dominik drobek (december 2009) observations of binary systems with pulsating components

Dominik Drobek (December 2009)

Observations of binary systems with pulsating components

Contents of this presentation:

• Pulsating stars:

• significance of n, l, m numbers

• radial and nonradial oscillations

• asteroseismology, construction of theoretical models

• Binary stars and determination of stellar parameters:

• optical binaries

• visual binaries with relative and absolute orbit

• orbital geometry and spectroscopic binaries

• detached eclipsing systems and light curve modelling

• summary

• Short, non-comprehensive list of binary systems with hot pulsating components.

Introduction - pulsating stars

• Pulsating star: star which exhibits intrinsic changes of brightness or radial velocity.

• This is caused by the existence of “pulsation mode” - a wave propagating in the stellar interior, causing periodic displacements of stellar matter.

• Nowadays, detection of stellar pulsations allows astrophysicists to construct accurate models of stellar interiors by means of asteroseismic modelling.

• Theory of stellar pulsations has been in very rapid developement in the twentieth century.

Introduction - pulsating stars: more details

• Pulsation modes are described by their:

• frequency (f) (or, equivalently, period - P)

• radial order (n)

• degree (l)

• azimuthal order (m), |m|<=l

• Numbers n, l, m describe the distortion of the star, caused by propagation of a mode.

• There are two types of pulsation modes: p-modes (restoring force is pressure) and g-modes (restoring force is buoyancy).

Pulsating stars: radial oscillations

• If l=0, then pulsations are radial: star maintains spherical shape throughout pulsation cycle (examples include Cepheids and RR Lyr stars).

• n describes the amount of node lines in radial direction (in the picture below, n=2)

• n=0 corresponds to the fundamental mode, n=1 to the first overtone, and so on.

Pulsating stars: nonradial oscillations

• More general case of l>0 corresponds to nonradial oscillations: stellar surface is divided into sectors, adjacent sectors are reciprocal in phase (l=3 in the picture below).

• l is the total number of nodal lines on the stellar surface, m is the number of meridional nodal lines on the surface.

• Modes with nonzero value of m represent waves travelling around the star (m<0 corresponds to retrograde motion).

zonal mode (|m|=0)

tesseral modesectoral mode:

(|m|=l)

tesseral mode

Pulsating stars: asteroseismology

• Pulsation modes give us information about stellar interiors, but different modes propagate in different parts of the star.

• In order to make use of the pulsations, we must identify the modes - find their n, l, m numbers.

• Identification may be either photometric (e.g. from amplitudes of modes in different filters) or spectroscopic (e.g. from the distortions of spectral lines’ profiles).

• Construction of seismic models: we compute sets of many models of stellar interiors, and find the model which reproduces the observed frequencies most accurately.

Pulsating stars: asteroseismology

• Models depend on many parameters:

• theoretical (opacities, mixing length, overshooting, ...)

• astrophysical (mass, radius, chemical composition, ...)

Stellar mass can be found if star is a component of binary system. Stellar radius can be found if star is in the eclipsing system.

• Determine some of the astrophysical parameters of the star under consideration to narrow down the search for correct model.

• Detect more frequencies on a star (the more frequencies, the better).

How can we improve accuracy of our models?

• Improve the theory to refine theoretical parameters.

Introduction - binary systems

• Binary system: two stars exhibiting orbital motion around common center of mass (barycenter).

• A large fraction of stars in the Universe is in binary or multiple systems. This presentation focuses on binary systems only.

• Determination of components’ masses and radii is possible only when we have enough observational data about the binary system.

• Next slides present which parameters can be derived from various types of binary systems.

Types of binary systems: optical

• Optical binary system: two stars which, just by chance, are very close to each other in the sky.

• They are not gravitationally bound, and therefore cannot be used to determine stellar parameters.

• Examples include 1 + 2 Tauri, Mizar + Alcor, and many others.

• However, it may turn out that one of the optical components is an unresolved “true” (gravitationally bound) binary system.

• Example: with an advent of telescopic observations, Mizar was discovered to be a visual binary system (Mizar A + Mizar B).

• Furthermore, both Mizar A and B are spectroscopic binary systems.

Types of binary systems: visual

• Visual binary system: system of two gravitationally bound stars, in which the angular separation is large enough so that both components can be resolved.

Determination of relative visual orbit:

• Telescope with large aperture is desired (large aperture gives better angular resolution).

• Historically, such measurements were made using filar micrometers. CCD cameras or interferometers are used today.

A - primary, B - secondary component

N - North direction, W - West direction

, - position angle and angular separation

• For a given moment of time, one has to measure two angles: position angle and angular separation.

Binary systems: relative visual orbit

• Angle is very small, so it can be treated as a line segment. (,) are polar coordinates of the secondary component in a given moment of time.

• Once enough observations are made, the relative orbit of secondary component can be plotted (primary is at the origin):

Relative orbit of 70 Oph, by J.E. Gore

• What information does the visual relative orbit give us?


• Consider Kepler’s third law:

a - length of semi-major axis, P - orbital period, M1,2 - masses of components

a is in astronomical units, P in years, M1,2 in solar masses

If we know a and P, we can calculate the total mass of binary system.

• At this point, there are two problems with the relative orbit:

• a is in arcseconds, not AU

• observed orbit it’s a projection of the real orbit onto a plane perpendicular to the line of sight:

blue line is the orbital plane of the binary system

angle i is the orbital inclination: if i=0O, system is viewed face-on, and if i=90O - edge-on.

• What are the consequences of such projection?


• The projected relative orbit does not obey Kepler’s laws! The primary is not at the focus of the ellipse, and the areal velocity is not constant.

• In order to use Kepler’s third law, we first have to know the true shape of relative orbit.

• Consider an elliptic cylinder over the projected relative orbit. There exists only one cross section of this cylinder for which Kepler’s laws are satisfied. If we find it, we will know the true shape of relative orbit.

• The details are well outside the scope of this presentation, but such an operation is possible to perform.

Thus we obtain elements of the relative visual orbit:

a’’ - semi-major axis, in arcseconds - argument of periastron

P - orbital period - longitude of line of nodes

e - eccentricity of orbit T0 - time of periastron passage

i - orbital inclination


• Now we know the true value of a’’. But it is still in arcseconds!

• There is not much we can do about it - we need to know the distance.

angle a’’ is very small, and the distance d is the inverse of parallax ’’,

thus:

• Now we can calculate the total mass of the system:

• Unfortunately, that’s all the information we can obtain if all we have is a relative visual orbit.

• Is it possible to calculate the mass of each component?

Binary systems: absolute visual orbit

• Instead of measuring the position of secondary component relative to the primary, we can measure positions of both components relative to the distant stars:

• Both stars move on a curved path which twists around the motion of barycenter (dotted horizontal line).

• Component’s relative displacement from the center of mass is inversely proportional to mass of component. That way, we can find the mass ratio q of the system.

• Together with the total mass of the system, this gives us enough information to calculate masses of both components.

Binary systems: absolute visual orbit

• In order to obtain the mass ratio, we could use spectroscopy (more information will follow).

Summary:

Obtaining stellar masses from visual orbits requires quite a lot of work:

• observations may take years,

• relative orbit has to be freed from the effect of inclination,

• one needs to know the distance (parallax) of the system.

Because of these limitations, this method has been used to obtain reliable values of masses for less than 100 binary systems.

Is there a better method?

Binary systems: spectroscopic binaries

• Periodic motion of stars around the common center of mass has one more important consequence: periodic change of radial velocities.

• The change in radial velocities causes an observable Doppler effect on stellar spectral lines:

• Sometimes it’s the only indication of star’s binary nature.

Binary systems: SB1 and SB2

• Spectroscopic binaries: stars which exhibit periodic displacement of their spectral lines owing to Doppler effect caused by orbital motion.

• Depending on components’ relative brightness, the observed spectrum will show the displacement of lines of one or both components (if a star is too faint, its lines will not be visible).

• If displacement of both components’ lines is visible, system is a double-lined spectroscopic binary (SB2). If all we observe is displacement of one component’s lines, system is a single-lined spectroscopic binary (SB1).

• From the displacements, one can calculate the corresponding radial velocities. Measuring radial velocities in time allows us to plot a radial velocity curve.

Radial velocity curves of QW Geminorum, by M. Richmond.


• In order to gain more insight, we have to look at the geometry of binary system:

• point S: orbiting star

• point B: center of mass

• point P: periastron

• point A: ascending node

• angle i: orbital inclination

• angle : true anomaly

• angle : argument of periastron

• vector r: radius vector

• vector z: projection of radius vector onto observer’s line of sight


• From the geometry of a binary system, one can see that .

• The orbital inclination manifests itself through the sin(i) term, and will be much more troublesome this time.

• The radial velocity can be written as , where is barycenter’s velocity.

• Using the following formulae known from celestial mechanics:

and

one can derive the following formula for radial velocity of an orbiting star:

• And finally, we denote .

• K is an amplitude of the radial velocity curve (to be clear: amplitude = half of the variability range).

Binary systems: SB1 spectroscopic binaries

• From those formulae we can immediately see that:

• If orbit is circular (e=0), then the RV curve is sinusoidal. For an elliptical orbit, the shape is more complicated.

• In the case of SB2 system, K1/K2 = a1/a2 = M2/M1, which gives us another way of measuring the mass ratio.

• From the formula for K we have

a is in kilometers, P is in days, K is in km/sec.

• Again, consider the third Kepler’s law:

• We can derive the formula for mass function:

K is in km/sec, P in days, M1,2 in solar masses.

• asin(i) and fm are the only pieces of information we can get from a single-lined binary - it’s definitely something, but we need separate values of M1 and M2.

Binary systems: SB2 spectroscopic binaries

• For a double-lined spectroscopic binary, the situation is much better: we write the formulae for asin(i) for both components, and add them together:

• Then, we use the third Kepler’s law to get the following:

• Finally, we use the relation K1/K2 = M2/M1 to obtain:

• For all of the above formulae we need to know the orbital eccentricity. We can derive it from one of the RV curves using Lehmann-Filhés method (outside the scope of this presentation).

• Everything would be perfect, except for those sin(i) terms. Is there anything we can do to remove them?

Binary systems: eclipsing systems

• If the orbital inclination is equal or close to 90 degrees, in the course of orbital motion one of the stars passes in front of another, causing an eclipse.

• Consider a very trivial example: e=0, i=90O. The primary (A) has a greater radius and surface brightness. System is detached and components are spherical in shape. The secondary (B) passes in front of primary:

• in time interval (t2-t1) star B travels the distance 2RB

• in time interval (t3-t1) star B travels the distance 2RA

• since the orbit is circular, we may write:

and from that: and

Binary systems: light curve modelling

• The previous example shows that we can try to create mathematical models for the light curves of the binary systems and obtain relative radii of the components.

• More sophisticated models may be created to take into account the orbital inclination, non-zero eccentricity, limb darkening and more effects.

• Computer programs which fit a model light curve to the observed one by means of nonlinear Least Squares Minimisation have been in use since 1960s.

• For more detailed description, see: Nelson & Davis, ApJ 174,617 (1972), Popper & Etzel, AJ 86,102 (1981).

• Some popular light curve modelling programs include: WD, WINK, EBOB, JKTBOP, Nightfall, and more.

• The bottom line is: we can use such codes to find orbital inclination, and also the radii of components relative to the orbital separation.

• Limitation: we can use such codes for detached eclipsing systems only - moments of contacts t1 - t4 have to be easily determined from observed light curve, stars should have spherical shape.

Binary systems parameters: summary

We can derive masses of binary system’s components in following cases:

• Visual binary with absolute orbit and parallax.

• Visual binary with relative orbit and parallax + SB1:

• from a’’ and ’’ we have a=a1+a2 in kilometers

• we find a1 from P, e, i and K1, then calculate a2

• a1/a2=K1/K2, calculate K2

• calculate M1 and M2 from K1, K2, P, e and i.

• Visual binary with relative orbit + SB2:

• we have 7 elements of relative orbit (we know the orbital inclination)

• from spectroscopic solution we have M1, M2, a1, a2

• since a=a1+a2 and we know a’’, we can also find the distance to the system

• Detached eclipsing system + SB2:

• get masses and semi-major axes multiplied by sin(i) from spectroscopy

• obtain sin(i) and relative radii of components from light curve modelling

• free masses and semi-major axes from sin(i), calculate RA and RB

Binary systems with pulsating components

What follows is a list of nine interesting eclipsing binaries with pulsating components. This list is by no means comprehensive, and is biased

towards binary systems with hot pulsating components ( Cep or SPB).

The list includes three promising eclipsing binaries with probable Cep components, which have recently been discovered in ASAS-3 data, but

have not been studied in detail yet.

16 Lac = EN Lac = HR 8725 = HD 216916

= 22h 56m 24s, = +41° 36’ 14’’

V = 5.58 mag, (B-V) = -0.14 mag, SpT: B2 IV

16 Lac (EN Lac)

• Struve and Bobrovnikoff (1925) discovered that 16 Lac is a single-lined spectroscopic binary (SB1).

• Walker (1951) discovered Cep type pulsations.

• Pulsation frequencies: f1= 5.91134 d-1, f2= 5.85286 d-1, f3 = 5.49990 d-1 (Fitch 1969)

• Orbital period: Porb = 12.097 d

• Primary eclipse was discovered by Jerzykiewicz et al. (1978).

• Spectroscopic elements found by le Contel et al. (1983)

• Components’ masses and radii were found by Pigulski and Jerzykiewicz (1988): M1 = 10.2±0.5 MSun, M2 = 1.29±0.06 MSun, R1 = 6.4±0.3 RSun, R2 = 1.2±0.3 RSun

• First attempt at mode identification: Dziembowski and Jerzykiewicz (1996).

• Spectroscopic mode identification: Aerts et al. (2002)

• Mass of primary from asteroseismology derived by Thoul et al. (2003)

16 Lac (EN Lac)

V381 Car = HD 92024

= 10h 36m 08s, = -58° 13’ 05’’

V = 9.03 mag, (B-V) = -0.04 mag, SpT: B1 III

V381 Car

• Cep type pulsations were discovered by Balona (1977).

• orbital period: Porb = 8.323 d

• Eclipses were discovered by Engelbrecht and Balona (1986), both primary and secondary are visible.

• Freyhammer et al. (2005) performed spectroscopic observations and analysis of all available photometric data. Parameters of the systems have been determined:

Distance to binary system is in good agreement with the distance of

NGC 3293 cluster (2.75±0.25 kpc) determined by Baume et al. (2003)

from isochrone fitting.

• There are 11 Cep type stars in NGC 3293 (Balona 1994).

V381 Car

Ori = HD 35411 = HR 1788

= 5h 24m 29s, = -02° 23’ 50’’

V = 3.38 mag, (B-V) = -0.17 mag, SpT: B0.5 V

Ori

• Component Ac revolves around the Aa-Ab pair with Porb = 9.51 years.

• B and C components are separated from A by 1.5’’ i 115’’, respectively.

• Components resolved interferometrically (McAlister 1976, De Mey 1996, Balega et al. 1999)

(Balega et al. 1999)

• Components Aa and Ab form a double-lined spectroscopic binary, which is also an ecliping system. Porb = 7.989 d.

• Radial velocity curves: Žižka, Beardsley (1981).

• Total eclipses discovered by Waelkens i Lampens (1988).

• Short term variability is attributed to Ab component.

• Controversy regarding variability: period is either 0.301 d or 0.432 d.

• Line profile variations of Ab with P = 0.13 d discovered by De Mey et al. (1996).

Ori

• Ori is a hierarchical quintuple system.

Shaula = Sco = HD 158926 = HR 6527

= 17h 33m 37s, = -37° 06’ 14’’

V = 1.62 mag, (B-V) = -0.14 mag, SpT: B2 IV

Sco

• Pulsations were discovered by Shobbrook and Lomb (1972), three frequencies: f1 = 4.6794 d-1, f2 = 9.3588 d-1, f3 = 0.0985 d-1

• Shobbrook and Lomb suggested existence of eclipses, that was confirmed by Uytterhoven et al. (2004) from the analysis of Hipparcos data.

• Definitive confirmation of eclipses from the photometric data from the WIRE satellite (Brunnt, Buzasi 2005).

• Line profile variations (e.g. Si III) with frequency f1 were observed.

Sco

• Two components are in close orbit (Porb = 5.593 d), third component revolves around the pair with Porb = 2.964 years.

• Sco is a triple stellar system.

V = 0.60 mag, (B-V) = -0.22 mag, SpT: B1 III

Agena = Cen = HD 122451 = HR 5267

= 14h 03m 49s, = -60° 22’ 23’’

Cen

• Cen is a double-lined spectroscopic binary. There are no eclipses.

• Large eccentricity (e = 0.81), Porb = 357.02 d (Ausseloos et al. 2002):

• Pulsations of primary component: f1 = 6.5148 d-1, f2 = 6.4136 d-1, f3 = 6.4952 d-1

• Both components could be Cep type variables.

• Visual orbit was determined interferometrically (Davis et al. 2005).

• Accurate masses of components have been found (M1 = M2 = 9.1 ± 0.3 MSun).

Cen

V539 Ara = HD 161783 = HR 6622

= 17h 50m 28s, = -53° 36’ 45’’

V = 5.92 mag, (B-V) = -0.08 mag, SpT: B3 V

V539 Ara

• Detached eclipsing binary + double-lined spectroscopic binary, Porb = 3.17 d.

• Parametrs of the system determined by Clausen (1996):

• Spectral types of components: B3 V + B4 V

• SPB pulsations: f1 = 0.7351 d-1, f2 = 0.5602 d-1, f3 = 0.9254 d-1, f4 = 0.3256 d-1

• The secondary component is pulsating, not the primary.

V539 Ara

ALS 2460 and ALS 2463

members of Stock 14 (OCL 865) open cluster:

= 11h 43.8m, = -61° 31’

age: 6 Myr (Moffat & Vogt 1975)

distance: 2.8 kpc (Fitzgerald & Miller 1983)

ALS 2460 = HD 101794 = Stock 14-13

• Photometric variability discovered by Hipparcos satellite (HIP 57106).

• Star classified as B1 IV e (Garrison et al. 1977).

• Weak emission in Balmer lines.

• At first ALS 2460 was classified as type Cas variable.

• short period variability discovered in ASAS-3 data by Pigulski and Pojmański (2008): f1 = 4.45494 d-1, f2 = 1.83952 d-1 (f1 - Cep, f2 - Eri, g mode (?))

• Eclipses discovered in ASAS-2 data by Pojmański, Porb = 1.4632 d.

Pigulski, Pojmański (2008)

ALS 2463 = HD 101838 = Stock 14-14

• MK classification: B1 III (Feast et al. 1961), B0.5/1 III (Houk, Cowley 1975), B1 II-III (Garrison et al. 1977), B0 III (Fitzgerald, Miller 1983)

• Orbital period: 5.41166 d, but one can’t rule out the possibility that the period is twice as long (in such case components would have similar masses).

• Pulsations of the primary: f1 = 3.12764 d-1, could be Cep.


ALS 4801 = HD 167003 = V4386 Sgr

• Photometric variability discovered by Hipparcos satellite (HIP 89404).

• Orbital period from ASAS-3 data: Porb = 10.79824 d:


• Frequencies found during analysis of out-of-eclipse ASAS-3 data: f1 = 5.37837 d-1, f2 = 6.77277 d-1, f3 = 7.01607 d-1, f4 = 7.54603 d-1. Primary component is likely a Cep variable.

• MK classification: B0.5 III (Hill et al. 1974), B1 II (Garrison et al. 1977), B1 Ib/II (Houk 1978).

Thank you for your attention.

dominik drobek (december 2009) observations of binary systems with pulsating components

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