Kepler's_laws_of_planetary_motion

# Kepler's laws of planetary motion  **Microsim — three.js (Wikitube framework):** *The orbits (Solar System explorer)* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/solar/Solar_System.html?view=orbits&embed=1" data-title="The orbits (Solar System explorer)"></div> *The Solar System explorer locked on this article's state (`?view=orbits`); every object and population of the [[PORTAL_Solar_System|Solar System portal]] has its own state in the same scene.*  **Kepler's laws of planetary motion** are three statements, published between 1609 and 1619, that fix the shape, the speed and the timing of a planet's [[Orbit|orbit]]: each planet moves on an [[Ellipse|ellipse]] with the [[Sun|Sun]] at one focus; the line from Sun to planet sweeps equal areas in equal times; and the square of the period is proportional to the cube of the semi-major axis.[^b077-grav] In the microsim below the reader sets two numbers, the [[Orbital_eccentricity|eccentricity]] e and the semi-major axis a, and three things respond at once: the orbit changes shape, a swept-area wedge keeps a constant area while the planet's speed follows the vis-viva relation v² = G·M·(2/r − 1/a), and a marker for the orbit drops onto a log–log chart of T against a where the eight planets already lie on a single straight line of slope 3/2, from T² = 4·π²·a³/(G·M). Presets put the reader on Mercury at e = 0.206 and on Halley's comet at e = 0.967. On the [[Physics|Physics]] flagship this article serves Part I — History at *Kepler's laws* (row P6). It is the hinge of the whole part: the point at which description of the sky stops being a fitting exercise — [[Deferent_and_epicycle|circles upon circles]] adjusted until they match — and becomes a law with no free parameters left to adjust. Kepler had no dynamics and no [[Gravity|gravitation]]; he had [[Tycho_Brahe|Tycho Brahe]]'s positions of [[Mars|Mars]], accurate to a couple of arcminutes, and the refusal to let an eight-arcminute residual pass. ## Comparison to Copernicus [[Nicolaus_Copernicus|Copernicus]] moved the [[Earth|Earth]], but he kept everything else. His planets still ride uniform circular motions, and reproducing the observations still requires small auxiliary circles — the *De revolutionibus* system uses of the order of thirty of them — so the machinery of [[Heliocentrism|heliocentrism]] is Ptolemaic machinery rebuilt around a different centre.[^b077-grav] Worse, that centre is not the Sun: Copernicus refers the orbits to the centre of the Earth's orbit, a geometrical point near the Sun but not on it, which leaves the Sun with no physical role at all. Kepler's three laws remove all of it. One ellipse replaces circle plus epicycle, so each planet needs two numbers of shape rather than a family of nested radii; the true Sun sits at a focus, and the empty focus has no body in it, which breaks the symmetry Copernicus had preserved; and the speed is no longer uniform about any point whatever. The third law then does something no version of the older scheme could: it ties the planets to one another. In Ptolemy and in Copernicus each planet's period is an independent datum; after 1619 it is a consequence of its distance. ## History Kepler joined Tycho Brahe at Benátky in 1600 and inherited the Mars data on Tycho's death the following year.[^gingerich] The *Astronomia nova* of 1609 records the eight-year fight that followed.[^astronomia] His best circular model — a "vicarious hypothesis" with an equant — reproduced Mars's longitudes to within about two arcminutes, well inside what anyone before Tycho could have detected, but it failed in latitude and left residuals of about eight arcminutes at the octants. Kepler's decision to treat those eight minutes as a fact rather than as noise is the methodological turn the rest depends on: Tycho's positions were good to roughly one to two arcminutes, so eight minutes was real.[^gingerich] He then tried ovals, found the correct area rule before he found the correct curve, and arrived at the ellipse in 1605. The harmonic law came much later, and by a different route: Kepler was pursuing musical ratios among the planetary speeds, and the relation T² ∝ a³ emerges in Book V of the *Harmonice mundi* of 1619, dated by him to a discovery on 15 May 1618.[^harmonice] ### As three laws Kepler never numbered them. He states the area rule and the ellipse as results inside the argument of the *Astronomia nova*, ten years before the harmonic law appears in a different book about music and polyhedra. The grouping into three numbered laws is later, consolidated in the eighteenth century as textbooks worked backwards from [[Isaac_Newton|Newton]],[^gingerich] and it reverses the historical order of discovery — the area law came first, the ellipse second, the harmonic law more than a decade later. The tidy triple is a retrospective tidying of a long and messy argument, but it is a fair one: the three statements are logically independent, and each fixes a different aspect of the motion. ## Formulary Written for an orbit of semi-major axis a, eccentricity e and period T about a central mass M, the three laws are compact enough to hold in one line each, and the microsim's HUD carries all three. ### First law The orbit is an ellipse with the Sun at one focus. In polar coordinates centred on that focus, r(θ) = p/(1 + e·cos θ), with p = a·(1 − e²), where p is the semi-latus rectum and θ the true anomaly measured from perihelion. The perihelion and aphelion distances are a·(1 − e) and a·(1 + e). For Earth, e = 0.0167 gives 0.9833 AU and 1.0167 AU, a variation of 3.3 per cent that has nothing to do with the seasons; for Halley's comet, e = 0.967 with a = 17.83 AU gives 0.586 AU and 35.08 AU (derived).[^jpl-elements][^halley] The ellipse is the [[Conservative_force|conservative]], bound case of the general conic solution; e = 1 is a parabola and e > 1 a hyperbola, which is why the same equation describes an interstellar visitor that never returns. ### Second law The radius vector sweeps equal areas in equal times: dA/dt is constant. Since dA/dt = ½·r²·(dθ/dt), the statement is that r²·(dθ/dt) = h is a constant of the motion, and h is [[Angular_momentum|angular momentum]] per unit mass. The second law is therefore not an empirical curiosity but conservation of angular momentum, which follows from the force being central — directed along the radius — whatever its strength.[^b076-gravity] This is the part of the sim the reader feels first. A wedge is shaded between the planet's position now and its position one-twentieth of a period ago; as the planet runs in toward perihelion the wedge becomes short and fat, and out near aphelion long and thin, with the shaded area readout never moving. The [[Velocity|speed]] follows from [[Conservation_of_energy|energy conservation]] — the sum of [[Kinetic_energy|kinetic]] and [[Potential_energy|gravitational potential energy]] is −G·M·m/(2a) on any orbit — as v² = G·M·(2/r − 1/a), and the ratio of the extreme speeds is exactly (1 + e)/(1 − e). At Mercury's e = 0.206 that is 1.52, giving 59.0 km/s at perihelion against 38.9 km/s at aphelion; at Halley's e = 0.967 it is 59.9, giving 54.6 km/s against 0.91 km/s (all derived).[^halley] Halley spends almost all of its 75 years crawling through the outer part of an orbit it crosses in weeks. ### Third law T² is proportional to a³, with the same constant of proportionality for every body orbiting the same primary. In SI units the constant is 4·π²/(G·M) = 2.975×10⁻¹⁹ s²/m³ for the Sun; in years and astronomical units it is simply 1 (derived).[^b077-grav] The log–log chart is the cleanest way to see it, because a power law becomes a straight line and the exponent becomes the slope: | planet | a (AU) | e | T (yr) | T²/a³ | |---|---|---|---|---| | Mercury | 0.3871 | 0.2056 | 0.2408 | 1.00003 | | Venus | 0.7233 | 0.0068 | 0.6152 | 1.00002 | | Earth | 1.0000 | 0.0167 | 1.0000 | 1.00003 | | Mars | 1.5237 | 0.0934 | 1.8808 | 1.00000 | | Jupiter | 5.2029 | 0.0484 | 11.863 | 0.99914 | | Saturn | 9.5367 | 0.0539 | 29.447 | 0.99978 | | Uranus | 19.189 | 0.0473 | 84.017 | 0.99900 | | Neptune | 30.070 | 0.0086 | 164.79 | 0.99878 | The ratio holds to better than one part in eight hundred across a range of eighty in distance, and the eccentricity column shows that the law does not care about it: [[Venus|Venus]] at e = 0.007 and Mercury at e = 0.206 sit on the same line.[^jpl-elements] The small deficit at [[Jupiter|Jupiter]] is not scatter. The exact relation is T² = 4·π²·a³/(G·(M + m)), and multiplying Jupiter's entry by (1 + m/M) with m/M = 9.55×10⁻⁴ returns 1.00010 (derived) — the planet's own mass, read off a chart of periods. ## Planetary acceleration Newton's route, set out in the *Principia* of 1687, runs the laws backwards: each is translated into a statement about acceleration, and the three together identify the [[Force|force]].[^newton] ### Acceleration vector In plane polar coordinates the acceleration of a body at (r, θ) is a = (r̈ − r·θ̇²)·r̂ + (r·θ̈ + 2·ṙ·θ̇)·θ̂, and the transverse bracket is (1/r)·d(r²·θ̇)/dt. Kepler's second law says r²·θ̇ is constant, so the transverse component vanishes identically and the acceleration is purely radial.[^b074-central] The second law alone, before any assumption about strength, establishes that whatever acts on a planet acts along the line to the Sun. ### Inverse-square law Substituting the first law into the radial component fixes the strength. With r = p/(1 + e·cos θ) and r²·θ̇ = h, differentiating twice gives r̈ − r·θ̇² = −h²/(p·r²): the radial acceleration falls off as the inverse square of the distance, and the constant h²/p belongs to the orbit.[^b074-central] The first two laws together therefore give an inverse-square attraction toward the Sun, but they do not yet say that the constant is the same for every planet. ### Newton's law of gravitation That is what the third law adds. For an ellipse the areal rate gives T = 2·π·a·b/h with b = a·√(1 − e²), and eliminating h shows h²/p = 4·π²·a³/T². By the third law a³/T² is one number for all the planets, so the constant in the inverse-square acceleration is a property of the Sun alone, G·M.[^b076-gravity] [[Newton's_laws_of_motion|Newton's third law of motion]] then forces the force to be symmetric between the two bodies, giving [[Newton's_law_of_universal_gravitation|F = G·m·M/r²]], and the two-body correction T² = 4·π²·a³/(G·(M + m)) appears — the departure the table shows for Jupiter.[^b080-idema] At that point Kepler's laws stop being laws and become theorems, and the constant that Kepler found among the planets becomes the same constant that governs [[Free_fall|free fall]] at the Earth's surface. ## Position as a function of time The laws give the orbit, but not the planet's place at a given date: the second law makes the angular rate vary, and the relation between time and angle has no closed form. Kepler's solution introduces two auxiliary angles between them, and the microsim uses the same chain to advance its planet each frame. ### Mean anomaly, <i>M</i> The mean anomaly is the angle a fictitious body would have covered moving uniformly on a circle in the same period: M = 2·π·(t − t₀)/T, with t₀ the time of perihelion. It is proportional to time by construction and carries no geometry, so it is the clock the other two angles are read against. ### Eccentric anomaly, <i>E</i> The eccentric anomaly is measured at the centre of the ellipse on the circumscribed circle of radius a. It is linked to the clock by Kepler's equation, M = E − e·sin(E), which is transcendental and cannot be inverted in elementary functions. Newton's method, E ← E − (E − e·sin E − M)/(1 − e·cos E), started at E = M, converges in about four iterations for planetary eccentricities (derived); for Mercury at M = 45° it gives E = 54.604°.[^b074-central] ### True anomaly, <i>θ</i> The true anomaly is the actual angle at the focus, from perihelion to the planet. It follows from E by tan(θ/2) = √((1 + e)/(1 − e))·tan(E/2), so for the Mercury example E = 54.604° gives θ = 64.906° (derived). The gap between M = 45° and θ = 64.9° is the second law made visible: a fifth of a period after perihelion the planet has already turned through more than a sixth of a full circle. ### Distance, <i>r</i> Finally r = a·(1 − e·cos E), which for the same example gives r = 0.8809·a (derived), or 0.341 AU. The chain M → E → θ → r is the standard machinery of orbital position, and it is unchanged from Kepler's own: everything after him has added perturbations and [[General_relativity|relativistic]] corrections around it, not replaced it. ## See also - [[Ellipse]] - [[Orbital_eccentricity]] - [[Johannes_Kepler]] - [[Tycho_Brahe]] - [[Newton's_law_of_universal_gravitation]] - [[Apparent_retrograde_motion]] - [[Orbit]] ## Explanatory notes - Kepler's three statements are logically independent: the second follows from any central force, the first and second together give the inverse square, and only the third makes the constant universal. - The semi-major axes and periods in the third-law table are J2000 mean elements. Rounded "current-value" planetary tables mix osculating and mean quantities and do not reproduce T²/a³ to better than about one per cent, so a consistent element set is needed before the constancy can be checked at all. - Footnote definitions for this page are collected under *References*. ## References [^b077-grav]: Ling, Samuel J.; Sanny, Jeff; Moebs, William (2016). *University Physics Volume 1*. OpenStax. Chapter 13, "Gravitation", pp. 611–664 (Kepler's three laws, the derivation of the third law from the inverse-square force, and orbital energy; page to pin), with Appendix D "Astronomical Data", pp. 885–886. Portal Book 077. https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-1 [^b076-gravity]: Gea-Banacloche, Julio (2019). *University Physics I: Classical Mechanics*. Chapter 12, "Gravity", pp. 239–270 (angular momentum and the area law; Kepler's third law from the central force; page to pin). Portal Book 076. https://open.umn.edu/opentextbooks/textbooks/university-physics-i-classical-mechanics [^b074-central]: Likharev, Konstantin K. (2013). *Essential Graduate Physics, Part CM: Classical Mechanics*. Chapter 3, pp. 33–54 (conservative two-body central forces: the effective potential, the conic solution, and the anomalies; page to pin). Portal Book 074. https://open.umn.edu/opentextbooks/textbooks/part-cm-classical-mechanics [^b080-idema]: Idema, Timon (2018). *Mechanics and Relativity*. Part I, "Classical mechanics", pp. 14–119 (the two-body problem, reduced mass, and the corrected third law; page to pin). Portal Book 080. https://open.umn.edu/opentextbooks/textbooks/mechanics-and-relativity [^astronomia]: Kepler, Johannes (1609). *Astronomia nova*. Prague. Chapters 19 and 40–60 (the vicarious hypothesis and its eight-arcminute residual, the area rule, and the ellipse; page to pin). [^harmonice]: Kepler, Johannes (1619). *Harmonice mundi*. Linz: Johann Planck. Book V, chapter 3, proposition 8 (the harmonic law, with Kepler's own date of 15 May 1618 for the discovery; page to pin). [^gingerich]: Gingerich, Owen (1993). *The Eye of Heaven: Ptolemy, Copernicus, Kepler*. New York: American Institute of Physics (the accuracy of Tycho's positions, the eight-minute residual, and the number of circles in the Copernican system; page to pin). [^jpl-elements]: J2000 mean orbital elements for the eight planets (semi-major axis, eccentricity and sidereal period) as tabulated in Standish, E. M.; Williams, J. G., "Orbital Ephemerides of the Sun, Moon, and Planets", in Urban, Sean E.; Seidelmann, P. Kenneth, eds. (2013), *Explanatory Supplement to the Astronomical Almanac*, 3rd ed., Mill Valley: University Science Books (page to pin). The T²/a³ column, the perihelion and aphelion speeds, and the Jupiter mass-correction figure are computed for this article from those elements and are not quoted from the source. [^halley]: Osculating elements of comet 1P/Halley (a ≈ 17.83 AU, e ≈ 0.967, period ≈ 75 years) as given by Yeomans, D. K.; Kiang, T. (1981), "The long-term motion of comet Halley", *Monthly Notices of the Royal Astronomical Society* 197 (pages to pin). The perihelion and aphelion distances and speeds quoted here are computed for this article from those elements. [^newton]: Newton, Isaac (1687). *Philosophiæ Naturalis Principia Mathematica*. London: Royal Society. Book I, Propositions 1–4 (the area law from a central force), Proposition 11 (the ellipse implies the inverse square) and Book III (the third law applied to the planets and their satellites); page to pin. ## General bibliography - Kepler, *Astronomia nova* (1609) and *Harmonice mundi* (1619), cited above. - Newton, *Principia* (1687), Books I and III. - Ling, Sanny and Moebs (2016), *University Physics Volume 1*, chapter 13, pp. 611–664. Portal Book 077. - Gea-Banacloche (2019), *University Physics I: Classical Mechanics*, chapter 12, pp. 239–270. Portal Book 076. - Likharev (2013), *Part CM: Classical Mechanics*, chapter 3, pp. 33–54. Portal Book 074. - Idema (2018), *Mechanics and Relativity*, Part I, pp. 14–119. Portal Book 080. ## External links - [*University Physics Volume 1*](https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-1), OpenStax — Portal Book 077, chapter 13 "Gravitation" - [*University Physics I: Classical Mechanics*](https://open.umn.edu/opentextbooks/textbooks/university-physics-i-classical-mechanics), Gea-Banacloche — Portal Book 076, chapter 12 "Gravity" - [*Part CM: Classical Mechanics*](https://open.umn.edu/opentextbooks/textbooks/part-cm-classical-mechanics), Likharev — Portal Book 074, chapter 3 on central forces  **Microsim — three.js (Wikitube framework):** *Kepler's laws of planetary motion* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Kepler's_laws_of_planetary_motion.html" data-title="Kepler's laws of planetary motion"></div> *Built from `MICROSIM_GUIDE/specs/sims/Kepler's_laws_of_planetary_motion.json`; part of the [[PORTAL_Matter|Matter portal]] spine (section sims and See-also variants).*  ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Kepler's_laws_of_planetary_motion) : [Wikitube](https://en.wikitube.io/wiki/Kepler's_laws_of_planetary_motion) · pinned revision [1374219016](https://en.wikipedia.org/w/index.php?oldid=1374219016) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. 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