Astronomy & Celestial Mechanics
the sky, computed
Concept

Focus — where it appears

One of the two interior points of a conic from which distances obey the defining relation, and the point the attracting mass occupies in an orbit. The other focus is empty, which is Kepler's first law stated as a piece of geometry.

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

An orbit at eccentricity 0.6. An orbit of eccentricity 0.6. The primary sits at a focus, offset from the centre by 0.6 of the semi-major axis, and the closest and furthest points differ by a factor of 4.00.

The orbit is an ellipse, and the Sun is not in the middle of it

Kepler's first law is usually drawn wrong. The interesting content is not the ellipse — it is the focus, and the fact that one of the two is empty.

orbits · The ellipse What an eccentricity does to the shape, and what it does to the offset. The fractional flattening 1 − b/a and the fractional focal offset c/a, against eccentricity. The offset is first order in e and the flattening is second, so at Earth's e = 0.0167 the outline is 0.014% from a circle while the Sun sits 1.67% of the semi-major axis off the centre — a factor of 120.

An orbit can look exactly like a circle and still not be one

Earth's orbit departs from a circle by fourteen parts in a hundred thousand. The Sun's offset from its centre is a hundred and twenty times larger, and everything interesting is in the offset.

orbits · The ellipse The three anomalies at E = 1.15 rad. The auxiliary circle construction. The eccentric anomaly E is measured at the centre, the true anomaly ν at the focus, and the mean anomaly M is time expressed as an angle. Here E = 1.150, ν = 1.827 and M = 0.602 radians, related by M = E − e sin E.

The position that has no formula, and is computed anyway

Kepler's equation relates where a body is to when it is there. It cannot be solved in elementary functions, Kepler said so, and nobody has managed it since — which has stopped nothing.

orbits · The ellipse Every orbit one force allows. Circle, ellipse, parabola and hyperbola, all sharing a focus and a closest approach. The eccentricity alone decides which one a body is on, and whether it returns.

Every orbit one force allows, and the number that picks between them

Circle, ellipse, parabola, hyperbola. A single inverse-square force permits exactly these four, and one number decides which — including whether the body ever comes back.

orbits · Conic sections One orbit seen from three angles. An orbit of eccentricity 0.5 with its periastron 40° from the line of nodes, seen face on and tilted by 60° and 75°, each panel centred on its apparent ellipse. Every projection is again an ellipse — of apparent eccentricity 0.88 and 0.97 — but only the face-on view has the star (red) at one of its foci (open circles). Tilted, the star sits 0.48 and 0.57 apparent semi-major axes from the nearer focus. The displacement is what measures the tilt.

An orbit seen at an angle keeps everything but its focus

No orbit outside the solar system is seen face on. It is seen as a shadow on the sky, and the shadow of an ellipse is an ellipse, swept at equal areas in equal times, indistinguishable in shape and timing from a real orbit. Only one thing gives the tilt away. The star the companion circles projects to where the true focus projects, and that is not where the apparent ellipse has its focus.

orbits · Conic sections

Named alongside it

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

EccentricityPeriapsisConic sectionsKepler's second lawOrbital elementsSemi-major axisAnalemmaAngular momentumAnomalyApparent orbitAstrometryCentral force

All concepts