Line Profiles of Magnetically Confined Winds

1
Line Profiles of Magnetically Confined Winds

• Stephanie Tonnesen

2
The Numerical Method

• The original r, ?, ? coordinates are aligned with
  the magnetic coordinate system.
• The z-axis is aligned with the magnetic poles
• In order to make my program more universally
  applicable I have allowed for the observers
  coordinate system to be tilted with respect to
  the magnetic coordinate system.
• In order to find the line of sight velocities
  necessary for the observer I need to do a
  coordinate transformation.
• vlos- (vx-mag)sin(tilt) (vz-mag)cos(tilt)
• I sum the emission of gridpoints that correspond
  to vlos values that go into the same bin. The
  bins are all the same range of velocity values.
  This histogram is my line profile.
• The emission is found by multiplying the volume
  of each element around my gridpoint by the
  emissivity constant by the density at the
  gridpoint squared. This causes the potential for
  noise in my line profile because normally at
  every point in the wind there would be a
  different emissivity. The finer my grid, the
  better the resolution in my line profile. My
  volume element size is very dependent on the
  number of both r and ? zones
• vol(i) ((redge(i)rstep)3 - (redge(i))3)(cos(ted
  ge(i))-cos(tedge(i)tstep))(pedge(i)pstep-pedge(
  i))/3
• Where the edge variables are the values that mark
  the sides of my volume elements, and the step
  variables are the size of the step between
  consecutive variables.

3
Smoothing

• Another change to the program that I made to
  smooth the profile had to do with seperating the
  emission into different velocity bins. A
  velocity was assigned to a bin no matter where it
  fell in the range, and all the emission from that
  gridvolume was added into that bin. This caused
  jagged peaks in my line profiles that were only
  the result of a small difference in the number of
  emission volumes put into each particular bin.
• The new method splits each velocity into the bin
  that it is in and the bin it is closest to. This
  is done by finding how close the velocity is to
  the center velocity of the two bins and splitting
  the emissivity into the two bins by the same
  fraction.
• You can see the results to the right.

4
There are other effects that cause noise, namely
the number of gridpoints.

• Until you use about 70 r gridpoints, there is a
  stair-step appearance to the profile that is
  reminiscent to a stack of the rectangles you get
  when looking at a single shell.
• The number of phi points has a large affect on
  the smoothness of the top of the profile, and you
  need about 60 phi points to get a smooth top.

5
Fitting the General Model

• The first task was to make sure that my program
  was able to match the analytic solution I had
  previously found for a spherically symmetric wind
  model.
• Note that the inner line is always the numerical
  solution.

6
Occultation

• Next I added occultation to make sure that as
  expected, it would only cause asymmetry in the
  profile.

7
Flared Equatorial Disk

• The MCWS model on which my line profiles will be
  based shows that the x-ray producing shocks occur
  around the magnetic equator, so I will cut off
  the emissivity around the poles.
• So the emission from the star only comes from an
  equatorial disk.

8
I can rotate my viewing axis to look at this
radially flowing wind from different angles.
This will give me different line profiles, both
from what emission exists at what doppler shifts,
and because different amounts of emitting wind
will be occulted.
0º (pole-on)
90º Along magnetic equator
45º
9
Future Work

• Currently I am working on adding a phi velocity
  component, and I will try and get more realistic
  equations to describe what may be happening in
  the phi component.
• Later I will add a theta velocity component.
• I will then postprocess Asifs simulations.
• Finally, I will add absorption into my program.