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Astron. Astrophys. 344, 687-695 (1999)

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2. Predicting temperature in a swept-up shell

In the jet-driven bow shock model, rotational CO transitions trace material that has been swept up behind the bow shock. In the MC93 model this swept-up gas forms an expanding, forward-moving shell in a momentum-conserving `snowplough' phase. By considering the dynamics of the shell, we can predict the temperature distribution in the CO outflow.

Following MC93, the shell contains the material swept up from the cavity along its path, and its momentum is determined by momentum transfer and thermal pressure at the bow shock. The thermal pressure is quickly dissipated close to the jet head and the shell behind the bow shock coasts outwards at a constant angle [FORMULA] to the jet direction (MC93). The shell slows as it continues to sweep up mass from the ambient medium, conserving momentum but losing energy in the form of radiation, in a way analoguous to the later stages of a supernova shell. By considering the energy balance (heating vs. cooling) in the shell, we determine the temperature of the shell.

We now look at the equations that govern the shell in the momentum-conserving phase. All shell quantities are calculated per unit length. The fundamental parameters of the model are given in Table 1. Fig. 1 illustrates the geometry of the model.

[FIGURE] Fig. 1. Shell model geometry, showing the ring of mass m coasting at angle [FORMULA] to the jet axis (after MC93).


[TABLE]

Table 1. Fundamental parameters of outflow shell model.


The evolution of jet length [FORMULA] with jet travel time [FORMULA] is found from the shock speed, given by MC93 as

[EQUATION]

where [FORMULA] is the ambient density, [FORMULA] the jet density, [FORMULA] the jet velocity and [FORMULA] the jet efficiency. We assume the power law density distribution

[EQUATION]

where [FORMULA] is the density at distance [FORMULA]. This is equivalent to a spherically symmetric radial density distribution as the outflow remains narrow [FORMULA]. We neglect the ambient density variation during the evolution of each shell section, taking the density equal to that at the point of origin of the section as in MC93. Each coasting shell section travels a distance which is short compared to the total length of the outflow.

The shell mass per unit length is

[EQUATION]

where r is the shell radius and [FORMULA] is the mass initially injected from the jet.

MC93 neglect [FORMULA], which is justified in their analysis as they consider only light jets which do not contribute much mass. Following measurements of [FORMULA] cm-3 in some jets (Bacciotti et al. 1995) we cannot neglect the mass input from the jet. The mass input rate to the shell from the jet equals the mass input from the jet minus the mass used in extending the jet length; per unit length of the shell this is

[EQUATION]

where [FORMULA] is the rate of advance of the jet head shock at the axis position [FORMULA] from which the shell element started.

The momentum equation is

[EQUATION]

where the initial momentum [FORMULA] in the direction [FORMULA] is related to the jet parameters by

[EQUATION]

For [FORMULA] close to 1 or [FORMULA] this approximates to the formula given in MC93. We neglect the distance that the shell travels under the power of thermal pressure at the bow shock. This distance is short - the shell is momentum driven within one jet radius of the shock (Chernin et al. 1994; Raga & Cabrit 1993) - and hardly affects the derived outflow shapes.

The kinetic energy of the shell is simply

[EQUATION]

To form an element of shell we consider that the jet travels to a distance [FORMULA] and the shell then propagates outwards at angle [FORMULA] to the jet axis. By solving the jet length equation (Eq. 1) for [FORMULA] and then the momentum equation (Eq. 5) for r, the radius r, velocity [FORMULA], and energy E of each shell element are found as a function of the jet travel time parameter [FORMULA]. The prescription for solving the equations to derive the energy input rate [FORMULA] is as follows. Firstly the momentum equation (Eq. 5) must be integrated to give [FORMULA] where t is the shell travel time from its origin at the jet axis. [FORMULA] then follows by differentiating. The energy equation (Eq. 7) gives E as a function of [FORMULA], and differentiating again gives [FORMULA]. Next, [FORMULA] for the progression of the shock is found by solving the jet length differential equation (Eq. 1). In the expression for [FORMULA], [FORMULA] and t are both functions of the on-axis origin of the shell, [FORMULA] and therefore of [FORMULA]: [FORMULA] is given by Eq. 2 upon substitution for [FORMULA], and [FORMULA] (the shell expansion time at each position is the total age of the outflow minus the time taken by the jet to reach the position on the z-axis from which the shell element was ejected).

We used Mathematica (Wolfram 1996) to solve the equations for [FORMULA] and the distance from the source [FORMULA]. These quantities can then can be plotted against each other.

[FIGURE] Fig. 2. Energy input rate to each element of the shell, as a function of distance from the source, for outflow parameters given in Sect. 2. The solid, short dashed and long dashed lines correspond to [FORMULA] and 2 density distributions, respectively.

The kinetic energy lost by the shell is radiated away in molecular lines, and the balance between the energy input rate and the line cooling determines the temperature of the shell. Some of the energy may also be lost to turbulence: we consider as limiting cases the equipartition case where thermal and turbulent energies are equal, and also the case where turbulence is negligible and all the energy goes into heating. To convert our energy input rates to temperatures, we use the results of Neufeld et al. (1995). They calculate the cooling rate for molecular gas with temperatures from 10 K upwards and densities above [FORMULA], appropriate for molecular outflows. The temperature which corresponds to a particular energy input rate [FORMULA] also varies with shell density n and with an optical depth parameter [FORMULA] which corresponds to the column density per unit velocity and determines the radiation transfer through the gas. Both of these parameters also vary along the length of the outflow. We take an average [FORMULA] at each position to be the column density divided by the total velocity range. The density n in the shell is not directly predicted by our dynamical model but is constrained to be more than the ambient number density by a factor of (shell volume)/(cavity volume). We take n to be a constant factor of 6 times the ambient density, 6 being the maximum compression possible across a strong shock in molecular gas (Dyson & Williams 1997). Note that the temperature varies only slowly with n: a change from [FORMULA] to [FORMULA] at [FORMULA] corresponds to a change of at most 0.2 in [FORMULA].

The fundamental parameters on which the model depends are given in Table 1. The parameters are independent so we cannot reduce their number by forming combinations. As there are nine parameters, it is not possible to explore all reasonable ranges of values in this paper. Instead we pick the values listed in Table 1, which are chosen from the literature to be representative of young outflows, and choose the ages to produce an outflow of the same length as L483 (see observations, Sect. 3) for easy comparison.

All choices of parameters result in energy deposition which increases towards the end of the jet. A rise in temperature with distance from the source is a general result of the bow shock driven shell scenario. This is because the energy input rate depends more heavily on time than on any other parameter, so the young parts of the shell nearer the bow shock have the most energy input and are hotter whereas the old sections near the star have slowed and are cooler.

The values taken for the parameters are as follows. For the jet parameters, jet density [FORMULA] from optical jet observations (Bacciotti et al. 1995); jet radius [FORMULA] pc or [FORMULA] at the L483 distance of 200 pc; jet transfer efficiency [FORMULA] (MC93); and jet speed [FORMULA]. For the ambient density, we consider [FORMULA],1 and 2 for constant, [FORMULA] and [FORMULA] density distributions with [FORMULA] at [FORMULA] (0.06 pc at 200 kpc). The age of the outflow (to get the [FORMULA] measured length of L483 assuming a plane-of-the-sky inclination and the other parameters as stated) is taken to be [FORMULA], [FORMULA] and [FORMULA] for [FORMULA],1 and 2 respectively.

Fig. 3 shows the temperature as a function of distance along the outflow for the outflow parameters given above and a [FORMULA] density distribution with [FORMULA],1 and 2. The temperatures towards the end of the jet are similar regardless of the ambient density distribution, reaching many hundreds of kelvin towards the end of the jet in all cases. The temperatures rise towards the jet head, with the steepest temperature gradient corresponding to the steepest density gradient. For the flat density distribution, temperatures never fall below 60 K. (For a [FORMULA] density distribution, temperatures fall unrealistically low near the source, because we have not included the usual ISM heating sources (microwave background radiation, cosmic rays) which would cause the temperatures to level off at [FORMULA] K in the inner parts.) Note that the kink in the curves at [FORMULA] K is due to structure in the cooling rates at around this temperature.

[FIGURE] Fig. 3. Temperature of element of the shell, as a function of distance from the source (in parsecs assuming a distance of 200 pc), with (top left ) [FORMULA] density distributions with [FORMULA] (solid), 1 (short dashed) and 2 (long dashed); (top right ) a [FORMULA] density distribution with (dashed) and without (solid) dust cooling; (bottom left ) a [FORMULA] density distribution with thermal energy only (solid) and thermal/turbulence equipartition (dashed); and (bottom right ) for a [FORMULA] density distribution with jet velocities of 50 km s-1 (bottom dashed), 100 km s-1 (solid), and 200 km s-1 (top dashed). Other outflow parameters are those given in Sect. 2.

The temperature predictions are plotted up to the bow shock (here 0.067 pc from the source) but the model fails close to the bow shock firstly because the shock speed may be sufficient to dissociate the gas, and the molecular cooling function is then not valid, and secondly, because the outflow is not momentum-conserving. In addition, until sufficient column density has been swept up ([FORMULA]), the CO will not be detected. The first shell segment is found a distance [FORMULA], or between 0.002 and 0.01 pc from the star in our models.

Collisions with dust grains become important for energy transfer at high densities [FORMULA]; these can either cool or heat the gas depending on the temperature of the dust. A full consideration of energy transfer to dust is beyond the scope of our model but we also show in Fig. 3 how a cool (10 K) dust population results in a faster falloff of temperature close to the source where densities are high. The figure illustrates the [FORMULA] density distribution case: flatter density distributions with lower central densities are less affected by dust cooling. Fig. 3 shows the lower temperatures which result from assuming half the energy input is lost to turbulence rather than heating, again for the [FORMULA] density distribution. The [FORMULA] and 2 density distributions are similarly affected.

General features of these models are that the temperature rises towards the end of the jet, reflecting the increased energy input, and that temperatures mid-outflow are of order 80 K, dropping to a few tens of kelvin close to the source and rising above 500 K within 0.01 pc of the end of the jet.

The model can produce hotter or colder outflows, if the energy input from the jet is increased or decreased. Temperature scales approximately linearly with the jet velocity: Fig. 3 shows the temperature distribution for [FORMULA] models calculated with jet velocities [FORMULA] and 200 km s -1. Other parameters which affect the energy input in a similar way to [FORMULA] are the jet area [FORMULA] and the jet density [FORMULA]. Increasing the angle between the shell velocity and the jet velocity, [FORMULA], also has a similar effect to raising [FORMULA]. Raising the ambient density produces older, cooler outflows for the same outflow length.

For comparison, both Lizano & Giovanardi (1995) and Cantó & Raga (1991) consider outflow models in which the molecular gas is entrained by mixing at the boundary between a jet/wind and its molecular surroundings. In the entrainment models, the kinetic energy from the wind is input along the whole length of the outflow, rather than being predominantly deposited at the jet head. Entrainment models therefore predict temperatures [FORMULA] K along the whole length of the jet. Similar temperatures are also reached in the shell model, but only towards the jet head where the jet energy is being deposited (Fig. 3). However, much of the jet energy is radiated at the bow shock rather than providing energy input to the shell, and the shell loses energy rapidly as it ploughs into the surrounding gas.

In addition, both entrainment models predict that the temperature should fall rather than rise towards the end of the outflow. In the Lizano & Giovanardi model, the temperature of the mixing layer rises at first and then falls further from the source. The turnover occurs about 20% along the outflow which they are modelling (L1551). The Cantó & Raga mixing layer temperatures decrease monotonically with distance from the source. This is the opposite gradient to what is predicted by the shell model.

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© European Southern Observatory (ESO) 1999

Online publication: March 18, 1999
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