Download The solution of the radiative transfer equation in axial symmetry

The solution of the radiative transfer equation
in axial symmetry
Daniela Korčáková and Jiřı́ Kubát
Astronomický ústav AV ČR, CZ-251 65 Ondřejov, Czech Republic
[email protected], [email protected]
Summary. We present a new method for the solution of the radiative transfer
equation in axial symmetry. The three dimensional problem is reduced to a two dimensional one using a set of independent planes. Our extended short characteristics
method is applied to the radiative transfer problem in each of the planes using polar
coordinates. The method is able to handle a velocity field, which is accounted for
using the Doppler shift between neighboring cells.
Our method is suitable for studying the stellar wind with rotation, where the
symmetry of the problem is naturally axial. Results of test calculations are shown
for the case of a stellar wind of a typical B star.
1 Introduction
Problems studied in the astrophysical radiative transfer are rather diverse.
There exist objects, like planetary nebulae, where the density is very low and
the medium is very transparent. On the other hand, some parts of galactic
nebulae can be opaque.
For the particular case of stars, their most important part is the stellar
atmosphere – a region where an optically thick star fades into an optically
thin interstellar medium. The values of physical quantities change by several
orders of magnitude there. The transfer problem must be solved from an optical depth of 106 to 10−6 . The geometrical scale of stellar atmospheres is
usually much larger than the mean free photon path and also much larger
than the thermalization length. As a consequence, long distance interaction
by radiation plays an extremely important role in the physical description
of the medium. These facts introduce problems. Some methods fail under
these conditions and become numerically unstable or they converge to the
wrong solution. Due to other inherent complexities (effects of a huge number of lines, absence of a thermodynamic equilibrium, etc. – see Kubát, this
volume), stellar atmospheres are usually being studied using one-dimensional
radiative transfer. However, for studying the various effects we must solve
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Daniela Korčáková and Jiřı́ Kubát
the radiative transfer equation (RTE) in a more general geometry than planeparallel or spherically symmetric ones. There are not too many methods available for solving the multidimensional problem. For optically thin regions, the
Monte Carlo method [21] may be used. On the other hand, in the optically
thick regions it’s possible to calculate using a method which uses the diffusion approximation [10]. Both of these methods are inappropriate for stellar
atmospheres.
The classical way for calculating a solution to this equation in more dimensions is the long characteristic method [2]. This method fully describes
the radiation field, but the necessary computer time is rather long. Due to
this reason, Kunasz & Auer [14] developed the short characteristic method,
which is the best out of the available multidimensional methods. There exist
several methods using short characteristics in a Cartesian grid [6, and references therein], 2D axially symmetric geometry [8], and a general method,
which is also able to solve the transfer equation in spherical and cyllidrical
grids [19, 20]. An efficient approach is using adaptive grids [7, 18]. An excellent
insight into astrophysical multidimensional radiative transfer can be achieved
from a paper of Auer [1].
Another possibility is to apply the finite element method, which has been
recently used for the problem of multidimensional radiative transfer by, e.g.,
Richling et al. [17]. However, the finite element method is not used very often due to its convergence problems. This annoyance has been overcome by
Dykema et al. [5], who used a modification of the finite element method,
namely the discontinuous finite element method.
For the study of a stellar wind, accretion discs, or stellar rotation we need
such a method for calculating a solution of the radiative transfer equation,
which is able to work in a more general geometry than the plane-parallel or
spherically symmetric ones. But, it isn’t necessary to include the whole three
dimensional space.
In this paper, we describe a new method for calculating a solution to
the radiative transfer problem in axial symmetry. The basic idea behind our
method is a solution of the transfer problem in separated planes. In these
planes, a combination of long and short characteristic methods is used. The
results are more accurate than in the plane-parallel and spherically symmetric
cases, and the calculation is faster than for the complete 3D problem. In
Section 3 we test this method for the case of a stellar wind of a main sequence
B star.
2 Technique
The basic idea of this method is the calculation of a solution of the radiative
transfer problem not in the whole star, but in separated planes intersecting
the star.
The solution of the radiative transfer equation in axial symmetry
3
Let us consider the spherical coordinate system (r, θ, φ). The axis of
symmetry is around θ = 0. We introduce the discretization of the radial
distance r and angle θ. Due to axial symmetry, the physical quantities don’t
vary with angle φ. The grid is chosen to give the best description of the
system. For example, for the study of stellar winds, the grid of angles θ can be
equidistant, but for accretion discs it must be finer near the equatorial plane.
We choose the grid in radial distances to be very similar to a 1D problem,
we want the points to be equidistant in the logarithm of the optical depth
scale. A good choice is about 5 points per decade. We assume the opacity and
emissivity of the stellar material to be known in the grid points. To reduce the
3D problem to 2D, we will not solve radiative transfer equation in this grid,
but in a set of “longitudinal” planes, intersecting the star parallel to the plane
φ = 0 (see Fig. 1). In every longitudinal plane we choose the polar coordinate
θ
r
φ
Fig. 1. Scheme of the set of longitudinal planes.
system and define a grid of concentric circles and radial lines. The grid of
concentric circles corresponds with a 3D grid in the planes, which intersect
the region of validity of a lower boundary condition. But, we must be careful in
other planes, where the grid chosen must imply the geometry of the problem
as well as the velocity field. We interpolate the opacity, emissivity, and the
source function to the new coordinate system. We solve the radiative transfer
equation in every longitudinal plane independently.
First, we solve the radiative transfer equation using the boundary condition at the end of the stellar atmosphere (the place where the star ends and
the interstellar medium begins) in the given longitudinal plane. Once the radiation field in the downward direction (towards stellar center) is known, we
can continue with the solution in the opposite direction – from the center of
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Daniela Korčáková and Jiřı́ Kubát
the star towards the upper boundary. We can assume the diffusion approximation at the inner boundary of the longitudinal planes, which intersect the
stellar center. In other planes (which do not intersect the stellar center), we
adopt as the lower boundary condition the intensity taken from the previously
calculated solution from the outer boundary to the center of the star.
To obtain the whole radiation field we take advantage of the symmetry of
the problem. We know the radiation field at the grid points in all directions
lying within the longitudinal planes. We choose the main plane to be the one
which intersects the stellar center. We obtain the radiation field in other directions by rotating the remaining chosen planes around the axis of symmetry
θ = 0. For the sake of this, we must choose the grid points in the planes at
the same distance from the equator (see Fig. 2).
β
β
Fig. 2. Scheme for the calculation of the whole radiation field.
The solution from the outer boundary to the central regions in the
longitudinal plane
The solution starts at the outer boundary (stellar surface), where we know the
boundary condition. In each grid point we choose three rays per quadrant (see
Fig. 3). Along these rays we solve the transfer equation. The angle distribution
of these rays may be the same as in the plane-parallel geometry, where the
angles are chosen to be the roots of Legendre polynomials in the interval (0, 1)
(µ = 0.8872983346, 0.5, 0.1127016654) to ensure better numerical accuracy of
angle integration [16, section 4.5]. We measure these angles from the normal
to the grid circle at the given point. As one can see in Fig. 3, the rays end
at the next grid circle. This means that it is possible that the rays intersect
some grid lines. The number of rays at a given point is sufficient, because the
The solution of the radiative transfer equation in axial symmetry
5
downward solution
radial grid line
zone
grid circle
Fig. 3. The scheme for the solution of the radiative transfer in the longitudinal
plane from the upper boundary to the central regions.
whole radiation field will be obtained by summing the information from all
longitudinal planes (see Fig. 2). So, the space description of the radiation field
is sufficient.
The rays connect grid circles with the possibility of intersecting with more
radial grid lines. We perform a linear interpolation of the source function
and opacity to obtain their values at points B and C and of the incoming
intensity for the value at point A (see Fig. 4). The optical depth difference
D
∆ s CD
∆τ CD
C
∆τ BC
B
A
∆ s BC
∆ s AB
∆τ AB
Fig. 4. The scheme for the integration along the ray.
∆τ is calculated along the ray between the individual intersection points (AB,
BC, and CD). We assume a linear dependence of the source function on
the optical depth between the intersection points. We solve the equation of
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Daniela Korčáková and Jiřı́ Kubát
radiative transfer between all intersection points along the ray. For the interval
AB the solution is
Z ∆τ(AB)
I(B) = I(A) e−∆τ(AB) +
S(t)e[−(∆τ(AB) −t)] dt.
(1)
0
The integration along the ray is performed in more steps due to two reasons.
First, in the nonorthogonal grid the parabolic interpolation fails and the linear interpolation between points AD is a better choice. Second, if there is a
velocity field presented, we can assume a constant velocity in the cells and
permit a change of a velocity only at the boundary of the cells. So, it’s possible
to solve the static equation of radiative transfer in the cells and to perform
the Lorentz transformation of frequency and intensity on the boundaries.
The solution from the central regions to the outer boundary in the
longitudinal plane
The upward solution is very similar to the previous step. From Fig. 1, one can
see that there exist planes which don’t intersect the inner boundary region.
For these planes we adopt the intensity calculated from the previous step
(solution from the upper boundary to the stellar center) as the lower boundary
condition. However, the solution of the transfer equation in the central grid
circle must be performed with care. We split these rays into two parts (AB
and BC – see Fig. 5), in which we solve the transfer problem separately. For
C
∆s BC
∆τ
B
BC
∆sAB
A
∆τ AB
Fig. 5. The scheme for solution of the radiative transfer equation in the central
region in the planes, which don’t intersect the region of validity of the diffusion
lower boundary condition.
planes, which intersect the region of validity of the diffusion approximation
(the stellar core), the situation is easier. We simply use the corresponding
lower boundary condition.
The solution of the radiative transfer equation in axial symmetry
7
Further solution is depicted in the Fig. 6. From the grid points we lead
three rays, using the same angles as in the previous case. In the intersec-
Fig. 6. The scheme for the solution from the center of the star to the upper boundary.
tion points we interpolate the opacity and the source function. Between these
points we calculate the optical depth and solve the transfer equation as in the
previous case of downward integration (see Eq. 1).
2.1 Velocity field
Let us consider a velocity field with a velocity gradient small enough to neglect
the aberration. This assumption is valid, if v/c 0.01 [9], [15]. We assume
the velocity field to be constant in every cell. At cell boundaries we perform
the Lorentz transformation of intensity and frequency and solve the static
equation within the cell. To ensure a correct treatment of the line transfer
we have to guarantee the frequency shift due to the motion is smaller than a
quarter of a Doppler halfwidth (see [11]). If it is not, we must make a finer
grid.
3 Test calculations
Tests of the method are performed for a main sequence B star with an effective
temperature of 17 000 K, a surface gravitational acceleration of log g = 4.12,
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Daniela Korčáková and Jiřı́ Kubát
and a radius of 3.26 R . From the hydrostatic spherically symmetric model
of the atmosphere [13] we obtain the state parameters, electron density and
temperature. For simplicity, we consider no incoming radiation at the outer
boundary and a diffusion approximation at the inner boundary.
We adopt the beta law as a velocity field of the stellar wind (see, e. g., [3])
v(r) = v∞
(
1− 1−
vR
v∞
β1
R
r
)β
,
(2)
with the parameter β = 1, the velocity in the photosphere v R = 100 km · s−1 ,
and the terminal velocity v∞ = 1000 km · s−1 . We adopt the Doppler profile
for hydrogen lines.
The results of the calculations are shown in Fig. 7. The Hα line profile
from the stellar wind (dotted line) is compared to the static profile (solid
line). Since we take the input parameters from the hydrostatic model, the
atmosphere is too thin for the P Cygni profile to be seen. For this reason, we
chose a different geometrical depth scale (r(d) = r(d − 1) · 1.0016) with the
same Teff , ne , and velocity at the corresponding grid points. This “extended”
atmosphere line profile is plotted using the dashed line and it has a P Cygni
shape.
1.2
’static’
’v1000’
’PCyg’
1.1
relative flux
1
0.9
0.8
0.7
0.6
0.5
4.564e+14
4.566e+14
4.568e+14
ν
4.57e+14
4.572e+14
Fig. 7. The line profiles obtained using our axially symmetric code. The solid line
indicates the static solution, the dotted one the calculation which includes the stellar
wind, and the dashed one is the P Cygni profile obtained from the extension of the
given atmosphere.
The solution of the radiative transfer equation in axial symmetry
9
3.1 The tests of the grid
The grid tests show more or less a linear dependence of the computing time
on the geometrical depth (see Fig. 8, left panel). The dependence of the computing time on the number of frequency points is shown in the right panel of
the same figure. Here, the fitting function is a polynomial of the third order.
700
1200
600
1000
500
800
t [s]
t [s]
400
600
300
400
200
200
100
0
100
200
300
400
500
600
D
700
800
900
1000
0
0
20
40
60
80
100
N
120
140
160
Fig. 8. The dependence of the computing time on the number of depth points (left
panel ) and on the number of frequency points (right panel ).
4 Conclusions
A new method for a solution of the radiative transfer equation in axial symmetry was presented. The transfer problem is solved at planes intersecting the
star independently. The radiative transfer equation is solved at every plane
using a combination of short and long characteristics methods. This allows
us to take into account the global character of the radiation field and the
necessary computing time is not too long.
The velocity field (with velocity gradients) is taken into account using a
Lorentz transformation of corresponding quantities at cell boundaries, which
also allows the treatment of nonmonotonic velocity fields.
We performed tests for a main sequence B star with an effective temperature of 17 000 K. In Fig. 7, Hα line profiles which include a velocity field
are presented. Since we adopted the input parameters, electron temperature
and density, from the hydrostatic model, the resulting line profiles don’t show
a P Cygni profile. The dependence of the computing time on the number
of geometrical depth points is linear, but the dependence on the number of
frequency points is a polynomial of third order.
This method is useful not only for the study of stellar winds and stellar
rotation, but also for axially symmetric planetary nebulae or accretion disks. It
takes advantage of the symmetry of a problem without the need of calculating
a complete 3D problem.
180
200
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Daniela Korčáková and Jiřı́ Kubát
Acknowledgements The authors would like to thank Adela Kawka for language
corrections. This research has made use of NASA’s Astrophysics Data System.
This work was supported by grants GA ČR 205/01/0656 and 205/04/ P224.
The Astronomical Institute Ondřejov is suported by projects K2043105.
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