Illustrated Physics
physics, drawn
Concept

Eccentricity — where it appears

How far an orbit departs from a circle: zero for a circle, between zero and one for an ellipse. It sets how much the distance, the speed and the depth of the gravitational potential vary around the orbit, and the Earth's 0.0167 swings the Sun's potential by 3.3 × 10⁻¹⁰ a year.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

The arrow the orbit cannot turn. A Kepler orbit of eccentricity 0.6, integrated for two revolutions, with the Laplace–Runge– Lenz vector constructed from the position and velocity at five points along it. Every one of the five is the same arrow: its length varies by 3.7e-11 over the whole run and its direction by 2.1e-10 radians. It points at the perihelion and its length is 0.600000, which is the orbit's eccentricity measured independently from the closest and furthest radii as 0.600000. Energy and angular momentum fix the size and shape of an orbit and say nothing about which way it points; this vector is the missing statement, and only an inverse square has one.

The arrow that says which way the orbit points

Energy and angular momentum fix the size and shape of an orbit and say nothing about its orientation. The inverse-square force has a third conserved quantity that supplies it — a vector pointing at the perihelion whose length is the eccentricity — and no other force law does.

astrophysics · Orbit stability Both clocks move, and their ratio does not. Two years of a clock's fractional frequency against a distant clock (upper panel), and of the ratio of two unlike clocks kept side by side (lower panel). Above, both clocks slow as the Earth nears the Sun, with an amplitude of 1.65·10⁻¹⁰, and if the redshift is universal the two curves are one curve. Below, the ratio: the flat line is what universality predicts, and the sinusoid is what a clock responding to the potential 10⁻⁶ more strongly than the other would produce — an annual term of 1.65·10⁻¹⁶, a million times smaller than the shift both clocks share and within reach of clocks that compare to parts in 10¹⁷.

The clocks that must all slow together

Every clock on the Earth runs slower in January than in July, by three parts in ten thousand million, because the orbit carries the planet deeper into the Sun's potential at perihelion. No clock on the Earth can see this, and that invisibility is the claim worth testing. If the redshift is a property of time rather than of clocks, two clocks built on different physics must slow by exactly the same fraction, and their ratio must not move with the seasons. A ratio that did move would mean the constants of nature depend on where they are measured.

astrophysics · Gravitational redshift The heat an eccentricity buys. The power tides dissipate inside Io against the eccentricity of its orbit, from (21/2)(k₂/Q) G M² R⁵ n e²/a⁶ with k₂/Q = 0.015: it grows as the square of the eccentricity. At Io's forced eccentricity, 0.0041, it is 9.3 × 10¹³ W, against about 1.0 × 10¹⁴ W measured from Io's infrared glow — the same to within the uncertainty of both — and against 4.7 × 10¹³ W for the whole of the Earth's internal heat, from a body forty-three times Io's volume. Halve the eccentricity and the heating falls fourfold; take it away and Io would freeze.

The moon kept molten by its neighbours

Io is the most volcanic body known, with hundreds of active vents and lava hotter than any erupting on the Earth, on a moon smaller than the Earth's own and far too small to have kept any heat from its formation. The heat is made continuously, by tides: Jupiter stretches Io, and because Io's orbit is slightly eccentric the stretch changes every day and a half, flexing the rock and warming it. Tides should have rounded the orbit off in a hundred thousand years and switched the heating off. They cannot, because two other moons keep tugging the orbit back into its oval, and the heat Io radiates is paid for by Jupiter's spin.

astrophysics · Self-gravity An orbit, and the circle its velocities make. Left: an orbit of eccentricity 0.6 round a central mass (dot), integrated step by step, with the velocity drawn as an arrow at twelve points equally spaced in time. Right: the same twelve velocities drawn from a single point, the origin of velocity. Their tips lie on a circle — the hodograph — of radius GM/h, whose centre is displaced from the origin by e times that radius, here 0.6: the integrated velocities stay on it to within one part in ten thousand. The fastest velocity, at the closest approach, is 1.60 times GM/h and the slowest, at the furthest point, 0.40 times. The tips are bunched near the slow end, because the body spends most of its time far away.

The velocity that always lies on a circle

A planet on an elliptical orbit speeds up and slows down, its velocity swinging round as it goes. Draw all its velocities from a single point and their tips fall exactly on a circle — not an ellipse, a circle — offset from the point by the orbit's eccentricity. Hamilton noticed this in 1846; it holds for every orbit under an inverse-square force and for no other force; and it turns the ellipse into something that can be proved with a ruler and compass, as Feynman did in a lecture that was lost for thirty years.

astrophysics · Orbit stability The tick an elliptical orbit adds to a clock. The periodic part of the time a satellite clock gains or loses against one on the ground, after its constant rate offset is removed, against hours from closest approach, for the two Galileo satellites stranded in 2014 on orbits of eccentricity 0.156 and period 12.9 hours, a GPS satellite at 0.01 and a Galileo satellite on its intended near-circular orbit. The stranded satellites' clocks run ahead and behind by ±367 ns every orbit: slow near closest approach, where they are deepest and fastest, fast near the far point. A GPS clock wobbles by ±23 ns, and a satellite on a circular orbit by almost nothing. The wobble is −2√(GMa) e sin E/c², fixed by the orbit alone, which is what makes it a test.

The satellites launched into the wrong orbit

In August 2014 two European navigation satellites were left on the wrong orbit by a frozen fuel line in their rocket's upper stage: an ellipse instead of a circle. For navigation they were nearly useless. For relativity they were a gift. A clock on an elliptical orbit dips deeper into the Earth's gravity and speeds up as it falls, then climbs and slows, and both effects make it lose time near the bottom of the ellipse and gain it back near the top — a wobble of 367 nanoseconds every 12.9 hours, set by the orbit alone. Watching that wobble for three years checked the gravitational slowing of clocks five times more closely than anything had in the forty years before.

relativity · Time dilation

Named alongside it

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

Angular momentumEquivalence principleGeneral relativityGravitational redshiftKepler orbitAtomic clockCentral forceClocksConservation lawsDegeneracyDissipationFine structure constant

All concepts