Measuring the Mass of Stars Physics 113 Goderya Chapter(s): 9 Learning 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, r A = r B. The more unequal the masses are, the more it shifts toward the more massive star.

Estimating Stellar Masses Recall Kepler’s 3rd Law: P y 2 = a AU 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 M A and M B different from 1 solar mass: M A + M B = a AU 3 ____ Py2Py2 (M A and M B in units of solar masses)

Examples: Estimating Mass a) Binary system with period of P = 32 years and separation of a = 16 AU: M A + M B = = 4 solar masses ____ 32 2 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) Time 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 M sun : ~ 30 million yr. 0.1 M sun : ~ 3 trillion yr Masses in units of solar masses Low masses High masses Mass L ~ M 3.5 t life ~ M -2.5

Maximum Masses of Main-Sequence Stars  Carinae M max ~ 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 M sun and 70 M sun star. Dramatic mass loss; major eruption in 1843 created double lobes.

Minimum Mass of Main-Sequence Stars M min = 0.08 M sun At masses below 0.08 M sun, stellar progenitors do not get hot enough to ignite thermonuclear fusion.  Brown Dwarfs Gliese 229B