Today in Astronomy 142 - University of Rochesterdmw/ast142/Lectures/Lect_03b.pdf · 24 January 2013 Astronomy 142, Spring 2013 1 Today in Astronomy 142: binary stars Binary-star systems

24 January 2013 Astronomy 142, Spring 2013 1

Today in Astronomy 142: binary stars

Binary-star systems. Direct measurements of stellar

mass, radius and temperature in eclipsing binary-star systems.

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).

http://adsabs.harvard.edu/abs/1999AJ....117.1490Phttp://adsabs.harvard.edu/abs/1998ApJ...502L..65S


Measurement of 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; stellar interfero-metry can be used in some of the larger, nearby cases. Mass: We can’t measure the mass of an isolated star; we need a test particle in “gravitational contact” with it. Most helpful test particles: binary star systems, though in

principle any multiple-star system could be probed for the radial velocities and periods required to measure masses.

Observations of certain binary star systems can also determine the radius and temperature of each member.

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


Binaries

Resolved visual binaries: see stars separately, measure orbital axes and radial velocities directly. There aren’t very many of these. The rest are unresolved. • At right 61 Cygni; Schlimmer 2009.

Note common proper motion. Astrometric binaries: only brighter

member seen, with periodic wobble in the track of its proper motion. The first system known to be binary was detected in this way (Sirius, by Bessel in 1844).

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

http://adsabs.harvard.edu/abs/2009JDSO....5..102S

Binaries (continued)

Spectroscopic binaries: unresolved 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.)

Binaries for which the separation is clearly larger than the stars are called detached. In semidetached or contact binaries, mass transfer may have modified the stars.


Platais et al. 2007

http://adsabs.harvard.edu/abs/2007A&A...461..509Phttp://adsabs.harvard.edu/abs/2007A&A...461..509Phttp://adsabs.harvard.edu/abs/2007A&A...461..509P

Eclipsing binary stars and orientation

If the distance between members of binary systems is small compared to their radii (as it is, typically), then the orbital axis must be very close to 90°. Example: consider two Sun-like stars orbiting each other 1

AU apart, viewed so they just barely eclipse each other in the view of a distant observer:


R

θ

AU 2

i

Orbital axis

To observer

2 2arcsin

AU AU2

2 AU1.561 radians 89.5

sin 0.99996

R R

Ri

i

θ

π

= ≅

= −

= = °=


Binary-star radial velocities

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

Binary-star radial velocities (continued)

If their orbits are circular, the radial velocity of each component will be sinusoidal in time, since this velocity tracks only one component of the motion:

The radial velocities of the

two stars are equal during eclipses.


( ) ( )

( )

( )

00

00

00

,

cos

sin

vtr t r tr

vtx t rrvty t rr

φ= =

=

=

Flux

Time

vr1v2v

PrimarySecondary Minima

P


Measurement of binary-star masses

Use radial-velocity measurements in Kepler’s Laws: #1: all binary stellar orbits are coplanar ellipses, each with one focus at the center of mass. 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=+

+π

b g b g


Measurement of binary-star masses (continued)

If orbital major axes (relative to center of mass) or radial velocities known, so is the ratio of masses: Furthermore, from Kepler’s third law (cf. HW#1), For unresolved binaries (the vast majority), we can measure only P and the two velocity amplitudes, so this is two equations in three unknowns:

21 2 2

2 1 1 1

r

r

vm a vm a v v

= = =

31 2

1 2 .2 sinP v vm m

G iπ+ + =

1 2, and sin .m m i


Measurement of binary-star masses (continued)

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, leaving us with two equations in two unknowns, Simulation of binary star orbits, useful in visualizing the

radial-velocity measurement situation: http://www.pas.rochester.edu/~dmw/ast142/Java/binary.htm . Courtesy of Terry Herter (Cornell).

1 2 and .m m

http://www.pas.rochester.edu/~dmw/ast142/Java/binary.htm


Measurement of stellar radius

Eclipses don’t just wink off and on instantaneously: the transitions are gradual. The shape of the system’s light curve is sensitive to the sizes of the stars. If one star can completely

block the light of the other, the “bottom” of the eclipse light curve will be flat.

The extent of the flat part of the curve is determined by the radius of the larger star.

The slope of the transitions is determined by the radius of the smaller star.

Time Fl

ux

1t 2t 3t 4t

2− 1− 0 1 2

0

0.5

1

Uniform brightness star (this calculation)Realistic (limb-darkened) starStraight line

Time, in units of Rs/vBr

ight

ess

of s

tar,

in u

nits

of i

ts to

tal f

lux

Measurement of stellar radius (continued)

If the light curve never flattens out, then the eclipse isn’t “total:” neither star can completely block the other.

Much can be made of the details of the shapes of the light curve at the onset (ingress) and end (egress) of the eclipse, to make out details such as the atmospheres or the uniformity each star’s brightness, but the transitions are nearly straight.


Measurement of stellar radius (continued)

The “corners” are usually labelled as indicated at right. During an eclipse, the stars move at speeds in opposite directions perpendicular to the line of sight, where are the two speeds measured from the orbital periods and radial- velocity amplitudes. Thus


Time Fl

ux

1t 2t 3t 4t

1 2 and ,v v

( )( )( )( )

1 2 2 1

1 2 3 1

2

2S

L

R v v t tR v v t t

= + −

= + −

1 2 and v v

Ingress Egress

Sf

Pf

Flux

Time

Measurement of stellar effective temperature

A useful measurement of the ratio of the stars’ Te comes from the relative depths of the primary (deeper) and secondary eclipses. When the stars are not eclipsed, their total flux is If the small star is hotter, the primary eclipse is when this star is behind the larger star:


2 20 0S S L Lf R I R Iπ π= +

20P L Lf R Iπ=

Measurement of stellar effective temperature (continued)

Then the secondary eclipse is when the small star passes in front of the larger one: Three equations, two unknowns. But usually these are used to construct one ratio, to remove uncertainties in distance r:


( )2 2 20 0S L S L S Sf R R I R Iπ π= − +

( )

( )

2 2 20 0 0

2 2 2 2 20 0 0 0

2 20 0 0

22 2 2 000 0

P S S L L L L

S S S L L L S L S S

S S S S S

LS LL L L S L

f f R I R I R If f R I R I R R I R I

R I R I IIR IR I R R I

π π π

π π π π

π π

ππ π

− + −=

− + − − −

= = =− −

Measurement of stellar effective temperature (continued)

Note that the way we have defined the I0s,

Thus If r is known, the luminosity, and of the brighter star is determined accurately from the primary minimum flux, and thus Te of the fainter star from this ratio.


2 4 42

0 02 2 24

4 4e eR T TLf R I I

r r rπ σ σ

ππ π π

= = = ⇒ =

40

40

P S eS

S L eL

f f I Tf f I T−

= =−

( )1 424 ,eT L rπ σ=


Example

An eclipsing binary is observed to have a period of 25.8 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 for the ingress to progress from first contact to 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?


Example (continued)

25.8 years

1.04 km/s

11 km/s

vr

Flux

Time

Closeup of primary minimum

11.7 hours

164 days


Example (continued)

Answers Since 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

9.53 AU

s sPr vπ

= = ×

=

131.34 10 cm2

0.90 AU

10.4 AU

Pr v

r

π= = ×

=

=


Example (continued)

Masses:

Stellar radii (note: solar radius = ):

[ ][ ]

3

2AU

15.2 year

10.61.3 , 13.9

s

s s

s

rm m M

Pm m

m 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


Data on eclipsing binary stars

The most useful, ongoing, big compendium of detached, main-sequence, double-line eclipsing binary data is by Oleg Malkov and collaborators (1993, 1997, 2006, 2007). Much of the data come from many decades of work by Dan Popper. Those objects for which orbital velocities have been

measured comprise the fundamental reference data on the dependences of stellar parameters – temperature, radius, luminosity on their masses.

A strong trend emerges in plots of mass vs. any other quantity, involving all stars which are not outliers in the mass vs. radius plot. This trend, in whatever graph it appears, is called the main sequence, and its members are called the dwarf stars.

http://adsabs.harvard.edu/abs/1993BICDS..42...27Mhttp://adsabs.harvard.edu/abs/1997A&A...320...79Mhttp://adsabs.harvard.edu/abs/2006A&A...446..785Mhttp://adsabs.harvard.edu/abs/2007MNRAS.382.1073M


Radii of eclipsing binary stars

0.1

1

10

100

0.1 1 10 100

Radi

us (R

)

Mass (M

)

Detached binaries

Semidetached/contact binaries

Giants


Luminosities of eclipsing binary stars

0.0001

0.01

1

100

10000

1000000

0.1 1 10 100

Lum

inos

ity (L

)

Mass (M

)

Detached binaries



Effective temperatures of eclipsing binary stars

0

10000

20000

30000

40000

50000

0.1 1 10 100

Tem

pera

ture

(K)

Mass (M

)

Detached binaries


The H-R diagram

We can’t, of course, measure the masses of single stars, or binary stars in orbits of unknown orientation. Unfortunately this includes more than 99% of the stars in the sky. We can usually measure luminosity (from bolometric

magnitude) and temperature (from colors) though. The plot of this relation is called the Hertzsprung-Russell diagram.

The H-R diagram for detached eclipsing binaries is very similar to that of stars in general: it corresponds to what is obviously the main sequence in other samples of stars.

This correspondence allows us to infer the masses of single or non-eclipsing multiple stars.


H-R diagram for eclipsing binary stars

Usually the T axis is plotted backwards in H-R diagrams.


0.0001

0.01

1

100

10000

1000000

100010000

Lum

inos

ity (L

)

Effective temperature (K)

Detached binaries


Giants


Eclipsing binaries vs. nearby stars

Stars within 25 pc: Gliese & Jahreiss 1991.

0.0001

0.01

1

100

10000

1000000

100010000

Lum

inos

ity (L

)


Stars within 25 pc of the Sun

Detached binaries

http://adsabs.harvard.edu/abs/1995yCat.5070....0Ghttp://adsabs.harvard.edu/abs/1995yCat.5070....0Ghttp://adsabs.harvard.edu/abs/1995yCat.5070....0Ghttp://adsabs.harvard.edu/abs/1995yCat.5070....0Ghttp://adsabs.harvard.edu/abs/1995yCat.5070....0Ghttp://adsabs.harvard.edu/abs/1995yCat.5070....0G

Eclipsing binaries vs. young stars in clusters

X-ray selected Pleiades: Stauffer et al. 1994.


0.0001

0.01

1

100

10000

1000000

100010000

Lum

inos

ity (L

)


The Pleiades young open cluster

Detached binaries

http://adsabs.harvard.edu/abs/1994ApJS...91..625S

Eclipsing binaries vs. bright or very nearby stars

Bright/nearby: Allen’s Astrophysical Quantities (2000).


0.0001

0.01

1

100

10000

1000000

100010000

Lum

inos

ity (L

)


Nearest and brightest stars

Detached binaries

http://adsabs.harvard.edu/abs/2000asqu.book.....C

Today in Astronomy 142: binary starsMeasurement of stellar mass and radiusBinariesBinaries (continued)Eclipsing binary stars and orientationBinary-star radial velocitiesBinary-star radial velocities (continued)Measurement of binary-star massesMeasurement of binary-star masses (continued)Measurement of binary-star masses (continued)Measurement of stellar radiusMeasurement of stellar radius (continued)Measurement of stellar radius (continued)Measurement of stellar effective temperature Measurement of stellar effective temperature (continued)Measurement of stellar effective temperature (continued)ExampleExample (continued)Example (continued)Example (continued)Data on eclipsing binary starsRadii of eclipsing binary starsLuminosities of eclipsing binary starsEffective temperatures of eclipsing binary starsThe H-R diagramH-R diagram for eclipsing binary starsEclipsing binaries vs. nearby starsEclipsing binaries vs. young stars in clustersEclipsing binaries vs. bright or very nearby stars

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Today in Astronomy 142 - University of Rochesterdmw/ast142/Lectures/Lect_03b.pdf · 24 January 2013 Astronomy 142, Spring 2013 1 Today in Astronomy 142: binary stars Binary-star systems