## 2. Model descriptionThis 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 ( ## 2.1. Gas metallicityWe adopt a fixed He/H abundance ratio of 0.1. The abundance set (by
number) of metals relative to H is the following: ## 2.2. Surrounding medium and nuclear sourceThe 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 ()]. 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: where is the monochromatic ionizing energy
flux at the distance of the bowshock's head,
the ionization potential of H, the hydrogen
number density of the ambient medium gas and To avoid introducing poorly constrained parameters, we have assumed
that the ionization parameter 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 ## 2.3. GeometryThe 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: where where and . The sine of the angle between the normal to the interface and the Z-axis (see sketch in Fig. 1) is then:
## 2.4. Post-shock conditionsThe 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.
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 is displayed in Fig. 3.
## 2.4.1. Case without magnetic fieldWithout magnetic field, the derived post-shock conditions for a
perfect gas with a specific heat ratio are the
following (all quantities labeled with " where is the post-shock compression factor and the isothermal Mach number. ## 2.4.2. Case with magnetic fieldWe have assumed that both (normal to the bowshock surface) and (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): with the magnetic field parameter ranging
from 0 to 5 The Rankine-Hugoniot post-shock compression factor
is the positive real solution of the following
second order equation (all quantities labeled with " with The Rankine-Hugoniot post-shock pressure and magnetic field then yield: The adiabatic compression factor associated with the adiabatic conversion of the remaining normal kinetic energy into ram-pressure is computed by solving numerically the equation: with ( beeing the final post-shock density) The final post-shock pressure and magnetic field (all quantities
label with ## 2.5. Total pressure and velocity fieldsFollowing TDA, all the particles sharing the same curvilinear
abscissa where is the bowshock velocity. ## 2.6. Particle evolutionIn this section, we describe the step by step computation of the
evolution of the fluid which entered the bowshock through the
annulus of curvilinear abscissa
and length (quantities
refering to the time step of the evolution of
this particle are labeled using the upperscript
## 2.6.1. Temperature, density and magnetic field evolutionAfter 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: where Along the flow, the total pressure and the frozen-in magnetic field of the gas particle yield the relations: This gives the particle temperature evolution relation: Therefore, the temperature of the fluid particle at the time step is computed step by step using the recursive equation (similar relations can be derived for the pressure, density and normal magnetic field): where is the total net cooling rate of the
gas in erg s ## 2.6.2. Ionization stateThe 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: where is the column vector containing the
ionic abundances and is the matrix containing
the rates per ion (s 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
The elapsed time between two steps is estimated using the particle velocity and the curvilinear abciss change ( particle, time step). If is too long to consider 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 . ## 2.6.3. Adaptive stepsizeAs 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 samplingWe have furthermore used an adaptive stepsize for the bowshock
sampling. We have monitored two parameters, the velocity and the
## 2.8. Line fluxesTo derive the flux emitted by the particle ( ; corresponding to the coordinates of the bowshock apex) as it flows and cools, we need to know its depth (radial extension) and its volume . From mass conservation relations, we derive (): The line emissivities for a given temperature, density and ionic populations are computed using MAPPINGS IC. © European Southern Observatory (ESO) 1997 Online publication: June 30, 1998 |