Title : will be set by the publisher Editors : will be set by the publisher EAS Publications Series, Vol.?, 26 RADIATIVE TRANSFER IN AXIAL SYMMETRY Daniela Korčáková and Jiří Kubát Abstract. We present a method for solving the radiative transfer equation in axial symmetry both in static case and in moving media. A usage of this method is shown here for stellar atmospheres, stellar winds, and accretion discs. The effects of limb darkening and stellar rotation are discussed, too. Introduction The astrophysical radiative transfer problem is a highly nonlinear task, which has not been solved in its complexity until now. The analytical solution of this problem is impossible, but also the numerical calculations are very time consuming. Due to this reason we have to use simplifications, which are based on physical conditions of a given object. One of the key parts of this problem is the solution of the radiative transfer equation (RTE). For optically thick regions we can use the diffusion approximation method (Kneer & Heasley, 979) for the solution of the radiative transfer equation. In an optically thin region it is sometimes possible only to sum the radiation from various regions neglecting absorption, or we can use the Monte Carlo method (Boissé, 99), which is very fast in these situations. An additional problem is the velocity field. With the exception of supernovae and accretion discs around black holes we can ignore relativistic effects on the radiation field. But even if the problem is nonrelativistic, in most astrophysical situations its solution is not trivial, since the opacity and emissivity can change very much along a ray due to the Doppler shift. If the velocity gradient is sufficiently high, we can use the Sobolev approximation (Sobolev, 947). If the Sobolev condition is not fulfilled, we have to calculate with the additional term on the left hand side of the radiative transfer equation. For the solution of this non-sobolev problem we can use either the observer frame or the comoving frame, the latter being more general. Astronomický ústav, Akademie věd České republiky, CZ-25 65 Ondřejov, Czech Republic c EDP Sciences 26 DOI: (will be inserted later)
2 Title : will be set by the publisher We want to describe the main features of the stellar wind and stellar rotation. This allows us to use the approximation of axial symmetry, which is appropriate also for the disc geometry, as is shown later. The more detailed description of this method is published in Korčáková & Kubát (25a). Here we show some new applications of this method. 2 Description of the method The basic simplifying idea of our method is in the solution of the radiative transfer equation not in the whole star, but separately in planes intersecting the star. In every plane, the radiative transfer equation is solved using a combination of short and long characteristic method. The main advantage of the combination of characteristics is in better description of the radiation field than the short characteristic method and does not consume as much computing time as the long characteristic method. We are not going into details in this paper, since an extensive description as well as the comparison of this method with another independent models is published in Korčáková & Kubát (25a). We section a star into longitudinal planes (see Fig., left panel). The radiative θ radial grid lines rotation axis r φ concentric circles R Fig.. left panel: A stellar sectioning and the coordinate system of longitudinal planes. right panel: Obtaining the whole radiation field by rotating the longitudinal planes (which is possible due to the axial symmetry). transfer equation is solved in every plane independently. The whole radiation field is obtained by rotation of these planes around the rotation axis of the star (see Fig., right panel). In every longitudinal plane we introduce the polar coordinate system. The solution in the given plane must start from the upper (i.e. outer)
Korčáková & Kubát: Radiative Transfer in Axial Symmetry 3 boundary. In every grid point we choose several (up to 9) rays per quadrant (see Fig. 2, left panel). Along these rays we solve the radiative transfer equation τ(ab) I (B) = I (A) e τ (AB) + S(t)e [ ( τ(ab) t)] dt. (2.) The quantities used have their usual meaning. The interval AB is a section of the ray within each cell, in which the source function S is assumed to be a linear function of the optical depth. radial grid line α grid circle zone Fig. 2. The schema for the solution of the radiative transfer equation in longitudinal planes. The rays (characteristics) are denoted using the dashed line. left panel: downward (inward) solution, right panel: upward (outward) solution. If the intensity in the central grid circle is known, we must check whether the longitudinal plane intersects the lower (inner) boundary. If not, we must solve the radiative transfer also in the central circle of this plane. In this way we obtain complete downward radiation intensity. The upward solution is very similar to the downward one. In every grid circle we choose the rays under the same angles as before, which is depicted in the Fig. 2 (right panel). We can see from this figure why it is necessary to start from the upper boundary. Since some rays can start and end on the same circle, we need to know the intensity at the beginning of the characteristic (which is the downward intensity) to solve the equation 2. upwards. If velocity is present, we can assume that the velocity vector is constant in the given cell and that it changes only at the cell boundaries. In this assumption we can use the Lorentz invariance of the radiative transfer equation to solve the static equation in given cells. The transformation of frequency is performed only at the cell boundaries (it is possible to neglect the transformation of intensity in case of nonrelativistic velocities).
4 Title : will be set by the publisher 3 Selected results In this section we show some calculations based on our method, which were presented at this workshop. Since most of these results have already been published, we show here the main ideas only and refer to corresponding papers for details. 3. limb darkening Limb darkening can be used as one of the most suitable tests of our stellar atmosphere models, since it depends on temperature and density distribution in stellar atmosphere. Its exact knowledge affects the quality of information that can be obtained from interferometric observations, as well as determination of rotation velocities in stars (see next section 3.2). Limb darkening is frequency dependent, which is shown in the Fig. 3. Left limb darkening specific intensity.45.4..8 limb darkening in line.35.3.25.2.2 4.567e+4 4.566e+4 4.565e+4.4.6 x [R ].8 4.564e+4 relative intensity.6.4.2.998.2.4.6.8 x [R ] specific intensity 4e-5 5e-5 4e-5 continuum line center 3e-5 2e-5 e-5.5.5 x [r/r ] 2 2.5 3 4.567e4 4.566e4 4.565e4 4.564e4 specific intensity 3e-5 2e-5 e-5.5.5 2 2.5 3 x [r/r ] Fig. 3. Limb darkening for a thin stellar atmosphere (top) and for an extended stellar atmosphere (bottom). The x-axis is the distance from the center of the star in units of stellar radius (x = for the stellar radius). left panels: 3D plot of limb darkening. right panels: Intensity variation for the central line frequency. panels show the 3D plots of limb darkening. Right panels emphasize in detail an intensity dependence across the stellar disk for the central frequency of a line,
Korčáková & Kubát: Radiative Transfer in Axial Symmetry 5 where limb brightening instead of limb darkening is present. This is similar to the effect of flash spectra in solar chromosphere. This effect is more important for extended stellar atmospheres. The limb darkening (brightening) is important for the evaluation of rotation velocities of these stars, since the easiest method, convolution, can produce a huge error. The results in Fig. 3 are calculated for a model atmosphere of a hot B star with effective temperature T eff = 7 3 K, gravitation acceleration log g = 4.2, and radius 3.26R. The stellar radius is artificially distended for the case of the extended atmosphere. 3.2 stellar rotation Stellar rotation is one of the most important features in stars. Unfortunately, until now a lot of unresolved questions remain. In stellar spectra modelling one uses very often rather crude simplifications. The most commonly used one is the convolution of the static profile (H()) with a rotating profile (G()), F = H() G() = H( ) G( ) d. (3.) F c The rotation profile depends on limb darkening. Since limb darkening is strongly frequency dependent, the obtained results can suffer from a huge systematic error. In Fig. 4 we show the comparison of a profile obtained from our axially symmetrical code with the profile obtained using convolution 3.. For the latter case, the limb darkening is taken in the form (see Gray, 976) I(x) = ( ɛ) + ɛ( x 2 ) /2. (3.2) Here x means the distance from the center of star in units of stellar radius (x = for r = R, where R is the stellar radius). The calculations are presented for a cool B-type main sequence star. For more details about the chosen model and results see Korčáková & Kubát (25b). Since our geometrical approximation is axial symmetry, we can naturally include gravity darkening and differential rotation. Obtained line profiles for the case of thin stellar atmosphere (cool main-sequence B type star) and extended atmosphere are plotted in Fig. 5. A detailed description of the stellar atmosphere models is published in Korčáková & Kubát (25b). 3.3 stellar wind There is a longstanding question about the origin of the stellar wind of hot Be stars. To answer this question, we need a model, which is able to include both deep stellar atmosphere layers and the wind region. The presented method is able to solve the radiative transfer equation in a stellar atmosphere where the gradient of global velocity is very small (and the Sobolev approximation is not valid), as well as in the stellar wind region, where the velocity gradient can be large.
6 Title : will be set by the publisher.98 relative flux.96.94.92 ε = ε = "rot-grav-dif" "rot+grav+dif" 4.56e+4 4.565e+4 4.57e+4 4.575e+4 Fig. 4. The comparison of line profiles of cool B-type main sequence star calculated using convolution (ɛ = and ɛ =, see Eq. 3.2) with two line profiles obtained from our code. One presents the rigid body solution and the other one the case including gravity darkening and differential rotation..98.98 relative flux.96.94.92 rot-grav-dif rot-grav-dif-velocity rot+grav-dif rot-grav+dif rot+grav+dif rot+grav+dif-velocity relative flux.96.94.92.9 rot-grav-dif rot-grav-dif-velocity rot+grav-dif rot-grav+dif rot+grav+dif rot+grav+dif-velocity.9 4.56e+4 4.565e+4 4.57e+4 4.575e+4 4.56e+4 4.565e+4 4.57e+4 4.575e+4 Fig. 5. The Hα line profiles for a thin stellar atmosphere (left panel) and for an extended stellar atmosphere (right panel). The line profiles are calculated with gravity darkening (+grav), differential rotation (+dif ) included, or excluded ( dif, grav). The lines indicated by velocity are obtained by neglecting the velocity field in the solution of the radiative transfer equation. The velocity field is present only in the flux calculation in this case. For example, in the Fig. 6 we plotted the Hα line profiles for the beta wind velocity law (see, e.g., Lamers & Cassinelli, 999) v(r) = v { [ ( vr v ) ] β } β R (3.3) r for various β parameters (β =.5,, 2). The input stellar parameters are the same as for limb darkening (section 3.) with the velocity in the photosphere v R = 2 km s and a terminal velocity of v = 2 km s. Since our method is based on the Local Lorentz Transformation (see Korčáková & Kubát
Korčáková & Kubát: Radiative Transfer in Axial Symmetry 7 23) we can calculate the decelerating velocity field, which is also plotted in Fig. 6. More details about this model and solution are in Korčáková & Kubát (25a)..95 relative flux.9.85.8 β =.5 β = β = 2 deceleration 4.565e+4 4.57e+4 4.575e+4 Fig. 6. Profile of the Hα line for the case of stellar wind for three values of the parameter β =.5,, 2 (see Eq. 3.3). For comparison, the line profile affected by decelerating velocity field is also plotted (fine dotted line). Though we show results for the stellar wind of hot stars here, this method is applicable to cool stars (Korčáková et al., 24) as well. 3.4 accretion disc Since we have a flexibility in selection of the latitudinal grid (angle θ), it is possible to adapt our method also to accretion discs. The grid in Fig. 7 (left panel) allows to describe an optically thin or optically thick disc, as well as the polar wind and hot corona. The radiation from the underlying star and from the outer boundary region is naturally included here, too. We test this method for a case of the cataclysmic variable HT Cas. The model of this system is taken from Horne et al. (99), Williams (98), and Williams & Shipman (988). The detailed description of the input model, as well as some preliminary results are described in Korčáková et al. (25). Here we show only the intensity map of the accretion disc (see Fig. 7, right panel). 4 Conclusion A method of the solution of the radiative transfer equation in axial symmetry is presented here. Since the detailed description of this method, as well as its tests in a number of stellar atmosphere models appeared in Korčáková & Kubát (25a) already, we merely emphasized the main ideas of this method. The greatest advantage of this method is in the simultaneous solution of the radiative transfer
8 Title : will be set by the publisher longitudinal plane Fig. 7. left panel: The latitudinal grid for solving the radiative transfer problem in accretion discs. right panel: The intensity map of the accretion disc for the model corresponding to HT Cas. equation in static media and in media with velocity gradients. The case of large velocity gradients can be handled using a finer space grid. Results from the first calculations are presented here. This method is very useful for an accurate description of limb darkening. It allows to calculate line profiles from rotating stars with gravity darkening and differential rotation. This is very useful for rapidly rotating stars, where the convolution method cannot be used, and flux calculation from non-distorted stars is also inappropriate. For winds we can take advantage of the solution of the wind region together with the stellar atmosphere layers. Since the grid can be finer near the equator, it is possible to calculate the radiative transfer equation in disks. In this case we can include not only the radiation of the disk, but also the central object, the boundary region, hot corona, or fast polar wind, if it is present. We are now working on a generalization of our method, which will consist mainly of the possibility of treating additional opacity sources, and of improved treatment of NLTE calculations. The authors would like to thank Júlia Sokolovičová for her comments to the manuscript. This research was supported by grants 25/4/P224 (GA ČR) and B3635 (GA AV ČR). The Astronomical Institute Ondřejov is supported by a project AVZ35. References Boissé, P. 99, A&A 228, 483 Gray, D. F. 976, Observation and Analysis of Stellar Photospheres, John Wiley & Sons, New York Horne, K., Wood, J. H., & Stiening, R. F. 99, ApJ, 378, 27 Kneer, F., & Heasley, J. N. 979, A&A, 79, 4 Korčáková, D., Kubát, J. 23, A&A, 4, 49
Korčáková & Kubát: Radiative Transfer in Axial Symmetry 9 Korčáková, D., Kubát, J. 25a, A&A, 44, 75 Korčáková, D., Kubát, J. 25b, Mem S. A. It., in press Korčáková, D., Kubát, J., Krtička, J., & Šlechta, M. 24, in The A-Star Puzzle, IAU Symp. 224, J. Zverko, J. Žižňovský, S. J. Adelman & W. W. Weiss eds., Cambridge, Univ. Press, p. 533 Korčáková, D., Kubát, J., Kawka, A., 25, in 4th European Workshop on White Dwarfs, D. Koester & S. Moehler eds., ASP Conference Series, 334, 43 Lamers, H. J. G. L. M., & Cassinelli, J. P. 999, Introduction to Stellar Winds, Cambridge Univ. Press, Cambridge Sobolev, V. 946, Dvizhushchiesia obolochki zvedz, Leningr. Gos. Univ., Leningrad Williams, R., E. 98, ApJ, 235, 939 Williams, R., E., & Shipman, H. 988, ApJ, 326, 738