Magnetohydrodynamic equilibria are time-independent solutions of the full magnetohydrodynamic (MHD) equations. An important class are static equilibria without plasma flow. They are described by the magnetohydrostatic equations
j
×
B
=
∇
p
+
ρ
∇
Ψ
,
∇
×
B
=
μ
0
j
,
∇
·
B
=
0.
B
is the magnetic field,
j
the electric current density,
p
the plasma pressure,
ρ
the mass density,
Ψ
the gravitational potential, and
µ
0
the permeability of free space. Under equilibrium conditions, the Lorentz force
j
×
B
is compensated by the plasma pressure gradient force and the gravity force.
Despite the apparent simplicity of these equations, it is extremely difficult to find exact solutions due to their intrinsic nonlinearity. The problem is greatly simplified for effectively two-dimensional configurations with a translational or axial symmetry. The magnetohydrostatic (MHS) equations can then be transformed into a single nonlinear partial differential equation, the Grad–Shafranov equation. This approach is popular as a first approximation to model, for example, planetary magnetospheres, solar and stellar coronae, and astrophysical and fusion plasmas.
For systems without symmetry, one has to solve the full equations in three dimensions, which requires numerically expensive computer programs. Boundary conditions for these systems can often be deduced from measurements. In several astrophysical plasmas (e.g., the solar corona), the magnetic pressure is orders of magnitudes higher than the plasma pressure, which allows a neglect of the plasma pressure in lowest order. If gravity is also negligible, Equation 1 then implies a force-free equilibrium in which the Lorentz force vanishes.
Generalizations of MHS equilibria are stationary equilibria including a stationary plasma flow (e.g., stellar winds in astrophysics). It is also possible to compute MHD equilibria in rotating systems (e.g., rotating magnetospheres, rotating stellar coronae) by incorporating the centrifugal force. MHD equilibrium theory is useful for studying physical systems that slowly evolve in time. In this case, while one has an equilibrium at each time step, the configuration changes, often in response to temporal changes of the measured boundary conditions (e.g., the magnetic field of the Sun for modeling the corona) or of external sources (e.g., mass loading in planetary magnetospheres). Finally, MHD equilibria can be used as initial conditions for time-dependent MHD simulations. This article reviews the various analytical solutions and numerical techniques to compute MHD equilibria, as well as applications to the Sun, planetary magnetospheres, space, and laboratory plasmas.

### Article

## Magnetohydrodynamic Equilibria

### Thomas Wiegelmann

### Article

## The Nuclear Physics of Neutron Stars

### Jorge Piekarewicz

Neutron stars—compact objects with masses similar to that of our Sun but radii comparable to the size of a city—contain the densest form of matter in the universe that can be probed in terrestrial laboratories as well as in earth- and space-based observatories. The historical detection of gravitational waves from a binary neutron star merger has opened the new era of multimessenger astronomy and has propelled neutron stars to the center of a variety of disciplines, such as astrophysics, general relativity, nuclear physics, and particle physics. The main input required to study the structure of neutron stars is the pressure support generated by its constituents against gravitational collapse. These include neutrons, protons, electrons, and perhaps even more exotic constituents. As such, nuclear physics plays a prominent role in elucidating the fascinating structure, dynamics, and composition of neutron stars.