Astronomy & Celestial Mechanics
the sky, computed
Concept

Retrograde orbit — where it appears

An orbit traversed in the opposite sense to the rotation or orbital motion of the body it is compared with, with inclination above 90°. For small bodies it marks capture or a violent past rather than formation in place.

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

An inner orbit's inclination, with and without the octupole. The inclination of an inner orbit to the plane of a distant perturber against time, in units of the quadrupole timescale, starting at 65 degrees with a nearly circular orbit, for octupole strengths ε = 0 and 0.03. At ε = 0 — a circular outer orbit — the inclination oscillates in the regular Kozai–Lidov cycle and never crosses ninety degrees, because the angular momentum along the outer orbit's axis is conserved and cannot change sign. With the outer orbit eccentric, the inclination drifts from cycle to cycle and crosses ninety degrees, the orbit turning from prograde to retrograde, first at t ≈ 120, 2 times in all within the time shown. The integration conserves the secular energy to 7·10⁻⁶.

An orbit that turns over

A distant third body makes an inner orbit trade inclination for eccentricity in slow, regular cycles, and one quantity — the inner orbit's angular momentum along the outer orbit's axis — keeps the trade bounded and forbids the orbit from ever tipping past ninety degrees. That conservation holds only if the outer orbit is a circle. Make it eccentric and the quantity drifts, the cycles stop repeating, and at a moment of extreme eccentricity the inner orbit turns over, from prograde to retrograde, with its pericentre pressed a thousand times closer than the regular cycle allowed.

orbits · Kozai–Lidov The velocities that decide whether Jupiter keeps a visitor. A velocity diagram in Jupiter's orbital plane, in km/s, with the Sun's frame at the origin. Jupiter moves at 13.06 km/s (the arrow along the axis). The dashed circle is the escape speed from the Sun at Jupiter's distance, 18.47 km/s: a body whose heliocentric velocity ends inside it is bound. A visitor arriving with 3 km/s to spare reaches Jupiter's distance at 18.71 km/s, just outside the circle; here its velocity makes 150° with Jupiter's, so it meets Jupiter at 30.7 km/s. A flyby keeps that relative speed and only turns it, by at most 81° for a pass grazing the cloud tops — so the possible outcomes lie on the drawn arc of a circle of radius 30.7 km/s about Jupiter's velocity. The part of the arc inside the escape circle (darker) is the set of flybys that leave the visitor bound to the Sun; the slowest outcome is 17.7 km/s. Capture needs the visitor to arrive barely faster than escape, so that the turn of its velocity relative to Jupiter can take the small excess away.

Jupiter keeps only the visitors already nearly bound

An interstellar object passing Jupiter can lose enough speed to stay, but one flyby can only turn the visitor's velocity relative to the planet, never shorten it, and the turn removes at most a few kilometres a second from its speed around the Sun. So capture is common for visitors that arrive with less than a kilometre a second to spare and almost impossible for the speeds the three known visitors had. A census of the slow tail of the interstellar population, and the orbits a capture leaves — more than a third of them backwards — follow from the same arithmetic.

orbits · Hyperbolic orbits

Named alongside it

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

The B-planeChaosConserved quantityEscape velocityGravitational captureGravity assistHierarchical tripleHyperbolic excess speedHyperbolic orbitInterstellar objectKozai–LidovOctupole

All concepts