- 19-52 Aubrey Truman
- An Introduction to the Equatorial Orbitals of Toy Neutron Stars
Oct 4, 19
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Abstract. In modelling the behaviour of ring systems for neutron stars in a non-relativistic setting one has to combine Newton's inverse square law of force of gravity with the Lorentz force due to the magnetic dipole moment in what we refer to as the KLMN equation. Here we investigate this problem using analytical dynamics revealing a strong link between Weierstrass functions and equatorial orbitals.
The asymptotics of our solutions reveal some new physics involving unstable circular orbits with possible long term effects - doubly asymptotic spirals, the hyperbolic equivalent of Keplerian ellipses in this setting. The corresponding orbitals can only arise when the discriminant of a certain quartic, $Q, \Delta_4=0$. When this occurs simultaneously with $g_3$, its quartic invariant, changing sign, the physics changes dramatically. Here we introduce dimensionless variables Z and W for this problem, $\Delta_4(Z,W)=0$, is then an algebraic plane curve of degree 6 whose singularities reveal such changes in the physics. The use of polar coordinates displays this singularity structure very clearly.
Our first result shows that a time-change in the equations of motion leads to a complete solution of this problem in terms of Weierstrass functions. Since Weierstrass's results underlie this entire work we present this result first. Legendre's work leads to a complete solution. Both results reinforce the physical importance of the condition, $\Delta_4=0$, our final flourish being provided by Jacobi with his theta function.