Line Profiles of Magnetically Confined Winds Stephanie Tonnesen Swarthmore College

The Numerical Method The original r, ,  coordinates are aligned with the magnetic coordinate system with the z-axis aligned with the magnetic poles The observer’s coordinate system can be tilted with respect to the magnetic coordinate system. In order to find the line of sight velocities for the observer I need to do a coordinate transformation. v los =- (v x-mag )*sin(tilt) +(v z-mag )*cos(tilt) I sum the emission of each spacial zone corresponding to a given v los value. This histogram of emission vs. v los is the line profile. The emission is found by multiplying the volume of each element around my gridpoint by the emissivity constant and the density squared at the gridpoint. This causes noise in the line profile because normally at every location within a given zone there would be a slightly different velocity. The finer my spatial grid, the less noise in my line profile. The volume element size is: vol(i) =((r edge (i)+r step ) 3 - (r edge (i)) 3 )*(cos(t edge (i))-cos(t edge (i)+t step ))*(p edge (i)+p step -p edge (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.

The Numerical Method The original r, ,  coordinates are aligned with the magnetic coordinate system with the z-axis aligned with the magnetic poles For every gridpoint there is a corresponding radial velocity based on the beta velocity law, a density, and an emissivity. There is also a surrounding volume element. The volume element size is: vol(i) =((r edge (i)+r step ) 3 - (r edge (i)) 3 )*(cos(t edge (i))-cos(t edge (i)+t step ))*(p edge (i)+p step -p edge (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. By multiplying the emissivity and the volume I can find the emission for each spatial element. The observer’s coordinate system can be tilted with respect to the magnetic coordinate system. Every point is switched to cartesian coordinates to make the transformation, whether I am transforming gridpoints or their corresponding velocity values. The coordinate transform to find the line of sight velocities for the observer is: v los =- (v x-mag )*sin(tilt) +(v z-mag )*cos(tilt) I sum the emission of each spatial zone corresponding to a given v los value. This histogram of emission vs. v los is the line profile. Because I am unable to give every location in every spatial zone its own velocity, there can be noise in the profile from assigning the same velocity to an entire volume element. The finer my grid, the less noise is cause by this problem.

Smoothing Another smoothing technique involves separating the emission into different velocity bins. In the upper profile, all of the emission from a volume element was consigned to a bin with no consideration for where the particular velocity element fell between the bin boundaries. This caused jagged peaks in the profile. The new method splits the emission from an element according to the velocity’s proximity to the two closest bin centers. The emissivity is separated into the two bins by the same fraction as the distance between the velocity and each bin center. You can see the results to the right. Both of the line profiles have one r and phi gridpoint and 1000 theta gridpoints.

There are other effects that cause noise, namely the number of gridpoints. Until you use about 90 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. Graphed above is a single shell, 15, 30, and 90 shells. In a spherically symmetric wind, the number of theta gridpoints dictates the noise in the horizontal top of the profile. About 700 points gives a relatively smooth line. Graphed above are profiles with 100, 360, and 900 theta points. The number of phi points has a larger affect than theta points on the noise in the profile when the observer’s system is tilted with respect to the magnetic axis, and you need about 60 phi points to get a smooth top. Graphed above are profiles with 2, 20, and 60 phi points.

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. The analytic solution has two equations applied at different velocities: The analytic profile Numerical solution

There are two reasons for the discrepancy of the numerical and analytical solutions. The reason the wings are not alinged is that in the previous slide I only go to an outer shell distance of 10 stellar radii. If I were to integrate to infinity, the lines would match perfectly in the wings. With the outer emitting radius at 150 stellar radii, the bottom wings are a close match. There are 600 r gridpoints. The numerical solution does not reach the height of the analytical solution because of the discreteness of the shells. Although the user puts in the innermost shell at 1.5 stellar radii, the numerical solution uses that radius as the inner “edge” of the innermost volume element. Until the center of the volume element is very close to 1.5 stellar radii, the numerical solution will be shorter than the analytical solution. The outer radius is again 150 stellar radii, but the number of r gridpoints has risen to

Occultation Next I added occultation to make sure that as expected, it would only cause asymmetry in the profile. A point is occulted if: sqrt(x 2 + y 2 ) ≤ 1 where x and y are in the observer oriented system. Make sure that this has a Spherical star!

Flared Equatorial Disk The initial approximation to 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. The emission from the star only comes from an equatorial disk, but the equatorial wind remains fully radial.

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) 45º 90º Along magnetic equator

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 Asif’s simulations. Finally, I will add absorption into my program.

Using Keplerian rotation I am getting line profiles. These profiles have both a phi and a radial velocity. My phi velocity is proportional to the inverse square root of r. Here the observer is looking pole-on at a spherically emitting wind. One line is a purely radial velocity and the other has both radial and phi components. Any noise in the profile is from lack of gridpoints. Neither wind has any occultation. All that has been changed in this image is that occultation has been added to the line profile with the phi velocity component.

Once you start looking at an angle, things begin to get crazy: Pole-on Here is a plot with all of the angles on it. Every profile has 100 r, 150 theta, and 100 phi zones. They all are occulted.