Stars

• Stellar radii– Stefan-Boltzman law

• Measuring star masses

• If we know the distance to a star and can resolve its disk, then we can use trigonometry.

• However, this has been successful for a very small number of stars

How to measure the size of star?

• Light from single source will produce an interference pattern if passed through two slits.• The interference pattern for two sources, or one extended source, will be smeared out.

By accurately measuring the interference pattern caused by combining the light from two telescopes, one can estimate the size of the object or the separation of close binary stars.

• Need b > /, where = wavelength, = angular size.

How to measure the size of star?

For the spherical object, the total power radiated = the total luminosity is:

L = 4R2T4

T = temperature = Stephan-Boltzman constant = 5.6710-8 W/m2 ·K4

R = radius

Luminosity of a ‘Black Body’ Radiator

Suppose the radius of the Sun increased by a factor of 4 but the luminosity remained the same. How would the surface temperature of the Sun change?

Luminosity of a ‘Black Body’ Radiator

42

22

41

21 44 TRTRL

11

2

1

1

2

1

2

12 2

1

4

1TTT

R

RT

• If we know luminosity and temperature, then we can find the radius:

L = 4R2T4

• Small stars will have low luminosities unless they are very hot.

• Stars with low surface temperatures must be very large in order to have large luminosities.

Stars come in a variety of sizes

Masses of stars

• Essentially all of the mass measurements that we have for stars are for stars in binary systems – two stars orbiting each other.

• The mass of the stars can be measured from the orbital period and either the stellar velocities or the separation between the stars.

Kepler’s 3rd Law applied to Binary Stars

Where:

• G is gravitational constant

• G = 6.67·10-11 m3/kg-s2 in SI units

• m1, m2 are masses (kg)

• P is binary period (sec)

• a is semi-major axis of orbit (m)

23

21

2

)(

4Pa

mmG

Simplified form of Kepler’s 3rd law using convenient units

Where M in solar masses

a in AU

P in Earth years

Example: a = 0.05 AU, P = 1 day = 1/365 yr, M1 + M2 = 16.6 Msun

2

3

21 P

aMM

a is semi-major axis of orbit = half the length of yellow line

• Double star – a pair of stars located at nearly the same position in the night sky.– Optical double stars – stars that appear close together,

but are not physically conected.– Binary stars, or binaries – stars that are gravitationally

bound and orbit one another.

• Visual binaries – true binaries that can be observed as two distinct stars

• How to distinguish true binary stars systems?

Binary star systems

Visual Binary Star Krüger 60 (upper left hand corner)

About half of the stars visible in the night sky are part of multiple-star systems.

0.008"

Mizer-Alcor : A double-double-double system!

10 arcmin

14"

Mizar A+B

Alcor

Mizar A

Mizar B

Mizar A

(Binary, P = 20.5 days)

Note: Mizar B is also a binary

with period of 6 months!

Mizar observations using the NPOI

(Naval Prototype Optical Interferometer, near Flagstaff Arizona)

Determining masses of Mizar-A binary stars from observations of period, angular separation, distance

1. Distance (from parallax) d = 25.4 pc

3. Physical separation D = θ·d = 0.49 AU

2. Max. angular separation (NPOI meas.) = 0.0192"

4. Sum of masses (Kepler’s 3rd law)

5. Orbit shows a1 ~ a2 (NPOI meas.) so:

7.4056.0

)2/49.0(2

3

2

3

21 P

aMM

MMM

aMaM

3.221

2211

• Spectroscopic binaries – There are lines in the spectrum of almost every star.– These lines will be Doppler shifted by the motion of the star in the

binary.– The shifts for the two stars will be out of phase, one star is moving

towards us as the other is moving away.– Can determine binary nature by looking for motions of lines (versus

wavelength) in spectra.

What about unresolved binary systems?

1. Determine semi-major axis using observed velocity (V), period (P)

a1 a2

a = a1 + a2

Determining component masses of eclipsing binaries using velocity curves

2. Determine sum of masses using Kepler’s 3rd law

3. Determine mass ratio using a1, a2

4. Use sum, ratio to determine component masses

21

22

11

2

2

aaaP

av

P

av

2

3

21 P

aMM

22112211 or vMvMaMaM

We have been assuming that we see the binary system face on when imaging the orbit and edge-on when measuring the velocity.

In general, the orbit is tilted relative to our line of sight. The tilt, or inclination i, will affect the observed orbit trajectory and the observed velocities. In general, one needs both the trajectory and the velocity to completely determine the orbit or some independent means of determining the inclination.

Tilt of Binary Orbits

Light curves of eclipsing binaries provide detailed information about the two stars.

Light curves of eclipsing binaries provide detailed information about the two stars.

In general, need to simulate orbit (period and separation), sizes and temperatures of stars, and heating to accurately reproduce the orbital light curve. Often useful to obtain light curves in multiple wavebands.