Astronomy & Celestial Mechanics
the sky, computed
Concept

Apparent orbit — where it appears

The ellipse a binary companion traces on the sky, the true orbit projected onto the plane perpendicular to the line of sight. Projection keeps its shape as an ellipse and its timing but moves the primary away from the focus.

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

Also named here as visual binaries — the same set of essays touches all of them, so they are one junction rather than several.

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 How the mass error grows as the record shortens. The fractional error in the system's mass, in per cent, against the fraction of an orbital period covered by 40 positions each good to 1 per cent of the orbit's size, for eccentricities 0, 0.3, 0.6. A whole orbit gives the mass to about 0.5 per cent; half an orbit, 0.8; a fifth, 2.8; a tenth, 10 for a circular orbit. Below about half a period the error grows roughly as the inverse square of the coverage, and beyond a whole period more coverage hardly helps.

A mass read from part of an orbit

Kepler's third law weighs anything that orbits — if the orbit's size and period are both known. A survey that watches a pair of stars for less time than they take to go round sees an arc, and many ellipses of different sizes and periods fit the same arc, sliding along exactly the combination the mass depends on. The error in the mass grows roughly as the inverse square of the fraction of the orbit seen, and which part of the orbit was seen matters nearly as much as how much.

orbits · Conic sections

Named alongside it

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

AstrometryConic sectionsFocusKepler's second lawVisual binariesOrbital inclinationParallel projectionThiele innes

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