CALC-140

Eclipses of a geostationary satellite: season length, longest eclipse and shadow time per year

REGISTERED 2026-10-15. Drafted for CLM-0895 (lessons records draft I, commit 9e851b21), checked by an independent reader (FIX, all applied, round 2 PASS with VER ids at landing), landed at SBSP-coord's request. Script: lessons-geo-eclipse.py. verified: VER-2391 CONFIRMED 2026-10-15.

Feeds CLM-0895. Basis: minutes; days; per cent of a tropical year of 365.2422 days (8,765.8 h).

Why it was needed. CLM-0895 gives about 77 hours of full shadow a year (0.88 per cent) and about 88 hours with partial shadow (1.0 per cent). No CALC held the working, and CALC-099's own annual figure (0.58 per cent) used a rough season length and a rough half-maximum average. This record computes the season, the longest eclipse and the annual total from the geometry.

Inputs.

Model. The satellite moves on a circle of 42,164 km in the equatorial plane at one turn per sidereal day. In the base case the Sun moves at a uniform rate along the ecliptic, at its mean distance. Four shadow tests are made at each step: - cylinder: the satellite is behind the Earth and nearer than 6,378.137 km to the Earth-Sun line (the Sun as a point at infinity); - umbra (full shadow): the Earth's disc, seen from the satellite, hides the whole of the Sun's disc; - any shadow (umbra plus penumbra): the Earth's disc hides some part of the Sun's disc; - weighted: the fraction of the Sun's disc that is hidden, for a disc of uniform brightness.

Working, closed form. Seen from GEO, the Earth's angular radius is rho = asin(6378.137 / 42164) = 8.7005 deg and the Sun's is s = asin(695700 / 149597870.7) = 0.2665 deg. Relative to the Sun, the satellite turns at 360 deg per mean solar day, 15.000 deg per hour. An eclipse can occur only while the Sun's declination is less than the shadow's angular half-width a: a = rho for the cylinder, rho - s for the umbra and rho + s for any shadow. sin(declination) = sin(23.4 deg) x sin(longitude), so the season spans 2 x asin(sin a / sin 23.4 deg) deg of the Sun's longitude, at 360 / 365.2422 deg a day.

case a (deg) longest = 2a / 15 deg/h season half-width (deg) season (days) cylinder 8.7005 69.60 min 22.389 45.43 umbra 8.4341 67.47 min 21.673 43.98 any shadow 8.9670 71.74 min 23.108 46.89

The same shadow in kilometres, across the anti-Sun line at the GEO radius (similar triangles from the Sun's and the Earth's radii and the Earth-Sun distance):

umbra cone length 149,597,870.7 x 6378.137 / (695,700 - 6378.137) = 1,384,195 km umbra radius at GEO 6378.137 - 42164 x (695,700 - 6378.137) / 149,597,870.7 = 6,183.9 km penumbra radius at GEO 6378.137 + 42164 x (695,700 + 6378.137) / 149,597,870.7 = 6,576.0 km as angles from GEO asin(6183.9 / 42164) = 8.4335 deg -> 67.47 min; asin(6576.0 / 42164) = 8.9726 deg -> 71.78 min

So the umbra at GEO is about 194 km (3 per cent) narrower in radius than the Earth, and the cylinder of the Earth's full radius lies between the umbra and the penumbra.

On a given day the eclipse lasts 2 x acos(cos a / cos declination) / 15 h. For small angles this is the longest eclipse x sqrt(1 - (declination / a)^2). Its mean over a season is pi/4 = 0.785 of the longest, which gives a cross-check on the annual totals below:

cylinder 2 x 45.43 d x 0.7854 x 69.60 min = 82.8 h umbra 2 x 43.98 d x 0.7854 x 67.47 min = 77.7 h any shadow 2 x 46.89 d x 0.7854 x 71.74 min = 88.1 h

Working, simulation (script output, transcribed). One tropical year from the June solstice, in 5 s steps. The season span runs from the start of the first eclipse to the end of the last, so it is one day shorter than the number of eclipses.

base case: circular Earth orbit, tilt 23.4 deg, Earth radius 6378.137 km cylinder 91 eclipses a year; March 45 in 44.00 d, Sept 46 in 45.00 d; longest 69.6 min; 82.4 h = 0.940 per cent umbra 88 eclipses a year; March 44 in 43.00 d, Sept 44 in 43.00 d; longest 67.5 min; 77.4 h = 0.883 per cent any shadow 94 eclipses a year; March 47 in 46.00 d, Sept 47 in 46.00 d; longest 71.8 min; 87.7 h = 1.001 per cent weighted 82.5 h = 0.941 per cent

sensitivity cylinder umbra any shadow weighted eccentric Earth orbit, e 0.0167 82.4 h 0.940% 77.4 h 0.883% 87.6 h 1.000% 82.4 h 0.940% (any shadow: March 46 eclipses, Sept 48) tilt 23.4393 deg 82.2 h 0.938% 77.2 h 0.881% 87.6 h 0.999% 82.3 h 0.939% Earth radius 6378 km 82.4 h 0.940% 77.4 h 0.883% 87.7 h 1.001% 82.5 h 0.941%

mean eclipse / longest eclipse: umbra 77.4 h x 60 / (88 x 67.47 min) = 0.78 cylinder 82.4 h x 60 / (91 x 69.60 min) = 0.78 edge of the shadow, entry time: (71.74 - 67.47) / 2 = 2.1 min at equinox; 3.0 min at 6 deg declination; 5.5 min at 8 deg (closed form) storage for 2 GW through the longest eclipse: 2 GW x 67.47 min = 2.25 GWh (umbra); 2 GW x 69.60 min = 2.32 GWh (cylinder, CALC-099); 2 GW x 71.74 min = 2.39 GWh (any shadow)

Result. A geostationary satellite is in full shadow for at most 67.5 minutes a night, on 44 nights around each equinox (seasons of 44.0 days), for 77.4 hours a year: 0.88 per cent. Counting partial shadow, the figures are 71.7 minutes, 47 nights (46.9 days) and 87.7 hours: 1.00 per cent. With the Sun as a point the figures are 69.6 minutes, 45 to 46 nights (45.4 days) and 82.4 hours: 0.94 per cent. The hidden fraction of the Sun, summed over the year, also comes to 0.94 per cent.

Against CLM-0895. The record's 77 h (0.88 per cent) is the umbra case and its 88 h (1.0 per cent) is the any-shadow case. Both reproduce. Its 70-minute maximum is the cylinder case (69.6 min), not the umbra. Its 44-day season is the umbra season. The record's mean of 0.77 of the maximum is 0.78 here.

Against held sources (paraphrased; text layer unless marked render).

Where this fails. - The Earth is a sphere of its equatorial radius. Its flattening, and its atmosphere, which dims sunlight a little beyond the solid edge, are not modelled. Neither effect is sized here. - The Sun is a disc of uniform brightness. Limb darkening would move the weighted total slightly towards the cylinder value. It is not sized here. - The satellite is a point on an exact circle at zero inclination. A real one drifts in inclination unless it is corrected (CLM-1437 limits). The Moon's shadow and the shadows of other satellites are not counted. - The base case moves the Sun at a uniform rate. The eccentric case shifts the any-shadow count between seasons (46 in March, 48 in September) but changes the annual totals by 0.1 h or less. - A season count depends on where the year starts and on the satellite's longitude, so the number of eclipses in one season can differ by one from the figures above. The longitude also moves each annual total by up to 0.1 h. The any-shadow share runs from 0.9996 to 1.0009 per cent, so it is 1.00 per cent to the precision the model holds. - The input constants other than the GEO radius and the 23.4 deg tilt are not from held sources. The sensitivity rows show that the held values (6378 km, 23.4 deg) and the unsourced ones agree to 0.2 h a year. - None of this says how long a satellite's output is off. Shutdown and restart add time on each side of an eclipse (SRC-466, CLM-1081), and output may also stop in partial shadow, which depends on the design.

Script. lessons-geo-eclipse.py, run with Python 3.14.0 and NumPy 2.4.4. Part A is the closed form; part B is the 5 s simulation and the three sensitivity runs. The output above is transcribed from one run.

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