Escape burn delta-v calculator from a circular orbit

An escape burn from a circular orbit does not require adding the full local escape speed. The spacecraft already has circular orbital velocity, so the ideal prograde impulse is the difference between escape speed and circular speed at the same radius. This calculator evaluates that difference from the central body's standard gravitational parameter and the orbit's centre-to-spacecraft radius. It reports both component speeds, the required delta-v in metres per second and kilometres per second, and the assumptions behind the result. The calculation is deterministic, uses no network or stored lookup table, and rejects a zero or negative orbital radius rather than returning an undefined physical result.

Why escape-burn delta-v is a difference between two speeds

A spacecraft in a circular orbit is already moving sideways at circular speed, given by the square root of the standard gravitational parameter divided by orbital radius. At that same location, the speed for a parabolic escape trajectory is the square root of twice the gravitational parameter divided by radius. A tangential prograde impulse raises the spacecraft's specific orbital energy from the negative value of the circular orbit to zero, the boundary between a bound ellipse and an unbound hyperbola.

Therefore the ideal escape-burn delta-v is escape speed minus circular speed, not escape speed by itself. Algebraically, this is also circular speed multiplied by the square root of two minus one. The subtraction matters: treating the spacecraft as stationary at its orbital radius would greatly overstate the propulsive requirement and ignore the useful kinetic energy it already possesses.

This capability displays circular speed and escape speed alongside the difference so that students, analysts, and software tests can inspect every important number. The result describes a zero-hyperbolic-excess departure, meaning speed approaches zero infinitely far from the central body in this two-body model. It is the minimum ideal impulsive departure, not a complete mission budget or launch-vehicle performance prediction.

Choose consistent inputs and interpret the response

Provide mu, the central body's standard gravitational parameter, in cubic metres per second squared, and radius, the distance from the body's centre to the spacecraft, in metres. Radius is not altitude above the surface. If a problem gives altitude, add the body's reference radius before using the calculator.

For Earth, a commonly used gravitational parameter is about 3.986004418 times ten to the fourteen cubic metres per second squared. A spacecraft four hundred kilometres above a reference Earth radius of roughly 6,371 kilometres therefore uses an orbital radius near 6,771 kilometres, converted to 6,771,000 metres. All numeric outputs use deterministic twelve-decimal-place rounding, although source data and the ideal model usually justify fewer significant figures.

Missing, nonnumeric, non-finite, or out-of-range inputs are rejected. The gravitational parameter must be positive, and the orbital radius must be strictly greater than zero because division by zero and a negative radial distance have no meaning in this calculation.

Understand the idealisations before using the number

The calculation assumes an instantaneous prograde burn from a perfectly circular orbit in a Newtonian two-body system. Real engines apply thrust over time, so a long burn changes position and direction while propellant is being expended and can introduce gravity losses. The result also ignores atmosphere, oblateness, rotation of the central body, perturbations from other bodies, navigation margins, steering losses, residuals, and the propellant needed for attitude control or trajectory correction.

It targets a parabolic escape with zero speed at infinity. Interplanetary missions usually require a specified positive hyperbolic excess speed, often written v-infinity, to enter a transfer trajectory after leaving the planet's sphere of influence; that departure burn is larger and needs a different equation. Likewise, this value is not the delta-v from a planetary surface, because a surface vehicle is not initially in a circular orbit and must pay ascent and loss terms.

Use this calculator for quick feasibility estimates, classroom mechanics, independent checks of orbital software, and the escape line in a preliminary delta-v budget. Then apply appropriate mission-specific modelling and margins before making engineering decisions. The browser calculation uses the same pure solver, helping repeated checks remain consistent across environments.