# 14.6 Binaries and Stellar Mass

With the typical separations of visual binaries, it takes many years of observation to measure an accurate orbit. It is worth the effort, however, because binaries give us the most direct way to measure stellar mass. Physical binaries also offer good laboratories for studying stellar evolution. Recall that mass is the single most important property of a star. The mass determines the internal structure, the rate of evolution, and the destiny of a star.

Two supermassive stars orbiting one another in the open cluster Pismis 24. Click here for original source URL.

Kepler’s third law of planetary motion explains the relationship between the period of an orbit (P) and the semi-major axis of the orbit, or mean distance of the planet from the Sun (a). In mathematical terms, we write this as P^{2} ∝ a^{3}. From this equation, it follows that planets farther away from the Sun travel on slower orbits.

Newton came up with a powerful and generalized form of Kepler's law that applies to any two bodies in orbit around each other. When any two objects in space are in orbit around each other, the period of revolution (the time it takes to complete an orbit) increases as the distance between them increases and as the sum of the masses of the two objects decreases. If M_{A} and M_{B} are the masses of the two stars in solar masses, and if we express the period P in years and the semi-major axis a in astronomical units, the equation is:

P^{2} = a^{3} / (M_{A} + M_{B})

You can see if M_{A} is one solar mass like the Sun and M_{B} is much smaller than M_{A} (as would be the case for any planet in our solar system), this equation reduces to P^{2} = a^{3}, which is the familiar form of Kepler's law in the solar system.) We can rearrange this equation to get:

M_{A} + M_{B} = a^{3} / P^{2}

This is a form we can use to calculate star masses. It does of course lead to the situation where we only measure accurate masses for stars in binary systems, so we have to assume that they appropriately represent the whole popluation of stars. This is often the situation in astronomy; a method or tool is useful but only in a limited arena and inference has to be used to draw a broader conclusion.

Binary stars are quite common. For many of these binary stars astronomers can measure both the period of revolution and the distance between the two stars. Thus they can calculate the sum of the masses, designated M_{A} + M_{B}, where A and B designate the two stars. But they want to know each individual mass, not the sum of the two. Newton showed that in a system of orbiting bodies, each body orbits around an imaginary point called the center of mass and that by measuring the distance of each star from the center of mass, we can measure the ratio of the masses. With both the sum of the masses and the ratio of the masses, astronomers can get each individual mass.

This Hubble Space Telescope image shows Sirius A, the brightest star in our nighttime sky, along with its faint, tiny stellar companion, Sirius B. Astronomers overexposed the image of Sirius A [at centre] so that the dim Sirius B [tiny dot at lower left] could be seen. The cross-shaped diffraction spikes and concentric rings around Sirius A, and the small ring around Sirius B, are artifacts produced within the telescope's imaging system. The two stars revolve around each other every 50 years. Sirius A, only 8.6 light-years from Earth, is the fifth closest star system known. Click here for original source URL.

The double star Sirius consists of the bright star Sirius A (which has the greatest apparent brightness of any star in the night sky) and the faint star Sirius B. The separation of the two stars is 20 A.U. and the orbit takes 51 years. We can use the equation above to calculate that M_{A} + M_{B} = 20^{3}/51^{2} = 3.1 times the mass of the Sun. The ratio of the distances of each star from the center of mass indicates that A is twice as massive as B, so M_{A}/M_{B} = 2. From these facts, we can deduce that Sirius A has a mass of 2 solar masses and Sirius B has a mass of 1 solar mass.

The bright star Alpha Centaurs orbits a fainter companion with a period of 80 years and a semi-major axis of 23 A.U. We can calculate that M_{A} + M_{B} = 23^{3}/80^{2} = 1.9 times the mass of the Sun. In this pair, the stars are at nearly equal distances from the center of mass, and M_{A}/M_{B} = 1.2. We deduce that M_{A} = 1.1 solar masses and M_{B} = 0.9 solar masses. The case of Alpha Centauri is complicated by the presence of a third star — a very low-mass star of 0.1 solar masses at a distance of about 10,000 A.U. from the more massive pair. But we can easily use Newton’s law of gravity to show that the influence of this third star is negligible. The gravity force is proportional to the masses and inversely proportional to the square of the distance between the masses. We know that the third star is 10 times less massive than A or B, and 10,000/23 = 430 times farther away from them than they are from each other. Therefore, the force of the third star on A or B is 430 x 10 = 4300 times smaller than the force that keep A and B orbiting each other. We can safely ignore the complication of the third star.