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Effective (Roche) potential along the line of centres, in units of the separation
Radial-velocity curves of both stars (the observer sees the orbit at the chosen inclination)
How much of its Roche lobe each star fills, through one orbit (red line = overflow)
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How to use
Choose a preset (Alpha Centauri, Sirius, Algol, a cataclysmic variable...) or set masses, radii, separation (logarithmic slider), eccentricity and inclination. The yellow and blue teardrops are the Roche lobes; when a star's radius exceeds its lobe, gas streams through L1 (the Algol and cataclysmic-variable presets do this). Watch the third graph for eccentric orbits: the lobes breathe with the separation. The radial-velocity graph shows what a spectrograph would record. Drag to orbit, wheel to zoom. Space pauses, R resets.
The stars, accretion disc, Lagrange markers, protractor ring and telescope were modelled in Blender (baked textures); the rubber sheet and the lobes are computed from the same Roche potential as the 2D lab.
The science
Two bodies, one ellipse. Two stars of mass M1, M2 orbit their common centre of mass. The relative orbit is a Kepler ellipse of semi-major axis a and eccentricity e, and each star follows a scaled copy of it: star 1 at -M2/(M1+M2) and star 2 at +M1/(M1+M2) of the separation vector. With a in AU, masses in solar masses and time in years, Kepler's third law is simply
P^2 = a^3 / (M1 + M2)
The position at time t comes from Kepler's equation M = E - e sin E with M = 2 pi t / P, then x = a (cos E - e), y = a sqrt(1-e^2) sin E, separation d = a (1 - e cos E). The stars move fastest at pericentre and slowest at apocentre (Kepler's second law).
Radial velocities. A spectrograph measures only the speed along the line of sight. For an orbit seen at inclination i (90 degrees = edge-on) the semi-amplitude of star 1 is
K1 = (M2 / (M1+M2)) (2 pi a sin i) / (P sqrt(1-e^2)), K2 = K1 M1 / M2, f(M) = P K1^3 (1-e^2)^(3/2) / (2 pi G) = M2^3 sin^3 i / (M1+M2)^2
The mass function f(M) is what a single-lined spectroscopic binary gives directly; it is a lower bound for the companion's mass.
Roche lobes. In the frame rotating with the stars (angular speed omega^2 = G (M1+M2) / d^3) a test particle feels the effective potential
Phi = - G M1 / r1 - G M2 / r2 - (1/2) omega^2 (x^2 + y^2)
Its stationary points are the five Lagrange points: L1 between the stars, L2 and L3 outside, L4 and L5 at the corners of equilateral triangles. The equipotential through L1 forms two touching teardrops, the Roche lobes. A star that grows past its lobe loses gas through L1 to its companion: this is how semi-detached systems such as Algol and cataclysmic variables work. The lobe radius is well fitted by Eggleton's formula
r_L / a = 0.49 q^(2/3) / ( 0.6 q^(2/3) + ln (1 + q^(1/3)) ), q = M(star) / M(other)
What the pictures show. The lobes are drawn for the instantaneous separation, so in an eccentric orbit they swell at apocentre and shrink at pericentre: a star can fill its lobe only near pericentre (see the third graph). The stream from L1 is a schematic arc, not a hydrodynamic solution.
Honest simplifications. (1) Point masses and spherical stars: no tides, no rotation, no mass loss or mass transfer feedback, no relativity. (2) The Roche geometry assumes a circular orbit and synchronous rotation and is applied quasi-statically at the instantaneous separation for eccentric orbits. (3) The overflow stream is illustrative. (4) Presets use rounded literature values (Alpha Centauri, Sirius, Algol, a generic cataclysmic variable) for orientation only. (5) Inclination only affects the radial velocities; the top-down picture is the true orbital plane.
Checks run in the page. L1 for equal masses against the midpoint; Kepler's third law from the period of an RK4 integration of the two-body equations; the integrated separation after 3.3 orbits against the solution of Kepler's equation; the integrator's energy drift; the mass function recovered from the simulated radial-velocity amplitude against its definition; and Eggleton's formula against the volume of the equipotential through L1 for q = 0.3 and q = 3.