Formation of a magnetosphere for gas giants / small stars This question got me thinking about this.

Jupiter has, and presumably super-Jupiters can have a strong magnetospheres.  A solid metallic core and rotating material around the core, creates a magnetic field and all 4 gas planets in our solar-system have magnetic fields, though Jupiter's is by far the strongest.   Source

As I understand it, a planetary magnetic field requires a solid core and a slight variation in rotation where the liquid outer core rotates in relation to the solid inner core.  The planet's speed of rotation might not be necessary, but it could be a factor.

During the hot period of formation, gas giants might not form magnetic fields but when they cool down enough to have a solid core, they probably would have them.

So, the question is, is there a rough mass for which planet-wide magnetic field formation becomes unstable, where the core of the heavy Jupiter or small or old brown dwarf takes a very very long time to solidify, say, many billions of years?   

I'd think the high temperature of formation and additional heat from fusion in a Brown Dwarf wouldn't be a good situation for that kind of star-wide magnetic field to form due too inevitable convection and probably no solid core.   That kind of star would probably form multiple magnetic dipoles like our sun has, but perhaps in an older brown dwarf a single magnetic field might be possible.

White dwarfs have very strong magnetic fields, so pressure obviously isn't an issue, in fact, I think high pressure can create a stronger magnetic field.
We do not know whether Jupiter has a solid core. That is one of the primary goals of the Juno mission.

In general it is clear that a solid core is not required to generate magnetic fields. The Sun does so and magnetic fields have been measured (with average surface field strengths of around 3 kG) on very low mass stars. Incidentally, the Sun's magnetic field does have a large scale dipole component.

More recently, there have been observations of cyclotron and auroral emission suggesting that even old, cold brown dwarfs have strong, organised magnetic fields. See for example Kuznetsov et al. (2012); Hallinan et al.(2015) and references therein.

http://arxiv.org/abs/1111.7019
http://arxiv.org/abs/1507.08739

At the moment, nobody can say for sure how exactly these fields are generated, sustained and organised.

In fact you could do worse than look at a paper that popped up at the top of today's astro-ph listings by Williams et al.  The abstract begins:

"The well-studied M9 dwarf TVLM 513-46546 is a rapid rotator (P_rot ~ 2 hr) hosting a stable, dipolar magnetic field of ~3 kG surface strength."

http://arxiv.org/abs/1511.05559