ISS

Predicting a pass means knowing where the station will be, where that puts it relative to you, and whether you will be able to see it. The three questions are handled separately, and the third is the one where mistakes are easiest.

1. 01

   Where the station will be

   le service répond aux instants futurs, jusqu'à 24 h, balayage au pas de 60 s, 30 instants par requête

   There is no orbit propagator on board, and that is deliberate. The service accepts a request for the position at a future instant, up to twenty-four hours ahead, so the window can simply be scanned minute by minute. A whole day costs forty-nine requests, against a stated quota of three hundred and fifty per five minutes. The scan runs only at startup, at midnight, and on demand.

2. 02

   The angle between you and the point below it

   γ = 2·arcsin( √( sin²(Δφ⁄2) + cos φ₁ · cos φ₂ · sin²(Δλ⁄2) ) )

   This is the angle at the Earth's centre between your home and the point directly under the station. It uses the haversine formula rather than the direct cosine, which loses all precision when two points are close, which is exactly when the station passes overhead. A test exercises it on two points a metre apart.

3. 03

   The elevation above the horizon

   h = arctan2( cos γ − R ⁄ (R + a) , sin γ )

   This is what the eye sees, and so the quantity that decides everything. The two-argument arctangent is used rather than a division, which would give infinity at the zenith, where the sine vanishes. The device setting acts on this height, ten degrees by default: below that, a pass is swallowed by roofs and trees.

4. 04

   The two distances, which are not comparable

   d = √( R² + (R + a)² − 2 · R · (R + a) · cos γ )

   The distance to the station is the chord through space, given by the law of cosines. The ground distance is the arc travelled over the surface, R times the angle. When the station is far, the arc is longer than the chord, which is surprising on the tile but perfectly correct: at a hundred degrees apart there are nearly eleven thousand kilometres of ground for ten thousand as the crow flies.

5. 05

   The check that validates the whole geometry

   rayon de visibilité = R · arccos( R ⁄ (R + a) ) ≈ 2 257 km à 420 km d'altitude

   This is the radius of the circle from which the station stands above the horizon. The service publishes a footprint of its own, the diameter of that very circle, so the two can be compared. The measured gap is 1.34 kilometres out of 2,258, six hundredths of one per cent, which matches the choice of Earth radius, mean here and equatorial there. The geometry is thus validated without having to take the service at its word.

6. 06

   Whether the station is lit, and the trap to avoid

   la station est éclairée si γ(station, point subsolaire) < 90° + arccos( R ⁄ (R + a) )

   The service publishes a field called “visibility”, and trusting it would be a mistake. It describes the ground under the station, not the station: it answers “daylight” when it is day below, even though the station is in full sunlight and perfectly visible to a distant observer in the dark. Relying on it led to the conclusion that no pass was visible over twenty-four hours; computing the illumination from the subsolar point, which the service also supplies, brought out two passes, both in the small hours, which is indeed when the station is seen at this time of year. The Earth's shadow is treated as a cylinder, and the altitude adds some twenty degrees to the half-angle.

7. 07

   How dark your sky is

   hauteur du Soleil chez l'observateur < −6°, le crépuscule civil

   This is the condition the service does not provide and which has to be computed here: a lit station crossing a daylit sky stays invisible. The Sun's elevation comes from the NOAA equations, the same as in the Ephemerides app, and the two implementations were compared: they agree to a ten-thousandth of a degree on a value read in operation.

8. 08

   Cutting the series into passes

   un passage = les instants consécutifs où h > 0 ; retenu si son sommet atteint la hauteur minimale

   The scanned series is cut into runs of consecutive instants where the station stands above the horizon. Each run gives a pass, of which the start, the end, the peak and its azimuth are kept. Those whose peak falls short of the configured height are discarded, and a pass is declared visible only if one of its instants meets all three conditions at once. The distance card, for its part, fires only on a downward crossing: otherwise it would fire again at every reading for as long as the station stayed inside the radius.