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The Sun and the Stars The Sun and the Stars Dr Matt Burleigh The Sun and the Stars Binary stars: Most stars are found in binary or multiple systems. Binary star systems consist of 2 stars which are gravitationally bound with each star orbiting a common centre of mass. We can distinguish between several different types Apparent – chance alignment – not true binaries Visual – resolved binaries (individual components can be separated visually) >1” ,generally long orbital periods Astrometric – unresolved, companion identified by stellar wobble Spectroscopic – unresolved, other component revealed by period shift in spectral lines Spectrum – unresolved – spectral decomposition reveals two stellar components Eclipsing- systems which show periodic dips in their apparent brightness (systems may also be visual, astrometric or spectroscopic) [check out eclipsing binary simulator at http://astro.unl.edu/naap/ebs/animations/ebs.html] The most important of these are visual, spectroscopic and eclipsing binaries Why are binaries important? because analysis of their orbits allow us to determine the masses of the individual stars, their radii and shape (particularly in eclipsing systems), and the physical characteristics of the systems (separations, periods) . Dr Matt Burleigh The Sun and the Stars Visual binaries – in the optical, require separations of > 1” from the ground, otherwise components are unresolved. Examples – alpha-Cen A and B, Sirius A and B The angular separations and orbital paths are only apparent because in general the orbit is inclined to the plane of the sky, so we see the orbit in projection Measuring the displacement of the primary relative to the apparent focus, yields the orbital inclination, i, the true ellipticity e, and the true semi-major axis a” Dr Matt Burleigh The Sun and the Stars e.g. consider the following: m2 r2 a1 a2 r1 m1 From Kepler’s III law we have (a1 a 2 ) 3 G(m1 m2 ) 4 P2 2 where m1 and m2 are the masses of the 2 components, and a1 ,a2 are the semi-major axes of their orbits In the case of the Earth-Sun system, mSun>>mEarth, and the common centre of mass is located within the stellar radius, i.e. a1>>a2 Expressing the masses in solar masses and orbital radii in AU, then P has units of years, and thus the general form of Kepler’s third law can be written: m1 m2 a3 P2 If we express the separation between the binary components in seconds of arc, ”, then a " " q "3 m1 + m2 = "3 2 p P Dr Matt Burleigh The Sun and the Stars Since m1r1 m2 r2 , where r1+r2=a to determine the individual masses, we must find the relative distance of each star from the centre of mass of the system. NB In proper motion, the centre of mass travels along a straight line relative to background stars (see e.g. Sirius A and B) Proper motion of the visual binary Sirius A and B relative to background stars Dr Matt Burleigh The Sun and the Stars Spectroscopic binaries Two unresolved stars, separation 1AU, Period ~ hours to months, inclination i>0. Binaries exhibit lines (in absorption or emission) that show periodic variations. Systems may be: (i) single-lined (only one component displays lines) or (ii) double-lined (both components display lines) Lines are shifted in wavelength by an amount relative to the rest-wavelength 0, ,blueward (star approaching), and redward (star receding) [doppler effect], such that 0 0 0 vr c Detection of shifts limited by spectral resolution for two stars vr ~ km/s for a planet/star vr ~ m/s Dr Matt Burleigh Spectroscopic Binaries Dr Matt Burleigh The Sun and the Stars Radial Velocity curves Constructed by converting wavelength shifts to velocity shifts as a function of time, folded on the orbital period e.g. Radial velocity curves for nearby binary stars **Radial velocity curve for a hot Jupiter** Dr Matt Burleigh The Sun and the Stars The simplest radial velocity curves are from those systems viewed edge-on (i=90 degrees). They appear sinusoidal with opposite phases e.g. In this case, each star orbits around the centre of mass with orbital period P, so r1 v1 P 2 and r2 v2 P 2 The ratio of the stellar masses is given by The relative semi-major axis is and m1 m2 m1 r2 v2 m2 r1 v1 a r1 r2 a3 P2 The system is completely determined!! Dr Matt Burleigh The Sun and the Stars This rarely ever happens because (i) The system may be single lined (can only determine P and r1) (ii) Unless the system is also eclipsing we don’t know the inclination 3 m 2 sin 3 i ( m1 m 2 ) 2 If (i) is true then we can only quote the mass function Why? Recall then So m1 m2 a3 P2 and m1r1 m2 r2 mr (m1 m2 ) P (r1 r2 ) r1 1 1 m2 2 3 3 m r13 1 1 m2 3 m m1 r1 2 m 2 3 3 r1 '3 (m1 m2 ) 3 sin 3 im 2 3 r1' = r1 sini i is the inclination m3 sin3 i (r1' )3 f (m1, m2 ) = 2 = (m1 + m2 )2 P 2 NB if the primary mass m1 can be obtained from the spectral type, the system can be solved. More generally the system will be inclined. If the radial velocity curve is sinusoidal, we know we are dealing with circular orbits in which case we measure the projected velocity Vr sini for each component. In the case of elliptical orbits, the velocity curves are no longer sinusoidal. Although radial velocity curves are mirror images, they may have differing amplitudes Dr Matt Burleigh The Sun and the Stars Eclipsing binaries Close binary systems (small separations and short periods) in which one star passes in front of the other periodically blocking some of the light. For each orbit there will be two eclipses, a primary eclipse (when the primary star is eclipsed by the secondary and a secondary eclipse wherein the primary passes in front of the secondary (by convention, the hotter star is designated the primary, the cooler star the secondary). Eclipses can be either total or partial e.g. SV Cam HIP 59683 Dr Matt Burleigh Eclipsing Binaries Dr Matt Burleigh The Sun and the Stars Note that the type of eclipse observed, depends upon the orbital eccentricity and inclination and the stellar radii and surface temperatures. Rc Rp Conditions for eclipse: (i) (ii) cos i R p Rc R p Rc cos i R p Rc (iii) cos i R p Rc no eclipse partial eclipse Rp+Rc total and annular eclipse NB =90-i From timing the points of contact we can estimate the relative stellar radii Rp /a, and Rc /a From the relative depths of the eclipses we can estimate the relative effective surface temperatures Tp /Tc Dr Matt Burleigh Accreting binaries • Cataclysmic variables consist of a white dwarf and a cool secondary (usually an M dwarf) • Periods of 1.5 to a few hours • Material is accreted via Roche Lobe Overflow into a disc surrounding the white dwarf • Occasionally the disc suffers a thermonuclear detonation when too much material has accumulated • Observed as novae • See also Xray Binaries (accretion onto a neutron star or a black hole, eg Sco X-1) Dr Matt Burleigh Accretion in Binaries Dr Matt Burleigh The Sun and the Stars Additional notes – derivation of Keplers III law 2 Gm1 m2 mv mv 1 1 2 2 2 r1 r2 (r1 r2 ) 2 Balance between gravity and centripetal force Relocate to frame of one of the masses and replace mass with reduced mass Since r r1 r2 ,then m1 m2 m1 m2 Gm1 m2 v 2 r r2 The period of the orbit T, is So, T 2r v Gm1 m2 (2r ) 2 r2 rT 2 and T2 4 2 r 3 Gm1 m2 therefore T2 4 2 r 3 G (m1 m2 ) Dr Matt Burleigh