Escape Velocity Calculator

In theory, escape velocity is independent of launch direction (straight up, at an angle, or even horizontal) if there is no atmosphere. What matters is the total kinetic energy, which depends only on the speed. In practice, launching horizontally places you in orbit below escape velocity, and launching upward (or at any angle) allows you to reach escape velocity with the same speed.

Circular orbital velocity at the surface is v_orb = sqrt(GM/r) = v_esc / sqrt(2). Orbital velocity is about 0.707 times escape velocity. To leave the orbit and escape, you need to increase your speed by a factor of sqrt(2) — about 41% more. For Earth, orbital velocity at the surface (ignoring atmosphere) would be 7.91 km/s, while escape velocity is 11.19 km/s.

Gas molecules in a planet's upper atmosphere have a Maxwell-Boltzmann distribution of speeds. Molecules moving faster than about 1/6 of the escape velocity at the exosphere can escape — a process called Jeans escape. For this reason, hydrogen (lightest gas, highest thermal velocity) escapes most easily. Earth retains N2 and O2 but has lost most primordial hydrogen and helium over geological time.

From Earth's surface, escaping the Solar System requires overcoming both Earth's gravity and the Sun's gravity. The combined escape speed is about 42.1 km/s from Earth's orbit — but starting from Earth's surface you need to reach 16.6 km/s (the hyperbolic excess speed) after escaping Earth's gravity. The Voyager probes exceeded this speed using gravity assists from Jupiter and Saturn.

The cosmic velocities are defined as: first (7.91 km/s) — circular orbit at Earth's surface; second (11.19 km/s) — escape from Earth; third (16.6 km/s) — escape from the Solar System from Earth's orbit; fourth — escape from the Milky Way. Rocket missions to the outer planets and beyond must exceed the third cosmic velocity.

The formula v_esc = c coincidentally gives the correct radius for a black hole (the Schwarzschild radius), but this is a coincidence. In Newtonian mechanics, light is not affected by gravity in the same way, and the concept of a black hole does not follow consistently from Newtonian physics. The correct treatment requires general relativity, where the event horizon is a geometric feature of curved spacetime, not a classical velocity barrier.

Rocket designers must account for escape velocity when designing launch vehicles. The delta-v (change in velocity) budget for a mission begins with the surface escape velocity. For Mars missions, the lower Martian gravity (escape velocity 5.03 km/s) makes launch from the surface much easier than from Earth. This is why a Mars sample return mission can use a much smaller rocket than Earth-based launches.

The escape velocity from the Milky Way depends on position and the total (including dark matter) mass distribution. From the Sun's location (about 8.5 kpc from the Galactic center), the escape velocity is roughly 550 km/s. Stars moving faster than this are hypervelocity stars that have been ejected, often by gravitational interactions with the central supermassive black hole.

Voyager 1, launched in 1977, was given enough velocity to escape the Solar System through a combination of launch speed and gravity assists from Jupiter and Saturn. It passed the solar escape velocity long ago and is now traveling at about 17 km/s (about 3.6 AU/year) in interstellar space. Its heliocentric speed exceeds the local solar escape velocity, confirming it is on a hyperbolic escape trajectory.