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Kepler’s third law states that:[22]: 3 [failed verification]
The ratio of the square of an object’s orbital period with the cube of the semi-major axis of its orbit is the same for all objects orbiting the same primary.
Or symbolically: {\displaystyle T^{2}\propto a^{3},}
where {\displaystyle T} is the object’s orbital period and {\displaystyle a} is the semi-major axis of its orbit.
This captures the relationship between the distance of planets from the Sun, and their orbital periods.
Kepler enunciated in 1619[14] this third law in a laborious attempt to determine what he viewed as the “music of the spheres” according to precise laws, and express it in terms of musical notation.[25] It was therefore known as the harmonic law.[26] The original form of this law (referring to not the semi-major axis, but rather a “mean distance”) holds true only for planets with small eccentricities near zero.[27]
Using Newton’s law of gravitation (published 1687), this relation can be found in the case of a circular orbit by setting the centripetal force equal to the gravitational force: {\displaystyle mr\omega ^{2}=G{\frac {mM}{r^{2}}}.} Then, expressing the angular velocity ω in terms of the orbital period {\displaystyle T} and then rearranging, Kepler’s third law is obtained: {\displaystyle mr\left({\frac {2\pi }{T}}\right)^{2}=G{\frac {mM}{r^{2}}}\implies T^{2}=\left({\frac {4\pi ^{2}}{GM}}\right)r^{3}\implies T^{2}\propto r^{3}.}
A more detailed derivation can be done with general elliptical orbits, instead of circles, as well as orbiting the center of mass, instead of just the large mass. This results in replacing a circular radius {\displaystyle r} with the semi-major axis {\displaystyle a} of the elliptical relative motion of one mass relative to the other, as well as replacing the large mass {\displaystyle M} with {\displaystyle M+m}. However, with planet masses being so much smaller than the Sun, this correction is often ignored. The full corresponding formula is {\displaystyle {\frac {a^{3}}{T^{2}}}={\frac {G(M+m)}{4\pi ^{2}}}\approx {\frac {GM}{4\pi ^{2}}}\approx 7.496\times 10^{-6}{\frac {{\text{AU}}^{3}}{{\text{days}}^{2}}}{\text{ is constant}},} where {\displaystyle M} is the mass of the Sun, {\displaystyle m} is the mass of the planet, {\displaystyle G} is the gravitational constant, {\displaystyle T} is the orbital period and {\displaystyle a} is the elliptical semi-major axis, and {\displaystyle {\text{AU}}} is the astronomical unit – the average distance from earth to the sun.
The following table shows the data used by Kepler to empirically derive his law:
Data used by Kepler (1618)
Planet Mean distance
to sun (AU) Period
(days) {\displaystyle {\frac {R^{3}}{T^{2}}}} (10−6 AU3/day2)
Mercury 0.389 87.77 7.64
Venus 0.724 224.70 7.52
Earth 1 365.25 7.50
Mars 1.524 686.95 7.50
Jupiter 5.20 4332.62 7.49
Saturn 9.510 10759.2 7.43
Kepler became aware of John Napier’s recent invention of logarithms and log–log graphs before he discovered the pattern.[28]
Upon finding this pattern Kepler wrote:[29]
I first believed I was dreaming… But it is absolutely certain and exact that the ratio which exists between the period times of any two planets is precisely the ratio of the 3/2th power of the mean distance.
— translated from Harmonies of the World by Kepler (1619)
Log–log plot of period T vs. semi-major axis a (average of aphelion and perihelion) of some Solar System orbits (crosses denoting Kepler’s values) showing that a3/T2 is constant (green line)
For comparison, here are modern estimates:[citation needed]
Modern data
Planet Semi-major
axis (AU) Period
(days) {\displaystyle {\frac {a^{3}}{T^{2}}}} (10−6 AU3/day2)
Mercury 0.38710 87.9693 7.496
Venus 0.72333 224.7008 7.496
Earth 1 365.2564 7.496
Mars 1.52366 686.9796 7.495
Jupiter 5.20336 4332.8201 7.504
Saturn 9.53707 10775.599 7.498
Uranus 19.1913 30687.153 7.506
Neptune 30.0690 60190.03 7.504
Planetary acceleration
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