Intrinsic
polarization
of Wolf
Rayet stars,
S.
Abdellaoui,
J. Krti£ka,
P. Kurfürst
Introduction
Model
Simulation
Results
Conclusion
1/19
Intrinsic polarization of Wolf Rayet stars,
hydrodynamic model and rotational eect
S. Abdellaoui, J. Krti£ka, P. Kurfürst
Masaryk University
February 28, 2022
Intrinsic
polarization
of Wolf
Rayet stars,
S.
Abdellaoui,
J. Krti£ka,
P. Kurfürst
Introduction
Model
Simulation
Results
Conclusion
2/19
Motivation
Wolf Rayet (WR) stars are thought to be the late
evolutionary stage of some massive stars,
Collapsar of fast rotating WR to black hole could generate
a long gamma ray burst (Woosley 1993)
Spectropolarimetry observation introcuded by Stevance et
al. 2018,
Intrinsic
polarization
of Wolf
Rayet stars,
S.
Abdellaoui,
J. Krti£ka,
P. Kurfürst
Introduction
Model
Simulation
Results
Conclusion
3/19
1 Introduction
2 Model Simulation
3 Results
4 Conclusion
Intrinsic
polarization
of Wolf
Rayet stars,
S.
Abdellaoui,
J. Krti£ka,
P. Kurfürst
Introduction
Model
Simulation
Results
Conclusion
3/19
WR groups
Figure: WR124 in the constellation Sagittarius is ejecting masses into
interstellar medium (Credit: Yves et al, and NASA)
Based on their emission line, WR stars are divided into
three main spectroscopic groups (WN, WC, WO),
WN stars show dominating nitrogen emission lines,
Intrinsic
polarization
of Wolf
Rayet stars,
S.
Abdellaoui,
J. Krti£ka,
P. Kurfürst
Introduction
Model
Simulation
Results
Conclusion
4/19
WR
3000 3500 4000 4500 5000 5500 6000 6500 7000 7500
zářivýtok
λ [A]
Hα+HeII
HeII+
Hβ
HeII
HeII HeII
NV
NIV
NIII
WN6
WN7
3000 3500 4000 4500 5000 5500 6000 6500 7000 7500
zářivýtok
λ [A]
HαHeII CIII+IV +HeII
WC5
WC7
Figure: Intensity as a function of wavelength for WR spectra
However WC stars are dominated by carbon and He
emission lines with absence of H and N,
WO stars are rare compare to WN, and WC, they have
oxygen emission lines with some carbon emission lines,
Intrinsic
polarization
of Wolf
Rayet stars,
S.
Abdellaoui,
J. Krti£ka,
P. Kurfürst
Introduction
Model
Simulation
Results
Conclusion
5/19
HR diagram
Figure: Hertzsprung-Russell diagram for massive stars(From Varsha et
al. 2019)
Intrinsic
polarization
of Wolf
Rayet stars,
S.
Abdellaoui,
J. Krti£ka,
P. Kurfürst
Introduction
Model
Simulation
Results
Conclusion
6/19
Rotation eect
Rapid rotation aects the structure and lives of many stars,
The rotation could break the sphericity and lead to
axisymmetric wind density (Owocki et al. 1996),
If the electron scattering takes place in this axisymmetric
envelope, the intrinsic polarization occurs,
The polarization is perpendicular to the scattering plane,
The scattering geometry is described in terms of the star's
reference frame,
And the angle of the scattering plane to the observer is
considered in terms of (θ,φ) and the inclination (i).
Intrinsic
polarization
of Wolf
Rayet stars,
S.
Abdellaoui,
J. Krti£ka,
P. Kurfürst
Introduction
Model
Simulation
Results
Conclusion
7/19
Hydrodynamic
Solving hydrodynamic model equations in spherical coordinate
system with oblate boundary conditions, using the VH1 code
(Blondin 1990) coupled the radiative subroutine (Owocki et al
1994)
Mass conservation equation ∂tρ + (ρ.u) = 0
Momentum conservation ρ(∂tu + (u. )u) = − P + ρf ext
Equation of state for isothermal wind P = c2
s ρ
where ρ is density, u = (ur, uθ, uφ) is the velocity in 3 direction,
P is the gas pressure and f ext is the external force.
In line-driven wind the external force is the sum of the eective
gravity with line radiative force.
The line force is expressed as:
glines(r) =
kσ1−α
Th
Wδρ(r)α−δcvth Ω
(n (n.v))α
nI(n, r)dΩ
Intrinsic
polarization
of Wolf
Rayet stars,
S.
Abdellaoui,
J. Krti£ka,
P. Kurfürst
Introduction
Model
Simulation
Results
Conclusion
8/19
Stellar parameters
The mass of the two star is 7M , and the terminal velocity is
5000km/s
WR log(L/L ) T (kK) R (R ) log( ˙M/1M yr−1)
93b 5.30 160 0.58 -5
102 5.45 210 0.39 -4.92
Base density is xed proportional to ˙M/csR2 and subsonic
outow for the velocity at lower boundary, and outow at
upper boundary
Reective boundary in latitudinal direction
Half sphere was considered for computational domain
Intrinsic
polarization
of Wolf
Rayet stars,
S.
Abdellaoui,
J. Krti£ka,
P. Kurfürst
Introduction
Model
Simulation
Results
Conclusion
9/19
Wind density contours
2 4 6 8 10
r/R
0
20
40
60
80
100
120
140
160
180
Co-latitutde(deg)
-11.000
(a) Radial force only
2 4 6 8 10
r/R
0
20
40
60
80
100
120
140
160
180
Co-latitutde(deg)
-11.200
(b) +Gravity darkening
Figure: Log of density as a function of r and θ
Radial force only leads to the formation of an equatorial
disk (Wind Compressed Disk),
Adding non radial forces and the gravity darkening the disk
is suppressed and the matter leaves the star from the polar
Intrinsic
polarization
of Wolf
Rayet stars,
S.
Abdellaoui,
J. Krti£ka,
P. Kurfürst
Introduction
Model
Simulation
Results
Conclusion
10/19
Polarization computation
The polarization of electromagnetic radiation is described by
Stokes vector S
S =




I
Q
U
V



 (1)
I is the radiation intensity, Q and U describe the linear
polarization, and V for circular polarization.
The degree of total and linear polarization, P and PL can be
expressed by Stokes's parameteres
P = Q2 + U2 + V2/I (2)
PL = Q2 + U2/I (3)
Intrinsic
polarization
of Wolf
Rayet stars,
S.
Abdellaoui,
J. Krti£ka,
P. Kurfürst
Introduction
Model
Simulation
Results
Conclusion
11/19
Polarization
Because of the geometry is spherical V = 0, and U = 0 left
two variables Q and I,
we have to solve the radiative transfer equation in order to
compute the radiation intensity Iand as a result get P!!
Call to Monte Carlo method is necessary,
Intrinsic
polarization
of Wolf
Rayet stars,
S.
Abdellaoui,
J. Krti£ka,
P. Kurfürst
Introduction
Model
Simulation
Results
Conclusion
12/19
Polarization
The intrinsic polarization is calculated according to the relation
(Brown et al 1977, Cassinelli et al 1987):
PR =
3
16
σT sin2
i
∞
R
1
−1
n(r, µ)(1 − 3µ2
)D(r) dr dµ.
Where n is the electron number density, µ = cos(θ), and
D = 1 − (R
r )2 is the depolarization eect introduced by
Cassinelli 1987
Brown et al 1977, showed that the polarization is positive if it is
obtained from oblate geometry and negative if it is calculated
from prolate structure.
Intrinsic
polarization
of Wolf
Rayet stars,
S.
Abdellaoui,
J. Krti£ka,
P. Kurfürst
Introduction
Model
Simulation
Results
Conclusion
13/19
Observation
Stevance et al 2018 derived the upper limit for the intrinsic
polarization of the two stars and found that PR < 0.077%
(Vrot < 324km/s) for WR93b and PR < 0.057%
(Vrot < 234km/s) for WR102.
Due to limitation of the hydrodynamic we were not able to run
for lower rotation velocities in order to compare with the
observation.
Using the relation of Wood et al 1993, we got an estimate of
the polarization for the two stars, where PWR93b ≈ 8.6% and
PWR102 ≈ 18%.
P0 =
3σT
32π
˙M
2πmpRv∞
,
Intrinsic
polarization
of Wolf
Rayet stars,
S.
Abdellaoui,
J. Krti£ka,
P. Kurfürst
Introduction
Model
Simulation
Results
Conclusion
14/19
Comparison
In order to check the hydrodynamic model we compared with an
analytic approximation of wind compressed disk (WCD) model
theory,
0 10 20 30 40 50 60 70 80 90
Inclination (deg)
4
2
0
2
4
6
8
10
Polarization(%)
WCD
Hydro with radial
Hydro with full model
Analytical gravity
0 10 20 30 40 50 60 70 80 90
Inclination (deg)
0
5
10
15
Polarization(%)
WCD
Hydro with radial force
Hydro with full model
Analytical gravity
Figure: Comparison between analytical and hydrodynamic model of
the polarization of WR93b and WR102, Vrot = 900 km/s
Intrinsic
polarization
of Wolf
Rayet stars,
S.
Abdellaoui,
J. Krti£ka,
P. Kurfürst
Introduction
Model
Simulation
Results
Conclusion
15/19
Full hydrodynamic model
To avoid the staircasing at the lower boundary conditions and to
ensure the optically thin regime we integrated from 1.2R .
0 10 20 30 40 50 60 70 80 90
Inclination (deg)
2.5
2.0
1.5
1.0
0.5
0.0
Polarization(%)
1100
900
700
500
0 10 20 30 40 50 60 70 80 90
Inclination (deg)
1.2
1.0
0.8
0.6
0.4
0.2
0.0
Polarization(%)
1400
1100
900
700
500
Figure: Polarization of WR93b and WR102 as function of inclination
for dierent rotational velocities
Intrinsic
polarization
of Wolf
Rayet stars,
S.
Abdellaoui,
J. Krti£ka,
P. Kurfürst
Introduction
Model
Simulation
Results
Conclusion
16/19
Discussion
The model with radial force only, oblate wind density
distribution, gave a positive polarization,
The model with non radial forces, prolate wind structure,
leads to negative polarization,
Increasing the rotational velocity will increase the
polarization in absolute value,
The polarization reaches its maximum when viewed edge-on
Intrinsic
polarization
of Wolf
Rayet stars,
S.
Abdellaoui,
J. Krti£ka,
P. Kurfürst
Introduction
Model
Simulation
Results
Conclusion
17/19
Angular momentum
WR PR% Vrot(Stevance's model) Vrot(Current model) log(j/1cm2s−1)
93b 0.077 324 277 17.88
102 0.057 234 444 17.85
Threshold of angular momentum of collapsar is
log(j/1cm2s−1) > 16
The obtained angular momentum exceed the threshold
Intrinsic
polarization
of Wolf
Rayet stars,
S.
Abdellaoui,
J. Krti£ka,
P. Kurfürst
Introduction
Model
Simulation
Results
Conclusion
18/19
Conslusion
The intrinsic polarization was obtained by integrating the
density distribution for dierent rotational velocities.
Radial force only leads to oblate density distribution as a
result the polarization >0,
Including non radial forces and gravity darkening leads to
prolate density distrubition, the polarization becomes <0,
The derived upper limit of angular momentum is
comparable to the one from the observation,
Thank you for your attention
19/19