gravity, Bekenstein-Hawking entropy, Wien peak wavelength, thermal energy, and CMB ratio share the same input.
• • NIST CODATA 2018 constants: G, c, hbar, k_B, M_sun, and the Wien displacement constant match published values to four or more significant figures.
• • Solar mass or kilogram input: Quote M87* in solar masses without first dividing by 1.989e30; switch to kilograms for primordial and Earth-mass exercises.
• • CMB comparison built in: The log10 ratio of the Hawking temperature to the 2.725 K cosmic microwave background is reported as the final row.
• • Worked-example friendly: Defaults are set to a 10-solar-mass stellar black hole and the worked examples cover the one-solar-mass textbook reference.

For one-off homework the calculator gives a defensible numerical answer in seconds; for research-style checks it gives consistent black-body outputs from one mass without re-typing the constants.

If you also need the Schwarzschild lifetime or the density, the same input flows into the dedicated black hole geometry calculator, so the two tools share the same NIST constants.

Three input factors change every output, and the resulting electromagnetic band depends on how small the mass is. The caveats below describe where the model is reliable and where it stops being exact.

• • The calculator assumes a non-rotating uncharged Schwarzschild black hole; Kerr or Reissner-Nordstrom holes need extra inputs (angular momentum or charge).
• • The Wien peak wavelength assumes a perfect black-body. Real Hawking radiation is grey-body corrected, so the true peak sits at a slightly shorter wavelength.
• • Near or below the Planck mass the formulas are extrapolations; quantum gravity is expected to modify the Hawking temperature at the final evaporation stage.

Hawking radiation has never been observed directly, so the temperature and entropy outputs are theoretical predictions of general relativity combined with quantum field theory in curved spacetime.

If you want to read the Hawking temperature as a thermal population, the same Boltzmann constant can be turned into a Boltzmann factor for any chosen energy.

According to NASA - Black Holes, isolated black holes emit Hawking radiation as a thermal black-body and evaporate on a timescale that scales as the cube of the mass, far below the cosmic microwave background for any astrophysical mass.

According to Fixsen (2009) - The Temperature of the Cosmic Microwave Background, the cosmic microwave background monopole temperature is 2.72548 K, rounded to 2.725 K here as the benchmark against the Hawking temperature.

To read the Hawking temperature as a thermal population weight exp(-E / k_B T_H), the Boltzmann Factor Calculator uses the same Boltzmann constant to convert T_H into a probability for a particle of any chosen energy.