Download mhd simulation of spherical accretion to a star in the

PROPAGATION OF MAGNETARS
THROUGH THE INTERSTELLAR MEDIUM
Olga Toropina,
Space Research Institute, Moscow, Russia
[In collaboration with Marina Romanova & Richard Lovelace]
[Department of Astronomy, Cornell University, USA]
I. INTRODUCTION - NEUTRON STARS EVOLUTION STAGES
Rotating magnetized neutron stars pass through different stages in their evolution:
Ejector – a rapidly rotating (P<1s) magnetized neutron star is active as a radiopulsar.
The NS spins down owing to the wind of magnetic field and relativistic particles from the
region of the light cylinder RL (RA > RL).
Propeller – after the NS spins-down sufficiently, the relativistic wind is then suppressed
by the inflowing matter RL > RA. Until RC<RA, the centrifugal force prevents accretion,
NS rejects an incoming matter (RC<RA< RL).
Accretor – NS rotates slowly, matter can accrete onto star surface (RA < RC , RA < RL).
Georotator – NS moves fast through the interstellar medium (RA > Rасс).
rV2/2 = B2/8p
RA =2 x 1012 cm, , B=1015 G, V=100 км/с, n=1 cm-3
Rасс = 2GM* / (cs2 + v2) =4 x 1013 M/v1002
RC =(GM/W2)1/3 = 2 x 108 P12/3 cm
RL=cP/2p = 5 x 109 P1 cm
• Alfven radius (magnetospheric radius):
• Accretion radius:
• Corotation radius:
• Light cylinder radius:
I. INTRODUCTION - NEUTRON STARS EVOLUTION STAGES
Some neutron stars with anomalously strong magnetic field
B ~ 1014 – 1016 G
are called Magnetars (Duncan & Thompson 1992, 1995)
There is suggestion (based on observations) that
soft gamma-ray repeaters (SGRs)
and
long-period pulsars in supernova remnants
are magnetars (Kulkarni & Frail; Kouveliotou et al)
The estimated birthrate of SGRs is ~ 10% of ordinary pulsars (Kulkarni &
Frail; Kouveliotou et al). Thus, magnetars may constitute a nonnegligible percentage of
neutron stars (unless their magnetic field decays rapidly)
Magnetars pass through their evolution stages much faster than classical pulsars.
I. INTRODUCTION - NEUTRON STARS EVOLUTION STAGES
We investigate the propagation of magnetars through the interstellar medium (ISM).
We consider two cases:
(1) Slow rotating magnetar, when
Rасс ~ RA
gravitational focusing is important,
matter can accrete onto star surface
(2) Magnetar is rotating in the propeller regime, when
RC < RA
rotating star’s magnetic field rejects an incoming matter.
II. MHD SIMULATION OF ACCRETION TO A MAGNETIZED STAR
We consider the equation system for resistive MHD (Landau, Lifshitz 1960):
7/5, 5/3
Non-relativistic, axisymmetric resistive MHD code is used. The code incorporates the methods of local
iterations and flux-corrected transport. We assume axisymmetry, but calculate all three components of v
and B. We use the vector potential A so that the magnetic field B =  x A automatically satisfies
 • B = 0. We use a cylindrical, inertial coordinate system (r,f,z) with the z-axis parallel to the star's
dipole moment m and rotation axis W. The equatorial plane is treated as symmetry plane.
II. GEOMETRY OF THE MHD SIMULATION REGION
m - the magnetic moment of the star;
W* - the angular velocity of the star;
R* - the radius of the star,
Rmax, Zmax - the limits of the computational
region: R* << Rmax
The size of the region was taken to be less
then the sonic radius of the Bondi flow
Rs=[(5-3g)/4]RB, thus matter inflows
supersonically to the computational region.
The inflow rate is taken to be the Bondi
accretion rate: MBondi= 4p l (GM*)2 r /c3,
where l =0.625 for g=7/5.
.
The incoming matter is assumed to be
unmagnetized.
A uniform (r,z) grid with 513 x 513 cells was
used.
There is only one analytical solution for spherical accretion onto non-magnetic center – Bondi 1952. We
use this Bondi solution as a boundary condition. In simulations of accretion to an unmagnetized star, we
observed that matter accretes supersonically at the Bondi rate. The velocity and density variations were
those of the Bondi solution, and no discontinuities were observed in the simulation region.
II. GEOMETRY OF THE MHD SIMULATION REGION
.
We use a cylindrical inertial coordinate system (r,f, z), with origin at the star’s center. Z-axis
is parallel to the velocity of the ISM at large distances. The dipole magnetic moment of the
star m is parallel or antiparallel to the z-axis. Supersonic inflow with Mach number M . The
incoming matter is assumed to be unmagnetized. g = 5/3
III. ACCRETION ONTO SLOW ROTATING AND SLOW MOVING STAR
1. THE STRUCTURE OF THE ACCRETION FLOW
Two distinct regions separated by a shock wave are observed. One is the external region where matter inflows with
the Bondi rate and the density and velocity agree well with the Bondi (1952) solution. The second is the internal
region, where the flow is strongly influenced by the stellar magnetic field.
The background scale represents the density of the flow and the solid lines the poloidal magnetic field lines. The length
of the arrows is proportional to flow speed. The thick solid line shows the Alfve´n surface. The dashed line shows the
sonic surface. The flow becomes strongly anisotropic close to the dipole.
III. ACCRETION ONTO SLOW ROTATING STAR AND SLOW MOVING STAR
2. THE STRUCTURE OF THE ACCRETION FLOW INSIDE OF MAGNETOSPHERE
The anisotropy of the flow is evident. The plasma flows along the magnetic field lines to form polar accretion
columns. The density in the accretion columns is 3–4 orders of magnitude larger than that in the equatorial plane.
The background scale represents the density of the flow and the solid lines the poloidal magnetic field lines. The
thick solid line shows the Alfveґn surface.
III. ACCRETION ONTO SLOW ROTATING STAR AND SLOW MOVING STAR
3. TWO DISTINCT REGIONS SEPARATED BY A SHOCK WAVE
Mass accretion rate through the spheres of radii R at time t=1.7tff .
R* is the radius of the star and Rsh is the equatorial radius of the shock.
One can see two regions separated by the shock wave with different accretion rate. Inside shock wave the
accretion rate is smaller than MBondi and approximately constant out to the radius of the shock (Rsh),
which is propagating outward. This means that a new matter flow has been established around the
magnetized star and that it is stationary. At the shock wave R = Rsh, the accretion rate jumps up to the
Bondi rate.
III. ACCRETION ONTO SLOW ROTATING STAR AND SLOW MOVING STAR
5. DEPENDENCE OF THE ACCRETION RATE ON MAGNETIC FIELD OF THE STAR
Dependence of the accretion rate on the magnetic moment ~ m -3
Differential mass accretion rate per unit solid angle dM/dW as a function of the polar angle Q for
different magnetic moments m at a radius 2R*. The dashed line corresponds to Bondi accretion onto a
nonmagnetized star.
IV. ACCRETION ONTO SLOW ROTATING AND MOVING STAR, (RA < Racc )
Gravitational focusing is important.
Magnetic field of the star acts as an obstacle for the flow; a conical shock wave forms.
Results of simulations of accretion to a magnetized star with magnetic field at Mach number M = 3 and g
= 5/3. Poloidal magnetic B field lines and velocity vectors are shown. The background represents
logarithm of density. The thick line represents the Alfven surface.
IV. ACCRETION ONTO SLOW ROTATING AND MOVING STAR, (RA < Racc )
DENSITY, VELOSITY AND ENERGY DISTRIBUTION
IV. ACCRETION ONTO SLOW ROTATING AND MOVING STAR, (RA ~ Racc )
Gravitational focusing is less important.
Magnetic field of the star acts as an obstacle for the flow; a conical shock wave forms.
Results of simulations of accretion to a magnetized star with magnetic field at Mach number M = 3 and g
= 5/3. Poloidal magnetic B field lines and velocity vectors are shown. The background represents
logarithm of density. The thick line represents the Alfven surface.
IV. ACCRETION ONTO SLOW ROTATING AND MOVING STAR, (RA ~ Racc )
ENERGY DISTRIBUTION IN MAGNETOTAIL
IV. ACCRETION ONTO SLOW ROTATING AND MOVING STAR, (RA ~ Racc )
DEPENDENCE OF THE ACCRETION RATE ON MAGNETIC FIELD OF THE STAR
Dependence of the accretion rate on the magnetic moment for M=3, normalized to MBHL.
IV. ACCRETION ONTO SLOW ROTATING AND FAST MOVING STAR, (RA > Racc )
Gravitational focusing is not important.
Magnetic field of the star acts as an obstacle for the flow; but bow shock is narrow.
Magnetic lines form long magnetotail.
Results of simulations of accretion to a magnetized star with magnetic field at Mach number M = 10 and
g = 5/3. Poloidal magnetic B field lines and velocity vectors are shown. The background represents
logarithm of density. The thick line represents the Alfven surface.
IV. ACCRETION ONTO SLOW ROTATING AND FAST MOVING STAR, (RA > Racc )
DENSITY AND MAGNETIC FIELD VARIATIONS AT DIFFERENT MACH NUMBERS
- Density in the magnetotail is low
- Magnetic field in the magnetotail reduced gradually
V. SPHERICAL ACCRETION ONTO STAR IN THE PROPELLER REGIME
1. THE STRUCTURE OF THE ACCRETION FLOW
Matter flow in the "propeller" regime for a star rotating at W*=0.5 WK* after 7 rotation periods of the star. The axes are
measured in units of the star's radius. The background represents the density and the length of the arrows is
proportional to the poloidal velocity. The thin solid lines are magnetic field lines.
Two distinct regions separated by a shock wave are observed. One is the external region where matter inflows with
the Bondi rate and the density and velocity agree well with the Bondi (1952) solution. The second is the internal
region, where the flow is strongly influenced by the stellar magnetic field and rotation.
V. SPHERICAL ACCRETION ONTO STAR IN THE PROPELLER REGIME
2. THE STRUCTURE OF THE ACCRETION FLOW INSIDE OF MAGNETOSPHERE
Spherical accretion to a rapidly rotating star. Matter flow in the "propeller" regime for a star rotating at W*=0.5 WK*
A new regime of matter flow inside the expanding shock wave. The axes are measured in units of the star's radius. The
background represents the density and the length of the arrows is proportional to the poloidal velocity. The thin solid
lines are magnetic field lines. The bold line represents the Alfven surface. Dotted line shows sonic surface.
V. SPHERICAL ACCRETION ONTO STAR IN THE PROPELLER REGIME
2. THE STRUCTURE OF THE ACCRETION FLOW INSIDE OF MAGNETOSPHERE
Matter flow in the "propeller" regime for a star rotating at W*=0.5 WK* . The background represents angular
velocity W*= vf (r,z)/r. The axes are measured in units of the star's radius. A new regime of matter flow forms
inside the expanding shock wave. The rapidly rotating magnetosphere expels matter outward in the equatorial
region.
V. SPHERICAL ACCRETION ONTO STAR IN THE PROPELLER REGIME
2. THE STRUCTURE OF THE ACCRETION FLOW INSIDE OF MAGNETOSPHERE
A new regime of matter flow inside the expanding shock wave. Figure shows the streamlines of the matter flow. Matter
free-falls along the field lines going into the poles of the star. Some matter that flows close to the z-axis accretes onto the
surface of the star. However, matter more remote from the z-axis comes close to the star, is deflected by the rotating
magnetic field, and then moves outward in the equatorial plane.
THE STRUCTURE OF THE ACCRETION FLOW FOR A STAR ROTATING AT W*=0.5 WK*
VI. ACCRETION ONTO MOVING STAR IN THE PROPELLER REGIME
1. THE STRUCTURE OF THE ACCRETION FLOW FOR MOVING STAR AT M=1
289 x 865
g = 5/3
The structure of flow depends on velocity of moving star v. Example of matter flow for a star rotating at
W*=0.7 WK and Mach number M=v/cs=1. Accretion radius Rасс=2GM* / (cs2 + v2) is large then
magnetospheric radius RA. A star can capture a matter, the flow looks like in the spherical case with
equatorial outflows. Velocities of star v and outflow vp are the same order, the bow shock is wide.
The axes are measured in units of Bondi radius. The background represents the density and the length
of the arrows is proportional to the poloidal velocity.
VI. ACCRETION ONTO MOVING STAR IN THE PROPELLER REGIME
1. THE STRUCTURE OF THE ACCRETION FLOW FOR MOVING STAR AT M=1
289 x 865, g = 5/3
V. ACCRETION ONTO MOVING STAR IN THE PROPELLER REGIME
2. THE STRUCTURE OF THE ACCRETION FLOW FOR FAST MOVING STAR, M=3
289 x 865
g = 5/3
Example of matter flow for a star rotating at W*=0.7 WK and Mach number M=3. Rасс ~ RA
The magnetic field of the star acts as an obstacle for the flow, most of the inflowing matter is kept away
from the star and a conical shock wave forms. At larger distances the field is stretched by the flow,
forming long magnetotail. The rapidly rotating magnetosphere expels matter outward in the equatorial
region. This matter first flows radially outward, then along Z-direction.
V. ACCRETION ONTO MOVING STAR IN THE PROPELLER REGIME
2. THE STRUCTURE OF THE ACCRETION FLOW FOR FAST MOVING STAR, M=3
289 x 865, g = 5/3
V. ACCRETION ONTO MOVING STAR IN THE PROPELLER REGIME
3. AN ANGULAR MOMENTUM FLUX
Rapidly rotating star looses an angular momentum and spins down. We can estimate the total angular
momentum loss rate from the star by evaluating the integral over the surface around the star's
magnetosphere:
Left panel: Angular momentum flow connected with matter.
Right panel: Angular momentum flow connected with magnetic field.
V. ACCRETION ONTO MOVING STAR IN THE PROPELLER REGIME
4. AN ANGULAR MOMENTUM EVOLUTION
The total angular momentum flux around the magnetosphere (solid line) becomes constant
approximately after 10-15 rotation periods of the star. As the matter is passing the angular
momentum flux in tail (dotted line) is increasing up to value of flux around the magnetosphere and
becomes constant. Figure shows that total flux across section z=0.6 becomes constant and equal to
flux around the magnetosphere approximately after 26 rotation periods.
V. ACCRETION ONTO MOVING STAR IN THE PROPELLER REGIME
DEPENDENCE OF THE ANGULAR MOMENTUM FLUX ON DIFFERENT PARAMETERS
The summary of scaling laws:
V. ACCRETION ONTO MOVING STAR IN THE PROPELLER REGIME
5. SPIN-DOWN OF MAGNETARS
We can estimate spin-down of magnetar due to the propeller effect for typically parameters:
M = 1.4 M☼= 2.8 x 1033 g,
R = 106 km,
B = 1015 Gs,
P = 105 s,
n∞ = 1 cm-3,
cs = 30 km/s.
.
.The characteristic time of spin-down is DT= L*/L,
where
.L*=I* W* - an angular momentum of a star
L - an angular momentum loss rate, which is obtained by evaluating the integral:
V. ACCRETION ONTO MOVING STAR IN THE PROPELLER REGIME
5. SPIN-DOWN OF MAGNETARS
The characteristic time of spin-down is
For periods P* ~ 103 s, which correspond to beginning of the propeller stage, the
evolution scale will be DT = 103 years, while at period P* ~ 106 s corresponding to
the end of propeller stage DT = 3 x 104 years. Thus we see that magnetars are
expected to spin down very fast at the propeller stage.
This time-scale however may be much larger if diffusivity is very small.
VLT observations by Kerkwijk and Kulkarni
Vela pulsar X-ray, Chandra data