Joel E. Tohline
tohline@rouge.phys.lsu.edu
As was pointed out earlier, there is no particular reason why one should guess ahead of time that the equilibrium properties of any rotating, selfgravitating configuration should be describable in terms of analytical functions. As luck would have it, however, the gravitational potential at the surface of and inside an homogeneous spheroid is expressible in terms of analytical functions. (The potential is constant on concentric spheroidal surfaces that generally have a different axis ratio from the spheroidal mass distribution.) Furthermore, the gradient of the gravitational potential is separable in cylindrical coordinates, proving to be a simple linear function of both v and z.
If the spheroid is uniformly rotating, this behavior conspires nicely with the behavior of the centrifugal acceleration  which also will be a linear function of v (see, in particular, Eq. [I.A.20])  to permit an analytical (and integrable) prescription of the pressure gradient. Not surprisingly, it resembles the functional form of the pressure gradient that is required to balance the gravitational force in uniformdensity spheres. 
As a consequence of this good fortune, the equilibrium structure of a uniformly rotating, uniformdensity (n = 0), axisymmetric configuration can be shown to be precisely an oblate spheroid whose internal properties are describable in terms of analytical expressions. These expressions were first derived by Colin Maclaurin (1742) in A Treatise on Fluxions, and have been enumerated in many subsequent publications (cf. Tassoul 1978; Chandrasekhar 1987).
Let a_{1} be the equatorial radius and a_{3} the polar radius of a uniformdensity object whose surface is defined precisely by an oblate spheroid. The degree of flattening of the object is conveniently parameterized in terms of the object's eccentricity,
The total mass of such a spheroid is,
For the purposes of normalization, we will find it useful to define,
which is the radius of a spherical, n = 0 polytrope that has the same density and mass as the spheroid.
In an accompanying discussion entitled, "Properties of Homogeneous Ellipsoids," an expression [III.R.1] is given for the gravitational potential F(x) at an internal point or on the surface of an homogeneous ellipsoid with semiaxes (x,y,z) = (a_{1},a_{2},a_{3}). For an homogeneous, oblate spheroid in which a_{1} = a_{2} > a_{3}, this analytical expression defining the potential reduces to the form, 
where, as defined on the accompanying page, the coefficients A_{1}, A_{3}, and I_{BT} are functions only of the spheroid's eccentricity.
According to the accompanying introductory discussion, for axisymmetric configurations the Poisson equation takes the following form in cylindrical coordinates:
[Equation III.F.13]

From the above expression [III.G.4] for the gravitational potential, we can deduce immediately that the gradient of the gravitational potential at any point on the surface or inside a uniformdensity spheroid is: 
Show that if we choose to express the properties of Maclaurin spheroids in terms of the system of T1 Coordinates instead of the system of cylindrical coordinates, and define the T1 Coordinate parameter q such that,
q^{2} º A_{3}/A_{1},
[Equation III.G.6]
the gradient in the gravitational potential takes the relatively simple form:
[Equation III.G.7]
In other words,
[Equation III.G.8]
and,
¶_{x2}F
= 0.
[Equation III.G.9]

In addition to deriving a gravitational potential that satisfies the Poisson equation for our chosen density distribution, we also must determine a pressure distribution P(v,z) and a (constant) angular velocity of rotation w that satisfies the conditions derived earlier for hydrostatic equilibrium in uniformly rotating, axisymmetric configurations, namely:
(This last expression is a combination of equations [III.F.22 & III.F.24].) 
By following a procedure analogous to the one that is used in numerical selfconsistentfield algorithms, derive analytical expressions for the equilibrium angular velocity w and equilibrium pressure distribution inside a Maclaurin spheroid. Specifically,
w^{2} = 2p G r [A_{1} 
(1e^{2})A_{3}],
[Equation III.G.10]
and,
[Equation III.G.11]
where,



Notice that in terms of the dimensionless angular velocity
[Equation III.G.13]
=
EFE, Chapter 5, Eqs. (4) & (6)
Tassoul '78, § 4.5, Eq. (52) and § 10.2, Eq. (12)
there is a unique relationship between the star's eccentricity and its equilibrium angular velocity. In the accompanying figure taken directly from EFE, w^{2}/pGr is plotted as a function of e.
EFE Figure 5  
This figure, taken directly from p. 79 of Chandrasekhar 1987, plots the squared angular velocity^{1} as a function of the figure eccentricity along the Maclaurin spheroid sequence (and along the Jacobi ellipsoid sequence, described elsewhere). Numerical values of w^{2} along the Maclaurin sequence also can be obtained from the accompanying Table [III.F.1] or Application [IV.B.1] . 
A
figure taken from
Tassoul (1978)
illustrates how w^{2}/2pGr varies as a function of
t_{rot}, the ratio of
rotational to gravitational potential energy (derived below).
Tassoul's Figure 4.2  
This figure, taken directly from p. 88 of Tassoul (1978), plots the squared angular velocity and the total angular momentum along the Maclaurin sequence, as functions of the energy ratio t = E_{rot}/E_{grav}. Numerical values of J and w^{2} along the Maclaurin sequence also can be obtained from the accompanying Table [III.F.1] or Application [IV.B.1] . 
From either of these two figures, it is clear that W does not increase without limit. Instead, it reaches a maximum W = W_{max} » 0.45 at a value of e » 0.9 and t_{rot} » 0.25. (One application below provides a means by which equation [III.G.13] can be evaluated numerically for any choice of equilibrium eccentricity, and the following homework exercise provides a precise determination of W_{max}.)
As the figure from EFE illustrates, W does not increase without limit but, instead, reaches a maximum, W = W_{max}, at a value of e » 0.9. At what precise value of the eccentricity e_{max} (and at what corresponding maximum t_{rot}) does W = W_{max}? What is the precise value of W_{max}?

To date, no one has inverted equation [III.G.13] in closed form to derive an analytical expression for e as a function of W, but e(W) can be determined numerically. Obviously from the accompanying figure, it is not possible to construct an equilibrium star with W^{2} > W^{2}_{max}, and at W_{max} there is a single value of the eccentricity, e_{max}, for which an equilibrium exists. But, as was first noticed by Simpson (1743), at any value of W selected in the range 0 £ W < W_{max}, two possible equilibrium figures of different eccentricities can be found. (One application below provides a means by which equation [III.G.13] can be evaluated numerically to determine what pair of equilibrium eccentricities are permitted for any choice of angular velocity W < W_{max}, and the preceeding homework exercise provides a precise determination of e_{max}.) 
Drawing on the general definition of the moment of inertia tensor of a uniformdensity ellipsoid [III.R.6], we know that the moment of inertia of a uniformdensity, oblate spheroid about its symmetry axis is, 
[Equation III.G.14]
[Equation III.G.15]
which, dimensionally, has units of angular momentum; and
which has units of energy. Then the total angular momentum of a Maclaurin spheroid is,
its rotational kinetic energy is,
and its gravitational potential energy is,
Hence, the ratio of rotational to gravitational potential energy in a Maclaurin spheroid is,
[Equation III.G.20]
In many of the classical discussions of ellipsoidal figures of equilibrium, the behavior of the function J(e) also usually is illustrated graphically [along with illustrations of the function w(e)]. This is presumably because total angular momentum generally proves to be an important physical parameter of the system. The dimensionless ratio J/J_{norm} is plotted as a function of e in the accompanying figure taken directly from EFE,
EFE Figure 6  
This figure, taken directly from p. 79 of Chandrasekhar 1987, plots the normalized angular momentum J/J_{norm} as a function of the figure eccentricity along the Maclaurin spheroid sequence (and along the Jacobi ellipsoid sequence, described elsewhere). Numerical values of J/J_{norm} along the Maclaurin sequence also can be obtained from column 3 of the accompanying Table [III.F.1] or calculated using the accompanying Application [IV.B.1] . 
and as a function of t_{rot} in a figure taken from Tassoul (1978).
APPLICATIONS  
III.G.1

Table III.F.1 illustrates how various physical parameters vary with e and t_{rot} along the Maclaurin sequence. Through the accompanying application, numerical values of the same set of physical parameters can be calculated for any choice of e or any selected range of eccentricities.  
III.G.2

If you would like to determine the pair of eccentricities at which an equilibrium configuration can be constructed for a given choice of W (or for a range of angular velocities), select Application 2. [NOT YET AVAILABLE] 
^{1}Utilizing the definitions of P_{0} and a_{mean}, it is clear that the leading coefficient in the pressure function P(v,z) (i.e., the pressure at the center of the spheroid) can be written in the form,
[Equation III.G.12']
Tassoul (1978) incorrectly states that this coefficient is 2pGr^{2} a_{3}^{2} A_{3}; too large by a factor of 2.
Home Page  Preface  Context  Applications  Appendices  Search Index 