Editing IVAJ/Level1 (section)

==<font color="#0000DD">Interpretation of Results</font>==

[[Image:Figure2_Tohline.jpg |border|center|600px|Figure 2.  Screenshot of the [http://www.vistrails.org VisTrails] spreadsheet after we used five different values of &Delta;&Omega; to execute our customized workflow.  Each 3D-rendered image displays eight equatorial-plane streamlines and a pair of isodensity surfaces (lower density surface colored blue; higher density surface colored red) that outline the structure of both stars as well as the connecting mass-transfer stream.  The propagation time is the same for all eight streamlines; along six streamlines, [http://www.vistrails.org VisTrails] carries out the integration in both directions from the location marked by a small colored sphere.]]
<center>'''Figure 2''' (scroll over image for caption)</center>

Figure 2 displays 3D renderings of the flow from one of our binary mass-transfer simulations as generated by our customized [http://www.vistrails.org VisTrails] workflow.  We have generated images assuming five different frame rotation frequencies, as specified by <math>\Delta\Omega</math>.  Aside from labeling <math>\Delta\Omega</math> values under each image, we produced Figure 2 by simply taking a screenshot of the [http://www.vistrails.org VisTrails] interactive spreadsheet.  The spreadsheet feature has proven to be extremely useful in this analysis because it facilitates the side-by-side comparison of scenes that [http://www.vistrails.org VisTrails] has rendered using different parameter values.  And, although we can't demonstrate it here in print, [http://www.vistrails.org VisTrails] lets users zoom, pan, and interactively rotate all 3D-rendered scenes simultaneously.

We'd like to determine which value of <math>\Delta\Omega</math> provides the best measure of the binary star system's true orbital period.  As expected, for all five choices of <math>\Delta\Omega</math>, we found the highest velocities (marked by the longest streamlines) along the relatively low-density mass-transfer stream that connects the two stars.  Material from the donor star (in the lower half of each rendered image in Figure 2) flows toward its stellar companion, reaching supersonic velocities before impacting the companion.  An oblique shock front &#150; whose location is delineated by kinks in the pink, blue, and orange streamlines &#150; terminates the component of motion perpendicular to the companion's surface.  Motion transverse to the shock becomes orbital motion in a thick, low-density disk that surrounds the companion star.

For all five choices of <math>\Delta\Omega</math>, the flow's behavior in the vicinity of the mass-transfer stream very closely resembles the behavior that Stephen Lubow and Frank Shu predicted more than 30 years ago.  Figure 3b shows the magnified view of this region of the flow from our simulation assuming <math>\Delta\Omega = - 0.041</math>.  We've reoriented this magnified image and numbered the streamlines to facilitate comparison with the Lubow and Shu illustration, which we've reprinted with permission here (see Figure 3a).  Rather than conducting a fully self-consistent 3D simulation &#150; which was computationally impractical at the time &#150; Lubow and Shu used a mathematical perturbation analysis to estimate what the flow should look like in the vicinity of the "L1" Lagrange point, as viewed from a frame of reference rotating with the correct instantaneous orbital frequency, <math>\Omega_\mathrm{frame}</math>.  The close resemblance between our 3D simulation results in the vicinity of the L1 Lagrange point, and the behavior that Lubow and Shu predicted provides a useful point of verification for our work.

Each star's center of mass should lie near the center of the highest density region inside each star (outlined by the nearly spherical, red isodensity surfaces in Figure 2).  When we've assigned <math>\Delta\Omega</math> a value that properly identifies the frequency at which the centers of mass of the two stars are orbiting one another, we should see very little residual motion near the donor star's center &#150; that is, streamlines rendered in white and green in Figure 2 should be quite short.  Furthermore, we expect that this residual motion should translate into concave streamline segments, mapping out simple circular motion around the center of the donor star.  When we've identified the correct value of <math>\Delta\Omega</math>, we also should expect the returning streamline nearest the mass-transfer stream (colored yellow) to remain inside the donor.  With these ideas in mind, we judge <math>\Delta\Omega = - 0.041</math>.

Finally, we note that as the pink and blue streamlines curve around the companion star, they extend outside the companion's disk (as outlined by the blue isodensity surface) in the rendered images with the most negative specified values of <math>\Delta\Omega</math>.  These two streamlines appear to align most neatly with the distribution of material in the disk in Figure 2a, that is, for <math>\Delta\Omega = + 0.02</math>.  This suggests that there's a characteristic frequency associated with motion in the companion's disk that's different from the binary orbital frequency.

[[Image:Figure3_Tohline.jpg |border|center|600px|Figure 3.  A magnified view of multiple streamlines in the region of the flow where the mass-transfer stream originates.  (a = left panel) Reproduction of Figure 3 from the published work of Stephen Lubow and Frank Shu (used with permission).  (b = right panel) A magnified segment of the flow depicted in Figure 2, image D (&Delta;&Omega; = -0.041) from this work; we've numbered the colored streamlines to aid in our comparison with the Lubow and Shu image.]]
<center>'''Figure 3''' (scroll over image for caption)</center>