Eclipsing binary

An eclipsing binary has an orbital orientation that lets one star pass in front of the other as seen by the observer, producing periodic changes in received radiative flux.

For a circular edge-on eclipsing binary with central passage, define $T_{+}=t_4-t_1$ between external contacts and $T_{-}=t_3-t_2$ between internal contacts, with $R_L\ge R_S$. The projected separation from conjunction is $a|\sin [2\pi (t-t_c)/P]|$. At contact it equals $R_L+R_S$ or $R_L-R_S$, proving the title formula. Taking half the sum and difference recovers the radii. When durations are small compared with $P$, this becomes $R_{L,S}=v_{\mathrm{rel}}(T_{+}\pm T_{-})/4$ with $v_{\mathrm{rel}}=2\pi a/P$. A nonzero impact parameter changes the contact chords.

For a detached double-lined binary, radial velocities supply an orbital length scale times $\sin i$, while eclipse geometry supplies inclination and relative radii. Together they determine physical stellar radii. Comparing them with angular radii from a calibrated surface-brightness/colour relation gives distance. The orbital scale is geometric, but the angular-radius calibration, extinction and third light add astrophysical assumptions.^[ref]

An Algol variable is an eclipsing binary with well-separated eclipse minima and relatively little variation between eclipses. This light-curve classification includes detached systems as well as interacting Algol binaries; it does not by itself imply Roche-lobe overflow.