Radiative Transfer in Axial Symmetry

Radiative Transfer in Axial Symmetry
Daniela Korčáková
kor@sunstel.asu.cas.cz
Astronomical Institute of the Academy of Sciences, CZ
collaborators:
T. Nagel, K. Werner, V. Suleymanov, V. Votruba,
J. Kubát
picture: http://chandra.harvard.edu/photo

Outline
Description of the method
Method applicability
limb darkening
stellar rotation
stellar wind
discs

description of the method
method preview
– axial symmetry
– LTE, NLTE
– hydrogen
– parallel version (in final test phase)
– input – n e (r, θ), T(r, θ), v(r, θ)
χ(r, θ), S(r, θ)
– output – line profile, intensity map
– Korčáková & Kubát, 2005, A&A 440, 715

asic idea
– solution of the radiative transfer equation in
separate planes
– polar coordinates in every plane
– combination of the short and long
characteristic methods
– velocity field – Lorentz invariance of RTE

longitudinal planes
whole radiation field
rotation
axis

upper boundary condition
radial grid line
grid circle
zone

integration
I (B) = I (A) e −∆τ (AB)
+
∫ ∆τ(AB)
0
S(t)e [−(∆τ (AB)−t)] dt
S(t):
– bound-bound
– bound-free
– free-free
– Thomson
scattering
A
∆s CD
∆τ CD
C
∆τ
B ∆s
BC
∆τ AB
∆
s
AB
BC
D

velocity field
d, th
v = const.
d−1, th−1
∆v
∆v
v = const. v = const.
d−1, th
v = const.
d, th+1
v = const.
d−1, th+1

solution in the central region
C
∆s BC
∆τ
BC
A
B
∆τ
AB
∆s
AB

lower boundary condition

mean intensity
rotation
axis
– the grid is defined
by global
properties
– advantage – better
description in outer
regions, where the
radiation field is
strongly
anisotropic

– disadvantage – not possible to use an area
method
– HEALPix (Hierarchical Equal Area isoLatitude
Pixelization)
Gorski, Hivon, Banday, Wandelt, Hansen, Reinecke,
Bartelmann, 2005, ApJ 622, 759
http://healpix.jpl.nasa.gov/

parallelization
– MPI
– memory × computational time ×
communication
– formal solution of the radiative transfer
equation is splitted into the nodes by lines
– solution of the NLTE rate equations is
distributed to the nodes by frequencies at the
given point
– in final test phase

advantages
– a better description of the global character of
the radiation field than by the short
characteristic method
– not so time consuming as the long
characteristic method
– arbitrary velocity field
disadvantages
– high velocity gradients =⇒ a finer grid
– computing time ∼ f 2 (f – number of frequency
points)

method applicability
– limb darkening
– stellar rotation
– gravity darkening, differential rotation
– stellar wind
– optically thin + optically thick regions
– polar + equatorial regions
– Be, B[e] stars, . . .
– discs
– cataclysmic variables, protostellar discs
– central object + surrounding medium
– protoplanetary nebulae

limb darkening
the specific intensity
4e−05
3e−05
2e−05
1e−05
0
0
1
x [r/R ∗ ]
2
3
4.567e+14
4.566e+14
4.565e+14 ν
4.564e+14

stellar rotation
– rapidly rotating stars
– gravity darkening
– differential rotation

extended rapidly rotating atmosphere
1
0.98
relative flux
0.96
0.94
0.92
0.9
rot−grav−dif
rot+grav−dif
rot−grav+dif
rot+grav+dif
4.56e+14 4.565e+14 4.57e+14 4.575e+14
ν

stellar wind
– optically thick atmospheric regions +
optically thin wind
– polar + equatorial region
– static layers + fast outer region
– stellar wind + rotation
– Be, B[e] stars

– Nv line 1242 Å
Krtička, Korčáková, Kubát, 2008, ASPC 388, 191
1
relative flux
0.9
0.8
0.7
0.6
one−component
four−component
1235 1240 1245 1250
λ[Å]

discs
– protostellar discs
– cataclysmic variables =
central white dwarf + transition region
+ disc + hot corona
– disc – optically thick or thin
– impossible to include a hot spot

technique
– structure, χ and S from AcDc code
Nagel et al., 2004, A&A, 428, 109
– disc = set of concentric rings
– consistent solution of:
– radiative transfer equation
– hydrostatic equation
– energy balance equation
– rate equation
– equation of charge and particle conservation
– 2D grid
– interpolation of S and χ to the new grid
Steffen, 1990, A&A, 239, 443
– Kepler rotation law
– radiative transfer using the 2.5D code

choice of the systems
– the quiescent phase – optically thinner =⇒
effects connected with the velocity field better
visible
– SS Cyg
– Hα
– Hγ
– an AM CVn system
– HeI 4923 Å

SS Cyg
– an intermediate polar
M ∗ wd
(0.81 ± 0.19)M ⊙
q ∗ = M K /M wd 0.683 ± 0.012
a ∗
R ∗∗
wd
P ∗
i ∗
(1.36 ± 0.11) × 10 11 cm
0.011R ⊙
0.27512973 days
45 ◦ − 56 ◦
∗ Bitner et al., 2007, ApJ, 662, 564
∗∗ Wood, 1995, LNP, 443, 41
– model – H + He – 8 rings;
Ṁ∈< 1 × 10 −11 , 1 × 10 −9 > M ⊙ yr −1

SS Cyg – quiescent phase – Hα
y [R wd ]
5
4
3
2
log(χ )
-2 0
-4
-6
-8
-10
-12
-14
-16
-18
1
0
5
0 10 20 30 40 50 60 r tidal
x [R wd ] 4
y [R wd ]
3
2
log(S)
-2
-3
-4
-5
-6
-7
-8
-9
1
0
0 10 20 30 40 50 60
x [R wd ]
r tidal

SS Cyg – quiescent phase – Hα
relative flux
2.8
2.6
2.4
2.2
2
1.8
1.6
1.4
1.2
1
i=0 o RTE
flux
relative flux
1.04
1.03
1.02
1.01
1
i=85 o RTE
flux
4.55e+14 4.6e+14
ν[Hz]
4.55e+14 4.6e+14
ν[Hz]
relative flux
1.15
1.1
1.05
i=70 o RTE
flux
relative flux
1.15
1.1
1.05
i=90 o RTE
flux
1
1
4.55e+14 4.6e+14
ν[Hz]
4.55e+14 4.6e+14
ν[Hz]

SS Cyg – quiescent phase – Hγ
y [R wd ]
5
4
3
2
log(χ )
-2
-4
-6
-8
-10
-12
-14
-16
1
0
5
0 10 20 30 40 50 60 r tidal
x [R wd ] 4
y [R wd ]
3
2
log(S)
-2
-3
-4
-5
-6
-7
-8
-9
1
0
0 10 20 30 40 50 60
x [R wd ]
r tidal

SS Cyg – quiescent phase – Hγ
2
1.04
relative flux
1.8
1.6
1.4
1.2
i=0 o RTE
flux
relative flux
1.03
1.02
1.01
i=85.0 o
RTE
flux
1
6.85e+14 6.9e+14 6.95e+14
ν[Hz]
1
6.85e+14 6.9e+14 6.95e+14
ν[Hz]
relative flux
1.08
1.06
1.04
1.02
i=70 o RTE
flux
relative flux
1.1
1.05
1
i=90 o RTE
flux
1
6.85e+14 6.9e+14 6.95e+14
ν[Hz]
0.95
6.85e+14 6.9e+14 6.95e+14
ν[Hz]

an AM CVn star
– P orb < 1 hour =⇒ compact donors
– nature
– white dwarf (neutron star) + white dwarf
– white dwarf (neutron star) + helium core burning
star
– white dwarf (neutron star) + main-sequence star
– model (GP Com)
– quiescence phase
– 9 rings
– Ṁ∼ 10 −11 M ⊙ yr −1
– HeI 4923Å

an AM CVn star – HeI 4923Å
y [R wd ]
0.15
0.1
0.05
log(χ )
-2
-4
-6
-8
-10
-12
-14
-16
-18
0.15
0
0 2 4 6 8 10 12
x [R 0.1
wd ]
0.05
y [R wd ]
log(S)
-2
-3
-4
-5
-6
-7
-8
-9
0
0 2 4 6 8 10 12
x [R wd ]

an AM CVn star – HeI 4923Å
relative flux
31
21
11
i=0 o RTE
flux
relative flux
1.3
1.2
1.1
i=89.0 o
RTE
flux
1
6.05e+14 6.1e+14 6.15e+14
ν[Hz]
1
6.05e+14 6.1e+14 6.15e+14
ν[Hz]
1.6
2.2
relative flux
1.5
1.4
1.3
1.2
1.1
1
i=70 o RTE
flux
6.05e+14 6.1e+14 6.15e+14
ν[Hz]
relative flux
2
1.8
1.6
1.4
1.2
1
i=90 o RTE
flux
6.05e+14 6.1e+14 6.15e+14
ν[Hz]

– face-on view
– rotation velocity field has a negligible influence
– static disc approximation for the radiative transfer
is valid with high accuracy
– edge-on view
– important difference
v rot [km/s]
y [R wd ]
5
4
3
2
2500
2000
1500
1000
500
0
1
0
0 10 20 30 40 50 60
x [R wd ]
r tidal

Where it can play the role?
– edge-on view – opening angle ∼ 10 ◦
– not so small probability
– self-shielding disc (e.g. V348 Pup)
– occulting systems – some UX UMa or SW Ser
systems
– low luminosity
– dust is not important in most CVs (in contrast to
polars)
– extended area
– wind
– important corona – polars, intermediate polars

conclusion
– axial symmetry + wind + rotation =⇒
– rapidly rotating stars
– extended stellar atmospheres
– stellar winds
– discs