Measuring the Masses of Stars:

Binary Stars

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Binary StarsMore 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 Masscenter 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.

Remember: What is Kepler’s 3rd law?(Py = orbital period in years; aAU =

average distance in AU)

1. Py = aAU

2. Py2 = aAU

3. Py = aAU2

4. Py2 = aAU

3

5. Py3 = aAU

2

Estimating Stellar MassesRewrite Kepler’s 3. Law as

1 = aAU3 / Py

2

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: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.

If we know that two stars are orbiting each other once every 5 years, at an average separation of 5

AU, and we know that one of the stars has a mass of 3 solar masses, what is the mass of the second star?

1. 1 solar mass.

2. 2 solar masses.

3. 3 solar masses.

4. 4 solar masses.

5. 5 solar masses.

MA + MB = 53/52 = 5

MA + MB = 5 and MA = 3

=> MB = 2

Spectroscopic Binaries

Usually, binary separation a can not be measured directly

because the stars are too close to each other.

But a lower limit (i.e. smallest possible value) on the masses can still be

estimated in the most common case:

Spectroscopic Binaries:

Approaching

Receding

Center of Mass

A B

Wavelength

Flu

x

Assume the two stars A and B have the same absorption line in

their spectra. They are so close to one another that we can only see

them as one star, and we can only measure their combined

spectrum. What will we see in the combined spectrum at the instant

illustrated above?

0

Spectroscopic Binaries

1. We will see one line exactly at wavelength .

2. We will see a blue shifted line from star A and a red shifted line from star B.

3. We will see a red shifted line from star A and a blue shifted line from star B.

4. We will see no absorption line.

5. We will see an emission line at wavelength.

What will we see in the spectrum?

Approaching

Receding

Center of Mass

A B

Spectroscopic Binaries

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

Masses of Stars in the

Hertzsprung-Russell Diagram

0.5

18

6

3

1.7

1.0

0.8

40

Masses in units of solar masses

Low m

asses

High masses

Mass

The higher a star’s mass, the more luminous 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.

Masses of Stars in the Hertzsprung-Russell Diagram

0.5

18

6

3

1.7

1.0

0.8

40

Low m

asses

High masses

Mass

What is the typical mass of the type of stars that show the strongest Balmer

lines in their spectra?

1. 0.5 solar masses.

2. 1 solar mass.

3. 3 solar masses.

4. 10 solar masses.

5. 40 solar masses.

Maximum Masses of Main-Sequence Stars

Carinae

Mmax ~ 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 enouth to

ignite thermonuclear fusion.

→ Brown Dwarfs

Gliese 229B

1. Infrared.

2. Optical

3. Ultraviolet.

4. X-rays.

5. Gamma-rays.

Considering that brown dwarfs are much colder than main-sequence stars, in which wavelength band do you think they could be most easily observed?

Brown DwarfsHard to find because they are very faint

and cool; emit mostly in the infrared.

Many have been detected in star forming regions like the Orion Nebula.