Concept

Kepler equation — where it appears

The relation M = E − e sin E between the mean anomaly, which grows uniformly with time, and the eccentric anomaly, which fixes the position on the ellipse. It has no closed-form solution, so finding where a body is at a given time means solving it numerically.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

$A fall into the Sun from a standing start. The distance of a body released at rest 1 AU from the Sun, as a fraction of its starting distance, against time as a fraction of the fall's total duration, 64.6 days: the exact radial orbit (solid) and a fall at the constant acceleration felt at the start (dashed). The exact fall begins exactly like the constant one and finishes far faster, because the pull grows as the inverse square of the shrinking distance: it reaches the Sun at 79 per cent of the time the constant pull would take. The curve is a cycloid, and the total time is a fifth of the circular orbital period at the starting distance — exactly 1/(4√2) of it — because a straight fall is half of an ellipse whose semi-major axis is half the starting distance.$

A fall is half an orbit with no width

Stop a planet dead and it falls into the Sun, in a time that is not a quarter of its year or a half but exactly one over four times the square root of two. The fall is an ellipse squeezed to a line, it spends four-fifths of its duration in the outer half of its path, and in a uniform cloud every shell of it arrives at the centre at the same moment — which is where the free-fall time of star formation comes from.

orbits · Conic sections $Four numbers that carry a position and a velocity round an orbit. The Lagrange coefficients f, g, ḟ and ġ — g and ḟ scaled by the mean motion so that all four are pure numbers — against time as a fraction of the period, for an ellipse of eccentricity 0.6 and periapsis 0.5 AU, each computed from one universal-variable solve starting at periapsis. The new position is f times the old position plus g times the old velocity, and the new velocity ḟ times the old position plus ġ times the old velocity: the state at any time is a linear combination of the state at the start, with no semi-major axis, eccentricity or anomaly formed. f and ġ return to one and g and ḟ to zero after a full period, which is the orbit closing. At every one of the 401 instants the combination f ġ − ḟ g equals one to 6·10⁻¹⁵, which is the conservation of angular momentum written as the area of a parallelogram.$

Four numbers that carry a state without an orbit

Once the universal variable is solved, a body's position and velocity at any later time are a linear combination of its position and velocity now, with four coefficients and no orbital elements formed at all. The four obey one identity exactly — the conservation of angular momentum, written as an area — and its failure measures the error a calculation has made without knowing the answer.

orbits · Universal variables

Errors the identity sees and errors it cannot. Propagations of an ellipse of eccentricity 0.6 through 0.37 of a period, spoiled four ways at sizes from 10⁻¹³ to a hundredth, plotted by their true relative position error across and by how far their coefficients miss f ġ − ḟ g = 1 up (both logarithmic; the bottom row is a miss below the arithmetic's resolution). A solve stopped early and coefficients jostled at random break the identity in proportion to the damage. A wrong time step and a wrong gravitational parameter do not break it at all, at any size — errors up to 0.009 of the position sit on the floor — because each produces an exact Kepler map, for a different question. One equation on four numbers leaves three directions unchecked, and two of them are the two inputs every propagation depends on.

A check that cannot see the clock

The four Lagrange coefficients that carry an orbit's state forward satisfy one identity exactly, and a propagation that breaks it is wrong by about as much as it breaks it. But one equation on four numbers leaves three directions unchecked, and two of them are the inputs every propagation depends on: the time and the strength of gravity. A state carried forward for the wrong time, or under the wrong gravitational parameter, satisfies the identity perfectly — and a solver can be written so that its own errors take exactly that form.

orbits · Universal variables

Named alongside it

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

Angular momentum Lagrange coefficients Stumpff functions Universal variable Cycloid Eccentricity Ephemeris Free-fall time Gravitational collapse Gravitational parameter Kepler's third law Orbit determination

All concepts