Escape Velocity Calculator – Break Free of Any Gravity Well
The Escape Velocity Calculator works out the minimum speed an unpowered object needs at a given distance from a celestial body in order to leave for good and never fall back. Pick a planet, moon, star or exoplanet from the presets — or enter your own mass, radius, surface gravity or density — and the tool returns the escape velocity, the matching circular orbital velocity, the launch energy for a payload and a full step-by-step derivation.
Where the Formula Comes From
Escape happens when a projectile's kinetic energy exactly cancels its gravitational binding energy. Setting ½mv² = GMm/r and cancelling the projectile mass m from both sides gives the central relation:
v_e = √(2GM / r)
G gravitational constant, 6.6743 × 10⁻¹¹ m³/(kg·s²)
M mass of the attracting body (kg)
r distance from the body's centre to the launch point (m)Two consequences follow immediately. Escape velocity does not depend on the mass of the escaping object — a pebble and a spacecraft need exactly the same speed. And it has no direction: any launch angle that misses the surface works, because energy is a scalar. Only the required energy scales with payload mass, which is why the calculator reports that separately.
Four Ways to Describe the Same Body
Mass and radius are the usual inputs, but the identical result follows from surface gravity via v_e = √(2gr), or from mean density via v_e = r·√(8πGρ/3). The calculator accepts whichever pair you have and reports the implied values for the rest, so you can cross-check a body built for worldbuilding against real planetary data. A world with two Earth masses packed into 1.5 Earth radii, for instance, comes out at roughly 12.9 km/s.
Escape Velocity Versus Orbital Velocity
At the same radius, the circular orbital speed is always v_o = v_e/√2, about 71% of the escape speed. That single factor explains a lot of spaceflight: from Earth's surface escape needs 11.19 km/s while low orbit needs 7.91 km/s, and a craft already circling at 400 km altitude needs only about 3.2 km/s more to leave entirely rather than the full surface figure. Because v_e falls off as 1/√r, climbing from the ground to the ISS altitude only trims the escape requirement by roughly 3%.
Atmospheres, Black Holes and the Limits of the Model
Escape velocity also decides which gases a world keeps. A body holds an atmosphere over geological time when v_e is more than about six times the thermal speed √(3kT/m)of the gas. At Earth's 288 K surface temperature that keeps nitrogen and oxygen easily while hydrogen leaks away; feed in the far hotter exosphere temperature of around 1000 K instead and helium escapes as well, which is exactly what happens. Cold, low-gravity Titan holds a thick nitrogen atmosphere at 94 K, while the Moon — a similar escape speed but much warmer days — holds essentially nothing.
Push the radius down far enough and the formula returns the speed of light. That happens at the Schwarzschild radius r_s = 2GM/c²— about 8.9 mm for Earth's mass, or 3 km for the Sun's. The calculator reports r_s and the compactness ratio r_s/r with every result and refuses any launch radius at or inside the horizon, where escape is impossible.
What the Number Does Not Mean
Escape velocity applies to an unpowered projectile given one push, like a cannonball. A rocket under continuous thrust can leave at any speed it likes, so long as it keeps burning fuel — the 11.2 km/s figure is an energy requirement, not a speed a launch vehicle must hit. Real missions also pay for atmospheric drag, gravity losses and steering, so a practical launch budget runs closer to 9.4 km/s of delta-v just to reach low orbit. The tool assumes a spherical, non-rotating body and ignores the pull of every other object, which is why leaving the Solar System from Earth's surface needs about 16.6 km/s once the Sun is included in the sum.