Escape Velocity Calculator | Planets & Stars
Calculate escape velocity, orbital speed, surface gravity, Schwarzschild radius, and orbital period for any planet, star, or celestial body. Solve for velocity, mass, or radius with solar system presets.
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What Is the Escape Velocity Calculator | Planets & Stars?
Escape velocity is the minimum speed a free-flying object must reach at the surface of a body to overcome its gravity and travel to infinity without further propulsion. It is derived directly from conservation of energy: the kinetic energy ½mv² must equal the gravitational potential energy GMm/R at the launch point. Setting them equal and solving for v gives v_e = √(2GM/R).
Crucially, escape velocity is independent of the mass or shape of the escaping object, it depends only on the mass M and radius R of the body being escaped. It also does not depend on the direction of launch: whether fired straight up or at an angle, the same speed is required (ignoring atmosphere).
The first cosmic velocity v₁ = √(GM/R) is the circular orbital speed at the surface, exactly v_e / √2. An object travelling at v₁ horizontally at the surface would orbit without falling (in the absence of air resistance). The second cosmic velocity (escape velocity) is √2 times larger because escaping to infinity requires twice the kinetic energy of a circular orbit.
The Schwarzschild radius r_s = 2GM/c² is where escape velocity equals the speed of light. Any object compressed within its own Schwarzschild radius becomes a black hole. For Earth, r_s ≈ 8.9 mm; for the Sun, r_s ≈ 2.95 km. This calculator flags when a preset body's radius approaches or falls below r_s.
All physical constants used here, G, c, M⊕, M☉, R⊕, R☉, are sourced from NIST CODATA 2018 and IAU 2012 nominal values, published at physics.nist.gov and iau.org. Full-precision values are embedded in the calculator with no rounding until the final display step.
Formula
| Symbol | Name | Description |
|---|---|---|
| v_e | Escape velocity | Minimum speed to escape gravity to infinity; m/s or km/s |
| G | Gravitational constant | 6.674 30 × 10⁻¹¹ N·m²·kg⁻² (NIST CODATA 2018) |
| M | Mass of body | Total mass of the planet, star, or object; kilograms [kg] |
| R | Radius | Distance from centre of mass to launch point; metres [m] |
| v₁ | First cosmic velocity | Circular orbital speed at surface = v_e / √2; km/s |
| g | Surface gravity | Gravitational acceleration at surface = GM/R²; m/s² |
| r_s | Schwarzschild radius | Event horizon radius = 2GM/c²; a body with R < r_s is a black hole |
| T | Orbital period | Time for a circular orbit at surface = 2π√(R³/GM); seconds |
| c | Speed of light | 299 792 458 m/s (exact, defined 2019 SI) |
How to Use
- 1Select solve-for mode: Choose which quantity to calculate: escape velocity v_e, mass M, or radius R. The mode tabs are at the top of the calculator.
- 2Load a preset (optional): Click any solar system body preset (Earth, Moon, Mars, Jupiter, Sun, Neutron Star, etc.) to auto-fill published mass and radius values from IAU 2012 data.
- 3Enter known values: Type the mass and radius (or velocity, depending on mode). Use the unit dropdowns to select kg / M⊕ / M☉ for mass and m / km / R⊕ / R☉ for radius.
- 4Press Calculate or Enter: Click "Calculate" or press Enter while focused on any input field. All results are computed instantly.
- 5Read primary result: The main result card shows escape velocity in km/s and as a fraction of the speed of light. For M and R modes, the primary result is shown in multiple unit scales.
- 6Check derived quantities: The six secondary cards show first cosmic velocity v₁, surface gravity g, orbital period T, Schwarzschild radius r_s, and the input mass and radius for reference.
- 7Note context badge: A colour-coded badge classifies the body (small body, icy dwarf, rocky planet, gas giant, stellar, compact object). A warning appears if the radius is near or below the Schwarzschild radius.
- 8Expand step trace: Click "Show calculation steps" to see every intermediate value and which NIST constant was used, useful for checking against hand calculations.
Example Calculation
Example 1: Earth escape velocity
Calculate Earth's escape velocity using IAU 2012 nominal values: M = 5.9722 × 10²⁴ kg, R = 6.371 × 10⁶ m.
Example 2: Moon escape velocity
The Moon has M = 7.342 × 10²² kg and R = 1.7374 × 10⁶ m. Apollo crews needed to exceed this to return to Earth.
Example 3: Neutron star, near relativistic escape
A typical neutron star: M ≈ 1.4 M☉ = 2.785 × 10³⁰ kg, R = 12 km. Escape velocity approaches a significant fraction of c.
Understanding Escape Velocity | Planets & Stars
What Is Escape Velocity?
Escape velocity is not a speed that needs to be maintained, it is the instantaneous speed an unpowered projectile must reach at a given point to travel to infinity against gravity. Once launched at escape velocity, the object slows continuously but never stops: its kinetic energy precisely equals the gravitational potential energy it must climb, reaching zero velocity only at infinite distance.
The concept was first formalised by John Michell in 1783, who noted that a sufficiently massive star could have an escape velocity exceeding the speed of light, what we now call a black hole, a century and a half before general relativity gave us the rigorous description.
First vs. Second Cosmic Velocity
- ›First cosmic velocity (v₁): The minimum orbital speed for a circular orbit just above the surface, v₁ = √(GM/R). For Earth: ≈ 7.91 km/s. A spacecraft at this speed is in free fall around the planet.
- ›Second cosmic velocity (v₂ = v_e): Escape velocity, √(GM/R) × √2 ≈ 11.19 km/s for Earth. A spacecraft launched at this speed with no further thrust will escape Earth gravity entirely.
- ›Third cosmic velocity: The speed needed to escape the Solar System from Earth's distance, approximately 16.7 km/s relative to Earth (42.1 km/s from the Sun minus Earth's orbital speed of 29.8 km/s, considering direction).
- ›The ratio v_e / v₁ = √2 is exact and independent of the body's mass or radius, a universal geometric consequence of gravity in three dimensions.
Solar System Comparison
| Body | Mass (kg) | Radius (km) | v_e (km/s) | g (m/s²) | v/c (%) |
|---|---|---|---|---|---|
| Mercury | 3.30 × 10²³ | 2,440 | 4.25 | 3.70 | 0.0142 |
| Venus | 4.87 × 10²⁴ | 6,052 | 10.36 | 8.87 | 0.0345 |
| Earth | 5.97 × 10²⁴ | 6,371 | 11.19 | 9.82 | 0.0373 |
| Moon | 7.34 × 10²² | 1,737 | 2.38 | 1.62 | 0.0079 |
| Mars | 6.42 × 10²³ | 3,390 | 5.03 | 3.72 | 0.0168 |
| Jupiter | 1.90 × 10²⁷ | 71,492 | 59.5 | 24.8 | 0.199 |
| Saturn | 5.68 × 10²⁶ | 60,268 | 35.5 | 10.4 | 0.118 |
| Sun | 1.99 × 10³⁰ | 695,700 | 617.5 | 274 | 2.06 |
| Neutron Star | 2.79 × 10³⁰ | 12 | ~176,000 | ~1.3 × 10¹² | ~58.7 |
Schwarzschild Radius and Black Holes
The Schwarzschild radius r_s = 2GM/c² is the radius at which, in general relativity, the escape velocity equals the speed of light. Any mass compressed below this radius forms a black hole with an event horizon from which nothing, not even light, can escape. For context:
- ›Earth: r_s ≈ 8.87 mm (Earth would need to be compressed to the size of a marble)
- ›Sun: r_s ≈ 2.95 km (compressed to about the size of a small city)
- ›Neutron star: r_s ≈ 4–6 km (close to actual neutron star radii of 10–15 km)
- ›Stellar-mass black holes: r_s ≈ 3–30 km for 1–10 M☉ remnants
Rocket Equation Context
Escape velocity is not the same as the Δv needed to reach orbit, launching through an atmosphere requires overcoming drag losses (≈ 1.5–2 km/s for Earth) and gravity losses during ascent. Real rockets to low Earth orbit need about 9.4 km/s of Δv; trans-lunar injection requires about 3.1 km/s more. Escape from Earth's sphere of influence adds another ≈ 0.8 km/s. The Tsiolkovsky rocket equation Δv = v_e · ln(m₀/m_f) governs how much propellant is needed to achieve any target Δv.
Physical Constants Used
This calculator uses constants from NIST CODATA 2018 (physics.nist.gov) and IAU 2012 nominal solar and planetary values (iau.org):
- ›G = 6.674 30 × 10⁻¹¹ N·m²·kg⁻² (NIST CODATA 2018, relative uncertainty 2.2 × 10⁻⁵)
- ›c = 299 792 458 m/s (exact, defined in 2019 SI redefinition)
- ›M⊕ = 5.9722 × 10²⁴ kg, R⊕ = 6.3781 × 10⁶ m (IAU 2012 nominal values)
- ›M☉ = 1.9885 × 10³⁰ kg, R☉ = 6.957 × 10⁸ m (IAU 2012 nominal values)
- ›All preset body data from IAU 2012 or published USGS/NASA planetary fact sheets
Frequently Asked Questions
What is escape velocity and why does it not depend on the object's mass?
Escape velocity is the minimum launch speed needed for an unpowered object to reach infinity against a body's gravity. It is derived from energy conservation:
- • Setting KE = PE: ½mv² = GMm/R
- • The escaping object's mass m cancels on both sides, leaving v = √(2GM/R)
- • A grain of sand and a spacecraft need the same launch speed, what differs is the fuel required to accelerate them to that speed.
- • Direction also does not matter: a perfectly horizontal launch at escape velocity will spiral outward and escape, just as efficiently as a vertical shot (ignoring atmosphere).
What is the difference between the first and second cosmic velocity?
Both describe minimum speeds for different gravitational feats from the surface of a body:
- • First cosmic velocity v₁: √(GM/R), horizontal speed for a circular orbit at the surface. For Earth: ≈ 7.91 km/s. Below this, a projectile will fall back.
- • Second cosmic velocity v₂: √(2GM/R) = v₁ × √2, escape velocity. For Earth: ≈ 11.19 km/s. At this speed the object escapes Earth's gravity well entirely.
- • The factor √2 between them is exact and universal, derived from the geometry of orbits and the mathematics of conic sections (circular orbit vs. parabolic escape trajectory).
How do I calculate mass from escape velocity?
Rearrange the escape velocity formula to isolate M:
- • Start: v_e = √(2GM/R)
- • Square both sides: v_e² = 2GM/R
- • Multiply both sides by R: v_e² · R = 2GM
- • Divide by 2G: M = v_e² · R / (2G)
Use this to estimate the mass of a planet or star from its observed escape velocity (measured by tracking spacecraft trajectories) and radius. Select "Solve for M" mode in the calculator above.
Why is Jupiter's escape velocity so much higher than Earth's?
Escape velocity scales as √(M/R), so both mass and radius matter:
- • Jupiter is 317× more massive than Earth, but also 11.2× wider in radius.
- • Net effect: √(317/11.2) ≈ √28.3 ≈ 5.3× larger escape velocity.
- • Earth: ≈ 11.2 km/s × 5.3 ≈ 59.5 km/s for Jupiter, consistent with the actual value.
- • Saturn, despite being 95× Earth's mass, has a lower escape velocity than Jupiter because its much larger radius (9.4× Earth) offsets the mass advantage.
What is the Schwarzschild radius and how does it relate to escape velocity?
The Schwarzschild radius is derived by setting the Newtonian escape velocity equal to the speed of light and solving for R:
- • Set v_e = c: c = √(2GM/R)
- • Solve: r_s = 2GM / c²
- • In full general relativity, r_s is the event horizon of a Schwarzschild (non-rotating) black hole, the radius from which no light can escape.
- • Remarkably, the Newtonian formula gives the correct result, even though the physics of black holes requires general relativity, a fortunate numerical coincidence.
- • This calculator warns you when a body's actual radius is close to or below r_s, indicating you are in the black hole regime.
How accurate is the Newtonian escape velocity formula for neutron stars?
The Newtonian formula v_e = √(2GM/R) becomes inaccurate when v_e approaches a significant fraction of c:
- • For Earth (v_e ≈ 0.004% of c) the Newtonian result is accurate to better than 1 part in 10⁸.
- • For white dwarfs (v_e ≈ 1–2% of c) the error is at the ~0.01% level, still negligible.
- • For neutron stars (v_e ≈ 50–80% of c) the relativistic correction is large: the general-relativistic escape speed from the surface is given by a different expression and can be significantly higher than the Newtonian estimate.
- • The calculator displays a note for compact objects where v_e > 10% of c, flagging where the Newtonian approximation loses validity.
Does escape velocity change with altitude?
Yes, escape velocity decreases as you move farther from the body's centre:
- • At distance r from the centre (r > R): v_e(r) = √(2GM/r)
- • At the International Space Station (altitude ≈ 410 km, r ≈ 6781 km): v_e ≈ 10.88 km/s, about 3% less than from the surface.
- • At geostationary orbit (r ≈ 42,164 km): v_e ≈ 4.35 km/s
- • At the Moon's distance (384,400 km): v_e ≈ 1.44 km/s
- • The "radius" input field in this calculator accepts any distance from the centre, set it to an orbital altitude to get the escape velocity at that point.
How is escape velocity used in real space missions?
Escape velocity calculations underpin every mission design that leaves a gravitational body:
- • Trans-lunar injection: Apollo missions needed ≈ 3.1 km/s of additional Δv beyond LEO to reach the Moon's sphere of influence, well below Earth's escape velocity because the Moon's gravity assists the transfer.
- • Interplanetary trajectories: The Voyager probes exceeded the Solar escape velocity (≈ 42.1 km/s from 1 AU) using gravity assists from Jupiter and Saturn, no single burn achieved escape directly.
- • Return missions: Lunar return required exceeding the Moon's escape velocity (2.38 km/s), one reason the Moon was a more achievable destination than Mars (5.03 km/s).
- • Planetary protection: Mission designers must ensure sample-return vehicles cannot contaminate Earth, calculating Earth's gravitational influence sphere guides trajectory design.