Lecture 3 1

Today in Astronomy 328: binary stars

Binary-star systems.Direct measurements of stellar mass and radius in eclipsing binary-star systems.

At right: two young binary star systems in the Taurus star-forming region, CoKu Tau 1 (top) and HK Tau/c (bottom), by Deborah Padgett and Karl Stapelfeldt with the HST (STScI/NASA).

Lecture 3 2

Stellar mass and radius

Radius of isolated stars: Stars are so distant compared to their size that normal telescopes cannot make images of their surfaces or measurements of their sizes; this requires stellar interferometry.Mass: measure speeds, sizes and orientations of orbits in gravitationally-bound multiple star systems, most helpfully in binary star systems.

Observations of certain binary star systems can also help in the determination of radius and temperature.

There are enough nearby stars to do this for the full range of stellar types.

Lecture 3 3

Binaries

Resolved visual binaries: see stars separately, measure orbital axes and radial velocities directly. There aren’t very many of these. (Example: Sirius A and B.)Astrometric binaries: only brighter member seen, with periodic wobble in the track of its proper motion.Spectroscopic binaries: unresolved (relatively close) binaries told apart by periodically oscillating Doppler shifts in spectral lines. Periods = days to years.

Spectrum binaries: orbital periods longer than period of known observations.Eclipsing binaries: orbits seen nearly edge on, so that the stars actually eclipse one another. (Most useful.)

Sirius A and B in X rays (NASA/CfA/CXO)

Lecture 3 4

Measurements of stellar radial velocities with the Doppler effect

Radial velocity : the component of velocity along the line of sight.Doppler effect: shift in wavelength of light due to motion of its source with respect to the observer.

Positive (negative) radial velocity leads to longer (shorter) wavelength than the rest wavelength.To measure small radial velocities, a light source with a very narrow range of wavelengths, like a spectral line, must be used.

restobserved 0

rest 0

rvc

λ λ λ λλ λ

− −= =

rv

Lecture 3 5

Determination of binary-star masses using Kepler’s Laws

#1: all binary stellar orbits are coplanar ellipses, each with one focus at the center of mass.

The stars and the center of mass are collinear, of course.Most binary orbits turn out to have very low eccentricity (are nearly circular).

#2: the position vector from the center of mass to either star sweeps out equal areas in equal times. #3: the square of the period is proportional to the cube of the sum of the orbit semimajor axes, and inversely proportional to the sum of the stellar masses:

PG m m

a a22

1 21 2

34=

++

πb g b g

Lecture 3 6

Binary stellar orbit simulations

There are some useful computer simulations that you can run, written in Java by Prof. Terry Herter (Cornell U.) and his students, athttp://instruct1.cit.cornell.edu/courses/astro101/java/simulations.htmCheck out the simulations of:

Binary orbits with spectraEclipsing binary with light curve

Lecture 3 7

Eclipsing binary stellar system

Flux

Time

vr1v

2v

PrimarySecondary Minima

P

Lecture 3 8

Stellar masses determined for binary systems

If orbital major axes (relative to center of mass) or radial velocities known, so is the ratio of masses:

If furthermore the period and sum of major axis lengths known, Kepler’s third law can the used with this relation to solve for the two masses separately.

21 2 2

2 1 1 1

r

r

vm a vm a v v

= = =

Lecture 3 9

Stellar masses determined for binary systems (continued)

If only the radial velocity amplitudes v1 and v2 are known, the sum of masses is (from Kepler’s third law)

You’ll prove this in Homework #2.

If orientation of the orbit with respect to the line of sight isknown, this allows separate determination of the masses; that’s why eclipsing binaries are so important (if the system eclipses, we must be viewing the orbital plane very close to edge on: sin i is very close to 1).

31 2

1 2 .2 sin

P v vm mG iπ

+⎛ ⎞+ = ⎜ ⎟⎝ ⎠

Lecture 3 10

Stellar radii determined for totally-eclipsing binary systems

Duration of eclipses and shape of light curve can be used to determine sizes (radii) of stars:

Relative depth of primary and secondary brightness minima of eclipses can be used to determine the ratio of effective temperatures of the stars.

( )

( )

1 22 1

1 23 1

2

2

sv vR t t

v vR t t

+= −

+= −

TimeFl

ux1t 2t 3t 4t

Lecture 3 11

Example

An eclipsing binary is observed to have a period of 8.6 years. The two components have radial velocity amplitudes of 11.0 and 1.04 km/s and sinusoidal variation of radial velocity with time. The eclipse minima are flat-bottomed and 164 days long. It takes 11.7 hours from first contact to reach the eclipse minimum.

What is the orbital inclination?What are the orbital radii?What are the masses of the stars?What are the radii of the stars?

Lecture 3 12

Example (continued)

8.6 years

1.04km/s

11km/s

vr

Flux

Time

Closeup ofprimary minimum

11.7hours

164days

Lecture 3 13

Example (continued)

AnswersSince it eclipses, the orbits must be observed nearly edge on; since the radial velocities are sinusoidal the orbits must be nearly circular.Orbital radii:

11 10.61.04

s

s

vmm v

= = =

141.42 10 cm2

=9.5 AU

s sPr vπ

= = ×

131.34 10 cm2

=0.90 AU

10.4 AU

Pr v

r

π= = ×

=

Lecture 3 14

Example (continued)

Masses:

Stellar radii (note: solar radius = ):

3

2 15.2

10.61.3 , 13.9

s

s s

s

rm m MP

m mm M m M

+ = =

+ =⇒ = =

(Kepler’s third law)

(previous result)

( )3 12369

sv vR t t

R

+= −

=

( )

( )( )

2 1

-1

10

26.02 km s 11.7 hr

7.6 10 cm 1.1

ss

v vR t t

R

+= −

=

= × =

6 96 1010. × cm

Lecture 3 15

Data on eclipsing binary stars

Latest big compendium of eclipsing binary data is by O. Malkov. See following slides.This, and vast amounts of other data, can be found on line at the NASA Astrophysics Data Center:

http://adc.gsfc.nasa.gov/adc.html

Why do the graphs appear as they do? That’s what we’ll try to figure out, as we study stellar structure during the next few lectures.

Lecture 3 16

Luminosities of eclipsing binary stars (Malkov 1993)

0.1 1 10Mass (solar masses)

10-410-310-210-1100101102103104105106

Lum

inos

ity (s

olar

lum

inos

ities

)

Lecture 3 17

Radii of eclipsing binary stars (Malkov 1993)

0.1 1 10Mass (solar masses)

0.1

1

10

Radi

us (s

olar

radi

i)

Lecture 3 18

Effective temperatures of eclipsing binary stars (Malkov 1993)

0.1 1 10Mass (solar masses)

2

3

4567

104

2

3

45

Effe

ctiv

e te

mpe

ratu

re (K

)

Lecture 3 19

Hertzsprung-Russell (H-R) diagram for eclipsing binary stars (Malkov 1993)

40000 30000 20000 10000 0Effective temperature (K)

10-3

10-2

10-1

100

101

102

103

104

105

106

Lum

inos

ity (s

olar

lum

inos

ities

)

Lecture 3 20

H-R diagram for binaries and other nearby stars

Stars within 25 parsecs of the Sun (Gliese and Jahreiss1991)Nearest and Brightest stars (Allen 1973)Pleiades X-ray sources (Stauffer et al. 1994)Binaries with measured temperature and luminosity (Malkov1993)

1 105 1 104 1 1031 10 5

1 10 4

1 10 3

0.01

0.1

1

10

100

1 103

1 104

1 105

1 106

Effective temperature (K)

Lum

inos

ity (s

olar

lum

inos

ities

)