L1448

4 Background to models

 

4.1 Boost factor and column density ratios

The LWS and SWS data sets contain lines from different molecules. In the LWS data, we detect the cool CO gas, rotationally excited to the high-J levels J>14. In the SWS data, we detect several H₂ emission line fluxes (see Tables 2 and 3).

A major obstacle in the analysis is that the LWS beam was much larger than the SWS aperture. The LWS data had a circular beam with an area of 5027arcsec² whereas the SWS aperture is rectangular and 280arcsec² in size in the wavelength range from 2 $\mu$m to 12 $\mu$m. We expect, however, that the spatial extent of the high-J CO and and the rotational H₂ to be comparable, both arising from cool gas. Therefore, if the cool gas extends outside the SWS aperture, we expect the measured fluxes of the CO lines to be boosted. Hence, we shall plot the observed fluxes, but apply a boost factor to the model CO fluxes to facilitate a comparison. From the K-band images presented here for the 1-0S(1) H₂line, we estimate a boost factor of $B({\rm CO})=3{-}10$. Note that the model boost factors encapsulate our ignorance not only due to the ratio of the apertures but also due to the CO abundance. We shall demonstrate that models exist for which plausible boost factors are indeed predicted and abnormally high or low CO abundances are unnecessary.

The fluxes of detectable H₂ lines typically span an order of magnitude. The derived column densities in the upper levels of the responsible transitions, however, span a range of up to five orders of magnitude even just for the ISO data, whereas errors on individual points are only 20%. Therefore, to display and fit the models, it is absolutely indispensible to normalise the column densities. This is traditionally done by dividing by the columns from a 2000K slab of a gas in which the level populations are in thermodynamic equilibrium (e.g. Brand et al. 1988). Although, for the ISO data, a 1000K slab would be more appropriate, we remain with the 2000K normalisation in order to avoid confusion. Hence, in the Column Density Ratio diagrams, we actually display for each line the quantity

$$\log \left[\frac{CDR}{CDR\,(2000~{\rm K})}\right] = {\log} \frac{N}{N_{\rm o}} - {\log} \frac{N_{2000}}{N_{\rm o}}$$ (1)

where N is the observed column in the upper energy level, N₂₀₀₀ is the column of an arbitrary slab at 2000K, and $N_{\rm o}$ is a constant chosen so that the column derived from a specific line is unity. For ground-based observations, usually the upper level of the 1-0S(1) line is used. Here, we employ the 0-0S(5) line.

 

 

Table 3: Observed lines in L1448.

Element	Transition	$\lambda_0 [\mu {\rm m}]$	S3*	S2*	S1*	C*	N1*	N2*
H2	0-0S(7)	5.511	<2.2
H2	0-0S(6)	6.108	<2.5	<2
H2	0-0S(5)	6.910	$10.0\pm1.4$	$6.0\pm2.6$			$3.7\pm1.8$	$5.4\pm3.2$
H2	0-0S(4)	8.025	$2.7\pm0.5$	$2.9\pm0.6$		$1.2\pm0.5$	$2.2\pm0.7$	$2.2\pm0.6$
H2	0-0S(3)	9.665	$8.5\pm0.7$	$6.9\pm0.9$	<1.5	$1.9\pm0.5$	$5.5\pm0.8$	$5.2\pm0.7$
H2	0-0S(2)	12.279	$5\pm2$	<5		<3	<3	$9.0\pm2.1$
H2	0-0S(1)	17.035	$2.3\pm0.5$	$3.4\pm0.4$	<1.4	$1.0\pm0.3$	$2.3\pm0.4$	$3.0\pm0.4$
[SiII]	2P3/2-2P1/2	34.815					<2	$5.9\pm1$
[OI]	3P1-3P2	63.184	$7.5\pm2.8$	$5.5\pm1.7$	$17.3\pm1.8$	$25.8\pm4.8$	$43.4\pm6$	64.7$\pm$3.9
o-H2O	221-110	108.073
CO	24-23	108.763	[0ex]$\}$$2\pm0.6$	[0ex]$\}$<3	[0ex]$\}$$7.1\pm1.5$	[0ex]$\}$$5.3\pm2$	[0ex]$\}$$6.5\pm1.5$	[0ex]$\}$$4.4\pm1.8$
CO	23-22	113.458
o-H2O	414-303	113.537	[0ex]$\}$$2.9\pm1$	[0ex]$\}$$3.8\pm1.1$	[0ex]$\}$$4.1\pm1.7$	[0ex]$\}$$7.3\pm2$	[0ex]$\}$$12.5\pm1$	[0ex]$\}$$5.1\pm1.1$
CO	22-21	118.581				<5
CO	21-20	124.193				<5
CO	20-19	130.369				$4.4\pm0.5$
CO	19-18	137.196				$4.3\pm0.6$	$6.7\pm1$	$5.0\pm1$
CO	18-17	144.784		<1.5	$3.6\pm1$	$6.9\pm0.5$	$5.6\pm1.5$	$5.4\pm0.4$
[OI]	3P0-3P1	145.525				$4.2\pm0.6$		$3.4\pm0.9$
CO	17-16	153.267	<1.4	<1.5	$4.4\pm1$	$7.4\pm1.7$	$15.0\pm3$	$15.0\pm2.1$
[CII]	2P3/2-2P1/2	157.741	$8.9\pm1.3$	$7.8\pm1.7$	$11.8\pm2$	$7.2\pm0.8$	$11.2\pm2.4$	$10.5\pm1.5$
CO	16-15	162.812	<2.5	<1.3	$7.2\pm1.6$	$7.1\pm2.1$	$4.9\pm1.5$	$13.2\pm2.2$
CO	15-14	173.631	<3.1	<5	$5.0\pm1.5$	$6.4\pm2.0$	$9.0\pm2$	$6.0\pm2.0$
o-H2O	303-212	174.626	$4.1\pm1.6$	$5.0\pm1.8$	$7.5\pm1.5$	$9.4\pm1.8$	$9.0\pm2$	<5
o-H2O	212-101	179.527	$6.2\pm1.7$	$6.9\pm2$	$9.7\pm1.3$	$17.5\pm1.9$	$8.2\pm1.8$	<5
CO	14-13	185.999			<7	$8.3\pm1.6$	$10.6\pm2.4$	$7.2\pm2.8$

$^{{\rm *}}$

Fluxes in L1448 are in 10^-16Wm^-2.

4.2 Model range

Models predictions relevant to near-infrared observations of warm molecular gas (i.e. 1500 K-4000 K) are discussed by Eislöffel et al. (2000). Here, we classify the models in the context of cool molecular gas (i.e. 300 K-1500 K).

Model 1TEMP. A reservoir of constant temperature gas, assuming LTE, produces a straight line on the log(CDR) diagram. The gradient yields the excitation temperature. The model is unrealistic for the cool gas since the cooling time at $\sim$ 300-1500K is less than a year. There is no mechanism to maintain a specific clump of gas at these elevated temperatures. The rotational CO emission diagrams for 1TEMP were published by McKee et al. (1982).

Model NTEMP. More than one co-existing constant temperature component is equally implausible since there are no equilibrium temperatures or multi-phase media expected in the temperature range in which H₂ is detected. Two component fits can appear successful when column densities are plotted (e.g. Everett et al. 1995 for OMC-1), but clearly fail the more sensitive CDR analysis (Burton & Haas 1997).

Model CSHOCK. A planar C-type shock, in which ion-neutral friction heats the gas in a thick shock layer (Draine et al. 1983). The predicted CDRs are similar to 1TEMP but with slight curvature at low T_j (see Fig.3 of Smith et al. 2000). The excitation temperature of a C-shock is sensitive to the density, ion fraction, magnetic field strength, magnetic field direction, shock speed and oxygen chemistry (Smith & Brand 1990). These parameters cannot be uniquely extracted through modelling. The CO predictions were presented by Smith (1991). A maximum CO flux occurs at some J-value, depending on the density and maximum temperature.

Models CT/CTF Time-dependent C-shocks. C-shocks with moderate to high Alfvén numbers (>5 for a transverse field) are unstable (Model CTU) (Wardle 1990). Instability alters the resulting CDR behaviour by increasing the quantities of hot and cold gas, steadily increasing the CDR curvature (Fig.16 of Mac Low & Smith 1997). C-shocks may also require excessive formation times (Model CTF). The time to reach the steady state CDR (Fig.12 of Smith & Mac Low 1997) beginning from a jump shock structure is about one flow time.

Model CBOW. A supersonic flow past an obstacle creates a curved shock surface. Each part of the surface will excite the molecules according to the local conditions. This yields convex CDR curves in which the excitation increases with upper energy level. For a C-type shock surface, the curve is independent of the bow speed provided the speed exceeds the molecular breakdown speed (i.e. the bow apex corresponds to a hot dissociated cap and the H₂ emission is distributed in fixed warm, cool and cold components in the wings). The CDR curves still depend on the bow geometry, departure from LTE, and the oxygen abundance (Smith et al. 1991a, 1991b). Highly aerodynamic bows have been suggested by the overall lower excitation in CepheusE (Eislöffel et al. 1996). The CO diagram does not display a maximum in the high-J range (due to the cool molecular component) (Smith 1991).

Model CABSORBER. In the shock absorber model, a C-type bow with a high upstream Alfvén speed generates wide H₂ profiles (Smith et al. 1991b). This model is able to accelerate molecules to high speed without dissociating them. Relatively less cool gas results in a maximum in the CO diagram at high-J (Smith 1991).

Model TURB. Supersonic turbulence is generated by a jet, wind or in the wake of a curved shock. The turbulence leads to shocks which then dissipate, interact and disperse. The shock spectrum is an exponential function of velocity and time which suggests that weak shocks dominate the molecular excitation properties (Smith et al. 2000). Both J-type, TURBJ, and C-type, TURBC, versions would predict low excitation (Smith et al. 2000).

Model STR. A quasi-steady velocity cascade, termed a "Supersonic Turbulent Reactor'' dissipates energy in shocks at the same rate as it is injected. Uniform driven turbulence generates a power-law spectrum of shocks following an inverse-square root law with the shock jump speed. The molecular excitation, still to be calculated, would probably be high.

Model JSLOW. Non-dissociative planar radiative shocks generate high excitation CDR curves ( 3000-3600K). Lower excitations are only possible with a high magnetic field or a carefully chosen shock velocity (Smith 1994a). The JSLOW model requires reasonably high ion/neutral fraction ($\chi$>10^-5). A CO diagram will be presented here.

Model JFAST. A dissociative front is followed by cooling and molecule reformation at $T \sim500$K. No detailed CDRs are available. However, this model is characterised by high 2-1S(1)/1-0S(1) ratio, low intensities and cascade signatures (i.e. strong fluoresence lines, Hollenbach & McKee 1989). Also immediately distinguished by the strong atomic hydrogen emission (including the infrared lines Br$\gamma$, P$\beta$ etc.). CO diagrams show maxima typically at $J\sim 15{-}25$ (Hollenbach & McKee 1989) for densities above 10⁴cm^-3.

Model JBOW. A curved J-type shock in which molecular emission is dominated by extended wings of Model JSLOW type (Smith 1994a). To reduce the excitation to observed levels often requires extremely long weak shock sections.

Model FLO. X-ray and UV radiation heats and excites the molecules by UV pumping, via electron collisions and by direct dissociation and consequent reformation. A fluorescent spectrum is produced. Each vibrational level produces a distinct CDR curve (e.g. McCartney et al. 1999, Black & van Dishoeck 1987). High-J CO emission lines are not expected from photodissociation regions since the CO layer lies deeper into the cloud where the temperatures are low. Mid-J CO can, however, be very strong, as observed, if the PDR is highly clumped.

Model FLY. UV radiation from nearby shocks excites the molecules by Lyman resonance excitations (Black & van Dishoeck 1987). Model FLY may be relevant to bow shocks, with the Ly$\alpha$ emission generated in strong shocks across the bow cap (Fernandes & Brand 1995).

Model PREC. Magnetic precursors may lead slow planar J-type shock provided the ion fraction is low. Precursor development, found and modelled within Model CTU, produces quasi-linear CDRs (Smith & Mac Low 1997). This is the "neutral transformation'' stage (Stage 3) which can last for a time of order of 100-1000 years. In fast shocks, pre-shock ionisation inhibits a magnetic precursor.

L1448