Physics & Mechanics

Hawking Radiation Temperature Calculator

Calculate the temperature of Hawking radiation emitted from a black hole based on its mass using Stephen Hawking's formula.

Hawking Temperature
6.1684 × 10⁻⁸
Luminosity (Power Output)9.0028 × 10⁻²⁹ W
Evaporation Time2.0974 × 10⁶⁷ Years
Event Horizon Radius2,954.127 m

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The Evaporation of Black Holes

In 1974, Stephen Hawking shocked the physics world by proving that black holes aren't completely black. Due to the bizarre rules of quantum mechanics operating right at the edge of the event horizon, black holes actually emit a very faint glow of thermal radiation, now known as Hawking Radiation.

Because they are emitting energy, they must be losing mass ($E=mc^2$). If a black hole doesn't consume new matter to replace what it radiates away, it will eventually shrink and "evaporate" entirely in a massive flash of gamma rays.

The Strange Rules of Temperature

Hawking's equation reveals an inverse relationship: the heavier the black hole, the colder it is.

  • A supermassive black hole at the center of a galaxy is incredibly cold (billionths of a degree above absolute zero).
  • A tiny, microscopic black hole would be blindingly hot, burning up almost instantly.

The Formula

T=c38πGMkB\begin{aligned} T = \frac{\hbar \cdot c^3}{8 \cdot \pi \cdot G \cdot M \cdot k_B} \end{aligned}

Where:
T=
Hawking Temperature (Kelvin, K)
\hbar=
Reduced Planck Constant
c=
Speed of Light
G=
Gravitational Constant
M=
Mass of the black hole
kBk_B=
Boltzmann Constant

Example Calculation

Calculate the temperature of a "stellar-mass" black hole that is exactly $5$ times heavier than our sun ($Mass \approx 10^{31} , \text{kg}$).

(Note: The constants $\hbar, c, G, k_B$ combine to form a constant multiplier. For quick math, $T \approx 1.227 \times 10^{23} / Mass$)

  1. Divide Constant by Mass: $1.227 \times 10^{23} / 10^{31} = 1.227 \times 10^{-8} , \text{Kelvin}$.

The temperature of a standard black hole is a tiny fraction of one billionth of a degree above absolute zero.

Frequently Asked Questions

It doesn't. Empty space is filled with 'virtual particles' that constantly pop into existence and annihilate each other. If a pair pops into existence right on the event horizon, one might fall in while the other escapes. The escaping particle becomes real Hawking radiation, stealing mass from the black hole to pay its 'energy debt'.

No. For any normal black hole, the Hawking temperature is far colder than the ambient temperature of deep space ($2.7 , \text{K}$ from the Cosmic Microwave Background). The black hole is absorbing more heat from empty space than it is emitting, making the radiation impossible to detect.

An unimaginably long time. A black hole the size of our sun would take about $10^{67}$ years to evaporate. For context, the entire universe is only $1.3 \times 10^{10}$ years old.