Black-hole scale explorer

For an isolated, nonrotating, uncharged black hole, mass sets a characteristic horizon scale. Choose a mass below and compare the radius with the diameter. The calculation is a model of a Schwarzschild black hole, not a way to establish that an observed object actually has a horizon.

Predict first. If the mass doubles, does the horizon radius double, quadruple, or stay unchanged?

Calculate a horizon scale

Try a challenge

Twice the mass: what happens to the horizon?

Predict how the radius and diameter change from 10 to 20 Suns, then compare both calculations. Use the mass buttons below to test the pair.

10 nominal solar masses

Radius: 29.53 km
Diameter: 59.07 km

20 nominal solar masses

Radius: 59.07 km
Diameter: 118.1 km

Reveal the explanation

Both radius and diameter double. The Schwarzschild horizon radius is proportional to mass; diameter is twice radius. These are horizon scales for isolated, nonrotating, uncharged black holes, not observed bright-ring diameters or a simulation of how black holes form.

Explore · choose a mass and calculate its horizon scale.

The static result below is for ten solar masses. Controls activate when the lab loads. If they remain unavailable, reload the lab or use the worked examples below.

Choose the mass

Mass in nominal solar masses

Enter a number from 0.0000000001 to 100 billion. A value of 1 uses the Sun’s nominal mass parameter.

M☉

Earth1 Sun10 Suns20 Suns4 million Suns

Calculate the horizonReset to 10 Suns

Idealized Schwarzschild result

Horizon scale

Radius

29.53 km

Diameter

59.07 km

For 10 nominal solar masses: radius 29.53 km; diameter 59.07 km.

The horizon is a boundary in spacetime, not a material surface. This model assumes an isolated, uncharged, nonrotating black hole; a calculated scale is not by itself observational evidence for an event horizon.

What the number means

The radius is the areal radius of the Schwarzschild event horizon. Doubling the mass doubles both radius and diameter; it does not describe a solid ball or the full glowing region around an astrophysical black hole.

Return to the first black-hole journey →

The presets are mass comparisons. Choosing the Earth or Sun preset does not turn either object into a black hole or assert that it is one. The calculation asks what horizon scale a Schwarzschild black hole with that mass parameter would have.

Work the ten-solar-mass example

The model gives

$$r_s=\frac{2GM}{c^2},\qquad d_s=2r_s.$$

Here $M$ is mass, $G$ is the gravitational constant, and $c$ is the speed of light. A sphere of areal radius $r_s$ has area $4\pi r_s^2$. The reported diameter is twice that scale; it is not a measured ruler distance through the interior.

Using the nominal solar mass parameter gives a radius of approximately $2.95325$ kilometers per nominal solar mass. Ten nominal solar masses therefore give a radius of $29.5325$ kilometers and a diameter of $59.0650$ kilometers. The display rounds those to $29.53$ and $59.07$ kilometers.

Mass parameter ratio to the nominal Sun	Horizon radius	Twice that radius
1	2.953 km	5.907 km
10	29.53 km	59.07 km
1,000,000	2.953 million km	5.907 million km

Prediction answer and exact conversion

The radius doubles. Mass appears to the first power, and the other quantities are fixed.

The calculator uses the exact nominal parameter $(GM)_\odot^{\rm N}=1.3271244\times10^{20}\ {\rm m^3\,s^{-2}}$ and exact SI speed of light $c=299\,792\,458\ {\rm m\,s^{-1}}$. The input is the ratio $\mathcal M=GM/(GM)_\odot^{\rm N}$, so

$$r_s=\mathcal M\,\frac{2(GM)_\odot^{\rm N}}{c^2}.$$

A nominal conversion constant is defined exactly for convenient conversion; it is not a claim that every measurement of the actual Sun’s mass is exact. See IAU 2015, Resolution B3 (PDF) and BIPM, SI Brochure, § 2.2 (PDF).

Keep three sizes separate

The horizon scale is not the radius of a material surface. It is also not the size of the bright emission ring in a telescope image. Emitting gas, light bending, viewing geometry, and rotation affect the image. Real astrophysical black holes are generally modeled with rotation, which this one-parameter calculation leaves out.

The Newtonian trick of setting an escape speed equal to $c$ produces the same algebraic scale. It does not supply the relativistic definition of an event horizon. For that distinction, read A Black Hole Is Not Defined by Newtonian Escape Speed.

Test your interpretation

An astronomer estimates an object’s mass to be five solar masses. Does the calculated $14.8$-kilometer radius show that the object must have an event horizon?

Answer

No. It states the horizon scale that an isolated Schwarzschild black hole with that mass would have. Establishing the nature and compactness of an observed object requires independent astronomical evidence, and the real object may not satisfy the model’s assumptions.

For educators

Preparation and objective. Use proportional reasoning to distinguish a calculated horizon scale from evidence that a black hole exists. In ten minutes, compare one, ten and twenty nominal solar masses. Expected radii are about 2.95, 29.5 and 59.1 km; the corresponding diameter is always twice the radius. In a twenty-minute extension, change units and compare an Earth-mass preset with an astronomical black-hole mass estimate. Explain why unit changes preserve the physical result and why a hypothetical preset is not a claim about the real Earth. The static table and formula support paper work. Correct the misconception that a calculated radius or an image’s bright ring directly establishes an event horizon.

Continue to Black Holes: Start Here for the evidence, or Solar-Mass Geometric Conversions for the deeper unit treatment.

References

• Bureau International des Poids et Mesures. The International System of Units (SI), 9th edition, § 2.2. DOI. Official PDF.
• International Astronomical Union. “Resolution B3 on Recommended Nominal Conversion Constants for Selected Solar and Planetary Properties.” XXIX General Assembly, 2015. Official PDF.

Download the original calculation module to inspect or reproduce the numerical result.