Presets
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Approach
In the planet frame the speed before and after is the same, only the direction changes. In the Sun frame the planet's orbital velocity is added, so a turn can raise or lower the speed.
Planet and solar tide
Model checks
The science
What is modelled
A massless probe approaches a planet on a circular, coplanar orbit around the Sun. Motion is integrated in the planet-centred frame with the planet's gravity and, optionally, the difference between the Sun's pull on the probe and on the planet (the tidal term). Because the planet itself moves, the probe's speed and energy relative to the Sun change during the encounter although they do not change in the planet frame.
Equations
- Planet-frame acceleration:
a = -μ r/r³ + [ g(rp + r) - g(rp) ], withg(x) = -μSun x/|x|³. - Two-body hyperbola: periapsis
q = b² / (√(b² + (μ/v∞²)²) + μ/v∞²)and turn angleδ = 2 arctan(μ / (b v∞²)). - Energy about the Sun:
ε = v²/2 - μSun/r; the largest possible speed change is 2 v∞.
Checks run in the page
- Closest approach against the analytic periapsis.
- Conservation of the two-body energy along the path.
- The measured turn angle against 2 arctan(μ/(b v²)), allowing for the finite start distance.
- The speed change seen from the Sun stays below 2 v∞.
The checks switch the solar tide off so that the exact two-body formulas apply.
Honest simplifications
Circular coplanar planet orbit, a massless probe, no atmosphere, no oblateness, no other planets or moons, no spacecraft thrust. The top view draws the in-plane motion only (the 3D page also allows an inclined approach).
What to try
- Compare Jupiter pass behind and pass in front: the probe gains speed on one side of the planet and loses it on the other.
- Move the impact parameter close to zero: the turn grows until the probe hits the clouds.
- Choose Earth: the same geometry bends the path far less, because its gravity is much weaker.