Gravity Assist Velocity Rotation

Rotate a planet-frame excess-velocity vector and transform it back to the heliocentric frame under an ideal unpowered flyby.

Description

An ideal unpowered gravity assist preserves the magnitude of planet-relative excess velocity while rotating its direction.

An ideal unpowered gravity assist preserves the magnitude of planet-relative excess velocity while rotating its direction. Adding the planet's velocity can nevertheless change heliocentric speed and energy.^[1]

Rotate a planet-frame excess-velocity vector and transform it back to the heliocentric frame under an ideal unpowered flyby.

When to use Gravity Assist Velocity Rotation

Use this tool to rotate a planet-frame excess-velocity vector and transform it back to the heliocentric frame under an ideal unpowered flyby.

Prepare the orbital state or model parameters in one reference frame. Confirm compatible distance/time units and the stated angular and gravitational-parameter conventions.

How to use Gravity Assist Velocity Rotation and understand its results

Prepare the orbital state or model parameters in one reference frame. Confirm compatible distance/time units and the stated angular and gravitational-parameter conventions.
Enter incomingPlanetRelativeVelocity, planetHeliocentricVelocity, turnAxis, turnAngleDegrees using the field table and calculation rules, then run the tool.
Read outgoingHeliocentricVelocity, heliocentricSpeedGain, specificEnergyChange. The explanation below defines the units, signs, ordering and any flags or placeholder values.
If the tool reports an input issue, correct the values and run it again. Apply the stated assumptions when interpreting a successfully returned result.

Input field	Meaning	Value type; see rules below	Requirement
incomingPlanetRelativeVelocity	Incoming planet relative velocity	list of number	Required
planetHeliocentricVelocity	Planet heliocentric velocity	list of number	Required
turnAxis	Turn axis	list of number	Required
turnAngleDegrees	Turn angle degrees	number	Required

Enter finite three-component velocity vectors in one frame, a nonzero three-component turn axis and a signed turn within ±180 degrees. Rodrigues rotation normalizes the axis. The tool returns outgoing heliocentric velocity, outgoing-minus-incoming heliocentric speed and signed kinetic energy per mass change, using squared velocity units for energy.

vOutHelio = rotate(vInPlanet,axis,turn) + vPlanetHelio

Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.

Output field	Meaning	Value type
outgoingHeliocentricVelocity	Outgoing heliocentric velocity	list of number
heliocentricSpeedGain	Heliocentric speed gain	number
specificEnergyChange	Specific energy change	number

json

{
  "input": {
    "incomingPlanetRelativeVelocity": [
      1,
      0,
      0
    ],
    "planetHeliocentricVelocity": [
      0,
      1,
      0
    ],
    "turnAxis": [
      0,
      0,
      1
    ],
    "turnAngleDegrees": 90
  },
  "output": {
    "outgoingHeliocentricVelocity": [
      6.123233995736766e-17,
      2,
      0
    ],
    "heliocentricSpeedGain": 0.5857864376269049,
    "specificEnergyChange": 1
  }
}

Limitations and assumptions

The turn is supplied, not derived from encounter parameters. It does not check B-plane geometry, periapsis clearance or atmospheric effects. All vectors must share axes and velocity units.

Numeric inputs must be finite. Follow the calculation-specific rules for overflow, underflow and exceptional cases. Lists are limited to 100,000 entries; the rules above may require smaller limits. Missing required fields, extra fields, wrong value types and violations of the calculation rules are input errors.

Alternative or Complementary methods

Use Impact Parameter Flyby Turn: Compute ideal hyperbolic flyby turn and periapsis from impact parameter and excess speed, with cancellation-resistant periapsis arithmetic. For a complementary task, use Powered Flyby Velocity Budget: Budget a local powered-flyby velocity vector change and specific kinetic-energy change from supplied periapsis speeds and angle; no B-plane trajectory is solved.

References

Gravity Assist Velocity Rotation — tool.io

Similar or alternative tools

Impact Parameter Flyby Turn
Compute ideal hyperbolic flyby turn and periapsis from impact parameter and excess speed, with cancellation-resistant periapsis arithmetic.
Powered Flyby Velocity Budget
Budget a local powered-flyby velocity vector change and specific kinetic-energy change from supplied periapsis speeds and angle; no B-plane trajectory is solved.