Theory The model I am using is that of a line driven wind that is confined by magnetic fields out to the alfven radius, where the energy from the velocity of the wind is greater than the energy of the magnetic fields. At this point the magnetic fields become open field lines flowing from the star. This model forces the hot emitting wind to be in particular areas, namely near the equator where the channeled wind collides. These shocks heat to wind to X-ray emitting temperatures. In addition to the more polar wind that is cool because it is never shocked, there is a cool equatorial plane of wind material that has been heated by the shock, and then emitted enough X-rays to cool down. Once the material is cool, it is pushed into a thin plane by the new flow of material. Both of these cool areas of the wind can absorb emitted light that would otherwise reach the observer. The important theory or assumption behind my work is that the shapes of the line profiles can be closely mimicked by only considering kinematics and dynamics. This allows me to ignore specific atomic structure considerations and any effects from the magnetic field other than containing and directing the wind flow. There are three factors that I take into consideration in my program. The first is the Doppler shift of the wind, the second is absorption and occultation, and the third is the misalignment of the magnetic and rotational axes. The Doppler shift of light is the perceived lengthening or shortening of its wavelength depending on whether the emitting object emitting is moving towards or away from the observer. Because stars emit in all directions, an emission line that is centered at some wavelength becomes broadened in either direction relative to the velocity of the visible outflow. However, some of the outflow may not be visible to an observer. The most obvious reason is that the star itself will block some of the wind. The wind it blocks is emitting the most redshifted light, so that should effect the symmetry of the line profile. Cool wind can also block or attenuate the light emitted by hotter wind. This is because the cool wind has atoms in a low energy state, and high energy light passing through the material may ionize or excite the atom rather than reaching the observer. Depending on the opacity of the wind and what emission is being attenuated, this can have different affects on the line profile shape and symmetry. The last important factor to take into account is the misalignment of the magnetic and rotational axes. This means that the observer is constantly viewing the star from a different angle, so different light is being blocked by occultation and absorption at different times. This effect does not have direct results, but rather changes what wind is redshifted and blueshifted, and what emission is occulted or absorbed by cooler winds. For example, when looking down from a magnetic pole, the cooling disk will attenuate much of the light from the lower hemisphere of the wind, and the polar wind will attenuate the emission from the upper equatorial shock. Occultation from the star may not have as much of an effect on the profile because hot wind is rare in the polar regions. However, when the observer is looking down the magnetic equator, much of the hot, highly redshifted wind at the equator is occulted by the star, and the cool equatorial disk and polar wind’s opacity is unimportant because little emission passes through that wind. Although it is the star that is rotating so that the orientation of the magnetic field is periodically changing with respect to the viewer, the effect is the same as thinking that the “eye” of the observer is circling around the star. This can be seen in the following figure. (show a figure of the misaligned magnetic and rotational axes, and the eye at different angles.) MATCHING THE PROGRAM TO THEORY (? Theory? How about known stuff??) The goal of my analytic model is to split a large spherical volume around a theoretical “star” into a grid. At the center point of each zone I can find the density, velocity and emissivity of the outflowing material. In addition, I can find the volume of each zone. Using this information I can create a line profile whose general shape should fit a large number of specific emission lines that are observed from a hot star wind. My program begins the gridding process by assigning evenly spaced r, theta, and phi grid points to a coordinate system aligned with the magnetic axis. This creates a 3D grid on which all of my calculations are based. Every gridpoint has a corresponding radial velocity based on the beta velocity law, which is: v(r) = v∞(1-R*/r)β Every point also has an assigned density that is proportional to 1/r2, and an emissivity that is directly proportional to the density. There is also a surrounding volume element. The volume element size is: vol(i)=((redge(i)+rstep)3 - (redge(i))3)(cos(tedge(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. I find them by finding the midpoint between actual gridpoints in all three dimensions. The “step” variables are the size of the step between consecutive edge variables, and are the same throughout the grid. By multiplying the volume and emissivity from each grid element I am able to find the total emission from each spatial element. In order to spread these emission values amongst their different velocities, I need to organize them according to line of sight velocity. Furthermore, because the magnetic and rotational axes do not need to match the observer does not always view the system from the same angle. Thus I need to define a “tilt” number, which is the number of radians the observer’s z-axis rotates around the y-axis from the magnetic field’s z-axis. A rotation about the y-axis is all that is needed because my system is azimuthally symmetric. Now, in order to find the line of sight velocities for any viewing angle, I need to perform the following coordinate transformation: vlos=- (vx-mag)*sin(tilt) +(vz-mag)*cos(tilt) where the velocities used in the calculation are the velocities in cartesian coordinates in the magnetic field coordinate system. Finally, I sum the emission of each spatial zone corresponding to a given vlos value. This histogram of emission vs. vlos 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 caused by this problem. The first task in this project was to make sure that the code on which the rest of my project would be based would duplicate the known line profile of an outwardly accelerating spherically symmetric wind without occultation from the inner star. First I duplicated a single spherical shell around a star. This should emit a line profile that is a rectangle centered on the wavelength emitted by material moving at zero velocity with respect to the observer and whose edges are at the wavelengths as seen by the observer when emitted by material moving at the speed of the shell directly towards and away from the observer. This widening of the profile is due to the first factor I take into account in my calculations, Doppler Broadening. Because many of my zones would be moving at slightly different velocities, I chose a number of velocity bins in which to sum my line of sight velocities. My first attempt simply summed all of the emission from a zone into the bin into whose limits the velocity of the center point fell. This did not take into account the possibility of a velocity being close to the boundary of a bin, and resulted in a jagged profile that was not close to rectangular (fig 1). By using a linear interpolation in which the emission is split according to the center gridpoint velocity’s proximity to the two nearest bin centers. The emission is separated into the two bins by the same fraction as the distance between the velocity and each bin center. Figure 1 shows two line profiles that display the difference the interpolation makes. Using the interpolation, the program produced the expected box shape. The next analytic solution is that of a complete wind with continuous layers of accelerating shells. The line profile is many rectangles stacked on top of one another. The maximum width is from the outermost layer where the wind has reached its terminal velocity, and the minimum width is that of the flat top, whose limits are defined by the velocity of the innermost emitting wind. It is a smooth curve connecting the innermost top edge to the outer bottom edge. (do I need to do this, or should I show a figure, OR should I just say, see fig 2, which will be a comparison of the analytic model to my numerical simulation?). Originally my solution did not match the analytic solution, as shown in figure 2. The reason that the wings were not aligned is that in my solution, I only went out to 10 stellar radii. Going out to 150 stellar radii matches the wings. The reason that the top of my program’s line profile does not match that of the analytic model is that I am not using enough shells. This means that although both the analytic solution and my program consider the innermost shell to begin at 1.5 stellar radii, the velocity of the analytic solution calculates the velocity at 1.5 stellar radii while my program uses the velocity of the center of that gridzone. When I use 2000 shells going from 1.5 to 150 stellar radii, my profile is a close match to the analytic profile. The analytic profile Fig. 2: This is the comparison of my numerical solution to the analytic solution. My program goes from 1.5 to 10 stellar radii in blank steps. Numerical solution Fig. 3: On the left the radius of the wind has been expanded from 10 to 150 stellar radii, and the number of shells is 600. Here the wings have matched, although the top of my program’s output is still lower than that of the analytic solution. The figure on the right has the same outer limit, but has 2000 shells. This has moved the velocity of the innermost shell of my program close to the analytic solution’s velocity at exactly 1.5 stellar radii, and thus the tops of the profiles are closely matched. Finding the best fit to the analytic profile meant finding the number of grid points that are necessary in the three coordinates, r, theta and phi, that cause a smooth profile. The number of shells, or r grid points, is important to the height and to the appearance of the sides of the profile. Until you use about ninety gridpoints in a wind to 10 stellar radii, there is a stair-step appearance to the profile because it is a stack of rectangles from looking at discrete shells rather than an integrated curve from a continuous structure. In a system viewed from the pole, the number of theta gridpoints dictates the noise in the horizontal top of the profile. About 700 points are necessary for a smooth profile. Once the observer’s view has moved from the magnetic pole, the number of phi points becomes more important, and you need about 60 points for a smooth top. Fig. 4: Moving from left to right, the first profile shows a single shell, 15, 30, and 90 shells. The second compares profiles with 100, 360, and 900 theta points, and the third graphs profiles using 2, 20, and 60 phi points. The next element of my model that I assume has an impact is occultation by the star. In an otherwise spherically symmetric emitting wind, this would only block the most redshifted emission, thus causing an asymmetric profile. In my program, a zone is occulted if its center gridpoint has both a negative z component in the observer’s coordinate system and √(x2 + y2)≤1. (I need to see if the figure I have comparing an occulted and non-occulted spherically symmetric wind uses a spherical star) My program often switches between Cartesian and polar coordinates in order to facilitate considering different viewing angles. This is because I consider a spherically symmetric wind with a radial outflow, but calculate my line profile based on the velocity in the zdirection of the observer oriented axis. Below is a figure showing the magnetic axis and the observer oriented axis. Note from above that the line of sight velocity is calculated using the Cartesian velocities from the magnetic axis. However, my velocity equation calculates a radial velocity. I may at some point also want to take a phi or theta velocity into account. There are three steps to changing a vector in polar coordinates into a vector in Cartesian coordinates. First, consider a vector in spherical polar coordinates. v = vr · r + vθ · θ + vΦ · Φ Now I need to know what r, θ, and Φ are in Cartesian coordinates. r = x · cosΦ · sinθ + y · sinΦ · sinθ + z · cosθ θ = x · cosθ · cosΦ + y · cosθ · sinΦ - z · sinθ Φ = -x · sinΦ + y · cosΦ Finally, I just need to sum the Cartesian components of each of these polar coordinates to find the Cartesian vectors. vx = vr · cosΦ · sinθ + vθ · cosθ · cosΦ – vΦ · sinΦ vy = vr · sinΦ · sinθ + vθ · cosθ · sinΦ + vΦ · cosΦ vz = vr · cosθ - vθ · sinθ I can then calculate the line of sight velocity using the previous equation. As expected, in a spherically symmetric wind, the line profiles from any angle are exactly the same. These checks used every factor that I had described in the Theory chapter, and found that my original program used them correctly to produce line profiles of known shapes.