Escape Velocity Calculator
Compute the escape velocity from a gravitational body using vesc = √(2GM / r). Solve for escape velocity, body mass, or distance from centre. Step-by-step math, a live gravity-well schematic with the escape trajectory, and companion metrics including orbital velocity, surface gravity, and Schwarzschild radius.
Escape velocity is a speed, not a direction
Contrary to intuition, escape velocity does not require launching straight up. Any object travelling at vesc in any direction — as long as its path does not intersect the body — will escape to infinity, provided no other forces act. This is why orbital launches aim mostly horizontally: reaching vesc horizontally is far more energy-efficient than climbing vertically, since the atmosphere is thin at altitude and the vehicle can use Earth's rotation as a head start.
Escape velocity assumes no further thrust
The formula describes the speed needed to escape using only the kinetic energy an object already has at distance r. A rocket that continues firing its engine can leave the gravitational field at any speed. Similarly, a body moving at less than vesc will follow an elliptical, parabolic, or hyperbolic trajectory depending on its energy — only speeds ≥ vesc lead to escape. At exactly vesc, the trajectory is parabolic and the object reaches infinity with zero residual speed.
Surface escape velocity scales with √(M/R)
For an object of uniform density ρ, mass scales as R³, so escape velocity scales as R√(ρ). This means that for the same density, a larger body has higher escape velocity. Compressing matter to higher density (a white dwarf or neutron star) dramatically increases escape velocity: a neutron star's escape velocity is about 0.6c, and a black hole's escape velocity at its event horizon is exactly c. This is why gravitational redshift and relativistic effects become important for compact objects.
Atmospheric retention depends on vesc
A planet retains a gas if the mean thermal speed of its molecules is much smaller than the escape velocity. Earth's escape velocity is 11.2 km/s while the mean thermal speed of hydrogen at Earth's upper atmosphere is about 2 km/s. Over geological time, the high-energy tail of the Maxwell distribution can still escape, which is why Earth has lost most of its hydrogen and helium. Titan, with escape velocity 2.6 km/s, retains a thick nitrogen atmosphere because nitrogen is heavier and the temperature is lower.
How to Use This Calculator
The escape velocity formula links the body mass, distance from centre, and escape speed. Pick which quantity you want the tool to solve, then fill in the other two.
1. Choose what to solve for
Pick vesc, M, or r from the toggle. The calculator rearranges vesc = √(2GM/r) automatically for whichever you choose.
2. Use realistic presets
Presets cover Solar System bodies: Moon, Mercury, Mars, Earth, Jupiter, and the Sun, with both mass and surface radius values available in the pill rows.
3. Use SI units throughout
Mass in kilograms, distance in metres, velocity in metres per second. G = 6.67430×10⁻¹¹ m³·kg⁻¹·s⁻² is a fundamental constant. The hints show Earth-mass and km/s equivalents.
4. Read the result and verdict
The result card shows escape velocity in m/s plus the km/s equivalent. A colour-coded verdict classifies the body from asteroid-class through terrestrial, Jovian, stellar, and relativistic (near light speed).
5. Check the schematic and charts
The schematic shows a gravity well with the body at the bottom, the escape vector, and the radius. The chart plots vesc vs. distance for the current mass, with the current point highlighted.
6. Switch to Advanced for depth
Advanced mode adds a chart of vesc vs. mass at fixed r, a sensitivity check on a +1% input change, and companion metrics including orbital velocity, surface gravity, Schwarzschild radius, and thermal escape temperature for various gases.
What This Tool Offers
Built around the standard escape-velocity formula vesc = √(2GM/r), the fundamental relationship between mass, distance, and the speed needed to break free of gravity.
Three-way solving
Solve for escape velocity, body mass, or distance from centre. The calculator rearranges the formula for whichever variable you choose.
Live gravity-well schematic
A potential-well diagram with the gravitating body, escape vector, radius marker, and a readout of the full calculation.
Body-class verdict
Each result is classified from asteroid-class through terrestrial, Jovian, stellar, compact, and relativistic (fraction of c) with a coloured badge.
Companion metrics
Orbital velocity, surface gravity, Schwarzschild radius, fraction of light speed, and thermal escape temperature for H₂, He, N₂, and O₂.
Dual comparison charts
Escape velocity vs. distance at fixed mass, plus an advanced escape velocity vs. mass chart at fixed distance.
Reference tables
Solar System escape velocities, atmospheric retention criteria, and a full variable reference table with physical constants.
Reference Tables & Conversions
Escape velocity classes, Solar System data, and gravitational constant reference.
Escape velocity class lookup
Solar System Escape Velocities
Surface escape velocity for planets, moons, and the Sun, with companion orbital velocity.
| Body | Mass (kg) | Radius (km) | vesc (km/s) | vorb (km/s) |
|---|
Atmospheric Retention Criteria
A planet retains a gas if the escape velocity exceeds about 6× the mean thermal speed of the gas.
| Body | vesc (km/s) | Retains H₂? | Retains He? | Retains N₂? | Retains O₂? |
|---|---|---|---|---|---|
| Moon | 2.38 | No | No | No | No |
| Mercury | 4.25 | No | No | No | No |
| Mars | 5.03 | No | No | Partial | Partial |
| Venus | 10.36 | No | No | Yes | Yes |
| Earth | 11.19 | No | No | Yes | Yes |
| Uranus | 21.3 | Partial | Yes | Yes | Yes |
| Neptune | 23.5 | Partial | Yes | Yes | Yes |
| Saturn | 35.5 | Yes | Yes | Yes | Yes |
| Jupiter | 59.5 | Yes | Yes | Yes | Yes |
| Sun | 617.5 | Yes | Yes | Yes | Yes |
Physical Constants
| Symbol | Name | Value | Units |
|---|---|---|---|
| G | Gravitational constant | 6.67430×10⁻¹¹ | m³·kg⁻¹·s⁻² |
| c | Speed of light in vacuum | 2.99792458×10⁸ | m/s |
| kB | Boltzmann constant | 1.380649×10⁻²³ | J/K |
| mH | Atomic mass unit | 1.66053906660×10⁻²⁷ | kg |
| M⊕ | Earth mass | 5.972×10²⁴ | kg |
| R⊕ | Earth radius | 6.371×10⁶ | m |
| AU | Astronomical unit | 1.495978707×10¹¹ | m |
Variable Reference
| Symbol | Meaning | Typical range | Units |
|---|---|---|---|
| vesc | Escape velocity | 10² – 10⁸ | m/s |
| G | Gravitational constant | 6.67430×10⁻¹¹ | m³·kg⁻¹·s⁻² |
| M | Body mass | 10¹⁵ – 10³⁶ | kg |
| r | Distance from centre | 10³ – 10¹⁵ | m |
| vorb | Circular orbital velocity | vesc/√2 | m/s |
| g | Surface gravity | GM/r² | m/s² |
What Does vesc = √(2GM / r) Measure?
Escape velocity is the minimum speed an object must have at a given distance from a gravitating body to escape its gravitational influence entirely — reaching infinity with zero residual speed and no further propulsion. It is a fundamental quantity in orbital mechanics, planetary science, and astrophysics, and it determines everything from the energy required to launch a spacecraft, to which planets retain which gases in their atmospheres, to the size of black holes.
Escape Velocity Formula
The defining relationship is:
Rearranged to solve for the other two unknowns:
r = 2GM / vesc²
The circular orbital velocity at the same distance is vesc / √2:
The Schwarzschild radius — the radius at which escape velocity equals c — is:
Escape Velocity Variables
| Symbol | Meaning | Notes |
|---|---|---|
| vesc | Escape velocity | Minimum speed to escape to infinity |
| G | Gravitational constant | 6.67430×10⁻¹¹ m³·kg⁻¹·s⁻² |
| M | Body mass | Mass enclosed within radius r |
| r | Distance from centre | Radius at the point of evaluation |
| vorb | Circular orbital velocity | vesc / √2 ≈ 0.707 vesc |
| rs | Schwarzschild radius | 2GM/c², event horizon of a black hole |
How Each Variable Affects Escape Velocity
Escape velocity scales with the square root of mass and inversely with the square root of distance. Doubling the body mass increases escape velocity by a factor of √2 ≈ 1.414. Doubling the distance reduces it by the same factor. This inverse-square-root dependence means that escape velocity falls off slowly with altitude — at 4× Earth's radius, escape velocity is still 11.186 / 2 = 5.59 km/s. It only reaches zero at infinity.
The formula assumes the mass M is concentrated within radius r and is spherically symmetric. For a point inside a uniform sphere, only the mass enclosed within r contributes, and the formula must be modified. For most astronomical applications — planets, moons, stars — the surface radius is used, where M is the full body mass.
Escape Velocity Calculation Examples
vesc = √(2 × 6.674×10⁻¹¹ × 5.972×10²⁴ / 6.371×10⁶)
vesc = √(7.971×10¹⁴ / 6.371×10⁶) = √(1.251×10⁸)
vesc = 11,186 m/s = 11.186 km/s
r = 2GM / vesc² = (2 × 6.674×10⁻¹¹ × 5.972×10²⁴) / (11186)²
r = 7.971×10¹⁴ / 1.251×10⁸ = 6.371×10⁶ m = Earth's surface radius
M = vesc² · r / (2G) = (6.175×10⁵)² × 6.957×10⁸ / (2 × 6.674×10⁻¹¹)
M = 1.989×10³⁰ kg = 1 solar mass
vorb = vesc / √2 = 11,186 / 1.4142 = 7,910 m/s = 7.91 km/s
This is the speed of a low-Earth-orbit satellite.
Frequently Asked Questions
Escape velocity depends only on the gravitating body (through M and r) because the object's own mass cancels out of the energy balance. The kinetic energy needed is ½mv², and the gravitational potential energy to be overcome is GMm/r. Setting them equal: ½mv² = GMm/r, and m cancels, giving v = √(2GM/r). This means a feather and a rocket need the same escape velocity — though in practice the rocket needs far more energy to overcome atmospheric drag and to achieve that speed.
Yes, and many things do. Spacecraft on interplanetary missions exceed Earth's escape velocity by a wide margin — the Voyager probes left the Solar System at about 17 km/s relative to the Sun. What matters is that the object reaches escape velocity at some point with no further propulsion. A rocket that continues to fire its engines can reach infinity at any speed. Conversely, an object moving slower than escape velocity but with continuous thrust can also escape — the formula only applies to ballistic (unpowered) trajectories.
At the event horizon (Schwarzschild radius), the escape velocity equals the speed of light. For a black hole of mass M, rs = 2GM/c². For the Sun's mass, rs ≈ 2.95 km. For Earth's mass, rs ≈ 8.87 mm. Inside the event horizon, escape velocity exceeds c, and no signal — not even light — can escape. This is the defining property of a black hole. The Newtonian formula vesc = √(2GM/r) happens to give the correct Schwarzschild radius when vesc = c, even though a full general-relativistic treatment is required for a rigorous derivation.
A planet retains a gas if its escape velocity is much greater than the mean thermal speed of the gas molecules, which is vth = √(3kBT/m). A useful rule of thumb is that the planet retains the gas if vesc ≥ 6vth. For Earth (vesc = 11.2 km/s), this means hydrogen (vth ≈ 1.9 km/s at 300 K) is only marginally retained, and indeed Earth has lost most of its hydrogen over geological time. Heavier gases like N₂ and O₂ are firmly retained. For the Moon (vesc = 2.4 km/s), even heavy gases escape, which is why the Moon has no atmosphere.
Yes — escape velocity is a scalar magnitude, independent of direction. An object launched horizontally at escape velocity will escape just as surely as one launched vertically, provided it doesn't hit the ground first. In practice, launching horizontally is far more energy-efficient because the vehicle can reach orbital velocity without first climbing out of the atmosphere. This is why real launch vehicles begin pitching over shortly after liftoff, gradually transitioning from vertical to nearly horizontal flight.
Orbital velocity is the speed needed to maintain a circular orbit at distance r: vorb = √(GM/r). Escape velocity is the speed needed to leave the gravitational field entirely: vesc = √(2GM/r). The ratio is always vesc / vorb = √2 ≈ 1.414. So for Earth at the surface, vorb = 7.91 km/s and vesc = 11.19 km/s. The difference in energy is exactly a factor of 2 — escaping requires twice the kinetic energy of orbiting at the same altitude.
Exactly the speed of light, c = 299,792,458 m/s, by definition. The Schwarzschild radius rs = 2GM/c² is the radius at which the Newtonian escape velocity equals c. Inside rs, the Newtonian formula would give vesc > c, which is physically impossible — a signal of the breakdown of Newtonian gravity. A full general-relativistic treatment shows that no timelike or null geodesic can escape from within rs, which is the rigorous statement that a black hole is black.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| v_esc | Escape velocity | m/s |
| G | Gravitational constant | |
| M | Mass of body | kg |
| r | Distance from center | m |
What it means
Escape velocity is the minimum initial speed required for an object to break free from the gravitational influence of a massive body (like a planet or star) without any additional thrust. It is derived from the conservation of energy, balancing kinetic energy against gravitational potential energy. For Earth, v_esc ≈ 11.2 km/s. This concept is essential for launching spacecraft, for understanding planetary atmospheres (gases with molecular speeds exceeding escape velocity are lost), and for studying the formation of celestial bodies. In astrophysics, it determines the size of a black hole’s event horizon (Schwarzschild radius). Understanding escape velocity is critical for astronauts, engineers, and planetary scientists to design missions and to interpret atmospheric evolution.
Worked example
Escape Velocity – Two Detailed Examples
Real‑World| Parameter | Value |
|---|---|
| G (m³/kg·s²) | 6.67430e-11 |
| M (kg) | 5.972e24 |
| r (m) | 6.371e6 |
| Parameter | Value |
|---|---|
| G | 6.67430e-11 |
| M | 7.342e22 |
| r | 1.737e6 |
Common mistakes
- Escape velocity: The minimum speed to escape a gravitational field.
- Units: G in m³/(kg·s²), M in kg, r in metres → v in m/s.
- r: The distance from the centre of the mass – not the surface radius if starting from altitude.
- Atmospheric effects: Does not account for air resistance – applies to vacuum.
- Energy conservation: Derived from kinetic = gravitational potential – ensure sign conventions.
Applications
Escape velocity, v_esc = √(2GM/r), is the minimum speed required for an object to break free from the gravitational pull of a celestial body without further propulsion. This concept is critical for space exploration: rockets must reach Earth's escape velocity (about 11.2 km/s) to leave our planet. Astronomers use it to assess whether a planet can retain an atmosphere – if the average molecular speed exceeds escape velocity, the atmosphere will be lost over time. By calculating escape velocity, engineers design launchers and planetary missions. It also helps in understanding the formation of planetary systems. This formula is a cornerstone of astrophysics, relating mass, radius, and gravitational binding.
- Spacecraft launch and interplanetary mission planning
- Atmospheric retention and evolution studies (planetary habitability)
- Determination of black hole event horizons (Schwarzschild radius)
- Design of gravitational assist manoeuvres
- Understanding of celestial body mass and size
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