geometric eclipse probability

Why the Geometric Eclipse Probability Scales as

A binary star eclipses only if its orbital plane is viewed sufficiently close to edge-on.

For a circular orbit, BIN01 introduces the order-of-magnitude relation

(1)

when

(1)

Here

• R₁ and R₂ are the stellar radii,
• a is the relative orbital separation for a circular binary,
• P_eclipse is the probability that a randomly oriented binary has an eclipse geometry.

The central idea is simple:

(2)

For random orbital orientations, the fraction of allowed viewing directions turns out to be proportional to the fractional size

(3)

Under the ideal assumptions of a circular orbit, spherical stars, and random orientations, the result is actually stronger than an order-of-magnitude estimate:

(2)

provided

(4)

The approximate symbol in the introductory BIN01 discussion is useful because real binary stars can have eccentric orbits, distorted stars, detection thresholds, third Light, and incomplete observational coverage.

1 Orbital inclination

Define orbital inclination i so that

i	= 0∘		face-on,	(5)
i	= 90∘		edge-on.	(6)

A face-on binary shows the orbit nearly in its own plane on the sky and cannot produce ordinary stellar eclipses.

An edge-on binary has one star pass directly in front of the other at conjunction.

The eclipse question is therefore a question about how close i must be to

(7)

2 Circular orbit geometry at conjunction

Consider a circular relative orbit of radius a.

At conjunction, the relative separation vector is directed as nearly as possible along the observer’s line of sight.

If the orbit is exactly edge-on,

(8)

and the sky-plane separation of the stellar centers at conjunction is zero.

If the orbit is tilted away from edge-on, the projected separation at conjunction is

(3)

The symbol b_phys here is a physical sky-plane offset, not a dimensionless impact parameter.

For inclinations between

(9)

we can write simply

(4)

3 The eclipse criterion

The projected stellar disks overlap if the projected center-to-center separation is smaller than the sum of their radii:

(5)

Insert Equation (4):

(10)

Divide by a:

(6)

Thus an eclipse occurs only for inclinations sufficiently close to edge-on.

4 Minimum inclination for an eclipse

Define the limiting inclination i_min by equality in Equation (6):

(7)

Therefore

(8)

An eclipse occurs when

(11)

It is often useful to describe the same geometry by the small angular departure from exactly edge-on:

(12)

At the eclipse boundary,

(13)

Since

(14)

we have

(9)

If

(15)

then δ_max is small and

(16)

with δ_max in radians.

Thus

(10)

This is the first intuitive reason the eclipse probability should scale as stellar size divided by orbital size.

5 Random orbital orientations

The missing step is probability.

A population of randomly oriented orbital planes does not have a uniform distribution in inclination i.

The orbital angular-momentum direction is isotropic on a sphere.

The solid-angle element is

(17)

After folding the physically equivalent orientations i and 180^∘− i, it is sufficient to consider

(18)

The normalized inclination probability density is

(11)

The normalization follows because

(19)

Equivalently,

(12)

This is the shortest route to the geometric eclipse probability.

6 Integrate over the eclipsing inclinations

The probability of an eclipse is the probability that

(20)

Therefore

Peclipse	= ∫ iminπ∕2 sin idi	(21)
	= iminπ∕2	(22)
	= cos imin.	(23)

Using Equation (7),

(13)

This is the desired result.

Notice that no small-angle approximation was required in the final probability integral.

For the ideal circular geometry, Equation (13) follows exactly from isotropic orientation statistics.

7 The same result from uniform

Equation (12) provides an even faster derivation.

Let

(24)

For random orientations,

(25)

is uniform on the interval

(26)

The eclipse condition is

(27)

The probability is simply the length of the allowed interval divided by the length of the full interval:

(28)

This is often the most efficient derivation used in transit and eclipsing-binary calculations.

8 Orientation-band interpretation

There is also a useful geometric interpretation on the sphere of possible orbital-pole directions.

Eclipses occur when the orbit Normal lies in a narrow band around orientations corresponding to an edge-on orbit.

The half-width of that band is δ_max.

The fractional area of the band is

(14)

From Equation (9),

(29)

so the same result follows.

9 Why BIN01 called it an order-of-magnitude probability

For the ideal circular model,

(30)

is a clean geometric result.

Real observations introduce additional effects:

• eccentricity changes the star-to-star separation at conjunction,
• stellar shapes can be distorted by tides and rotation,
• limb darkening changes the light-curve morphology,
• third light dilutes eclipse depth,
• a grazing overlap can be too shallow to detect,
• survey cadence and observing baseline can miss an eclipse,
• stellar variability can hide a shallow eclipse.

Thus it is useful in an introductory survey article to write

(31)

as an order-of-magnitude selection rule.

The present derivation shows the ideal geometric foundation beneath that estimate.

10 Grazing versus total eclipses

Equation (5) describes any overlap of the two projected stellar disks.

For a total eclipse of the smaller stellar disk, the projected center separation must satisfy

(15)

For a circular orbit with random orientation, the corresponding geometric probability is

(16)

provided the smaller star can be completely hidden by the larger disk.

The probability of a grazing-or-partial eclipse is therefore associated with the orientation interval between

(32)

and

(33)

For equal-radius stars,

(34)

so a geometrically complete disappearance of one equal-sized disk occurs only for perfect projected alignment.

11 Dimensionless impact parameter

It is common to normalize the projected separation by a stellar radius.

For example, relative to star 1 define

(17)

Then the any-overlap condition becomes

(18)

This form is closely related to the impact-parameter notation used in transit and eclipsing-binary light-curve modeling.

12 Example 1: the BIN01 estimate

Consider two stars with approximately one solar radius each:

(35)

Using

(36)

we have

(37)

Let the circular orbital separation be

(38)

Then

Peclipse	=	(39)
	= 0.186.	(40)

Therefore

(19)

The limiting inclination is

imin	= cos −1(0.186)	(41)
	≈ 79.3∘.	(42)

Thus this binary eclipses if it is viewed within about

(43)

of exactly edge-on.

13 Example 2: widening the orbit

Keep the same two solar-radius stars but increase the orbital separation to

(44)

Then

(45)

Therefore

(46)

At

(47)

the probability falls to approximately

(48)

This is a strong geometric selection effect favoring close eclipsing binaries.

14 Connect eclipse probability to orbital period

Kepler’s third law gives

(49)

Therefore

(50)

Insert this scaling into

(51)

Then

(20)

For broadly similar stellar masses and radii,

(21)

Short-period binaries are therefore much more likely to be found as eclipsing systems even before observational cadence and signal-to-noise effects are considered.

15 Eccentric binaries

For an eccentric orbit, the star-to-star separation is not constant.

The radial separation is

(22)

where

• a is the relative semimajor axis,
• e is eccentricity,
• ν is true anomaly.

The eclipse probability therefore depends on the separation at conjunction rather than simply on a.

For nearly edge-on geometry, a common conjunction approximation gives

(23)

for one conjunction and

(24)

for the opposite conjunction, with the labeling depending on the adopted argument-of-periastron convention.

Replacing a by the conjunction separation gives

(25)

and

(26)

These formulas show that an eccentric binary is more likely to eclipse at a conjunction occurring near the smaller orbital separation.

16 Primary and secondary eclipses

For a circular orbit, the separation is the same at both conjunctions.

If the geometry permits one stellar disk to overlap the other at one conjunction, the corresponding opposite conjunction also has an overlap geometry.

However, the two events need not have equal photometric detectability.

The primary and secondary eclipse depths depend on

• stellar surface brightnesses,
• radii,
• passband,
• limb darkening,
• third light,
• eclipse chord.

A geometrically allowed secondary eclipse can therefore be too shallow to identify observationally.

For eccentric binaries, the two conjunctions occur at different separations, so their geometric eclipse conditions can differ.

17 Geometric probability is not survey detection probability

Equation (13) answers only a geometric question:

(52)

A survey detection probability contains additional factors.

Schematically,

(27)

This distinction becomes essential when inferring the intrinsic binary population from an eclipsing-binary catalog.

18 Why this is a selection effect

Suppose two populations contain the same intrinsic number of binaries but one population has systematically smaller orbital separations.

Even if the orbital orientations are equally random, the close population will contribute more eclipsing systems because

(53)

An observed eclipsing-binary sample is therefore not a random sample of all binaries.

It is geometrically weighted toward

• smaller separations,
• shorter periods,
• larger stellar radii relative to the orbit,
• favorable eccentric conjunction geometries.

Correcting for this selection function is a major theme in modern population studies.

19 Common mistakes

1. Assuming orbital inclination is uniformly distributed in i rather than uniformly distributed in cos i.
2. Forgetting that edge-on corresponds to i = 90^∘ in the convention used here.
3. Using R₁ alone when either stellar disk can contribute to the overlap condition.
4. Using the difference of radii for any eclipse rather than for total coverage of the smaller disk.
5. Confusing physical projected separation with dimensionless impact parameter.
6. Treating a as the instantaneous separation in an eccentric orbit.
7. Assuming the introductory circular formula remains exact for eccentric binaries.
8. Confusing geometric eclipse probability with survey detection probability.
9. Assuming a geometrically allowed secondary eclipse must be photometrically detectable.
10. Forgetting that the eclipse selection function strongly favors short-period systems.

20 Practice exercises

1. Starting from circular-orbit geometry, derive b_phys = a cos i at conjunction.
2. Derive the eclipse criterion cos i ≤ (R₁ + R₂)∕a.
3. Show that random orbital orientations imply p(i) = sin i on the interval from zero to ninety degrees.
4. Integrate the inclination probability density to derive P_eclipse = (R₁ + R₂)∕a.
5. Derive the same result by showing that cos i is uniformly distributed.
6. A circular binary has R₁ = R₂ = R_⊙ and a = 0.10 AU. Estimate its eclipse probability.
7. For the preceding binary, find the minimum inclination required for any eclipse.
8. Derive the total-eclipse probability |R₁ − R₂|∕a for spherical stars in a circular orbit.
9. Use Kepler’s third law to derive the scaling P_eclipse ∝ P^−2∕3 when stellar masses and radii are held approximately fixed.
10. Explain why geometric eclipse probability and survey detection probability are not the same quantity.

21 Summary

For a circular binary, the projected center separation at conjunction is

(54)

Any eclipse requires

(55)

Thus

(56)

Random orbital orientations imply that cos i is uniformly distributed.

Therefore the fraction of orientations satisfying the eclipse condition is

(57)

For

(58)

the angular window around edge-on viewing is approximately

(59)

in radians.

The formula is therefore both a direct geometric probability and an intuitive statement that an eclipsing system must lie within a narrow orientation band whose angular width is set by stellar size divided by orbital size.

References

References