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Aberration: where a star at angle theta in the star frame appears in the ship frame
Alpha Centauri (4.37 ly) flip-and-burn for accelerations from 0.25 g to 5 g: ODE integration and bisection (dots) against the closed form (lines).
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How to use
Fly a starship at a large fraction of the speed of light and see what special relativity does to the night sky. Pick a preset (cruise speeds, or a real 1 g trip to Alpha Centauri), then change the camera or drag the sliders. Space pauses, R restarts.
- Try 1: aberration. Cruise at 0.5c, then 0.9c. Every star appears to slide toward the direction you are flying: at 0.9c half of all stars crowd into a cone about 25 degrees wide ahead.
- Try 2: colour and brightness. Switch the Doppler colour or brightness off and on. Stars ahead turn blue-white and brighter (D greater than 1), stars astern turn red and fade; at high speed the rear sky goes dark.
- Try 3: all-sky map. Camera All-sky map: ahead is the centre, the edge is straight astern. The dashed orange circle is the headlight cone of half-angle acos(beta), the star-frame cone that contains half of everything the ship radiates.
- Try 4: time dilation. Preset 1 g to Alpha Centauri: ship clock 3.6 years against 6.0 years on Earth, with a peak speed of 0.95c. Press Run experiment to repeat it for other accelerations with numerical integration.
- Try 5: from outside. Camera Outside: the ship is shortened by 1/gamma and passes through the lattice of gates. The faint cone ahead is the headlight beam.
Numerical checks run in the page; they are listed below the graphs.
The upper panel is the sky seen from the ship (azimuthal map, ahead at the centre, straight astern at the edge); the lower panel is a Minkowski diagram of the flight with the ship worldline, the light cone and the ship simultaneity line. Buttons choose which panel fills the canvas.
The science
Two inertial frames: the star frame S (the stars and gates are at rest) and the ship frame S' moving at speed v = βc along the line of flight. A source seen in S at angle θ from the direction of motion is seen from the ship at θ'.
cosθ' = (cosθ + β) / (1 + β cosθ) (relativistic aberration)
D = ν'/ν = γ(1 + β cosθ) = 1 / (γ(1 - β cosθ')) γ = 1/√(1 - β²)
Solid angle and brightness. The map squeezes the sky: dΩ'/dΩ = 1/D². The ratio Iν/ν³ of specific intensity to frequency cubed is invariant, so the bolometric surface brightness scales as D&sup4; and the flux from an unresolved point source such as a star as D². Stars are drawn with flux × D² and the colour of a blackbody at temperature D×T. Lit surfaces (planets, station) use a tint of a 5800 K blackbody shifted by D and brightness D3.5, a compromise between the band-limited D³ and the bolometric D&sup4;. This is an approximation: a real camera would integrate each object's spectrum through the eye's response.
Sky fractions. Stars at θ = 90 degrees in S appear at cosθ' = β, so the fraction of the sky visible in the ship's forward hemisphere is (1 + β)/2 and half of all stars lie inside the forward cone of half-angle acos β. The same angle bounds the headlight effect: half of the light an isotropic lamp on the ship emits ends up inside a star-frame cone of half-angle acos β.
Constant proper acceleration. With acceleration a felt aboard (1 g = 9.81 m/s² = 1.032 ly/yr²), rapidity φ = aτ/c and
β = tanh φ t = (c/a) sinh φ x = (c²/a)(cosh φ - 1) γ = cosh φ
A flip-and-burn trip over distance d accelerates to d/2 and brakes at the same rate: τtotal = (2c/a) acosh(1 + ad/2c²) and ttotal = (2c/a) sinh(τtotala/2c). For Alpha Centauri (4.37 ly) at 1 g this gives 3.58 years aboard, 6.00 years in the star frame, peak speed 0.952c.
How the picture is made. The 3D scene is the rest-frame scene seen from the ship position. A vertex shader takes each vertex direction, applies the aberration map to its angle from the direction of flight (keeping its distance and azimuth) and projects it with a perspective, a 180-degree or a 360-degree azimuthal camera; the fragment shader applies the Doppler tint and brightness. That is exactly what a camera at rest would see in a static scene, observed by a moving viewer; no light-travel-time effect of moving objects is needed because everything in the scene is at rest in S. Meshes are subdivided so straight edges bend correctly, though long flat triangles are only approximated. The cockpit frame is drawn without any transform (it is at rest in S').
Honest simplifications. Stars are a random catalogue at infinity with invented temperatures and brightness, not real star positions; there is no parallax, no real spectra, no eye model and no dark-adaptation. The outside views show the instantaneous length contraction of the ship by 1/γ; an actual photograph would show the Terrell-Penrose rotation instead. Gates, station, planets and the track length are a compressed diorama (the track is 1000 scene units whatever the mission distance); the readouts (time, distance) are the physical values. Interstellar gas, radiation damage and the energy needed (kinetic energy (γ - 1)mc²) are not modelled.
Numerical checks in this page. (1) RK4 integration of dβ/dt = a(1 - β²)3/2 against the closed forms for ship time and distance; (2) the Alpha Centauri trip: integrated half-way distance and the 3.6 / 6.0 year values; (3) the aberration formula and Doppler factor against an explicit Lorentz boost of a photon four-momentum; (4) the solid-angle ratio against a finite-difference Jacobian of the boosted map; (5) sampling of the sphere for the forward-sky fraction (1 + β)/2 and the headlight cone.