Stellar 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
wrote:
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.

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

On Jul 5, 4:24 pm, tadchem 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

tadchem wrote in
:

On Jul 5, 5:48*pm, John Schutkeker
wrote:
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.

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?

On Jul 6, 8:23*am, John Schutkeker
wrote:
tadchem wrote :





On Jul 5, 5:48*pm, John Schutkeker
wrote:
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.

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.

Tom Davidson
Richmond, VA

On Jul 5, 7:12*pm, Andrew Usher wrote:
On Jul 5, 4:24 pm, tadchem 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.

Equation of state for hydrogen plasma in Jupiter (not as extreme as
intra-stellar environments):
http://www3.interscience.wiley.com/j...TRY=1&SRETRY=0

Tom Davidson
Richmond, VA

tadchem wrote in
:

On Jul 6, 8:23*am, John Schutkeker
wrote:
tadchem wrote
innews:6dd26197-65fb-4e5d-85bd-662578
:





On Jul 5, 5:48*pm, John Schutkeker
wrote:
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 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

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.

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.

tadchem wrote in
:

On Jul 6, 8:23*am, John Schutkeker
wrote:
tadchem wrote
innews:6dd26197-65fb-4e5d-85bd-662578
:

On Jul 5, 5:48*pm, John Schutkeker
wrote:
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 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

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.

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 ,
John Schutkeker 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
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.)

(Steve Willner) wrote in
:

In article ,
John Schutkeker 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.)

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.