2. Formulation of the problem
Here we use the same 1D slab geometry as in AH and also denote the different models in the same way (see Table 1 in AH). We also chose the temperature K in order to separate the inner and outer regions of our prominence models. We assume a steady inflow of hot plasma through this boundary. This flow has to stream along the magnetic field lines, resulting in an inflow of enthalpy and ionisation energy through this boundary. The formula which allows us to calculate this flow is adopted from that given by Chae et al. (1997):
where F is the flux in x - direction, I is the ionisation energy, v the flow velocity along the field at the boundary and the field vector at this boundary. The mass flow is
Table 1. Summary of physical quantities, the different models are denoted in the same way as in AH.
We also have
where is the total hydrogen density (i.e. neutral plus ionised particles), i the ionisation degree, the hydrogen ionisation energy per atom amounting to erg, p the gas pressure and the density. As in AH we have taken a hydrogen plasma with 10% helium added and we have neglected the effects due to helium ionisation. With these definitions Eq. (1) can be rewritten as:
The amount of energy which is available for heating is the difference of this flow at the surface and the flow near the center. Since mass conservation of the flow in a steady state gives
we then obtain
From the models of AH one sees that at the surface and near the center.Taking a central value of , we obtain an upper limit of
where is the central temperature of the prominence. It is interesting to note that for these parameters the enthalpy contribution is about erg compared to the ionisation energy of erg.
Our non-LTE radiative transfer models were calculated under the assumption of magneto-hydrostatic equilibrium. But the present considerations require a non-vanishing inflow velocity. Therefore, using the AH - type models is not entirely self-consistent. But the flow velocities are subsonic in the hot (K) corona, therefore from Eq. (7) and from the fact that the gas pressure has to increase towards the cooler region we find that the flows are highly subsonic inside the prominence. This then means that dynamic contributions to the pressure term can be completely neglected and our equilibrium models are good approximations.
The question of the gradual mass increase in the prominence resulting from this inflow will be discussed later.
© European Southern Observatory (ESO) 2000
Online publication: June 20, 2000