Ellipsoidal Variation: Reading Tidally Distorted Stars in a Binary Light Curve

What ellipsoidal variation is: the physical basis

Bring two stars close together and each raises a tidal bulge on the other, exactly as the Moon raises tides on Earth's oceans. In a tight binary the effect is enormous: the star deforms into a triaxial ellipsoid, elongated along the line connecting the two bodies. Crucially, that long axis is locked to the orbital geometry, always pointing at the companion.

As the system orbits, the observer's sightline sweeps around this figure. Twice each orbit — at both quadratures — we look at the star broadside and see its largest projected area, so the flux peaks. At conjunctions (superior and inferior), we look nearly down the long axis, the projected area shrinks, and the flux dips. Because the pattern repeats twice per revolution, the leading Fourier term is cos(2φ) at half the orbital period.

• No eclipse is required — the effect is visible at low inclination.
• The star's varying projected area is the dominant driver; gravity darkening (cooler, dimmer bulge tips) and limb darkening modulate the exact amplitude.

The mechanism and the governing amplitude relation

The workhorse expression comes from Morris & Naftilan (1993) and Kopal's earlier tidal theory. The fractional flux amplitude of the leading harmonic is approximately:

ΔF/F ≈ α_ellip × q × (R/a)³ × sin²i

where q = M₂/M₁ is the mass ratio, R the primary radius, a the semi-major axis, i the inclination, and α_ellip = 0.15 (15 + u)(1 + g) / (3 − u) is a coefficient of order unity set by the linear limb-darkening coefficient u and the gravity-darkening exponent g.

The scaling is the key insight. The tidal distortion itself scales as (R/a)³ — the same inverse-cube falloff as the tidal force — and the observable amplitude is directly proportional to the companion mass through q. Substituting Kepler's third law (a³ ∝ M P²) shows amplitude ∝ M₂ / P², so short-period, massive companions give the strongest signal. Because α_ellip absorbs the messy atmospheric physics, the amplitude is a remarkably clean mass diagnostic.

Characteristic numbers and a worked example

Consider a Sun-like star (R = 1 R☉) with a hot Jupiter (M₂ = 1 M_J ≈ 0.001 M☉) on a 2-day orbit. The semi-major axis is a ≈ 0.032 AU ≈ 6.9 R☉, so (R/a)³ ≈ (1/6.9)³ ≈ 3.0×10⁻³. With q ≈ 10⁻³, α_ellip ≈ 1, and sin²i ≈ 1:

ΔF/F ≈ 1 × 10⁻³ × 3.0×10⁻³ ≈ 3×10⁻⁶ ≈ a few parts per million.

That is why detecting planetary ellipsoidal signals demanded space photometry — Kepler reaches ~10 ppm precision on bright stars. Real hot-Jupiter systems land at ~10–60 ppm.

• Stellar-mass companion (q ~ 0.5–1): amplitude jumps to several percent — easily seen from the ground.
• Compact-object companion: a dark 10 M☉ black hole orbiting a giant can produce amplitudes of 0.05–0.2 mag.
• Period dependence: halve the period and, at fixed masses, the amplitude roughly quadruples.

How it is observed and detected

Ellipsoidal variation shows up as a smooth, quasi-sinusoidal modulation in a folded light curve, with two maxima and two minima per orbit. Two subtle diagnostics identify it and separate it from cousins:

• Unequal maxima and minima: beaming and reflection add in-phase P-period terms, so the two peaks differ slightly in height — a fingerprint of a real orbiting companion rather than starspots.
• Harmonic structure: a Fourier fit isolates the cos(2φ) coefficient as the ellipsoidal amplitude.

The landmark technique is the BEER method (BEaming, Ellipsoidal, Reflection), developed by Simchon Faigler and Tsevi Mazeh at Tel Aviv University (2011). By fitting all three modulations to non-eclipsing Kepler and CoRoT light curves, BEER recovers companion masses without any eclipse or radial velocity. Its flagship result was Kepler-76b (Faigler et al. 2013), a ~2.0 M_J hot Jupiter found photometrically and later confirmed spectroscopically. TESS and Gaia now extend the method across the whole sky.

Comparison with related phenomena

Ellipsoidal variation is easily confused with several look-alikes, and distinguishing them is where the physics pays off:

• Eclipsing binaries: require i ≈ 90° and show sharp, flat-bottomed dips. Ellipsoidal signals are smooth sinusoids and survive at low inclination — indeed they are strongest where eclipses are absent.
• Doppler beaming: a relativistic effect modulating at the full period P (one max), proportional to radial velocity K, not to (R/a)³. In hot Jupiters it competes with the ellipsoidal term at the ppm level.
• Reflection/emission: also a P-period cosine, tracing the companion's illuminated face; combined with an offset hotspot it reveals atmospheric superrotation.
• Rotational spot modulation & pulsations: occur at the rotation or pulsation period, not phase-locked to a companion, and lack the characteristic twice-per-orbit symmetry.

The formal variable-star class is the ELL / ELV (ellipsoidal) type. Contact and semidetached binaries blur into this regime, and principal-component analysis is now used to tell pure ellipsoidals apart from overflowing systems.

Significance, famous cases, and open questions

Ellipsoidal variation has become a premier tool for weighing the invisible. Because amplitude ∝ q with no requirement that the companion emit light, a large ellipsoidal signal on a single-lined star flags a massive, dark secondary — a white dwarf, neutron star, or black hole.

• Gaia DR3 released ~6306 short-period ellipsoidal variables selected for large amplitudes, using a modified minimum mass ratio derived from the amplitude alone (no primary mass needed) to shortlist compact-object candidates.
• Gaia BH1 (a dormant ~9.6 M☉ black hole with a G-star companion) and Gaia BH3 (a 32.7 M☉ black hole — the most massive stellar black hole known in the Milky Way) exemplify the dark-companion hunt this signature enables.
• Kepler-76b's mismatch between beaming- and ellipsoidal-derived masses revealed an eastward-shifted hotspot (superrotation), phase-shifted by ~10°.

Open issues remain: gravity-darkening exponents for convective envelopes are uncertain at the ~10% level, higher-order tidal harmonics matter for the tightest binaries, and cleanly separating pure ellipsoidals from contact binaries in survey data is still an active classification problem.

The three orbital phase-curve effects that reveal non-transiting companions (the BEER modulations), plus their periods, physical drivers, and mass sensitivity.

Effect	Period / phase	Physical cause	Amplitude scaling
Ellipsoidal (E)	P/2, cos(2φ), two maxima	Tidal distortion → changing projected area	∝ q (R/a)³ sin²i (∝ M₂, ∝ 1/P²)
Beaming / Doppler (B)	P, sine, one max	Relativistic boost of light from the moving primary	∝ (K/c) ∝ M₂ sin i / P^(1/3)
Reflection / emission (R)	P, cos(φ), one max	Starlight reflected + reradiated by companion	∝ (R_p/a)² × albedo
Eclipse (transit)	P, sharp dips	Companion physically blocks the star's disk	depth ∝ (R₂/R₁)² (needs i ≈ 90°)