Astronomy & Celestial Mechanics
the sky, computed
Generator

The kepler-third generator

Period against size for the planets, around the Sun
Period against size for the planets, around the Sun. Orbital period against semi-major axis on logarithmic axes. The line has slope exactly three-halves — the harmonic law — and the measured bodies sit on it.

Orbital period against semi-major axis on logarithmic axes. The line has slope exactly three-halves — the harmonic law — and the measured bodies sit on it.

5 essays call kepler-third. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Every figure listed here is the same construction drawn at different numbers, so a correction to one is a correction to all of them.

Period against size for the planets, around the Sun. Orbital period against semi-major axis on logarithmic axes. The line has slope exactly three-halves — the harmonic law — and the measured bodies sit on it. Orbits

The law that links period to size, and weighs everything

Kepler found that the square of the period goes as the cube of the orbit. Newton found the constant of proportionality, and that constant is a mass — which is how every mass in astronomy has been obtained since.

 The mass correction against the mass. Each planet's departure from the massless harmonic law, against its own mass in solar units, on logarithmic axes. The exact law puts every point on the diagonal. Jupiter and Saturn are the only planets whose mass correction is larger than the perturbations from everything else, and Saturn's measured departure has the opposite sign. Orbits

The third law is wrong by the mass of the planet

Kepler's harmonic law says the square of the period goes as the cube of the size. Newton's version has one more term in it, and the term is the orbiting body's own mass — negligible for a planet, decisive for a binary star, and the reason a period can be converted into a mass at all.

 Mars through 8.3 months of sky. The geocentric ecliptic longitude and latitude of Mars over 252 days — 8.3 months — centred on opposition, computed as the direction of P − E with both orbits taken as circles: the Earth's of radius 1 AU, the planet's of 1.5237 AU inclined 1.850°. The motion reverses for 72.7 days and backs up 15.94° of longitude, and that interval is centred on opposition to within 0.0000° of longitude. The track closes on itself: over 149 days the planet visits the same point of the sky twice, and the loop it encloses is 15.9° long and 2.94° tall. Longitude and latitude are at different scales: a degree of latitude is drawn 4.6 times a degree of longitude, because the sweep of 51° in longitude and 4.5° in latitude will not share a scale on one page. Circular orbits mean one loop for every apparition, and that is the real cost of this figure: Mars is 0.525 AU away at the opposition drawn here, its true distance at opposition varies with its eccentricity, and the real loops differ in size from one apparition to the next because of it. The marks are 20 days apart, and they crowd where the motion stops. The observed sky

The loop a planet does not make

Mars stops in the sky, backs up for ten weeks, and goes on. Neither orbit reverses anywhere — the loop belongs to the difference of two position vectors seen from one of them, and its width and duration are fixed by the ratio of the two radii and by nothing else.

 A doubling rule, and the planet that broke it. Each planet's real semi-major axis against the value the Titius–Bode rule 0.4 + 0.3 × 2ⁿ gives it, both in astronomical units on logarithmic axes; points on the diagonal are fitted exactly. The rule gives the Earth 1.0 by construction and fits Venus, Mars, Jupiter and Saturn within a few per cent. It needed the value n = 3 left empty until Ceres was found in its slot in 1801, and it predicted Uranus, discovered in 1781, within 2%. Neptune, discovered in 1846 by a calculation that assumed the rule, sits 22% inside its predicted place, and Pluto occupies the place Neptune should have had. Mercury is given n = −∞, which is a free choice made to fit it. The labels give each body's departure from the rule. Orbits

A spacing law that random orbits obey

The planets' distances roughly double from one to the next, and a rule written down in 1766 predicted the asteroid belt and Uranus before either was found. It failed on Neptune, and the reason it worked at all turns out to be that almost any set of orbits forbidden to crowd each other looks geometric on a logarithmic scale — as a test against thousands of random systems shows.

 A moonlet's period measures its primary's density. The orbital period of a small satellite against the radius of its orbit in units of the primary's radius, both logarithmic, for primaries of bulk density 1, 2, 3.5 g/cm³. The size of the asteroid cancels: a moonlet at five radii from a body of 2 g/cm³ takes 26 hours whether the body is a kilometre or a hundred across. At one radius the curves give the spin period at which a strengthless rubble pile flies apart — 3.3, 2.3 and 1.8 hours. Didymos–Dimorphos, at 3.1 radii and 11.92 hours, gives 2.4 g/cm³; 1999 KW4, at 3.9 radii and 17.42 hours, gives 2.1 g/cm³. Orbits

A moonlet that weighs a pile of rubble

Put Kepler's third law round an asteroid and its size cancels: a moonlet's period depends only on how many of the asteroid's radii away it orbits and on the asteroid's density. So the first asteroid moon ever seen, a mile-wide speck beside Ida, turned a picture into a density — and the density came out too low for solid rock. A fifth to a half of the asteroids weighed this way is empty space.

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