masses of orbiting stars

OLIVEROSENTERO

SON

THE MASSES OF ORBITING

STARS

TYPES OF BINARY STARS

Apparent Double Star – two stars lying in the same direction but not in orbit around

each other.

Visual Binaries – because we can see two separate stars and their individual motion.

As each star moves along its orbit, it alternately moves toward and then away

from the Earth. This motion creates a Doppler shift, and

so the spectrum of the star pair shows two sets of spectral lines that shift relative

to each other.

While one star’s spectral lines shift to longer wavelengths (because the star is

moving away from us), the other’s spectral lines shift to shorter wavelengths (because it is approaching us). Then, in a

cyclic fashion, half an orbit later, the pattern reverses.

SPECTROSCOPIC BINARIES

Eclipsing Binaries -are pairs of stars whose orbits are oriented exactly edge-on to our line of sight; so the stars eclipse each other sequentially.

MEASURING STELLAR

MASSES WITH BINARY STARS

Astronomers can find the mass of a stellar pair using a modified form of Kepler’s third law.

Kepler demonstrated that the time required for a planet to orbit the Sun is related to its distance from the Sun.

If P is the orbital period and a is the semimajor axis(half the long dimension) of the planet’s orbit, then P2 = a3, a relation called Kepler’s third law.

Newton discovered that Kepler’s third law could be generalized to apply to any two bodies in orbit around each other.the sum of their masses,

the sum of their masses, MA + MB, can be found from the period of their orbit and their separation:

• where aAU is expressed in astronomical unitsPyr in years• the masses are in solar masses

• This relationship is our basic tool for measuring stellar masses

• It may take many years to observe theentire orbit, but eventually we can determine P, the time required for the stars to complete an orbit.• From the plot of the orbit, and with

knowledge of the stars’ distance from the Sun, astronomers next measure the semimajor axis, a, of the orbit of one star about the other.

for example:the orbit of the two stars that compose the visual binary star Alpha Centauri

orbital period of 68 yearsa semimajor axis of 20.6 AU

That is, the combined mass of the stars is 1.9 times the Sun’s mass.

THE CENTER MASS

Kepler’s law allows us to find the combined mass of two stars that orbit each other.Additional analysis of stars’ orbits allows us to find their individual masses.

We can tell their masses relative to each other by the amount each star moves.If the two stars are equal in mass, they will move the same amount.

Pairs of stars do not have such disparities in mass, though, so they orbit a point more nearly equidistant between them called the center of mass,

If one star has mass MA and is orbiting at a distance aA from the center of mass, and the other has mass MB and is orbiting at a distance aB MA × aA = MB × aB.

The larger mass has the smaller distance from the center of mass.

If two stars are equal in mass (MA = MB), then aA = aB, and they will orbit a point exactly halfway between them.

On the other hand, if star B is two times less massive than star A (MB = ½ × MA), then star B will orbit two times farther from the center of mass than its orbital companion (aB = 2 × aA)

For spectroscopic binaries, we can compare the relative speed of each star.Each orbits the center of mass in the same period of time, but the more massive one has less distance to move in its orbit.

for example, we might detect star A moving toward us at 2 kilometers per second while star B moves away from us at 8 kilometers per second

for example, we might detect star A moving toward us at 2 kilometers per second while star B moves away from us at 8 kilometers per second

Half an orbit later we see star A moving away from us at 2 kilometers per second while star B moves toward us at 8 kilometers per second. This shows us that star B is moving 4 (= 8/2) times faster and farther than star A. Therefore, star B is four times less massive than star A.

As two stars orbit their center of mass, the more massive star completes a smaller orbit about the center of mass and therefore travels at a slower speed than the less massive star. The Doppler shift of the more massive star is also smaller in proportion to the stars’ relative masses.

measurements show that star A orbits about 0.7 times as far from the center of mass as star B. So aA = 0.7 × aB. We therefore know thatMB = 0.7 × MA. From the previous section we also know that their combined mass equals 1.9 solar masses: MA + MB = 1.9 M .⊙

MA + 0.7 × MA = 1.7 × MA = 1.9 M .⊙

Alpha Centauri

So Alpha Centauri A must be about 1.1 solar masses, while Alpha Centauri B is about 0.8 solar masses.

astronomers have determined that most stars have masses from about 0.1 to 30 M⊙

A few rare stars are even more massive, ranging up to more than 100 M⊙

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