hawking radiation calculator

What is Hawking radiation?

Hawking radiation is a quantum effect predicted by Stephen Hawking in 1974. In simple terms, black holes are not perfectly black: they emit thermal radiation due to quantum field effects near the event horizon. As energy is emitted, the black hole loses mass over time.

The surprising result is that black holes have a temperature, and that temperature depends inversely on mass. Big black holes are colder than the cosmic microwave background and gain mass overall in realistic environments, while tiny black holes would be very hot and evaporate quickly.

What this calculator computes

Given a black hole mass, the calculator estimates:

• Schwarzschild radius (event horizon size)
• Hawking temperature in Kelvin
• Thermal energy scale in electron-volts (eV)
• Radiated power in watts
• Mass loss rate from radiation alone
• Evaporation lifetime (idealized full evaporation time)

Equations used

For a non-rotating, uncharged black hole (Schwarzschild case), standard leading-order formulas are:

• r_s = 2GM / c²
• T_H = ħc³ / (8πGMk_B)
• P = ħc⁶ / (15360πG²M²)
• τ = 5120πG²M³ / (ħc⁴)

Here, G is Newton’s gravitational constant, c is the speed of light, ħ is reduced Planck’s constant, and k_B is Boltzmann’s constant.

How to interpret the results

1) Larger mass means lower temperature

Temperature scales like 1/M. A solar-mass black hole has an extremely tiny Hawking temperature (far below 1 Kelvin), effectively undetectable against cosmic background radiation.

2) Power drops quickly as mass increases

Radiated power scales like 1/M². Very massive black holes emit almost no Hawking radiation per second, while very small hypothetical black holes would emit much more strongly.

3) Lifetime rises dramatically with mass

Evaporation time scales like M³. Doubling mass multiplies lifetime by eight. This is why astrophysical black holes survive for times far longer than the current age of the universe.

Quick examples

• 1 solar mass black hole: ultracold and unbelievably long-lived.
• 10¹² kg black hole: hotter and more luminous, but still tiny on everyday power scales.
• Near-Planck masses: formulas become less reliable, and full quantum gravity is needed.

Limits and caveats

This calculator is educational and uses simplified formulas. Real black holes can spin (Kerr), carry charge (Reissner–Nordström, usually negligible astrophysically), and interact with surrounding matter and radiation fields. Also, near the final evaporation stage, semiclassical approximations may break down.

Still, these equations are the standard first pass used in physics education and provide strong intuition for how black hole mass controls temperature, power, and longevity.