Re: [Phys-L] field of an ellipsoidal distribution

[David Bowman <David_Bowman@georgetowncollege.edu>]

Regarding Don's question about the Somigliana equation I gave & some more elaboration:

> On August 8, 2015 at 5:14 pm, David Bowman wrote:
>
> > "Another consequence of this result is that the surface gravity
> > g as a function of geodetic latitude, θ must acquire the form:
> >
> > g(θ) = g_0*(1 + k*sin^2(θ))/sqrt(1 - e^2 *sin^2(θ))
> >
> > where g_0 is the surface gravity on the equator, e is the
> > spheroid's eccentricity, and k is a nonnegative constant that
> > depends on how the mass is distributed internally.  If the
> > internal mass distribution has a constant density then k is
> > zero.  For Earth k = 0.00193185 indicating that the Earth is
> > denser in its inside than on its outside.  Also Earth's values
> > for the other two parameters are e = 0.081819191 &
> > g_0 = 9.7803268 m/s^2.  I believe the the above equation for
> > g(θ) is called the Somigliana equation."
>
> The equation doesn't seem to apply to a spherical (e = 0),
> constant density (k = 0) object which is rotating, since you
> get:
>
> g(theta) = g_0
>
> which is correct if the object is not rotating but shows no
> dependence of g on latitude (theta) due to the rotation.   Is k
> not zero for a rotating object even if the density is constant?
> Or ???
>
> Don
>
> Dr. Donald G. Polvani

As JD so aptly pointed out, the equation is *only* for an object that is made of a deformable effectively *fluidic-like* material that can't resist a static shear in equilibrium and actually *is* in mechanical equilibrium, so its surface is an equi-potential surface in a frame rotating with the object where all the material of that object rotates together with the same common angular velocity.  If the object is also stipulated to have a uniform density, and if such a system assumes a spherical shape then it is not rotating.  If it *is* rotating then it is not a sphere sphere but is an ellipsoid.

BTW, in case anyone is interested in some more of the details I'll give here the formulas for the values of the constants g_0 and k in terms of the eccentricity, e and a dimensionless parameter relating the rotation rate to the mass & size for the *non*uniform density case.  Because the formulas are so messy I decided to break them down into parts with some major expression parts being given letter names to make writing and reading of them more manageable.  Also, because of my past problems with posting content with Greek letters I'll spell them out below just to be safe.

Let A == (2/3)*([omega]^2 * a^3)/(G*M) .

Here [omega] is the common rotation rate, a is the equatorial radius, M is the total mass, and G is Newton's universal gravitational constant.  The parameter A is a dimensionless parameter that sort of measures the ratio of the typical strength of the centrifugal field to the typical Newtonian gravitational field at the surface.

The following named dimensionless quantities are utilized mathematically strictly for formula-simplification purposes only.

Let B == (1-e^2)^(3/2) * arcsin(e)/e -1 +(4/3)*e^2
Let C == (1-(2/3)*e^2)*arcsin(e)/e - sqrt(1-e^2)
Let S == 1 - A*B/C

Then the formulas for g_0 and k are given by:

g_0 = G*M*S/(sqrt(1-e^2)*a^2)

k = (e^2)*((e^2)*A/(3*C) -1)/S .

In the special constant uniform density case the parameter A can be rewritten in terms of the density [rho] instead of the separate (no longer independent) parameters a & M.  In that case we have

A = ([omega]^2)/(2*[pi]*G*[rho]*sqrt(1-e^2)) .

Also in the uniform density case there is an additional constraint on the value of A coming from the requirement that the internal field solution match the external one on the outer surface of the object.  This constraint requires that (e^2)*A/(3*C) = 1 , (i.e. k = 0), or equivalently,

([omega]^2)/(2*[pi]*G*[rho]) =
               (3/e^2 -2)*sqrt(1-e^2)*arcsin(e)/e - 3/e^2 + 3 .

A remarkable thing about this uniform density case equation, which purports to implicitly determine the value of e from the [omega]^2)/[rho] ratio, is that the e-dependent RHS has a maximum value over the allowed 0 <= e < 1 interval of (RHS)_max = 0.224665706 when e = 0.929955685.  This means that for a given density the object can't spin any faster than some maximum spin rate given by [omega]_max = 1.188114583*sqrt(G*[rho]) in the uniform density case.  Not only that, because the RHS has a maximum as a function of e, this means that for slower rotation rates than the maximum value there are actually two different e values possible for the same spin rate (one on each side of the maximum).  Of course because the two solutions each have wildly different eccentricities that same spin rate corresponds to two wildly different angular momenta because of the different moments of inertia.  If we wrote the above equation in terms of the angular momentum instead of the spin rate the corresponding RHS would be monotone rising as a function of e rather than being 2 to 1 with a maximum between the two different e values for the same [omega] value.

But this maximum [omega], double e-value stuff is unrealistic and kind of moot anyway because, according to the Chandrasekhar paper to which JD earlier referred, it was shown back in the 19th century that the large eccentricity solution is unphysical because its equilibrium is unstable against tiny pertubations, and no actual object could possibly settle down to that shape.  It was also shown that even on the low eccentricity side (with [omega] below the maximum value) the low eccentricity solution also goes unstable at e >= 0.81267 (as [omega] & e are increased) well before the maximum is reached.  When that happens the stable solution branches off from the oblate spheroid solution becoming a triaxial ellipsoid.  A lot more much more complicated behaviors develop at even higher spin rates and eccentricities, and the Chandrasekhar paper categorizes them and gives some "phase diagrams" showing which kinds of solutions are stable in which parameter regimes.

A really weird thing about that stuff is that some of the solutions (the ones found by Dedekind) actually have the object's triaxial ellipsoidal figure remaining *stationary* in the *inertial* frame as the object's mass rotates around through it at a constant rate, continuously deforming to maintain the fixed ellipsoidal shape as it goes, (kind of like how mountain wave clouds form and remain stationary on the leeward side of a mountain range as the steady winds continue to blow right through them).

Dave Bowman