Measuring the Mass of StarsPhysics 113 Goderya

Chapter(s): 9Learning Outcomes:

Binary Stars

More than 50 % of all stars in our Milky Way

are not single stars, but belong to binaries:

Pairs or multiple systems of stars which

orbit their common center of mass.

If we can measure and understand their orbital

motion, we can

estimate the stellar masses.

The Center of Mass

center of mass = balance point of the system.Both masses equal => center of mass is in the middle, rA = rB.

The more unequal the masses are, the more it shifts toward the more massive star.

Estimating Stellar Masses

Recall Kepler’s 3rd Law:

Py2 = aAU

3

Valid for the Solar system: star with 1 solar mass in the center.

We find almost the same law for binary stars with masses MA and MB different

from 1 solar mass:

MA + MB = aAU

3 ____ Py

2

(MA and MB in units of solar masses)

Examples: Estimating Mass

a) Binary system with period of P = 32 years and separation of a = 16 AU:

MA + MB = = 4 solar masses.163____322

b) Any binary system with a combination of period P and separation a that obeys Kepler’s

3. Law must have a total mass of 1 solar mass.

Visual Binaries

The ideal case:

Both stars can be seen directly, and

their separation and relative motion can be followed directly.

Spectroscopic Binaries

Usually, binary separation a can not be measured directly

because the stars are too close to each other.

A limit on the separation and thus the masses can

be inferred in the most common case:

Spectroscopic Binaries

Spectroscopic Binaries (2)The approaching star produces blue shifted lines; the receding star produces red shifted lines in the spectrum.

Doppler shift Measurement of radial velocities

Estimate of separation a

Estimate of masses

Spectroscopic Binaries (3)T

ime

Typical sequence of spectra from a spectroscopic binary system

Eclipsing Binaries

Usually, inclination angle of binary systems is

unknown uncertainty in mass estimates.

Special case:

Eclipsing Binaries

Here, we know that we are looking at the

system edge-on!

Eclipsing Binaries (2)

Peculiar “double-dip” light curve

Example: VW Cephei

Eclipsing Binaries (3)

From the light curve of Algol, we can infer that the system contains two stars of very different surface temperature, orbiting in a slightly inclined plane.

Example:

Algol in the constellation of Perseus

The Light Curve of Algol

Masses of Stars in the Hertzsprung-Russell Diagram

The higher a star’s mass, the more luminous

(brighter) it is:

High-mass stars have much shorter lives than

low-mass stars:

Sun: ~ 10 billion yr.10 Msun: ~ 30 million yr.0.1 Msun: ~ 3 trillion yr.

0.5

18

6

31.7

1.00.8

40

Masses in units of solar masses

Low

masses

High masses

Mass

L ~ M3.5

tlife ~ M-2.5

Maximum Masses of Main-Sequence Stars

Carinae

Mmax ~ 50 - 100 solar masses

a) More massive clouds fragment into smaller pieces during star formation.

b) Very massive stars lose mass in strong stellar winds

Example: Carinae: Binary system of a 60 Msun and 70 Msun star. Dramatic mass loss; major eruption in 1843 created double lobes.

Minimum Mass of Main-Sequence Stars

Mmin = 0.08 Msun

At masses below 0.08 Msun, stellar progenitors do not get hot enough to ignite thermonuclear fusion.

Brown Dwarfs

Gliese 229B