We begin our discussion with a short general description of the complete evolution of the jet-cloud interaction, that can be summarized in three steps (see Fig. 3 for a visualization of the basic features of the three steps for the case and ):
The jet hits the cloud, forming a strong shock, the post-shock region becomes hot and blows up, because of its increasing pressure; the jet material is conveyed in a back-flow that squeezes the jet itself. During this process the cloud material is compressed and heated by the shock, at the head the temperature is very high ( K), while on the jet sides it is lower K, so that it can cool down, to reach the observed line emission conditions. It is in this region, forming a layer around the jet, that the narrow line emission can originate. Our analysis will therefore concentrate on the properties of this region. During this first phase, in which the jet crosses the cloud, the layer is accelerated by the strong inside pressure and cools down, its density thereby increases (see the leftmost panels in Fig. 3).
The second phase begins when the jet is completely out of the cloud, the compressed emitting material reaches a quasi-steady state, during which the emission is almost constant, the inside pressure begins to decay, but the emitting layer is still accelerated. From Fig. 3 (central panels), we can see that the material in the layer has been compressed, its maximum density has increased, while its temperature has decreased. The maximum density is found now at temperatures around K, and its velocity has also increased.
In the third phase, the inside pressure has decayed and the emitting layer begins to slow down, the jet flows freely through the cloud and also the emission decreases, eventually disappearing. From Fig. 3 (rightmost panels), we see a decrease in density and velocity, while almost all the layer is found at K.
The efficient formation of the line emitting region will therefore depend on the efficiency of radiation during the jet crossing of the cloud. We will then introduce two typical timescales, the cloud crossing time and the radiative time , whose ratio will be a fundamental parameter for determining the evolution of the narrow line emitting layer. Following analytical treatments of the jet-ambient interaction we define the cloud crossing time as , where we have assumed, for the jet head velocity in the cloud, the steady velocity obtained from the 1-D momentum balance in a medium with (see, e.g., Cioffi & Blondin 1992, Norman et al. 1982). In this way we are actually overestimating the crossing time, since our situation is not steady, however this value is sufficiently accurate for our purposes. Concerning the radiative time, its definition is given in Sect. 4.3, however, we must notice that for its evaluation we have to assume a value for the temperature, in the following considerations we have taken K, that is the average of the typical post-shock temperature in the region of our interest, this choice is properly done for all jets with and , while it is overestimated for the low velocity cases, that means that for those cases are shorter than the real ones. We have then defined as the ratio between crossing and radiative time scales and this, as said before, is an important parameter for the interpretation of the results.
As a first step in our analysis, we have performed an exploration of the parameter space. As discussed before, we reduced our parameters to and the initial . In Table 1 we report, for each pair of their values, typical values of density, expansion velocity and temperature of the emitting material and the value of . The density is the median value of density distribution weighted on the emissivity function (that is proportional to ), while velocity and temperature are those corresponding to this density value. All the quantities are evaluated at , this choice is due to the fact that during this period the expansion velocity of the emitting material increases rapidly reaching a maximum and then decreases monotonically, so that, if the expansion velocity does not match the observational constraint within this time, it never will, and the case will not be of interest for our analysis. Radiation must therefore act efficiently during this time, in order to create the needed conditions for radiation, and this poses a lower limit on the value of . On the basis of the values reported in this table we choose the most promising cases for our investigation.
Table 1. Parameters of the simulations
Considering the first column we can immediately realize that jets at low velocity cannot reach conditions comparable to those observed. The values of for these simulations are high, meaning that radiation is very efficient. On the other hand, the jet momentum is low and cannot drive the emitting material at high velocities. For the case we have, in fact, high densities in accord with the high value of , but very low velocities. For this reason we did not perform simulations for the other two cases of higher density, since jets would produce stronger and cooler compression practically at rest, very far from the observational scenario.
Looking at the high velocity case, we see that, in the case of small inhomogeneities, has a very low value and, therefore, radiation is inefficient. The jet is very energetic and sweeps the cloud, before radiation becomes effective and so it does not form any condensation (the velocity reported for this case is therefore meaningless). Increasing the cloud density, we increase also the value of : the maximum density increases, but it is still quite low. Only for the high density cloud (), we get values of density and velocity in agreement with observations.
Regarding the intermediate velocity, the values of are : radiation is efficient and thus the emitting layer can reach sufficiently high densities. Only in the lighter cloud case, however, the velocity is comparable to the observed values.
From this exploration of the parameter space we can conclude that the observed conditions can be matched only for a narrow range of parameters and that the properties of the emitting layer depend essentially only on one parameter, the ratio between the radiative timescale and the cloud crossing timescale . For low values of (), radiation is inefficient and the densities in the layer are too low. For higher values of () we find, on the other hand, that the velocity of the emitting layer becomes too small. This is because the cloud density is high and the jet momentum flux is too small to impart to it a large enough velocity. Only for a narrow range of values of we can match the observed conditions and, in Table 2, we have translated these limits into limits on velocity range at different cloud densities.
Table 2. Ranges of jet velocities that can match the observed properties
5.1. Case of
In this subsection we will discuss in more detail the case that best matches the observational scenario. We begin our discussion showing, Fig. 4, a gray-scale image with a snapshot of the density distribution at and three small panels showing enlargements of the region of interaction between jet and cloud referred to density, temperature and the expansion velocity of emitting gas. The proper physical condition for emission are reached in a thin layer of compressed cloud material, whose width and mass grow in time as the shocked cloud material cools down.
The detailed physical properties of this line emitting region are reported in Fig. 5, where we have represented the behavior of density, temperature and velocity along radial cuts through this layer. We note that the proper conditions are matched in a layer of width pc.
How the properties of the material contained in this thin layer compare with the physical conditions of gas of the NLR? To answer the question we plot in Fig. 6 the temporal behavior of the mean expansion velocity (panel a) and mass (panel b) of the emitting material shell at two different density limits. We see that from the time when the jet touches the cloud until , when a strong interaction between the jet head and the cloud takes place, the cloud material is accelerated and the quantity of emitting material increases; after this interval the jet flows, essentially freely, across the cloud without any further acceleration of the compressed material shell and the accelerated cloud material slows down monotonically.
Notice that the mean expansion velocity, relative to an observer, lies, for the denser material, in the range (since one must consider twice the mean expansion velocity), that is in good agreement with the velocity deduced by the line widths detected. Looking more in detail at the emitting mass, we see that its growth begins some time after the jet has initiated to drill its way into the cloud, and this delay corresponds to the cooling time of the shocked material. We also note that, after , the jet continues to sweep out material laterally at a pace that is higher for the lighter material, the total mass exceeds at ys and this would correspond to an luminosity of erg s-1 which, considering also the possibility of having simultaneously several active clouds, is consistent with the observed values.
As the interaction is effective over a timescale much longer than the jet will quickly propagate into the low density inter-cloud medium and it will reach any other cloud lying on its path. Thus more than one cloud will be effectively interacting with the jet at any time. Each will display a behaviour typical of its evolutionary stage and the total emitting mass must be considered as the total over all clouds. Furthermore, this will naturally reproduce the jet-like morphology of the NLR.
As discussed in the Introduction, the source of ionization of the NLR is still matter of debate. While the NLR gas is certainly illuminated by the nuclear source, its interaction with the radio jet also produces regions of high temperature and density which radiates ionizing photons. In this paragraph we derive the conversion efficiency of the jet kinetic power into energy radiated in ionizing photons. To estimate the ionizing energy flux we integrated radiative losses over all regions where K as above this temperature most of them correspond to the production of photons with energy higher than the hydrogen ionization threshold.
In Table 3 we summarize our results reporting the kinetic power referred to the three different velocities (, where , and A are respectively the density, the velocity and the transverse section of the jet) and the conversion efficiency at peak and after 2 for all the cases considered.
Table 3. High frequency radiative power and radiative efficiencies
The peak efficiency reaches in one case a value as high as 10% but it is usually 0.1 - 2%. However, over the interaction, the typical value of (well represented by its value after 2 ) is much lower . Faster jets have lower efficiency than slower jets and this conspires in producing a very similar amount of energy radiated in ionizing photons, , in all cases.
In Seyfert galaxies (Koski 1978). The minimum ionizing photon luminosity required to produce a given line emission luminosity corresponds to the limiting case in which all ionizing photons are absorbed and all photons have an energy very close to the hydrogen ionization threshold . In this situation
where is the probability that any recombination will result in the emission of an H photon.
It appears that, even in this most favourable scenario, the radiation produced in shocks can only represent a small fraction of the overall ionization budget of the NLR, particularly as sources with high radio luminosity (in which usually radio-jets are found) also have the highest line luminosity (e.g. Whittle 1985).
Nonetheless, in the most promising case examined above ( and ) at the peak of the conversion efficiency the radiated energy is and it is substained over a crossing time, years. Shock ionization may thus produce important ionization effects which, however, can be only both local and transient.
© European Southern Observatory (ESO) 2000
Online publication: March 28, 2000