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Astron. Astrophys. 333, 687-701 (1998)

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6. Results of calculations

In this paper we discuss the results of calculations done for the shock waves with upstream velocities [FORMULA] propagating through the unperturbed hydrogen gas of temperature [FORMULA] and density [FORMULA] ([FORMULA]). In total we computed 46 models with upstream velocity increment of [FORMULA]. The outer boundary of the preshock region, where the gas is assumed to be unperturbed, is set at [FORMULA]  cm. Calculations were done for the two-level hydrogen atom, the first atomic state being treated in non-LTE. The radiation transfer equation was solved for the both Lyman and Balmer continua ([FORMULA]). Thus, the frequency point nearest to the Lyman edge frequency [FORMULA] was set at [FORMULA]. More extensive calculations for the larger number of bound atomic states L and hydrogen continua [FORMULA] as well as for various temperatures [FORMULA] and densities [FORMULA] of the unperturbed gas will be given in the forthcoming paper.

The most fascinating feature of radiative shock waves is that they demonstrate the strong interaction between gas material flows and the radiation field which they produce. This interplay is best seen from the plots of the divergence of radiative flux as a function of the spatial coordinate X. One of such plots is shown in Fig. 3. By definition, the divergence of radiative flux is negative, if the fluid absorbs more energy than emits and, therefore, is heated. And conversely, when [FORMULA], the gas radiatively cools since it radiates more energy than it absorbs.

[FIGURE] Fig. 3. The divergence of radiative flux against the spatial coordinate in the shock wave model with [FORMULA]

As is seen in Fig. 3, the divergence of radiative flux is always negative in the preshock zone, the departure of [FORMULA] from zero gradually increasing while the gas approaches the discontinuous jump. Heating of the precursor gas material is due to absorption of the Lyman continuum radiation, hence the region, where the divergence of radiative flux [FORMULA] perceptibly deviates from zero, extends over a few units of the optical depth at frequency [FORMULA].

The properties of the radiation field do not change appreciably across the discontinuous jump. The spatial resolution of our shock wave models near the discontinuous jump is limited by the space interval [FORMULA]  cm. The change of [FORMULA] and [FORMULA] over this interval increases with increasing upstream velocity [FORMULA] but does not exceed 0.3% for [FORMULA]. Thus, just behind the discontinuous jump the divergence of radiative flux goes on to gradually decrease and reaches the minimum in the vicinity of the maximum of the electron temperature [FORMULA]. The rapid growth of the divergence of radiative flux behind its minimum implies that the gas material flows into the radiatively cooling zone. For the shock wave model with [FORMULA] the distance between maximum and minimum of [FORMULA] is [FORMULA]  cm and corresponds to the optical depth between these layers of [FORMULA]. It should be noted that both the minimum and maximum of the divergence of radiative flux as well as other properties of the shock wave very strongly (for some variables nearly exponentially) depend on the upstream velocity [FORMULA].

6.1. The radiative precursor

At the outer boundary of the preshock region the first level of the hydrogen atom is approximately in LTE since the Lyman continuum radiation emerging from the postshock region is negligible. Both the Saha-Boltzmann relation and equations of statistical equilibrium give nearly the same (within a few per cent) the hydrogen ionization degree of [FORMULA]. The main sources of opacity are bound-free transitions in the Lyman continuum (i.e. at frequencies [FORMULA]) and at lower frequencies the Rayleigh scattering by hydrogen atoms in the ground state. The free-free opacity and the Thomson scattering are negligible. With approaching to the discontinuous jump the hydrogen ionization degree increases, so that the both free-free opacity and Thomson scattering increase but nevertheless even for [FORMULA] they remain negligible within the entire preshock region.

The radiative precursor is revealed as the part of the preshock region where the hydrogen gas is heated and is ionized by the Lyman continuum radiation emerging from the postshock region. The temperature of heavy particles (neutral hydrogen atoms and hydrogen ions) [FORMULA] remains constant ([FORMULA]) in the radiative precursor. In Fig. 4 are shown the plots of the electron temperature [FORMULA] and the hydrogen ionization degree [FORMULA] as a function of the distance from the discontinuous jump for the shock wave models with upstream velocities [FORMULA], 45, 50 and [FORMULA]. As is seen, the perceptible heating and ionization occur at distances from the discontinuous jump smaller than [FORMULA]. The distance [FORMULA] corresponds to the optical depth of [FORMULA]. Thus, for [FORMULA] the geometrical thickness of the radiative precursor approximately does not depend on the upstream velocity.

[FIGURE] Fig. 4. The hydrogen ionization degree [FORMULA] (upper panel) and the electron temperature [FORMULA] (lower panel) in the preshock region of the shock wave models with upstream velocities [FORMULA], 45, 50 and [FORMULA]

Within the radiative precursor the hydrogen gas and radiation field are in strong departure from LTE. In particular, ionization of hydrogen atoms is mainly due to radiative transitions, whereas collisional recombinations appreciably exceed collisional ionizations. Fig. 5 shows the rates of radiative transitions [FORMULA] and [FORMULA] as well as the collisional rates [FORMULA] and [FORMULA] [FORMULA] as a function of X for the shock wave model with [FORMULA]. The precursor transition rates are very sensitive to the upstream velocity [FORMULA]. For example, for upstream velocities [FORMULA] the rates of ionizations from the ground state increase in the ranges [FORMULA] and [FORMULA], respectively. However, notwithstanding such a strong dependence on [FORMULA], for all shock wave models the spatial dependencies of transition rates were found to be qualitatively similar to those displayed in Fig. 5.

[FIGURE] Fig. 5. The rates of ionizations (solid lines) and recombinations (dashed lines) against the distance from the discontinuous jump in the pre-shock region of the shock wave model with [FORMULA]

The growth of the hydrogen ionization degree with approaching to the discontinuous jump is accompanied by the increase of the gas pressure gradient. The corresponding decrease of the gas material velocity becomes, however, perceptible only for [FORMULA]. For example, at the inner boundary of the preshock region the relative decrease of U is [FORMULA] for [FORMULA] and is [FORMULA] for [FORMULA].

Of great interest are the physical conditions at the inner boundary of the preshock region. For the models considered in the present study the spatial coordinate of this boundary is [FORMULA]  cm. For upstream velocities [FORMULA] both the hydrogen ionization degree and the electron temperature nearly do not change within the preshock region. These quantities show the perceptible dependence on upstream velocity only for [FORMULA]. In Table 1 we give the hydrogen ionization degree [FORMULA], the electron temperature [FORMULA] as well as the transition rates at the inner boundary of the preshock region. For the sake of convenience the upstream velocity [FORMULA] is given in [FORMULA].


[TABLE]

Table 1. The properties of the preshock inner boundary


The electron temperature just ahead the discontinuous jump increases with increasing upstream velocity nearly exponentially. For the shock wave models of the present study this dependence can be approximately expressed due to the following fitting formula

[EQUATION]

where the upstream velocity [FORMULA] is expressed, for the sake of convenience, in [FORMULA].

As is seen in Table 1, the dominating process just ahead the discontinuous jump is photoionization and, therefore, the shortest relaxation time in the radiative precursor is that of photoionizations from the ground state [FORMULA]. The photoionization relaxation time gradually decreases with increasing upstream velocity from [FORMULA]  s for [FORMULA] to [FORMULA]  s for [FORMULA]. Comparing these relaxation times with the time needed for gas to flow through the precursor [FORMULA] we find that for models with upstream velocities [FORMULA] the ratio of the photoionization relaxation time to the hydrodynamic time is [FORMULA] and only for [FORMULA] the photoionization relaxation time becomes nearly comparable with hydrodynamic time: [FORMULA]. Because for establishment of the statistical equilibrium this ratio should be [FORMULA], the ground state populations of the hydrogen atom are in strong departure from the statistical equilibrium and the hydrogen ionization degree cannot be described in assumption of statistical equilibrium. For example, at the inner boundary of the preshock region of the shock wave model with [FORMULA] the hydrogen ionization degree is [FORMULA], whereas solution of the equations of statistical equilibrium gives [FORMULA].

6.2. The thermalization and recombination zones

Behind the discontinuous jump the translational kinetic energy of heavy particles is redistributed among various degrees of freedom characterized by different relaxation times. The fastest relaxation process is the translational kinetic energy exchange in elastic collisions of electrons with neutral atoms and ions. Another relaxation process is excitation of bound atomic states and ionization of hydrogen atoms. Both excitation and ionization need, however, much more collisions than translational kinetic energy exchange (Stupochenko et al. 1967), so that just behind the discontinuous jump the electron temperature gradually increases, whereas the hydrogen ionization degree remains nearly constant. Note that although the bound-bound transitions were not considered in our model, the excitation of atomic states is taken into account as a result of bound-free transitions.

In Fig. 6 are shown the electron temperature [FORMULA] and the temperature of heavy particles [FORMULA] as a function of distance from the discontinuous jump for shock wave models with upstream velocities from [FORMULA] to [FORMULA]. As is seen, the characteristic time of the electron temperature growth rapidly decreases with increasing upstream velocity. Furthermore, for [FORMULA] there is a temperature plateau within of which both temperatures approximately do not change until they begin to decrease. The existence of the temperature plateau is due to the fact that for upstream velocities [FORMULA] the electrons acquire the energy from heavy particles due to elastic collisions with neutral hydrogen atoms. The rate of energy gain in such collisions is very small and gradually decreases when the electron temperature [FORMULA] approaches the temperature of heavy particles [FORMULA]. The temperature plateau appears when the electron energy gain in elastic collisions with neutral hydrogen atoms becomes almost negligible. At upstream velocities [FORMULA] the electron temperature plateau is ended by the slight bump, the bump being wider and higher with increasing [FORMULA].

[FIGURE] Fig. 6. The electron temperature [FORMULA] (solid lines) and temperature of heavy particles [FORMULA] (dashed lines) as a function of distance from the discontinuous jump in the postshock region

In the second and third columns of Table 2 are given both the maximum value of the electron temperature in the postshock region as well as the distance of this point from the discontiuous jump expressed in cm. In the first column of Table 2 we give the upstream velocity expressed, for the sake of convenience, in [FORMULA]. The electron temperature peak was not found for the shock wave models with [FORMULA] and in these cases we give only the electron temperature of the plateau. The electron temperature maximum can be approximately considered as a point where temperatures of heavy particles and free electrons equilize. Thus, the width of the relaxation zone where both temperatures equalize decreases by a factor of [FORMULA] for the upstream velocity increasing from [FORMULA] to [FORMULA]. It is of interest to note that for upstream velocities [FORMULA] the temperature of heavy particles [FORMULA] remains nearly constant until the electron temperature begins to increase just before its drop. This is due to the fact that at small upstream velocities the fractional abundance of free electrons is so small that they cannot perceptibly affect the gas of heavy particles.


[TABLE]

Table 2. Properties of the postshock region


The translational energy exchange between heavy particles and electrons is due to elastic collisions of electrons with both neutral hydrogen atoms and hydrogen ions. The cross section of elastic collisions with hydrogen ions is much larger than that of electrons with neutral atoms and, therefore, the translational energy gain by electrons from heavy particles is strongly dependent on the hydrogen ionization degree. The rates of energy gain by electrons in both these processes are shown in Fig. 7. The abrupt decrease of [FORMULA] and [FORMULA] occurs when both temperatures equalize.

[FIGURE] Fig. 7. The rates of energy gain by electrons in elastic collisions with hydrogen ions (solid lines) and neutral hydrogen atoms (dashed lines)

For shock wave models considered in the present study the rate of energy gain by electrons in elastic collisions with ions exceeds that in elastic collisions with neutral atoms only when the hydrogen ionization degree is [FORMULA]. As is seen in Fig. 7, for upstream velocities [FORMULA], the electron temperature equalizes with temperature of heavy particles only due to elastic collisions of electrons with neutral atoms. Comparing with lower panel of Fig. 6 one sees that when the electron temperature reaches the plateau, the rate of energy gain [FORMULA] decreases by nearly two orders of magnitude.

At upstream velocities [FORMULA], the energy gain in elastic collisions with ions begins to dominate only just before the temperature drop. It is this increase of [FORMULA] that is responsible for the electron temperature peak near the end of the electron temperature plateau. The translational energy gain by electrons in elastic collisions with hydrogen ions becomes completely dominating for [FORMULA]. The gradual decrease of [FORMULA] just behind the discontinuous jump in the shock wave models with [FORMULA] (see upper panel of Fig. 7) is due to the temperature dependence of the equipartition time given in Eq. (34).

In Fig. 8 are shown the plots of the hydrogen ionization degree in the postshock region of the shock wave models with upstream velocities [FORMULA], 50 and [FORMULA]. Comparing with Fig. 6 one sees that the maximum of the hydrogen ionization occurs at much larger distances from the discontinuous jump than the maximum of the electron temperature. Very approximately the distances of both these maxima can be expressed as power functions of the upstream velocity:

[EQUATION]

In these approximate expressions, for the sake of convenience, the distances [FORMULA] and [FORMULA] are expressed in km whereas the upstream velocity is expressed in [FORMULA]. The maximum values of [FORMULA] as well as the corresponding distances from the discontinuous jump expressed in cm are given in the fourth and fifth colums of Table 2, respectively. According to the schematical division of the shock wave noted above the maximum of [FORMULA] can be considered as the boundary between the thermalization and recombination zones.

[FIGURE] Fig. 8. The hydrogen ionization degree as a function of the distance from the discontinuous jump in the shock wave models with upstream velocities [FORMULA], 50 and [FORMULA]

The large distance between maxima of the electron temperature and the hydrogen ionization degree implies that the degree of freedom associated with ionization of hydrogen atoms is frozen in comparison with excitation of translational motions. As a result, the gas flows through the maximum of [FORMULA] at smaller heat capacity [FORMULA] than it would be in equilibrium. Thus, when the perceptible fraction of hydrogen atoms is ionized, the gas density begins to increase. At larger distances the gas density goes on to increase due to the radiative cooling (see Fig. 9).

[FIGURE] Fig. 9. The postshock gas density [FORMULA] against the distance from the discontinuous jump. The numbers at the curves indicate the upstream velocity [FORMULA] in [FORMULA]

The total ionization rate of hydrogen atoms is a sum of photoionizations and collisional ionizations, so that it depends on the number density of free electrons. Because no ionization occurs across the discontinuous jump, the total ionization rate behind the discontinuous jump strongly depends on the ionization in the radiative precursor. To demonstrate this dependence, in Fig. 10 are shown the rates of ionizations and recombinations, both collisional and radiative, in the postshock regions of the shock waves with upstream velocities 30 and [FORMULA].

[FIGURE] Fig. 10. The rates of ionizations (solid lines) and recombinations (dashed lines) against the distance from the discontinuous jump X in the postshock region. Upper panel: [FORMULA] ; lower panel: [FORMULA]

For the shock wave model with [FORMULA] the hydrogen ionization degree at the discontinuous jump is [FORMULA], therefore just behind the discontinuous jump the role of collisional ionizations is negligible and the number density of free electrons increases initially only due to photoionizations. For [FORMULA] the hydrogen ionization degree at the discontinuous jump is [FORMULA]. Free electrons created in the precursor play a role of the seeds producing yet more electrons and leading to the electron avalanche.

Compared to the preshock region where the contribution of opacity sources nearly does not depend on the upstream velocity, the gradual growth of the hydrogen ionization degree behind the discontinuous jump leads to the appreciable dependence of postshock opacities on the upstream velocity [FORMULA]. For [FORMULA] the postshock opacity in the Balmer continuum is mainly due to Rayleigh scattering by neutral hydrogen atoms. Only near the Balmer edge, the opacity due to bound-free transitions from the second level becomes most important. With increasing upstream velocity the role of the bound-free transitions from the second level increases and for [FORMULA] this opacity mechanism becomes dominating within the whole range of the Balmer continuum. The role of the Thomson scattering is somewhat perceptible only near the Balmer edge, whereas the free-free opacity can be neglected.

6.3. The radiation field

Although we did not include into the system of ordinary differential equations (20) - (24) the terms with radiation energy density [FORMULA] and radiation pressure [FORMULA], these quantities were evaluated as

[EQUATION]

[EQUATION]

together with total radiative flux [FORMULA] and divergence of the radiative flux [FORMULA] each time when the radiation transfer equation was solved. For the fixed spatial coordinate X the ratio of the radiation energy density to the total translational kinetic energy gradually increases with increasing upstream velocity and is in the range [FORMULA] for [FORMULA]. The ratio [FORMULA] is highest at the inner boundary of the preshock region because just behind the discontinuous jump the total translational kinetic energy increases by more than an order of magnitude, whereas the total change of [FORMULA] within the shock wave does not exceed 30%. Because the most of the energy flux is contained in the radiative flux [FORMULA], we find that [FORMULA] for [FORMULA]. Thus, our assumption that the radiation energy and radiation pressure can be neglected is enough good.

In Fig. 11 are shown the dependencies of [FORMULA] on the distance from the discontinuous jump for the both preshock and postshock regions of the model with [FORMULA]. The increase of [FORMULA] in the preshock region with approaching to the discontinuous jump is due to the radiative heating by the layers absorbing the Lyman continuum radiation. Behind the discontinuous jump the radiation energy density reaches the maximum in the hydrogen recombination zone.

[FIGURE] Fig. 11. The total radiation energy density as a function of distance from the discontinuous jump in the shock wave model with [FORMULA]

The most of the shock wave radiation, by definition, is produced in the layers where the divergence of radiative flux [FORMULA] reaches the maximum. These layers locate nearly at the same distance from the discontinuous jump as the maximum of the hydrogen ionization degree. The maximum of [FORMULA] exponentially increases with increasing upstream velocity and for [FORMULA] is given by

[EQUATION]

where the upstream velocity is expressed in [FORMULA]. Maximum values of [FORMULA] are given in the sixth column of Table 2.

As was noted above the shock wave models have a small optical depth in the Balmer continuum ([FORMULA]) and at the same time they are opaque for the Lyman continuum photons. Therefore, the radiation emerging from the both surfaces of the slab is mostly within the Balmer continuum whereas the Lyman continuum radiation is transported only within the narrow zone surrounding the discontinuous jump. In the lower panel of Fig. 12 for the shock wave model with upstream velocity [FORMULA] is shown the Lyman continuum radiative flux [FORMULA] as a function of the spatial coordinate X. This dependence is qualitatively similar to that of the divergence of radiative flux shown in Fig. 3. The negative flux corresponds to the radiation propagating in the direction of the preshock region. The optical depth between minimum and maximum of [FORMULA] is [FORMULA] for [FORMULA] and slowly decreases from [FORMULA] for [FORMULA] to [FORMULA] for [FORMULA]. This decrease is mostly due to the growth of the hydrogen ionization degree in the postshock region. The ratios of the maximum radiation flux transported within the Lyman continuum to the total radiation flux are given in the last column of Table 2.

[FIGURE] Fig. 12. The radiative flux within the Lyman continuum [FORMULA] (lower panel) and the total radiative flux [FORMULA] (upper panel) against the spatial coordinate X for the shock wave model with [FORMULA]

In the upper panel of Fig. 12 for the same model is shown the total radiative flux [FORMULA]. This plot displays only the very vicinity of the discontinuous jump. In the preshock region, beyond the radiative precursor, the total radiative flux [FORMULA] remains constant since the divergence of radiative flux is [FORMULA]. In the postshock region the total radiative flux gradually increases and asymtotically tends to the same value as in the preshock but with the opposite sign. For all shock wave models we obtained that the total radiative fluxes emerging from both surfaces of the slab are [FORMULA].

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© European Southern Observatory (ESO) 1998

Online publication: April 20, 1998
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