Astronomy & Celestial Mechanics
the sky, computed
Series

Sidereal time — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Why the solar day is 3m 55.91s longer than the sidereal one. Two positions of the Earth one day apart, with the direction of a fixed star and the direction of the Sun marked at each. After one sidereal day — 23h 56m 4.0905s — the Earth has turned through exactly 360° and the star is back on the meridian; the Sun is not, because the Earth has moved 0.9856° along its orbit, and the further 0.9856° of turning takes 3m 55.91s. That is the whole of the difference: 366.2422 turns against the stars in the 365.2422 solar days of a year, one more turn than the 365.2422 against the Sun. Nothing here is to scale. The orbital arc is drawn at 40°, which exaggerates the real 0.9856° by 41 times, and the Earth's disc is about 4073 times too large for its orbit. The two star sight lines are drawn parallel because they are: a star's distance cannot be put on the same page as an orbit.

    The day that is four minutes short

    A star crosses the meridian 3 minutes 55.91 seconds earlier each night, and the same star slips 3 minutes 56.56 seconds a day against a civil clock. Those are two different numbers, and the gap between them is the one extra turn the Earth makes against the stars in every year.

    part 1 · sky
  2. A sidereal clock that runs up to 1.15 seconds ahead or behind, with the Moon's node. The equation of the equinoxes from 1990 to 2030: apparent sidereal time, counted from the true equinox, minus mean sidereal time, counted from an equinox that only precesses. It is the nutation in longitude times the cosine of the obliquity, and it is drawn here from the four largest nutation terms, which carry it to about a hundredth of a second. The dominant term follows the Moon's node round its 18.61-year cycle with an amplitude of ±1.052 s; riding on it are a half-yearly term of ±0.081 s from the Sun and a fortnightly one of ±0.014 s from the Moon. Over this span the sum runs from −1.147 to 1.146 s. None of it is the Earth's rotation: it is the zero point of the clock moving, because the zero point is the intersection of the equator with the ecliptic and the equator nods.

    A clock whose zero is moving

    Sidereal time is counted from the equinox, and the equinox does not stay put. Its steady drift makes the sidereal day eight milliseconds short, its acceleration puts a quadratic term into the formula for sidereal time, and the Moon makes it nod by a second every nineteen years. The Earth rotation angle removes all three by counting from a point defined not to move along the equator.

    part 2 · sky
  3. When the Orion Nebula can be observed from latitude 52°, through a year. Every night of 2027, from local noon to the following noon, for a station at latitude 52°. The shaded cells are the times at which the Orion Nebula (right ascension 5.59 h, declination −5.4°) stands at least 20° above the horizon while the Sun is more than 18° below it. The two outer curves are the start and end of astronomical darkness, which never comes on 64 nights around midsummer; the diagonal line is the object's transit, which arrives four minutes earlier each night and wraps through the whole day once a year. It crosses local midnight on 15 Dec, when the object stands opposite the Sun. The shaded season is the stretch of the year either side of that date in which the transit falls inside the dark hours. From this latitude the object transits at 32.6° altitude; it is observable on 181 nights for at least an hour, for 6.3 hours on the best of them (3 Jan), and for 860 hours in the year.

    A right ascension is a date

    An object can be observed only when two clocks agree — the sidereal clock that brings it high in the sky and the solar clock that makes the sky dark. The two drift apart by one turn a year, so every right ascension has a season, every latitude gives that season a different length, and an object that never sets is best observed at the opposite time of year from the one its right ascension names.

    part 3 · sky
  4. When a target can be observed from a telescope that cannot look near the Sun. The days of the year (across) on which a target at ecliptic longitude 0° and a given ecliptic latitude (up) can be observed by a telescope allowed to point only between 85° and 135° from the Sun, the constraint of a large infrared telescope at the Sun–Earth L2 point. The Sun moves along the ecliptic once a year, and the allowed band sweeps round with it. A target on the ecliptic is inside the band twice a year, for about 51 days each time, a little before and a little after its opposition — never at it, since opposition is 180° from the Sun. The two windows widen with latitude and merge into one above 45°, where the elongation at opposition no longer exceeds 135°; within 5° of either ecliptic pole, where the elongation at conjunction no longer falls below 85°, the target is never outside the band at all, which is the continuous viewing zone. The pattern depends on ecliptic, not equatorial, coordinates: the season of a space telescope is set by the Sun's path, and the Earth's axis has no part in it.

    A season without a night

    A telescope in space has no horizon and no night, and it can still only look at most of the sky for part of the year. Its season is a band of solar elongation swept round the ecliptic by the Sun, so an object on the ecliptic can be seen for two stretches of about seven weeks and never at opposition, while the few degrees round the ecliptic poles can be watched all year. From low orbit the Earth takes the night's place, and the zones it never hides are carried round the sky every eight weeks by the planet's equatorial bulge.

    part 4 · sky
  5. A year that asks for more than a day on some days. The observing time requested per day through the year, with each target's request spread evenly over the days it can be seen, for the same programme filling 90 per cent of the year. The telescope supplies one day per day (dashed). On 104 days the visible targets ask for more than that, peaking at 1.6 days per day when the Galactic centre is in its window; on the rest there is slack. The annual total fits; the seasons do not.

    A schedule written by where the Sun is

    A space telescope that must keep the Sun at a fixed range of angles sees each part of the sky in windows set by the date, and the sky its users want is not spread evenly. A schedule can fill around a crowded window as long as the crowd fits inside it, by sending everything else to other dates. Once the demand for one region exceeds the share of the year that region is visible, no choice of proposals elsewhere can recover the excess — the limit is each window, not the total.

    part 5 · sky

All series