Sofia Kovalevskaya

Leigh Ellison

Saturn's Rings

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The last of the three papers which she submitted to the University of Göttingen in July of 1874 was an investigation into the form of Saturn's rings. Whilst she was working with Weierstrass at the time, it is likely that she wrote this particular paper on her own initiative as he never really showed any particular interest for this particular subject. A character trait which she shared with her teacher meant that she was prone to lose interest in a problem when she had solved it to her own satisfaction. This helps to explain why many of the results contained in this paper are merely suggested and not actually proved. Had she provided the relevant theorems giving the conditions under which her technique could be applied, this work, might have been one of the outstanding early papers in this area.

This paper dealt with the stability of motion of liquid ring-shaped bodies. Sofia was to prove that Saturn's rings were egg-shaped ovals, symmetric about a single point. This work was later made obsolete by the proof showing that the rings actually consist of discrete particles and not a continuous liquid. Her work in this paper contained much that was astutely done however, and is of interest for that reason.

Working with the assumption made by Laplace that the rings were elliptic, Sofia was first to compute the potential of the ring at the point with cylindrical coordinates (\rho1, θ1, z1) = (1 - cos(t, θ1, aφ(t1)). This is the most difficult part of the problem and saw Sofia use Gauss's divergence theorem. She was to break the potential of the ring up into terms and apply the series expansion for the complete elliptic integral. V = 1/2\int_{-\infty}\infty\int_{-\infty}\infty cos θ dœÉ gives the gravitational potential at the point P1 of a solid body occupying the region B, where the integral is taken over the boundaries of this region. Since she was interested in a ring-shaped body, Sofia set , and parameterised the surface by angles and ranging from 0 to 2p.
x = (1 - a cos t) cos ψ
y = (1 - a cos t) sin ψ
z = aφ(t)

with similar expressions for P1 in terms of t1 and ψ1.

Having calculated the potential energy she then went on to express the terms making up the total energy as Fourier series:
M(ρ12+ z12)-1/2= m0+ m1cos t1+ m2cos t1+ ...

before writing the energy equation V + M(ρ12+ z12)-1/2+ 1/2n2ρ12 as the system:
V0= m0= 1/2n2(1 + 1/2a2) - C = 0
V1+ m1- n2a = 0
V2+ m2+ 1/2n2a2= 0
Vj+ mj= 0 , j = 3,4, ...

where Vj and mj are expressed in terms of β's. This system of equations was then solved for n, C, and the β's. The method suggested by Sofia for dealing with the problems which arose due to the infinite set of unknowns in each of these equations was really the most important part of the entire paper.

Sofia said that she was deterred from doing more precise calculations on this subject,
not only by the difficulty of the calculation, but also by the fact that due to Maxwell's research it has become doubtful whether Laplace's view of the structure of the rings of Saturn is acceptable.

Sofia did not regard this work as being of particular importance, and only published it having been hired by the University of Stockholm and feeling that she should publish as much as possible. It would seem that her heart was not really in this work, and it seems to have only been included in her dissertation at the last moment.

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