- In words,
Kepler's 3rd law
states that the square
of an orbital period
of a planet
is proportional to the
cube
of the
mean orbital radius (AKA
semi-major axis)
of its elliptical orbits
around the
Sun.
In natural units,
the formula is
P_years**2 = R_AU**3 ,

where AU stands for astronomical unit (AU) ≡ 1.49597870700*10**11 m ≅ 1.496*10**11 m. Of course, for the Earth, we have 1 = 1 since the natural units are natural for us Earthlings. - Kepler's 3rd law
was remarkable in showing there was a connection between
the specific geometry of
orbits
(i.e., ellipses)
and kinematics (i.e., which is concerned with
the description of motion).
- Isaac Newton (1643--1727) in his
*Principia*(1687) derived Kepler's 3rd law from Newtonian physics (i.e., Newton's 3 laws of motion, Newton's law of universal gravitation, etc.) thereby connecting kinematics and dynamics (which is concerned with the causes of motion). The fact that gravity is an inverse-square law is a key factor in the derivation. - Note that
Kepler's 3rd law
only holds exactly for
gravitationally bound
two-body systems.
However, for planetary systems
and
satellite systems (AKA planet-moon systems),
it is usually the case that
the host star and each
planet
in the first case
and the host planet and each
moon in the second case
approximate a
gravitationally bound
two-body system
to high accuracy/precision.
- The Newtonian physics
formula for
Kepler's 3rd law is
P = [(2π)/sqrt[G(M+m)]]*R**(3/2) ≅ [(2π)/sqrt[GM]]*R**(3/2) for m << M ,

gravitational constant G = 6.67430(15)*10**(-11) (MKS units), M is the mass of the larger spherically symmetric astro-body, and m is the smaller spherically symmetric astro-body. In natural units, the second version of the above formula becomesP_years**2 = [1/sqrt(M/M_☉)]R_AU**3 ,

where the solar mass M_☉ = 1.98855*10**30 kg. - Johannes Kepler (1571--1630) published
Kepler's 1st law
and
Kepler's 2nd law
in
*Astronomia nova*(New Astronomy) (1609). Kepler's 3rd law was discovered later and published in*Epitome Astronomiae Copernicanae*(Epitome of Copernican Astronomy) (1618--1621).

Caption: Kepler's 3rd law illustrated.

Features:

Image link: Itself.

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