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Astron. Astrophys. 322, 73-85 (1997)

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2. Model description

This model, based on TDA's one, is a simple Lagrangian description of the bowshock structures which are expected to occur in active galaxies. The bowshock is assumed to be stationary and axi-symmetric. We will therefore use the cylindrical coordinates (R, Z, [FORMULA]), the Z-axis being the axis of symmetry. All quantities are given in the bowshock surface frame.

2.1. Gas metallicity

We adopt a fixed He/H abundance ratio of 0.1. The abundance set (by number) of metals relative to H is the following:
[FORMULA],
respectively. The metallicity [FORMULA] is simply a scaling factor. Unless specified otherwise, we assume [FORMULA], which corresponds to the solar metallicity abundance set of Anders & Grevesse (1989). We assume the gas is dust free as we expect that dust would be rapidly destroyed in the very hot post-shock zone considered here.

2.2. Surrounding medium and nuclear source

The surrounding medium in which the bowshock propagates (which in most of the cases will be the ISM of the host galaxy) is assumed to have reached thermal and ionization equilibria and to be homogeneous with a filling factor of unity. It is illuminated by a nuclear ionizing source [power law with a spectral index [FORMULA] ([FORMULA])]. The ionization parameter, a measure of the excitation level of the ionized gas, is defined as the ratio of the density of ionizing photons to the gas hydrogen number density:

[EQUATION]

where [FORMULA] is the monochromatic ionizing energy flux at the distance of the bowshock's head, [FORMULA] the ionization potential of H, [FORMULA] the hydrogen number density of the ambient medium gas and c the speed of light. For definiteness, we adopt the canonical index [FORMULA] as in TDA.

To avoid introducing poorly constrained parameters, we have assumed that the ionization parameter [FORMULA] of the unshocked gas is constant (i.e. we neither account for spatial variation in geometrical dilution of the nuclear radiation, nor for additional ionizing radiation generated in situ by the hot shocked gas [see Sect. 4.5.2]). Once the abundances and [FORMULA] are set, this leaves us with the two remaining free parameters [FORMULA] and [FORMULA]. Note that in the low density regime and in the absence of magnetic field, line ratios are weakly dependent on the explicit values of [FORMULA] and [FORMULA]. The equilibrium ionization balance of the ambient medium and its temperature are computed using MAPPINGS IC (see Appendix A).

2.3. Geometry

The layer of shocked gas is assumed to be thin compared to the characteristic size of the bowshock. Therefore, we do not distinguish between the bowshock surface and the interface with the radio material. We use the same fixed geometry than TDA for the bowshock surface (see Fig. 1), with a profile:

[EQUATION]

where C and [FORMULA] (input parameters for the model) are defined in TDA (the first one has been labeled C instead of B, to avoid confusion with the magnetic field). The curvilinear abscissa S along the bowshock surface is then derived using:

[EQUATION]

where [FORMULA] and [FORMULA]. The sine of the angle [FORMULA] between the normal to the interface and the Z-axis (see sketch in Fig. 1) is then:

[EQUATION]

[FIGURE] Fig. 1. Drawing of the model geometry in the bowshock frame (the ambient medium moves in the Z -direction with the velocity [FORMULA]). An annular fluid particle (the [FORMULA]) enters the bowshock at the curvilinear abscissa [FORMULA] and then flows along the shock surface. At the [FORMULA] time step of its evolution, it has reached the curvilinear abscissa [FORMULA].

2.4. Post-shock conditions

The oblique shock geometry is shown in Fig. 2. In our stationary model, the gas velocity component normal to the interface must be zero after the shock. We have assumed that this occured in two steps. First, just behind the shock front, the conversion into pressure of the momentum component normal to the interface is partial and yields the usual Rankine-Hugoniot relations for an oblique, adiabatic shock with or without frozen-in magnetic field. In a subsequent step, all the remaining momentum normal to the interface is converted adiabatically (as in the first stage), instead of isothermally as assumed in TDA. This requires that the transverse move during this conversion remains sufficiently small that we can neglect any longitudinal changes in the physical parameters. Note, that this holds as long as the thin shock layer assumption does.

[FIGURE] Fig. 2. Sketch of the [FORMULA] fluid cell of the [FORMULA] particle.

Once the post-shock pressure and density known, the post-shock temperature is computed using the perfect gas law. The ionic populations are assumed to remain unchanged through the shock interface (thin shock layer assumption), leading to an out of equilibrium state consisting of a very high temperature but low excitation (relative to equilibrium case) gas. A plot of the post-shock temperature and density as a function of the isothermal Mach number [FORMULA] is displayed in Fig. 3.

[FIGURE] Fig. 3. Post-shock temperature (left panel) and hydrogen number density (right panel) as a function of the isothermal Mach number [FORMULA] derived from TDA's model (dashed line) and from our set of equations (solid line). The initial density, temperature and magnetic parameter were, 1 cm-3, [FORMULA]  K and 3 µG cm [FORMULA]. The specific heat ratio of the gas was [FORMULA] =5/3. Note, at high Mach numbers, the systematic shift between the results of the two models. This is due to our choice of adiabatical conversion of the residual normal momentum instead of the isothermal conversion assumed by TDA. In the small Mach number range (typically [FORMULA]), TDA's strong shock assumption breaks down which explains why the two curves rapidly separate.

2.4.1. Case without magnetic field

Without magnetic field, the derived post-shock conditions for a perfect gas with a specific heat ratio [FORMULA] are the following (all quantities labeled with "o " refer to the unperturbed ambient medium; [FORMULA] is the plasmon velocity):

[EQUATION]

where [FORMULA] is the post-shock compression factor and [FORMULA] the isothermal Mach number.

2.4.2. Case with magnetic field

We have assumed that both [FORMULA] (normal to the bowshock surface) and [FORMULA] (longitudinal to the bowshock surface) ambient magnetic field components have the same amplitude along the whole bow shape. They have been taken to be (Dopita & Sutherland 1995):

[EQUATION]

with the magnetic field parameter [FORMULA] ranging from 0 to 5 µG cm [FORMULA].

The Rankine-Hugoniot post-shock compression factor [FORMULA] is the positive real solution of the following second order equation (all quantities labeled with "o " refer to the unperturbed surrounding medium):

[EQUATION]

with

[EQUATION]

The Rankine-Hugoniot post-shock pressure and magnetic field then yield:

[EQUATION]

The adiabatic compression factor [FORMULA] associated with the adiabatic conversion of the remaining normal kinetic energy into ram-pressure is computed by solving numerically the equation:

[EQUATION]

with ([FORMULA] beeing the final post-shock density)

[EQUATION]

The final post-shock pressure and magnetic field (all quantities label with PS refer to the final post-shock conditions) yield:

[EQUATION]

2.5. Total pressure and velocity fields

Following TDA, all the particles sharing the same curvilinear abscissa S are assumed to be in mechanical equilibrium. We can therefore derive the total pressure [FORMULA] (sum of the gas and magnetic pressures) and velocity fields (along the bowshock surface) using the post-shock total pressure [FORMULA] and the conservation laws of longitudinal momentum and mass. This gives:

[EQUATION]

where [FORMULA] is the bowshock velocity.

2.6. Particle evolution

In this section, we describe the step by step computation of the evolution of the fluid which entered the bowshock through the [FORMULA] annulus of curvilinear abscissa [FORMULA] and length [FORMULA] (quantities refering to the [FORMULA] time step of the evolution of this [FORMULA] particle are labeled using the upperscript i and the subscript j).

2.6.1. Temperature, density and magnetic field evolution

After the shock discontinuity, the particle flows along the surface of the bowshock. We assume that it does not interact thermally with the other particles. Writing the internal energy density conservation law for the particle (Shapiro et al. 1992) and using the perfect gas equation of state, we derive the relation used by TDA to compute the temperature evolution:

[EQUATION]

where dQ is the net cooling of the gas (losses minus gains).

Along the flow, the total pressure and the frozen-in magnetic field of the gas particle yield the relations:

[EQUATION]

This gives the particle temperature evolution relation:

[EQUATION]

Therefore, the temperature of the [FORMULA] fluid particle at the [FORMULA] time step is computed step by step using the recursive equation (similar relations can be derived for the pressure, density and normal magnetic field):

[EQUATION]

where [FORMULA] is the total net cooling rate of the gas in erg s-1  cm-3. As we do not assume ionization equilibrium but compute explicitly the temporal evolution of the ionization balance using MAPPINGS IC routines, the cooling rate depends on the particle's ionization 'history' as well as on its temperature and density. Note that all these relations assume that the mean molecular weight of the gas particle remains constant during the step. In fact, its change is computed a posteriori, once the new ionization state of the fluid known.

2.6.2. Ionization state

The ionization state of the particles is computed with MAPPINGS IC by solving the time dependent ionization balance set of equations (Binette et al. 1985). Briefly, the heavy element ionization balance equations are expressed in a matrix form:

[EQUATION]

where [FORMULA] is the column vector containing the ionic abundances and [FORMULA] is the matrix containing the rates per ion (s-1) corresponding to the various changes in ionization stage allowed for the atomic element under consideration (see Appendix A). If we assume that the rates remain constant during some timestep t, then the ionic abundances at the end of the time step, [FORMULA], are given in terms of the abundances at the start, [FORMULA], by the matrix equation:

[EQUATION]

In the case of hydrogen, however, it is necessary to solve the ionization balance separately and in an analytical form since the electron density (used in determining the recombination and collisional ionization rates) is itself directly proportional to the fractional ionization of hydrogen. The mode assumes the on-the-spot approximation (Osterbrock 1989) in the processing of the local diffuse field in the interest of simplicity (see Sect. 4.5.2).

The elapsed time [FORMULA] between two steps is estimated using the particle velocity [FORMULA] and the curvilinear abciss change [FORMULA] ([FORMULA] particle, [FORMULA] time step). If [FORMULA] is too long to consider [FORMULA] as a constant matrix (as a result of rapid variations in temperature, total or electronic density), the timestep is divided into smaller time intervals with successive updating of [FORMULA].

2.6.3. Adaptive stepsize

As the fluid flows along the bowshock surface, it crosses various regions characterized by very different evolution time scales. To save computation time without loosing accuracy, we have used a varying curvilinear abscissa stepsize. The stepsize control is achieved through a maximum allowed change rate based on four tolerance parameters (temperature, total pressure, velocity and mass density times the velocity). We have also added an absolute maximal change in velocity which is set using the expected velocity sampling of the final model output. The setting procedure of these tolerance parameters is described in Sect. 4.1.

2.7. Bowshock sampling

We have furthermore used an adaptive stepsize for the bowshock sampling. We have monitored two parameters, the velocity and the Z position of the sharp cooling zone (see Sect. 3.2). This was done to ensure that the final samplings in velocity and in Z were high enough for the astrophysical purpose of the modeling.

2.8. Line fluxes

To derive the flux emitted by the [FORMULA] particle ([FORMULA] ; [FORMULA] corresponding to the coordinates of the bowshock apex) as it flows and cools, we need to know its depth (radial extension) [FORMULA] and its volume [FORMULA]. From mass conservation relations, we derive ([FORMULA]):

[EQUATION]

The line emissivities for a given temperature, density and ionic populations are computed using MAPPINGS IC.

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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