Double stars and binary stars

Builds on: Sections 3.1 and 3.7
Reading time: 25 min
Exercises: 3
Olympiad problems: 1

In the handle of the Big Dipper, sharp eyes see two stars close together, Mizar and the fainter Alcor, about 12′ apart. A small telescope splits Mizar itself into two stars 14″ apart. In 1889 the spectrum of the brighter of those two, Mizar A, was found to show each line doubled at some times and single at others, repeating every 20.5 days. Mizar A is itself two stars, too close for any ordinary telescope to separate. We can watch another kind of pair with the eye alone. Every 2.87 days the star Algol, in Perseus, fades from magnitude 2.1 to 3.4 for a few hours, because a dimmer companion passes in front of it. In this section we see what makes two stars a binary and how each kind is found.

8.1.1Double stars and true binaries

Before you readMake a guess first

What fraction of stars like the Sun have at least one companion star?

Two stars that look close together on the sky can be a chance alignment of stars at very different distances, called an optical double. A binary star is a pair held together by their gravity, each star moving round their common centre of mass ((3.16)). Three tests tell them apart. The two stars of a binary are at the same distance within the size of their orbit, so they have the same parallax. They move across the sky together, with the same proper motion, apart from their small orbital motion. Over years or centuries, the line joining them turns as they orbit.

8.1.2Four ways a binary shows itself

How we find a binary depends on how far apart its stars are and how its orbit is tilted to our line of sight.

In a visual binary both stars are seen separately in a telescope, and over the years we can trace the orbit of one round the other. These are wide pairs with periods of years to centuries, such as Sirius A and B, or α Centauri A and B.
In an astrometric binary only one star is seen, but its path across the sky wobbles about a straight line as it orbits an unseen companion. In 1844 Friedrich Bessel found such a wobble in Sirius and predicted its companion, which was first seen in 1862.
In a spectroscopic binary the stars are too close to separate, but their spectral lines shift back and forth with the Doppler effect ((S.30)) as each star moves towards us and then away. If both stars' lines are seen, as in Mizar A, it is double-lined, and if only the brighter star's lines are seen, it is single-lined.
In an eclipsing binary we see the orbit nearly edge on, and each star passes in front of the other once in every period, so the total light dips regularly, as in Algol.

One binary can belong to more than one kind. Close pairs are usually spectroscopic, and those seen nearly edge on are also eclipsing. Wide pairs are visual, and they have long periods and small orbital speeds, so their Doppler shifts are hard to measure.

Figure 8.1Two stars round their centre of mass

Mass of 2 over mass of 1Eccentricity

e

Watch

B lies at
Speeds

The figure shows two stars, 1 and 2, on ellipses round their centre of mass B, always on opposite sides of it and with the dashed line between them passing through it. The heavier star 1 moves on the smaller ellipse, by the ratio of the masses. The figure can be set to , whose companion is a white dwarf of about one solar mass, or to , two stars a little like the Sun.

8.1.3Size, period and mass

For a binary, Kepler's third law ((3.16)) has the sum of the two masses in it. In the units of the Solar System, the astronomical unit, the year and the mass of the Sun, the factor $4 π^{2} / G$ becomes 1, since Earth has $a = 1$ au and $T = 1$ year round one solar mass.

ResultKepler's third law for a binary(8.1)

\begin{aligned} M_{1} + M_{2} & = \frac{a^{3}}{P^{2}} \\ θ & = \frac{a}{d} \end{aligned}

$M_{1}, M_{2}$	the masses of the two stars, in solar masses
$a$	the semi-major axis of the orbit of one star relative to the other, in astronomical units
$P$	the period, in years
$θ$	the angle on the sky that $a$ covers, in arcseconds, when the orbit is seen face on
$d$	the distance, in parsecs

The second line is the definition of the parsec, by which 1 au at 1 pc covers 1″. Measuring

P

and

a

gives only the total mass. Splitting it into the two masses needs the size or the speed of each star's own orbit round the centre of mass, as in Sections 8.2 and 8.3.

Quick checkWeighing α Centauri

α Centauri A and B have a period of 79.9 years and a relative orbit with semi-major axis 23.4 au. What is their total mass?

solar masses

Example 8.1Why Mizar A cannot be split

Mizar A is two stars of about 2.4 and 2.5 solar masses with a period of 20.5 days, at a distance of 26 pc. How far apart do the two stars appear?

Solution

The period is $20.5 / 365.25 = 0.0561$ years, so by (8.1), $a^{3} = 4.9 \times {0.0561}^{2} = 0.0154$ , and $a = 0.249$ au. At 26 pc this covers $θ = 0.249 / 26 = 0.0096 ″$ .

AnswerThe stars are about 0.01″ apart, about a hundredth of the best resolution from the ground on a typical night, so only their spectra show that they are two. Interferometers, which combine telescopes far apart, have since resolved the pair.

Example 8.2How often a binary eclipsesfinish it

Two stars like the Sun, each of radius $R_{⊙} = 4.65 \times 10^{- 3}$ au, orbit each other on a circle of radius $a = 0.1$ au. From a direction chosen at random, how likely is it that we see eclipses?

The start

Eclipses happen when one star passes in front of the other as seen from Earth. The line of sight then lies within an angle of the orbital plane whose sine is $(R_{1} + R_{2}) / a$ , so the orbit must have $\cos i$ less than $(R_{1} + R_{2}) / a$ , where $i$ is the angle between the orbit's axis and our line of sight. For directions chosen at random, $\cos i$ is equally likely to take any value from 0 to 1.

Your stepNow find the chance that

\cos i

is less than

(R_{1} + R_{2}) / a

Common errorsSection 8.1

Two stars close together on the sky need not be a binary. They are a binary only if they share a distance and a proper motion, or if we see them orbit.
The period and the size of the orbit give only the sum of the masses. One more measurement is needed to split it between the stars.
The $a$ in (8.1) is the semi-major axis of one star's orbit relative to the other, the sum of the sizes of the two orbits round the centre of mass, and not the size of either of them.

ExercisesSection 8.13 questions

8.1.1
Two stars of one solar mass each orbit each other with a separation of 1 au. What is their period?
years
8.1.2
A binary 100 pc away has a relative orbit of semi-major axis 50 au, seen face on. How far apart do its stars appear at most?
″
8.1.3
A pair of stars has a period of 3 days. How is it most likely to be found?

Olympiad problemsPast IOAA problems, restated1 problem

2011Is Proxima bound to α Centauri?30 points

Adapted from IOAA 2011 (Katowice), theory problem 3. We have restated it in our own words and kept the original data. The solution was last checked on 21 October 2026.

A planetarium program gives these data for two stars, each of about one solar mass.

Star	1	2
Right ascension	14h 29m 44.95s	14h 39m 39.39s
Declination	−62° 40′ 46.14″	−60° 50′ 22.10″
Distance	1.2953 pc	1.3475 pc
Proper motion in right ascension	−3.776″ per year	−3.600″ per year
Proper motion in declination	+0.95″ per year	+0.77″ per year

The proper motions in right ascension are already multiplied by the cosine of the declination, so they are true angles on the sky. Using these data, decide whether the two stars are bound to each other by gravity. Write YES if they are and NO if they are not.

I solved it without the solution

AnswerNO. The relative speed, at least 1.58 km/s, is more than three times the escape speed of 0.49 km/s at their separation of about 15 000 au. The official solution leaves out the factor $\cos δ$ in the right-ascension difference, getting 3.09° for the separation, and quotes a speed of about 15 km/s, ten times too large, though its answer is the same. The stars are Proxima and α Centauri. More precise distances and radial velocities measured since suggest that Proxima is in fact bound to the pair, on an orbit lasting more than half a million years, so the answer belongs to the data given.