Introduction – spinning polytropes

In our continued progression toward decreasing symmetry, we now arrive at the most individualistic systems of all. Previously we discussed flattened systems which were mainly thin disks. But those cannot represent the large observed class of elliptical galaxies. Ellipticals show moderate rotation, and as a result they have less global coherence than either spirals or spherical non-rotating systems. Pressure, rotation and internal currents all combine to determine a variety of structures and shapes.

Historically, some of the great eighteenth, nineteenth- and early-twentieth century mathematicians began to develop this subject by rigorously analyzing certain simple, highly specialized cases. Magnificent though these achievements were – and they often led to important mathematical insights – they also raised the stature of certain models so high that their astronomical importance became greatly exaggerated. All these models were built from uniform, incompressible fluids. The first of these non-spherical self-gravitating models were Maclaurin's (mid-eighteenth- century) spheroids. They have a stately uniform rotation, with no internal motions. Gravity is balanced by incompressibility and rotation. This balance can exist only for certain values of the spin ratio (58.1), nearly equal to unity. Two types of Maclaurin spheroids, with small and large eccentricities, are possible equilibrium figures.

So simple and satisfying seemed Maclaurin's results, that nearly a century went by before Jacobi realized that other equilibrium figures exist.