A Tool to Test the Disentangling Technique

Transcription

1 Synthetic Spectra of OB-Type Binary Stars A Tool to Test the Disentangling Technique Submitted for the degree of M.Sc. Astronomy University of Sussex July 1996 By Robert I. Hynes

2 Abstract Synthetic spectra of six OB-type close binaries are constructed based on Kurucz (1991) model atmospheres and realistic estimates of the signal-to-noise ratios to be expected using the Danish 1.54m telescope at La Silla, Chile. These spectra are used to test the ability of the disentangling technique to determine accurate system parameters. We find that for four of the systems, masses within an accuracy of 1% should be attainable. Extensive use is made of the option to estimate realistic errors from the scatter between multiple data sets differing only in noise. We find that these errors may be significantly larger than those estimated by examining the curvature of the residual surface. A variety of models of rectification errors are tested. Provided such errors are kept within %, the effects on the disentangling analysis are small enough to be ignored. Finally we examine the scale of distortion and proximity effects and test one model for representing them in terms of a varying light ratio. We find that for plausible variations, the effects on the deduced parameters are negligible. The suitability of the disentangling technique for the determination of system parameters is considered. We conclude that the technique does have some advantages over rivals such as cross-correlation, but that more work is needed before it can be applied with confidence. 1

3 Contents Abstract...1 Contents Introduction The importance of binary systems The determination of stellar masses The need to study OB binaries The difficulties of studying OB binaries Techniques for analysing binary spectra Line shifts Fourier cross-correlation Tomography Disentangling Properties of the stars Magnitudes Orbital parameters Spectra The observing site and equipment Observing conditions at La Silla The 1.54m Danish Telescope Calibration with observed count rates Producing the synthetic spectra Radial velocity shifts Selection of phases Producing a composite spectrum Noise Sample results...43

4 6. Disentangling the spectra to determine system parameters The approach to disentangling Optimisation of system parameters Estimating errors Preliminary results Expected errors Exposure time needed The number of spectra needed Optimum phase distribution Rectification of the spectra Local rectification errors Large scale rectification errors Disentangling the orders separately Effects of distortion and proximity Preliminary estimates Variations in luminosity ratio Spectral variations Discussion and conclusions...79 Acknowledgements...8 Appendix I: Response curves for the DFOSC instrumentation...83 Appendix II: Noise-free, synthetic spectra of the components...86 (a) Overall views...86 (b) Detailed spectrum of GL Car A...9 Appendix III: Radial velocity and light curves...94 Appendix IV: Roche lobe filling of the stars References

5 1. Introduction 1.1 The importance of binary systems Binary stars are of importance to astronomers for many reasons. Historically, they have provided evidence that our understanding of Newtonian mechanics and gravity is valid beyond the solar system and today apsidal motion in binary systems is being used to test the validity of the successor to Newton s theory; general relativity. Their mere existence and abundance provides constraints on theories of star formation, as any valid theory must explain how binary, and multiple, stars can form. In their later stages of evolution interacting binaries provide a valuable laboratory in which to study white dwarfs, neutron stars and possibly black holes. An exciting possibility that is just emerging is that it may be possible to use OB binaries as distance indicators to Local Group galaxies (Giménez et al., 1994). Perhaps most significantly for our understanding of stars, binary systems present the only direct way to measure the mass of stars, and the best way to obtain their radii. They can be classified according to several schemes. For our purposes, the most useful is to consider how they reveal their binary nature. The earliest type of binary stars to be positively identified as such were visual binaries; systems where both components are visible and resolved. Periods are typically from decades to centuries, so it is often possible to trace out their orbits around each other. This reveals valuable information such as the inclination and eccentricity of the system. Given absolute, rather than just relative, positions then the ratio of orbital semi-major axes, and hence the mass ratio, can be determined. Periods can usually be measured, or at least estimated if they are too long to observe in full. Unless we know the distance to the system, however, we cannot find the scale of the orbit, and so the absolute masses of the components remain unknown. 4

6 For some binaries, we see the orbit almost exactly edge-on. It is then possible to see eclipses of one star by the other. This shows up as periodic dips in the brightness of the system; the light curve. The timing of these eclipses provides the most accurate way to measure the period of a binary. Furthermore, by study of the relative timing of primary and secondary eclipses, together with their durations, depths and shapes, it is possible to determine some or all of the following information. a) The inclination of the orbital plane to the line of sight. b) e cos ω - a combination of the eccentricity of the orbit, e, and the longitude of periastron, ω. The latter is defined in 5.1; it specifies the direction of the line of centres of the stars at periastron (closest approach). Some close binaries exhibit what is termed apsidal motion - a slow change of ω over time. For example, the eccentric binary GL Car has an apsidal motion period of 5. years (Giménez and Clausen, 1986) corresponding to ω increasing by 14.3 per year. This effect arises because the stars are neither point masses, nor perfectly spherical and hence the potential in which they move is not strictly Keplerian, leading to orbits that do not close. Analysis of apsidal motion can yield e sin ω, allowing us to solve for e and ω individually. c) The radii of the two stars as fractions of their separation. d) The ratio of the temperatures of the stars. Individual temperatures may be deduced from multicolour photometry and/or spectroscopy. e) Limb darkening can be measured for systems showing total or very deep eclipses. The one thing that is absent is an absolute scale; we cannot determine the separation, absolute radii or velocities only from the light curve, and hence we cannot find the masses. 5

7 Most binaries are too close to resolve their components separately and are not inclined edgeon; instead we must seek information in their spectra. We see that for some stars, many spectral lines are in fact doubles, with two components moving back and forward over time. This occurs because the star is an unresolved binary; a spectroscopic binary. The stars are orbiting about each other and so except for the rare case where we are viewing the orbit face on, the stars will have some time-dependant component of motion along the line of sight. This leads to time-dependent Doppler shifts of the spectral lines, giving the observed spectrum. Such systems in which we see two sets of lines are known as double-lined spectroscopic binaries. There are also single-lined spectroscopic binaries in which we only see a single set of lines, but this is moving back and forward just like a single component of a double-lined spectrum. In this case, we deduce that the system is a binary in which one component is too faint to contribute noticeably to the spectrum; it is detected only by its effect on the motion of the companion. For spectroscopic binaries it is possible to measure the orbital velocities from the observed Doppler shifts, and by plotting a radial velocity curve we can, in principle, also obtain the eccentricity and orientation of the orbit. These together with the period of the system allow us to set lower limits on the masses of the components. The only problem remaining is that we do not know the inclination of the orbit; this cannot be determined from velocity measurements alone. Thus, if we see small radial velocities, we do not know whether this is because the masses are really quite small, or whether we are seeing the system face on so that the radial velocity is only a small component of the overall motion. It should be clear that to determine full system parameters is impossible for a system which only fits into one of these groups; we require information of at least two types to complete our knowledge. In this respect, visual binaries are of little value. Because of their large separations, it becomes very unlikely that we will observe an eclipse, and because of their long periods, orbital velocities are small, if not undetectable. Furthermore, large 6

8 uncertainties in distance for most systems limits severely the quality of masses obtained in this way. In contrast, binaries which are both spectroscopic and eclipsing are, if not common, at least relatively abundant. This is aided by the fact that to see eclipses, we must observe at high inclination, enabling us to also see maximum radial velocities. Such binaries do enable us to deduce full system parameters, and most crucially, the masses and radii of the two components Finally, if the information obtained from a binary system is to be of relevance to the study of isolated stars, it is also clear that we need to study binaries whose components behave as if they were isolated. This is only the case for detached systems which have not in the past undergone a mass transfer phase. To conclude, to determine masses and radii that can be used to test models of the evolution of isolated stars we require detached, double-lined, eclipsing binaries. This is the conclusion of Anderson (1991a) who takes as a necessary objective masses and radii of accuracy ~1-%. Data with uncertainty of 5% or greater yields few useful constraints on theoretical models. Conversely, data of better than 1% accuracy are unnecessary as uncertainties in other parameters, such as metal abundances then dominate. Given such data, how can it be used? Among some of the possibilities are the testing of models of main-sequence evolution, examining the effects of varying chemical composition and the testing of opacity tables and models of convection. Accurate study of binary stars is thus crucial to our further understanding of the structure and evolution of stars, which in turn make up most of the visible mass of the universe. 1. The determination of stellar masses Let us now briefly examine how we can determine masses from the observed data. We assume that the photometric observations have enabled us to determine an accurate 7

9 ephemeris for the system; a time corresponding to a known orbital phase (typically primary eclipse), T, together with the period, P. This then allows us to assign a phase to any observation at a known time. From the light curve we also obtain the inclination, i. The eccentricity, e, is either taken to be zero (if the orbit is known to be circular) or is measured using information from the light curve and apsidal motion studies. We further assume that suitable spectroscopic analysis has been done, utilising the photometric data, to yield the velocity semi-amplitudes, K 1 and K. Ways to achieve this are discussed in. Our approach draws on Heintz (1978) and Böhm-Vitense (1989). We take as our starting point, Kepler s 3 rd Law, together with expressions for K for an elliptical orbit, and relationship between the mass and velocity ratios, M + M = 1 ( a a ) + 4π, (1-1) P G 1 3 K i = πa P i sini ( 1 e ) 1 (1-) and M M 1 K1 = (1-3) K We then rearrange (1-) and eliminate a 1, from (1-1), M + M = 1 3 ( K + K ) P( 1 e ) 1 3 πgsin i 3 (1-4) Using relation (1-3) now allows us to eliminate either M 1 or M. We thus obtain, ( + ) ( 1 ) M K K K P e 1 1 = 3 πgsin i 3 (1-5) We have thus obtained an expression for M 1 in terms of observed or fundamental quantities. A similar expression exists for M. If P is measured in days, M in M sun and K 1 and K in kms -1 then, 8

10 M 1 = ( + ) ( 1 ) K K K P e sin i 3 and M = ( + ) ( 1 ) K K K P e sin i 3 (1-6) We now proceed to consider how errors propagate through these formulae; we will need these results in 7.1 to determine the errors in mass expected, given estimated errors in K 1 and K. We first calculate the fractional error on K 1 +K, as follows, 1 σ 1+ σ 1 σ = + K + K K + K K + K 1 1 (1-7) We then deduce the fractional error on M 1 and M, ( M1) σ σ σ 1 4 M = K K + K 1 and ( M) σ σ σ 1 1 M = K1 K1 + K (1-8) This assumes, of course, that the uncertainty in P, e and i can be neglected. P will be well known and both 1-e and sin 3 i are typically close enough to unity to be only weakly dependant on e and i. 1.3 The need to study OB binaries The study of eclipsing, spectroscopic binaries has made considerable progress in the last two decades, with key advances including the introduction of CCD s and the development of cross-correlation analysis. The result is that for most areas of the H-R diagram we now have a sizeable body of accurate parameters, including masses and radii, with which to constrain stellar theories. The current position of research in this area is well reviewed by Anderson (1991a,b). There remain a few areas for which data is much less satisfactory. One such area is the hottest end of the Hertzsprung Russell diagram; O and early B type stars. 9

11 Popper s review (1980) lists twenty systems of type B5 or earlier, including detached, semidetached and probable contact systems. He concludes, however, that for only four of these can we be confident that the masses obtained are accurate even to within 15%. In response to this poor state of affairs, Hilditch and Bell (1987) sought to extend the body of available data. They list thirty-one systems (including all of Popper s twenty), of which sixteen are detached. Typical standard errors on the masses are 10%. Of these only eight, together with the system EM Car are judged by Anderson (1991a) to be of sufficient accuracy (see 1.1) to be included in his review. Extending the sample of OB stars with accurate masses and radii is thus a very worthwhile observational objective and one that can be expected to yield valuable input to the understanding of hot stars. 1.4 The difficulties of studying OB binaries Let us now ask why is it so difficult to obtain accurate masses for early-type binaries? Clearly one important reason is their relative paucity; early-type stars are intrinsically rare, lying at the extreme end of the stellar distribution function. There are difficulties beyond this, however, which are inherent in the spectra of these stars. The O and early B-type stars listed by Hilditch and Bell (1987) range in temperature from ~15,000 K (DI Her B, B5) to ~38,000 K (V38 Cyg A, O7.3). Within this range of temperatures the dominant lines are the hydrogen Balmer lines and the diffuse helium lines at 406 Å and 4471 Å; other lines are present, but weak. These lines, however, are strongly Stark broadened, giving extended wings that often overlap with adjacent lines. The sample spectra in Appendix II illustrate this problem. It arises because the electric fields of passing ions can significantly perturb, and hence broaden, the energy levels of hydrogen and helium 10

12 atoms. The weaker lines, while narrower intrinsically, are broadened by the high rotational velocities commonly encountered in early-type stars, making them so dilute that they are easily lost in noise. He I (4387) He I (4471) Hγ (4340) Fig. 1.1 A section of a synthetic spectrum of the B1 V binary DW Car, near quadrature (i.e. maximum relative radial velocities. As a consequence, line blending is a major problem in these spectra. This takes several forms. (See Anderson, 1991a for discussion; 1991b gives more detail.) Firstly there may be significant blending of two lines within the spectrum of a single star. This effect, whilst undesirable, is not a major problem, as it does not dependent on phase; the spectrum of each component is fixed. More of a problem, however, is blending between the component spectra, either of the same line if its width is larger than the separation of the two components, or of two different lines which happen to be shifted into the same vicinity. This is illustrated amply by Popper (1981) who collects microphotometer tracings of spectra of 6 OB-type binaries most of which show blending. In fig. 1.1 we show a region of a synthetic spectrum of DW Car (B1 V), at a phase near maximum relative velocity. We see that the Hγ Balmer line and the diffuse Helium line at 4471 Å both show serious blending between the 11

13 components. The weaker Helium line at 4387 Å shows minimal blending, but even for this line, the problem is not completely absent. At phases of smaller relative velocities, the problem will clearly be more serious. As Anderson discusses, and as is taken up more fully in, this can lead to substantial systematic errors in parameters determined using heavily blended lines. Only with the appearance of higher quality spectra allowing measurements of the much weaker unblended lines, and with the development of more sophisticated data analysis techniques such as cross-correlation and disentangling are these difficulties beginning to be overcome. 1

14 . Techniques for analysing binary spectra We now consider the techniques that can be used to extract information from the spectra of OB-type binaries..1 Line shifts Historically, radial velocity measurements have been made from photographic films or plates by measuring the displacements of individual spectral lines. This could be done visually or by oscilloscopic scanning. Either way, however, the problem of line blending, as discussed in 1.4, leads to severe underestimates in most measurements made in this way. Anderson (1991a,b) concludes that a reasonable estimate of the systematic error in masses is 30-35% for measurements based on the hydrogen lines and 10% if the diffuse helium lines are used. While these errors can be avoided by using unblended lines (the narrow helium lines, or metallic lines), due to the high rotational velocities involved, these lines tend to be washed out, so the results then suffer more from noise.. Fourier cross-correlation Fourier cross-correlation is a more sophisticated way of analysing digitised spectra. The technique was introduced by Simkin (1974); and further developed by Tonry and Davis (1979). We here outline the important characteristics of the technique qualitatively. We take some estimate of what the unshifted stellar spectrum is, the template, and then shift this to give a best fit to the observed spectrum. This may be done over a large area of the spectrum, or just a single line. The template may use theoretical line profiles, but is more commonly a real spectrum of a single star. The analysis yields a cross-correlation function, the c.c.f.. This measures the correlation between the observed and template spectra as a function of shift in logarithmic wavelength (i.e. component velocity). For a single-lined binary, this will be a single-peaked function. It 13

15 is generally not a simple function, and whilst the centre resembles a Gaussian, the wings show significant structure, including prominent side-lobes (secondary maxima). A doublelined binary will give a double-peaked c.c.f., with the two peaks arising from the velocities of the two components. In this case, the two peaks will not be exactly at the velocities of the components; the double-peaked c.c.f. can be thought of as a blend of two single-peaked c.c.f.s. It is then necessary to assume a form for the individual c.c.f.s, e.g. Gaussian, in order to resolve the overall c.c.f. into components and determine the correct component velocities. Difficulties arise because, as noted, the c.c.f. is not a Gaussian and so we are, to some extent, fitting dissimilar functions. It is found for spectra showing significant blending (Anderson, 1991b) that when only a narrow spectral range is used, interference between the side lobes can lead to systematic errors. Popper and Hill (1991) construct synthetic binary spectra from photographic spectra of isolated OB stars and find that conventional cross-correlation analysis leads to average overestimates of the velocity semi-amplitudes by % for the primary component and 3% for the secondary. Hill and Holmgren (1995) avoid these problems by careful consideration of the form of the c.c.f.; they cross-correlate the spectrum of the isolated star 10 Lac with a rotationally broadened (~150 kms -1 ) version of itself to obtain a realistic individual c.c.f. which can then be used to measure the binary c.c.f. instead of a Gaussian. An extension of the technique is two-dimensional cross-correlation (Zucker and Mazeh, 1994). This is an obvious development, but one which required excessive computation time until Zucker and Mazeh were able to develop an algorithm to greatly reduce this time and render the technique useable. Instead of just defining one template spectrum, we now provide two - one for each star. This has the immediate advantage of providing a much better model for pairs of dissimilar stars. We then calculate the correlation between the observed spectrum and a composite of the two templates for a range of wavelength shifts of both 14

16 templates. Since we now have two wavelength shifts to optimise, the c.c.f. defines a surface and the maximum of the surface tells us the velocities of the two components. The chief advantage of the method is that the c.c.f. is now single-peaked, so we completely bypass problems caused by the two components of a one-dimensional c.c.f. blending into one another. This should avoid the problems of systematic errors and also allows us to measure accurate velocities even for phases when the relative velocity of the stars is so small that the one-dimensional c.c.f. is unresolved. In conclusion, cross-correlation is a powerful and versatile technique, and can be applied to both single-line and double-line spectroscopic binaries. It has also been used to determine galactic redshifts (e.g. Tonry and Davis, 1979). The method can be susceptible to systematic errors, but several recent developments allow these problems to be avoided..3 Tomography Primary spectrum Secondary spectrum Fig..1 Representing the combining of two component spectra in tomographic terms. We now take a brief historical digression to consider another technique used for spectral analysis which does not yield the radial velocities. Instead, tomography is used to reconstruct the component spectra from a set of composite spectra. Bagnuolo and Gies (1991) discuss 15

17 the method and illustrate its application to the O-type binary AO Cas. In general terms, tomography involves seeking to reconstruct an object in depth from a number of views of the object. We can, however, think of separating a binary star spectrum in the same way. The analogy is illustrated in fig..1. The composite spectrum consists of contributions from the two component spectra, shifted according to radial velocity. We can imagine laying the two spectra one behind the other, as shown. Any combination of radial velocities of the stars can then be represented by looking at the two component spectra from a particular angle; the angle depending on their relative radial velocities, and hence their phase. A set of spectra of different phases can thus be thought of as a set of views of the two component spectra from different angles. We need to know the radial velocities (possibly from cross-correlation analysis) in order to map phases to angles. Obtaining the two separate components is then a straightforward tomographic problem. The solution is typically by an iterative least squares method. We assume an initial form for the component spectra, together with a luminosity ratio. This allows us to predict what the observed spectra should look like. Comparing the observed spectra with the predictions then allows us to apply corrections to the component spectra and refine them iteratively, until, hopefully, convergence is reached. The technique does have difficulties. Care has to be taken in implementing the algorithm in a way which ensures that it does in fact converge to a solution rather than oscillating. Difficulties also arise because in regions of the spectrum where there are strong spectral lines, the luminosity ratio is poorly defined and varies rapidly. This leads to disruption of the secondary spectrum in the vicinity of strong lines in the primary (Maxted et al., 1994). It has been suggested by Zucker and Mazeh (1994) that tomography could profitably be combined with two-dimensional cross-correlation. We begin by taking best estimate 16

18 template spectra for the two components and perform a two-dimensional cross-correlation to obtain a first approximation to the radial velocities for each observed spectrum. We then perform a tomographic analysis using these radial velocities together with our templates as first approximations to the component spectra. This allows us to improve our templates, which can then be used for a more accurate cross-correlation analysis. The process is iterated until the results (hopefully) converge..4 Disentangling Disentangling grew out of work on tomography. The technique was first demonstrated by Simon and Sturm (1994) who applied it to the early B-type binary, V453 Cygni. It has subsequently also been used on the O-type systems, DH Cephei (Sturm and Simon, 1994) and Y Cygni (Simon et al., 1994) and the F-type system, DM Vir (Maxted, 1996a). It is clear from this brief list that the technique is still in its early stages, and remains to be tested fully, both on synthetic spectra and by comparison with previous studies of the same stars. While the application of the technique is a complex problem, the essential ideas are very straightforward, and similar in some respects to tomography. We can think of each observed spectrum as being a linear combination of the spectra of the two components, shifted according to their radial velocities. This can be represented in matrix form. We form the column vector, x, by concatenating the two component spectra, x A and x B, and the column vector, b, by concatenation of the observed spectra, b 1, b,, b n. Then we represent the transformation that maps x onto b by a matrix M. b1 NA, NB, b NA, NB, = b N N n 1 1 An, Bn, x x A B (-1) Then each submatrix, N A,i determines how the elements of the spectrum of star A contribute to the observed spectrum b i and similarly for N B,i. It will be clear that the matrix M will be 17

19 very large. For example if we have twenty 3000 pixel spectra, and represent the component spectra with the same resolution, we would expect a 60,000 column, 6000 row matrix, albeit a very sparse one, as each element of the observed spectrum will only receive a contribution from one part of each component spectrum. Nonetheless, there does exist, in principle, an inverse matrix which will map the observed spectra into two, separated component spectra. Disentangling determines this matrix. The original motivation for using the technique was that the resulting component spectra are ideally suited to quantitative analysis and calibration of stellar models in a way which is not possible using the unseparated, blended spectra. In addition, however, since the principal factors determining the elements of M are the orbital parameters of the binary, a best fit solution for the inverse matrix will yield these parameters rather precisely. In particular, as found by Simon and Sturm (1994) and confirmed by our results discussed later, there appears to be no need for any systematic corrections of the type discussed by Popper and Hill (1991). Disentangling thus emerges as a strong competitor to cross-correlation analysis, with clear advantages over it for heavily blended spectra for which cross-correlation is susceptible to systematic errors. In this work, in line with Popper and Hill s suggestion (1991), we test the disentangling technique by constructing and then analysing synthetic binary spectra. 18

20 3. Properties of the stars The synthetic spectra were produced to reproduce observations made by Dr. P.F.L. Maxted at La Silla, Chile. Six stars were; all are close, early B-type eclipsing binaries in the southern sky. They are listed in table 3.1. HD number Right ascension (000.0) Declination (000.0) DW Carinae h 41 m 15 s GL Carinae h 14 m 14. s LZ Centauri h 50 m 3.46 s V346 Centauri h 4 m 50.4 s AC Velantis h 46 m 18.4 s NSV h 37 m s Table 3.1: The stars to be studied 3.1 Magnitudes To estimate signal-to-noise ratios, we would like a homogeneous set of UBV magnitudes for these stars. Such data is not available, so it was necessary to use a variety of sources, together with some extrapolation Observed data The General Catalogue of Variable Stars (Kholopov et al., 1985) and Smithsonian Astrophysical Observatory Star Catalogue (Whipple et al., 1966) were used as a starting point. Where more precise data were available, UBV photometry especially, these were preferred, although they are not guaranteed to be more accurate. All the data that was available, together with the adopted spectral type and luminosity class, are listed in table 3.. The latter were taken, in the absence of other information, from the Simbad database operated at CDS, Strasbourg. They are generally uncertain to within at least ±1 classification Colour correction For those stars for which UBV photometry was available (GL Car, V346 Cen, NSV 5783), (B-V) and (U-B) values as given were used to deduce the U and B magnitudes listed. For the 19

21 other stars, only a V or even photographic magnitude (approximately equivalent to the B band) was available. For these tabulated intrinsic (B-V) 0 and (U-B) 0 colour indices (Deutschman et al., 1976) for the estimated spectral type and luminosity class were used. These were then corrected for interstellar absorption as follows. Measured magnitudes DW Car m pg = 9.60 Measured colours (UBV) GL Car B = 9.73 V = 9.81 (B-V)= (U-B)= LZ Cen V = 8.10 V = 8.09 V346 Cen B = 8.48 V = 8.55 (B-V)= (U-B)= AC Vel V = 8.60 V = 8.88 NSV 5783 m pg = 8.50 m pv = 8.70 V = 8.67 (B-V)= -0.0 (U-B)= Table 3.: Observed properties of stars Spectral type & luminosity class B1 V B3 V B1 III B1 V B3 IV B5 V Source Kholopov et al. (1985) Maxted (1995) Kholopov et al. (1985) Nicolet (1978) Simbad (1995) Kolopov et al. (1985) Vaz et al. (1995) Kholopov et al. (1985) Nicolet (1978) Giménez et al. (1986) Kholopov et al. (1985) Wolf and Kern (1983) Simbad (1995) Whipple et al. (1966) Deutschmann et al. (1976) Simbad (1995) Firstly a comparison was made between the observed and predicted magnitudes for those systems which have been studied in the Johnson system. The tacit assumption was made that since the systems being considered are all of comparable apparent magnitude and luminosity, they will experience comparable reddening. Whilst this is may be a poor assumption, the effects of reddening are small, and so its consequences are minimal. It was found that for the three systems that have been studied, on average, (B-V) ~ 0.7, (U-B) ~ This is in poor agreement with the expected reddening relation: ( U B) 07. (3-1) ( B V ) 0

22 It was decided, however, to follow the empirical data that was known, given insufficient data for a more thorough treatment. The resulting above atmosphere magnitudes are listed in table 3.3. Above atmosphere Below atmosphere U B V U B V DW Car GL Car LZ Cen V346 Cen AC Vel NSV Table 3.3: UBV magnitudes of stars above and below the atmosphere Atmospheric extinction The observations were carried out at the European Southern Observatory, described in 4.1. The atmospheric extinctions listed there for a zenith angle of 30 were applied to obtain the final, below atmosphere magnitudes also listed in table 3.3. These are the magnitudes that were used in determining the expected signal-to-noise ratios in Uncertainties in the magnitudes These UBV magnitudes have had to be patched together from a variety of measurements of variable accuracy, with corrections based on uncertain assumptions. An error of 0. in a magnitude will lead to a corresponding error of 0% in the predicted number of photon counts and hence 10% in the signal-to-noise ratio; see It is likely that other uncertainties, such as those in instrumental properties will be as large, or larger, than this. 3. Orbital parameters In order to combine the spectra it is necessary to be able to calculate a radial velocity curve for each system. To do this, several orbital parameters are needed; these are the velocity semi-amplitudes, K 1 and K, the orbital eccentricity, e and the longitude of periastron, ω. Most of these are known and straightforward. The longitude of periastron required 1

23 modification for GL Car, as this system has a short apsidal motion period of 5. years (Giménez and Clausen, 1986). Thus their value of ω 0 =66.1 ± 0.3 (HJD ± ) had to be updated by nearly a full apsidal motion period. Note that to combine the spectra, we do not need to use the period as all spectra are specified in terms of phase. Also, systemic velocity, γ, is not necessary as the disentangling procedure is completely independent of an overall shift in velocity. The parameters used are listed in table 3.4. It was decided that for convenience, phases would not be specified relative to the primary minimum of the light curve, the convention for eclipsing binary studies. Instead, they would be measured from periastron, a much more natural system for the calculation of radial velocities. For systems with circular orbits, ω was taken to be zero, and so the reference point becomes the ascending node. The difference, φ, to be subtracted from to our phases to convert them to the conventional notation is also tabulated in table 3.4. K 1 K e ω( ) φ (kms -1 ) (kms -1 ) DW Car GL Car LZ Cen V346 Cen AC Vel NSV Table 3.4: Orbital parameters 3.3 Spectra Synthetic spectra for the individual components were generated by a two stage process. Firstly, Kurucz model atmosphere spectra (Kurucz, 1991) based on a grid of values of log g and T eff, listed in table 3.5, were provided by Dr. C.S. Jeffery of the University of St. Andrews. These spectra cover the range Å. They are based on a normalised, flat continuum and include lines from HI, HeI/II, CII/III, Mg II, Ca II and Si II/III/IV; most prominent are the Balmer lines of hydrogen. Then to obtain spectra appropriate for the particular stars under study, these were rotationally broadened by Dr. P.F.L. Maxted, with

24 interpolation between the log g and T eff values as necessary. The values of log g, T eff and v rot sin i assumed are given in table 3.6, together with the luminosity ratio to be used in combining the spectra. Unless otherwise noted, data is taken from Clausen (1995). 3.5 / K 4.0 / K 4.5 / K 3.5 / 4000 K 4.0 / 4000 K 4.5 / 4000 K 3.5 / K 4.0 / K 4.5 / K 4.0 / K 4.5 / K Table 3.5: Available spectra (log g / T eff ) Log g T eff (K) v rot sin i L /L 1 (kms -1 ) DW Carinae A (165) DW Carinae B (165) GL Carinae A (100) GL Carinae B (100) LZ Centauri A LZ Centauri B V346 Centauri A V346 Centauri B AC Velantis A (85) AC Velantis B (85) NSV 5783 A (00) NSV 5783 B (00) Table 3.6: Relevant physical properties of the stars The rotational velocities listed above require a little comment; spectroscopic measurement of rotational velocity has only been carried out for two of the systems - LZ Cen (Vaz et al., 1995) and V346 Cen (Giménez et al., 1986). For the other systems, it was necessary to estimate likely velocities. For close binaries of this type, an estimate can be obtained by using the synchronous rotational velocity; in the case of the systems with elliptical orbits, an average velocity was used. These assumptions lead to an underestimate of the actual rotational velocity for two reasons: a) Full synchronisation may not have been completely reached. For example, in the case of LZ Cen, the closest binary in the sample, Vaz et al. (1995) find that the stars may rotate marginally faster than synchronously, whilst Giménez et al. (1986) report that both components of V346 Cen rotate much faster than synchronously, consistent with an orbit that has not yet been circularised. 3

25 b) For elliptical systems, synchronisation appears to take place at the periastron angular velocity, which is faster than the average angular velocity. An underestimate of the rotational velocity can be expected to make the lines easier to resolve than will in fact be the case, and so will lead also to an underestimate of the errors to be expected. So this analysis is useful in identifying whether a system is worth further study given the resources available; it does not guarantee that this study will be successful. The above arguments clearly do not apply in the case of NSV In this case, the calculated synchronous velocities would be less than 0 kms -1. Since this system has a longer period and tidal effects are consequently much weaker than in the other cases, we cannot expect significant synchronisation to have occurred. The value of 00 kms -1 assumed is of the order to be expected for non-synchronised early B-type stars. As will emerge more fully later, the value of the rotational velocity of NSV 5783 is crucial in determining whether any worthwhile results can be obtained. Since a value of 00 kms -1 is larger than either of the orbital velocity semi-amplitudes we can expect serious problems in resolving the two components. An example of a typical synthetic spectrum is shown in fig. 3.1; this is the spectrum of the primary component of GL Car. Note that the region below ~3800 Å is not expected to be realistic, as the Balmer series has only been calculated as far as Hι. This simplification, together with the Balmer discontinuity itself will seriously distort the spectrum in this region. The full set of component spectra is included in Appendix II(a). In Appendix II(b) the spectrum of GL Car A is shown enlarged. Its major lines have been identified with the aid of Walborn and Fitzpatrick (1990) and Moore (197). 4

26 Fig. 3.1 The synthetic spectrum of the primary component of GL Car. 5

27 4. The observing site and equipment The observations were carried out at La Silla, the European Southern Observatory (ESO) site in Chile, from the 6 th to 13 th March The Danish 1.54m telescope was used, coupled with the Danish Faint Object Spectrograph and Camera (DFOSC). 4.1 Observing conditions at La Silla The relevant details of the site are summarised in table 4.1. They are taken from the ESO User s Manual (Schwarz and Melnick, 1993). U (3600 Å) B (4400Å) V (5500 Å) Extinction per air mass (mags) Extinction, 30 from zenith Sky brightness, no Moon (mags. arcsec - ) Sky brightness, full Moon Median seeing (arcsec) 0.85 Table 4.1 Observing conditions at the ESO site. The sky brightness at full Moon was determined using the guidelines within the SIGNAL program (Benn 199). These suggest a maximum lunar correction of 4 m arcsec -. These are appropriate worst case values to use for this observing run as full Moon fell on 5 th March Given the seeing of ~0.85, a slit width of 0.8 was chosen for estimating the amount of light to enter the slit. This was done using the LIGHT_IN_SLIT program (Benn 1996), with the slit assumed to be oriented vertically to eliminate the effects of differential refraction. This yielded an estimate of 66% of the light of the star being collected. Whilst a wider slit would give a stronger signal, it would also reduce the spectral resolution. The value chosen is a sensible compromise. 6

28 4. The 1.54m Danish Telescope Details of the telescope and the instrumentation used are given in table 4.. U (3600 Å) B (4400Å) V (5500 Å) 1.54m telescope Useful area (m ) 1.87 DFOSC Transmissions:- General optics Grism # Grism # Grisms #9 and #10 (combined) Dispersions (Å mm -1 ):- Grism #6 110 Grism #7 110 Grism #9 6 Grism # Ford-Loral CCD Efficiency Readout noise (electrons per pixel) 7. Pixel size, physical (µm) 15 Pixel size, angular (arcsec) 0.4 Table 4. Properties of the telescope and instrumentation. Information on the telescope is taken from Schwarz and Melnick (1993). There are two mirrors between source and spectrograph; in line with the SIGNAL program defaults, a reflective efficiency of 85% per mirror was assumed. Data on the Danish Faint Object Spectrograph and Camera (DFOSC) is drawn from Anderson (1996). Although it has other capabilities, e.g. direct imaging, for our purposes it is a grism based echelle spectrograph. Eleven grisms are available; two were chosen to give peak transmission in the blue range of the spectrum that is of chief interest for the stars under study. These were the very high resolution echelle grism, #9, together with the course resolution cross dispersing grism, #10. In addition, we require information on grisms #6 and #7 for calibration purposes (see 4.3). 7

29 Details of the Ford-Loral CCD can be found in Anderson (1996). The readout noise is taken from Storm (1996). When coupled with echelle grism #9, the wavelength scale is 0.39 Å per pixel. Transmissions and efficiencies are represented in table 4. by values at the centres of the Johnson U. B and V bands. Full response curves over the spectral range of interest can be found in Appendix I. Note that the peak transmission of the echelle orders has been used in the table, so there will be regions, corresponding to the ends of the orders, where the transmission is significantly less than this. 4.3 Calibration with observed count rates The Ford-Loral CCD has been tested with grisms #6 and #7 on the star LTT 318 (Anderson, 1996). This has V = , (B-V) = +0.0, (U-B) = (Hamuy et al., 199), giving us below atmosphere magnitudes, assuming a zenith angle of 30, of U = 1.06, B = 1.33, V = 11.99, using the method discussed in 3.. The measured count rates, in e - s -1 Å -1 are shown in table 4.3. These measurements were made using a 10 slit which should ensure that all light from the star is used. These results will be compared with predicted count rates in 5.4. to estimate and correct for a systematic error in our predictions. Wavelength (Å) Grism # Grism # Table 4.3 Measured count rates, in e - s -1 Å -1 for the star LTT 318 using DFOSC with the Ford-Loral CCD. 8

30 5. Producing the synthetic spectra The composite spectra were produced (from the component spectra) using a FORTRAN program, SYNTHESIS, written for this project. The operation of the program can be divided into several stages, as outlined in fig Begin Read in system parameters Read in primary spectrum Read in secondary spectrum Determine spectral range Calculate radial velocity Add spectra Add noise to composite spectrum Write composite spectrum Repeat for each phase End Fig. 5.1 Overall flow of the SYNTHESIS program 9

31 All spectra are stored as one-dimensional FITS files (Wells et al., 1981), binned logarithmically. The wavelength calibration is then specified by two FITS keywords: W0 specifies the base logarithmic wavelength (log to base 10 of the lowest wavelength in Ångstroms) and WPC specifies the increment in logarithmic wavelength per pixel. In handling the files, extensive use was made of the FITSIO library (Pence, 1995) of FORTRAN subroutines for interfacing with FITS files. The implementation of the disentangling algorithm requires all of the spectra to have the same spectral range and scaling. We thus determine a common spectral range that will always lie within both component spectra for all phases. This is then used for all the composite spectra. An additional constraint is imposed that the spectra should not extend too near to the Balmer limit. This is achieved with a cut off at 3800 Å 1. There is also an alternate version of the program which allows the spectral range to be specified manually. This is useful for concentrating on a narrower window of the spectrum. The resolution of the composite spectra is based on that expected with the DFOSC system (0.39 Å; see 4.), not that of the higher resolution component spectra. 5.1 Radial velocity shifts To produce the composite spectra, it is necessary to know the wavelength shifts of the two component spectra as a function of orbital phase, φ. This is readily determined given the radial velocity, v R, of each star as a function of phase., i.e. the radial velocity curve. An elliptical orbit is shown in fig. 5.. We refer phases to the periastron point, P; this is separated from the ascending node, L, by angle, ω, the longitude of periastron. For a circular orbit, periastron is undefined so we instead take the passage of the primary through the 1 In view of subsequent work on edge effects ( 8.3.3), this is probably a bad choice as it lies well on one of the Balmer lines; a better choice would have been a point between two of the lines. 30

32 ascending node as our point of reference. The angle between the star B and our reference point is ν, the true anomaly. For a circular orbit, this will simply be πφ, but for an elliptical orbit, the situation is more complicated. The method used to calculate ν, and hence v R, adapted from Heintz (1978) is outlined below. Fig. 5. Illustration of the key features of an elliptical binary orbit. We show the relative orbit of star B about star A. The projection of the observer s line of sight onto the orbital plane is also shown. We first compute the mean anomaly, M, from the orbital phase, φ (relative to periastron), M = πφ (5-1) Next, we must obtain the eccentric anomaly, E, by solving Kepler s equation, E esin E= M (5-) This is solved iteratively, using the first approximation, 1 E0 = M+ esin M+ e sin M (5-3) 31

33 and successive approximations, E E E e sin ' = E M 1 ecose (5-4) From the eccentric anomaly we proceed to calculate the true anomaly, ν, ν= tan e E tan 1 e where 0 ν<π (5-5) The radial velocity is then given by v [ ( ) R = γ + K ecosω+ cos ν+ ω ] (5-6) where K is taken to be negative for the secondary star. For our case, we ignore γ as this does not affect the disentangling procedure. In fig. 5.3, we show a sample radial velocity curve generated in this way for the eccentric binary V346 Cen. A full set of curves is included in Appendix III. Fig. 5.3 Radial velocity curve for V346 Cen. The zero of phase is taken to be the periastron point. 3

34 5. Selection of phases 5..1 Eclipses Before discussing how these radial velocities are used to produce composite spectra, we take a brief digression to consider which phases we should reproduce. Initial experiments with a regular phase distribution (e.g. 0.00, 0.05, 0.10, 0.15, etc.) were felt to be unrealistic, as the sampling of phases obtained influences the quality of the results of disentangling the spectra (see 7.4) and real data would not be this regular. It was also desirable to avoid phases of eclipse, as these are not suitable for use in disentangling. The final approach adopted, until the observed phases were known, was to determine the phase ranges within eclipse and then choose random phases, uniformly distributed over the remaining range. This range was determined by producing simple model light curves, which can be expected to identify correctly the onset and end of eclipse (ignoring distortions of the stars), but will not accurately reproduce the eclipse profile, or any ellipsoidal variation outside of eclipse (see, for example, the light curve of LZ Cen obtained by Vaz et al., 1995). They are adequate for our purposes. The method is an extension of that used in 5.1 to determine radial velocity curves, and as far as (5-9) follows Duffett-Smith (1981) We have obtained the eccentric anomaly (5-3,4) and the true anomaly, ν, (5-5). We proceed to calculate the radius vector, r, in the orbital plane as a fraction of the semi-major axis to be r = 1 ecos E (5-7) The position angle, θ, relative to the ascending node, is θ= tan sin( ν+ ω) cos cos( ν+ ω) 1 i (5-8) The apparent separation, ρ, of the two stars, again as a fraction of the semi-major axis, is then ρ= rcos( ν+ ω) cosθ (5-9) 33

35 We thus know the separation of the two stars as a fraction of the semi-major axis. The fractional radii of the stars are given is table 5.1 (Clausen, 1995). Fig. 5.4 Illustration of the geometry of the eclipse of star by star 1. We have drawn the line of centres horizontally for convenience, as it is only the apparent separation of the stars which matters, not the position angle. Primary radius Secondary radius DW Car GL Car LZ Cen V346 Cen AC Vel NSV Table 5.1 Fractional radii of the stars in the sample. We now calculate the fractional area eclipsed. The geometry is illustrated in fig Star is eclipsed by star 1. Given r 1, r and ρ, we use the cosine rule to determine angles θ 1 and θ. The area of star that is eclipsed, A, is clearly the sum of segments AP 1 B and AP B. Segment AP 1 B is given by AP B = 1 1 r 1 1 r1 1 r1 1 θ cosθ sin θ (5-10) Segment AP B is given similarly. We thus find 1 A = r r θ1 sinθ1 θ sin θ (5-11) 34