FINAL YEAR SEMINAR, OLD PROGRAM

ECLIPSING AND SPECTROSCOPIC BINARY STARS

Author: Franci GorjupMentor: Prof. dr. Tomaž Zwitter

Ljubljana, December 2013

Abstract:First, we will see, what kind of binary star systems we encounter, and where different categories come from. Then we will briefly examine Doppler effect, as it plays an important role in todays astronomy. We will also examine mass function and demonstrate how it can be used in binary star systems. Than

we will finally go into detail, what eclipsing and spectroscopic binaries are and why are they so important.

CONTENTS:

1 INTRODUCTION.........................................................................................................................3

2 DIFFERENT TYPES OB BINARY STARS.................................................................................3

3 GEOMETRY.................................................................................................................................5

3.1 DOPPLER EFFECT..............................................................................................................5

3.2 BINARY MASS FUNCTION................................................................................................6

4 ECLIPSING BINARIES...............................................................................................................7

4.1 SPETROSCOPICS BINARIES.............................................................................................7

4.2 ECLIPSING BINARIES........................................................................................................7

4.3 THREE DIFFERENT TYPES OF EB...................................................................................9

5 ACQUSITION OF DATA...........................................................................................................11

6 CONCLUSION...........................................................................................................................12

REFERENCES.........................................................................................................................................13

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1 INTRODUCTION

People often believe that stars are lone objects. Reality is different as observations clearly indicate frequently this is not the case. According to estimats, more than half of stars are members of star systems composed of two or even more stars, which move around a common center of mass. Often, our attention is drawn to binary star systems as they allow us to get to know some of the fundamental star properties [1]. We should mention that even multiple star systems often act as binary star systems. For example, in triple star systems, usually we have a relatively compact binary star system and the third star at a much larger separation circling around a common center of mass with the binary [2]. This can be easily understood as such hierarchical multiple systems are stable in time, while multiple systems with components at similar separations are not.Based on observation technique, we categorize binary star systems into four major types – visual, astrometric binary, spectroscopic binary and eclipsing binary system. In this seminar the latter will be discussed in more detail than others [3].

2 DIFFERENT TYPES OF BINARY STARS

Before we start talking about eclipsing binary stars, we will look at some basic properties of other types of binary stars. Binary stars can be categorized in different ways. For example, if one star is closely linked to its companion in a binary system, we call these types of stars interacting binary stars and are divided into three main types. Detached, semi-detached and over-contact binary stars. In this seminar we will categorize binary stars depending the method of detection (how we figure out, that the observed system is actually a binary star system).Sometimes it seems that two stars are in a binary system, but they can actually have no connection whatsoever. They just happen to be along the same line of sight as viewed from the Earth . This virtual binary stars are called optical doubles [4]. Real binary stars with gravitationally bound components are divided into four categories. If we can discern two stars moving on an elliptical orbit around each other, we call them visual binaries [4]. When we observe a star that wobbles on the sky, moving here and there, we can assume it is actually a binary system [4]. One of the stars is significantly brighter than the other and completely overshines it's companion. We call this type of object astrometric binaries [1]. Next type are spectroscopic binary stars. We detect these by looking at their spectra. One way is seeing two sets of spectral lines with variable separation due to projected motion of binary components along the line of sight. The other case is an apparent single star but with periodically changing radial velocity. Similarly as with astrometric binaries the other component may be much dimmer and so escapes our detection [2]. The last type are eclipsing binaries. As the name already indicates, binary systems of this type have periodical eclipses, which we detect as measured brightness minima. We measure this with photometry. Another name for this type of stars is photometric binary stars. The latter is a broader category which includes also stars with periodically varying flux but with no eclipses – variation can be a consequence of the fact that stellar surface brightness can depend on direction, so that it shows us a brighter or

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dimmer face as it rotates along binary orbit [2].These types are not mutually exclusive – spectroscopic binary stars can also be eclipsing for example. And some types are more useful for us to learn about stars in general and their life than others [1].

Figure 1: Binary star system can be of different types. One example is Mizar, which is an optical double (in the widest view: top), a visual binary (at larger magnification: middle) and two

spectroscopic binaries (deduced from spectroscopic analysis: bottom) [4]. Mizar is a quadruple system composed of two binary star systems. Mizar and star Alcor form a new binary system. The distance between the Mizar and Earth is approximately 80 light years. Mizar can be found in constellation of

Big Dipper.

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3 GEOMETRY

Before we start examining spectroscopic and eclipsing binaries, lets have a look at some fundamental laws of physics, needed for better understanding. These are binary mass function and Doppler effect.

3.1 DOPPLER EFFECTDoppler effect is crucial to determe stellar radial velocity [5]. Bound-bound electron transitions in gases in the outer parts of star's atmosphere give rise to absorption lines in stellar spectra [6]. Because of the radial movement of an observed star, spectral lines shift, according to Doppler effect. We can only measure radial component of velocity this way as Doppler effect has no influence on tangential component. We also note that the velocities involved are much smaller than the speed of light, so relativistic effects are negligible and we can use simple equation connecting radial velocity, rest wavelength and wavelength shift [5]:

(1)

c is the speed of light, vRad is radial velocity, Δλ (Δλ= λMeasured - λ) is wavelength shift and λ is rest wavelength (the same as if there was no wavelength shift) [5].Wavelength variation is caused by relative movement between us and a star. Shifts are usually small. When Δλ is positive, that means the distance between us and a star is increasing, we call that red shift. If it's negative, that means the distance is decreasing, we call it blue shift [7]. Radial velocities are usually drawn as a function of phase, defined as a time since the last primary star passage in front of the secondary star measured in units of orbital period. When orbits are circular, we get a sinusoidal variation of radial velocity with time as velocities of stars in such system do not change in size but only in direction [7]. Inclination of orbit i also influences the radial velocity. Inclination is angle between the plane of the orbit and the plane of the sky. On a graph this is seen as reduced amplitude of radial velocity for a factor sin(i). A smaller inclination, i.e. a binary with an orbital plane close to the plane of the sky, implies a smaller amplitude of radial velocity [7]. But orbits are often elliptical. In that case graphs are no longer sinusoidal, but they become skewed, because star velocities are no longer constant. In this case the shape of the radial velocity curve is influenced also by orientation of the elliptical orbit within the orbital plane [7].

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Δλλ

=vRad

c

Figure 2: Graph of primary (blue symbols) radial velocity and secondary radial velocity (yellow symbols) of an eclipsing binary star AR~Aur in Auriga [8].

3.2 BINARY MASS FUNCTION

In most binary star systems we see just one component, because the other star is too dim due to large luminosity differences between the two components. As an extreme case the dim component can be a white dwarf, neutron star or a black hole. In such cases we can measure only the orbital period P and the projected semi-major axis a·sin(i) of the visible star. Here i is the inclination angle, defined above [9]. We cannot determine individual masses of the system components for such binaries [2]. But we can derive a convenient quantity f(m1,m2), which is called a mass function. Using equations of motion and Kepler's laws we get [9]:

(2)

G is gravitational constant, m1 and m2 are the masses of both stars [9]. Even if we do not know the mass m1 of the more luminous star 1 or inclination angle (i) the value of the mass function sets a lower limit for m2 , since the right side of equation is always less than m2. In this way we can determine what kind of object the secondary star is. For example if it is rather massive but still not visible, it is bound to be a compact object. Possibilities, which are judged from the derived mass m2 can be a black hole, a neutron star or a white dwarf. If we can use the spectrum of the visible star 1 to judge its mass, and if the presence of photometric eclipses or light curve variation limits the range of acceptable inclination angles, we can use the value of the mass function to determine the exact mass of the invisible compact component [1].An example is a spectroscopic binary A0620-00. A short orbital period of 7.75 hours and a large amplitude of the visible star radial velocity of 433 km/s implies the value of the mass function of 3

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f (m1,m2)=K1

3 P2πG

=m2

3sin (i)3

(m1+m2)2

Solar masses. But when further constraints on inclination and nature of the visible star are taken into account we derive that a visible star of 0.15 MSun < M1 < 0.38 MSun has an invisible companion of 3.30 MSun < M2 < 4.24 MSun. Such a large mass of the invisible companion implies that it is a black hole [10].

4 ECLIPSING BINARIES

Because most of eclipsing binary stars are also spectroscopic binaries, we will first examine the latter.

4.1 SPECTROSCOPIC BINARIES

Sometimes with even the best telescopes we cannot determine if the object examined is a single star or a binary system, as the object is too far away to hope to see the components as separate points on the sky. An answer to that is spectroscopy [2]. The light that is collected is sent to an diffraction grating, which disperses the light according to its wavelength – the result is called a spectrum. The star's movement around a common center of mass causes spectral lines to shift from their rest frame wavelengths, but only if the star's direction of movement is not perpendicular to the line of sight [1]. If one star is not significantly more luminous than the other, we can observe spectra of both stars. Spectral lines of a star moving towards us are blueshifted and of a star moving away from us are redshifted [1]. This is the case, when we can observe spectra of both stars. A system like that is a called double-lined spectroscopic binary [1]. If one star is much brighter than the other, the darker star is overwhelmed and we can observe just one star. This is called a single-lined spectroscopic binary [1]. In this case a periodically varying Doppler shift of the spectral lines also reveals that we have a binary system, but informations about the system are fewer [1]. When inclination angle is large enough and the stars are close enough to each other, that means spectroscopic binary star is also an eclipsing binary [3].

4.2 ECLIPSING BINARIES

When observing stars, we can notice some have time-dependent flux, which is caused by various mechanisms. When it is caused by physical mechanisms and luminosity is actually changing, like in pulsating stars for example, such objects are intrinsic variables. But when geometry of a system is to blame, we deal with extrinsic variables. Eclipsing binary stars are an example of the latter case [3]. Eclipsing binary stars are binaries with orbital plane oriented very close to the line of sight of the observer. So we have two stars circling around a common center of mass, periodically eclipsing each other. This causes periodic variation of measured flux from the binary system [1]. Time dependent magnitude curve, acquired by photometry is called the light curve [2]. Eclipsing probability depends on distance between the stars. The smaller the orbit, the greater is the probability of an eclipse [3]. Eclipsing binary systems have inter-star distances which are only a few time larger than sizes of stars. Orbital periods are usually less than 10 days, usually even less than a day. They may however reach several years in some rare cases where the inclination angle is very close to 90 degrees [3]. Short periods and subsequently short distances between stars can mistakenly lead a

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reader to a conclusion that these stars are actually interacting binary stars. Although small distances are a characteristic of both systems, interacting binary stars are a special group in which life of one star is closely linked to its companion, e.g. through occasional mass transfer between the components [3].As mentioned before, if an inclination is near 90° the system can be both a spectroscopic binary and an eclipsing binary [2]. Even if the inclination is just 75° an estimate assuming that our line of sight is exactly in the orbital plane still gives an acceptable value for the total mass of the system [1]. In over-contact binaries, eclipses can occur even at much smaller angles (35°). These are of course just rare exceptions [3]. Inclination angle can be well estimated from light curve, in particular from the shape of eclipses. When eclipses are total, so that one of the stars gets completely hidden behind the other one, eclipses with approximately flat bottom will be observed in light curve (for both primary and secondary eclipse). But when we are dealing with partial eclipses, eclipses on a light curve are V shaped because the eclipsed area of one of the stars keeps changing during the whole eclipse. Partial eclipses imply that the inclination is smaller than 90° [1].

Figure 3: Graphs represent shape of time-dependent flux for partial (above) and total eclipse (below) . In both cases, blue star is the hotter component [11].

If we compare duration of an eclipse to orbital period we can estimate sizes of both stars in comparison to their separation [4]. Timing of primary and secondary eclipses allow us to estimate even orbit eccentricity. If the secondary eclipse occurs exactly at mid-point between primary eclipses the orbits are circular or at least eclipses happen when we look exactly along the major axis of the elliptic orbit [4]. In case of total eclipse, we can get star radius from elapsed time between eclipse start and occurrence of totality. And from decrease in total observed flux, we get relative effective temperatures of both stars [1]. When examining eclipsing binaries, we often use a combination of photometry and spectroscopy [3]. That way we get as most information. Examining spectra of both stars is especially useful to derive masses, chemical composition of both stars and their effective temperatures [4], while photometry fixes sizes of both stars and helps with the ratio of effective temperatures [3].

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When examining variation of measured brightness in a collection of eclipsing binaries, we quickly realize their light curves can be divided in three different categories. These categories are Algol, β Lyrae in W Ursae Majoris [4]. This categorization is corresponds to proximity category of both stars, as measured by their relative size compared to separation. Eclipsing binaries can be detached systems, semi-detached systems or over-contact systems. Other parameters also influence the measured magnitude – limb darkening, surface reflection in rotating stars also gravity darkening. Mass transfer from one star to another can affect surface temperature of stars. If the stars are close enough, their shape becomes ellipsoidal, caused by gravitational pull. One star may even fill its critical Roche lobe which is an equipotential surface on the verge of mass transfer. Consequently, a star like that has different brighntess in different parts of its surface[4]. All these effects affect the light curve and should be therefore taken into account when computing parameters of a system. This is accomplished by proper use of the physical modelling. We choose a model that fits the observed experimental data as closely as possible and then extract physical parameters [4]. But this procedure can be also very complicated, because different combinations of physical parameters may yield very similar experimental data.

4.3 THREE DIFERENT TYPES OF EB

Let's see how we categorize eclipsing binaries based on variation of the light curve. As mentioned before, different types are Algol, β Lyrae and W Ursae Majoris [4]. Acronyms are in use: EA (Algol), EB (β Lyrae) and E UMa (W Ursae Majoris) [3].

In Algol, light curve has a significant primary minimum and not so significant secondary minimum [4]. Most of the time measured brightness is constant, as EA are usually detached systems. So most of the time (except during eclipses) both stars are 'visible' and total luminosity is a sum of each of the star's luminosity [4]. When the bigger star (usually a cool giant) covers the small, hotter star, primary minimum occurs. And secondary minimum occurs when the situation is reverse [4]. That is the case in visible part of the spectrum. In infrared, the spectrum is continuously variable and the secondary minimum is relatively deep. This proves that a hotter star (radiating more in visual, blue part of the spectrum) is relatively smaller than its companion (usually a cool giant, radiating mostly in the infrared) [3].

Figure 4: Graph shows a light curve for Algol-type eclipsing binary. In this particular case we have Algol star in constellation Perseus, after which eclipsing binaries of this type are named [12].

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β Lyrae type are binary stars, named after a binary system in constellation Lyra. These systems are usually semi-detached [3]. That means that one star fills its Roche lobe and becomes drop-like shaped [4]. Therefore the total luminosity is fluctuating continuously, which differs from Algol type where total luminosity is mostly constant [4]. Light curve has two minima and the primary minimum is significantly deeper than the secondary one [3]. In binary systems like this, mass transfer from the primary to the secondary star is possible. This affects the light curve [4].

Figure 5: Graph shows a light curve for β Lyrae star in Lyra [13].

The third type are W Ursae Majoris type eclipsing stars [4]. These are over-contact systems. They form when both stars overfill their Roche lobes and generate a common envelope. Magnitude of such stars is constantly fluctuating, minima are of similar depth and of rounded shape.

Figure 6: Graph shows a light curve for W Ursae Majoris eclipsing binary star. Eclipsing binaries, which have light-curve as seen above, are W Ursa Majoris type [14].

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5 ACQUSITION OF DATA

If we wish to derive parameters of stars in a binary system, this is how we do it. First we choose an object, for which we already know it is a close binary star. Such objects can be found in various catalogs found on the web, Hipparcos catalog for example, which is found on VizieR [15].

http://vizier.u-strasbg.fr/viz-bin/VizieR

We choose an object that is visible from our geographic latitude and in the part of the year when we want to observe it.Nearly all binaries found in the catalogs already have a measured light curve. This way we don't have to engage in photometric measurements, which take a lot of time. We can of course redo the measurements to see if they match the already measured ones. If we want to solve a system (determine mass of individual components) we also need spectroscopic measurements.Some binaries some of the radial velocity measurements are already available in the literature. In this case we need to cover the part of the orbital phase where no measurements exist. When determining radial velocities synthetic spectra can be very helpful. We find a synthetic spectrum that fits our measured spectrum best. Parameters describing our synthetic spectrum (temperature, surface gravity, metallicity, rotational velocity...) are then taken as approximate parameters of our measured spectrum. That is of course not exact as parameters often correlate.Doppler shift between the measured and synthetic spectrum gives us radial velocity. This is not the radial velocity of individual components (of eclipsing binary stars) but radial movement of the system relative to the observer. Determining radial velocities of individual components is done as follows. Chemical composition of both stars is probably identical or very similar as binary stars are often born together. So we fit best the measured spectrum using the same composition for both stars. The remaining free parameters are velocities of both components, which are determined by seeking the best fit to of synthetic spectra to measured one. Software package called IRAF (Image Reduction and Analysis Facility) from National Optical and Astronomical Observatories is frequentlyused for this task. Once we have the light curve and radial velocity graph, we can determine mass and other parameters of eclipsing binary star. There are different computer programs intended for modeling eclipsing binary stars that can help us. One of these is PHOEBE (PHysics Of Eclipsing BinariEs), which was built on the top of the classical and widely used Wilson-Devinney code [16] [17].

http://phoebe-project.org/

ftp://ftp.astro.ufl.edu/pub/wilson/

As before we find a model that best fits our light curve and radial velocity measurements and through that determine parameters like masses of both stars, radius of the stars and even appearance of the observed eclipsing binary.

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6 CONCLUSION:Binary systems are space laboratories used for determination of stellar characteristics. We have learned that there are different types of binary systems, often even combinations of different types. Amongst these types, eclipsing binaries are especially interesting as they allow us to gather the maximum amount of information. With photometric measurements and spectroscopy we can extract a great deal of information about stars and star systems. Kepler's laws and Doppler effect are of course a must.

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REFERENCES:

[1] B. W. Carroll, D. A. Ostline, An Introduction to Modern Astrophysics, (Addison-Wesley Publishing Company, 1996)

[2] H. Karttunen, P. Kröger, H. Oja, M. Proutanen, K. J. Donner, Fundamental Astronomy, (Springer Verlag, 2007)

[3] J. Kallrath, E. F. Milone, Eclipsing Binary Stars: Modeling and Analysis, (Springer Verlag, 1997)

[4] F. H. Shu, The physical universe: an introduction to astronomy, (Oxford University Press, 1985)

[5] http://books.google.si/booksid=JdWHgT13w0AC&printsec=frontcover&source=gbs_ge_summar yr&cad=0#v=onepage&q&f=false (28. 5. 2013)

[6] http://spiff.rit.edu/classes/phys240/lectures/expand/expand.html (28 .5 .2013)

[7] http://books.google.si/booksid=2QVmiMW0O0MC&pg=PA86&lpg=PA86&dq=radial+velocity+w avelength+shift&source=bl&ots=47zLqIfehb&sig=98hbVmyIq3mP6kgvyNIRhEzJdlA&hl=en&sa =X&ei=EbmiUfzQCZLe7Aaz3IGwAQ&redir_esc=y#v=onepage&q=radial%20velocity%20wave length%20shift&f=false (28. 5. 2013)

[8] http://inspirehep.net/record/855053/plots (28. 5. 2013)

[9] http://astronomy.swin.edu.au/cms/astro/cosmos/b/Binary+Mass+Function (27. 11. 2013)

[10] http://arxiv.org/pdf/1311.5412v1.pdf (27. 11. 2013)

[11] http://www.oocities.org/wormdoggydog/chapter11.html (28. 5. 2013)

[12] http://csep10.phys.utk.edu/astr162/lect/binaries/algol.html (3. 12. 2013)

[13] https://www.google.com/searchq=beta+lyrae+binary&client=ubuntu&hs=zdt&channel=fs&sourc e=lnms&tbm=isch&sa=X&ei=yvqdUqqHH8WWtAaM7ICYCA&ved=0CAkQ_AUoAQ&biw=1 301&bih=678#facrc=_&imgdii=_&imgrc=2oJRKc9qV5ISUM%3A%3BWCh8YYqlQODvD%3B http%253A%252F%252Fmedia.skyandtelescope.com%252Fimages%252Ftop12_04_l.gif%3Bhtt p%253A%252F%252Fwww.skyandtelescope.com%252Fobserving%252Fobjects%252Fvariables tars%252F19211684.html%3B392%3B375 (3. 12. 2013)

[14] http://faculty.randolphcollege.edu/tmichalik/eclpvar.htm (3. 12. 2013)

[15] http://vizier.u-strasbg.fr/viz-bin/VizieR (27. 11. 2013)

[16] http://phoebe-project.org/ (27. 11. 2013)

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[17] ftp://ftp.astro.ufl.edu/pub/wilson/ (27. 11. 2013)

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