{"id":1889,"date":"2026-09-15T07:15:04","date_gmt":"2026-09-15T07:15:04","guid":{"rendered":"https:\/\/physicsfundamentalsinfo.com\/blog\/?p=1889"},"modified":"2026-09-15T07:15:06","modified_gmt":"2026-09-15T07:15:06","slug":"keplers-laws","status":"publish","type":"post","link":"https:\/\/physicsfundamentalsinfo.com\/blog\/mechanics\/keplers-laws\/","title":{"rendered":"Kepler’s Laws of Planetary Motion"},"content":{"rendered":"\n
Definition<\/div>

\nKepler’s laws are three rules for how planets orbit the Sun: each moves on an ellipse with the Sun at one focus, the Sun\u2013planet line sweeps equal areas in equal times, and the period squared is proportional to the semi-major axis cubed. In years and astronomical units, T\u00b2 = a\u00b3. Newton derived all three from gravity.\n<\/p><\/div>\n\n

Earth is about 5 million kilometres closer to the Sun in early January than in early July, and it travels faster in January to match. Mercury’s swing is far wider, and a comet’s can carry it from inside Earth’s orbit to beyond Neptune’s. None of it is random.<\/p>\n\n

Three short rules account for every one of those motions. Johannes Kepler extracted them from Tycho Brahe’s naked-eye observations, above all of Mars, long before anyone knew why they worked, and they still give each planet’s period to within a tenth of a per cent from the size of its orbit alone. This guide builds each law up with scale diagrams, JPL data for all eight planets, seven solved examples and a lab in which you can reshape an orbit yourself.<\/p>\n\n

What Are Kepler’s Laws?<\/h2>\n\n

Kepler’s laws are three rules that fix a planet’s orbit completely: its shape, its changing speed and the time it takes to go round once. Stated in full, they read:<\/p>\n\n

    \n
  1. First law, the law of ellipses.<\/strong> Every planet moves on an ellipse, with the Sun at one of the ellipse’s two foci rather than at its centre.<\/li>\n
  2. Second law, the law of equal areas.<\/strong> The line joining the Sun to a planet sweeps out equal areas in equal intervals of time, so the planet speeds up as it nears the Sun and slows as it pulls away.<\/li>\n
  3. Third law, the law of periods.<\/strong> The square of the orbital period T is proportional to the cube of the semi-major axis a, the orbit’s half-length. For anything circling the Sun, with T in years and a in astronomical units, T\u00b2 = a\u00b3.<\/li>\n<\/ol>\n\n

    Each law answers a different question. The first draws the track, the second sets the timetable along it, and the third tells you how long one lap takes. That third answer depends on the size of the orbit and the mass of the star and on nothing else: stretch a circle into a long, thin ellipse with the same semi-major axis and the year does not get a second longer.<\/p>\n\n

    Kepler found these rules in data, not in theory. The explanation came decades later from Isaac Newton, whose laws of motion<\/a>, combined with an inverse-square pull of gravity, turn all three into consequences of a single force. The same rules therefore govern moons, artificial satellites and planets round other stars, provided you swap in the mass of whatever sits at the focus; for binary stars, whose masses are comparable, M becomes the sum of the two (see Newton’s correction below).<\/p>\n\n

    The Formulas Behind Kepler’s Laws<\/h2>\n\n

    Each law has a compact mathematical form. The first is the equation of an ellipse measured from the Sun, which gives the distance r at every angle \u03b8 round the orbit, together with the two extreme distances:<\/p>\n\n

    r = a(1 – e\u00b2) \/ (1 + e cos \u03b8)<\/div>\n\n
    q = a(1 – e) and Q = a(1 + e)<\/div>\n\n

    The second law says the area A swept by the Sun\u2013planet line grows at a steady rate, covering the whole ellipse, of area \u03c0ab, once per period:<\/p>\n\n

    dA\/dt = \u03c0ab \/ T = constant<\/div>\n\n

    The third law, in the form Newton gave it, links the period to the semi-major axis and the central mass:<\/p>\n\n

    T\u00b2 = 4\u03c0\u00b2a\u00b3 \/ (GM)<\/div>\n\n
    \n\n\n\n\n
    Symbol<\/th>\nQuantity<\/th>\nSI unit<\/th>\nNotes<\/th>\n<\/tr>\n<\/thead>\n
    r<\/td>distance from the Sun (the occupied focus) to the planet<\/td>m<\/td>changes all the way round an elliptical orbit<\/td><\/tr>\n
    \u03b8<\/td>angle of the planet from perihelion, measured at the Sun<\/td>rad (or \u00b0)<\/td>0 at perihelion, 180\u00b0 at aphelion<\/td><\/tr>\n
    a<\/td>semi-major axis<\/td>m<\/td>half the ellipse’s longest diameter; also the average of q and Q<\/td><\/tr>\n
    b<\/td>semi-minor axis<\/td>m<\/td>b = a \u00d7 sqrt(1 – e\u00b2)<\/td><\/tr>\n
    e<\/td>eccentricity<\/td>none<\/td>0 for a circle; between 0 and 1 for an ellipse<\/td><\/tr>\n
    q<\/td>perihelion distance, the closest approach<\/td>m<\/td>q = a(1 – e)<\/td><\/tr>\n
    Q<\/td>aphelion distance, the farthest point<\/td>m<\/td>Q = a(1 + e)<\/td><\/tr>\n
    A<\/td>area swept by the Sun\u2013planet line<\/td>m\u00b2<\/td>grows at the same rate all the way round<\/td><\/tr>\n
    T<\/td>orbital period<\/td>s<\/td>years are handier for planets<\/td><\/tr>\n
    G<\/td>gravitational constant<\/td>N m\u00b2 kg-2<\/sup><\/td>6.67430 \u00d7 10-11<\/sup> (CODATA)<\/td><\/tr>\n
    M<\/td>mass of the central body<\/td>kg<\/td>about 1.988 \u00d7 1030<\/sup> kg for the Sun<\/td><\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n\n

    To solve the third law for whichever of T, a or M you are missing, such as a moon’s period from its distance or a star’s mass from a planet’s year, use the Kepler’s third law calculator<\/a>, which takes AU, kilometres, days and solar masses as readily as SI units.<\/p>\n\n

    Round the Sun the constants disappear. A year is Earth’s period, and an astronomical unit is Earth’s semi-major axis to within three parts in a million, so 4\u03c0\u00b2\/(GM) for the Sun comes out as 1 year2<\/sup> per AU3<\/sup> to about four parts in 100,000. The third law becomes the version the planet table and the lab use:<\/p>\n\n

    T\u00b2 = a\u00b3 (T in years, a in AU, orbiting 1 solar mass)<\/div>\n\n

    Around a star of M solar masses it becomes T\u00b2 = a\u00b3\/M, so a star twice as heavy as the Sun carries a planet round a 1 AU circle in 0.7071 years. A common student slip is to use a satellite’s height above the ground, or an orbit’s full diameter, as a. Orbital distances are measured from the centre of the body being orbited, so for a satellite in a circular orbit a is the planet’s radius plus the altitude.<\/p>\n\n

    The lab below puts all three laws on one screen. It opens on an orbit with a = 1 AU and e = 0.5 round a star of one solar mass: the period reads 1.000 years, the planet passes perihelion at 51.59 km\/s and aphelion at 17.20 km\/s, and the ellipse is cut into 8 sectors of 0.3401 AU2<\/sup> each, every one swept in 45.66 days.<\/p>\n\n

    Drag the eccentricity slider and the shape and the speeds change while the period stays where it is. Drag the semi-major axis and the orbit’s marker slides along the gold line of the log\u2013log chart, and the planet buttons load JPL’s semi-major axis and eccentricity for Mercury through Neptune.<\/p>\n\n

    <\/span>Kepler's Laws Lab<\/span><\/div>