2. Resonance line scattering polarization in planar equatorial disks
In deriving the emission profiles from equatorial disks, several assumptions are made, as follows:
This restrictive set of assumptions allows to focus on the effects of the parameter, viewing inclination i, and the disk velocity field for the polarized line profiles.
The notation and formalism of Paper II are adopted. The new considerations revolve around the polarized flux and . By symmetry, the Stokes flux is zero for an expanding disk since it is left-right symmetric with respect to the viewing sightline. However, for rotation, the disk is back-front symmetric only if the star is approximated as a point source of illumination. Occultation by the finite sized star breaks that symmetry, and so a net flux is generally to be expected for a rotating disk, which will be shown below.
2.1. General expressions for optically thin polarized line profiles in equatorial disks
Assuming optically thin line emission, the Stokes fluxes , , and observed at frequency in the profile corresponding to velocity shift is given by a volume integral over the isovelocity zone. Consider a star of luminosity , distance D, and radius . Light emanating from the star is scattered in a disk with surface density . The disk is permitted to have a velocity field of the form , where is some appropriate normalization constant for the speed distributions and , the angle i is the viewing inclination of the disk, and cylindrical coordinates are used. Introducing a normalized cylindrical radius , the Stokes fluxes will then be
The angle describes the locus of points in the disk for which (see Paper II). The variable is a scaling parameter of the disk surface density so that is a function of only. For a resonance line at wavelength with a cross section , the various constants have been collected into the parameter . The normalized Eddington moments and are and divided by . Note that the value of will vary from line to line.
2.2. Line profiles from disks in constant expansion or rotation
Assuming either constant expansion or rotation, the disk velocity field takes the form of and , or and . Thus and are either 0 or 1, and the observed Doppler shift toward the observer depends only on azimuth as
In the following, analytic expressions for the polarized emission profiles from the disk are derived for resonance line scattering. The simple case of a point source of illumination is derived first, followed by a consideration of the finite star effects of stellar occultation and finite star depolarization.
2.2.1. The point star approximation
Since only pure expansion or pure rotation is considered, the denominator involving the disk velocity field ultimately reduces to .
The total polarized line profile is defined to be
The "0" subscript on has been dropped. Clearly the numerator can be positive or negative. Polarization is normally positive definite, but here the sign is used to indicate the position angle orientation of the polarization, with positive (+) signifying a polarization parallel to the disk symmetry axis and negative (-) indicating one that is orthogonal to that axis. Note that the polarization in Eq. (9) is normalized by the emission line flux only - the continuum or "direct" contribution to the flux by the star has not been included. Fig. 1 shows plots of polarized line profiles from Eq. (9) as fractional polarizations for various values of and viewing inclinations i. Expanding disks in (a) and (c) give polarized profiles that are concave down in shape, but rotating disks (b) and (d) are concave up.
In the case of low spectral resolution data, it is useful to consider the line integrated polarization
which is a function of and inclination i only. Fig. 2 shows the run of plotted against . The curves are for different values of ranging from 0 to 1 in steps of 0.2. Generally, one will know from atomic physics the value for a given line transition, so that a plot like Fig. 2 could then be used to determine the disk viewing inclination from a measurement of the total line polarization. A similar consideration can be used by measuring the continuum electron scattering polarization; however, the electron scattering optical depth must be known (see Brown & McLean 1977). The advantage of the resonance line is that the line optical depth (if optically thin) will cancel out when taking the ratio of polarized line flux to total line flux (i.e., not including the continuum emission). A disadvantage is that it can be difficult to set the continuum level, and so the line flux measurement may have substantial error. Moreover, the effects of absorption, stellar occultation, and finite star depolarization have so far been neglected.
2.2.2. Finite star effects
If the scattering region extends down to the stellar radius, then occultation cannot be ignored. The lower boundary to the integration over cylindrical radius in Eqs. (1)-(3) is now a function of the Doppler shift in the profile and the viewing inclination. The problem is to determine where rays intersect the disk if tangent to the stellar photosphere and to relate that locus of points to the corresponding line-of-sight Doppler shift. The geometrical solution has been discussed by Fox & Brown (1991) who derived
which traces the projection of the stellar limb onto the disk. It is straightforward to relate to using Eq. (4) for the dependence of Doppler shift on azimuth . For constant expansion, one has ; for pure rotation the solution is .
Another finite star effect, Cassinelli et al. (1987) discussed how the continuum polarization arising from electron scattering can be reduced at small radii where the star cannot be treated as a point source. The reduction of polarization owes to the more nearly isotropic distribution of incident starlight at the scatterer. This same effect holds for the case of resonance line scattering, since the anisotropic scattering is like that of free electrons. Following Cassinelli et al., the depolarization effect is contained within and . Consider for example a uniformly bright stellar disk, the Eddington factors take on the familiar forms of and , where . If , then and making the polarization contribution in Eqs. (2)-(3) vanish.
Even with occultation and finite star depolarization, the expressions for the Stokes parameters for the emission profile shape can be derived analytically in the case of constant expansion or rotation. For an expanding disk, occultation affects only the redshifted profile for disk material receding from the observer on the far side of the disk. The blueshifted emission does not suffer from occultation, and the expressions for the flux and polarization (noting that by symmetry) are
For the redshifted side of the profile, the flux and polarization are given by
Note that if , corresponding to or , the expressions for the red wing reduce to the same form as that for the blue wing, as required.
In the case of rotation, one must bear in mind that the isovelocity zones are left-right symmetric about the line-of-sight to the star. Occultation blocks emission from the far side of the disk, so this means that the redshifted and blueshifted profiles are affected equally. Moreover, there now exists a net Stokes flux. In normalized form, the Stokes fluxes are given by
Polarized line profiles based on expressions (12)-(18) are shown in Fig. 3 and Fig. 4, the latter being for Stokes is only relevant to rotating disks. The format is the same as in Fig. 1. Note that the polarized profiles for the expanding disk case are asymmetric about line center, whereas those for a rotating disk are symmetric. Also in Fig. 4, the profiles are antisymmetric, hence the integrated flux across the entire line is zero.
2.3. Line profiles from disks in linear expansion or Keplerian rotation
Having derived analytic results for simplified cases to highlight the consequences of anisotropic scattering in disks and the various finite star effects, line profiles for more realistic disk velocity fields are computed in this section. Several assumptions remain, such as axisymmetry and the lines being optically thin, but now absorption of starlight by the intervening disk is also included. Only two specific velocity fields are considered: linear expansion that is truncated at a finite radius and Keplerian rotation.
2.3.1. The case of linear expansion
Line profiles have been computed for a disk that expands linearly as
where represents a transition from linear expansion to constant expansion. Using this velocity parametrization, the isovelocity zones reduce to lines of constant in the region of acceleration (see Fig. 1 of Paper I for a sketch detailing the and coordinates). The surface number density of the disk becomes
Fig. 5 shows line profiles for linear expansion using a value of , beyond which the contribution to the emission by the constant expansion flow is ignored. Results are given for and three different viewing inclinations of , , and from top to bottom. The latter is chosen because a planar disk viewed perfectly edge-on degenerates to a line as projected against the plane of the sky, an artifact of the disk having no vertical extent. Recall that is the flux of line emission only, hence the vertical axis is , for the stellar continuum flux outside the line frequencies. At right is the Stokes profile now shown as . (In previous figures, the polarization was given relative to the line emission only, whereas here it is shown relative to the total emission.) Again note that since , the Stokes flux represents the total polarized flux of the line emission.
The total flux emission profiles have already been discussed in Paper II. Basically the blueshifted absorption does little to affect the emission profile; however, stellar occultation of the far side of the disk substantially changes the redshifted profile shape, as is evident from its asymmetry about line center. In contrast, the polarized profile is mostly symmetric, although not exactly. It generally maintains the concave down appearance, at least in its central portion, as in the analytic cases of the previous section. Note that all the profiles show similar widths in the plots, but recall that , hence in velocity, frequency, or wavelength units, the emission and polarized profiles in Fig. 5 would appear progressively broader from top to bottom (i.e., for constant ).
2.3.2. The case of Keplerian rotation
The velocity field used for a rotating disk is the Keplerian prescription with
where is the equatorial rotation speed. In the case of linear expansion, the isovelocity zones reduce to lines of constant z, but for rotation the topology of the zones is more complicated, with a kind of nested loop pattern. A plot of isovelocity zones is shown in Fig. 2 of Wood & Brown (1994a).
As in Paper II, the disk density is assumed to be of the form
with q positive. So as to make comparison with the expansion case more direct, line profiles have been computed using . For Be star disks, q-values in the range of 2-3 are found from analyses of the disk IR emission (e.g., Waters 1986) and H line profile fitting (e.g., Hummel 2000).
Line profiles from a Keplerian disk are shown in Fig. 6 and Fig. 7, in the same format as Fig. 5 for the expanding disk case. The profiles from a rotating disk look radically different from those of an expanding disk: both the total emission profiles and the polarized profiles are generally double-peaked. Notice that the central part of the Stokes profiles are concave down. Finally, the profiles are symmetric about line center.
The profiles are displayed in Fig. 7. As in the simplified analytic case, the profiles are antisymmetric about line center, thus yielding no net flux as integrated across the entire line profile. Also, for nearly edge-on viewing perspectives, approaches zero at each point in the profile.
© European Southern Observatory (ESO) 2000
Online publication: December 5, 2000