Parameter determination of a binary star system

Diploma Thesis, 2001
71 Pages, Grade: 1,0




1 Review of Binary Stars
1.1 Introduction
1.2 Observational Classification of Binary Stars
1.3 Visual Binaries
1.3.1 Orbital Elements
1.3.2 Mass Determination
1.4 Astrometric Binaries
1.5 Spectroscopic Binaries
1.5.1 Orbital Elements
1.5.2 Double-lined Spectroscopic Binary Star
1.5.3 Single-lined Spectroscopic Binary Star
1.6 Eclipsing Binaries
1.6.1 Phenomenological Classification
1.6.2 Morphological Classification
1.7 The Importance of Data Derived from Eclipsing Binaries

2 Spectroscopy
2.1 Astrophysical Spectra
2.2 Radial Velocities
2.3 Spectrophotometry
2.4 Line-Profile Analysis

3 Photometry
3.1 Photoelectric Photometry
3.2 Two-Star Photometers
3.3 Photoelectric Observations
3.4 Imaging Data

4 IRAF - Image Reduction and Analysis Facility
4.1 A Short Introduction to Analysis of Single Dispersion Spectra
4.1.1 Philosophy and Practice of IRAF
4.1.2 IRAF Set-up
4.1.3 Aperture Extraction
4.1.4 Wavelength Calibration
4.1.5 Flux Calibration
4.2 Reducing Echelle Spectra
4.2.1 Introduction
4.2.2 Processing Details

5 Binary System UV Leonis
5.1 UV Leonis
5.2 The 182-cm Telescope at Cima Ekar (Asiago)
5.2.1 Instrumentation of the Asiago 182-cm Telescope
5.3 The Pika Telescopes at Crni Vrh Observatory
5.3.1 Crni Vrh Observatory
5.3.2 The ACIT Imaging System
5.3.3 The AIT Imaging System
5.4 Spectroscopical and Photometrical Results
5.4.1 Parameter Determination of UV Leonis
5.4.2 The Radial Velocity Curve of UV Leonis
5.4.3 The B and V Light Curves of UV Leonis
5.4.4 Discussion Epilogue

List of Figures

1.1 Orbital elements of a binary system

1.2 Synthetic Algol-type light curve

1.3 Synthetic /3 Lyrae-type light curve

1.4 Synthetic W UMa-type light curve

1.5 Corotating coordinates for a binary star system

1.6 The effective gravitational potential

1.7 Roche potential and shape of a detached binary system

1.8 Roche potential and shape of a semi-detached binary system

1.9 Roche potential and shape of a (over-)contact binary system

2.1 Terminology for spectral lines

4.1 Blaze and Echelle grating

4.2 A twodimensional picture of an Echelle spectrum

4.3 Identification of the apertures with the star of interest

4.4 Fitting the tracing to the spectrum

4.5 The science spectrum calibrated in wavelength

4.6 An expanded plot of an echelle order examining the background regions

4.7 A plot of the comparison spectrum

4.8 Initial wavelength fitting for an echelle spectrum with residuals versus pixel number

4.9 Initial wavelength fitting with residuals versus order number

4.10 A second pass through the fitting of the dispersion function

4.11 The final fit

4.12 Continuum normalization

4.13 Examination of the extracted and calibrated spectra

5.1 The radial velocity curve of UV Leonis

5.2 The B arid V light curves of UV Leonis

5.3 Image of UV Leonis


The focus of this work about binary star systems is on model applications and selections of an algorithm to determine the parameters of individual binaries. It consists of four major parts: a general introduction to the theory of binary stars (Chapter 1); introductory material about spectroscopy and photometry (Chapter 2 and 3); a description of IRAF (Imaging Reduction and Analysis Facility) - this is a software program which allows the reduction and the analy­sis of gathered data of astronomical objects - and its application on parameter determination of such a binary system (Chapter 4); and the presentation of the two observatories Crni Vrh and Asiago, and of the spectroscopical and photo- metrical results of the binary system UV Leonis (Chapter 5).

Chapter 1 presents the ideas and concepts that are necessary to understand the interaction of binary systems and to get a physical overview of the field. First, the classical classification of binary stars based on observational results is given, followed by brief descriptions of each type of these systems. In particular, the issues of what can be derived from spectroscopic and eclipsing binary stars and why these data are relevant to astrophysics in general are considered. Further, the general concept of parameter determination for each type of binary systems and of equipotential surfaces in case of eclipsing binaries is introduced.

Since binary star analysts need understanding of observational data, some back­ground on the data base and methods of data acquisition is necessary. There­fore, Chapter 2 and 3 review the two most common methods of data analysis, spectroscopy and photometry. The intention is to summarize aspects of obser­vational astronomy relevant to light curve acquisition and modeling.

Chapter 2 begins with an overview of astrophysical spectra and their two most important features: radiative transfer and spectral lines. In additioxr, the charac­teristics of radial velocities and the advantages of spectrophotometry are shortly explained. Finally, the most basic mechanisms of spectral line broadening, which originate in the star itself, are enumerated.

Chapter 3 intends to clarify the importance of photometry in optical astron­omy. It reaches from photoelectric photometry, still the most precise and ac­curate means of obtaining flux measurements, via two-star photometers and the description of the process of standardization of photoelectric data to CCD (“charge couple device”) photometry, whose main advantage is that it can detect many sources simultaneously and that detection of faint sources is possible even in the presence of bright sources.

Chapter 4 focuses on the data reducing and analyzing system IRAF, which in­cludes a good selection of programs for general image processing and graphics, a large number of programs for the reduction and the analysis of optical and infrared astronomy data. First, a brief introduction to the analysis of single dis­persion spectra, which deals with aperture extraction, wavelength calibration, and flux calibration of these types of spectra, then, to the reduction of echelle spectra is given.

The second part is more extensively described, starting with a short explanation of echelle spectra followed by the processing details, which are the same as in the case of single dispersion spectra, because it is a crucial part of this work. The process done by IR.AF is illustrated with figures shown in this chapter, where the evaluated data was gathered at the Asiago Observatory (observed with an echelle spectrograph).

In the beginning of the last chapter a short description of the characteristics of the binary system UV Leonis and of both observatories, Crni Vrh Observatory and Asiago Observatory at Cima Ekar, where all spectroscopical and photomet- rical data of UV Leonis used for this work were gathered, is presented. Finally, the obtained results, the parameter determination, the radial velocity curve, and the B and V light curves of UV Leonis, are shown.

Chapter 1 Review of Binary Stars

1.1 Introduction

Binary stars are important, first, because they are numerous. The observed fre­quency of spectroscopic binaries detected in the galactic halo is not significantly different from that in the disk, despite differences in kinematic properties and chemical composition.

It is approximately 20% but the actual frequency is higher because many bina­ries remain undetected. In the solar neighbourhood the frequency is more than 50% - because of the advantage of proximity so that proper motion variations can be detected and several stars are in fact multiple systems.

The second reason for the importance of binaries is that they are the primary source of our knowledge of the fundamental properties of stars, for example, the direct determination of the mass of any astronomical object. This requires mea­surable gravitational interaction between at least two objects (galaxy-galaxy, star-star, star-planet, planet-satellite).

In galaxy-galaxy interactions, the distances and separations are so large that no detectable motion on the plane of the sky is possible. In star-planet, interactions only the star’s motions are detectable, and the properties of that star must be assumed, mainly on the basis of previous binary star studies, in order to deduce the properties of the planet.

In star-star interactions, the variations in position and velocity caused by or­bital motion are detectable for a wide range of stellar separations. It is often the case that both stars may be studied in any of several ways, depending on their distance, brightnesses, and motions.

Other basic properties of stars and of the systems they constitute can be deter­mined through analysis of observational data, depending on the observational technique by which the interaction is studied.

Observational Classification of Binary Stars

1. Optical double:

These systems are not true binaries but simply lie along the same line of sight. As a consequence of their large physical separations, they are not gravitationally bound and are not useful in determining stellar masses.

2. Physical double:

These systems are true binaries in gravitationally bound orbits. There are different types of such systems and the most common ones are listed below.

(a) Visual binary:

the two stars are seen as separate light sources that orbit one another with the passage of time.

(b) Astrometric binary:

the physical association of two stars is inferred, although only one star is actually observed, because the proper motion of the visible star wobbles in the sky.

(c) Spectroscopic binary:

a physical pair is inferred from spectroscopic observations which show a periodic variation of the Doppler shift of the spectral lines. If the spectral lines of both stars are observed, the system is a double-lined spectroscopic binary. If the spectral lines of only one component are observed, the system is a single-lined spectroscopic binary.

(d) Eclipsing binary:

a bound pair of stars is deduced from periodic changes of the total light from the system that can be interpreted in terms of eclipses of one star by the other. An analysis of the light curve can often yield an estimate of the inclination of the orbit of the system relative to the line of sight.

Three types of systems can provide us with mass determinations:

- visual binary with analysis of proper motions, if the distance to the system is known,
- visual binary with analysis of proper motions, if the radial velocities of the two components are known, and
- double-lined spectroscopic and eclipsing binary with analysis of light and velocity curves.

1.3 Visual Binaries

Both stars in the binary can be resolved independently in a telescopic eyepiece and, assuming that the orbital period is not prohibitively long, it is possible to monitor the motion of each member of the system and to determine the component masses m1 and m2.

They are derived from Kepler’s third law and the moment equation

illustration not visible in this excerpt

where [Abbildung in dieser Leseprobe nicht enthalten] and [Abbildung in dieser Leseprobe nicht enthalten] are the semi-major axis of the absolute orbits of the components about a common center of mass. In most cases, only the relative orbit and its semi-major axis, [Abbildung in dieser Leseprobe nicht enthalten] can be determined.

These systems provide important information about the angular separation of the stars from their mutual center of mass. The resolving power, or the ability to resolve both components, can be described mathematically by

illustration not visible in this excerpt

where A is the minimum angular separation in radians, D is the aperture of the telescope, and A is the wavelength in the same units.

This quantity is, in fact, the central radius of the diffraction disk. If the distance to the binary is also known, the linear separations of the stars can then be calculated.

Resolution achievable by telescopes on the ground is, however, always limited by atmospheric seeing and not by diffraction. Active optics is necessary to push the seeing limit significantly below 1 arc-sec. In space, instrumental resolution is the limiting condition.

The repaired Hubble Space Telescope, for example, permits direct viewing of both the separation and rough surface details of the Pluto-Charon system. Direct angular measurements of some of the largest of the sky’s bright stars are now possible.

The following less direct but more effective techniques also permit high angular resolution:

- Lunar occultations:

The edge of the Moon occasionally occults a star or stellar system within the maximum range of its declination (about±28°). Analysis of the resulting diffraction pattern intensities can determine binary star separations and even the diameters of stars.

- Phase interferometry:

Around 1920, Michelson and Pease determined the sizes of bright red-giant stars with the help of a phase interferometer.

The practical limit to angular resolution with this method was about 0,01 arc-sec, and was set by two factors: mechanical flexure of the interferometer arm and atmospheric seeing.

- Intensity interferometry:

Beginning in the 1950s, the diameters of blue stars were measured with an intensity interferometer at a multiple telescope observatory at Narrabri, Australia. The technique involves determining the correlation between the light received by several collectors.

A similar method is now attempted on a much larger scale by the interferometric mode of the VLT (Very Large Telescope) at ESO.

- Speckle interferometry:

Speckle observations involve the determination of pattern parameters in which the atmosphere acts as a diffusing screen. Although it is called interferometry, it is rather a correlation method. This method has proven very fruitful for visual binary work.

Long-baseline interferometry permits the resolution of many spectroscopic binaries. As is the case for all well-determined visual binaries, coupled with high precision radial velocity data, the parameters can yield all the geometric elements of the orbits.

Interferometric observations from space offer many advantages, among them a spectral range from the far-ultraviolet to the far-infrared.

This means the possibility of observations of objects such as protostar binaries which radiate in the far infrared.

1.3.1 Orbital Elements

We distinguish between absolute and relative orbits. Orbits with absolute orbital semi-major axis [Abbildung in dieser Leseprobe nicht enthalten] and U2 have the origin of coordinates at the system barycenter, while orbits with the relative orbital semi-major axis a describe the motion with respect to the center of its companion star.

Absolute and relative orbits are coupled by

illustration not visible in this excerpt

and the moment equation (1.1).

From the so-called seven classical orbital elements the following four determine the true orbit and the motion in it:

illustration not visible in this excerpt

The other three elements describe the projection of the true orbit onto the plane-of-sky and where the orbit lies relatively to the observer:

illustration not visible in this excerpt

1.3.2 Mass Determination

From the orbital data, the orientation of the orbits and the mutual center of mass can be determined, providing knowledge of the ratio of the stars’ masses.

If the distance to the system is also known, from trigonometric parallax for instance, the linear separation of the stars can be determined, leading to the individual masses of the stars in the system.

- Mass sum:

The general form of Kepler’s third law

illustration not visible in this excerpt

gives the sum of masses of the stars (G is the gravitational constant), provided that the semimajor axis of the relative orbit of the two stars is known.

The semimajor axis[Abbildung in dieser Leseprobe nicht enthalten] can be determined directly only if the distance from the observer to the binary star system, d, has been determined.

- Mass ratio:

The ratio of masses may be found from the ratio of the angular separations of the stars from the center of mass from Equation (1.1)

illustration not visible in this excerpt

where [Abbildung in dieser Leseprobe nicht enthalten] and 02 are the semi-major axis of the ellipses.

The angles subtended by the semi-major axis are

illustration not visible in this excerpt

where [Abbildung in dieser Leseprobe nicht enthalten] and [Abbildung in dieser Leseprobe nicht enthalten]are measured in radians.

Substituting for these expressiones, the mass ratio simply becomes

illustration not visible in this excerpt

Even if the distance to the star system is not known, the mass ratio may still be determined. Note that since only the ratio of the subtended angles is needed, [Abbildung in dieser Leseprobe nicht enthalten]and [Abbildung in dieser Leseprobe nicht enthalten]may be expressed in arc seconds, the unit typically used for angular measurements in astronomy.

- Individual masses:

Assuming that the distance d is known, the sum of the masses, mi + m2, may be combined with the mass ratio, toi/to2, to give each mass separately.

This process is complicated somewhat by the proper motion of the center of mass, by Earth’s motion around the Sun, and by the fact that most orbits are not conveniently oriented with their planes perpendicular to the line of sight of the observer.

1.4 Astrometric Binaries

If one member of a binary system is significantly brighter than the other, it may not be possible to observe both members directly. In such a case the existence of the unseen member may be deduced by observing the oscillatory motion of the visible component.

Since Newton’s first law requires that a constant velocity be maintained by a mass unless a force is acting upon it, such an oscillatory behavior indicates that another mass is present.

The motion of the main component about the center of mass of this system can be detected with the measurement of its proper motion.

1.5 Spectroscopic Binaries

If the period of a binary system is not prohibitively long and if the orbital motion has a component along the line of sight, a periodic shift in spectral lines will be observable. The Doppler effect causes the spectral lines of a star to be shifted from their rest frame wavelengths if that star has a nonzero radial velocity. Since the stars in a binary system are constantly in motion about their mutual center of mass, there must necessarily be periodic shifts in the wavelength of every spectral line of each star (unless the orbital plane is exactly perpendicular to the line of sight).

It is also apparent that when the lines of one component are blueshifted (the star is moving toward the observer), the lines of the other must be redshifted (the star is moving away), relative to the wavelengths that would be produced if the stars were moving with the constant velocity of the center of mass.

The stronger the shift in the wavelength,[Abbildung in dieser Leseprobe nicht enthalten], the bigger the radial velocity, v, of the object

illustration not visible in this excerpt

where c is the light speed and A the zero position of the line.

Assuming that the luminosities of each component are comparable, both spectra will be observable (a so-called double-lined spectroscopic binary star). However, if one star is much more luminous than the other, then the spectrum of the less luminous companion will be overwhelmed and only a single set of periodic spectral lines will be seen (a so-called single-lined spectroscopic binary star). In either situation, the existence of a binary star system is revealed.

Nevertheless, it may be that the orbital period is so long that the time variation of wavelengths of spectral lines may not be apparent.

In any case, if one star is not overwhelmingly more luminous than its companion and if it is not possible to resolve each star separately, it may still be possible to recognize the object as a binary system by observing the superimposed and oppositely Doppler-shifted spectra.

Even if the Doppler shifts are not significant (if the orbital plane is perpendicular to the line of sight, for instance), it may still be possible to detect two sets of superimposed spectra if they originate from stars of differing spectral classes.

1.5.1 Orbital Elements

When we consider spectroscopic binaries we must take into account that the determination of orbital elements is a little bit more difficult than in case of visual binaries.

First, it is only possible to measure the radial velocity component of velocity, therefore the inclination remains unknown.

The elements

illustration not visible in this excerpt

cannot be separated.

We only obtain the expression a sin«.

Second, the position angle of the line of nodes, D, is not determinable. Also, the center of mass may move at a nonzero radial velocity with respect to Solar barycenter, so we introduce

illustration not visible in this excerpt

The individual masses of both stars in spectroscopic binaries can only be determined under special conditions. The lines of both components must be seen in the spectrum, and the system must be simultaneously an eclipsing binary.

1.5.2 Double-lined Spectroscopic Binary Star

Since the individual members of the system cannot be resolved, the techniques used to determine the orientation and eccentricity of the orbits of visual binaries are not applicable.

Also, the inclination angle, i, clearly plays a role in the solution obtained for the stars’ masses because it directly influences the measured radial velocities. In fact, the actual measured radial velocities depend upon the positions of the stars at that instant; if the directions of motion of the stars happen to be perpendicular to the line of sight, then the observed radial velocities will be zero.

For a star system having circular orbits, the speed of each star is constant. If the plane of their orbits lies in the line of sight of the observer (with an angle of inclination i = 90°), then the measured radial velocities produce sinusoidal velocity curves.

Changing the orbital inclination does not alter the shape of the velocity curves; it merely changes their amplitudes by the factor sin[Abbildung in dieser Leseprobe nicht enthalten].

When the eccentricity of the orbits is not zero, the observed velocity curves become skewed. The exact shapes of the curves also depend strongly on the orientation of the orbits with respect to the observer, even for a given inclination angle.

In reality, many spectroscopic binaries possess nearly circular orbits, simplifying the analysis of the system somewhat.

If we assume that the orbital eccentricity is very small (e <C 1), then the velocities of the stars are essentially constant and given by

illustration not visible in this excerpt

for stars of mass m1 and m2, respectively.

Solving for [Abbildung in dieser Leseprobe nicht enthalten] and a2 and substituting into Equation (1.3), we find that the mass ratio of the two stars becomes

illustration not visible in this excerpt

Recalling that

illustration not visible in this excerpt

the previous equation can be written in terms of the observed radial velocity amplitudes rather than actual orbital velocities

illustration not visible in this excerpt

As is the situation with visual binaries, the ratio of the stellar masses can be determined without knowing the angle of inclination. However, as is also the case with visual binaries, determination of the sum of masses requires that the angle of inclination be known.

Replacing a with

illustration not visible in this excerpt

in Kepler’s third law (1.2) and solving for the sum of masses, we have

illustration not visible in this excerpt

Writing the actual radial velocities in terms of the observed values, we get finally the mass sum of a double-lined spectroscopic binary

illustration not visible in this excerpt

If the value of inclination, i. is not known, only the lower limit to the total mass of the system can be obtained.

1.5.3 Single-lined Spectroscopic Binary Star

We see from Equation (1.5) that the sum of masses can be obtained only if both observed radial velocities, v1r and v2r, are measurable. Unfortunately, this is not always the case.

If one star is much brighter than its companion, the spectrum of the fainter member will be overwhelmed. Such a single-lined spectroscopic binary system shows only the spectrum of the brighter component while the spectrum of the fainter one is not observable.

Equation (1.4) allows v2r to be replaced by the stellar mass ratio, giving a quantity that is dependent on both of the system masses and the inclination. Substituting, Equation (1.5) becomes

illustration not visible in this excerpt

and rearranging terms gives the mass function

illustration not visible in this excerpt

The right-hand side of this expression depends only on readily observable quantities: period and radial velocity amplitude. The value of the mass function presents a lower limit to the mass m2 of the unseen star. This fact is used to prove the existence of massive faint objects, such as stellar black holes. Since the spectrum of only one star is available, Equation (1.4) cannot provide any information about mass ratios.

As a result, the mass function is useful only for statistical studies or if an estimate of the mass of at least one component of the system already exists. Even if both radial velocities are measurable, it is not possible to get exact values for m\ and m2 without knowing the inclination, i. Since stars can be grouped according to their effective temperatures and luminosities, and assuming that there is a relationship between these quantities and mass, then a statistical mass estimation for each class may be found by choosing an appropriately averaged value for sin3 i.

However, since no Doppler shift will be produced if i = 90°, it is more likely that a spectroscopic binary star system will be discovered if i differs significantly from 0°.

Evaluating masses of binaries has shown the existence of a well-defined mass-luminosity relation for the large majority of stars in the sky.

1.6 Eclipsing Binaries

For binaries that have orbital planes oriented approximately along the line of sight of the observer, one star may periodically pass in front of the other, blocking the light of the eclipsed component. Such a system is recognizable by regular variations in the amount of light received at the telescope.

Observations of these light curves not only reveal the presence of two stars but can also provide information about relative effective temperatures and radii of each component from the amount of light decrease and the length of the eclipse.

Eclipsing binaries establish a special class of variable stars. While eruptive, pulsating, rotating, and cataclysmic variables are said to be intrinsic variables caused by different physical mechanisms, eclipsing binaries are extrinsic variables requiring models including both astrophysics and geometry.

The smaller the orbit relative to the sizes of the stars, the greater the likelihood of eclipses. Historically, considerations concerning the likelihood (probability) of eclipses lead to a connection between eclipsing binaries and “close binaries”. In the early days, a so-called close binary was defined as a binary with component radii not small compared to the stars’ separation. This definition was later replaced by a more physical definition related to the evolution of the components.

Eclipsing binary studies often involve the combination of photometric (light curve) and spectroscopic (radial velocity curve) data. Analysis of the light curve yields, in principle, the orbital inclination and eccentricity, relative stellar sizes and shapes, the mass ratio in a few cases, the ratio of surface brightnesses, and brightness distributions of the components among other quantities.

If radial velocities are available, the masses arid semi-major axis may also be determinable. Many other parameters describing the system and component stars may be determined, in principle, if the light curve data have high enough precision and the stars do not differ very much from the assumed model.

The prediction of the information content of particular light curves has been a major topic of concern in binary star studies; the light curve is also the basis of the determination of the orbit of such systems which is much more complicated than it is for spectroscopic binaries because of the greater number of physical parameters.

Orbital elements and other physical parameters

A complete solution involves at least 19 physical parameters from which 8 are a function of the wavelength. In addition to the 7 parameters of the relative orbit, [Abbildung in dieser Leseprobe nicht enthalten] there are 6 parameters for specification of radii, luminosities and limb-darkening (of both components), and other 6 parameters in case of close systems like ellipticity, coefficients of gravitational darkening (magnitude varies with local gravity), or reflection effect.

The full determination of absolute eclipsing binary parameters requires both a light curve and a radial velocity curve for each component.

Eclipsing binary stars are informative objects because they allow photometry and spectroscopy to be combined effectively. Eclipsing, double-lined systems are rare but very valuable. If the data quality is high and the binary configuration is well conditioned, we have a fundamental source of information about sizes, masses, luminosities, and distances or parallaxes of stars. Many other parameters can be determined from precise light curve data if the configuration fulfills certain requirements (e.g., by having complete eclipses). Because such stars may be found over the full range of ages, they also tell how stars evolve - at least in binary stars.

1.6.1 Phenomenological Classification

In general, there are three different types of binary systems:

illustration not visible in this excerpt

The light curves of this type are typically almost flat-topped, suggesting that effects due to the proximity of the components are small, with a large difference between the depths of the two minima. Indeed, in some wavelengths the secondary minimum may be undetectable, and there may be an increase in light near the expected phase of secondary minimum due to the “reflection effect”.

illustration not visible in this excerpt


These light curves, on the other hand, are continuoslv variable, characteristic of tidally distorted components (both components are not any longer spherical, isotropically radiating objects), and with a large difference in depths of minima indicating components of quite different surface brightness.

W Ursae Majoris-type

The light curves of W UMa-type are also continously variable, but with only a small difference in the depths of the minima.

illustration not visible in this excerpt

Figure 1.4: Synthetic W UMa-type light curve.

Analysis of the Light Curve

From the light curve we readily obtain the orbital period, width, shape and depth of the primary and secondary minimum. There are also some effects which disturb the light curve of such systems. The most important ones are listed below:

- Reflection effect

In a binary system the presence of a companion star leads to an increased radiative brightness on the side that faces toward the companion.

The cause is heating by radiation from the companion and that in turn leads to an increase of local surface temperature.

Since heating caused by mutual irradiation is the physical cause, it is somewhat misleading to use the expression mentioned above.

However, in very hot binaries a considerable fraction of the incident radiation is simply scattered by free electrons, so in that case the term reflection effect is reasonably appropriate.

For binaries whose components have similar temperatures and are close to but not actually over-contact, it may be necessary to consider multiple reflection. The first star heats the second one, and the now warmer second star then heats the first star more than otherwise expected because of its own raised temperature.

This process is iterative, leading to higher temperatures on the facing hemispheres.

- Limb-darkening

There is ample evidence from the Sun and other stars that the surface brightness varies from mid-disk to the limb. The so-called center-to-limb variation is the dependence of intensity on angular distance from the surface normal. It arises because temperature increases with depth in stellar atmospheres, and the line-of-sight at the limb does not penetrate into high-temperature regions as it does close to the disk center.

- Gravity brightening

A rotating star differs from a nonrotating star in both shape, local surface gravity acceleration, and surface brightness. It develops oblateness and a pole-to-equator variation in surface brightness.

This phenomenon, for radiative envelopes, has its origin in the temperature gradient in and near the surface.

The atmosphere is assumed to be locally plane-parallel (that implies we need to consider only one geometrical dimension), but irradiated from below by a radiative flux varying across the stellar surface.

The result is that this flux is proportional to the effective surface gravity acceleration at the given point of the surface. Gravity brightening is thus described by a relation between the local effective temperature (or the local bolomctric flux) and the local surface gravity acceleration.

- Variation of period

Not all systems have stable periods of light variation.

- rotation of the line of apsides:

This effect arises because the gravitational attraction of the companion deviates from a 1/r2 force law because of general-relativistic corrections.

We have already remarked that a 1/r2 force law (Newtonian gravity for spherical bodies) yields an ellipse as the most general geometric form for a bound orbit. The long axis of this ellipse joins the point of closest approach and the point of greatest separation, points which are called “apsides”.

The Newtonian theory of binary stars predicts that this long axis, or “line of apsides”, should remain fixed in direction in an inertial frame.

In Einstein’s theory, however, the gravitational force between two mass points does not vary exactly as 1/r2; consequently, the line of apsides of an eccentric orbit should rotate slowly in space.

— decrease in the orbital period:

Newton’s theory applied to the bound orbit of two mass points predicts that the binary period should remain absolutely constant. In general relativity, however, two stars which revolve about one another suffer accelerations and such accelerations generate gravitational radiation.


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Parameter determination of a binary star system
Technical University of Graz
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Double star, Parameter, Determination
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DI Mag Fabian Prilasnig (Author), 2001, Parameter determination of a binary star system, Munich, GRIN Verlag,


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