## Magnetic fields in astrophysics: an electronic 'textbooklet'

Magnetic fields play an important role in many objects in the universe, from the Sun with its spots and the magnetically heated corona visible during a solar eclipse, to pulsars and the spectacular 'jets' from black holes and protostars. The behaviour of the magnetic field in these objects, however, is very different from experience at home or in physics class, since in astrophysical objects magnetic field lines are 'tied' to an ionized gas. The theory for such magnetic fields, called magnetohydrodynamics (MHD), is explained in a concise textbook published online. It emphasizes understanding of MHD by visualization of the flows and forces as they take place in a magnetized fluid. To this end, the text also includes a number of small video clips of basic MHD flows.

 Fig. 1: Field lines in the head of a magnetic jet. (Rainer Moll, MPA) Fig. 2: A fluid flow stretching a bundle of field lines (mp4 movie) (Merel van 't Hoff, MPA)

In physical processes where magnetic fields are present, one generally also has electric fields, currents and charge densities. Mathematically speaking, one has to deal with the full set of Maxwell's equations plus the equations of motion for the particles making up the plasma - the domain of plasma physics. Luckily, for most flows seen in astronomical objects, however, this complexity is rarely necessary. The electrical conductivity of an ionized gas makes MHD an extremely accurate approximation. Compared with ordinary fluid mechanics, only the magnetic field needs to be included explicitly in the theory. The other electromagnetic quantities can be evaluated afterwards; they are neither needed for a proper description, nor of much use for physical understanding. Thanks to this simplification it has become possible to include magnetic fields realistically in numerical simulations, for example of extragalactic jets (Fig. 1).

The price to be paid is that we have to give up some of our intuitions about the way electric and magnetic fields work. Our experience is dominated by processes taking place in the Earth's electrically insulating atmosphere (in copper wires, batteries, induction coils etc.). Most astrophysical processes on the other hand happen in an ionised gas, such as in a star, the solar wind, or the intergalactic medium.

Because of the strong coupling between the magnetic field and the electrically conducting gas, MHD flows behave more or less like visco-elastic but otherwise ordinary fluids. This makes MHD an eminently visualizable theory (for an example see the video clip in Fig. 2), which also motivates the approach used in the textbooklet. The first chapter (only 36 pages) is a concise introduction including exercises. The exercises are important as illustrations of the points made in the text (especially the less intuitive ones). Almost all are mathematically unchallenging, though some do require a background in undergraduate physics. This is the 'essential' part. The supplement in chapter 2 contains further explanations, more specialized topics and occasional connections to topics somewhat outside the scope of MHD.

#### Reference

H. C. Spruit, "Essential Magnetohydrodynamics for Astrophysics",
arXiv:1301.5572