Astronomy & Celestial Mechanics
the sky, computed
Concept

Critical inclination — where it appears

The tilt of about 39.23 degrees past which a distant perturber can drive a circular orbit to high eccentricity. It is arccos of the square root of three fifths, a number with no astronomy in it at all, and below it the mechanism does nothing for any perturber whatever.

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

An eccentricity and an inclination trading, at 65° of mutual tilt. The secular equations integrated from a nearly circular orbit (e = 0.02) inclined at 65° to a distant perturber's plane, over three oscillations. Above, the eccentricity; below, the inclination, with the constant √(1−e²)cos i drawn as the flat line it is. The eccentricity climbs to 0.8380 and the inclination falls to 39.25° at the same instant, and neither is a coincidence: the product is fixed, so one can only rise as the other falls. That floor is the same for every starting tilt — at maximum eccentricity j = √(5/3)Θ, so cos i = √(3/5) and the inclination arrives at 39.23° whether the orbit began at 50° or at 89°. The closed form for a circular start is e_max = √(1 − (5/3)cos²i₀) = 0.8380, which contains nothing about the perturber — not its mass, not its distance. Those set the clock and not the amplitude, and the period here is 4.83 Kozai times. What the figure cannot show is what happens at the top of the cycle in a real system: at e = 0.838 the pericentre is 0.1620 of the semi-major axis, where tides, general relativity or a stellar surface all intervene, and the quadrupole picture ends.

An inclination that turns into an eccentricity

A distant companion cannot change an orbit's size or its energy. It can take a circular orbit tilted past 39.23 degrees and drive it to an eccentricity near one, and back, over and over — and the companion's mass and distance set only the clock.

orbits · Kozai–Lidov A thousandth of the field, and all of the precession. Left: the Earth's figure against a sphere of the same equatorial radius, with the flattening drawn 28× its true value. The real difference between the equatorial and polar radii is 21.4 km on 6378 — 1 part in 298 — which at this size would be 0.5 pixels and invisible, so the drawing is a schematic and the number is here instead. Right: the two components of the J₂ perturbation at the surface, each as a fraction of the monopole μ/r², both differentiated from the potential rather than quoted. The radial one strengthens the inward pull by 1.62×10⁻³ over the equator, where the extra mass is, and weakens it by 3.25×10⁻³ over the poles, vanishing at ±35.26° where P₂ does. The transverse one is zero at the equator and at both poles and peaks at ±45°, at 1.62×10⁻³ — and that is the component that does the work. It pulls an inclined orbit back towards the equatorial plane, which is a torque about the line of nodes, and a torque applied to something already turning moves it sideways rather than back. Averaged over an orbit the pair leave a, e and i untouched and turn the whole plane instead, which is why a term a thousandth of the field is the largest single perturbation on almost every satellite ever flown.

The Earth's shape, read off a satellite's node

The Earth is a thousandth of a part from being a sphere, and that thousandth turns every satellite's orbital plane. Vanguard 1 measured it in 1959 — and one retrograde inclination turns the plane at exactly the rate the Sun moves, which is a perturbation used as a design constraint rather than corrected for.

gravitation · Oblateness The figure of eight a geosynchronous orbit draws at 5°, 15°, 30° of inclination. The ground track over one sidereal day of a circular orbit whose period is exactly a sidereal day, at inclinations of 5, 15, 30°, centred on its own mean longitude. It is not a point. The latitude swings to ±i and back twice a day, and the longitude falls behind and then runs ahead of the Earth's rotation, because the rate at which an inclined orbit gains longitude is u̇ cos i / cos²φ: slowest at the nodes, where part of the motion is north–south, and fastest at the extremes of latitude, where all of it is eastward and a degree of longitude is shorter — so the track closes as a figure of eight. Its half-width in longitude is ±0.109° at 5°, ±0.993° at 15°, ±4.117° at 30° — exactly arcsin(tan²(i/2)) — against the small-inclination form i²/4 = ±0.109°, ±0.982°, ±3.927°: quadratic in the inclination, so the eight is tall and very thin. The longitude axis is stretched 8 times relative to the latitude axis, and without that stretch every one of these curves would be drawn as a vertical line. This is why a few degrees of inclination, which a geostationary operator spends most of its propellant preventing, moves the satellite a long way north and south and almost not at all east and west.

A stationary satellite that draws a figure of eight

A satellite with a period of exactly one sidereal day returns over the same ground every day, but only an orbit in the equator with no eccentricity returns over a single point. A few degrees of tilt draw a figure of eight, a little eccentricity a swing in longitude, and the two together draw the figure the Sun draws in the sky over a year.

spaceflight · Ground tracks

Named alongside it

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

AnalemmaApsidal precessionArgument of perigeeEccentricityEquation of timeGeoidGeostationary orbitGeosynchronous orbitGround trackHierarchical tripleHigh-eccentricity migrationInclination

All concepts