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Astron. Astrophys. 343, 615-623 (1999)

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2. Equilibrium models

In the thin flux tube approximation the force balance tangential [FORMULA] and normal [FORMULA] to the flux tube at [FORMULA] is given by (the subscript `0' denotes the equilibrium values)

[EQUATION]

[EQUATION]

where the curvature [FORMULA] of the flux tube follows from

[EQUATION]

The pressure balance in lateral direction is given by

[EQUATION]

The usual notation of variables is adopted, a subscript `e' denotes external quantities, a hat unit vectors. As equation of state we used the ideal gas law

[EQUATION]

with the Boltzmann constant k, the mass of the proton [FORMULA] and the number of particles per nucleon [FORMULA]. In our models we adopted [FORMULA] according to the solar chemical composition.

These equations are solved in a cartesian coordinate frame, where the path of the flux tube is given by [FORMULA]. The gravitational field [FORMULA] is considered as constant with only a vertical component [FORMULA].

The corona is assumed to be isothermal, leading to

[EQUATION]

with the pressure scale height [FORMULA] and the pressure [FORMULA] at the level [FORMULA]. This assumption does not consider the temperature decrease towards the photosphere. As the transition layer is very thin, this is a good approximation for our investigation. However, the transition to the photosphere is important for the stability analysis (photospheric boundary condition) which is discussed in detail in Sect. 3.2.

The equilibrium models calculated by Cramphorn and Degenhardt include an energy balance between radiative losses, heat conduction and coronal heating:

[EQUATION]

where the heat flux [FORMULA] is given by Spitzer (1962)

[EQUATION]

with [FORMULA]. The radiative losses are determined by loss functions given by Hildner (1974), Cox & Tucker (1969) and Kuin & Poland (1991). The coronal heating is chosen as

[EQUATION]

with a constant h. This energy balance leads to temperature profiles with mainly constant temperatures in the central part of the prominence, a steep gradient in the transition region to the coronal part and a slowly increasing temperature in the corona as shown in Fig. 2.

[FIGURE] Fig. 2. Temperature profile inside the flux tube for a prominence model with self-consistent energy balance. The central part with low temperatures is shown enlarged in the left plot. The prominence has a width of about [FORMULA].

Using this energy and force balance basic parameters of prominences can be reproduced except for the width which is smaller than observed. Widths bigger than about [FORMULA] can only be achieved by using very small density contrasts and small central temperatures. But the prominence width has an important influence on the stability, so that the restriction on these models is not very reasonable. In order to analyse wider prominences we worked with a model in which the temperature profile inside the flux tube is prescribed analytically and approximately represents the self-consistent treatment mentioned above. These models are only in magnetohydrostatic force equilibrium and we only investigate the dynamical stability due to adiabatic perturbations. We remark that for the stability analysis only the temperature profiles of the equilibrium model are important and not the energy equations used for obtaining these profiles.

The internal temperature profile of the model is described by

[EQUATION]

with the half width b of the prominence, the width s of the transition region, the prominence central temperature [FORMULA] and the external temperature [FORMULA]. This temperature profile shows (for appropriate values of s) the steep gradient in the transition region and is a good approximation for the central part of the prominence. In the coronal part of the flux tube the temperature inside the flux tube is equal to [FORMULA]. This is the simplest possible choice but fortunately the results are not strongly dependent on the coronal temperature profile, as long as the temperature is close to the external temperature. This is discussed in detail later.

The influence of an external magnetic field depends mainly on the particular configuration. We used a common arch like potential field

[EQUATION]

[EQUATION]

where [FORMULA] denotes the external pressure scale height and [FORMULA] the field strength at the level [FORMULA]. This field is a reasonable approximation for the external background field of prominences with normal polarity we want to describe with our model. Another reason for taking this field is the fact that the stability formalism we want to apply was derived for the case without external magnetic field. But it is possible to include this field by a formal substitution we describe in the following lines. A useful property of this field structure is that the field lines are mainly parallel to the path of the flux tube and that the magnetic pressure [FORMULA] has the same dependence on height as the external gas pressure [FORMULA]. This enables an analytical treatment of the external field by the substitution

[EQUATION]

where [FORMULA] and [FORMULA] satisfy the equation

[EQUATION]

because of the same height dependence of [FORMULA] and [FORMULA]. By making use of this property it is possible to write Eqs. (2) and (4) in the form

[EQUATION]

and

[EQUATION]

which is mathematically equivalent to the case without an external magnetic field, but corresponds to a solution with another effective density contrast between prominence and corona.

The effective central density contrast of the prominence is given by

[EQUATION]

which is smaller than the real density contrast [FORMULA] and is dependent on the external plasma beta [FORMULA]. The real density contrast of prominences is about 100. Taking an external magnetic field of about [FORMULA] and an external gas pressure of [FORMULA], we receive effective density contrasts in the range of [FORMULA].

Thus using the above described external magnetic field it is sufficient to solve the flux tube equations without an external magnetic field, but keeping in mind that the density contrast is modified by Eq. (18).

If the internal temperature profile, the external plasma beta, external temperature and the (real) density contrast are fixed, the path of the flux tube is unaffected by changing the external gas pressure. The other physical values scale linear with the external gas pressure so that changing the external gas pressure does not produce new solutions. Thus it is possible to reduce the free parameters [FORMULA], [FORMULA] and [FORMULA] of the model in the following way:

[EQUATION]

The other free parameters are:

  • external temperature: [FORMULA]

  • internal temperature profile: [FORMULA], b, s

  • height of prominence: [FORMULA]

For solving the equations in the cartesian coordinates the following relations are useful. As independent variable we use the horizontal distance [FORMULA], related to the arclength [FORMULA] and height [FORMULA] by [FORMULA].

In the cartesian coordinate frame we get for the Frénet basis ([FORMULA])

[EQUATION]

The curvature of the flux tube is given by

[EQUATION]

Using these expressions, Eqs. (1) and (16) can be written as

[EQUATION]

and

[EQUATION]

Using Eqs. (5), (6) and (10) these equations can be integrated with standard routines for ODE's.

A typical prominence model is shown in Fig. 3. Pressure and density profiles of the central part are shown on an enlarged scale. Note that the density contrast inside the flux tube is larger than the density contrast between the prominence and the external corona.

[FIGURE] Fig. 3a and b. Equilibrium model for a typical prominence with a prescribed temperature profile. The height is [FORMULA], the width [FORMULA], the central temperature [FORMULA] and the effective density contrast between prominence and corona 10. The prominence is embedded in an isothermal corona of [FORMULA]. a  Path of the flux tube. The cool and dense prominence matter rests in the dip at the very center. b  Pressure (solid line, left axis) and density (dotted line, right axis) profile, enlarged.

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© European Southern Observatory (ESO) 1999

Online publication: March 1, 1999
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