East Tennessee State University

From the SelectedWorks of Richard Ignace

December 10, 2010

The Hanle Effect as a Diagnostic of MagneticFields in Stellar Envelopes. V. Thin Lines fromKeplerian Disks.Richard Ignace, East Tennessee State University

Available at: https://works.bepress.com/richard_ignace/43/

The Astrophysical Journal, 725:1040–1052, 2010 December 10 doi:10.1088/0004-637X/725/1/1040C© 2010. The American Astronomical Society. All rights reserved. Printed in the U.S.A.

THE HANLE EFFECT AS A DIAGNOSTIC OF MAGNETIC FIELDS IN STELLAR ENVELOPES. V.THIN LINES FROM KEPLERIAN DISKS

R. Ignace

Department of Physics and Astronomy, East Tennessee State University, Johnson City, TN 37614, USA; ignace@etsu.eduReceived 2010 July 5; accepted 2010 October 10; published 2010 November 22

ABSTRACT

This paper focuses on the polarized profiles of resonance scattering lines that form in magnetized disks. Opticallythin lines from Keplerian planar disks are considered. Model line profiles are calculated for simple field topologiesof axial fields (i.e., vertical to the disk plane) and toroidal fields (i.e., purely azimuthal). A scheme for discerningfield strengths and geometries in disks is developed based on Stokes Q − U diagrams for the run of polarizationacross line profiles that are Doppler-broadened by the disk rotation. A discussion of the Hanle effect for magnetizeddisks in which the magnetorotational instability (MRI) is operating is also presented. Given that the MRI has atendency to mix the vector field orientation, it may be difficult to detect the disk fields with the longitudinal Zeemaneffect, since the amplitude of the circularly polarized signal scales with the net magnetic flux in the direction ofthe observer. The Hanle effect does not suffer from this impediment, and so a multi-line analysis could be used toconstrain field strengths in disks dominated by the MRI.

Key words: accretion, accretion disks – circumstellar matter – line: profiles – magnetic fields – polarization –stars: magnetic field

1. INTRODUCTION

Spectropolarimetry continues to be a valuable technique fora broad range of astrophysical studies (e.g., Adamson et al.2005; Bastian 2010), including applications for circumstellarenvelopes. Advances in technology and access to larger tele-scopes mean an ever growing body of high quality spectropo-larimetric data. It is therefore important that the arsenal of diag-nostic methods and theoretical models in different astrophysicalscenarios keep pace. This paper represents the fifth in a seriesdevoted toward developing the Hanle effect as tool for measur-ing magnetic fields in circumstellar media from resonance linescattering polarization. The observational requirements for theeffects examined in this series are ambitious: high signal-to-noise (S/N) data and high spectral resolving power. However,these demands are being met, as illustrated by Harrington &Kuhn (2009a) in a spectropolarimetric survey of circumstellardisks at Hα.

There are numerous effects that can influence the polarizationacross resolved lines. A number of researchers have investigatedthe effects of line opacity for polarization from Thomson scat-tering (Wood et al. 1993; Harries 2000; Vink et al. 2005; Wang& Wheeler 2008; Hole et al. 2010). Scattering by resonancelines can generate polarization similar to dipole scattering (e.g.,Ignace 2000a). With high S/N and high spectral resolution,Harrington & Kuhn (2009b) have identified a new polarizingeffect for lines that coincides with line absorption. An explana-tion for this previously unobserved effect in stars is discussedby Kuhn et al. (2007) and is attributed to the same dichroicprocesses detailed by Trujillo Bueno & Landi Degl’Innocenti(1997) for interpreting polarizations in some solar spectral lines.Generally associated with circular polarization, the Zeeman ef-fect has received acute attention of late as a result of techniquesthat co-add many lines (Donati et al. 1997). The method has beenused successfully in many studies (see the review of Donati &Landstreet 2009). In relation to massive stars, the technique hasled to the detection of magnetism in several stars (e.g., Donatiet al. 2002, 2006a, 2006b; Neiner et al. 2003; Grunhut et al.2009). In fact, there exists a large collaborative effort called the

MiMeS1 collaboration (e.g., Wade et al. 2009) to increase thesample of massive stars with well-studied magnetic fields.

Another method that has been used productively in studiesof solar magnetic fields involves the Hanle effect (Hanle 1924).This is a weak Zeeman effect that operates when the Zeemansplitting is roughly comparable to the natural line broadening.Monographs that deal with atomic physics and polarized radia-tive transfer provide excellent treatments of the Hanle effect,including Stenflo (1994) and Landi Degl’Innocenti & Landolfi(2004). This contribution extends a series of theoretical papersthat has enlarged the scope of its general use with circumstel-lar envelopes. Building on work developed in the solar physicscommunity, and using expressions for resonance line scatteringwith the Hanle effect (e.g., from Stenflo 1994), Ignace et al.(1997, Paper I) provided an introduction of the Hanle effect foruse in studies of circumstellar envelopes. Ignace et al. (1999,Paper II) considered its use in simplified models of disks withconstant radial expansion or solid body rotation as pedagogicexamples. Both of these papers approximated the star as a pointsource of illumination. Ignace (2001, Paper III) then incorpo-rated the finite source depolarization factor of Cassinelli et al.(1987) into the treatment of the source functions for the Hanleeffect in circumstellar envelopes. Finally, Ignace et al. (2004,Paper IV) calculated the Hanle effect in P Cygni wind linesusing an approximate treatment for line optical depth effects.

The focus of this fifth paper in the series is the Hanle effect inpolarized lines from Keplerian disks, of relevance for accretiondisks and Be star disks (e.g., Cranmer 2009). As in previouspapers of this series, the Sobolev approximation (Sobolev 1960)for optically thin scattering lines is adopted for exploring the runof polarization with velocity shift across line profiles. This paperadopts the notation laid out in Paper II. The results of Paper IIare expanded for Keplerian rotation with the inclusion of finitesource depolarization (ala Paper III) and stellar occultation.Resonance line scattering polarization from such disks were

1 http://www.physics.queensu.ca/∼wade/mimes/MiMes_Magnetism_in_Massive_Stars.html

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No. 1, 2010 HANLE EFFECT AS A DIAGNOSTIC OF MAGNETIC FIELDS IN STELLAR ENVELOPES. V. 1041

Figure 1. Illustration of the relation between observer and stellar coordinatereference frames. The observer is in the direction of Z; the symmetry axis forthe disk is Z∗. A scattering point is in direction r . This point lies in the equatorialdisk, hence ϕ = π/2.

explored in Ignace (2000a) for unmagnetized disks withoutconsideration of the Hanle effect.

Drawing on these previous works, methods for computingthe polarization of line profiles will be briefly reviewed inSection 2. In Section 3, results for the Hanle effect from disklines are described. There are three applications that will beconsidered: purely axial fields (i.e., normal to the disk plane),purely toroidal fields (i.e., azimuthal in the disk plane), and fieldtopologies that can arise from the magnetorotational instabilityor MRI (Balbus & Hawley 1991). Conclusions of this studyand observational prospects are presented in Section 4. TheAppendix details analytic derivations for special cases of theHanle effect in Keplerian disks.

2. THIN SCATTERING LINES IN KEPLERIAN DISKS

2.1. Coordinate Systems of the Model

The focus of this paper is thin scattering lines from a planarcircumstellar disk in which the gas obeys circular Keplerianmotion. To describe this geometry and the line scatteringpolarization, a number of coordinates will be needed, whichare introduced here.

1. A Cartesian coordinate system for the star is assigned to be(X∗, Y∗, Z∗). The Z∗-axis is the rotation axis of the star anddisk.

2. Another one for the observer is (X, Y,Z). The observer islocated at large positive distance along the Z-axis.

3. Spherical coordinates in the star frame are (r, ϑ, ϕ).Cylindrical coordinates are identified by (�,ϕ,Z∗).

4. Cylindrical coordinates in the observer frame are (p,ψ,Z).5. Spherical angular coordinates defined in a frame of the local

magnetic field at any point will be (θ, φ). The latter systemis needed to evaluate the Hanle effect.

6. The Z-axis is taken to be inclined to the Z∗-axis by anangle i.

7. The circumstellar disk assumed to be axisymmetric andexists entirely in the equatorial plane of the star, and so it islocated at ϑ = π/2. The scattering angle between a pointin the disk to the observer is signified by χ .

Figure 1 shows a spherical triangle used in relating theobserver and star coordinate systems.

2.2. Line Velocity Shifts

The Sobolev approximation is employed to describe thepolarimetric line profiles. The Sobolev approach relies onidentifying “isovelocity zones,” which are the locus of pointsthat share the same Doppler shift for a distant observer. TheDoppler shift in frequency of any point in the scattering volumeis determined by

ΔνZ = −νulvZ

c= − vZ

λul, (1)

where νul is the frequency of the line transition, and vZ is theline-of-sight velocity shift given by

vZ = −�v · Z. (2)

A Keplerian disk follows a velocity profile of the form

�v = vϕ ϕ = v0

√R0

rϕ, (3)

where R0 is the innermost radius of the disk with tangentialspeed v0 at that location.

It is convenient to define dimensionless variables for distancesand velocities. For the normalized cylindrical radius in the planardisk, � = r/R0 is used. The line-of-sight velocity shift thenbecomes

vZ = v0 �−1/2 sin i sin ϕ. (4)

A normalized velocity is also introduced with wz =vZ/(v0 sin i).

The solution for the isovelocity zones in terms of � (wz, ϕ)is thus given by

� = sin2 ϕ

w2z

= sin2 ϕ

sin2 ϕ0. (5)

This describes a loop path that starts at R0 at an azimuth ϕ0on the front side of the disk, where sin ϕ0 = wz, and then endsagain at R0 and π −ϕ0. The loop extends to a maximum distanceat azimuth ϕ = π/2 with a value of � = 1/w2

z . The systemis left–right mirror symmetric, with redshifted velocities on oneside and blueshifted ones on the other. The convention hereis that the disk rotates counterclockwise as seen from above(i.e., “prograde”). An example is displayed in Figure 2. Withthe arrow toward the observer as the reference for ϕ, redshiftedvelocities come from the interval 0 < ϕ < π , and blueshiftedones come from π < ϕ < 2π .

2.3. Line Flux

For optically thin scattering, the observed flux of scatteredlight at a given velocity shift in the line is determined by a vol-ume integral along the associated isovelocity loop. To describethe polarization, a Stokes vector prescription is adopted, withstandard I,Q,U, V notation. Referring back to Figure 1, anedge-on disk with i = 90◦ would yield intensities with Q > 0and U = 0. In this case “north” is chosen to be along the di-rection of +Z∗, and Q > 0 corresponds to oscillations of theelectric vectors of the light being preferentially parallel to north.

For unresolved disks the corresponding observed fluxes are�Fν = (FI ,FQ,FU , 0), where it is assumed that the Stokes-V

flux is zero for the problem at hand.

1042 IGNACE Vol. 725

Figure 2. Illustration of isovelocity zones for a Keplerian disk. The hatchedregion at center is the star, assumed to rotate ccw, as indicated by the arrow attop. In this example an observer is located in the equatorial plane. Red curveson the right are for redshifted velocities (dashed), with the smallest loop havingthe greatest speed. Curves on the left are for blueshifts (dotted). Emission atline center arises from the vertical solid line.

(A color version of this figure is available in the online journal.)

Generally following the notation of Paper II and using resultsfrom Ignace (2000a) for Keplerian disks, the disk is assumed tohave a power-law surface number density given by

Σ(� ) = Σ0 �−m, (6)

for power-law exponent m and inner value Σ0 at � = 1. Notethat in the case of the Be stars, values of m range from 3 to 4 (e.g.,Waters 1986; Lee et al. 1991; Porter 1999; Jones et al. 2008),and m = 3.5 will be adopted in example cases. Implicit is thatthe lines of interest form from the dominant ion. Temperatureand ionization variations or different disk densities could beincluded in the model, but the intent of this contribution is tohighlight the line polarization and Hanle effect, and so a simplepower-law density is adopted as a baseline for the analysis.

The Stokes fluxes for thin lines from this disk are given by

�Fν(wz) = τ0 F0

∫�h(�,ϕ)

� 2−m

wϕ(� )

d�

| cos ϕ| . (7)

The denominator in the integrand arises from the Sobolevapproximation, with wϕ = �−1/2. The factor | cos ϕ| will

more conveniently be written as√

1 − w2z� in what fol-

lows. The vector �h represents the “scattering factor” with�h = (hI , hQ, hU , hV ). It is this factor that incorporates thedipole scattering of resonance line polarization, the impact ofmagnetic fields via the Hanle effect, and the finite depolarizationfactor. Finally, the scaling factors outside the integral are fromEquations (24) and (25) of Paper II, with

τ0 = σl Σ0 λul

4π vrot sin i, (8)

where σl is the frequency integrated line cross section, and

F0 = Lν,∗4π d2

, (9)

where Lν,∗ is the specific luminosity of the star and d is thedistance to the star from Earth.

2.4. Solution for Non-magnetic Scattering Polarization

Before progressing to a consideration of the Hanle effect,there is value in formulating the solution for the line scatteringpolarization in the zero-field case. This is because the Hanleeffect primarily modifies the scattering polarization, meaningthat a good understanding of the non-magnetic case is anecessary reference for understanding the Hanle effect. Manypapers have dealt with line formation from Keplerian disks.This work draws in particular on the work of Huang (1961) andRybicki & Hummer (1983) for planar disks, and also Ignace(2000a) for scattering polarization without the Hanle effect.

There are two main cases that are explored here: the simplisticcase of illumination by a point source and then the more realisticcase involving the effects of the finite size of the illuminatingstar. Analytic and semi-analytic solutions are described in theAppendix. The value of the point source approximation is thatit gives context for how finite stellar size effects modify theline scattering polarization. The only portion of the point sourceapproach that is reproduced here from Appendix A.1 is theexpression for isotropic scattering of starlight by a planar diskin the Sobolev approximation.

For isotropic scattering of light from a point star, one has�h = (1, 0, 0, 0). Only the Stokes-I flux survives in Equation (7).It is convenient to introduce a change of variable for computingthe integration of that equation, with t = �−1. Then the fluxbecomes

FI = τ0 F0 × 2∫ 1

t0(wz)

tm−1

√t − t0

dt, (10)

where t0 = w2z . The appearance of the factor of two arises

because the integration for the front half of the loop (from ϕ0 toϕ = π/2) is the same as for the back half.

The Appendix details analytic cases for the above case. Theaddition of polarization and finite source effects amount toinserting multiplicative “weighting” functions that modify theabove integrand that describes the emission contribution as afunction of location along a loop.

There are several finite source effects that can be included,such as finite star depolarization (Cassinelli et al. 1987), stellaroccultation effects (e.g., Fox & Brown 1991), limb darkeningeffects (Brown et al. 1989), and absorption of starlight by thedisk (Ignace 2000a). Only two of these effects will be consideredhere: finite star depolarization and stellar occultation of the disk.Limb darkening could be included; however, limb darkeningmainly “softens” the finite star depolarization factor, making itslightly less severe. Thus, the inclusion of limb darkening doesnot seem to be a pressing issue for illustrating the Hanle effectin disks.

Absorption by the disk does affect the shape of the Stokes-Iprofile; however, in the current treatment it does not influence �hwhich determines the line scattering polarization. Consequently,absorption affects the continuum level of direct starlight at theline, but not the scattered Stokes fluxes FI , FQ, or FU . On theother hand, a photospheric absorption line would alter the lineshapes of the scattered flux profiles; however, results presented

No. 1, 2010 HANLE EFFECT AS A DIAGNOSTIC OF MAGNETIC FIELDS IN STELLAR ENVELOPES. V. 1043

here will be in terms of ratios of scattered fluxes, qsν = FQ/FI

and usν = FU/FI , for which the influence of a photospheric line

will cancel.These ratios qs

ν and usν should be considered as polarimetric

“efficiencies.” They are not generally what an observer wouldactually measure, because the denominator involves only thescattered light of the Stokes-I flux. Instead an observer wouldnormally measure FQ, FU , and Ftot = F0 + FI , the latter beingthe sum of the direct starlight (first term) and the scattered light(second term). In this case qtot = FQ/(F0 + FI ) ≈ FQ/F0,and likewise for the Stokes-U polarization, assuming that thescattered flux in the line is small compared to the direct fluxby the star itself. As a result, (qtot, utot) = (qs

ν, usν) × (FI /F0).

It is worth noting that the qsν and us

ν “efficiency” profiles areindependent of the line optical depth τ0 in the thin limit, sinceFI , FQ, and FU scale linearly with τ0.

Equation (C1) gives the scattering function components of �hin the case of no magnetic field with the star treated as a point starof illumination. Modifications to those functions that allow forscattering by a finite sized and uniformly bright central star weredetermined in Paper III. For the case of zero magnetic field, thescattering function is shown explicitly with factors arising fromthe finite stellar size. These factors also appear in the scatteringfunction with the Hanle effect in the same positions as withoutthe Hanle effect. The vector components of �h are

�h =

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

hI = W +3

8E1 μ∗

[1

3(1 − 3 cos2 i) + sin2 i cos 2ϕ

]

hQ = 3

8E1 μ∗ [sin2 i − (1 + cos2 i) cos 2ϕ]

hU = 3

8E1 μ∗ cos i sin 2ϕ,

(11)

whereW = 4� 2 W (� ), (12)

with the dilution factor W (� ) = 0.5 (1 − μ∗), and

μ∗ =√

1 − �−2 (13)

is the finite depolarization factor when there is no limb darken-ing. The factors W and μ∗ represent corrections to the scatteringfunction that account for the effects of finite star size in relationto the incident radiation field at a scattering point around thestar.

Scattering by resonance lines can be approximated as partdipole and part isotropic. The parameter E1 is a factor withvalues from 0 to 1 that represents the extent to which aresonance line scatters like a dipole radiator (see Chandrasekhar1960). Only the dipole portion of the scattered light contributesto observable polarization. In the solar literature, it is morecommon to use the notation W2 for the fraction of scatteredlight that is dipole-like (e.g., Stenflo 1978).

Accounting for stellar occultation breaks the back-frontsymmetry of the isovelocity loops. The front loop still has lowerand upper limits of t0 and 1, but the back half has limits of t0and t∗ � 1. Here t∗ is associated with the minimum value of� = �∗ at the back-projected limb of the star corresponding toa maximum azimuth of ϕ = ϕ∗. An expression for �∗ is givenin Ignace (2000a), expressed here as

t∗ =√

1 − sin2 i cos2 ϕ∗ =√

cos2 i − w2z sin2 i /t∗. (14)

Figure 3. Line profiles for a disk with no magnetic field. Upper left: areanormalized Stokes-I line profiles for viewing inclinations of i = 0◦ (red),i = 30◦ (green), i = 45◦ (dark blue), i = 60◦ (light blue), and i = 90◦(magenta). Lower left: Stokes-Q profiles plotted as the ratio of the Q-flux tothe scattered I-flux as fractional polarization (not percent). The yellow dashedline is a guide for zero polarization. Lower right: like the Q-profiles, Stokes-Uprofiles shown as fractional polarizations, which arise solely from the effect ofstellar occultation. Upper right: a Q − U diagram across the profile.

(A color version of this figure is available in the online journal.)

This expression provides a cubic relation in t∗ as a function offixed viewing inclination i and velocity shift wz.

Numerical results for polarized profiles without the Hanleeffect are shown in Figure 3. This figure and the ones that willfollow use m = 3.5 and E1 = 0.5. Figure 3 should be comparedto Figure 9 for the point star case. For point star illumination,the maximum polarization occurs at the line wings and FU ≡ 0by symmetry. This changes dramatically when the effects of thefinite stellar size are included.

In Figure 3, each panel has five curves color-coded fordifferent viewing inclinations: sin2 i = 0 is red, sin2 i = 0.25is green, sin2 i = 0.5 is deep blue, sin2 i = 0.75 is lightblue, and sin2 i = 1.0 is magenta. The values correspond toviewing inclination angles of i = 0◦, 30◦, 45◦, 60◦, and 90◦,respectively. The upper left panel displays FI profiles thatare normalized to have unit area. In the lower right panel forthe qs

ν polarization, the finite depolarization factor shifts peakpolarization to lower velocity shifts as compared to the pointstar case, and qs

ν remains symmetric about line center. Stellaroccultation leads to the survival of a small us

ν polarization asshown in the lower right panel. The upper right panel is a qs

ν–usν

plot across the respective line profiles. Note that the scalesin qs

ν and usν differ by about an order of magnitude. These

small loops signify position angle rotations across the observedpolarized line profiles, with polarization position angle ψP givenby tan 2ψP = us

ν/qsν .

We next turn our attention to the Hanle effect. The re-sults of this section will prove valuable in following how theHanle effect modifies the run of polarization across the lineprofile.

1044 IGNACE Vol. 725

3. THE HANLE EFFECT IN KEPLERIAN DISKS

The Hanle effect applies to resonance line scattering andcan be interpreted in semi-classical terms as a precession of anoscillating emitter that occurs over the radiative lifetime of theline emission. An angular quantity can be defined to representthe effective amount of precession, as given by

tan α2 = 2 gL ωL

Aul= B

BHan, (15)

where ωL is the angular Larmor frequency, Aul is the Einsteinradiative rate for the transition of interest, gL is the Lande factorof the upper level, B is the magnetic field strength in Gauss, andBHan is the Hanle field sensitivity defined as

BHan = 56.9A9

gLG, (16)

with A9 being the radiative rate normalized to 109 Hz. TheHanle effect manifests itself in the scattering physics with theappearance of cosine and sine functions of the angle α2 (and arelated quantity α1 = 0.5α2) within the scattering functions �h.Note that with no magnetic field, B = 0 and α2 = 0. The limitof a high Larmor frequency yields α2 ≈ π/2. In this latter case,the Hanle effect is said to be “saturated.”

There are a few rules of thumb to help understand the Hanleeffect in polarized lines. First, modifications to the polarizedprofile will depend on the orientation of the field in relation toboth the direction of incoming radiation and outgoing radiation(i.e., the observer). Second, there is no Hanle effect when theincoming radiation field is symmetric about the field direction.These first two “rules” indicate that there is no Hanle effect forradial magnetic fields (assuming a spherically symmetric starand no diffuse radiation) and that only non-radial componentsof the field contribute to modifying the polarization. However,the third point is that in spite of this, the angle α2 is sensitive tothe total field strength: both radial and non-radial components,at the site of scattering. Fourth, and finally, the Hanle effect issensitive only to the field topology in the saturated limit, not thefield strength.

With these rules of thumb, consideration of the Hanle effectin three particular cases follow: a purely axial field, a purelytoroidal field, and last a scenario involving fields as arising fromthe operation of MRI in disks.

3.1. Axial Magnetic Field

In this case an axial magnetic field is envisioned as threadingthe disk perpendicular to the equatorial plane, hence �B =BZ∗(� ) Z∗. As for the field distribution throughout the disk,the following is adopted as a conservative limiting case:

BZ∗ = B0 �−1, (17)

and so the Hanle angle α2 is determined by

tan α2 = B0

BHan�−1 ≡ b0 t, (18)

where b0 = B0/BHan for B0 a field strength at the inner diskradius. It is important to note that if B0 BHan, then the Hanleeffect is weak everywhere in the disk. If B0 � BHan, then theinner disk will be in the saturated limit; however, there willbe a transition to a weak Hanle effect at some radius in disk,

Figure 4. Hanle effect for a disk threaded by a purely axial magnetic field. Leftis for qs

ν and right is for usν . Profiles are shown for different viewing inclinations

with i = 30◦ at top, then i = 45◦, i = 60◦, and i = 90◦ at bottom. The colorsare for different field strengths with b0 = 0.01 (red), 0.1 (green), 1.0 (dark blue),10 (light blue), and 100 (magenta).

(A color version of this figure is available in the online journal.)

characteristically where � = B0/BHan. As a general rule, for agiven magnetized disk, different lines will have different valuesof BHan, and thus will be sensitive to different aspects of the fieldat different locations in the disk.

Expressions for �h in the point star limit are given inAppendix A.3. Revisions to those expressions for finite stardepolarization and stellar occultation are the same as inEquation (11). Results are shown in Figure 4. Profiles of qs

ν

and usν are displayed at left and right, respectively. Panels from

top to bottom are now for different viewing inclinations ofsin2 i = 0.25, 0.5, 0.75, and 1.0. Different colors are for dif-ferent values of b0 with b0 = 0.01 (red), 0.1 (green), 1.0 (darkblue), 10 (light blue), and 100 (violet).

The Stokes-Q profiles are symmetric about line center,whereas the U profiles evolve from weak and anti-symmetric,owing to stellar occultation, to strong and mostly symmetric,to a null profile at the strongly saturated limit. In the point starcase, the saturated limit yields us

ν = 0 and qsν = constant for

all velocity shifts. With finite source effects, qsν is mostly flat

throughout the central portion of the profile, but goes to zeroin the wings. In the presence of limb darkening, the qs

ν profilewould become more flattened toward the wings.

Figure 5 displays qsν–us

ν curves across the line profile. Notethat the two axes have different scales: variations in us

ν aresmaller than for qs

ν . The colored curves are the same cases asshown in Figure 4. Viewing inclinations are indicated. For theedge-on case of i = 90◦, the curves are all degenerate withus

ν = 0. The red loop is for a very weak field and is seen to betop–bottom symmetric in this space. As the field is increased,qs

ν and usν profiles become individually more nearly symmetric.

In the qsν–us

ν space, this results in curves with only small loops,for example the modest field case with the dark blue curve. Asthe disk enters the saturated limit, us

ν drops to zero, qsν becomes

somewhat flat-topped in appearance (again, except at the linewings), which degenerates mainly to a point in the qs

ν–usν space

No. 1, 2010 HANLE EFFECT AS A DIAGNOSTIC OF MAGNETIC FIELDS IN STELLAR ENVELOPES. V. 1045

Figure 5. Q − U diagram for the polarized profiles shown in Figure 4 with thesame color scheme in relation to results for different values of b0. The strongly“saturated” case of b0 = 100 is present in magenta. This case has a modestrange in qs

ν from 0 to about 0.1, but with usν ≈ 0 everywhere, the profile is only

a short horizontal line in the Q − U plane and difficult to see in this figure. Forthe edge-on case, us

ν ≡ 0 and all profiles degenerate to horizontal lines from 0to a maximum value of qs

ν that depends on b0.

(A color version of this figure is available in the online journal.)

for most velocity shifts, with extension to zero polarization onlyfor the line wings.

3.2. Toroidal Magnetic Field

A completely azimuthal field configuration has also beenconsidered. As in the axial field case, the toroidal field strengthis assumed to decrease inversely proportional to � , and soEquation (18) remains valid. This means that differences in themodel polarized profiles between the axial and toroidal fieldconfigurations arise because the Hanle effect is sensitive to thevector field orientation.

As with previous considerations, some analytic results arederivable in the point star limit, and these are detailed inAppendix A.4 for a disk with a toroidal field. With a toroidalfield, the geometry associated with determining the Hanleeffect is more complex than in axial field case. Geometricalrelationships between the various angles defining the scatteringproblem with a toroidal field in a disk were given in the appendixof Paper II and will not be repeated here. Results for thecalculation of polarized profiles, including the effects of thestar’s finite extent, are shown in Figure 6. This figure is presentedin the same manner as Figure 4 for the axial field case: qs

ν is atleft and us

ν at right; the panels are for different values of sin2 i;the colors are for different field strengths characterized by b0.

Note the marked differences in the line polarization betweenthe toroidal case and that of an axial field. The Stokes-U profilesare always antisymmetric. Like the axial case, qs

ν profiles aresymmetric, but the behavior is quite different. For example,the edge-on disk case at lower left indicates that the profilepolarization completely changes sign, from positive definite atevery velocity shift when B = 0 to everywhere negative in thesaturated limit. This implies a 90◦ rotation in position angle

Figure 6. Similar to Figure 4 but now for a toroidal field instead of an axial one.From top to bottom, panels are for the same viewing inclinations, and the colorscheme is for the same values of b0. The qs

ν profiles are symmetric althoughdiffering in details from the axial field case. More striking is that the us

ν profileis decidedly antisymmetric for the Bϕ case, whereas profiles tend toward beingsymmetric in the axial field case.

(A color version of this figure is available in the online journal.)

between these two limiting cases for the entire profile. Forintermediate field strengths, there is a position angle rotationacross the profile that occurs at different velocity shifts. Inthe axial field case, there is never a position angle rotationfor an edge-on disk. For the qs

ν–usν shapes, differences in the

polarizations are especially clear, with results for the toroidalfield in Figure 7 to be compared against those for an axial fieldin Figure 5.

What is the source of these differences between the axialfield and the toroidal one? The field orientation with respect tothe viewer is clearly the key to the interpretation. For the axialcase, the field at every point in the disk has some componentof the magnetic vector directed toward the observer (if seenat i < 90◦) or away from the observer (if seen at i > 90◦).Thus, an axial field leads to a net projected magnetic flux ofone sign or the other, which is true for every isovelocity loop.The toroidal field is manifestly different, since one side of thedisk has components toward the observer and the other side hasthem away. For an axisymmetric toroidal field, the projectednet magnetic flux is identically zero for the entire disk, but isoppositely signed in isovelocity loops for blueshifted velocitiesversus redshifted ones. In terms of the semi-classical precessiondescription, flipping the field by 180◦ amounts to a precession inthe opposite direction. In the axial field case, the precession ofthe radiating oscillator is uniform—entirely clockwise (cw) orcounterclockwise (ccw). Both cw and ccw precessions occur fora toroidal field configuration, with one precession occurring inhalf the line profile and the opposite for the other half.

To illustrate this effect, Figure 8 shows line profile results for atoroidal field that now goes in the other direction as compared toFigure 6. The qs

ν profiles are nearly the same, but the usν profiles

are nearly mirror images of one another. Slight differences area result of the stellar occultation, which does not flip when thefield orientation is reversed.

1046 IGNACE Vol. 725

Figure 7. Similar to Figure 5 but now for a toroidal field instead of an axial one.The combination of antisymmetric us

ν with symmetric qsν leads to Q − U loops

that are top–bottom symmetric in this space. Note that the axis scale is differentfor us

ν as compared to qsν .

(A color version of this figure is available in the online journal.)

3.3. Magnetorotational Instability

A large literature has developed over the last 20 years inrelation to the MRI (Balbus & Hawley 1991, 1998). There isan interesting history about this effect (e.g., Balbus 2003). Theinstability can be illustrated through an analogy to two massesin a differentially rotating disk that are slightly offset from eachother along a radius. These two masses are connected by a springto represent the effect of an axial magnetic field. The end resultis that the two masses on different orbits seek to increase theirdisplacement from one another. Coupling by the spring leads toa runaway situation ensues.

Key for the context of magnetic diagnostics is how thisinstability impacts the field topology and strength throughoutthe disk. The mass-spring analogy above has built into it theassumption of flux freezing. The separation of the masses alongwith the differential rotation would appear to evolve the axialfield through the disk into a toroidal one. Different researchershave studied the operation of the MRI in accretion disks throughboth semi-analytic work and numerical simulations (e.g., Balbus& Hawley 1991, 1992; Hawley & Balbus 1991, 1992; Hawleyet al. 1995; Hawley 2000; Fromang & Stone 2009; Lesur &Longaretti 2009; Maheswaran & Cassinelli 2009).

The goals of the MRI simulations are to understand better thephysics of angular momentum transport through disks and diskstructure. However, this paper seeks better insight into whetherand how disk magnetism might be directly detected, with afocus on the Zeeman and Hanle effects for spectral lines. Ignace& Gayley (2008) reported on a simplistic calculation of theZeeman effect and the Hanle effect for a Keplerian disk witha purely toroidal field. Here, application of the Hanle effect toKeplerian disks has been developed in a more complete way.But before discussing its application to the MRI scenario, it isworth commenting on the conclusions of Ignace & Gayley forthe use of the Zeeman effect.

Figure 8. Same cases as shown in Figure 6 except now for a toroidal fieldwith the opposite sense of rotation. Note how the us

ν profiles are essentially thereverse of those shown in Figure 6.

(A color version of this figure is available in the online journal.)

The MRI leads to a field topology that consists of a toroidalcomponent and a randomized component. The toroidal compo-nents exist in mixed polarity, by which it is meant that somesectors run cw and others run ccw. For the weak longitudinalZeeman effect, the Stokes-V flux scales with the net magneticflux associated with a spatial resolution element (e.g., Mathys2002). With bulk motions frequency (or wavelength or velocity)resolution ultimately maps to geometrical zones at the source(an example of this involving the Sobolev approximation wasdetailed by Gayley & Ignace 2010 for spherical winds). The keypoint is that if both the randomized field and the polarity of thetoroidal component changes on small scales, then the flux ofcircularly polarized light arising from the Zeeman effect will bestrongly suppressed owing to little net magnetic flux.

The issue of variations of the field on “small scales” must beevaluated with care. In the Sobolev approximation for purelyKeplerian rotation, we have seen that the isovelocity zones are“loop” structures. These loops extend out to large radius forlow velocity shifts, but they can be quite small at high velocityshifts. The highest velocity shifts degenerate to a pair of pointsfor either limb of the star at the projected stellar equator! Moreimportantly, every isovelocity zone intersects with the innermostradius of the disk. (If the disk extended down to the star, thiswould be the photospheric radius.) A steep power-law densityensures that the bulk of emission or scattered light comes frominner disk radii. As a result, the Zeeman effect would be mostsensitive to a magnetic field at these locations, and thus “smallscales” refers to turnovers in the field direction that are smallcompared to the inner radius of the disk. Consequently, thedetection of the Zeeman effect from a disk with a given spectralresolution and density distribution constrains the characteristicspatial wavelengths at which the field turns over with radius fromthe star, azimuth around the star, and vertical height through thedisk.

A detection of the Zeeman effect in a disk has previouslybeen reported by Donati et al. (2005) in the case of FU Ori.

No. 1, 2010 HANLE EFFECT AS A DIAGNOSTIC OF MAGNETIC FIELDS IN STELLAR ENVELOPES. V. 1047

This is thought to be an accretion disk with a disk wind, asevidenced by some lines showing P Cygni absorption. However,the circular polarization profile is argued as being associatedwith the innermost region of the rotating disk where the fieldis of kilogauss strengths. The detection implies a net magneticflux per spectral resolution element, and thus sets limits on theturnover (or “tangledness”) length scale for the disk field, ifindeed the MRI is operating in this case.

It is interesting to consider signals that could result withthe Hanle effect. The Hanle effect can operate in regionswhere magnetic fields are “tangled” or randomized. This meansthat spatial averages of 〈 �B〉 tend toward zero although 〈B2〉does not, such as is the case for the MRI mechanism. Anextensive literature exists for the Hanle effect with randomfields in applications to solar studies (e.g., Frisch et al. 2009and references therein). Here I simply want to outline some ofthe limiting behavior in applications to disks.

Consider a Keplerian disk that contains everywhere a trulyrandomized magnetic field. If the field is weak at all locations,meaning that B BHan, then of course a qs

ν line profile resultsas in the case of no Hanle effect. But if the field is strong, suchthat the denser regions of the disk are largely saturated, then thebehavior is much different. With different field orientations,one expects that us

ν will yield a null profile, by symmetryconsiderations. However, the qs

ν profile is different.As a specific example, consider the resulting FQ line from an

edge-on disk. Without a field, the polarization at line centerwould be zero, owing to forward scattering of unpolarizedstarlight. With a randomized field, the polarization will stilltend toward zero at line center. In the point star limit, somenet polarization is expected to survive in the line wings. Thispolarization will be significantly reduced in comparison to thezero field case. If E1 were unity, the polarization at the line wingswould be 100%, since the scattering geometry is 90◦. In analogywith considerations of scattering polarization off the solar limb,a reduction in polarization by a factor of five for isotropicallydistributed fields should be expected (see Stenflo 1982).

Now consider the introduction of a sustained toroidal fieldcomponent. Of course, toroidal fields were considered in theprevious section. Now, however, the toroidal field has polarityflips within the disk—Bϕ is alternately cw or ccw at essentiallyrandom points within the isovelocity zones. What this means isthat there is a sign change in the direction of Larmor precessionin the classical picture of a harmonic oscillator. The effect of thisis to drive the us

ν signal to zero faster than if the toroidal fieldhad one sense of polarity. In the saturated limit, the qs

ν profileis unchanged, because the surviving polarized signal does notdepend on polarity at all. This is quite different from the Zeemaneffect that is sensitive to the net magnetic flux. For the Zeemaneffect, the circular polarization will be suppressed when Bϕ

switches polarity on small spatial scales.

4. CONCLUSIONS

Polarized line profile shapes from magnetized Keplerian diskshave been calculated under a number of simplifying assump-tions: the disk is geometrically thin; the scattering lines areoptically thin; primarily simple fields were considered (axial ortoroidal); and no account was taken of photospheric absorptionlines. On the other hand, the model profiles do include finitesource depolarization and the effects of stellar occultation. Thepresentation of results focused on the polarimetric “efficiencies”qs

ν and usν with a description of how to identify the occurrence

of the Hanle effect in scattering lines from disks. In addition, adiscussion was presented for the Hanle effect from a magnetizeddisk in which the MRI mechanism is operating.

One of the main conclusions from this work is that axialand toroidal fields in disks are easily distinguishable throughan analysis of qs

ν–usν figures for the run of Stokes polarizations

across line profiles. Although a usν profile does exist even without

the Hanle effect, owing to stellar occultation, its amplitude isquite small. The strongest us

ν profiles result when much of theinner disk, where most of the scattered light is produced, hasvalues of B/BHan of order a few. If the inner disk is mostly inthe saturated limit of the Hanle effect, the us

ν profile becomes anull profile for both axial and toroidal fields.

One should bear in mind that the polarized efficienciesare upper limits to the polarizations that would actually bemeasured, since the efficiencies are with reference to thescattered light only and do not take account of the direct starlightfrom the system. Since this starlight is expected to be largelyunpolarized, its contribution acts to “dilute” the polarizationsubstantially below the efficiency levels reported here. Forexample, if the line is relatively weak at 20% of the continuumlevel, then the expected measured polarizations would havefractional values at about 1/5 of the efficiency values, resultingin line polarizations of around 1%–2% for qs

ν and 0.5% or lessfor us

ν .As percent polarizations, such values would appear to be

easily measurable, but in practice there are several challenges.First, a spectral resolution yielding several points across thepolarized profile is needed. Circumstellar disks have rotationspeeds of order 500 km s−1. A requisite velocity resolution ofperhaps 50 km s−1 would then be needed, implying a resolvingpower of λ/Δλ ≈ 6000. Harrington & Kuhn (2009a) havedemonstrated that such resolving powers can be achieved inspectropolarimetry; however, the next requirement is that offinding suitable scattering lines.

The very interesting effects seen in the sample of Harring-ton & Kuhn (2009a) exploit a relatively new effect of en-hanced polarization that is coincident with regions of higherline absorption, a consequence of optical pumping effects. Forthe Hanle effect, or even for non-magnetic resonance scatter-ing, lines that are predominantly scattering are needed. Forhot star disks, such as the disks of Be stars, resonance scat-tering lines are generally to be found at UV wavelengths.This requires space-borne spectropolarimeters. Although theWisconsin Ultraviolet Photo-Polarimeter Experiment (WUPPE)obtained exciting new results from UV polarimetry, its resolvingpower was only about 200 (Nordsieck et al. 1994). A new UVspectropolarimeter called the Far-Ultraviolet SpectroPolarime-ter (FUSP) is a sounding rocket payload that will have a resolv-ing power of about 1800 (Nordsieck 1999). Although possiblytoo low for circumstellar disks, it will be suitable for study-ing scattering line polarizations from high velocity stellar windsources.

As a matter of practical analysis, how should spectropolari-metric data best be managed to measure a Hanle effect? Plottingthe velocity shifted polarizations in the qs

ν–usν space appears to

be most promising. The sequencing of the analysis for scatteringlines from disk sources might proceed as follows.

1. Subtract off the continuum polarization in the vicinity ofthe line of interest. This will counter the effects of bothinterstellar polarization and any other broadband sourcepolarization, such as may arise from Thomson scattering inthe disk.

1048 IGNACE Vol. 725

2. Plot the Q and U line fluxes, not fractional or percentpolarizations that would be derived through normalizationby the total I-flux. The reason for plotting polarized fluxesis that many common UV resonance lines are doublets (e.g.,N v, Si iv, and C iv UV resonance doublets). The shorterwavelength component (“blue”) has E1 = 0.5, and the longwavelength component (“red”) has E1 = 0 (e.g., Table 1 ofPaper II). If the lines are thin, then the polarized flux fromthe red line will not be influenced by the blue one. However,if the lines are sufficiently closely spaced, then FI will bea blend, and normalization by that blend would artificiallyskew the Q − U figure shape, making interpretation moredifficult. If the doublet components are well separated, thenrelative polarizations could also be used in what follows.

3. Determine whether the resultant figure for the line polar-ization shows any axis of symmetry. If so, then either (a)there is no Hanle effect or (b) there is a Hanle effect with adominant toroidal component. For an axis of symmetry, arotation of the figure from observer Q − U axes to a sourceset of axes Q′ – U ′ could be accomplished with a Muellerrotation matrix. After rectifying the figure in this way, therelative amplitude of FU should be compared to FQ. IfFU FQ, then there is likely little Hanle effect or the diskis in the saturated limit of the Hanle effect. If FU ∼ FQ,then a Hanle effect is required with b0 ≈ few in the diskwhere the bulk of scattered light is produced. This meansB ≈ BHan.

4. If there is no symmetry axis to the Q − U figure, thena Hanle effect involving an axial field is most likely theculprit. Recall that a radial field could be present, but thiswould give no Hanle effect by itself.

5. For identifying a field distribution arising from MRI, thingsare more complicated. If the Hanle effect is in operation,then the Stokes-U flux is likely driven to zero, even ifb0 ∼ 1. A netFQ profile should survive; however, it may notbe symmetric. Using an oversimplification, each isovelocityzone can be considered to have a different effective b0 value.This is already the case in the pure axial or toroidal field casethat is axisymmetric, but the variation of the effective b0with velocity shift is monotonic (in a flux-weighted sense).With MRI, one expects non-monotonicity from fluctuationsof the field strength. This amounts to the introduction ofamplitude fluctuations across the polarized line.

There are complications to the above approach. For example,in the case of the Be star disks, there is substantial evidencefor one-armed spiral density wave patterns that make the diskdensity non-axisymmetric (e.g., Okazaki 1997; Papaloizou &Savonije 2006). For thin Thomson scattering, there is littleobservational consequence of this effect which is mainly aredistribution of disk scatterers in a point antisymmetric pattern(e.g., Ignace 2000b). However, with resolved line profiles, thenon-axisymmetric density distribution will produce asymmetryin the qs

ν profile and will be a new source of net usν signal,

also asymmetric. Certainly it will be important to model the FI

profile self-consistently along with the polarized profiles.Note that there may be some concern about the choice adopted

for the radial distribution of the field strength through thedisk. The choice of | �B| ∝ �−1 is perhaps the most shallowdistribution that one could expect. A steeper gradient of thefield strength will restrict the operation of the Hanle effect toa more restricted range of radii in the disk. If the interval inradius where b0 � 1 becomes narrow in relation to where most

of the scattered light is produced, then the Hanle effect will beirrelevant for the observed polarization.

What are the next steps in formulating better diagnosticsof the density and magnetic field structure in disks? Most ofthe work considered in this series of papers has focused onoptically thin lines in an attempt to gain a better understandingof how the Hanle effect influences line polarizations that formin circumstellar media. In Paper IV, the issue of optical deptheffects for P Cygni lines from stellar winds were considered bytreating regions of high optical as contributing no polarizationat all. Although simplistic, insight was gained into how opticaldepth effects provide an additional spatial “filter” in terms ofwhere bulk of line polarization will be produced. Naturally,rigorous radiative transfer and a more realistic disk model(i.e., not simply planar) are desperately needed to extend theconsiderations of this paper. Even with thin line scattering, itwould be useful to explore how non-axisymmetric disk models,such as the one-armed spiral density wave pattern for Be disks,will modify the line polarization. Moreover, winds driven offmagnetized disks (e.g., Konigl 1989; Proga et al. 1998, 2000)have been ignored entirely in this work. These are all newcalculations that will need to be considered in the future.

The author is indebted to the anonymous referee whosecomments have improved this manuscript. Appreciation isexpressed to Joe Cassinelli, Ken Gayley, and Gary Henson forengaging conversations about line polarizations and the Zeemanand Hanle effects. This work was supported by a grant from theNational Science Foundation (AST-0807664).

APPENDIX

SPECIAL CASES FOR THE LINE PROFILE SHAPES

In the appendix, a number of instructional cases are consid-ered for the polarized line profile shapes from a Keplerian diskwhen the illuminating star is treated as a point source. Thismeans that both stellar occultation and the finite star depolar-ization factor are ignored. A consequence of this approximationis that a non-zero us

ν profile can only result from the Hanle ef-fect. Before considering polarized line profiles, Stokes-I profileshapes are derived for the case of isotropic scattering. Thesesolutions form the base emissivity function from which the po-larized lines are constructed.

A.1. The Case of Isotropic Scattering

Isotropic scattering corresponds to E1 = 0, and it means thereis no polarization from resonance line scattering. Of course,that also means there is no Hanle effect, regardless of the fieldstrength.

Even though there is no Hanle effect, the isotropic caseis useful to explore as a reference for the production of theStokes-I line shape. The integrand for the line emission as afunction of velocity shift in the observed line represents thecontributions by the disk density and the Sobolev effect for theprofile shape. Allowing for E1 �= 0 and the Hanle effect simplyrepresents new weighting functions for non-isotropic scatteringthat multiply the integrand from the isotropic case.

The flux of line emission at normalized Doppler shift wz is

F isoν = τ0 F0 × 2

∫ 1

t0

tm−1

√t − t0

dt, (B1)

No. 1, 2010 HANLE EFFECT AS A DIAGNOSTIC OF MAGNETIC FIELDS IN STELLAR ENVELOPES. V. 1049

Figure 9. Analytic line profile shapes from isotropic scattering for the density exponent parameter m = 3 at left and m = 4 at right. The flux profiles are shown attop and qs

ν profiles at bottom. The different curves are for different viewing inclinations, with sin2 i = 0.0, 0.25, 0.5, 0.75, and 1.0. In each case E1 = 0.5. The fluxprofiles are normalized to the total line flux in the isotropic case. The normalized isotropic line is plotted in light blue in the upper panels.

(A color version of this figure is available in the online journal.)

where the factor of 2 arises from the back-front symmetry of theintegration along the isovelocity zone. As a reminder, t = �−1

and t0 = w2z .

Again, the preceding expression is only valid in the point starapproximation. The power-law exponent m is from the surfacedensity distribution that is assumed to be a power law of theform Σ = Σ0 �−m. This formulation leads to symmetric double-peaked line profile shapes for m > 2. Larger values of m resultin profiles that have more pronounced double-horns at greatervelocity shifts from line center.

For integer values of m, the integral is analytic, and solutionsfor m = 3 and m = 4 are given here by way of example. Form = 3 the result is

F isoν (3) = τ0 F0 × 2

∫ 1

t0

t2

√t − t0

dt,

= τ0 F0 × 4

15

(3 + 4w2

z + 8w4z

) √1 − w2

z , (B2)

and for m = 4,

F isoν (4) = τ0 F0 × 2

∫ 1

t0

t3

√t − t0

dt,

= τ0 F0 × 4

7

√1 − w2

z +6

7w2

z F isoν (3). (B3)

Note that increasing values of m also result in profile shapes oflower amplitude. In fact, it is possible to solve for the integratedlight from a resonance scattering line in some special cases.Defining

Ftot(m) =∫ +1

−1F iso

ν dwz, (B4)

a few selected results are Ftot(1) = 4π · τ0F0, Ftot(2) =5π/3 · τ0 F0, Ftot(3) = 2π/3 · τ0 F0, and Ftot(4) = π/2 · τ0 F0.

It would appear that with τ0 = constant, different values ofFtot, and thus lines of different brightness levels, could result.

This seems counterintuitive if lines characterized by differentvalues of m have the same optical depth. However, τ0 dependsonly on the density at the inner portion of the disk Σ0, not thevalue of m. Thus, when the line is optically thin, the appropriateoptical depth to use is one that is angle averaged, similarin spirit to τ in Brown & McLean (1977) for optically thinelectron scattering. Hence use of a new optical depth parameterTl = τ0 × τ0F0/Ftot(m) would ensure that lines of different mvalues with isotropic scattering will have the same total lineemission (i.e., “area under the curve”) even though they havedifferent profile shapes.

A.2. The Case of Resonance Scattering with B = 0

For resonance line scattering with E1 �= 0 but with B = 0everywhere, the vector scattering function �h greatly simplifies.Following Paper II, we have that δ = ϕ, α2 = 0, C = cos 2ϕ,D = sin 2ϕ, and ψs = 0. The components of the phase functionbecome

�h =

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

hI = 1 +3

8E1

[1

3(1 − 3 cos2 i) + sin2 i cos 2ϕ

]

hQ = 3

8E1 [sin2 i − (1 + cos2 i) cos 2ϕ]

hU = 3

8E1 cos i sin 2ϕ.

(C1)

The Stokes flux of scattered light is

�F sc = τ0 F0 × 2∫ 1

t0

tm−1

√t − t0

�h dt. (C2)

Note that care must be taken in dealing with terms that are oddand even in ϕ between angles of 0 and π . For example, hUis odd around the loop, ensuring that FU = 0. Accountingfor the odd/even terms, and using the fact that sin2 ϕ =t0/t = w2

z/t , solutions for the scattered flux in Stokes-I and

1050 IGNACE Vol. 725

Stokes-Q are

FIsc =

(1 +

1

8E1 − 3

8E1 cos 2i

)F iso

ν (m)

− 3

4E1 sin2 i t0 F iso

ν (m − 1), (C3)

FQsc = − 3

4E1 cos2 i F iso

ν (m)

+3

4E1 (1 + cos2 i) t0 F iso

ν (m − 1). (C4)

Note at the extrema of the line wings, F isoν (m) = F iso

ν (m′) andt0 = 1, and qs

ν = 3E1/(4 − E1) always.The resultant polarization across the line is entirely in Stokes-

Q when the observer’s reference axes are aligned with thesymmetry axis of the star. With E1 = 0.5, Figure 9 shows a plotof F sc

I for m = 3 (left) and m = 4 (right) at different viewinginclination angles of i = 0◦, 30◦, 45◦, 60◦, and 90◦ along thetop panels. The profiles are normalized with respect to the totalemission produced if the line had been isotropically scattering.At bottom is the relative fractional polarization qs

ν = F scQ/F sc

I

for m = 3 and m = 4 for the same viewing inclinations. Notethat at the edges of the line, qs

ν = 3/7 for E1 = 0.5, independentof the viewing inclination, as expected under the point starapproximation.

The goal here is to illustrate the polarimetric “efficiency.” Theactual measured fractional polarization would be much smallerowing to dilution by direct starlight. These efficiency curves arerelatively smooth functions of velocity shift. This smoothnessis partly due to the fact that dipole scattering is a fairly slowlyvarying function of location around the disk and also becauseisovelocity loops sample a range of scattering angles.

A.3. The Case of an Axial Field B = BZ∗

For an axial magnetic field with �B = BZ∗ (� ) Z∗, we havethat ψs = 0 and θs = i. The scattering phase functions are quitesimilar to the zero field case, except that now the Hanle effectappears in factors in the functions C and D. Using Paper II, thephase functions are given by

�h =

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

hI = 1 +3

8E1

[1

3(1 − 3 cos2 i) + sin2 i C(�,ϕ)

]

hQ = 3

8E1[sin2 i − (1 + cos2 i) C(�,ϕ)]

hU = 3

4E1 cos i D(�,ϕ),

(D1)

where

C(�,ϕ) = cos2 α2 cos 2ϕ − 1

2sin 2α2 sin 2ϕ (D2)

D(�,ϕ) = − cos2 α2 sin 2ϕ − 1

2sin 2α2 cos 2ϕ, (D3)

where

cos2 α2 = 1

1 + b20 t2

, (D4)

and

cos α2 sin α2 = b0 t

1 + b20 t2

. (D5)

Solutions for the vector Stokes flux are no longer analytic.With the Hanle effect, FU �= 0 except for B = 0 or in thesaturated limit. In the latter case of b0 � 1 everywhere, ananalytic solution can be obtained, which is given by

F scI =

(1 +

1

8E1 − 3

8E1 cos 2i

)F iso

ν (m) (D6)

F scQ = 3

4E1 sin2 i F iso

ν (m). (D7)

Note that in this limit the relative polarization becomes

qsν = F sc

Q/F scI = 3E1 sin2 i

8 + E1 (1 − 3 cos2 i), (D8)

which is a constant across the profile and a function of viewinginclination only. This means that the polarized profile is flat-topped.

Figure 10 displays profiles for the axial field case with E1 =0.5 at a fixed value of sin2 i = 0.4 but with different field valuesat the inner disk radius of � = 1, of log b0 = −2,−1, 0, +2, +4to achieve a large dynamic range in Hanle ratios throughout thedisk. As b0 increases, the location where B = BHan movesoutward to � = b0. The upper left panel in Figure 8 showsnormalized profiles of F sc

I ; the lower panels show the fractionalpolarizations qs

ν and usν ; and the upper right shows the qs

ν–usν

shapes. The color sequencing is the same as in Figure 5. In largepart the effect of an axial field is to rotate and foreshorten theQ − U segments relative to the zero field case.

A.4. The Case of a Toroidal Field B = Bϕ

For a toroidal magnetic field with �B = Bϕ ϕ, the sphericalgeometry for the scattering problem is moderately complex. Itis not possible to write down simple complete expressions forthe phase scattering functions. But as demonstrated in Paper II,there are still some special cases that are analytic. For example,in the saturated limit, F sc

U = 0, and the I and Q fluxes become

F scI =

(1 +

1

8E1

)F iso

ν (m) − 3

8E1 sin2 i t0 F iso

ν (m − 1)

(E1)

F scQ = −3

8E1

[F iso

ν (m) − (1 + cos2 i) t0 F isoν (m − 1)

].

(E2)

For a disk viewed edge-on, solutions for the Stokes fluxescannot be derived analytically; however, the scattering functionssimplify to

�h =

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

hI = 1 +3

8E1 (1 − 3 sin2 ϕ) +

3

8E1 cos2 ϕ cos2 α2,

hQ = −3

8E1 cos2 ϕ +

3

8E1 (1 + sin2 ϕ) cos2 α2,

hU = −3

8E1 sin ϕ sin 2α2.

(E3)Examples of scattering and polarized profiles are shown inFigure 11 as well as Q − U plots across the polarized profiles.

No. 1, 2010 HANLE EFFECT AS A DIAGNOSTIC OF MAGNETIC FIELDS IN STELLAR ENVELOPES. V. 1051

Figure 10. Similar to Figure 5, here for the case of an axial field in the point star approximation.

(A color version of this figure is available in the online journal.)

Figure 11. Similar to Figure 10, here for the case of a toroidal field in the point star approximation.

(A color version of this figure is available in the online journal.)

1052 IGNACE Vol. 725

In the previous section, sin2 i = 0.4 was used for an axial fieldto give a significant signal in us

ν . For this toroidal field case, anedge-on disk with sin2 i = 1.0 was used. The style of this figureis the same as in Figure 10. A major distinctive in relation to adisk with an axial field is that a toroidal field leads to us

ν profilesthat are antisymmetric instead of symmetric.

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