Astronomy & Celestial Mechanics
the sky, computed
The observed sky

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.

Assumes Seasons and Celestial sphere.

16 min read 6 figures Scales that defeat intuitionThe sky as a clock

The seasons come from a tilt of 23.44°, and so does everything else on a school atlas’s key. The tropics, the polar circles, the midnight sun, the day the Sun stands in a well at Syene — all five zones and all four boundary lines are one number, used twice.

The two uses look different and are the same inequality. The Sun’s declination runs between +ε+\varepsilon and −ε-\varepsilon over a year, so the Sun stands overhead at noon somewhere within ∣φ∣≤ε|\varphi| \le \varepsilon, and fails to rise at all on some day wherever ∣φ∣≥90°−ε|\varphi| \ge 90° - \varepsilon. The first reads the obliquity from the equator and the second from the pole.

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°.
Fig. 1 The Sun’s highest and lowest noon altitude against latitude. The upper curve is noon on the summer solstice, 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 reaches 90° is the tropic at 23.44°; where the lower reaches 0° is the polar circle at 66.56°. The areas are the part that is not intuitive: the tropics hold 39.8 per cent of the Earth’s surface, the temperate zones 52.0, and the polar caps only 8.3.

Where the areas come from

The fraction of a sphere between two latitudes is half the difference of their sines. That is Archimedes’ result — the area of a spherical zone equals the area of the cylinder it projects onto — and it is one of the oldest theorems in the subject and one of the least intuitive.

It says that a band of given angular width near the equator has far more area than the same band near a pole, and it says so with a factor that runs from one to nothing. A degree of latitude at the equator carries cos⁡0°=1\cos 0° = 1 of relative area; a degree at 60° carries a half; a degree at 80° carries a sixth.

So the tropics, a band reaching only a quarter of the way from the equator to the pole, hold two fifths of the Earth’s surface. The polar regions, reaching three quarters of the way out, hold a twelfth.

The Sun can stand directly overhead on nearly two fifths of the Earth, which is the statement most people find surprising, and it follows from a theorem about cylinders rather than from anything astronomical.

Five zones, one angle. The Sun's highest and lowest noon altitude against latitude, for an obliquity of 15°. 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 15° in each direction, which is why one angle fixes both boundaries. Where the upper curve reaches 90° is the tropic, at 15.00° — 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 75.00° — 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 25.9 per cent of the Earth's surface, the temperate zones 70.7 per cent, and the polar caps only 3.4. The zone where the Sun can be overhead is 7.6 times the area of the zone where it can fail to rise, and both boundaries are the same 15.00°.
Fig. 2 The same construction for an obliquity of 15°. Both boundaries move inward from their own ends — the tropic to 15° and the polar circle to 75° — and the temperate zone widens at the expense of both. The tropics now hold 25.9 per cent of the surface and the polar caps 3.4. The two boundaries are not independent and never were: they are ε\varepsilon and 90°−ε90° - \varepsilon, so lowering the obliquity shrinks both zones and one number decides both.

The same theorem carries a consequence for how much sunlight a planet intercepts, and it is the reason the tropics dominate the energy budget as thoroughly as they do. A planet presents a disc of area πR2\pi R^2 to the Sun and has a surface of 4πR24\pi R^2, so the mean insolation is a quarter of the solar constant — and that quarter is not spread evenly. Weighting each latitude by its area and by the cosine of its zenith angle, the tropics collect about half of everything the Earth receives.

Two fifths of the area collecting half of the energy is a modest concentration, and its modesty is the point. The equator-to-pole contrast in annual insolation is a factor of 2.4, which is far smaller than the temperature contrast the Earth actually has. The difference between the two is what atmospheric and oceanic heat transport does, and the size of the job it has to do is set by this geometry.

Five zones, one angle. The Sun's highest and lowest noon altitude against latitude, for an obliquity of 5°. 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 5° in each direction, which is why one angle fixes both boundaries. Where the upper curve reaches 90° is the tropic, at 5.00° — 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 85.00° — 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 8.7 per cent of the Earth's surface, the temperate zones 90.9 per cent, and the polar caps only 0.4. The zone where the Sun can be overhead is 22.9 times the area of the zone where it can fail to rise, and both boundaries are the same 5.00°.
Fig. 3 An obliquity of 5°, near Mercury’s and Jupiter’s. The tropics collapse to a narrow band holding 8.7 per cent of the surface and the polar caps to 0.4 per cent, and the temperate zone runs almost from pole to pole. A planet like this has almost no seasons and a very sharp climate gradient, because the annual insolation falls monotonically and steeply from equator to pole with nothing to even it out. The Earth’s 23° is enough to give the high latitudes a summer, and that summer is most of what makes them habitable.

What each boundary actually means

The two definitions sound parallel and are not quite. It is worth separating them.

The tropic is a statement about a single day. At exactly 23.44° the Sun reaches the zenith at noon on one day of the year and never again. Inside the tropics it does so on two days, separated by an interval that shrinks to zero at the tropic itself and grows to half a year at the equator.

The polar circle is a statement about a single day too, in the same way: at exactly 66.56° the Sun fails to clear the horizon on one day and rises on every other. Poleward of it the number of such days grows, to half a year at the pole.

So both are the boundary case of a phenomenon rather than its beginning, which is why neither is a place where anything changes abruptly. A traveller crossing the Arctic Circle does not enter a different regime; they enter a regime in which one day a year has no sunrise, and the day after crossing, two.

The gradient is steepest at the boundary and shallow in the interior, which is the opposite of what a line on a map suggests. At 67° there are about three days of polar night a year; at 70° about twenty-five; at 80° about four months.

Five zones, one angle. The Sun's highest and lowest noon altitude against latitude, for an obliquity of 32°. 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 32° in each direction, which is why one angle fixes both boundaries. Where the upper curve reaches 90° is the tropic, at 32.00° — 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 58.00° — 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 53.0 per cent of the Earth's surface, the temperate zones 31.8 per cent, and the polar caps only 15.2. The zone where the Sun can be overhead is 3.5 times the area of the zone where it can fail to rise, and both boundaries are the same 32.00°.
Fig. 4 An obliquity of 32°, roughly Saturn’s. The tropics swell to 53 per cent of the surface, the polar caps to 15, and the temperate zone is squeezed to under a third. At this tilt the tropic and the polar circle are only 26° of latitude apart, and a planet at an obliquity above 45° would have them cross: the polar circle would sit equatorward of the tropic, and there would be places where the Sun both stands overhead in summer and fails to rise in winter.

The planets that have crossed it

That crossing is not hypothetical. Uranus has an obliquity of 97.8°, which puts its “polar circle” at 90°−97.8°=−7.8°90° - 97.8° = -7.8° — a latitude in the other hemisphere — and the geometry becomes something a two-zone diagram cannot describe.

At that tilt, every point on the planet experiences polar day and polar night at some point in the orbit, and the poles receive more annual insolation than the equator does. The seasons are not a modulation of a day-night cycle; they replace it.

The threshold is 54°, where the annual mean insolation at the pole equals that at the equator. Above it a planet’s poles are its warmest regions on average, which inverts every climate intuition built on the Earth.

And obliquity is not a constant. The Earth’s own tilt varies between about 22.1° and 24.5° on a 41,000-year period, so the tropics and the polar circles migrate by about a degree and a half — some fifteen kilometres a year, which is fast enough that the markers placed on roads at the Arctic Circle are wrong within a decade of being installed.

The variation is small because the Moon holds the tilt steady, and a planet without one would have a very different figure here: the zones would migrate over tens of degrees rather than over one, and the classification would be a description of the present epoch rather than of the planet.

The one that is not a consequence of the tilt

Four of the five zones’ boundaries come from the obliquity. There is a fifth line on many maps that does not, and it is worth separating because it is often drawn in the same style.

The boundary of the midnight sun as experienced is not the polar circle. Atmospheric refraction lifts the Sun by about half a degree at the horizon, and the Sun’s own disc is a quarter of a degree in radius, so the Sun appears to rise when it is geometrically 0.833° below the horizon. That pushes the effective polar circle about a degree equatorward of the geometric one.

The same 0.833° sets where sunset is, and it is a fact about the atmosphere rather than about the orbit. A planet with no atmosphere has its polar circle exactly at 90°−ε90° - \varepsilon; the Earth’s is at about 65.7° in practice and 66.56° on the map.

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°.
Fig. 5 The Earth’s own case again, drawn to make the geometry explicit: the two curves are one function shifted, the zenith line and the horizon line are the two thresholds, and the boundaries are where each curve meets each line. The figure contains no atmosphere. Every altitude drawn is geometric, so the crossings are the mapped boundaries rather than the observed ones, and the difference between the two is about a degree of latitude at the polar circle and nothing at all at the tropic.

The four named parallels and the two that are not

There is a detail in the naming that is worth one paragraph because it is almost always got wrong on maps.

The Tropic of Cancer and the Tropic of Capricorn are named for the zodiacal constellations the Sun stood in at the solstices when the names were fixed, over two thousand years ago. Precession has since moved the equinoxes by nearly a full sign, so the June solstice Sun is now in Taurus and the December one in Sagittarius. The lines are correctly placed and misleadingly named.

The polar circles have no such problem because they were never named after anything in the sky — “Arctic” is from the Great Bear, which is a description of the direction rather than of the Sun’s position, and “Antarctic” is its negation.

So two of the four boundaries carry a two-thousand-year-old error in their names and none carries one in its position, which is a reasonable trade and is the reverse of what usually happens to astronomical terminology.

Why the tropic matters and the polar circle does not

The two boundaries have wildly unequal consequences, and the asymmetry is instructive.

The tropic marks where the Sun can be overhead, and being overhead matters because insolation falls as the cosine of the zenith angle. A surface tilted 30° from the incoming light receives 87 per cent of what a perpendicular one does; the loss is slow for small angles. So the tropics are not a region of sharply higher insolation than the subtropics — the cosine is flat near its maximum, and that flatness is why the tropics are broad and mild rather than narrow and extreme.

The polar circle marks where the Sun can fail to rise, and a day without sunrise still receives light — twilight, and sunlight scattered from the atmosphere above. The Sun being below the horizon is not the same as the sky being dark, and the polar circle is the boundary of the first rather than the second.

Neither line is a climate boundary and both are drawn on climate maps. The actual boundaries of the tropical and polar climates lie elsewhere, are not circles of latitude at all, and depend on ocean circulation, on altitude and on the distribution of land — none of which the obliquity decides.

Two more lines the same angle draws

The obliquity fixes two further boundaries that are not on atlases and are worth having, because both are places where a familiar phenomenon stops.

The latitude above which astronomical twilight never ends at midsummer is 90°−ε−18°=48.56°90° - \varepsilon - 18° = 48.56°, since the Sun’s lowest point on the solstice is φ−(90°−ε)\varphi - (90° - \varepsilon) below the horizon and the threshold for a dark sky is 18°. Everywhere north of Paris has at least one night a year with no astronomical darkness in it, which is the constraint that ends an observing season rather than a curiosity.

And the latitude at which the Sun’s noon altitude equals its own angular radius — where the Sun at its lowest just grazes the horizon at noon rather than clearing it — is the polar circle itself, to within the quarter degree the disc subtends. That quarter degree is the whole difference between “the Sun rises” and “part of the Sun rises”, and it is the reason the geometric and observed boundaries differ.

Both are computed from the same ε\varepsilon, both by subtraction, and neither is marked anywhere. The pattern is general: any threshold defined by a solar altitude becomes a latitude by subtracting it from 90°−ε90° - \varepsilon, and the atlas draws the two thresholds that happen to be 0° and 90°.

Five zones, one angle. The Sun's highest and lowest noon altitude against latitude, for an obliquity of 45°. 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 45° in each direction, which is why one angle fixes both boundaries. Where the upper curve reaches 90° is the tropic, at 45.00° — 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 45.00° — 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 70.7 per cent of the Earth's surface, the temperate zones 0.0 per cent, and the polar caps only 29.3. The zone where the Sun can be overhead is 2.4 times the area of the zone where it can fail to rise, and both boundaries are the same 45.00°.
Fig. 6 The critical case: an obliquity of exactly 45°, where the tropic and the polar circle coincide at the same latitude and the temperate zone vanishes. Every point on such a planet is either inside the tropics or inside the polar regions, with no band that is neither. Above 45° the two swap order and the zones overlap rather than tiling, which is where the classification stops describing anything — and roughly where a planet’s poles begin to receive more annual sunlight than its equator.

An angle known to milliarcseconds, and a sphere that is not one

The obliquity is measured, and it is among the best-determined angles in astronomy: 23.4392911° at the standard epoch, known to a few milliarcseconds from the positions of solar system bodies fitted over decades.

The zone areas are geometry and are exact given the obliquity.

The noon altitudes drawn are exact spherical trigonometry, h=90°−∣φ−δ∣h = 90° - |\varphi - \delta|, which is the special case of the general altitude formula at hour angle zero. Nothing in the drawing is fitted or approximated, which is unusual enough to be worth saying: a picture of the sky usually computes an observable from a model, and this one computes a definition from an angle.

What is idealised is the Earth. It is taken to be a sphere, and it is an oblate spheroid whose geographic latitude and geocentric latitude differ by up to 11 arcminutes. At the tropic that shifts the boundary by about a kilometre, which is below the resolution of any map that draws it and above the precision with which it is usually quoted.

Where the boundaries are on other worlds

Running the same construction on other bodies is cheap and the results are worth setting out, because they show how unusual the Earth’s arrangement is rather than how typical.

Mars has an obliquity of 25.2°, close enough to the Earth’s that its zones have nearly the same proportions — which is part of why its seasons are recognisable, though twice as long and far more extreme because of its eccentricity. Its obliquity is also far less stable, because it has no large moon to damp the variation, and it has ranged from near zero to above 60° over the last ten million years.

Venus has an obliquity of 177°, which is to say it is nearly upside down, so its effective tilt is 3° and it has essentially no seasons and no zones.

Jupiter’s 3.1° gives it the collapsed geometry drawn two figures above. Saturn’s 26.7° gives it zones much like the Earth’s, on a year of twenty-nine.

And Uranus is the one that breaks the scheme entirely, at 97.8°. Its tropic sits at 82° and its polar circle at −8-8°, which is not a latitude in the same hemisphere — so every point on the planet lies inside both, and the two zones that are supposed to be disjoint cover the whole surface twice.

The obliquity as a free parameter

Treating the tilt as an adjustable number, as three of the figures above do, is not idle. It is how the habitability of a planet’s surface is assessed when nothing about it is known except its orbit.

A planet’s obliquity is essentially never measurable directly. What can sometimes be measured is the angle between a star’s spin and its planet’s orbit, which is a different quantity and constrains the planet’s own tilt not at all. So the tilt enters a habitability estimate as a distribution rather than a value, and the estimate has to be run across it.

What the running shows is that the answer is not very sensitive. A planet at low obliquity has a steep pole-to-equator gradient and may freeze its poles permanently; one at high obliquity has violent seasons and may freeze its equator instead; and in between — which is most of the range — the total insolation is the same and only its distribution differs. The annual mean insolation at a latitude depends on the obliquity, and the global annual mean does not depend on it at all.

That last statement is exact and worth keeping. Averaged over a year and over the whole surface, a planet intercepts a quarter of the solar constant regardless of how it is tilted, because the tilt redistributes energy and does not collect any. Every consequence of obliquity is therefore a consequence of where the energy arrives and when, which is precisely what the daily integral and the seasonal lag are about.

The oldest measurement in the subject

The tropic is where this whole construction was first noticed, and the noticing is one of the few pieces of ancient astronomy that can be reconstructed exactly.

Eratosthenes was told that at Syene, on the summer solstice, a vertical stick cast no shadow and the Sun was reflected from the bottom of a well. At Alexandria on the same day a stick cast a shadow corresponding to about a fiftieth of a circle. From that he obtained the Earth’s circumference, and the measurement is famous.

What is less often said is that the first observation — the shadowless stick at Syene — is a measurement of the obliquity rather than of the Earth’s size. It says that Syene’s latitude equals the Sun’s solstitial declination, which is to say that Syene is on the tropic. Syene is at 24.09°N, which is 0.65° north of the tropic as it now stands and was within about 0.4° of it in 240 BC, since the obliquity was then slightly larger.

So the town was nearly on the line and not quite, and the well had a small shadow at solstice noon that the account does not mention. The measurement that followed did not depend on it.

Still open: nothing, and that is the point

There is no live question here. The obliquity is measured, the trigonometry is exact, and the zones follow.

What the zones supply instead is a correction to an intuition, and the intuition is worth stating because everything after it depends on it. A map suggests that the tropics are a narrow hot band and the poles a large cold one. The areas say the reverse — two fifths against a twelfth — and the reason is a theorem about spheres.

From here, noon stops being the question

The noon altitude is not what a climate answers to. What matters is the energy delivered over a whole day, which is the altitude integrated over however long the Sun is up — and those two quantities disagree about where summer is strongest.

Computing that integral gives an answer worth the arithmetic: on the day of the solstice the sunniest place on Earth is the pole.

Where this goes next

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. Two equilibria become four, and three worlds sit near the join. The Cassini equilibria of a spin axis, drawn against the ratio of its own precession rate to the rate at which its orbit plane turns, for an orbit inclination of 1.5 degrees. Each column of dots is the full set of obliquities at which the two precessions keep step at that ratio, found by root-finding rather than by tracing a remembered curve. Below α cos ε/|g| = 1.135 there are two such obliquities and above it there are four, and the figure checks both counts on either side of the join. The three marked bodies are placed by their own measured precession constants: the Earth with the Moon at 2.67, safely on the four-state side; the Earth without it at 0.86; and Mars at 1.06. Two of the three sit within a few tenths of the bifurcation, which is the whole reason their obliquities are not constants: near the join the equilibria are close together, the libration around them is wide, and a body pushed between neighbouring resonances wanders. The Moon's contribution to the Earth's precession constant is what moves the first mark away from that region, and the second mark is the same planet with that contribution removed. This is a two-frequency model of a many-frequency system, and the real chaos comes from the overlap of resonances it does not contain. A tilt that is not a constant The Earth's axis leans by 23.4 degrees, and that lean is what makes the seasons. It is also a dynamical variable with its own equilibria, its own resonances and its own chaos — and on Mars the same variable has swung between nearly zero and sixty degrees without anything having to happen. 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.

About the same objects

Not linked from either essay — found by the objects both name.

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

Axial tiltCelestial sphereDeclinationEquinoxInsolationLatitudeObliquityPolar daySolar declinationSolstice