# Black Hole Temperature Calculator

By Álvaro Díez
Last updated: Jun 30, 2021

The black hole temperature calculator allows you to calculate the black body temperature of a black hole in a dark, cold environment. A black hole is a massive object with very special properties, one of which is the relationship between its temperature and its mass when considered as a black body. It is this relationship that we explore in the calculator, and we hope that this simple tool will help you better understand the properties of black holes.

## Black hole as an almost perfect black body

To understand this black hole temperature calculator, we first need to explain what is a black body. A black body is a theoretical object in physics that absorbs all radiation that gets to it. The temperature of this black body depends on the amount of absorbed radiation, which in turn determines the frequency and intensity of the electromagnetic radiation that it emits. A black hole is an almost perfect black body, and, as such, follows similar rules of emission and absorption.

As a black body, a black hole has an emission pattern in the shape of a Stefan-Boltzmann distribution, as shown in the image below, which is also dependent on temperature. Since the temperature of a black hole (or its first approximation) depends on its mass, you can predict how a black hole will emit radiation, the so called Hawking radiation, just by knowing its mass and using the Stefan-Boltzmann law calculator. That calculator was built precisely for this purpose, allowing you to obtain the temperature (and thus the emission spectra) of a black hole by just knowing its mass. ## Black body temperature, Hawking radiation, and consequences

The emission mechanism of a black hole is a very complex process that was first theorised by Stephen Hawking. It involves complex quantum mechanics, the uncertainty principle, and the spontaneous creation of particle pairs. Don't worry, we don't fully understand it either. We can, however, provide a very simple and quick explanation - imagine a pair of particles created on the edge of the event horizon so that one of the particles can escape the black hole whilst the other gets trapped inside the black hole's gravitational field.

To learn more about this process and the event horizon, please have a look at the Wikipedia articles on Hawking Radiation and the Event Horizon, as well as at the Schwarzschild Radius Calculator, for its explanation on how the event horizon is defined and its importance. One of the surprising facts about Hawking radiation is that it increases as the size of the black hole decreases.

This result constitutes one of the most mind-boggling pieces of Hawking's work - the conservation of energy, together with the emission/radiation process, leads to only one possible conclusion (again, in a very simple and very incomplete statement): "All black holes evaporate and they evaporate faster as they get smaller".

## How the black hole temperature calculator works

This black hole temperature calculator provides a very quick and simple way of obtaining the temperature of a black hole from its mass, or vice versa:

`T = ℏ c³ / (8 * π * G * M * Kb)`

The relationship between the mass and temperature of a black hole is a very simple one, and is set out in the equation above, where all values, except for the mass (`M`) and the temperature (`T`), are universal constants. Let's see what each of the parameters refers to, as well as the values of the constants and typical values for the free parameters:

1. `ℏ` is the reduced Planck constant (h/2π), and has a value of 1.0545718001 × 10−34 Js in SI units.
2. `c` is the speed of light, equal to 299792458 m/s.
3. `π` is the mathematical constant with a value of 3.141592...
4. `Kb` is the Boltzmann constant; in SI units, its value is 1.380649 × 10−23 J/K.
5. `G` is the universal gravitational constant with a value of 6.67408 × 10−11 Nm2/kg2 in SI units.
6. `T` is the black hole temperature/black body temperature. For example, a black with a with `M = 1 Solar Mass` has a temperature of 0.06172 * 10-6 K.
7. `M` is the black hole mass, which, due to their typical size in the universe, is commonly expressed in solar masses.

To use the black hole temperature calculator, it is as simple as inputting either the mass or the temperature of the black hole, which will give you the other parameter. It is important to note that typical black holes have a mass of at least several solar masses, but it is also interesting to see what would happen in the case of much smaller micro black holes, which have been theorised but never experimentally observed, because of how fast they evaporate (amongst many other reasons).

As we can see from the example above, the black body temperature of a typical black hole is very low, that is they emit a very small amount of Hawking radiation, which explains why black holes in the universe seem to exist forever. It is also important to realise that this temperature-mass relationship only holds when there is no radiation absorbed by the black hole, i.e. in a dark and cold environment very similar to the darkest parts of the universe. In reality, many of the black holes found in the universe do absorb and interact with other massive objects, and hence their real temperature is different from the expected black body temperature.

Álvaro Díez
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