Astron. Astrophys. 325, 745-754 (1997)

4. The Ly radiation transport code

The variation of H-Ly resonance emission intensity with the line element counted along the line of sight in an emitting and re-absorbing interplanetary H-medium is determined by loss and gain processes of photons and is given by the well-known radiation transport equation. This equation is an integro-differential equation which usually is treated by introducing the optical depth (e.g. Mihalas 1978). The formal solution of the radiation transport equation can be obtained in the form of a Neumann series expansion (Courant & Hilbert 1968) with respect to scattering orders i, leading to the following representation (Fahr & Smid 1982, Fahr et al. 1986, Scherer & Fahr 1996).

with

If the intensity of the primary source (here the sun) is known, all higher scattering orders ( to ) at any position in space successively can be determined with the above equation. is the angle-dependent, partial redistribution function and describes the probability that a photon within the original frequency interval arriving from the solid angle will be absorbed and will be re-emitted within the frequency interval into the solid angle (see Hummer 1962, modified by Scherer & Fahr 1996).

For a solution of Eq. (1) and Eq. (2) one has to start from the sun as the central source of the H-Ly radiation and to use the thermodynamical conditions of the interplanetary hydrogen to specify the redistribution function (Scherer & Fahr 1996). Based on the analytical and numerical concept for solving the first and second order of Eq.(2), it can be shown that the second and all higher scattering orders only play a minor role for intensities registered at solar distances smaller than 5 AU when taking into account the actual local temperature and velocity of the hydrogen, and also the angle- and frequency- dependence of photon redistribution by the the scattering agent (Scherer & Fahr 1996).

In this case equation (Eq. 1) can then be approximated by

and one finally obtains for (Details in Scherer & Fahr 1996)

In this equation represents the total H-Ly flux of the solar disk, the H-Ly absorption cross section of hydrogen, are the appropriate parameter of a fit to the solar H-Ly emission line, are local quantities of the bulk velocity, radial temperature pattern and density distribution of the interplanetary hydrogen and is the parametrized line of sight for a detector at a given position with a view direction in space.

All remaining integrations in Eq. 4 have to be done numerically with the use of a theoretical model description of the interplanetary hydrogen distribution developed by Osterbart & Fahr (1992), Fahr & Osterbart (1993, 1995), (see below) and as result of these scattering calculations we obtain a specific spectrum for every position and viewing direction of the probe.

4.1. The density model by Osterbart and Fahr

Based on the Parker model (Parker 1963), used for the description of the heliospheric flows, Osterbart & Fahr (1992) use a gas kinetic approach by means of the Boltzmann equation to describe the heliospheric hydrogen distribution. In this approach they take into account the charge exchange processes of the hydrogen with the interface plasma. They determine the time-independent distribution function of the interplanetary hydrogen in the vicinity of the sun. Assuming a rotation symmetry with respect to the inflow direction of the interstellar medium, i.e.: f depends only on the radial solar distance r, the angle (measured against the inflow direction of the hydrogen) and the local velocity vector, meaning:

The earlier model by Osterbart & Fahr (1992) was recently modified by Kausch (1996). Within this refined and improved model it is possible to consider variable ratios of radiation force to gravitation force () especially important in the vicinity of the sun and needed for the reliable theoretical description of the GHRS HST H-Ly glow data.

By calculating the higher moments of the distribution function the density, velocity and temperature pattern of the interstellar hydrogen in the neighborhood of the sun is derived (Fig. 3).

The parameters of the interstellar hydrogen used here in the density model are: density cm^-3, temperature = 8000 K, inflow velocity = 26 kms^-1 and a termination shock located at 80 AU (more details in Osterbart & Fahr (1992)).

In Fig. 3 one sees a deceleration of the hydrogen at around 130 AU, i.e. the assumed termination shock. Inside the termination shock especially the slower moving hydrogen atoms are influenced by the solar wind. For passing a given distance slower moving atoms need more time than faster moving atoms. Although the ionisation cross section for hydrogen is only weakly dependent on the relevant velocities, a larger proportion of the slower-moving hydrogen is ionized than of the faster-moving hydrogen, because for equal distances covered, the former is affected by the solar wind for a longer time than the latter. Therefore, in the vicinity of the sun the low-velocity wing of the hydrogen distribution function is decreased in comparison to the high-velocity wing of the distribution function. So, the mean weighted value for the bulk velocity is shifted to higher values. This means that near the sun the bulk velocity evidently increases, and nothing is left of the deceleration of hydrogen seen at regions close to the shock.

Fig. 3. Local density, temperature and bulk velocity (axial component) values for derived with the density model by Osterbart & Fahr are shown as function of radial solar distance for three different angles i.e.: - = 0 ; - - - = 90 ; - - - = 180.

© European Southern Observatory (ESO) 1997

Online publication: April 28, 1998

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