Editing Appendix/Ramblings/Saturn (section)

=Binary Mass-Transfer=
In connection with our own efforts to realistically model dynamical, mass-transfer events in close binary systems, we have noticed the spontaneous development of standing waves in the equatorial regions of the accreting star.  As is illustrated in Figure 3, below, during each of the three cited model evolutions (A, B, and C) we have identified standing waves that are very clearly 3-sided (triangular; bottom row of Figure 3), 4-sided (box-shaped; middle row of Figure 3), or 5-sided (pentagonal; top row of Figure 3).  In association with our published discussion of each of these three evolutions &#8212; relevant links are provided at the top of each Figure column &#8212; an animation sequence has been provided that shows the model's time-evolutionary behavior.  (Evolution A has been followed through 15.3 P<sub>0</sub>; evolution B has been followed through 18.3 P<sub>0</sub>; and evolution C has been followed through 34.0 P<sub>0</sub>, where P<sub>0</sub> is the associated model's initial binary orbital period.) If you watch any one of these evolutions, you will see the smooth development of these nonlinear-amplitude, standing-wave structures in sequence.  Looking closely, at an appropriate time in each evolution, you should also be able to identify the development of a (low-amplitude) standing wave structure that has 6 sides; that is, a hexagonal standing wave.  We have wondered whether the physical processes that conspire to resonately excite a hexagonal-shaped standing wave in our binary mass-transfer simulations is related to the physical processes that are responsible for creating and ''sustaining'' the hexagonal-shaped storm in Saturn's northern hemisphere.


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  <th align="center" colspan="3">FIGURE 3: &nbsp; Nonlinear-Amplitude Distortions that Develop in Three Separate Model Evolutions</th>
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  <th align="center">Evolution A</th>
  <th align="center">Evolution B</th>
  <th align="center">Evolution C</th>
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[https://iopscience.iop.org/article/10.1086/522076/fulltext/71427.html MFTD (2007)]<br />
Model Q0.4D <br />[https://youtu.be/lFR5S_Fc-9w  YouTube Animation]:  video4.mpg<br />
(link is in caption of their [https://iopscience.iop.org/article/10.1086/522076/fulltext/71427.figures3.html Figure 3])
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[https://iopscience.iop.org/article/10.3847/1538-4365/aa5bde MFSCFEDT (2017)]<br />
Model Q0.4P_S1 <br />Animation:  apjsaa5bdef26_video.mpg<br />
(link is in caption of their [https://iopscience.iop.org/article/10.3847/1538-4365/aa5bde#apjsaa5bdef26 Figure 26])
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[https://iopscience.iop.org/article/10.3847/1538-4365/aa5bde MFSCFEDT (2017)]<br />
<sup>&dagger;</sup>Model Q0.5P_G1 <br />Animation:  apjsaa5bdef21_video.mpg<br />
(link is in caption of their [https://iopscience.iop.org/article/10.3847/1538-4365/aa5bde#apjsaa5bdef21 Figure 21])
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  <td align="center" bgcolor="black">[[File:Q04DcroppedD.png|300px]]</td>
  <td align="center" bgcolor="black">[[File:Q0.4P_S1croppedD.png|300px]]</td>
  <td align="center" bgcolor="black">[[File:Q0.5P_G1croppedD.png|300px]]</td>
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  <td align="center" bgcolor="black">[[File:Q04D_squareD.png|300px]]</td>
  <td align="center" bgcolor="black">[[File:Q0.4P_S1_squareD.png|300px]]</td>
  <td align="center" bgcolor="black">[[File:Q0.5P_G1_squareD.png|300px]]</td>
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  <td align="center" bgcolor="black">[[File:Q04D_triangleD.png|300px]]</td>
  <td align="center" bgcolor="black">[[File:Q0.4P_S1_triangleD.png|300px]]</td>
  <td align="center" bgcolor="black">[[File:Q0.5P_G1_triangleD.png|300px]]</td>
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  <td align="left" colspan="3"><sup>&dagger;</sup>The evolution identified here as Model Q0.5P_G1 was first discussed in &sect;5.2 of [https://ui.adsabs.harvard.edu/abs/2006ApJ...643..381D/abstract DMTF (2006)], wherein it was identified as Model Q0.5-Da; the caption to Figure 7 of that paper contains a link to an (mpeg_file = video3-2.mpg) animation that presents a 3D rendering of this model's evolution.</td>
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Several key references:
*  [Paper DMTF (2006)] [https://ui.adsabs.harvard.edu/abs/2006ApJ...643..381D/abstract M. C. R. D'Souza, P. M. Motl, J. E. Tohline &amp; J. Frank (2006)], ApJ, 643, p. 381:  ''Numerical Simulations of the Onset and Stability of dynamical Mass Transfer in Binaries''
*  [Paper RPA (2007)] [https://ui.adsabs.harvard.edu/abs/2007MNRAS.380..381R/abstract &Eacute;. Racine, E. S. Phinney &amp; P. Arras (2007)], MNRAS, 380, 381:  ''Non-dissipative tidal synchronization in accreting binary white dwarf systems''
*  [Paper MFTD (2007)] [https://ui.adsabs.harvard.edu/abs/2007ApJ...670.1314M/abstract P. M. Motl, J. Frank, J. E. Tohline &amp; M. C. R. D'Souza (2007)], ApJ, 670, p. 1314:  ''The Stability of Double White Dwarf Binaries Undergoing Direct-Impact Accretion''
*  [Paper MFSCFEDT (2017)] [https://ui.adsabs.harvard.edu/abs/2017ApJS..229...27M/abstract P. M. Motl, J. Frank, J. Staff, G. C. Clayton, C. L. Fryer, W. Even, S. Diehl &amp; J. E. Tohline (2017)], ApJSuppl., Vol. 229, Issue 2, article id. 27, 41 pp.:  ''A Comparison of Grid-based and SPH Binary Mass-transfer and Merger Simulations''

The following discussion has largely been extracted from &sect;3.1.4 (p.29) of [https://ui.adsabs.harvard.edu/abs/2017ApJS..229...27M/abstract MFSCFEDT (2017)]:  
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<font color="green">In &sect;4 of their paper, [https://ui.adsabs.harvard.edu/abs/2006ApJ...643..381D/abstract MFTD (2007)] point out that in the vicinity of the accretor some of the models developed nonlinear-amplitude "equatorial distortions with [azimuthal mode numbers] 6 &ge; m &ge; 3."</font>  As is illustrated by the trio of images displayed in the bottom row of Figure 3, above, at a certain point (or points) in each of these binary mass-transfer evolutions <font color="green">&hellip; the disk surrounding the accretor has a triangular shape &hellip;</font> presumably associated with the excitation of an m = 3 mode.  This trio of ''triangular shaped'' images come from, respectively, evolutionary times:  (A, B, C)<sub>m=3</sub> = (15.3P<sub>0</sub>, 17.50P<sub>0</sub>, 24.33P<sub>0</sub>).  Similarly, the trio of images displayed in the middle row of Figure 3 show the excitation of an m = 4 (box-shaped) mode at evolutionary times:  (A, B, C)<sub>m=4</sub> = (14.1P<sub>0</sub>, 16.44P<sub>0</sub>, 30.48P<sub>0</sub>).  And the trio of images displayed in the top row of Figure 3 show the excitation of an m = 5 (pentagonal-shaped) mode at evolutionary times:  (A, B, C)<sub>m=5</sub> = (13.1P<sub>0</sub>, 15.94P<sub>0</sub>, 32.72P<sub>0</sub>).


<font color="green">We suspect &hellip; that "standing-wave" azimuthal distortions of this type are routinely excited in</font> [these types of binary mass-transfer events] <font color="green">as a result of dynamical interactions between the mass-transfer stream and the accretion disk.  However, the distortions do not grow to nonlinear amplitude, and therefore are not visible to the eye, unless</font> [the mass-transfer rate] <font color="green">is sufficiently large.</font>
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The following discussion has largely been extracted from &sect;4 (esp. pp. 1322-1324) of [https://ui.adsabs.harvard.edu/abs/2007ApJ...670.1314M/abstract MFTD (2007)]:  
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<font color="green">&hellip; [https://ui.adsabs.harvard.edu/abs/2007MNRAS.380..381R/abstract RPA (2007)] discussed the possibility that a resonance condition between the orbital frequency and the eigenfrequencies of some of the generalized ''r''-modes in the accretor star saturates its spin and channels rotational kinetic energy into oscillation modes.  This translates into a dissipationless torque that is capable of returning the spin angular momentum back to the orbit and thus increasing the efficiency of tidal coupling.  The fact that in our nonlinear simulations the change in the spin of the accretor is coupled to the dynamics of the binary and that we see equatorial distortions with 6 &ge; m &ge; 3 &hellip; seem at first sight to suggest that these modes play a role.  However, when examined in detail, there are some aspects of the evolution that are inconsistent with [this] interpretation.</font>
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<li>&hellip; Strictly speaking, the [https://ui.adsabs.harvard.edu/abs/2007MNRAS.380..381R/abstract RPA (2007)] <font color="green">calculated mode frequencies and estimated spin saturation frequencies are valid for an accretor in solid body rotation, whereas the accretor in our simulations is differentially rotating and develops a prominent "accretion belt";</font></li>
<li><font color="green">the extremely high accretion rate and stream impact in our simulations are significant deviations from the conditions assumed in [https://ui.adsabs.harvard.edu/abs/2007MNRAS.380..381R/abstract RPA (2007)];</font></li>
<li><font color="green">our simulations place no restrictions on the number, amplitude, or character of modes present, whereas [https://ui.adsabs.harvard.edu/abs/2007MNRAS.380..381R/abstract RPA (2007)] only consider generalized ''r''-modes in the linear regime.</font></li>
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