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
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.
> 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
> 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.
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.
On Jul 5, 4:24 pm, tadchem <tadc...@comcast.net> wrote:
> 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.
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.
>> 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
>> 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.
> 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.
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?
> >> 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
> >> 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.
> > 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.
> 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?
My bad... Try "stiffened equation of state". <31 hits>
This will give you an entré into non-ideal fluid behavior.
> On Jul 5, 4:24 pm, tadchem <tadc...@comcast.net> wrote:
> > 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.
> 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
Non-ideality is significant even in "ordinary" densities. I have measured viscous flow of helium at millitorr pressures and ambient (~25° C) temperatures.
>> >> 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 th > e >> >> 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
>> >> 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.
>> > You might try a more realistic equation of state. The one you have >> > i > s >> > 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 wel > l >> > 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.
>> 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?
> My bad... > Try "stiffened equation of state". <31 hits>
> 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.
>> >> 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 th > e >> >> 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
>> >> 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.
>> > 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.
>> 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?
> My bad... > Try "stiffened equation of state". <31 hits>
> 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.
In article <Xns9AD2B538FAB62alsfaskldfjakls...@207.115.33.102>, John Schutkeker <jschutke...@sbcglobal.net.nospam> writes:
> 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.)
-- Steve Willner Phone 617-495-7123 swill...@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.)
> In article <Xns9AD2B538FAB62alsfaskldfjakls...@207.115.33.102>, > John Schutkeker <jschutke...@sbcglobal.net.nospam> writes: >> 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 i
