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Astron. Astrophys. 363, 1106-1114 (2000)

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2. Resonance line scattering polarization in planar equatorial disks

In deriving the emission profiles from equatorial disks, several assumptions are made, as follows:

  1. The intrinsic line profile is treated as a delta function in frequency.

  2. The line is optically thin.

  3. The disk is axisymmetric and planar at the equator, hence a delta function along the symmetry axis, [FORMULA] (see Figs. 1 and 2 of Paper II).

  4. Only pure radial expansion or rotation are treated.

This restrictive set of assumptions allows to focus on the effects of the [FORMULA] 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 [FORMULA] and [FORMULA]. By symmetry, the Stokes [FORMULA] 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 [FORMULA] 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 [FORMULA], [FORMULA], and [FORMULA] observed at frequency [FORMULA] in the profile corresponding to velocity shift [FORMULA] is given by a volume integral over the isovelocity zone. Consider a star of luminosity [FORMULA], distance D, and radius [FORMULA]. Light emanating from the star is scattered in a disk with surface density [FORMULA]. The disk is permitted to have a velocity field of the form [FORMULA], where [FORMULA] is some appropriate normalization constant for the speed distributions [FORMULA] and [FORMULA], the angle i is the viewing inclination of the disk, and cylindrical coordinates [FORMULA] are used. Introducing a normalized cylindrical radius [FORMULA], the Stokes fluxes will then be

[EQUATION]

[EQUATION]

and

[EQUATION]

The angle [FORMULA] describes the locus of points in the disk for which [FORMULA] (see Paper II). The variable [FORMULA] is a scaling parameter of the disk surface density so that [FORMULA] is a function of [FORMULA] only. For a resonance line at wavelength [FORMULA] with a cross section [FORMULA], the various constants have been collected into the parameter [FORMULA]. The normalized Eddington moments [FORMULA] and [FORMULA] are [FORMULA] and [FORMULA] divided by [FORMULA]. Note that the value of [FORMULA] 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 [FORMULA] and [FORMULA], or [FORMULA] and [FORMULA]. Thus [FORMULA] and [FORMULA] are either 0 or 1, and the observed Doppler shift toward the observer depends only on azimuth as

[EQUATION]

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

Treating the star as a point source of illumination implies [FORMULA]. Taking the disk density to be [FORMULA] and noting that isovelocity zones correspond to [FORMULA], the Stokes flux relations reduce to the forms

[EQUATION]

[EQUATION]

and

[EQUATION]

Since only pure expansion or pure rotation is considered, the denominator involving the disk velocity field ultimately reduces to [FORMULA].

The total polarized line profile is defined to be

[EQUATION]

which for [FORMULA] reduces to

[EQUATION]

The "0" subscript on [FORMULA] 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 [FORMULA] 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.

[FIGURE] Fig. 1a-d. A comparison of the polarized emission profiles (given as fractional polarizations) for equatorial planar disks in constant expansion (left panels a and c ) versus ones with constant rotation (right panels b and d ). The upper panels are for edge-on viewing perspectives with [FORMULA] but [FORMULA] allowed to vary as indicated. Lower panels are for [FORMULA] but different viewing inclinations. The primary morphological difference between the two cases is that expansion yields concave down profiles whereas rotation yields concave up profiles.

In the case of low spectral resolution data, it is useful to consider the line integrated polarization

[EQUATION]

which is a function of [FORMULA] and inclination i only. Fig. 2 shows the run of [FORMULA] plotted against [FORMULA]. The curves are for different values of [FORMULA] ranging from 0 to 1 in steps of 0.2. Generally, one will know from atomic physics the [FORMULA] 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.

[FIGURE] Fig. 2. Shown is the line integrated polarization [FORMULA] plotted as a function of [FORMULA], with each different curve corresponding to a value of [FORMULA] and 1.0. These curves are for the simplified case of a point source star and a disk in either constant expansion or rotation, yet the figure illustrates how resonance scattering polarization with even low spectral resolution data might be used to infer the disk viewing inclination. As long as the line is optically thin, these curves are independent of the line optical depth (see the text).

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

[EQUATION]

which traces the projection of the stellar limb onto the disk. It is straightforward to relate [FORMULA] to [FORMULA] using Eq. (4) for the dependence of Doppler shift on azimuth [FORMULA]. For constant expansion, one has [FORMULA]; for pure rotation the solution is [FORMULA].

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 [FORMULA] and [FORMULA]. Consider for example a uniformly bright stellar disk, the Eddington factors take on the familiar forms of [FORMULA] and [FORMULA], where [FORMULA]. If [FORMULA], then [FORMULA] and [FORMULA] 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 [FORMULA] by symmetry) are

[EQUATION]

For the redshifted side of the profile, the flux and polarization are given by

[EQUATION]

Note that if [FORMULA], corresponding to [FORMULA] or [FORMULA], 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 [FORMULA] flux. In normalized form, the Stokes fluxes are given by

[EQUATION]

[EQUATION]

[EQUATION]

Polarized line profiles based on expressions (12)-(18) are shown in Fig. 3 and Fig. 4, the latter being for Stokes [FORMULA] 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 [FORMULA] profiles are antisymmetric, hence the integrated [FORMULA] flux across the entire line is zero.

[FIGURE] Fig. 3a-d. As in Fig. 1, but now with both stellar occultation and finite star depolarization effects included. The overall effect is to reduce the peak polarizations that are obtained relative to the point star case, and also the polarized profiles for an expanding disk are asymmetric about line center. Note that only for expansion does [FORMULA], whereas a rotating disk also has a Stokes [FORMULA] profile owing to the effect of stellar occultation (see Fig. 4).

[FIGURE] Fig. 4a and b. The [FORMULA] profile corresponding to the rotating disk case of Fig. 3. Stellar occultation breaks the back-front symmetry associated with a rotating disk, hence a Stokes [FORMULA] flux remains. However, note that the [FORMULA] profile is antisymmetric about line center so that the line integrated [FORMULA] flux vanishes.

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

[EQUATION]

where [FORMULA] represents a transition from linear expansion to constant expansion. Using this velocity parametrization, the isovelocity zones reduce to lines of constant [FORMULA] in the region of acceleration (see Fig. 1 of Paper I for a sketch detailing the [FORMULA] and [FORMULA] coordinates). The surface number density of the disk becomes

[EQUATION]

Fig. 5 shows line profiles for linear expansion using a value of [FORMULA], beyond which the contribution to the emission by the constant expansion flow is ignored. Results are given for [FORMULA] and three different viewing inclinations of [FORMULA], [FORMULA], and [FORMULA] 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 [FORMULA] is the flux of line emission only, hence the vertical axis is [FORMULA], for [FORMULA] the stellar continuum flux outside the line frequencies. At right is the Stokes [FORMULA] profile now shown as [FORMULA]. (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 [FORMULA], the Stokes [FORMULA] flux represents the total polarized flux of the line emission.

[FIGURE] Fig. 5. Emission profiles (left) and polarized profiles (right) from a circumstellar disk undergoing linear expansion. From top to bottom, the viewing inclinations of the disk are [FORMULA], [FORMULA], and [FORMULA]. The profiles are asymmetric, especially the blueshifted peak is more prominent than that for the redshifted wing. The polarized profile is concave down at the line core. Note that in distinction to the preceding figures for the analytic examples, the polarized profiles here are shown as [FORMULA] divided by the total observed flux [FORMULA] for the line and [FORMULA] for the direct continuum starlight ([FORMULA] for expanding disks).

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 [FORMULA], 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 [FORMULA]).

2.3.2. The case of Keplerian rotation

The velocity field used for a rotating disk is the Keplerian prescription with

[EQUATION]

where [FORMULA] 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

[EQUATION]

with q positive. So as to make comparison with the expansion case more direct, line profiles have been computed using [FORMULA]. 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[FORMULA] 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 [FORMULA] profiles are concave down. Finally, the profiles are symmetric about line center.

[FIGURE] Fig. 6. Similar to Fig. 5, but now for a disk in Keplerian rotation with no radial motion. The profiles are symmetric, and the polarized profiles are concave up at the central region. Note that for rotating disks, stellar occultation leads to a net [FORMULA] polarization, which is shown in Fig. 7.

[FIGURE] Fig. 7. The [FORMULA] polarization from a disk executing Keplerian rotation. The [FORMULA] flux is antisymmetric about line center. For a nearly edge-on disk, [FORMULA] at each point in the profile, owing to the symmetry of the viewing orientation.

The [FORMULA] profiles are displayed in Fig. 7. As in the simplified analytic case, the profiles are antisymmetric about line center, thus yielding no net [FORMULA] flux as integrated across the entire line profile. Also, for nearly edge-on viewing perspectives, [FORMULA] approaches zero at each point in the profile.

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Online publication: December 5, 2000
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