Astronomy & Celestial Mechanics
the sky, computed
Concept

Latitude — where it appears

Angular distance north or south of the equator. For an observer it fixes the altitude of the celestial pole and the range of declinations that ever rise; for a launch site it is a hard lower bound on the orbital inclination any single launch can reach.

Named by 8 essays across 2 fields — each of them below, with the objects they name alongside it.

Twilight at latitude 52°. Solar altitude through the second half of the day at 52°, on June solstice, equinox, December solstice, with the four thresholds that define twilight marked. Sunset is at −0.833° rather than at 0° because the Sun's own semi-diameter and the atmosphere's refraction together lift it by that much when it is geometrically already down. Civil twilight ends at −6°, when the brightest stars appear and outdoor work stops; nautical at −12°, when the horizon can no longer be seen against the sky and a sextant becomes useless; astronomical at −18°, when the sky stops contributing to a photometric measurement. Their durations at equinox here are 34 min, 40 min, 42 min — twilight is not a fixed length, it is the reciprocal of how steeply the Sun descends, and the Sun descends at an angle of about 90° − φ to the horizon. At this latitude the curve for June solstice never reaches −18° at all: astronomical twilight does not end, and there is no astronomically dark night from about 48.6° upward.

Three definitions of night

Twilight ends at three different depression angles, and each threshold is a statement about what can no longer be done. Its length is not a duration but a rate — how fast the Sun goes down — and above one latitude the deepest of the three never arrives at all.

sky · Twilight Which inclinations a launch site can reach, and which it cannot. Orbital inclination against launch azimuth for three sites, from cos i = sin A cos φ. Due east is the only azimuth that gives the minimum, and that minimum is the latitude itself: Kourou 5.2°, Kennedy 28.5°, Baikonur 45.6°. Everything below the shaded line is unreachable from the highest-latitude site by any azimuth at all, and getting there costs a plane change afterwards — 5.94 km/s from a 400 km orbit to reach the equator from 45.6°. Baikonur in fact flies no lower than 51.6° rather than its 45.6°, and the extra 6.0° is overflight constraint rather than mechanics: the azimuth that would give 45.6° sends the spent stages over places they may not fall on.

A plane change paid at the worst speed there is

A launch reaches an inclination fixed by its latitude and its azimuth, and no azimuth reaches an inclination below the latitude. Getting there afterwards means turning at orbital speed, which is the most expensive place available — so a site's latitude is a floor no trajectory removes.

spaceflight · Plane change The extreme sunset and sunrise, neither of them on the December solstice. Sunset and sunrise in local mean solar time through the ninety days around the December solstice, at latitude 52°, computed from Kepler's equation and the tilt. Each is the solar noon — which is 12:00 minus the equation of time — plus or minus half the day arc, and the two ingredients have different stationary points. The day length is stationary at the solstice, exactly, because the declination is. The equation of time is not: it is changing at about 0.50 minutes a day there, moving solar noon steadily. So the sum is still changing after the day length has stopped, and the earliest sunset comes on 14 Dec — 9 days before the solstice on 23 Dec — while the latest sunrise comes on 31 Dec, 9 days after it. The shortest day is the solstice, and neither of its ends is extreme there. Nothing in this is an approximation or a correction: it is a sum of two functions with different stationary points, and the stationary point of a sum is not the stationary point of either.

The earliest sunset is not the shortest day

A sunset time is solar noon plus half the day length, and the two have different stationary points. The day length stops shortening at the solstice; the equation of time does not stop moving, so the sum keeps falling — and the earliest sunset comes days before the shortest day.

sky · Equation of time 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.

sky · Sidereal time An hour that was a twelfth of the daylight. The length of a daylight hour when the time from sunrise to sunset is divided into twelve, through a year, at Alexandria (31.2°N), Rome (41.9°N), London (51.5°N), Stockholm (59.3°N). At Alexandria the hour runs from 51 minutes at the winter solstice to 71 at the summer solstice; at Rome the hour runs from 46 minutes at the winter solstice to 76 at the summer solstice; at London the hour runs from 39 minutes at the winter solstice to 83 at the summer solstice; at Stockholm the hour runs from 30 minutes at the winter solstice to 93 at the summer solstice. The dashed line is sixty minutes, the length it has at both equinoxes everywhere. This is the hour of the ancient Mediterranean world and of medieval Europe until mechanical clocks: an hour defined by the Sun, which a sundial with suitably drawn lines reads exactly, and in which the equation of time does not exist, because nothing is being compared with a uniform clock. Sunrise, sunset and noon are each defined by the Sun, and the clock that would disagree with them had not been built.

An hour that stretched with the season

For most of recorded history an hour was a twelfth of the daylight — seventy-six minutes at a Roman midsummer and forty-six in midwinter — and a sundial read it exactly. In that system there was no equation of time, because nothing uniform was being compared with the Sun. The sixteen-minute correction became real only when the hour was made equal, and measurable only when clocks could keep time more steadily than the Sun by more than it.

sky · Equation of time Five zones, one angle. The Sun's highest and lowest noon altitude against latitude, for an obliquity of 23.4393°. The upper curve is noon on the summer solstice and the lower is noon on the winter one; they are the same function of latitude displaced by 23.4393° in each direction, which is why one angle fixes both boundaries. Where the upper curve reaches 90° is the tropic, at 23.44° — the Sun is overhead at noon there on exactly one day, and somewhere inside it on every other day of the year. Where the lower curve reaches 0° is the polar circle, at 66.56° — the Sun fails to clear the horizon on the winter solstice, and on more days the further poleward one goes. They are the same inequality: |φ| ≤ ε for the first and |φ| ≥ 90° − ε for the second, and an obliquity of zero would collapse the tropics to the equator and push the polar circles to the poles, leaving one zone. The areas are the part that is not intuitive. The fraction of a sphere between two latitudes is the difference of their sines, so the tropics — a band a quarter of the way to the pole — hold 39.8 per cent of the Earth's surface, the temperate zones 52.0 per cent, and the polar caps only 8.3. The zone where the Sun can be overhead is 4.8 times the area of the zone where it can fail to rise, and both boundaries are the same 23.44°.

Five zones, and one angle

The tropics are where the Sun can stand overhead; the polar circles are where it can fail to rise. Both boundaries are 23.44° measured from opposite ends, and the zone the Sun can reach is nearly five times the area of the zone it can miss.

sky · Seasons The sunniest place on Earth is the summer pole. Daily-mean insolation against latitude, on the June solstice, an equinox, the December solstice, computed from Q = (S₀/π)(ā/r)²[H₀ sin φ sin δ + cos φ cos δ sin H₀] with the half-day angle clipped to a whole day inside the polar circle. On the June solstice the north pole receives 524 W/m² and the equator receives 385 — a third more, at a place where the Sun never climbs above 23.4°. Both halves of the product move: the Sun is low, so each square metre catches little; and it never sets, so the catching goes on for twenty-four hours. The second wins. That is why the daily total and the noon altitude disagree about where summer is strongest, and the daily total is the one an ice sheet answers to. The two hemispheres are not equal either. The Earth is nearest the Sun in early January, so the December solstice delivers 559 W/m² to the south pole against the north pole's 524 in June — 6.7 per cent more. What this cannot show is any temperature. Antarctica is the coldest place on Earth and receives the most sunlight of anywhere on any day; between the insolation and the temperature sit an albedo, an altitude and an ocean.

The sunniest place is the summer pole

On the June solstice the north pole receives 524 watts per square metre averaged over the day, and the equator 385. The Sun there never climbs above 23.4° and never sets, and the second wins.

sky · Seasons The hottest month is not the sunniest. The annual cycle of insolation at latitude 45°, and the temperature three surfaces of different heat capacity answer it with — each curve scaled to its own peak, because what is being compared is timing and not degrees. A surface losing 2 W m⁻² per kelvin above equilibrium obeys C dT/dt = Q − λT, and for a sinusoidal forcing that is one line of algebra: the response lags by arctan(ωC/λ) and is reduced by (1 + (ωC/λ)²)^(−½). a continental interior, 2.4 m of water, lags by 45.6 days and swings by 71 per cent of what a massless surface would; a shallow shelf sea, 10 m of water, lags by 77.6 days and swings by 23 per cent of what a massless surface would; a deep ocean mixed layer, 50 m of water, lags by 88.5 days and swings by 4.8 per cent of what a massless surface would. The lag can never reach a quarter of a cycle — three months — because an arctangent cannot reach 90°, which is why no inhabited place has its warmest month in December in the north, and why the sea comes closest. And the two numbers are one number: the same ωC/λ that carries the phase towards the quarter cycle divides the amplitude, so a long lag is bought with a small swing and cannot be had any other way. The solstice falls on day 170 and the peak insolation with it; the warmest day at this latitude follows between 46 and 89 days later depending on what is underneath. What no curve here can show is the atmosphere's own transport, which carries heat sideways between the surfaces drawn and makes each one's effective capacity partly its neighbour's.

The hottest month is not the sunniest

A surface with a heat capacity answers a sinusoid late and small, and the two are the same number. The lag can never reach a quarter of a cycle — three months for a year, six hours for a day — because an arctangent cannot reach ninety degrees.

sky · Seasons

Named alongside it

The objects these essays reach for when they reach for this one.

ObliquitySolar declinationSolsticeAxial tiltDeclinationEquinoxInsolationCelestial sphereDay lengthEquation of timeHour angleMean solar time

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