Today in Astronomy 38: 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 1
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
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. Sirius A and B in X rays (NASA/CfA/CXO) 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.) Lecture 3 3
Measurements of stellar radial velocities with the Doppler effect Radial velocity v r : 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. λobserved λrest λ λ0 v = = r λ λ c rest 0 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. Lecture 3 4
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). #: 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 = 3 b g b g 1 1 4π Gm + m a + a Lecture 3 5
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, at http://instruct1.cit.cornell.edu/courses/astro101/java/simulations.htm Check out the simulations of: Binary orbits with spectra Eclipsing binary with light curve Lecture 3 6
Eclipsing binary stellar system Flux Secondary Primary Minima P Time v r v v 1 Lecture 3 7
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: m a v v = = = m a v v 1 r 1 1r 1 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. Lecture 3 8
Stellar masses determined for binary systems (continued) If only the radial velocity amplitudes v 1 and v are known, the sum of masses is (from Kepler s third law) m 3 P v1 + v 1 m. + = πg sini You ll prove this in Homework #. 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). Lecture 3 9
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: v1 + v Rs = t t v1 + v R = t t ( ) 1 ( ) 3 1 Flux t1 t t3 t 4 Relative depth of primary and secondary brightness minima of eclipses can be used to determine the ratio of effective temperatures of the stars. Time Lecture 3 10
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 11
Flux Example (continued) Closeup of primary minimum 8.6 years Time v r 1.04 km/s 11 km/s 11.7 hours 164 days Lecture 3 1
Answers Example (continued) 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: m m r s s vs 11 = = = 10.6 v 1.04 P 14 = vs = 1.4 10 cm π =9.5 AU r r P 13 = v = 1.34 10 cm π =0.90 AU = 10.4 AU Lecture 3 13
Example (continued) Masses: r ms + m = = 15. M P m + 10.6m = s s 3 m = 1.3 M, m = 13.9 M s (Kepler s third law) (previous result) Stellar radii (note: solar radius = 696. 10 10 cm): vs + v vs + v R s = ( t t1) R = t t -1 = 6.0 km s 11.7 hr = 369 R ( )( ) 10 = 7.6 10 cm = 1.1 R ( ) 3 1 Lecture 3 14
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 15
Luminosity (solar luminosities) 10 6 10 5 10 4 10 3 10 10 1 10 0 10-1 10-10 -3 10-4 Luminosities of eclipsing binary stars (Malkov 1993) 0.1 1 10 Mass (solar masses) Lecture 3 16
Radii of eclipsing binary stars (Malkov 1993) Radius (solar radii) 10 1 0.1 0.1 1 10 Mass (solar masses) Lecture 3 17
Effective temperatures of eclipsing binary stars (Malkov 1993) Effective temperature (K) 5 4 3 10 4 7 6 5 4 3 0.1 1 10 Mass (solar masses) Lecture 3 18
Hertzsprung-Russell (H-R) diagram for eclipsing binary stars (Malkov 1993) Luminosity (solar luminosities) 10 6 10 5 10 4 10 3 10 10 1 10 0 10-1 10-10 -3 40000 30000 0000 10000 0 Effective temperature (K) Lecture 3 19
H-R diagram for binaries and other nearby stars Luminosity (solar luminosities) 1 10 6 1 10 5 1 10 4 1 10 3 100 10 1 0.1 0.01 1 10 3 1 10 4 1 10 5 1 10 5 1 10 4 1 10 3 Effective temperature (K) Stars within 5 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) Lecture 3 0