--- title: Orbit summary: The path one body follows around another under gravity. For two bodies alone it is an ellipse, fixed by Kepler's three laws and described by six orbital elements. science_status: [observed, sim] categories: [Orbital mechanics, Physics concepts, Planetary systems] aliases: [Orbits, Kepler's laws, Orbital elements, Keplerian orbit, Kepler's equation, Vis-viva equation, Hill sphere, Roche limit] infobox: type: physics_concept name: Keplerian orbit definition: "The path of a body moving under the gravity of a single point mass: a conic section (ellipse, parabola or hyperbola) with the combined centre of mass at one focus." formulae: - name: "Kepler's third law, Newton's form" expression: "P^2 = 4 pi^2 a^3 / (G (M + m))" symbols: "P period (s); a semi-major axis (m); G = 6.674e-11 m^3 kg^-1 s^-2; M and m the two masses (kg)" - name: "Vis-viva equation" expression: "v^2 = G (M + m) (2/r - 1/a)" symbols: "v orbital speed (m/s); r present distance between the bodies (m)" - name: "Kepler's equation" expression: "M = E - e sin E" symbols: "M mean anomaly (rad), growing uniformly with time; E eccentric anomaly (rad); e eccentricity (dimensionless)" - name: "Hill radius" expression: "r_H = a (m / 3M)^(1/3)" symbols: "m the orbiting body's mass; M the central mass; a the orbit's semi-major axis" - name: "Fluid Roche limit" expression: "d = 2.44 R (rho_M / rho_m)^(1/3)" symbols: "R radius of the primary; rho_M and rho_m the densities of primary and satellite" - name: "Relativistic periapsis advance, per orbit" expression: "delta_omega = 6 pi G M / (c^2 a (1 - e^2))" symbols: "c = 299,792,458 m/s; result in radians per orbit" validity: "Exact for two point masses in Newtonian gravity. Real orbits drift slowly under the pull of other bodies, the central body's equatorial bulge and general relativity." worked_example: "Earth: a = 1.000 AU, e = 0.0167, P = 365.256 days; speed 30.29 km/s at perihelion in early January and 29.29 km/s at aphelion in early July." sim_use: "Every planet and moon in Pax Abyssi moves on a two-body Keplerian orbit, solved every tick from its elements at true scale." misconceptions: - "Earth's seasons come from its axial tilt; the 3% change in distance between perihelion and aphelion is far too small to drive them." - "Mercury's relativistic perihelion advance is 43 arcseconds per century; the other 532 arcseconds of its observed 575 come from the other planets and the Sun's oblateness." - "Titius-Bode spacing describes the pattern of one system; no law of orbital mechanics requires it." sim: entity: physics.orbit.keplerian refs: - id: murray1999 type: book author: [{family: Murray, given: Carl D.}, {family: Dermott, given: Stanley F.}] title: "Solar System Dynamics" publisher: Cambridge University Press issued: 1999 DOI: 10.1017/CBO9781139174817 - id: standish1992 type: webpage author: [{family: Standish, given: E. Myles}, {family: Williams, given: James G.}] title: "Keplerian Elements for Approximate Positions of the Major Planets" container-title: JPL Solar System Dynamics URL: https://ssd.jpl.nasa.gov/planets/approx_pos.html accessed: 2026-09-27 - id: park2021 type: article-journal author: [{family: Park, given: Ryan S.}, {family: Folkner, given: William M.}, {family: Williams, given: James G.}, {family: Boggs, given: Dale H.}] title: "The JPL Planetary and Lunar Ephemerides DE440 and DE441" container-title: The Astronomical Journal volume: 161 page: 105 issued: 2021 DOI: 10.3847/1538-3881/abd414 - id: will2014 type: article-journal author: [{family: Will, given: Clifford M.}] title: "The Confrontation between General Relativity and Experiment" container-title: Living Reviews in Relativity volume: 17 page: 4 issued: 2014 DOI: 10.12942/lrr-2014-4 - id: park2017 type: article-journal author: [{family: Park, given: Ryan S.}, {family: Folkner, given: William M.}, {family: Konopliv, given: Alexander S.}, {literal: "et al."}] title: "Precession of Mercury's Perihelion from Ranging to the MESSENGER Spacecraft" container-title: The Astronomical Journal volume: 153 page: 121 issued: 2017 DOI: 10.3847/1538-3881/aa5be2 - id: gravity2020 type: article-journal author: [{literal: GRAVITY Collaboration}, {family: Abuter, given: R.}] title: "Detection of the Schwarzschild precession in the orbit of the star S2 near the Galactic centre massive black hole" container-title: Astronomy & Astrophysics volume: 636 page: L5 issued: 2020 DOI: 10.1051/0004-6361/202037813 - id: gladman1996 type: article-journal author: [{family: Gladman, given: Brett}, {family: Quinn, given: D. Dane}, {family: Nicholson, given: Philip}, {family: Rand, given: Richard}] title: "Synchronous Locking of Tidally Evolving Satellites" container-title: Icarus volume: 122 page: 166-192 issued: 1996 DOI: 10.1006/icar.1996.0117 - id: holman1999 type: article-journal author: [{family: Holman, given: Matthew J.}, {family: Wiegert, given: Paul A.}] title: "Long-Term Stability of Planets in Binary Systems" container-title: The Astronomical Journal volume: 117 page: 621-628 issued: 1999 DOI: 10.1086/300695 - id: doyle2011 type: article-journal author: [{family: Doyle, given: Laurance R.}, {family: Carter, given: Joshua A.}, {family: Fabrycky, given: Daniel C.}, {literal: "et al."}] title: "Kepler-16: A Transiting Circumbinary Planet" container-title: Science volume: 333 page: 1602-1606 issued: 2011 DOI: 10.1126/science.1210923 images_wanted: - file: File:Orbital_elements_diagram.svg subject: "Diagram of the six orbital elements: a tilted ellipse over a reference plane, the line of nodes, the ascending node, the inclination i, the longitude of the ascending node (Omega) measured from the reference direction, the argument of periapsis (omega) measured in the orbit plane, the semi-major axis a, and the body at its true anomaly nu" source: other note: "shot list: to be drawn as a clean SVG diagram in the site's style; no existing agency image" - file: File:S2_Schwarzschild_precession_ESO.jpg subject: "Artist's impression of the rosette-shaped, precessing orbit of the star S2 around Sagittarius A*, with the effect exaggerated for clarity" source: eso page_url: https://www.eso.org/public/images/eso2006a/ credit: "ESO/L. Calçada" licence: "CC BY 4.0 (ESO usage terms, https://www.eso.org/public/copyright/)" --- An **orbit** is the path one body follows around another under their mutual gravity. When only two bodies matter, the path is a conic section with the pair's centre of mass at one focus: a circle or ellipse for a body that is bound, a parabola or hyperbola for one that is passing through. Johannes Kepler found the rules for planets from Tycho Brahe's observations early in the seventeenth century, Isaac Newton derived them from his law of gravitation, and six numbers, the orbital elements, are still how every planet, moon and spacecraft in the Solar System is catalogued :cite[murray1999]. ## Kepler's three laws Kepler published the first two laws in 1609 and the third in 1619 :cite[murray1999]. 1. **Each planet moves on an ellipse with the Sun at one focus.** The ellipse's size is its semi-major axis $a$, half its longest diameter, and its shape is its eccentricity $e$, from 0 for a circle toward 1 for a long, thin ellipse. The nearest point to the Sun, the perihelion, lies at $a(1-e)$ and the farthest, the aphelion, at $a(1+e)$. 2. **The line from the Sun to the planet sweeps out equal areas in equal times.** A planet therefore moves fastest at perihelion and slowest at aphelion. The law is conservation of angular momentum in geometric form. 3. **The square of the period is proportional to the cube of the semi-major axis.** In units of years and astronomical units around the Sun, $P^2 = a^3$. Jupiter, at 5.20 AU, takes $5.20^{3/2} = 11.9$ years. Newton's version of the third law holds for any pair of bodies and brings in their masses: $$ P^2 = \frac{4\pi^2 a^3}{G\,(M + m)} $$ Here $G$ is the gravitational constant, $6.674 \times 10^{-11}\ \mathrm{m^3\,kg^{-1}\,s^{-2}}$, and $M$ and $m$ are the two masses. Timing an orbit is therefore how astronomers weigh things. The period and size of the Moon's orbit give the mass of Earth; the orbits of stars around the centre of the Milky Way give the mass of [[Sagittarius A*]]; the wobble of a star around its planet gives the planet's mass. ## Speed along the orbit Energy conservation gives the speed at any point, in what is called the vis-viva equation: $$ v^2 = G\,(M + m)\left(\frac{2}{r} - \frac{1}{a}\right) $$ where $r$ is the present distance between the two bodies. On a circle, $r = a$ and the speed is $\sqrt{GM/a}$: 29.78 km/s for Earth. Earth's small eccentricity, 0.0167, lifts that to 30.29 km/s at perihelion in early January and lowers it to 29.29 km/s at aphelion in early July. Setting $a$ to infinity gives the escape speed, $\sqrt{2GM/r}$, 42.1 km/s at Earth's distance from the Sun. Halley's Comet, with $a \approx 17.8$ AU and $e \approx 0.967$, shows the extremes: it rounds the Sun at about 55 km/s at 0.59 AU and crawls at under 1 km/s near its aphelion, 35 AU out, beyond Neptune. ## The six orbital elements Two numbers give an ellipse's size and shape. Three more fix how it sits in space, and one says where the body is on it at a chosen moment, the epoch. | Element | Symbol | What it sets | |---|---|---| | Semi-major axis | $a$ | Size of the orbit, and through Kepler's third law its period | | Eccentricity | $e$ | Shape, from 0 (circle) toward 1 | | Inclination | $i$ | Tilt of the orbit to the reference plane; above 90 degrees the motion is retrograde | | Longitude of the ascending node | $\Omega$ | Where the orbit crosses the reference plane going north, measured from a reference direction | | Argument of periapsis | $\omega$ | Where the closest point lies, measured within the orbit from the ascending node | | Mean anomaly at epoch | $M_0$ | Where the body is at the epoch | For planets the reference plane is usually the ecliptic of the year 2000, J2000; for moons it is often the planet's equator. A set of elements is always tied to its epoch and its frame, and swapping frames without converting is a classic source of error :cite[murray1999]. ::figure{src="File:Orbital_elements_diagram.svg" size=wide alt="Diagram of an ellipse tilted above a reference plane, with the line of nodes, inclination, longitude of the ascending node, argument of periapsis and true anomaly marked" caption="Diagram: the six elements that place an orbit in space and a body on it."} ## Kepler's equation The second law says where a planet is at any time, but not in closed form. The practical route runs through two angles. The **mean anomaly** $M$ grows at a steady rate, $M = M_0 + n\,(t - t_0)$ with mean motion $n = 2\pi/P$, as if the planet moved on a circle at constant speed. The **eccentric anomaly** $E$ is a geometric angle measured from the ellipse's centre. They are linked by Kepler's equation: $$ M = E - e \sin E $$ There is no algebraic solution for $E$, so it is found by iteration. Newton's method converges quickly; Halley's method, which also uses the second derivative, converges faster still. Once $E$ is known, the distance is $r = a(1 - e\cos E)$ and the true anomaly $\nu$, the actual angle from perihelion, follows from $$ \tan\frac{\nu}{2} = \sqrt{\frac{1+e}{1-e}}\,\tan\frac{E}{2} $$ **Worked example: Mercury.** Take $e = 0.2056$ and $a = 0.3871$ AU, and ask where Mercury is a quarter of its 88-day year after perihelion, when $M = 90$ degrees (1.5708 rad). A standard first guess, $E_0 = M + e\sin M$, gives 1.7764 rad; one Halley step corrects it to $E = 1.7722$ rad (101.5 degrees), which satisfies the equation to about one part in a million. Then $r = 0.403$ AU and $\nu = 112.9$ degrees. In a quarter of its year Mercury has swept almost a third of the way round the Sun, because it moves fastest near perihelion, as the second law says it must. ## Where two-body orbits stop being enough ### Other bodies pull too In the real Solar System every planet tugs on every other, so orbital elements drift. The Earth-Moon barycentre's longitude of perihelion, for example, advances by about 0.32 degrees per century. Published sets of approximate elements therefore carry rates of change and a stated range of validity: the Standish and Williams elements, good to about 20 arcseconds for Earth between 1800 and 2050, are the familiar example :cite[standish1992]. For precise work, the planets are integrated together numerically, as in JPL's DE440 and DE441 ephemerides, fitted to decades of radar ranging, spacecraft tracking and lunar laser ranging :cite[park2021]. ### Hill spheres and Roche limits A moon can stay with its planet only within a region where the planet's gravity dominates the star's tides. Its size is roughly the **Hill radius**, $r_H \approx a\,(m/3M)^{1/3}$ :cite[murray1999]. Earth's is 0.01 AU, 1.5 million km; the Moon, at 384,400 km, sits about a quarter of the way out. Jupiter's is 0.355 AU, some 740 Jupiter radii, room for its whole retinue of distant captured moons. Stable orbits reach only part of the way to the Hill radius; see [[Natural satellite]]. At the other extreme, a moon held together only by its own gravity is pulled apart by tides inside the **Roche limit**, about $2.44\,R\,(\rho_M/\rho_m)^{1/3}$ for a fluid body, where $R$ and $\rho_M$ are the planet's radius and density and $\rho_m$ the moon's :cite[murray1999]. For icy particles at Saturn it comes to about 134,000 km, just inside the outer edge of the bright A ring at 136,800 km. Rigid bodies with internal strength survive somewhat closer. ### Tides lock spins Tides raised on a moon by its planet drain the moon's spin until it turns once per orbit, keeping one face toward the planet. The despinning time grows as the sixth power of the orbital distance, so close moons lock quickly and distant ones may never lock :cite[gladman1996]. Earth's Moon, Jupiter's four large moons and most regular moons in the Solar System are locked. ### Two suns Planets can orbit one star of a binary (an S-type orbit) or both (a P-type, or circumbinary, orbit). Holman and Wiegert fitted the stability boundaries from numerical experiments :cite[holman1999]. For two equal stars on a circular orbit, a planet around one star is safe out to about 0.27 of the binary's separation, and a circumbinary planet must stay beyond about 2.4 separations. Kepler-16 b, a Saturn-mass planet circling a pair of stars that orbit each other every 41 days, sits at 0.70 AU, just outside the roughly 0.65 AU limit that the fit gives for its binary :cite[doyle2011]. ### Relativity General relativity adds a small extra turn to every eccentric orbit. To first order, the periapsis advances by $$ \Delta\omega = \frac{6\pi G M}{c^2\,a\,(1 - e^2)} $$ radians per orbit. For Mercury that is 0.1035 arcseconds per orbit, or 42.98 arcseconds per century :cite[will2014]. Mercury's total perihelion advance, measured from ranging to the MESSENGER spacecraft, is 575.31 arcseconds per century; the other planets and the Sun's slight oblateness account for the rest, and Einstein's term closes the gap that Newtonian gravity left in the nineteenth century :cite[park2017]. Near a black hole the effect is large. The star S2, whose 16-year orbit brings it within about 120 AU of Sagittarius A*, advances by 12 arcminutes per orbit, a precession detected by the GRAVITY instrument in 2020 :cite[gravity2020]. ::figure{src="File:S2_Schwarzschild_precession_ESO.jpg" size=wide alt="Artist's impression of a star tracing a rosette of overlapping ellipses around a bright point" caption="Artist's concept: relativistic precession turns an orbit into a rosette. The effect is exaggerated here; for S2 it is 12 arcminutes per orbit."} ## How we know Kepler worked from naked-eye positions accurate to about an arcminute. Modern orbits come from a far wider base: radar and laser ranging to planets, the Moon and spacecraft; spacecraft tracking by Doppler shift; and astrometry of asteroids and moons from ground telescopes and from Gaia. JPL's ephemerides fit all of it together, and the fits are good enough to measure the Sun's oblateness and test relativity at the level of parts in $10^5$ :cite[park2021] :cite[park2017]. Outside the Solar System, orbits are read from a star's radial-velocity wobble, the timing of transits, and, for the stars around Sagittarius A*, direct imaging over decades. ## Notable orbits | Body | $a$ | $e$ | Period | Why it is notable | |---|---|---|---|---| | Earth | 1.000 AU | 0.0167 | 365.256 d | Defines the astronomical unit and the ecliptic | | Mercury | 0.387 AU | 0.206 | 88.0 d | The classic test of general relativity | | Halley's Comet | about 17.8 AU | about 0.967 | about 75 yr | A retrograde orbit (inclination about 162 degrees) reaching beyond Neptune | | Kepler-16 b | 0.705 AU | small | 229 d | The first fully characterised circumbinary planet | | S2 | about 1,000 AU | 0.88 | about 16 yr | Relativistic precession around a black hole | ::orrery{system=sol} :::callout{type=sim title="In Pax Abyssi"} Every planet and moon in Pax Abyssi moves on a two-body Keplerian orbit at true scale, in double precision, with its position recomputed every tick from its six elements. Kepler's equation is solved by Halley iteration, the method described above, and the game's C++ solver was checked against the Python original it was ported from: the two agree to within 0.8 mm across Sol's nine planets. Periods come from Kepler's third law and the star's mass. Sol carries 37 orbiting bodies: nine planets, Pluto included, and 28 real moons. The planets use J2000 elements; the moons were fitted to JPL Horizons ephemerides and are referenced to their planets' equators. The elements do not change with time, so the game has no secular drift, no resonant interactions between moons and no N-body integration. Where a real ephemeris would show the planets' orbits slowly turning, the game's stay fixed. The orrery draws each planet's Hill sphere, the Lagrange points and Roche lobes, and can plan a live Hohmann transfer: Earth to Mars needs burns of 2.94 and 2.65 km/s and takes 259 days. In generated systems, each [[Planetary system archetypes|system archetype]] sets its own planet spacing. A mutual-Hill-radius stability check, relativistic precession and the Holman-Wiegert binary limits exist in the simulation's code but do not yet act on any orbit in the game. ::: ## See also - [[Planetary system archetypes]] - [[Star system generation]] - [[Natural satellite]] - [[Asteroid belt]] - [[Sol]] - [[Black hole]] - [[Sagittarius A*]]