Presets
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Sail
Sail loading and target.
Steering
Cone angle of each competing sail.
Layers
Model checks
The science
Sail force and heliocentric motion
Photon pressure on an ideal flat, perfectly reflecting sail at cone angle α (between the sail normal and the Sun line) pushes along the sail normal n, with magnitude falling as the inverse square of the distance, like gravity. Defining the lightness number β = Frad/Fgrav at any distance:
d2r/dt2 = − μ r/r3 + β μ cos2α n / r2, β = 2 S1AU / (c σ μ/AU2) ≈ 1.53 / σ [g/m2]A prograde tilt (positive α) adds energy and spirals the sail outward; a retrograde tilt removes angular momentum and spirals it inward. The tangential part of the force, ∝ sinα cos2α, peaks at α = 35.26 deg. A purely radial sail (α = 0) just reduces the effective gravity to μ(1 − β).
The page integrates all three sails with RK4 (step 0.002 yr) and shows the arrival time at the target radius, the specific energy and the speed.
Honest simplifications
- Ideal flat reflective sail; real sails billow, absorb part of the light and age in the UV.
- Circular coplanar planet orbits are drawn only as references; planets do not attract the sails and no rendezvous is computed.
- The cone angle is constant for each sail (no optimal steering); good Mars transfers use a time-varying angle.
- Relativistic and photon-pressure-gradient effects are ignored.
What to try
- Run Spiral out to Mars: sails at different cone angles arrive at different times; the best angle is near 35 degrees only for the tangential force, not for the whole transfer.
- Run Radial sail: the sail starts at the circular speed of full gravity but feels only (1 - beta) of it, so it swings out on a wide ellipse or escapes.
- Raise beta to 0.5 and watch the spiral open up.
- Look at the fourth graph: the force falls with cos2, so steep angles give little thrust.