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I must give the endpoint of the hyperbola the Earth's velocity first. I can think of several velocities to use, the velocity of the Earth at that moment in its conic orbit around the Sun, or the velocity of a satellite in its own conic orbit around the Sun at the new distance, but there could be other options.

Assuming:

  • non-spinning SOIs - they only undergo translational motion along the planet's orbit, but direction of their axes relative to each other is always constant. That's the trivial approach; sun-synchronous orbits are impossible and tidally locked bodies will need spin equal to orbital period, instead of a flat 0, but calculations are made easier.
  • The "root" frame of reference is the Sun's SOI (Sun is assumed to be immobile).
  • We're not concerned about the motion of the craft, just motion of the conics.

The hyperbola exists in Earth's SOI and is boundstationary relative to it (speed relative to Earth translation-wise,SOI = 0); its endpoint is its part and distant stars rotation-wise. That's whatso, the craftendpoint's velocity is calculated relative to in the hyperbola trajectory. You don't spin the SOI relative"root" frame of reference will be equal to anything, ever (no daily, no synodic - axis X points towards vernal equinox always.)speed of Earth's SOI at that point - only translate it along the orbital path of the planet around the Sunthat is, Earth velocity.

That wayThis would become more complex in case of spinning SOIs, linearrequiring to trace the velocity of every point of the SOI is the sameedge relative to the Sun (and same as the planet's), so regardless of the point of escape you just add the planet's velocity, and the craft velocity, and use thatbut as initial velocity at the same point in the sun's SOI. Same approach on going "down to the planet" - substract the planet's velocity vector from the craft entering, and you have the speed in the planet's SOII understand this is not our case.

I must give the endpoint of the hyperbola the Earth's velocity first. I can think of several velocities to use, the velocity of the Earth at that moment in its conic orbit around the Sun, or the velocity of a satellite in its own conic orbit around the Sun at the new distance, but there could be other options.

The Earth's SOI is bound to Earth translation-wise, and distant stars rotation-wise. That's what the craft velocity is calculated relative to in the hyperbola trajectory. You don't spin the SOI relative to anything, ever (no daily, no synodic - axis X points towards vernal equinox always.) - only translate it along the orbital path of the planet around the Sun.

That way, linear velocity of every point of the SOI is the same relative to the Sun (and same as the planet's), so regardless of the point of escape you just add the planet's velocity, and the craft velocity, and use that as initial velocity at the same point in the sun's SOI. Same approach on going "down to the planet" - substract the planet's velocity vector from the craft entering, and you have the speed in the planet's SOI.

Assuming:

  • non-spinning SOIs - they only undergo translational motion along the planet's orbit, but direction of their axes relative to each other is always constant. That's the trivial approach; sun-synchronous orbits are impossible and tidally locked bodies will need spin equal to orbital period, instead of a flat 0, but calculations are made easier.
  • The "root" frame of reference is the Sun's SOI (Sun is assumed to be immobile).
  • We're not concerned about the motion of the craft, just motion of the conics.

The hyperbola exists in Earth's SOI and is stationary relative to it (speed relative to Earth SOI = 0); its endpoint is its part and so, the endpoint's velocity in the "root" frame of reference will be equal to speed of Earth's SOI at that point - that is, Earth velocity.

This would become more complex in case of spinning SOIs, requiring to trace the velocity of the SOI edge relative to Sun, but as I understand this is not our case.

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SF.
SF.
  • 56k
  • 12
  • 184
  • 347

I must give the endpoint of the hyperbola the Earth's velocity first. I can think of several velocities to use, the velocity of the Earth at that moment in its conic orbit around the Sun, or the velocity of a satellite in its own conic orbit around the Sun at the new distance, but there could be other options.

The Earth's SOI is bound to Earth translation-wise, and distant stars rotation-wise. That's what the craft velocity is calculated relative to in the hyperbola trajectory. You don't spin the SOI relative to anything, ever (no daily, no synodic - axis X points towards vernal equinox always.) - only translate it along the orbital path of the planet around the Sun.

That way, linear velocity of every point of the SOI is the same relative to the Sun (and same as the planet's), so regardless of the point of escape you just add the planet's velocity, and the craft velocity, and use that as initial velocity at the same point in the sun's SOI. Same approach on going "down to the planet" - substract the planet's velocity vector from the craft entering, and you have the speed in the planet's SOI.