Hawking Radiation Calculator

Black holes were so named because they were once thought to give off no radiation whatsoever, but Stephen Hawking showed that this is not the case. Black holes do produce radiation, with an intensity inversely proportional to the square of their mass. Since most black holes are thought to form from collapsed stars and are very massive, they give off very little radiation. However, a smaller hole (on the order of only a few billion tons) would radiate a great deal more, making it an excellent power source for an advanced civilization.

The series of input forms below (Javascript source) calculates various useful characteristics of a black hole and its emissions. The initial mass is set to a billion metric tons, the mass of a standard industrial neuble from The Collapsium. Specifying any quantity causes the others to be recalculated accordingly. The drop-down menus select the units of measure to be used for their corresponding input field.

Quantity Value Units Expression
Mass [ M ]
Radius [ R = M * 2G/c^2 ]
Surface area [ A = M^2 * 16 pi G^2/c^4 ]
Surface gravity [ kappa = 1/M * c^4 / 4G ]
Surface tides [ dkappa_R = 1/M^2 * c^6 / 4G^2 ]
Entropy (dimensionless) [ S = M^2 * 4 pi G / hbar c ln 10 ]
Temperature [ T = 1/M * hbar c^3/8k pi G ]
Luminosity [ L = 1/M^2 * hbar c^6/15360 pi G^2 ]
Lifetime [ t = M^3 * 5120 pi G^2/hbar c^4 ]


The radius of a Schwarzchild black hole of mass M is

[ R = 2G/c^2 * M ]

As per [Hawking 1974], the thermodynamic temperature of such a hole is

[ T = kappa/2pi = hbar c^3/8k pi G * 1/M ]

with brackets delimiting constants of multiplication. Its surface area is

[ A = 4 pi R^2 = 16 pi G^2/c^4 * M^2 ]

making the Hawking radiation luminosity at least

[ L = A sigma T^4 = hbar c^6/15360 pi G^2 * 1/M^2 ]

At a distance r from a hole with mass M, the incident radiation flux is therefore

[ Phi = L/4 pi r^2 = hbar c^2/61440 pi^2 G^2 * 1/M^2 r^2 ]

The amount of radiation actually intercepted by an object necessarily depends upon its exposed area.

The expression for L makes it possible to calculate the lifetime of a black hole of given initial mass M0, assuming no mass input. Luminosity means energy output, thus

[ -dE/dt = hbar c^6/15360 pi G^2 * 1/M^2 ]

And since dE = dM c2,

[ -dM/dt = hbar c^4/15360 pi G^2 * 1/M^2 ]

Separating variables and integrating:

[ t = 5120 pi G^2/hbar c^4 * M^3 ]

Calculating the various constants, this works out to

[ t = (M / M_sol)^3 * 2.099 * 10^67 yr ]

where [ M_sol ] is one solar mass, 2 × 1030 kg.

Clearly for a solar-mass hole the lifetime is essentially infinite. In fact, for a large enough hole (such that T < 2.726 Kelvin, or M > 0.75% the mass of the Earth) the hole will actually grow slightly by feeding on cosmic background radiation. Only when the universe cools below the hole's Hawking temperature will it start to shrink.