Stellar structure, magnetism and the variational principle

Abstract

Matter interacts through two long range forces: gravity and electromagnetism. While all matter contributes to the gravitational potential, electromagnetic effects were traditionally expected to cancel in large systems because positive and negative charges balance. Yet astrophysical objects clearly show long range electromagnetic behavior, so the cancellation cannot be perfect. This paper develops a model for stationary aggregation of matter into a star that consistently includes angular momentum and electromagnetic effects. We reformulate the standard polytropic stellar model as a variational problem and extend it to include the kinetic energy of rigid rotation and the electromagnetic interaction energy between oppositely charged baryonic matter. The electromagnetic contribution to the action is taken to be the minimal energy required to generate the stellar magnetic dipole moment. This energy has two parts: the pure electromagnetic contribution, expressible as a surface integral, and the free energy difference between magnetized and unmagnetized matter, obtained by analyzing a degenerate electron gas in a background of cold ions. Differential forms provide a convenient mathematical framework. The resulting model incorporates electromagnetic effects into stellar structure in a way consistent with linearized general relativity. Although the full system forms a complex open boundary problem, exact solutions exist under simplifying assumptions. The phase diagram predicted by the simplified model shows patterns that may motivate further study of the balance between matter, gravitation, and electromagnetism

Figures (10)

• Figure 1: Six different configurations of currents and paramagnetic layers producing the same magnetic dipole moment. Left: top: empty spherical coil, Left: middle: paramagnetic sphere with $\mu$=1000 filling spherical coil. Left: bottom: paramagnetic shell ($0.85r_o<r<r_o$) with $\mu$=5 magnetized by a coil on the inner surface, Right: top: paramagnetic sphere with radius 0.75$\,r_o$ and $\mu=10^3$ inside spherical coil with radius $r_o$, Right:middle: the same paramagnetic sphere as above but driven by current on the surface. Right: bottom: paramagnetic shell as to the left magnetized by a coil on the outer surface. Driving current is represented by blue (for incoming) and red (for outgoing) dots with diameter proportional to current. Black streamlines follow magnetic field lines, while the grayish logarithmically scaled contour lines and the blue to brown color shading plots indicate the distribution of $B^2$.
• Figure 2: Classical orbit corresponding to dynamical solution of $H_{class}$ for $\varepsilon>0$ (left) and $\varepsilon<0$ (right). Orbit starting point is denoted by a red dot. Arrows in magenta depict electric field (\ref{['EFieldB']}).
• Figure 3: Wave functions $\vert n,s\rangle$ for (n=10, s=100), (n=40, s=2000) and (n=80, s=4000) . Red points are at the center ${\tilde{R}}_c=\sqrt{2(n+s-1)}$ and green points mark the width at ${\tilde{R}}_c\pm \frac{3}{2}\sqrt{n}$ .
• Figure 4: The gas of ions (red) occupies a smaller cylinder than the gas of electrons (blue). From $n_e R_{ele}^2-n_i R_{ion}^2=0$ it follows: $(n_e-n_i)R_{ele}^2\approx 2n_i R_{ele}\delta R$, where $\delta R=R_{ele}-R_{ions}$.
• Figure 5: Energy density of magnetized degenerate gas of electrons for different values of magnetic field as a function of electron density. Dotted lines represent the part due to magnetization. The gray boxes indicate conduction electron density in iron and typical average electron density in white dwarfs.