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For two bodies of masses $M_1,M_2$, the relative Kepler orbit has semimajor axis $a$ and orbital period $P$ satisfying Kepler's third lawHere $a$ is the semimajor axis of the separation vector, not either star's individual barycentric orbit. In conventional stellar units the same relation is approximately $(P/\mathrm{yr}{)}^2=(a/\mathrm{AU}{)}^3/[(M_1+M_2)/M_{\odot }]$, with year, astronomical unit and solar mass as the units.

Kepler's third law is $P^2=4{\pi }^2{a}^3/[G(M_S+M_P)]$, with $a$ the semimajor axis of the relative orbit. For a circular edge-on orbit, the star's speed is $V_S=2\pi a_S/P$, and the centre of mass relation gives $a_S=a{M}_P/(M_S+M_P)$. Cubing the speed expression and eliminating $a$ yieldsFor $M_P\ll M_S$, this reduces toWithout the edge-on assumption, a factor ${\sin }^{3}i$ multiplies the numerator, explaining the minimum-mass degeneracy of the radial-velocity method.

Negligible spin and a circular orbit give $J=M_1{M}_2\sqrt{Ga/M}$. Fixed $M$ and $J$ imply $a(M_1{M}_2{)}^2$ is constant. Kepler's third law then gives the period-product invariant $P(M_1{M}_2{)}^3$.

A circumplanetary orbit whose orbital period equals the central planet's spin period. Its semi-major axis follows from Kepler's third law. Matching periods alone does not fix the satellite above one ground location; that additionally requires a planet-stationary orbit.

The semimajor axis $a_1$ of the pulsar's orbit about the binary barycenter, projected along the line of sight and divided by the speed of light. The orbital inclination $i$ is zero for a face-on orbit. Thus $x$ is a time, often quoted as the numerical number of light-seconds in $a_1\sin i$, not the true binary separation. The relative-orbit semimajor axis is instead $a=a_1+a_2$; conservation of momentum gives $a_1=a{m}_2/(m_1+m_2)$.

Combining Kepler's third law with that barycenter relation gives the timing form of the binary mass function, also valid for eccentric Kepler orbits:Choose $x=2\mathrm{s}$ and $P_b=8\mathrm{h}$ for an original example: $a_1\sin i=599,585\mathrm{km}$ and $f\simeq 0.0773M_{\odot }$, where $M_{\odot }$ is a solar mass. For a specified pulsar mass $m_1=1.4M_{\odot }$, the smallest allowed companion mass occurs at $\sin i=1$ and is approximately $0.698M_{\odot }$. A different assumed $m_1$ changes this bound. One measured $f$ does not determine both masses and orbital inclination.

The nominal semi-major axis of an exterior $(p+q):p$ mean-motion resonance follows from $(p+q)n_r=p{n}_p$ and Kepler's third law. Apsidal precession displaces the actual libration center because stationarity of the resonant argument requires $(p+q)n-p{n}_p-q\dot{\varpi }=0$.

If $R_2=R_L\simeq Ca(M_2/M{)}^{1/3}$, Kepler's third law givesThus a Roche lobe-filling star's period measures its inverse square-root mean mass density, largely independently of the companion mass in this approximation. A local mass-radius relation $R_2\propto M_2^{\zeta }$ gives $P\propto M_2^{(3\zeta -1)/2}$.

For contact with $R_d/R_{\odot }=M_d/M_{\odot }$ and $R_L/a=f(M_d/M{)}^{1/3}$, Kepler's third law gives $P_0=2\pi \sqrt{R_{\odot }^3/(f^{3}G{M}_{\odot })}$. The total binary mass cancels. The linear period relation depends on that particular equilibrium radius sequence; a donor with a different stellar radius response exponent follows a different mass-period relation.

Orbital dynamics gives the most direct stellar masses: a resolved binary orbit and distance determine the relative semimajor axis, and Kepler's third law gives the total mass. The barycentric size or radial-velocity amplitude ratio gives individual masses. An eclipsing binary inclination removes the sine-cubed projection factor from spectroscopic masses; a single-lined binary mass function alone does not uniquely determine both masses. Surface-gravity/radius and stellar-track estimates are model-dependent and are calibrated against dynamical binary measurements.

Half the peak-to-peak line-of-sight orbital velocity range of a star in a single-companion Kepler orbit. It equals $K=2\pi a_{\star }\sin i/[P\sqrt{1-e^2}]$, where $a_{\star }$ is the stellar barycentric semi-major axis, $P$ the orbital period, $i$ the sky-plane orbital inclination, and $e$ the orbital eccentricity. The Doppler spectroscopy formula follows from Kepler's third law for the relative orbit, $a^3=G(M_{\star }+M_p)P^2/(4{\pi }^2)$.

Assume Kepler's third law, a negligible planet mass, and $M_{\star }/M_{\odot }=R_{\star }/R_{\odot }$. Define $A_{\odot }=[(GM{)}_{\odot }{P}^2/(4{\pi }^2){]}^{1/3}/R_{\odot }$. The total duration of a circular planetary transit determines $a/R_{\star }$, giving the displayed stellar radius. Unknown orbital eccentricity, transit dilution or a different stellar mass-radius calibration introduces additional uncertainty.

A satellite remaining above a fixed point of a uniformly rotating spherical planet must have a prograde equatorial circular Kepler orbit with angular speed $2\pi /T$. Balancing gravitational acceleration and centripetal acceleration gives the displayed radius and speed $v=2\pi r/T$. The satellite's own mass cancels. A physically realizable external orbit additionally requires this radius to exceed the planet's radius.

For a small planet on a nearly circular Kepler orbit, a stellar chord at normalized transit impact parameter $b$ has length $2R_{\star }\sqrt{1-b^2}$. Dividing by transverse orbital speed $2\pi a/P$ gives the duration. A central transit has the longest duration within this approximation. Eliminating $a$ with Kepler's third law yields $P/D^3\ge {\pi }^{2}G{\rho }_{\star }/3$ for known mean stellar mass density. Finite planet radius, appreciable orbital eccentricity, and $R_{\star }/a$ corrections change the bound.