The Odd-Even Transit Depth Test: Catching Eclipsing-Binary False Positives

What the test is and its physical basis

A transiting planet blocks the same fraction of starlight on every orbit. The fractional dip, or transit depth, is set by the geometry: δ = (Rp/R★)², the square of the planet-to-star radius ratio. Because the planet does not change size or brightness between orbits, transit number 1, 3, 5, 7 (odd) and transit number 2, 4, 6, 8 (even) must all reach the same depth within the photometric noise.

An eclipsing binary breaks this symmetry in a subtle way. Detection algorithms like Box Least Squares (BLS) that scan for periodic dips frequently lock onto half the true orbital period of an EB, because the deep primary eclipse and the shallow secondary eclipse look like two evenly spaced events of a shorter period. Folded at that half-period, primaries land on the odd transits and secondaries on the even transits (or vice versa). Since two stars of different size and temperature block different amounts of light, the two eclipse depths differ — and that difference is exactly what the odd-even test measures.

The mechanism: why the depths split

Consider two stars, A (larger, hotter) and B (smaller, cooler). When B passes in front of A, it blocks a patch of A's bright surface — a deep primary eclipse. Half an orbit later A passes in front of B, blocking B's fainter surface — a shallow secondary eclipse. For circular orbits these are separated by exactly half the period.

• Primary depth scales roughly as (R_B/R_A)² × [1 − (surface-brightness ratio term)].
• Secondary depth scales with how much of the fainter star's light is removed — set by the ratio of effective temperatures, ~(T_B/T_A)⁴ via the Stefan–Boltzmann surface-brightness contrast.

Because these two depths depend on different combinations of radius and temperature, they are essentially never equal unless the stars are near-identical twins. Fold at the half-period and you interleave a deep and a shallow event — the machine reports one 'transit' depth that alternates. The odd-even statistic simply asks: is |δ_odd − δ_even| larger than the combined noise?

Key quantities and a worked example

The diagnostic is a signal-to-noise comparison. Define S = |δ_odd − δ_even| / √(σ_odd² + σ_even²), where each σ is the uncertainty on the separately fitted, phase-folded depth. Pipelines flag the candidate when S exceeds a threshold — 3σ in the Kepler Data Validation module, with ~5σ used as a more conservative cut in some TESS/ground-based surveys.

Worked example. Suppose a G star (R★ ≈ 1 R_sun) has a companion. Folded at the half-period, odd events show δ_odd = 1.20% and even events δ_even = 0.90%, each with σ ≈ 0.05%. Then S = |1.20 − 0.90| / √(0.05² + 0.05²) ≈ 0.30 / 0.071 ≈ 4.2σ — a clear fail. A genuine planet at ~1% depth implies Rp/R★ ≈ √0.01 = 0.1, i.e. a ~1 R_Jupiter body, so the depths look planetary; only the odd-even split reveals the two-star nature. Doubling the folding period then shows one deep and one shallow eclipse per cycle, confirming the EB.

How it is observed and applied in pipelines

The test lives inside automated vetting stages that process every Threshold Crossing Event (TCE):

• Kepler Data Validation (DV) — Twicken, Jenkins et al. built the odd-even depth and epoch comparison directly into the DV reports for the ~34,000 TCEs from the Q1–Q17 runs; it appears as a one-page summary metric.
• TESS SPOC and QLP — the same check runs on TESS TCEs, where background eclipsing binaries blended into the large 21-arcsecond pixels are the dominant false-positive source.
• NGTS, K2, ground-based surveys — odd-even flux asymmetry is a standard entry in vetting checklists and citizen-science tools like Planet Hunters.

Two companion statistics ride alongside it: the epoch statistic (do odd and even events arrive on the predicted schedule, or is there a timing offset signalling an eccentric EB?) and a direct secondary-eclipse search at phase 0.5. Together they form the classic 'is this an EB?' triage before expensive follow-up like radial velocities or high-resolution imaging.

How it compares to related vetting tests

The odd-even test is one member of a suite; each catches a different flavor of impostor:

• Secondary-eclipse detection — finds the shallow dip at phase 0.5. Complementary: an EB can fail odd-even even when its secondary is too shallow to detect alone, and vice versa.
• Centroid / difference-image analysis — checks whether the flux dip comes from the target pixel or a nearby background eclipsing binary. Odd-even catches unequal depths; centroids catch spatial offset.
• V-shape vs. U-shape — grazing EBs make V-shaped, depth-dependent ingress; planets make flatter-bottomed U-shapes.
• Radial velocity — the gold standard: a stellar companion induces km/s reflex motion, versus m/s for planets.

Its great virtue is cost: it needs only the discovery light curve, no new observations. Its blind spots are twins (near-equal stars give equal depths and sail through) and cases where the true period was correctly recovered, so primaries and secondaries never interleave.

Significance, limits, and open issues

Eclipsing binaries — foreground, background-blended, or hierarchical triples — are historically the largest single class of transit false positives. Radial-velocity and multicolor studies (e.g., Santerne and collaborators on Kepler giant-planet candidates) found false-positive rates of tens of percent for hot-Jupiter-sized signals, much of it EBs. Automated odd-even vetting removes a big fraction of these before humans ever look, which is why Kepler could statistically 'validate' thousands of small planets.

Open and debated issues remain. The test is weak against equal-mass twin binaries and against background EBs so faint that dilution shrinks the odd-even split below 3σ — a growing concern for TESS's large pixels. Choosing the threshold is a real trade: too strict (3σ) rejects genuine planets whose depths differ by chance or by stellar activity; too loose (5σ+) lets EBs slip through into occurrence-rate samples. Modern machine-learning vetters (Robovetter, ExoMiner, and newer multimodal networks) fold the odd-even statistic in as one input feature rather than a hard cut, letting the classifier weigh it against centroids, shape, and stellar parameters.

Genuine planetary transit vs. an eclipsing binary detected at half its true period

Property	Real planet	Eclipsing binary (period halved)
Odd vs. even depth	Equal to <1σ	Differ by 3–many σ
Physical cause of dip	Rp/R★ ratio, same each orbit	Primary vs. secondary eclipse of two stars
Typical depth	~0.01%–1% (100 ppm–1%)	Often 1%–50%, unequal by design
Secondary eclipse	Absent or tiny (ppm-level)	Present, ~half phase between primaries
Odd-even epoch offset	Zero	Can be non-zero if orbit eccentric
Correct interpretation	One transiting body	Two stars, true period = 2×folded period