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Spectral Modeling

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The Synthetic Stellar Spectral Codes

We create model spectra for high-gravity stellar atmospheres using the codes TLUSTY and SYNSPEC (http://nova.astro.umd.edu ; Hubeny, I. 1988, Comput. Phys. Commun., 52, 103; Hubeny, I.,& Lanz, T. 1995, ApJ, 439, 875).

Atmospheric structure is computed (using TLUSTY) assuming a H-He LTE atmosphere; the other species are added in the spectrum synthesis stage using SYNSPEC. For hot models (say T>40,000K) we switch the approximate NLTE treatment option in SYNSPEC (this allows us to consider and approximate NLTE treatment even for LTE models generated by TLUSTY). We generate photospheric models with effective temperatures ranging from 12,000K to 75,000K in increments of about 10 percent (e.g. 1,000K for T~15,000K and 5,000K for T~70,000K). As the WD mass in these systems is not always known, we choose values of Log(g) ranging between 7.5 and 9.5. We also vary the stellar rotational velocity Vrot sin(i) from a few km/s to 1000km/s. In order to fit the absorption features of the spectrum, we also vary the chemical abundances of C, N, S and Si. For any WD mass, there is a corresponding radius, or equivalently, one single value of Log(g). See the mass radius relation from Hamada, T., & Salpeter, E.E. 1961, APJ, 134, 683; or see Wood, M.A. 1995, in White Dwarfs, Proceedings of the 9th European Workshop on White Dwarfs, Lecture Notes in Physics, Vol.442, eds. Detlev Koester & Klaus Werner, Springer-Verlag, Berlin Heidelberg New York, p.41 Panei, J.A., Althaus, L.G., & Benvenuto, O.G. 2000, A&A, 353, 970; for different composition and non-zero temperature WDs.

The same suite of code is also used to generate spectra of accretion disks (Wade, R.A., & Hubeny, I. 1998, APJ, 509, 350) based on the standard model of Shakura, N.I., & Sunyaev, R.A., 1973, Astro.&Astroph. 24, 337. In the present work we use disk spectra from the grid of spectra generated by Wade, R.A., & Hubeny, I. 1998, APJ, 509, 350 as well as disk spectra that we generate. A detailed description of the procedure to generate such disk spectra is given in Wade, R.A., & Hubeny, I. 1998, APJ, 509, 350.

Modeling the ISM Hydrogen Absorption Lines

For most systems showing ISM atomic and molecular hydrogen absorption lines, we identify these lines in the figures to avoid confusing them with the WD lines. For some systems, however, ISM lines are deep and broad and we decided to model them, especially since some of the WD lines (such as S IV (1062.6A & 1073A) are located at almost the same wavelengths.

We model the ISM hydrogen absorption lines to assess the atomic and molecular column densities. This enables us to differentiate between the WD lines and the ISM lines, and helps improve the WD spectral fit. The ISM spectra models are generated using a program developed by P.E. Barrett. This program uses a custom spectral fitting package to estimate the temperature and density of the interstellar absorption lines of atomic and molecular hydrogen. The ISM model assumes that the temperature, bulk velocity, and turbulent velocity of the medium are the same for all atomic and molecular species, whereas the densities of atomic and molecular hydrogen, and the ratios of deuterium to hydrogen and metals (including helium) to hydrogen can be adjusted independently. The model uses atomic data of Morton, D.C. 2000, ApJS, 130, 403 and Morton, D.C. 2003, ApJS, 149, 205, and molecular data of Abgrall, H., Roueff, E., & Drira, I. 2000, AAS, 141, 297. The optical depth calculations of molecular hydrogen have been checked against those of Morton, D.C. 2003, ApJS, 149, 205. The ratios of metals to hydrogen and deuterium to hydrogen are fixed at 0 and 2.E-05, respectively, because of the low signal-to-noise ratio data. The wings of the atomic lines are used to estimate the density of atomic hydrogen and the depth of the unsaturated molecular lines for molecular hydrogen. The temperature and turbulent velocity of the medium are primarily determined from the lines of molecular hydrogen when the ISM temperatures are < 250K.

The ISM absorption features are best modeled and displayed when the theoretical ISM model (transmission values) is combined with a synthetic spectrum for the object (namely a WD synthetic spectrum).

Synthetic Spectral Model Fitting

Before carrying out a synthetic spectral fit of the spectra, we masked portions of the spectra with strong emission lines, strong ISM molecular absorption lines, detector noise and air glow. These regions of the spectra are somewhat different for each object and are not included in the fitting. The regions excluded from the fit are in blue in the figures. The excluded ISM quasi-molecular absorption lines are marked with vertical labels in the figures.

After having generated grids of models for each target, we use FIT a chi square minimization routine (e.g. Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P., Numerical Recipes in Fortran 77, The Art of Scientific Computing, Second Edition, 1992, Cambridge University Press), to compute the reduced chi square (chi square per number of degrees of freedom) and scale factor values for each model fit. While we use a chi square minimization technique, we do not blindly select the least chi square models, but we examine the models that best fit some of the features such as absorption lines and, when possible, the slope of the wings of the broad Lyman absorption features. When possible, we also select the models that are in agreement with the distance of the system (or its best estimate).

The flux level at 1000A (between Lyman delta and Lyman gamma) is close to zero for temperatures below 18,000K; at 30,000K it is about 50 percent of the continuum level at 1100A and it reaches 100 percent for T>45,000K. At higher temperature (T>50,000K) the spectrum becomes pretty flat and there is not much difference in the shape of the spectrum between (say) a 50,000K and a 80,000K model. When fitting the shape of the spectrum in such a manner, an accuracy of about 500-1,000K is obtained, due to the S/N. In theory, a fine tuning of the temperature (say to an accuracy of about 50K) can be carried out by fitting the flux levels such that the distance to the system (if known) is matched. However, the fitting to the distance depends strongly on the radius (and therefore the mass) of the WD. In most of the systems the mass of the WD and the distance are unknown and therefore we are unable to assess the temperature accurately. Furthermore, since the Lyman beta profile depends on both the temperature and gravity of the WD, the accuracy of the solution is further decreased as there is a degeneracy in the solution, namely the solution spreads over a region of the Log(g) and T parameter space. And last, reddening values are rarely known, therefore increasing even more the inaccuracy of assessing the temperature by scaling the synthetic flux to the observed flux. As a consequence, for each target we present more than one model fit.

The WD rotation (Vrot sin(i)) rate is determined by fitting the WD model to the spectrum while paying careful attention to the line profiles. We did not carry out separate fits to individual lines but rather tried to fit the lines and continuum in the same fit while paying careful attention to the absorption lines.

For each spectrum, when possible, we try to fit a single WD model, a single disk model, and a composite WD+disk model, assuming different reddening values. For the systems for which the WD mass is unknown, the distance is estimated using the maximum magnitude/period relation Warner, B. 1995, Cataclysmic Variable Stars (Cambridge: Cambridge Univ. Press); Harrison, T.E., Johnson, J.J., McArthur, B.E., Benedict, G.F., Szkody, P., Howell, S.B., Gelino, D.M. 2004, AJ, 127 460; or (when the period is unknown) the method described by Knigge, C. 2006, MNRAS, 373, 484 and Knigge, C. 2007, MNRAS, 382, 1982.

For the single WD model, we vary the temperature while first keeping the WD mass constant, starting at about 0.4Msun. Once the lowest chi square has been found for a given mass, we vary the projected rotational velocity, and possibly also the abundances, to further lower the chi square and obtain a best fit. Once the best fit has been found for that mass, we assume a slighly larger mass and again vary the temperature until the lowest chi square is found. We follow this procedure iteratively until we reach a mass of about 1.2Msun. The next step is to find, from all these lowest chi square models, the one which agrees best with the distance estimate, or the one which has the lowest chi square of all (if the constraint on the distance cannot be used). For the single disk model, we carry out a similar procedure by varying the mass accretion rate and inclination assuming discrete values of the WD mass, and then chose the least chi square model agreeing best with the distance. And last, we use the same procedure for the WD+disk composite modelng to find the best fit model.