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
24 January 2013 Astronomy 142, Spring 2013 2
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.
24 January 2013 Astronomy 142, Spring 2013 3
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.
24 January 2013 Astronomy 142, Spring 2013 4
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:
24 January 2013 Astronomy 142, Spring 2013 5
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
= ≅
= −
= = °=
24 January 2013 Astronomy 142, Spring 2013 6
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.
24 January 2013 Astronomy 142, Spring 2013 7
( ) ( )
( )
( )
00
00
00
,
cos
sin
vtr t r tr
vtx t rrvty t rr
φ= =
=
=
Flux
Time
vr1v2v
PrimarySecondary Minima
P
24 January 2013 Astronomy 142, Spring 2013 8
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
24 January 2013 Astronomy 142, Spring 2013 9
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
24 January 2013 Astronomy 142, Spring 2013 10
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
24 January 2013 Astronomy 142, Spring 2013 11
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.
24 January 2013 Astronomy 142, Spring 2013 12
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
24 January 2013 Astronomy 142, Spring 2013 13
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:
24 January 2013 Astronomy 142, Spring 2013 14
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:
24 January 2013 Astronomy 142, Spring 2013 15
( )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.
24 January 2013 Astronomy 142, Spring 2013 16
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=
24 January 2013 Astronomy 142, Spring 2013 17
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?
24 January 2013 Astronomy 142, Spring 2013 18
Example (continued)
25.8 years
1.04 km/s
11 km/s
vr
Flux
Time
Closeup of primary minimum
11.7 hours
164 days
24 January 2013 Astronomy 142, Spring 2013 19
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= = ×
=
=
24 January 2013 Astronomy 142, Spring 2013 20
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
24 January 2013 Astronomy 142, Spring 2013 21
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
24 January 2013 Astronomy 142, Spring 2013 22
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
24 January 2013 Astronomy 142, Spring 2013 23
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
Semidetached/contact binaries
24 January 2013 Astronomy 142, Spring 2013 24
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
Semidetached/contact 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.
24 January 2013 Astronomy 142, Spring 2013 25
H-R diagram for eclipsing binary stars
Usually the T axis is plotted backwards in H-R diagrams.
24 January 2013 Astronomy 142, Spring 2013 26
0.0001
0.01
1
100
10000
1000000
100010000
Lum
inos
ity (L
)
Effective temperature (K)
Detached binaries
Semidetached/contact binaries
Giants
24 January 2013 Astronomy 142, Spring 2013 27
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
)
Effective temperature (K)
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.
24 January 2013 Astronomy 142, Spring 2013 28
0.0001
0.01
1
100
10000
1000000
100010000
Lum
inos
ity (L
)
Effective temperature (K)
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).
24 January 2013 Astronomy 142, Spring 2013 29
0.0001
0.01
1
100
10000
1000000
100010000
Lum
inos
ity (L
)
Effective temperature (K)
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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