Astronomy & Celestial Mechanics
the sky, computed
Concept

Solar declination — where it appears

The Sun's angular distance north or south of the celestial equator, running between the obliquity's positive and negative values through a year. It is what fixes the length of the day at any latitude, and its two turning points are the solstices.

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

The Sun's altitude through the day at latitude 52°. Solar altitude against the hour of the day, at one latitude, for the solstices and the equinox. Where a curve crosses zero is sunrise or sunset, and the width between the crossings is the length of the day.

The Sun's path, and the tilt that makes the seasons

Summer is not when the Earth is closest to the Sun — that happens in January. It is when the Sun climbs higher and stays up longer, and both come from a 23.4° tilt.

sky · Seasons The equation of time, and its two causes. The difference between a sundial and a clock over the year, in minutes, computed from Kepler's equation and the tilt. The eccentricity term has one cycle a year and the obliquity term has two; their sum runs from −14.2 to 16.4 minutes.

The Sun is a bad clock, by up to sixteen minutes

Solar noon and twelve o'clock are not the same instant, and the discrepancy runs through a fixed annual cycle. It has two causes, one from the shape of the orbit and one from the tilt of the axis.

sky · Equation of time June sunlight at 65°N, against where perihelion sits. Daily-mean insolation at latitude 65 degrees north on the June solstice, at an obliquity of 23.44 degrees, against the longitude of perihelion measured from the March equinox — the angle that precesses right round in about twenty-one thousand years. Three eccentricities are drawn. The swing is ±10.0 per cent at e = 0.05 and ±1.0 per cent at e = 0.005, in proportion to the eccentricity, because the Sun–Earth distance on a fixed date carries e cos of the precession angle and that is first order. Over the same range of eccentricity the annual mean at this latitude moves by 0.12 per cent, because the annual mean carries 1/√(1−e²) and that is second order. The two together are the whole of the precession term in Milankovitch's theory: the eccentricity does almost nothing to how much sunlight the Earth receives and a great deal to when it arrives, and the ice sheets of the northern hemisphere respond to the summer they might melt in rather than to the year's total. It also explains why the precession signal disappears when the orbit is nearly circular: multiply a large angular swing by a vanishing eccentricity and there is nothing left, which is what the innermost curve here is.

An average that precession cannot move

Sunlight arrives as the inverse square of the distance and time passes as its square, so the two cancel exactly in a year's integral. The longitude of perihelion therefore changes the annual mean insolation at every latitude by precisely nothing — and changes June at 65°N by ten per cent.

orbits · Orbital averages The equation of time, and its two causes. The difference between a sundial and a clock over the year, in minutes, computed from Kepler's equation and the tilt. The eccentricity term has one cycle a year and the obliquity term has two; their sum runs from −14.2 to 16.4 minutes.

Two fictitious suns in sequence

The equation of time is usually described as one quantity with two causes. It is better read as two separate reductions, each with its own imaginary body — one that moves uniformly along the ecliptic and one that moves uniformly along the equator — and the second is not the first.

sky · Equation of time 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 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.

ObliquitySolsticeEquinoxInsolationAnalemmaAxial tiltEccentricityLatitudeDeclinationEquation of timeMean solar timeCelestial sphere

All concepts