Schwarzschild Radius
Calculate the Schwarzschild radius (event horizon) of any object from its mass using Rs = 2GM/c². Free online black hole calculator with charts, presets, and step-by-step breakdowns.
About This Calculator
The Schwarzschild Radius Calculator computes the event horizon radius of any massive object using the Schwarzschild formula Rs = 2GM/c². Named after the German physicist Karl Schwarzschild who first solved Einstein's field equations of general relativity in 1916, this calculator is an essential tool for astrophysics students, educators, and space enthusiasts exploring black hole physics.
The Schwarzschild radius represents the boundary — called the event horizon — where the escape velocity equals the speed of light. Any object compressed within its own Schwarzschild radius becomes a black hole, since not even light can escape its gravitational pull. The formula depends only on the mass of the object, making it remarkably simple: double the gravitational constant (6.674 × 10⁻¹¹ N·m²/kg²), multiply by the mass, and divide by the square of the speed of light (299,792,458 m/s). For a star with 10 times the mass of our Sun, the Schwarzschild radius is about 29.5 kilometers — roughly the size of a city, yet containing over 10 solar masses of matter.
How to Use
Enter the mass in solar masses (M☉) in the input field, or select a preset celestial body from the dropdown — options range from Earth (which would have an 8.87 mm Schwarzschild radius) to M87*, the supermassive black hole at the center of galaxy M87 with 6.5 billion solar masses. The calculator instantly computes the Schwarzschild radius in kilometers, the gravitational acceleration at the event horizon, and the average density of the black hole. Use the chart tabs to view a comparative bar chart of different black hole properties, or switch to the Comparison tab to see how the Schwarzschild radii of various celestial bodies stack up against each other.
Key Applications
The Schwarzschild radius is fundamental to understanding black hole physics, gravitational waves detected by LIGO and Virgo, and the behavior of matter under extreme gravitational fields. It helps astronomers classify compact objects — a neutron star typically has a radius of about 10 km, so any object of similar mass with a radius smaller than its Schwarzschild radius must be a black hole. The 2020 Nobel Prize in Physics was awarded for the discovery of a supermassive compact object at the center of our galaxy, Sgr A*, whose Schwarzschild radius was key to identifying it as a black hole.
Educational Value
This calculator is ideal for physics students learning general relativity, astronomy enthusiasts studying black holes, and educators demonstrating the relationship between mass and gravitational collapse. It also serves as a quick reference for researchers and science communicators needing accurate black hole parameters for presentations, articles, or classroom demonstrations. The comparison feature lets you explore the vast range of scales — from millimeter-scale Earth to supermassive black holes spanning billions of kilometers.
Frequently Asked Questions
What is the Schwarzschild radius?
The Schwarzschild radius is the distance from the center of a non-rotating black hole to the event horizon, where the escape velocity equals the speed of light. It is calculated using the formula Rs = 2GM/c², where G is the gravitational constant, M is the mass of the object, and c is the speed of light.
How do you calculate the Schwarzschild radius of a black hole?
To calculate the Schwarzschild radius, multiply the mass by twice the gravitational constant (2G), then divide by the square of the speed of light (c²). For a 10 solar mass black hole, the Schwarzschild radius is approximately 29.5 kilometers.
What is Earth's Schwarzschild radius?
Earth's Schwarzschild radius is about 8.87 millimeters. This means the entire mass of Earth would need to be compressed into a sphere smaller than a marble to form a black hole. For the Sun, the Schwarzschild radius is approximately 2.95 kilometers.
What is the event horizon of a black hole?
The event horizon is the boundary around a black hole beyond which nothing, not even light, can escape. For a non-rotating (Schwarzschild) black hole, the event horizon is located exactly at the Schwarzschild radius. Everything that crosses this boundary is lost forever and cannot communicate with the outside universe.
What types of black holes exist?
Stellar-mass black holes range from 3 to 100 solar masses and form from collapsing massive stars. Supermassive black holes contain millions to billions of solar masses and reside at galaxy centers, like Sgr A* (4.3 million solar masses) in the Milky Way and M87* (6.5 billion solar masses) in the M87 galaxy. Intermediate-mass and primordial black holes are also hypothesized.
How dense is a black hole at the event horizon?
Black hole density decreases as mass increases. A 10 solar mass black hole has density around 10^17 kg/m³, comparable to nuclear density. Supermassive black holes like Sgr A* have average density less than water (about 1 kg/m³), and the largest known black holes can have density lower than air.
Can objects other than black holes have a Schwarzschild radius?
Yes, every object with mass has a Schwarzschild radius, but for most objects it is far smaller than their actual size. Earth's Schwarzschild radius is only 8.87 mm, and the Sun's is 2.95 km. An object becomes a black hole only when it is compressed inside its own Schwarzschild radius.
What is the gravitational acceleration at the Schwarzschild radius?
The gravitational acceleration at the event horizon of a Schwarzschild black hole is given by g = GM/Rs². For a 10 solar mass black hole, this is about 1.5 × 10^12 m/s², or roughly 150 billion times Earth's gravity. For supermassive black holes, the surface gravity can be surprisingly low despite the enormous mass.