# Escape Velocity Calculator

> Computes the escape velocity √(2GM/r) and the circular orbital velocity √(GM/r) at the surface (or a given altitude) of Earth, the Moon, Mars, Jupiter, the Sun or a custom body defined by mass and radius, plus its surface gravity GM/R².

- Calculator id: `escape-velocity` · Category: Biology, Earth & Space (`science`) · Tool name: `calculate_escape_velocity`
- Canonical page: https://tttkmbb.com/science/escape-velocity · This document: https://tttkmbb.com/science/escape-velocity.md · JSON definition: https://tttkmbb.com/science/escape-velocity.json

## Purpose

Computes the escape velocity √(2GM/r) and the circular orbital velocity √(GM/r) at the surface (or a given altitude) of Earth, the Moon, Mars, Jupiter, the Sun or a custom body defined by mass and radius, plus its surface gravity GM/R².

**Use when:** You need the speed required to escape a planet or moon without further propulsion, the speed of a low circular orbit, or the surface gravity of a body from its mass and radius.

**Do not use when:** You need the period of an orbit of a given size (use orbital-period), the rocket delta-v including gravity and drag losses, or the weight of an object on a planet (use planet-weight).

## Input

| Parameter | Type | Unit | Required | Description |
| --- | --- | --- | --- | --- |
| `body` | enum: earth \| moon \| mars \| jupiter \| sun \| custom |  | required | Body to escape from; GM and mean radius are taken from the JPL/NASA constants. |
| `mass_kg` | number | kg | optional | Mass of a custom body; only used when body is custom. (> 0, max 1e+42) |
| `radius_m` | number | m | optional | Radius of a custom body in metres; only used when body is custom. (> 0, max 1000000000000000) |
| `altitude_m` | number | m | optional, default 0 | Height above the surface at which the speeds are evaluated (0 = surface). (min 0, max 1000000000000000) |

## Output

| Field | Type | Unit | Description |
| --- | --- | --- | --- |
| `escape_velocity_m_s` | number | m/s | √(2GM / r) with r = radius + altitude. |
| `escape_velocity_km_s` | number | km/s | Escape velocity / 1000. |
| `escape_velocity_km_h` | number | km/h | Escape velocity × 3.6. |
| `circular_orbital_velocity_km_s` | number | km/s | √(GM / r): escape velocity / √2. |
| `surface_gravity_m_s2` | number | m/s² | GM / R² at the surface (altitude 0). |
| `radius_used_km` | number | km | Radius + altitude. |
| `gm_m3_s2` | number | m³/s² | Standard gravitational parameter of the body. |

## Formula

`v_escape = √(2 × GM / r); v_orbit = √(GM / r); g = GM / R²; r = R + altitude_m; GM(custom) = 6.67430 × 10⁻¹¹ × mass_kg`

Energy conservation for a non-rotating, airless body (kinetic energy equals the gravitational binding energy). Constants: GM in m³/s² Earth 3.986004418 × 10¹⁴, Moon 4.9048695 × 10¹², Mars 4.282837 × 10¹³, Jupiter 1.26686534 × 10¹⁷, Sun 1.32712440018 × 10²⁰; mean radii 6,371 / 1,737.4 / 3,389.5 / 69,911 / 695,700 km. Real launches need extra speed for drag and gravity losses and gain from the body's rotation.

## Data Sources

- Wikipedia – Escape velocity — https://en.wikipedia.org/wiki/Escape_velocity (reference, retrieved 2026-09-24)
- NASA NSSDCA – Planetary Fact Sheet — https://nssdc.gsfc.nasa.gov/planetary/factsheet/ (government, retrieved 2026-09-24)
- NASA JPL Solar System Dynamics – Astrodynamic parameters (GM of the Sun and planets) — https://ssd.jpl.nasa.gov/astro_par.html (government, retrieved 2026-09-24)

Data freshness: `static`. Deterministic formula with fixed constants; results never go stale. Inputs supplied by the caller determine the output.

## API

- `GET https://tttkmbb.com/api/v1/calculate/escape-velocity?body=…`
- `POST https://tttkmbb.com/api/v1/calculate/escape-velocity` with JSON body `{"inputs": {…}}`
- Response: unified envelope (`success`, `request`, `result.values`, `result.units`, `sources`, `freshness`, `timestamp`, `next_actions`, `links`); see https://tttkmbb.com/docs/response-format.md
- Schema: https://tttkmbb.com/api/v1/calculators/escape-velocity · OpenAPI operationId `calculate_escape_velocity` in https://tttkmbb.com/openapi.json
- Authentication: none. Rate limit: fair use, see https://tttkmbb.com/docs/rate-limits.md.

## MCP

- Server: `https://tttkmbb.com/mcp` (Streamable HTTP, JSON-RPC 2.0, no auth)
- Tool:  `run_calculator` with `{"calculator_id": "escape-velocity", "inputs": {…}}`

## Example

- Earth, surface: inputs `{"body":"earth"}` → `{"escape_velocity_m_s":11186.1,"escape_velocity_km_s":11.186,"escape_velocity_km_h":40270,"circular_orbital_velocity_km_s":7.91,"surface_gravity_m_s2":9.82}`
- Moon, surface: inputs `{"body":"moon"}` → `{"escape_velocity_km_s":2.376,"circular_orbital_velocity_km_s":1.68,"surface_gravity_m_s2":1.625}`

```
GET https://tttkmbb.com/api/v1/calculate/escape-velocity?body=earth
```

## Limitations

You need the period of an orbit of a given size (use orbital-period), the rocket delta-v including gravity and drag losses, or the weight of an object on a planet (use planet-weight). Energy conservation for a non-rotating, airless body (kinetic energy equals the gravitational binding energy). Constants: GM in m³/s² Earth 3.986004418 × 10¹⁴, Moon 4.9048695 × 10¹², Mars 4.282837 × 10¹³, Jupiter 1.26686534 × 10¹⁷, Sun 1.32712440018 × 10²⁰; mean radii 6,371 / 1,737.4 / 3,389.5 / 69,911 / 695,700 km. Real launches need extra speed for drag and gravity losses and gain from the body's rotation. All values are computed from the formula above; no measurement or live data is involved.

## FAQ

**Why is 11.2 km/s different from the speed rockets reach?**

Escape velocity is the speed needed at the surface with no further thrust and no atmosphere; rockets accelerate gradually, lose speed to drag and gravity, and usually enter orbit (7.9 km/s at low altitude) rather than escaping.

**Why does the value change with altitude?**

Both speeds fall with distance from the centre as 1/√r: at 400 km above Earth the escape velocity is about 10.85 km/s and the orbital speed 7.67 km/s.

**How does surface gravity relate to escape velocity?**

v_escape = √(2 g R): two bodies with the same g but different radii have different escape speeds, which is why the Moon's small radius gives only 2.38 km/s.

## Related

- [Orbital Period Calculator](https://tttkmbb.com/science/orbital-period.md) — Period and speed of an orbit of any size around the same bodies.
- [Weight on Other Planets Calculator](https://tttkmbb.com/science/planet-weight.md) — Weight of an object under each body's surface gravity.
- [Kinetic Energy Calculator](https://tttkmbb.com/physics/kinetic-energy.md) — Kinetic energy needed at escape speed for a given mass.