# Solar Astrophysics, Modelling of Prominences, Coronal Mass Ejections, Equilibrium and Stability of Astrophysical Structures.

## Equilibrium and stability of astrophysical structures

### Equilibrium

Long lived structures may only exist if they are in what is called a 'stable equilibrium' i.e. forces are balanced. This may apply to mechanical forces e.g. for a prominence, the downward gravitational force is balanced by the upward magnetic force, or it can apply to thermal effects i.e. heat sources and sinks must balance.

### Stability

In addition, an equilibrium must be stable for it to exist for any reasonable time. An equilibrium is stable if forces close to equilibrium force the structure to return to equilibrium i.e. the structure will return to equilibrium if there are slight perturbations. An unstable equilibrium, however, is one where the forces close to equilibrium force the structure further away from the equilibrium. Thus a structure in an unstable equilibrium would evolve away from this equilibrium under even the smallest perturbation. Thus, structures in unstable equilibria are likely to be short-lived.

### Linear and non-linear stability

It is important to consider the difference between linear stability and non-linear stability. Linear stability refers to small perturbations (i.e. the situation very close to equilibrium) while non-linear stability refers to larger perturbations (i.e. the situation some distance from equilibrium).

Is it possible, for example, for a structure to be linearly stable but non-linearly unstable i.e. the structure will resist small perturbations but will evolve away from equilibrium when the perturbation is larger. In the opposite case, linear instability but non-linear stability, small perturbations will cause evolution to start but the non-linear terms will damp and stop this evolution. Of course, the linear and non-linear terms may act in a consistent manner.

One important consequence of non-linear stability is that perturbations in one direction may be stabilised while those in the other direction may grow. Thus, a structure may be able to cool but not to heat, or may be able to rise but not to fall.

### Thermal stability and instability

A thermal equilibrium consists of a balance between thermal conduction, radiative cooling and heat sources. Normally, conduction has a stabilising effect while the balance between heating and radiation can be de-stabilising, especially if the radiation is not a monotonically increasing function of the temperature. For example, in the solar corona, the radiation decreases with increasing temperature for temperatures around one hundred thousand to one million degrees. Structures at these temperatures may cool catastrophically, perhaps leading to prominence formation.

### Recent work

Most work on thermal stability has dealt with isothermal structures. Recent work however, has brought in the effects of non-isothermallity, dealing with both the linear and non-linear cases.

Another new direction is that of combining the effects of dynamics and second-order terms.

### Recent Papers

Nonlinear Thermal Stability in Two Dimensions with MH Ibanez

Non-linear Thermal Stability in Optically Thin Plasmas with MH Ibanez and E Sira.

### Solar Astrophysics

An important and fascinating element of the Sun is its outer atmosphere, also known as the corona. Although the surface of the Sun (photosphere) is at a temperature of around six thousand degrees, the corona is at a temperature of one or two million degrees. It exists in the plasma state i.e. atoms are stripped of their electrons. Because of this magnetic effects are of extreme importance and play a leading role in the structures described below.

### Modelling of prominences

Prominences are regions of cool, dense material embedded in the hotter corona. They are known to be associated with the magnetic field. Recent observations have shown them to possess fine structure rather than being 'blocks' of cool material. Prominences can last for up to six months and are often seen to erupt into space. Unanswered questions on prominences include the following.

#### How are they supported against gravity ?

It is believed that prominences exist in dips in the magnetic field i.e. there is an upwards magnetic tension force counteracting the downwards gravitational force. Two common configurations are the 'normal-polarity proninence' where the overlying magnetic field points in the same direction as the field in the prominence and the 'inverse-polarity prominence' where the opposite is true. Much work has been done on putting this on a firm mathematical basis. One line of work has been to incorporate the fibril structure. This has been achieved by dividing the prominence up into hot and cool regions in the horizontal direction and has been generalised by incorporating variations in the vertical direction.

#### How does a cool, dense region form from a less dense region ?

Forming a cool region at the centre of a hot region involves some process to cause heat to travel from the cooling region to the warmer surroundings. It is thought that the process of 'thermal instability' is responsible for this.

#### Why do prominences erupt if they have happily lasted for several months ?

If a prominence has existed in the same situation for weeks or even months, it may seem strange for it to begin erupting at speed. However, it is believed that the eruption is caused by either the equilibrium or the stability of the equilibrium being lost. This phenomenon has much in common with the eruption of coronal mass ejections.

### Coronal Mass Ejections

These are often observed above erupting prominences. In fact a configuration normally consists of three components, an erupting prominence, a void or cavity above it, and a shell or bubble above this, all erupting outwards. Normally eruption speeds start off low but then increase before gradually tending towards a constant velocity.

The question was addressed of how the configuration can remain in roughtly the same position for weeks or months and then change so rapidly. It is believed thatfor the 'life-time' of a prominence, it is in a stable equilibrium. As conditions change, the configuration will evolve towards a neighbouring stable equilibrium. This can happen at speeds close to quasi-static.

However, it is possible that the configuration can evolve towards a situation where there is no neighbouring equilibrium. In this case, a small perturbation will cause the system to move away from the equilibrium and the speed of this move will increase. Later, as the configuration (coronal mass ejection or CME) moves further away from the Sun and forces decrease, the speed levels off. This has been put on a mathematical basis.

### Equilibrium and stability of astrophysical structures

#### Equilibrium

Long lived structures may only exist if they are in what is called a 'stable equilibrium' i.e. forces are balanced. This may apply to mechanical forces e.g. for a prominence, the downward gravitational force is balanced by the upward magnetic force, or it can apply to thermal effects i.e. heat sources and sinks must balance.