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Space Forum / Astronomy / July 2008Stellar Hydrostatic Equilibrium with Differential Rotation
I followed the procedure at the following site, http://www.astro.utu.fi/~cflynn/Stars/l4.html and I believe that I've correctly added a centripetal force term to the derivation of the differential equation for stellar hydrostatic equilibrium. That equation relates dP/dr to d_rho/dr, where P is the hydrostatic pressure, rho is the mass density and r, of course, is the radial coordinate. I added the centripetal force, rho*w^2*r, as a second body force term in equation (11) on that site, making that equation into dP/dr=rho(r)*w^2*r*sin(theta)-G*m(r)*rho(r)/r^2, where w is omega, the rotational frequency of the star, which is allowed to vary with the latitude on the star, as w=w(theta). The middle part is pretty much just algebra, and the answer I got was (1/r^2)*d[(r^2/rho)(dP/dr)]/dr+4*pi*G*rho=[3w^2+2*w*r*dw/dr]*sin(theta). Actually, you can tell by the presence of a dw/dr term that I allowed w to vary as a function of both r and theta, but I have no idea whether the is necessary to understand the physics, so I kept it for completeness and just in case. If I were to put in the compressible equation of state for a fluid, P=K*rho^gamma, my result would be an improved Lane-Emden equation. I haven't done that yet, but so far it looks straightforward, and I plan to do it within the day or two. In summary, I need to know whether the above equations and derivation look familiar to the people in this group, and does anybody know if this particular approach to the problem has ever been taken before. I have to know whether I have just reinvented the wheel, so I can start thinking about whether to get the entire derivation published, rather than just the first and last equations. :b Any reasonable input would be greatly appreciated. TIA. On Jul 5, 5:48 pm, John Schutkeker <jschutke...@sbcglobal.net.nospam> wrote: > I followed the procedure at the following site, > [quoted text clipped - 36 lines] > > Any reasonable input would be greatly appreciated. TIA. You might try a more realistic equation of state. The one you have is an ideal thermodynamic gas equation with considerations made for heat capacities (gamma = Cp / Cv is the adiabatic index - the ratio of specific heats). This might be sufficient for gases at densities well below those found at the critical point, but at higher densities, the fluid becomes 'incompressible'. You might try a 'hardened' equation - one which has special considerations for high pressures, in which drho/dP is less variable at higher pressures. Tom Davidson Richmond, VA > You might try a more realistic equation of state. The one you have is > an ideal thermodynamic gas equation with considerations made for heat [quoted text clipped - 6 lines] > considerations for high pressures, in which drho/dP is less variable > at higher pressures. This is wrong - stars like the Sun are nearly ideal throughout, though with varying heat capacity in the ionisation zones. The deviation from ideality that does occur in stars at high density is well modeled by electron degeneracy pressure. Andrew Usher > > You might try a more realistic equation of state. The one you have is > > an ideal thermodynamic gas equation with considerations made for heat [quoted text clipped - 13 lines] > > Andrew Usher Non-ideality is significant even in "ordinary" densities. I have measured viscous flow of helium at millitorr pressures and ambient (~25° C) temperatures. Equation of state for hydrogen plasma in Jupiter (not as extreme as intra-stellar environments): http://www3.interscience.wiley.com/journal/114277408/abstract?CRETRY=1&SRETRY=0 Tom Davidson Richmond, VA > On Jul 5, 5:48 pm, John Schutkeker <jschutke...@sbcglobal.net.nospam> > wrote: [quoted text clipped - 55 lines] > considerations for high pressures, in which drho/dP is less variable > at higher pressures. I googled "hardened equation of state," but nothing came up. Can you write down that equation for gamma, or give me a reference to a place where I might look it up, either online or in a printed journal article? On Jul 6, 8:23 am, John Schutkeker <jschutke...@sbcglobal.net.nospam> wrote: > > On Jul 5, 5:48 pm, John Schutkeker <jschutke...@sbcglobal.net.nospam> > > wrote: [quoted text clipped - 60 lines] > write down that equation for gamma, or give me a reference to a place > where I might look it up, either online or in a printed journal article? My bad... Try "stiffened equation of state". <31 hits> This will give you an entré into non-ideal fluid behavior. Tom Davidson Richmond, VA > On Jul 6, 8:23 am, John Schutkeker <jschutke...@sbcglobal.net.nospam> > wrote: [quoted text clipped - 81 lines] > > This will give you an entré into non-ideal fluid behavior. That gives me a lot of hits, but I can't seem to find a tutorial on the basic material. Wikipedia is the only thing that's even close, but they only wrote down the equation for a stiffened state equation of an incompressible fluid. But since I'm using PV^gamma=const, I need it for a compressible fluid. The most focused search that I did was "tutorial compressible stiffened equation of state," but still all I get are references to actual papers with that phrase in them, for which I'd have to pay $35 to find out whether they do or don't. Let me know if you have any ideas for sites with tutorials on the beginning material of stiffened state equation for a compressible fluid. > On Jul 6, 8:23 am, John Schutkeker <jschutke...@sbcglobal.net.nospam> > wrote: [quoted text clipped - 79 lines] > > This will give you an entré into non-ideal fluid behavior. That gives me a lot of hits, but I can't seem to find a tutorial on the basic material. Wikipedia is the only thing that's even close, but they only wrote down the equation for a stiffened state equation of an incompressible fluid. But since I'm using PV^gamma=const, I need it for a compressible fluid. The most focused search that I did was "tutorial compressible stiffened equation of state," but still all I get are references to actual papers with that phrase in them, for which I'd have to pay $35 to find out whether they do or don't. Let me know if you have any ideas for sites with beginning material or tutorials on the of stiffened state equation for a compressible fluid. > I added the centripetal force, rho*w^2*r, as a second body force term in > equation (11) on that site, making that equation into > > dP/dr=rho(r)*w^2*r*sin(theta)-G*m(r)*rho(r)/r^2, > where w is omega, the rotational frequency of the star, which is allowed to > vary with the latitude on the star, as w=w(theta). What you want to do is add the centrifugal force on the right side. It looks to me, without checking a text, like you have it right if you change sin(theta) to cos(theta). (Centrifugal force is zero at the poles.) I've seen this sort of thing in text books. Centrifugal force is only important for rapidly-rotating stars. I vaguely remember having seen comments to the effect that one really needs 3d, not just 2d, models for those, but I don't know whether that's true or not. There are certainly stellar models for rapid rotators in the literature. While I suspect there is work yet to be done on the subject, it is not as if no one has thought about it before. You should try an ADS search if you haven't already. > If I were to put in the compressible equation of state for a fluid, As someone else mentioned, the ideal gas law is fine for all but the most extreme stars. (The high temperature overcomes the high density until degeneracy sets in.)  SignatureSteve Willner Phone 617-495-7123 swillner@cfa.harvard.edu Cambridge, MA 02138 USA (Please email your reply if you want to be sure I see it; include a valid Reply-To address to receive an acknowledgement. Commercial email may be sent to your ISP.) >> I added the centripetal force, rho*w^2*r, as a second body force term >> in equation (11) on that site, making that equation into [quoted text clipped - 7 lines] > you change sin(theta) to cos(theta). (Centrifugal force is zero at > the poles.) That's right, and sin(theta) is zero at the poles. I didn't swap the two. > I've seen this sort of thing in text books. Centrifugal force is > only important for rapidly-rotating stars. Let me know if you can quote a reference or find a link, because I live in chronic fear of reinventing the wheel. I also live in fear of inventing an irrelevant problem, and I fear that rapid rotators may only include white dwarves and neutron stars. Do we have the observational abilities to find these rapidly rotating ordinary stars in the galaxy, and if so, have we ever found any? The equation I derived is nonlinear and inhomogeneous, and I recently learned on sci.math that the sophomore techniques we learn for solving inhomogeneous ODE's, ie. superposing general and particular solutions, only work for linear ODE's. I fear that may be a dead end, but I have to go back there and ask if there are any known techniques. ;( > I vaguely remember having > seen comments to the effect that one really needs 3d, not just 2d, > models for those, but I don't know whether that's true or not. It's probably true, since the sun is so extraordinarily complex. Putting aside radial flows from thermal buoyancy, I believe that whatever phenomenon causes the differential rotation should also cause a corresponding radial flow. I'm not even sure we know what causes the differential rotation, in which case, we are obliged to assume the worst, until the contrary is reliably proven. AFAIK, there's no way to get observational information about radial flows, except perhaps in a very thin layer right beneath the surface, and maybe not even that much. Besides, it probably makes the problem so complex that it can only be solved by simulation. I don't like simulations, because they obscure the mathematical forms of the underlying equations. All models are mathematical approximations, of varying degree of accuracy, and everything theoretical we say is an approximation. We're making laminar arguments, but it's common knowledge solar flows are turbulent. Thus everything we write is such an extraordinarily gross approximation, and in fact, it's completely wrong, but that's no reason to stop work. Turbulent mathematics won't be solved any time soon, and lacking that solution, laminar theory is the only game in town. Theories are built up in stages, from the easy to the difficult, and before we can try to understand the 3D flow, we have to understand the 2D flow. If that's still incomplete, it must be finished before moving on. And even if 3D were finished, 2D would still have to be done, to fill in the empty gap. > There > are certainly stellar models for rapid rotators in the literature. > While I suspect there is work yet to be done on the subject, it is > not as if no one has thought about it before. You should try an ADS > search if you haven't already. There is one hydrostatic paper, using analytic continuation (whatever that is) to set up a numerical integration. It's got at least one good point I hadn't thought of, but it's completely different from my idea to solve the nonlinear equation with a series solution. I burned out on simulations when I was in my twenties, but series solutions should be standard procedure by now, in the august tradition of Bessel and Legendre. Computers have their uses, but they're not as great as everybody seems to think. Beyond that, the ADS papers all hydrodynamic, and with the exception of one paper that's all math, it looks like computer simulations dominate the library, too. The point of interest seems to be what instabilities arise as the spin rate increases, but I'd be more interested in just how the geometry changes, becoming oblate or prolate. However, it raises the question of whether rapidly rotating stars are by definition generally relativistic, and whether my analysis could find an application in an actual star. >> If I were to put in the compressible equation of state for a fluid, > > As someone else mentioned, the ideal gas law is fine for all but the > most extreme stars. (The high temperature overcomes the high > density until degeneracy sets in.) Honestly, I've got to say, Davidson's idea rings true that the equation of state must be stiffened. However, I doubt that anybody has any clue what stiffening equation describes a stellar plasma, nor what stiffening constant it would contain, nor how to keep the mess of the equations manageable if it were introduced. In other words, that idea may simply be too complex to accomplish, which is what I think you may really have been saying. In this work, it's hard to keep from going off on wild goose chases and becoming enchanted by the next insanely cool idea that appears on one's radar. Just as a general paradigm of working life, it's important to keep our equations comfortably simplified. The real trick is not to throw the baby out with the bathwater, by simplifying too much. One's work still has to be relevant, even if it is simplified. [> >> latitude on the star, as w=w(theta).] > That's right, and sin(theta) is zero at the poles. You must have meant w=colatitude, then. I'm sure you are doing your calculations right. > Let me know if you can quote a reference or find a link, because I live > in chronic fear of reinventing the wheel. I strongly suspect that's what you are doing. Here's a preprint of a recent article on Vega: http://arxiv.org/abs/0803.3145 (The article is published in ApJ, but I'm assuming you don't have easy access to that.) You will have to trace references backwards, but you can see the sort of work that's being done. > I also live in fear of inventing an irrelevant problem, and I fear that > rapid rotators may only include white dwarves and neutron stars. As noted in the article, Vega is a rapid rotator, and it's far from the only one known. > The equation I derived is nonlinear and inhomogeneous, and I recently > learned on sci.math that the sophomore techniques we learn for solving > inhomogeneous ODE's, ie. superposing general and particular solutions, > only work for linear ODE's. I fear that may be a dead end, but I have > to go back there and ask if there are any known techniques. All the work nowadays is numerical solutions. You have to do that for the opacity anyway, so you might as well just use numerical calculations for everything. > Putting aside radial flows from thermal buoyancy, I believe that > whatever phenomenon causes the differential rotation should also cause a > corresponding radial flow. There's convection (at least in some stars), but I'm not sure there are bulk radial flows. This is _very_ far outside my expertise, but I _think_ there are latitudinal bulk flows at least in some stars. Or at least I vaguely remember hearing something to the effect that there might be. By analogy with Hadley cells in Earth's atmosphere, I would think such circulation might well be important, but it could be that it isn't for some reason. > AFAIK, there's no way to get observational information about radial > flows, except perhaps in a very thin layer right beneath the surface, > and maybe not even that much. Helioseismology comes to mind, but I really don't know. Some stars show evidence of "dredge-up" of elements from the core, but whether this has anything to do with rapid rotation, I don't know. > We're > making laminar arguments, but it's common knowledge solar flows are > turbulent. Modern models consider turbulence, at least at some approximation. Or at least that's my understanding. See again "...outside my expertise!" > However, it raises the question of whether rapidly rotating stars are by > definition generally relativistic, Not at all, at least for main sequence stars. You will have to consider GR for neutron stars and possibly for white dwarfs (though I think not), but I'm not sure those stars qualify as rapid rotators. Sure, the periods are short, but the surface gravities are large, and the question is how centrifugal force compares with that. I don't know and don't have time to work it out, but it wouldn't be hard to do. > > the ideal gas law is fine for all but the > > most extreme stars. (The high temperature overcomes the high > > density until degeneracy sets in.) > > Honestly, I've got to say, Davidson's idea rings true that the equation > of state must be stiffened. I think you might want to put in some numbers. > However, I doubt that anybody has any clue > what stiffening equation describes a stellar plasma, I'd be astonished if the equation of state isn't known. Think of the applications for high temperature, high pressure plasmas and how much money has gone into both theory and experiment on the subject. It's conceivable that the best results are classified, but I'd expect usable results to be public. One piece of evidence is that solar models reproduce helioseismology results to exquisite accuracy, and they couldn't do that if the equation of state were wrong. If you are serious about working on this subject, I think you are going to need guidance from an expert. SignatureSteve Willner Phone 617-495-7123 swillner@cfa.harvard.edu Cambridge, MA 02138 USA (Please email your reply if you want to be sure I see it; include a valid Reply-To address to receive an acknowledgement. Commercial email may be sent to your ISP.) |
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