Keplerian orbit: position, velocity and the law of areas

The central mass, the semi-major axis, the eccentricity and the instant give, through Kepler’s equation, the position on the ellipse, the speed (vis-viva), the acceleration towards the occupied focus, the two apsides and the invariants h and ε. The “law of areas” view cuts the ellipse into twelve sectors of equal duration and equal area.

Kepler’s equation has no solution in elementary functions: the solver applies Newton’s method down to a residual below 10⁻¹² (nine iterations at worst at e = 0.95). Speed, apsides, swept area and invariants follow in closed form. G is the CODATA 2018 value, the one used by the gravitation and orbits tool.

Semi-major axis (a)

half the major axis of the ellipse. It alone sets the period (third law) and the energy ε = −μ/(2a).

Eccentricity (e)

the flattening of the orbit: 0 for a circle, close to 1 for a very elongated ellipse. The foci sit at distance a·e from the centre.

Anomalies (M and E)

two angles that locate the body on its orbit. M, the mean angle, grows linearly with time; E locates the true position on the ellipse. Kepler’s equation ties them together.

Periapsis and apoapsis

the two ends of the major axis: r_peri = a(1−e), the closest and fastest point; r_apo = a(1+e), the farthest and slowest. Conservation of h imposes r_peri·v_peri = r_apo·v_apo.

Areal velocity (dA/dt)

the area swept by the radius vector per unit time. It is constant and equals πab/T = h/2 (Kepler’s second law).

• Test body: m ≪ M, the central body is point-like and fixed at the occupied focus. The motion of both masses about the barycentre is not handled.
• Newtonian two-body gravitation: no J2 flattening, no atmospheric drag, no third body, no relativity.

Mercury’s perihelion advance, 43″ per century (Le Verrier, 1859), explained by general relativity (Einstein, 1915), is outside this model. The conversion 1 au = 149,597,870,700 m is exact by definition (IAU 2012 B2); the uncertainty of the results comes from G and from the inputs.

• 0 ≤ e ≤ 0.95practical · refusal

  The eccentricity is bounded between 0 (circle) and 0.95, the limit of the display and of the solver tests.

  Instead · For a near-circular orbit, enter e = 0; a near-parabolic or hyperbolic trajectory is outside this model.

On the chronicle, r/a runs from 1 − e to 1 + e; its amplitude is the eccentricity. The body spends most of its period far from the focus, where it moves slowly. For an equal duration T/12, the arc covered is long at periapsis and short at apoapsis, for the same area πab/12.

With the mean elements of Mercury proposed by the form (Standish table, JPL, valid 1800-2050) and M = GM of the Sun / G, the model gives T = 87.969 days, the observed period of Mercury, v_peri ≈ 59 km/s and v_apo ≈ 39 km/s.

1. IAU, Resolution B2 on the re-definition of the astronomical unit of length, 2012.

   Location: Resolution B2 (2012): 1 au = 149 597 870 700 m exactly · accessed 2026-08-25

Provisional · Sources and assumptions are provided; the documentary review is incomplete.

Physical model. The result follows the significant figures of the least precise input.

revision 1 · reviewed on 25 August 2026

Revision log

1. revision 1 · 2026-08-25 · scope change

   Tool creation: exact orbital state at time t on the elliptical branch (Kepler’s equation solved by Newton, residual below 10⁻¹²), apsides, invariants h and ε, acceleration μ/r², law of areas in twelve equal-duration sectors.

   First published result of the tool, with no earlier value to compare against.