Eclipsing binaries

Builds on: Sections 8.3 and 4.6
Reading time: 45 min
Exercises: 3
Olympiad problems: 2

Every 2.87 days the star Algol, the second brightest in Perseus, fades from magnitude 2.1 to 3.4 over about five hours, then brightens again over the next five. In 1783 John Goodricke, a young English astronomer, timed these fadings and suggested that a darker body passes in front of the star once in each period. He was right. Algol is an eclipsing binary, a hot blue star with a larger, cooler and fainter companion, and between the deep eclipses there is a slight dip, too small to see by eye, when the hot star hides the cool one. The shape of such a light curve tells us the sizes of both stars and how hot they are, which no telescope could see directly. In this section we read it.

8.4.1Two eclipses in every period

Before you readMake a guess first

An eclipsing binary has a large cool star and a small hot star. Which eclipse makes the deeper dip in its light?

In each period of an eclipsing binary there are two eclipses, one when each star passes in front of the other. The deeper one is the primary minimum and the shallower the secondary minimum. We name four moments in an eclipse, called contacts. At first contact the edges of the two discs touch and the light begins to fall. If one disc can fit wholly inside the other, at second contact it is entirely in front of or behind it and the light stops falling, and at third contact it begins to leave. At fourth contact the eclipse ends. Between second and third contact the light stays steady, and the minimum has a flat bottom. If the orbit is tilted enough that one disc never fits inside the other, the eclipse is partial and the minimum is a rounded V.

On a circular orbit seen edge on, the small star crosses the large one at the relative speed $v = v_{1} + v_{2}$ , the sum of the two orbital speeds, which a double-lined velocity curve gives ((8.4)). From first to fourth contact the small star moves the sum of the two diameters relative to the large one, and from second to third contact the difference.

ResultRadii and temperatures from eclipses(8.6)

\begin{aligned} R_{l} & = \frac{v (t_{e} + t_{f})}{4}, R_{s} = \frac{v (t_{e} - t_{f})}{4} \\ \frac{Δ F_{p}}{Δ F_{s}} & = \frac{J_{hot}}{J_{cool}} = {(\frac{T_{hot}}{T_{cool}})}^{4} \end{aligned}

$R_{l}, R_{s}$	the radii of the larger and the smaller star
$v$	the relative speed of the two stars, $v_{1} + v_{2}$
$t_{e}$	the length of the eclipse, from first to fourth contact
$t_{f}$	the length of the flat bottom, from second to third contact
$Δ F_{p}, Δ F_{s}$	the light lost at the primary and at the secondary minimum
$J, T$	the surface brightness and the temperature of each star

These hold for a circular orbit seen edge on, where both eclipses hide the same area. The second line uses the Stefan-Boltzmann law ((4.12)), by which the light from each square metre goes as

T^{4}

. Darkening towards the edges of the discs, which real stars show, rounds the corners of the curve, and careful fits allow for it.

Figure 8.4An eclipsing binary and its light curve

Radius of star 1Radius of star 2Temperature of 2 over 1Inclination

i

Watch

Primary minimum
Secondary minimum
Each eclipse lasts

The figure shows two stars, 1 and 2, as seen from Earth, with radii given as fractions of their separation, moving on a circular orbit drawn dotted. Below is the total light of the pair over one period, as the number of magnitudes it is fainter than at its brightest. At phase 0 star 2 passes in front of star 1, and at phase 0.5 behind it. The stars are drawn as discs of even brightness, so the curve has sharp corners at the contacts. Lowering the inclination makes the eclipses shorter, then partial, then stops them. The figure can be set to , , , or .

Quick checkThe size of the larger star

In an eclipsing binary seen edge on, the relative speed of the stars is 100 km/s. An eclipse lasts 10.0 hours from first to fourth contact, with a flat bottom lasting 2.0 hours. What is the radius of the larger star, in solar radii?

solar radii

Example 8.7Temperatures from two depths

On a circular orbit seen edge on, the primary minimum removes 50% of a binary's light and the secondary minimum 10%. How much hotter is the star hidden at the primary minimum than the other?

Solution

Both eclipses hide the same area, so by (8.6) the surface brightnesses are in the ratio of the light lost, $J_{hot} / J_{cool} = 50 / 10 = 5$ .

AnswerThe temperatures are in the ratio

5^{1 / 4} = 1.50

, so the star hidden at the primary minimum is half as hot again as its companion.

8.4.2Why eclipses give true masses

A velocity curve gives only $M \sin^{3} i$ ((8.5)). Eclipses happen only when the orbit is seen nearly edge on, so $\sin i$ is close to 1, and a fit to the light curve gives $i$ itself. A double-lined eclipsing binary therefore gives both masses and both radii, with no distance needed. Such binaries supply most of the accurate masses and radii of stars that we have.

Example 8.8How close to edge onfinish it

In a binary the two radii add up to 0.20 of the separation. If eclipses are seen, how small can $\sin^{3} i$ be?

The start

As in Section 8.1, eclipses need $\cos i$ less than $(R_{1} + R_{2}) / a = 0.20$ , so $i$ is more than $\arccos 0.20 = 78.5 °$ .

Your stepNow find

\sin^{3} i

at that inclination.

Common errorsSection 8.4

The deeper minimum comes when the hotter star is hidden, whatever the sizes of the stars. It does not mean that the larger star is in front.
The time from first to fourth contact gives the sum of the two radii, not the radius of the larger star alone.
The speed that sets the length of an eclipse is the relative speed $v_{1} + v_{2}$ , not the speed of either star round the centre of mass.
A flat-bottomed minimum does not always mean that a star is wholly hidden. When the smaller star passes in front of the larger, it covers only part of it, and the light is steady because the covered area stays the same.

ExercisesSection 8.43 questions

8.4.1
In a binary seen edge on with a relative speed of 150 km/s, an eclipse with no flat bottom lasts 6.0 hours from first to fourth contact. What is the sum of the radii of the two stars, in solar radii?
solar radii
8.4.2
In a binary on a circular orbit seen edge on, the primary minimum removes 60% of the light and the secondary minimum 15%. What is the ratio of the temperatures of the two stars?
8.4.3
Both minima of an eclipsing binary are rounded V shapes with no flat part. What does this tell us?

Olympiad problemsPast IOAA problems, restated2 problems

2008Radii and masses from an eclipse100 points

Adapted from IOAA 2008 (Bandung), theory problem 1. We have restated it in our own words and kept the original data. The solution was last checked on 21 October 2026.

An eclipsing binary has a period of 30 days. When the secondary star eclipses the primary star, the eclipse lasts 8 hours from first to last contact, and its flat bottom lasts 1 hour 18 minutes. Spectra show that the largest radial velocity of the primary star is 30 km/s and of the secondary star 40 km/s. The orbits are circular and the inclination is 90°. Find the radii and the masses of both stars, in solar radii and solar masses.

I solved it without the solution

AnswerThe primary star has a radius of 0.84 solar radii and a mass of 0.61 solar masses, and the secondary 0.61 solar radii and 0.46 solar masses.

2009Two stars shaped like rugby balls10 points

Adapted from IOAA 2009 (Tehran), theory problem 15. We have restated it in our own words and kept the original data. The solution was last checked on 21 October 2026.

A close eclipsing binary has two giant stars of the same size, pulled by each other's gravity into prolate spheroids with $a = 2 b$ , where $a$ and $b$ are the semi-major and semi-minor axes, and their long axes always lie along the line between them. A prolate spheroid is the shape made by turning an ellipse about its long axis, like a rugby ball. The orbit is seen edge on. Ignoring any differences in temperature over the surfaces and darkening towards the edges, find the amplitude of the variation of the light, in magnitudes.

I solved it without the solution

AnswerThe light varies by 1.5 magnitudes from brightest to faintest. The official solution writes $Δ m = - 1.5$ , but an amplitude is quoted as a positive number of magnitudes.