#### Transcript Lecture 6 - University of Waterloo

Lecture 5 Binary stars Binary stars • 85% of all stars in the Milky Way are part of multiple systems (binaries, triplets or more) • Some are close enough that they are able to transfer matter through tidal forces. These are close or contact binaries. Examples Two stars are separated by 3 A.U. One star is three times more massive than the other. Plot their orbits for e=0. Example: binary star system Two stars orbit each other with a measurable period of 2 years. Suppose the semimajor axes are measured to be a1=0.75 A.U. and a2=1.5 A.U. What are their masses? m1 r2 a2 m2 r1 a1 4 2 a 3 P G m1 m2 2 Visual binaries: mass determination A perfect mass estimate of both stars is possible if: 1. Both stars are visible 2. Their angular velocity is sufficiently high to allow a reasonable fraction of the orbit to be mapped 3. The distance to the system is known (e.g. via parallax) 4. The orbital plane is perpendicular to the line of sight Example: Sirius m1 r2 a2 m2 r1 a1 4 2 a 3 P G m1 m2 2 Sirius A and B is a visual binary: a period of 49.94 yr a parallax of p=0.377” The angular extent of its semimajor axis is a=aA+aB=5.52”. aA/aB=0.466 a Assume the plane of the orbit is in the plane of the sky: R R is the distance to the star. Visual binaries: inclination effects In general the plane of the orbit is not in the plane of the sky. Here is the true orbit Focii Visual binaries: inclination effects In general the plane of the orbit is not in the plane of the sky. Here is the true orbit, which defines the orbital plane Visual binaries: inclination effects In general the plane of the orbit is not in the plane of the sky. Now imagine this plane inclined against the plane of the sky with angle i: i Visual binaries: inclination effects In general the plane of the orbit is not in the plane of the sky. Now imagine this plane inclined against the plane of the sky with angle i: i 2acosi Instead of measuring a semimajor axis length a, you measure acosi where i is the inclination angle Visual binaries: inclination effects This projection distorts the ellipse: the centre of mass is not at the observed focus and the observed eccentricity is different from the true one. This makes it possible to determine i if the orbit is known precisely enough Visual binaries: inclination effects In practice we don’t measure a physical distance a, but rather an angular distance that we’ll call a. If a is the true angular distance, and a is the measured (projected distance) then: m1 a 2 a 2 cos i a 2 m2 a1 a1 cos i a1 So the ratio of the masses is independent of the inclination effect However, the sum of the masses is not: a cosi R 4 2 a 3 m1 m2 GP 2 3 4 2 aR GP 2 4 2 R 3 a 2 GP cos i 3 Example How does our answer for the mass of Sirius A and B depend on inclination? a cosi Sirius A and B is a visual binary: a period of 49.94 yr a parallax of p=0.377” The angular extent of its observed semimajor axis is a=5.52”. aA/aB=0.466 R Ra cos i 7.1 10 5 pc cos i a i 2acosi Example How does our answer for the mass of Sirius A and B depend on inclination? Ra cos i 7.1 10 5 pc cos i a mB 0.40 M Sun cos 3 i mB a A 0.466 m A aB a 1.5 1012 aB m 1 a A / aB cos i m A 2.1mB 4 2 a3 mB GP 2 1 aB / a A 0.84 M Sun cos 3 i Thus our answers are a lower limit on the mass of these stars. The measured inclination is actually i=43.5 degrees. So cos3i=0.38 and mA=2.2 Msun, mB=1.0 Msun Break z obs rest rest rest Spectroscopic binaries vr z if z 1 c Single-line spectroscopic binary: the absorption lines are redshifted or blueshifted as the star moves in its orbit Double-line spectroscopic binary: two sets of lines are visible Java applet: http://instruct1.cit.cornell.edu/courses/astro101/java/binary/binary.htm Spectroscopic binaries: circular orbits • If the orbit is in the plane of the sky (i=0) we observe no radial velocity. • Otherwise the radial velocities are a sinusoidal function of time. The minimum and maximum velocities (about the centre of mass velocity) are given by v max v sin i 1r 1 v2max r v2 sin i Spectroscopic binaries: circular orbits • We can therefore solve for both masses, depending only on the inclination angle i m1 m2 v max 2 2r v 2G sin 3 i max 2r Pv v max 1r max 2 2r v 2G sin 3 i max 1r Pv max 1r • In general it is not possible to uncover the inclination angle. However, for large samples of a given type of star it may be appropriate to take the average inclination to determine the average mass. Spectroscopic binaries: non-circular If the orbits are non-circular, the shape of the velocity curves becomes skewed in a way that depends on the orientation e.g. e=0.4, i=30°, axis rotation=45° • • • A sinusoidal light curve means orbits are close to circular From analysis of light curve it is possible to determine the eccentricity and orbit orientation, but not the inclination. In practice most orbits are circular because tidal interactions between the stars tend to circularize the orbits Java applet: http://instruct1.cit.cornell.edu/courses/astro101/java/binary/binary.htm Single-lined spectroscopic binaries In general, one star is much brighter than the other (remember faint stars are much more common than bright stars). This means only one set of absorption lines is visible in the spectrum. The Doppler motion of this single set of lines still indicates the presence of a binary system. We can still solve for a function of the two masses: m23 P v1max 3 r sin i 2G m1 m2 2 3 This is the mass function Eclipsing binaries A good estimate of the inclination i can be obtained in the case of eclipsing binaries, separated by distance d: R1+R2 To observer i cos i d R1 R2 d If d » R1+R2 (which is usually the case) then i~90 degrees Eclipsing binaries A good estimate of the inclination i can be obtained in the case of eclipsing binaries, separated by distance d: R1+R2 To observer cos i i R1 R2 d If d » R1+R2 (which is usually the case) then i~90 degrees d Assume i=90 degrees when in reality i=75 degrees. What is the error in sin3i? sin3(75)=0.9 So the error on the masses is only 10% if d > 3.9(R1+R2) Eclipsing binaries In the system just described, the eclipse just barely happens: To observer Face on So the amount of light blocked is not constant, and the light curve (total brightness as a function of time) looks something like this: Eclipsing binaries However, in the case of total eclipse the smaller star is completely obscured. In this case it is even more likely that the inclination is close to 90 degrees Face on To observer And the light curve shows constant minima: Eclipsing binaries In the case of a total eclipse we can also measure the radii of the stars, and the ratio of their effective temperatures If we assume i~90 degrees and circular orbits that are large relative to the stellar radius, then the radius of the smaller star is: And for the larger star: v rl tc t a 2 v rs tb t a 2 Where v is the relative velocity between the two stars Eclipsing binaries Ratio of effective temperatures (0) (1) (2) L 4R 2Te4 Ltotal Rs2Ts4 Rl2Tl 4 L1 Rl2Tl 4 L2 R R Tl R T 2 l 2 s Ltotal L1 Rs2Ts4 2 4 Ltotal L2 Rs Tl Ts Tl 4 4 2 4 s s Note that (1) will be the deepest minimum if Ts>Tl. (often the case since the brightest, largest stars are the cool supergiants) Alternatively (2) will be the deepest minimum if Tl>Ts Stellar masses • For select star systems, we can therefore measure the mass directly. • Luminosity is closely correlated with stellar mass Energy production rate is related to stellar mass If the available energy is proportional to mass, how do stellar lifetimes depend on their main sequence location? L M 2.5 L M5 The main sequence revisited • The main sequence is a mass sequence More massive stars are closer to the top-left (hot and bright) M=30MSun M=MSun M=0.2MSun The main sequence revisited • The main sequence is a mass sequence More massive stars are closer to the top-left (hot and bright) • Stars on the main sequence have radii 13 times that of the Sun Supergiants have R>100 RSun White dwarfs have R~0.01 RSun M=30MSun M=MSun M=0.2MSun Densities Since we know the stellar masses and radii, we can compute their average densities Sun: M 4 / 3R 3 1.99 1030 kg 4 / 36.96 10 1409 kg/m 3 8 m 3 Recall water has a density of 1000 kg/m3 Dry air at sea level: 1.3 kg/m3 Densities Since we know the stellar masses and radii, we can compute their average densities Supergiants (Betelgeuse): M 10M Sun R 1000 RSun Sun M R M Sun RSun 3 1409101000 1.4110 5 kg/m 3 3 or 1/100,000 times less dense than air. Densities Since we know the stellar masses and radii, we can compute their average densities White Dwarfs (Sirius B): M 0.6M Sun R 0.01RSun 6 105 Sun or 850,000 times denser than water.