Vol. Science · Astronomy · The shape

Orbital
eccentricity

Eccentricity is the number that turns a circle into a long ellipse. For a bound Keplerian orbit, 0 ≤ e < 1. Periapsis, the near end, is r_p = a(1 − e). Apoapsis, the far end, is r_a = a(1 + e). Their ratio is r_a / r_p = (1 + e) / (1 − e). When e = 0, both ends equal a and the path is a circle. As e approaches 1 the ellipse grows long and skinny, still bound, with the Sun at one focus. Speed is vis-viva, v = √[μ (2/r − 1/a)]. The same a and a smaller r make a larger speed, so periapsis is faster than apoapsis. Around the Sun, μ is the Sun Fact Sheet GM, 132,712 × 10⁶ km³/s². Earth’s orbital table prints a = 149.598 × 10⁶ km and e = 0.0167, so r_p = 147.100 × 10⁶ km, r_a = 152.096 × 10⁶ km, v_p = 30.286 km/s, and v_a = 29.291 km/s. The sheet lists 30.29 km/s and 29.29 km/s. Mercury’s e is 0.2056. Halley’s, on the comet fact sheet, is 0.967. The year still comes from a alone, which is the Kepler desk. The speed that does not come back is escape. The half-ellipse between two circles is Hohmann. A hyperbolic excess is C3. A flyby that turns a v_∞ is a gravity assist. Where a moon shreds is Roche. Where a planet dominates is Hill. The solar map is where the bodies sit on a date. This page is only the shape.

Classroom cartoon. One ellipse, one focus, no other planets pulling. Top-down from the north, so motion runs counterclockwise. The Sun is a mark, not to scale, unless the hint says the radius is drawn. Arrow length is speed, not a vector to fly. Inclination is omitted: Halley’s comet sheet lists 162.2°, and this drawing stays in one plane. Not a JPL Horizons ephemeris. Not a perturbation model. Not a mission.

The formulas

Words

Eccentricity

How far the orbit is from a circle. 0 is a circle. Toward 1, a long skinny ellipse, still closed. The focus sits a distance ae from the center, so e is also that offset divided by a.

Periapsis and apoapsis

The nearest and farthest points from the focus. Around the Sun the names are perihelion and aphelion. r_p = a(1−e) and r_a = a(1+e). This desk prints those formula ends, then sets the fact sheet’s printed distances beside them when a preset is on.

Why periapsis is faster

Vis-viva. v = √[μ (2/r − 1/a)]. Closer to the focus, 2/r is larger, so the speed is larger. The arrow at periapsis is the long one. The arrow at apoapsis shrinks by v_a / v_p.

Which a

Planet presets use the orbital-parameters semimajor axis in 10⁶ km, because μ is in km³/s². The J2000 AU line is labeled beside it and is not substituted into r_p = a(1−e). Halley has no kilometer line on the comet sheet. Its a is 17.94 AU times the IAU unit, 149,597,870.7 km.

What the ellipse leaves out

Other planets, the Sun’s finite size except as a warning when periapsis falls inside 695,700 km, inclination, and any date. Halley’s 162.2° tilt is a fact-sheet number this flat drawing does not show. Not Horizons. Not a mission. The escape desk is the speed that does not return. C3 is a hyperbolic excess. Hohmann assumes the circles this page is stretching.

Four printed orbits

Each row is a(1±e) and vis-viva at those ends, with the Sun’s GM. Planet a and e are the orbital-parameter tables. Halley’s a and e are the comet fact sheet, converted with the IAU astronomical unit. The sheet’s own perihelion, aphelion, and velocities are in the readout, not forced into these cells. Educational comparison, not an ephemeris.

Sources

Classroom picture of one ellipse. Earth’s a(1−e), with the Sun’s GM, is 30.286 km/s at periapsis and 29.291 km/s at apoapsis. The Earth Fact Sheet lists 30.29 km/s and 29.29 km/s. Point-mass, two-body. Not a Horizons ephemeris, not perturbations, and not a mission. The approximate AU ticks are landmarks, not fact-sheet lines. Earthrise preview: NASA / Bill Anders (public domain).

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