Schwarzschild radius calculator

What is the Schwarzschild radius?

It is the radius of the event horizon of a black hole that does not spin and carries no charge: r = 2GM ÷ c², with G the gravitational constant, M the mass and c the speed of light. Only the mass changes from one such black hole to another, so the radius is simply proportional to the mass. OpenStax Astronomy 2e puts it as about 3 km for one solar mass; this page gets 2.95 km (1.84 mi).

What is the Schwarzschild radius of the Earth and the Sun?

For the Sun, 2.95 km (1.84 mi). For the Earth, 8.87 mm, a little over a third of an inch (0.349 in). These are theoretical values: neither body has anywhere near enough mass to collapse into a black hole, and both are far larger than the radius at which they would become one.

Can the Sun become a black hole?

No. OpenStax Astronomy 2e notes that the Sun does not have enough mass to end its life as a black hole; the calculation only shows how small the Sun would have to be squeezed. A star has to be considerably more massive than the Sun to leave a black hole behind.

What happens to a clock near a black hole?

A clock that is held still at distance r from the center runs slow by the factor √(1 − r_s ÷ r) compared with a clock far away. At twice the Schwarzschild radius the factor is 0.7071, at ten times it is 0.9487, and at 1.01 times it is only 0.0995. The factor reaches zero at the horizon, so there is no such thing as a clock held still at or inside it.

What is the photon sphere, and what is the innermost stable orbit?

For a non-spinning black hole, light can travel on an unstable circular orbit at 1.5 times the Schwarzschild radius (3GM ÷ c²). Matter can orbit stably down to 3 times the radius (6GM ÷ c²), and inside that it spirals in. For one solar mass those are 4.43 km and 8.86 km. These figures are for the Schwarzschild solution only; spin changes them.

Is the surface gravity at the event horizon infinite?

For someone trying to hover just outside the horizon, the acceleration needed to avoid falling grows without limit as the horizon is approached, so no ordinary 'surface gravity in g' exists to quote there. A free-falling object feels nothing special in the instant it crosses a large horizon. The tide, the difference of gravity across your body, is what hurts: OpenStax says it kills you before the horizon for a black hole of a few solar masses and mainly after it for a supermassive one. This page therefore gives no gravity figure at the horizon.

Is a black hole always extremely dense?

The average density inside the horizon falls as the mass grows, because the radius grows in step with the mass while the volume grows with its cube. For ten solar masses the horizon radius is 29.53 km and the mean density 1.84 × 10¹⁷ kg/m³. A black hole of about 1.4 × 10⁸ solar masses has a mean density equal to that of water.

What the number means

Any mass has a Schwarzschild radius, but it only matters when the mass is squeezed inside it. Squeeze the Earth to 8.87 mm and it would be a black hole; as it is, its real radius is about 6.4 million meters. The horizon is the boundary after which nothing, light included, can get out to a faraway observer. A handy way to remember the formula is to set the escape speed √(2GM ÷ r) equal to the speed of light, which gives r = 2GM ÷ c². That is only a memory aid; the exact result for a non-rotating black hole comes from general relativity, and it happens to agree.

How the calculator works

The mass is converted to kilograms: a solar mass is 1.99 × 10³⁰ kg, from the IAU 2015 nominal value of GM for the Sun, 1.3271244 × 10²⁰ m³/s², divided by G = 6.6743 × 10⁻¹¹. Then r = 2GM ÷ c² with c = 299,792,458 m/s. The table adds the photon sphere (1.5 r), the innermost stable circular orbit (3 r), the average density M ÷ (4/3 π r³) and, if you enter a distance, the slowing of a clock that stays at that distance and the redshift of its light, which is 1 ÷ √(1 − r_s ÷ r) − 1. The distance is measured from the center in the Schwarzschild coordinate sense, which is the one that makes the circumference of a circle 2π times r.

Worked example

One solar mass: r = 2 × 6.67 × 10⁻¹¹ × 1.99 × 10³⁰ ÷ (299,792,458)² = 2,953.3 m, or 2.95 km. Its photon sphere is at 4.43 km and its innermost stable orbit at 8.86 km. The average density inside the horizon is 1.84 × 10¹⁹ kg/m³. Ten solar masses give 29.53 km, exactly ten times larger. A clock held at 2 r_s (5.91 km from a solar-mass black hole) ticks at 0.707 of the far-away rate; light climbing out from there is redshifted by z = 0.414.

Limits of this method

The formulas assume a black hole with no spin and no charge. Real astrophysical black holes spin, and for a spinning one the horizon, the photon orbits and the innermost stable orbit are all at different radii; the 3 r_s figure applies only to the non-rotating case (Carroll, equation 7.55). The clock factor describes an observer who is held at rest, which needs a rocket pushing ever harder as the horizon nears; a freely falling observer is a different case. Finally, a mass is a black hole only if it really is inside its Schwarzschild radius: the numbers for Earth and the Sun are thought experiments. For everyday gravitational time effects, such as the GPS clocks (about 45 microseconds per day fast from weaker gravity, about 7 slow from speed, OpenStax Astronomy 2e section 24.4), the effect is tiny because r is enormously larger than r_s.

Common mistakes

A frequent slip is to treat the Schwarzschild radius as the size of the black hole's 'surface': nothing is there, the horizon is a boundary in spacetime, not a material skin, and the matter that formed the hole is not found at that radius. Another is to enter the radius of a star in the mass field, or to mix up solar masses and kilograms; check the unit selector. People also expect the density to be the same for every black hole; it is not, and for the heaviest supermassive holes it is lower than that of water. And the clock factor applies to a hovering observer, not to one falling in.

Sources: OpenStax Astronomy 2e, section 24.5 “Black Holes” (Schwarzschild radius, about 3 km for the Sun, tidal forces) and section 24.4 “Time in General Relativity” (GPS clocks); S. Carroll, Lecture Notes on General Relativity, arXiv:gr-qc/9712019, equation 7.55 (smallest stable circular orbit, 6GM); J. Pinochet, arXiv:2609.17591, equations 64 and 77 (proper time and photon sphere); IAU 2015 Resolution B3, nominal solar and planetary constants (arXiv:1510.07674); G from CODATA. Related: escape velocity calculator, time dilation calculator, orbital period calculator, redshift and Hubble’s law calculator.