2 Binary stars85% 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.
3 ExamplesTwo stars are separated by 3 A.U. One star is three times more massive than the other. Plot their orbits for e=0.
4 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?
5 Visual binaries: mass determination A perfect mass estimate of both stars is possible if:Both stars are visibleTheir angular velocity is sufficiently high to allow a reasonable fraction of the orbit to be mappedThe distance to the system is known (e.g. via parallax)The orbital plane is perpendicular to the line of sight
6 Example: Sirius a R a R is the distance to the star. Sirius A and B is a visual binary:a period of yra parallax of p=0.377”The angular extent of its semimajor axis is a=aA+aB=5.52”.aA/aB=0.466Assume the plane of the orbit is in the plane of the sky:aRaR is the distance tothe star.
7 Visual binaries: inclination effects In general the plane of the orbit is not in the plane of the sky.Here is the true orbitFocii
8 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
9 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
10 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:iTrue major axis=2a2acosiInstead of measuring a semimajor axis length a, you measure acosiwhere i is the inclination angle
11 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
12 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:So the ratio of the masses is independent of the inclination effectHowever, the sum of the masses is not:a cosiRa
13 ExampleHow does our answer for the mass of Sirius A and B depend on inclination?a cosiSirius A and B is a visual binary:a period of yra parallax of p=0.377”The angular extent of its observed semimajor axis is a=5.52”.aA/aB=0.466Ri2acosia
14 ExampleHow does our answer for the mass of Sirius A and B depend on inclination?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
16 Spectroscopic binaries Single-line spectroscopic binary: the absorption lines are redshifted or blueshifted as the star moves in its orbitDouble-line spectroscopic binary: two sets of lines are visibleJava applet:
17 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
18 Spectroscopic binaries: circular orbits We can therefore solve for both masses, depending only on the inclination angle iIn 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.
19 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 orientatione.g. e=0.4, i=30°,axis rotation=45°A sinusoidal light curve means orbits are close to circularFrom 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 orbitsJava applet:
20 Example: z PhoenicisHere are raw observations of the velocities of two stars in a double-lined spectroscopic binary system, z Phoenicis:Julian date is a numerical date where each unit is 1 day
21 Example : z PhoenicisThis doesn’t look much like a sinewave!! The problem is that observations are taken at discrete points in time, and thus randomly sample the phase of variation.Find the period that gives the “best” curve when plotted against orbital phase:P=1.65 daysP= dP=1.68 daysPhasePhasePhase
22 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:This is the mass function
23 Example: z PhoenicisImagine only one of the velocity curves in this system was visible.v1=180 km/sThenIf we assume m1~m2 then m~4sin3i(1.38)=4.0 MSunsin3i(Recall from both velocity curves we recovered m1=1.9 MSunsin3i and m2=2.8 MSunsin3i )
24 Eclipsing binariesA good estimate of the inclination i can be obtained in the case of eclipsing binaries, separated by distance d:To observerR1+R2idIf d » R1+R2 (which is usually the case) then i~90 degrees
25 Eclipsing binariesA good estimate of the inclination i can be obtained in the case of eclipsing binaries, separated by distance d:To observerR1+R2idAssume i=90 degrees when in reality i=75 degrees. What is the error in sin3i?sin3(75)=0.9So the error on the masses is only 10% if d > 3.9(R1+R2)If d » R1+R2 (which is usually the case) then i~90 degrees
26 Eclipsing binariesIn the system just described, the eclipse just barely happens:Face onTo observerSo the amount of light blocked is not constant, and the light curve (total brightness as a function of time) looks something like this:
27 Eclipsing binariesHowever, 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 degreesFace onTo observerAnd the light curve shows constant minima:
28 Eclipsing binariesIn the case of a total eclipse we can also measure the radii of the stars, and the ratio of their effective temperaturesIf we assume i~90 degrees and circular orbits that are large relative to the stellar radius, then the radius of the smaller star is:Where v is the relative velocity between the two starsAnd for the larger star:
29 Eclipsing binaries Ratio of effective temperatures (0) (1) (2) 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
30 Stellar massesFor select star systems, we can therefore measure the mass directly.Luminosity is closely correlated with stellar massEnergy production rate is related to stellar massIf the available energy is proportional to mass, how do stellar lifetimes depend on their main sequence location?
31 The main sequence revisited The main sequence is a mass sequenceMore massive stars are closer to the top-left (hot and bright)M=30MSunM=MSunM=0.2MSun
32 The main sequence revisited The main sequence is a mass sequenceMore massive stars are closer to the top-left (hot and bright)Stars on the main sequence have radii 1-3 times that of the SunSupergiants have R>100 RSunWhite dwarfs have R~0.01 RSunM=30MSunM=MSunM=0.2MSun
33 DensitiesSince we know the stellar masses and radii, we can compute their average densitiesSun:Recall water has a density of 1000 kg/m3Dry air at sea level: 1.3 kg/m3
34 DensitiesSince we know the stellar masses and radii, we can compute their average densitiesSupergiants (Betelgeuse):or 1/100,000 times less dense than air.
35 DensitiesSince we know the stellar masses and radii, we can compute their average densitiesWhite Dwarfs (Sirius B):or 850,000 times denser than water.