Observations of binary systems with pulsating components
Dominik Drobek (December 2009)
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.
• Theory of stellar pulsations has been in very rapid developement
in the twentieth century.
• Nowadays, detection of stellar pulsations allows astrophysicists
to construct accurate models of stellar interiors by means of
asteroseismic modelling.
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 d 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).
zonal mode
(|m|=0)
tesseral mode
tesseral mode
sectoral mode:
(|m|=l)
• 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).
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, ...)
How can we improve accuracy of our models?
• Detect more frequencies on a star (the more frequencies, the
better).
• Improve the theory to refine theoretical parameters.
• Determine some of the astrophysical parameters of the star
under consideration to narrow down the search for correct model.
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.
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 q1 + q2 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).
• For a given moment of time, one has to measure two angles: position
angle and angular separation.
A - primary, B - secondary component
N - North direction, W - West direction
q, r - position angle and angular separation
• Historically, such measurements were made using filar micrometers.
CCD cameras or interferometers are used today.
Binary systems: relative visual orbit
• Angle r is very small, so it can be treated as a line segment. (q,r) 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?
Binary systems: relative visual orbit
• 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?
Binary systems: relative visual orbit
• 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
w - argument of periastron
P - orbital period
W - longitude of line of nodes
e - eccentricity of orbit
T0 - time of periastron passage
i - orbital inclination
Binary systems: relative visual orbit
• 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 p’’, 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.
Binary systems: spectroscopic binaries
• 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 u: true anomaly
• angle w: argument of periastron
• vector r: radius vector
• vector z: projection of radius vector
onto observer’s line of sight
Binary systems: spectroscopic binaries
• 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.
, where g is barycenter’s velocity.
• The radial velocity can be written as
• 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 + 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
• Visual binary with relative orbit and parallax + SB1:
• from a’’ and p’’ 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.
• 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 (b Cep or SPB).
The list includes three promising eclipsing binaries with probable b Cep
components, which have recently been discovered in ASAS-3 data, but
have not been studied in detail yet.
16 Lac (EN Lac)
16 Lac = EN Lac = HR 8725 = HD 216916
a = 22h 56m 24s, d = +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).
• Orbital period: Porb = 12.097 d
• Walker (1951) discovered b Cep type pulsations.
• Pulsation frequencies: f1= 5.91134 d-1, f2= 5.85286 d-1, f3 = 5.49990 d-1 (Fitch
1969)
• 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)
V381 Car
V381 Car = HD 92024
a = 10h 36m 08s, d = -58° 13’ 05’’
V = 9.03 mag, (B-V) = -0.04 mag, SpT: B1 III
V381 Car
• b Cep type pulsations were discovered by Balona (1977).
• Eclipses were discovered by Engelbrecht and Balona (1986), both primary and
secondary are visible.
• orbital period: Porb = 8.323 d
• 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 b Cep type stars in NGC 3293 (Balona 1994).
h Ori
h Ori = HD 35411 = HR 1788
a = 5h 24m 29s, d = -02° 23’ 50’’
V = 3.38 mag, (B-V) = -0.17 mag, SpT: B0.5 V
h Ori
• h Ori is a hierarchical quintuple system.
• 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).
• 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)
• 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).
l Sco
Shaula = l Sco = HD 158926 = HR 6527
a = 17h 33m 37s, d = -37° 06’ 14’’
V = 1.62 mag, (B-V) = -0.14 mag, SpT: B2 IV
l Sco
• l Sco is a triple stellar system.
• Two components are in close orbit (Porb = 5.593 d), third component revolves
around the pair with Porb = 2.964 years.
• 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
• Line profile variations (e.g. Si III) with frequency f1 were observed.
• 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).
b Cen
Agena = b Cen = HD 122451 = HR 5267
a = 14h 03m 49s, d = -60° 22’ 23’’
V = 0.60 mag, (B-V) = -0.22 mag, SpT: B1 III
b Cen
• b 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 b 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).
V539 Ara
V539 Ara = HD 161783 = HR 6622
a = 17h 50m 28s, d = -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.
ALS 2460 and ALS 2463
members of Stock 14 (OCL 865) open cluster:
a = 11h 43.8m, d = -61° 31’
age: 6 Myr (Moffat & Vogt 1975)
distance: 2.8 kpc (Fitzgerald & Miller 1983)
ALS 2460 = HD 101794 = Stock 14-13
• Star classified as B1 IV e (Garrison et al. 1977).
• Weak emission in Balmer lines.
• Photometric variability discovered by Hipparcos satellite (HIP 57106).
• At first ALS 2460 was classified as type g Cas variable.
• Eclipses discovered in ASAS-2 data by Pojmański, Porb = 1.4632 d.
Pigulski, Pojmański (2008)
• short period variability discovered in ASAS-3 data by Pigulski and Pojmański
(2008): f1 = 4.45494 d-1, f2 = 1.83952 d-1 (f1 - b Cep, f2 - l Eri, g mode (?))
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).
Pigulski, Pojmański (2008)
• Pulsations of the primary: f1 = 3.12764 d-1, could be b Cep.
ALS 4801 = HD 167003 = V4386 Sgr
• MK classification: B0.5 III (Hill et al. 1974), B1 II (Garrison et al. 1977),
B1 Ib/II (Houk 1978).
• Photometric variability discovered by Hipparcos satellite (HIP 89404).
• Orbital period from ASAS-3 data: Porb = 10.79824 d:
Pigulski, Pojmański (2008)
• 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 b Cep variable.
Thank you for your attention.