Astronomy 142 1

Today in Astronomy 142: 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).

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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.

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

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Binaries

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). The second is very difficult to see in the optical but is clearly resolved in the X-rays.

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

A1V + DA2, d=2.6pc

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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. Or a spectrum with many sharp features.

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.

! 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.

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Platais et al. 2007

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Eclipsing binaries

Is the combined total observed flux brightest when the smaller star is

1)  in front of the bigger star?

2)  behind the bigger star?

3)  Separated from the bigger star?

eclipsing binaries (continued)

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http://astro.unl.edu/naap/ebs/animations/ebs.html

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:

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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 close 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:

P 2 =(2⇡)2

G(m1 +m2)(a1 + a2)

3

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.

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http://outreach.atnf.csiro.au/education/senior/astrophysics/binary_types.html

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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),

For unresolved binaries (the vast majority), we can measure only P and the two velocity amplitudes, so this is two equations in three unknowns

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Measurement of stellar radius

!  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 depends on the radius of the larger star.

!  The slope of the transitions depends on the radius of the smaller star.

Time Fl

ux

t1 t2 t3 t4

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.

Ingress + Egress

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Time

Flux

t1 t2 t3 t4

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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.

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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)

Shown in Workshop #2.

If orientation of the orbit with respect to the line of sight is known, 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).

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Angular rotation rate and Rotation Period

For an asteroid in a circular orbit about the Sun

⌦ =p

GM/r3

For a binary system both objects orbit the center of mass at the same angular rotation rate

d✓

dt= ⌦

P =2⇡

⌦

Period is time it takes to go 2π

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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.

Time Fl

ux

t1 t2 t3 t4

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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?

Example (continued)

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8.3 years

11km/s

1.04km/s

Time

Flux

t1 t2 t3 t4

11.7 hours

164 days

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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 or semi-major axes:

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Example (continued)

! Masses:

! Stellar radii (note: solar radius = ):

(Kepler’s third law)

(previous result)

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OGLE survey (search for gravitational lenses) found a whole bunch of variable stars

An eclipsing binary found by the OGLE survey near the Galactic Center

OGLE field BWC. Star no: V19. Type: Eclipsing RA (2000.0):18:03:48.45 Dec (2000.0):-29:58:06.9 I magnitude:15.21 V-I color:0.95 I amplitude:1.07 Period (days):1.18572 JDhel at Min:2448723.6851 Class:EA

Light curve folded over the period.

http://bulge.princeton.edu/~ogle/

phase m

agni

tude

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:

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L = 4⇡R2S�BT

4S + 4⇡R2

L�BT4L

f =1

D2

⇥R2

S�BT4S +R2

L�BT4L

⇤

fP =1

D2

⇥R2

L�BT4L

⇤

Measurement of stellar effective temperatures (continued)

Then the secondary eclipse is when the small star passes in front of the larger one:

Three equations (for f, fS, fL), two unknowns (the temperatures). Construct one ratio, to remove uncertainties in distance D:

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fS =1

D2

⇥(R2

L �R2S)�BT

4L +R2

S�BT4S

⇤

f � fPf � fS

=R2

SR4S +R2

LR4L �R2

LR4L

R2ST

4S +R2

LR4L � (R2

L �R2S)T

4L �R2

ST4S

=

✓TS

TL

◆4

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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.

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Radii of eclipsing binary stars (Malkov)

Rad

ius

(Sol

ar r

adiu

s)

Giants

empirically M / R

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Luminosities of eclipsing binary stars

Lum

inos

ity (s

olar

)

L / M4

L / M3

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Effective temperatures of eclipsing binary stars Te

mpe

ratu

re (K

)

solar masses

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H-R diagram for binaries and other nearby stars

Stars within 25 parsecs of the Sun (Gliese and Jahreiss 1991) Nearest and Brightest stars (Allen 1973) Pleiades X-ray sources (Stauffer et al. 1994) Binaries with measured temperature and luminosity (Malkov 1993)

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The two body problem

Energy for two masses

And velocities

Note that this separates into two pieces, one for the center of mass, the other to describe the motion between the two masses

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Example, Detecting the motion of a star caused by a planet

The center of mass coordinate system is defined by

The motion of the star is approximately

Where a is the distance between the star and planet Note the mass of Jupiter is about 10-3 times that of the sun.

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Motion of a star caused by a Giant planet

For a star 10 pc away The angular separation is 0.005 AU/10pc ~ 2x10-9 radians ~ 4x10-4” The motion of a solar mass star 10 pc away by a Jupiter mass planet at 5AU is about 0.4mas. To detect this you need an astrometric grid that is good to better than this. This is perhaps detectable by GAIA

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The astrophysical importance of binaries

Observations of binaries allow accurate mass measurements. Original estimate of Pluto’s mass, for example, was about a factor of 10 off. Subsequently the fact that Pluto is a binary with Charon, allowed a much more accurate mass estimate. Likewise planets with satellites have accurate mass estimates. Likewise stellar binaries can be used to constrain the stellar models for the stars in the binaries. Extra solar planets are discovered in a way similar to finding binaries with a radial velocity study, just that the velocity shifts are about 1000 times lower than previously measured over long periods of time.

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The astrophysical importance of exoplanet transits

Mercury transits the Sun (R. Dantowitz)

Amateur Bruce Gary’s observations of exoplanet transit XO-1

Transits allow one to measure the radii, mass and so densities of both objects. Even though the total number of exoplanets that transit is few these measuresments are important. They allow one to determine whether the planets are gaseous or rocky.

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Common envelope eclipsing binaries

Eclipse never quite ends.

http://outreach.atnf.csiro.au/education/senior/astrophysics/binary_types.html

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Mass transfer systems

When do binaries transfer mass? Tidal overflow criterion. Tidal force from one must exceed the gravitational force of the other on itself. This criterion can be written as a ratio of densities. Close compact objects (white dwarfs, neutron stars, black holes) transfer mass becoming cataclysmic variables and X-ray binaries.

From physics.ubishops.ca/evolution/

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Condition for Tidal shredding

Can be applied to galaxies, clusters, stars, planets, comets …. Roche/Hill radii scale with mass to 1/3 power

FT ⇠ GM1R2

D3

Tidal force on the surface of M2 caused by M1 D Distance between M1 and M2

Self gravity of M2 R2 Radius of M2

FG ⇠ GM2

R22

FT > FG Tidal force exceeds self gravity on the surface of M2 M1

D3>

M2

R32 density like comparison

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Summary

! How to estimate stellar masses from radial velocity measurements in a stellar binary. ! How to estimate stellar radii from eclipsing binary light curves. ! The importance of eclipsing binaries in providing measurements of stellar masses and radii. ! Review of Tidal force