Radiative Transfer Solution

beray works in the coordinates defined by the plane of the sky, which will

be taken to define the xsky−ysky plane. The zsky co-ordinate, directed at the observer located at an infinite distance , then defines the rays along which the equation of radiative transfer needs to be solved. Before this can be done, however, the intersections and boundary conditions for each ray are needed. Figure 3.4 shows the relationship between the coordinate system defined by the sky plane, (xsky,ysky,zsky), that of the disk and star (xsky,ysky,zsky) and the inclination angle i.

The bulk of the changes to beray involve changing the routines which

determine if a ray terminates on the stellar surface (because of its distorted shape) and the local conditions at the end of such rays (as Teff and logg now vary over the surface). Consider the ray defined by (xsky, ysky).

In the case of a spherical star, it is easy to determine if this ray termi- nates on the stellar surface. If x2

sky +ysky2 is less the stellar radius, then the ray intersects the star. If it is larger, the ray misses the star, and if it is equal, the ray is tangent to the star. The intersection point can be found by zsky =

p

r2

y

Figure 3.4: Diagram showing the relationship between the coordinate systems (xsky,ysky,zsky), (xsky,ysky,zsky) and the inclination angle,i. xsky/xstar is coming out of the page.

set the stellar intensity correctly: the surface inclination, µ, and the local projected surface velocity, vz. These are easily found in the spherical case as

vz =−(xstar/rstar)vsini andµ=zstar/rstar, where vsiniis the projected equa- torial velocity of the star. Gravity and temperatures are also need but these are constant over the stellar surface.

In the case of a rapidly rotating, distorted star, all of the above quantities are still needed, but are more complex to compute. For example, even finding if a ray terminates on the star requires many more calculations. There are a few are simple cases:

• When the radius on the sky, ie. in the xsky−ysky plane, r2D sky given by q

x2

sky+ysky2 is more than requat, the equatorial radius, the ray cannot intersect the star as it is beyond the largest possible radius of the star.

Ysky Xsky=Xstar Xsky p Zstar Zsky Ysky edge Xsky Zsky Ystar Xsky Ysky Ysky p = Rp Re p sky X Ysky p p sky X Equator Pole θ edge Zstar Ysky = A: Sky View B: Cut Through C: Cut Through

Figure 3.5: Figure A shows a top down view of the star withzsky pointing out of the page. Figure B is a cut though the star at some ysky. The location of

θ1 _{has been highlighted. Figure C is a cut though the star at some x}

sky, the overhanging and underhanging regions at the edge of the star.

In this case the (stellar co-ordinates) of the intersection are

sinθstar = cos(3 arccos(ω shape frac r2D sky −3rp )−4π) ω_fracshape (3.19) cosθstar = p 1−sin2θstar (3.20)

zstar =rp x(ωfracshape,sinθstar) cosθstar (3.21)

However, unlike these simple cases, the general cases have no algebraic so- lution and the intersection must be determined numerically. A straightforward and stable method for root finding is the bisection method. It is of sufficient speed for our purposes because with each step, the error in the approximation is halved. However to implement the bisection method, a bound on the root is required, namely two points along the ray, one “above” and outside the star and one inside. The search for the stellar surface occurs along a ray looking

4. Find the stellar radius, R(θguess_star ).

5. If R(θguess_star )2 _{> R}guess2 _{then the guess position is within the star. If}

R(θ_starguess)2 _{< R}guess2_{, the guess position is outside the star.}

Choosing a location outside the star is simple by taking z_skyguess = 2rp. The difficulty is finding a point on the ray within the star because this only occurs if there is an intersection. For the bulk of the rays which intersect the star (all them when i = 0 or i = 90), the zsky = 0 plane is within the star while only the edges of the star hang over or under thezsky = 0 plane, the overhang- ing and underhanging regions,– see illustration C of Figure 3.5. For a given ray (xsky, ysky), if the point (xsky, ysky,0) is within the star, the search occurs between zsky = 0 and zsky = 2rp.

If the point (xsky, ysky,0) is not within the star the ray may intersect the star in the over hang or not intersect it at all. For these rays it must be determined if an intersection occurs. The edge of the star y_skyedge and its height above (or below) thezsky = 0 plane is found for the ray. This search occurs in the xsky-zsky plane though the coordinate θstar betweenθ1, the lowest possible value ofθstar for a particular xsky, given by

sin(θ1) = cos(3 arccos(ω shape frac |xstar| −3rp )−4π) ωshape_frac (3.22)

(as shown in Figure 3.5 B), and θ2 which is slightly greater than π/2. The bisection method is used to find the location where µis zero, which indicates the surface is tangent to the zsky-axis and thus the edge of the star for that

xstar. For each θguessstar , the remaining coordinates are determined by:

φ(θ_starguess) = arccos( xstar

R(θguess_star ) sin(θguess_star ), (3.23)

y(θ_starguess) =R(θguess_star ) sin(θ_starguess) sin(φ(θ_starguess)), (3.24) and

z(θguess_star ) = R(θ_starguess) cos(θguess_star ). (3.25) Once yedge_sky has been found if|ysky|< yedge_sky the ray intersects the star. For the portion of the star which over-hangs or under-hangs the zsky = 0 plane, a new lower bound is needed. zsky is found using the bisection method betweenz_skyedge and 2rp if ysky <0 or between −zskyedge and 0 if ysky >0.

Once an intersection is located, it is straightforward to calculate the quan- tities on the stellar surface required to implement the boundary condition for the integration of the transfer equation:

R =px2

star+ystar2 +zstar2 (3.26)

~g =−GM R2 ( xstar R , ystar R , zstar R ) + (Ωshape 2

xstar,Ωshape2ystar,0) (3.27)

µ= ~g |~g| ·zˆsky = (gx, gy, gz) p g2 x+gy2+gz2 ·(0,−sin(i),cos(i)) (3.28) log|~g|= log q g2 x+g2y+gz2 (3.29) Tef f = (Cω|~g|) 1 4 (3.30)

| | g_x+g_y+g_z ~ g2 =−GM R2 ( xstar R , ystar R , zstar R ) (3.35)

+(Ωtemp2xstar,Ωtemp2ystar,0) (3.36) log|~g|= log q g22 x+g22y +g22z (3.37) Tef f = (Cω|g~2|) 1 4 (3.38)

vzstar = ˆzsky ·~vstar =−sin(i)xstarΩtemp. (3.39)

textcolorsarahIn either case, full gravitational darkening or spherical gravita- tional darkening, when rays do not intersect the star, the boundary condition for the transfer equations is taken to beI = 0, reflecting no incident radiation on the stellar disk.

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The Thermal Structure of

Gravitationally-Darkened Disks

A version of this chapter has been published as McGill, M.A., Sigut, T.A.A., & Jones, C.E. 2011, ApJ 743, 111-115.

A classical Be-star is a non-supergiant B star that possesses a gaseous equatorial disk (Porter & Rivinus, 2003). This circumstellar material produces an emission line spectrum including a prominent Hα emission line. Some Be stars, for example γ Cassiopeia, produce a rich emission line spectrum with many hydrogen lines and some metal lines. Be star disks also produce a continuum excess due to bound-free and free-free emission. This excess has been observed at a variety of wavelengths from the visible to the radio (Cot´e & Waters, 1987; Waters & Cot´e, 1987). Linear polarization of up to 2% has been observed in the continuum emission and is caused by electron scattering in the non-spherical circumstellar envelope (McLean & Brown, 1978; Poeckert et al., 1979). Be star disks have been partially resolved by long-

baseline interferometric measurements, initially in the radio (Doughterty & Taylor, 1992), and later at infrared and optical wavelengths (Stee & Bittar, 2001; Tycner, 2004).

The stellar absorption lines of Be stars indicate rapid rotation of the central star, and this property is thought to play an important role in the release of material into the disk (Porter, 1996). There is strong evidence that Be star disks are in Keplerian rotation (Hummel & Vrancken, 2000; Oudmaijer et al., 2008); however, it is still uncertain if the equatorial rotation speeds of Be stars are close to “critical”, i.e. close to the Keplerian orbital speed at the inner edge of the disk (Cranmer, 2005). In addition, another mechanism must somehow allow for the wide range of behaviours observed in Be stars: some systems have exhibited essentially stable Hαemission for as long as they have been observed, while others are highly variable with timescales ranging from less than a day to decades (Porter & Rivinus, 2003).

A rapidly rotating star is changed in two ways: the stellar surface is dis- torted with the equatorial radius becoming larger than the polar radius, and the stellar surface temperature acquires a dependence on stellar latitude with cooler gas at the equator compared to the hotter pole. These phenomena to- gether are commonly called gravitational darkening. The rotational distortion and surface intensity variation with latitude have direct observational support from interferometric observations, including the stars Achernar (Domiciano de Souza et al., 2003) and Altair (Monnier et al., 2007).

Because the equatorial region of the photosphere is both the fastest rotating and the dimmest, it can be difficult to detect this maximally Doppler-shifted region in integrated light. This problem can be compounded by the obscura- tion of the equatorial regions by disk material. As a result, the bounds on Be star rotation rates remain contentious (Chauville et al., 2001; Townsend et al.,

tempts to match the observed distributions using a parametrized distribution of equatorial velocities. He concludes that early-type Be stars, O7–B3.5, must rotate at rates significantly less then their critical speeds (peaking at 40-60%) while later type Be stars, B3.5–A0, rotate much closer to their critical speeds (peaking at 70-90%).

Previously, the effects of gravitational darkening have been included in disk models for the stars Achernar, or α Eridani, (Carciofi et al., 2008) and

ζ Tauri (Carciofi et al., 2009). However, the rotating star was assumed to be a spheroid rather than following a Roche profile (see Section 4.1). In the literature, gravitational darkening has been included only as a fit parameter in models of individual stars. A comprehensive study of the how the inclusion of gravitational darkening affects models of Be star disks has not yet been performed.

In this paper, we systematically examine the effect of gravitational darken- ing on the thermal structure of a circumstellar disk. We restrict our investiga- tion to changes in the energy reaching the disk and the resulting changes in the temperature distribution. Section 4.1 outlines the basic theory behind gravi- tational darkening and also gives computational details. Section 4.2 describes the stellar and disk models chosen for this investigation. Results are presented in Section 4.3 where the consequences of rotation on the energy reaching the circumstellar disk (Section 4.3.1) and the corresponding changes in the disk

temperature structure (Section 4.3.2) are investigated. In Section 4.4, we ex- amine the two principle ingredients of gravitational darkening (temperature variation with latitude and distortion of the stellar surface) separately in or- der to gauge their relative importance. We also evaluate the effectiveness of the simpler spherical gravitational darkening approximation in which the shape distortion produced by rapid rotation is ignored. Section 4.5 gives the conclusions, and in Appendix A, we discuss the subtle aspects of how varying the equatorial radius can change the inherent properties of our disk models.

4.1

Theory of Gravitational Darkening