Astron. Astrophys. 342, 233-256 (1999)

6. Time scales and stability

One of the primary purposes of the work presented in this paper is to examine whether substantial star formation has occurred, or is likely to occur, in the fingers of the Eagle nebula. The IR observations indicate that there are no young stellar objects embedded inside the dense fingertips. This is corroborated by the relatively narrow line widths present in the molecular line observations, indicating a lack of outflow activity. By examining the time scales associated with the various physical processes acting on the clouds, we hope to create a plausible picture of their eventual fate.

The fingers are estimated to have number densities typically n(H ₂ ) 2 10⁴ cm^-3, but in the clumps located near the tips, the density increases to n(H ₂) 2 10⁵ cm^-3. For the purpose of the following discussion, we adopt a simple model for a dense clump at a fingertip which is assumed to be spherical, with a radius r = 0.085 pc, and a mass of M 31 . We further assume that the ionising sources are located 2 pc away from the fingertip, and produces 2 10⁵⁰ photons s^-1. It is observed that the peak submm continuum emission from the clumps is located 0.1 pc deeper into the fingers than their photoionised front surfaces.

A substantial amount of theoretical work has been carried out to try to understand the evolution of dense, gaseous clumps that are being subjected to the ionising radiation of nearby OB stars. A discussion of propagating ionisation fronts is presented in Spitzer (1968), and a number of more recent papers have been written on the subject, though with emphasis on the radiatively driven implosion (RDI) stage (Bertoldi 1989, Lefloch & Lazareff 1994, 1995), and the cometary globule stage (Bertoldi & McKee 1990).

The analysis of propagating ionisation fronts indicates that there are two distinct types, which are characterised by their propagation speed. The R -type (`Rarified') ionisation fronts travel through a low density medium with a velocity of , where c <sub>i</sub> = sound speed of the ionised medium. Typically, c <sub>i</sub> 11.4 km s <sup>-1</sup>. The second type of ionisation front is referred to as D -type (`Dense'), and these travel through denser gas with a velocity of , where c _I = the ound speed in the pre-ionised material and . Often a dense gaseous clump will not satisfy the conditions necessary for either an R -type or a D -type front when initially subjected to a strong ionising source. A front formed under these circumstances is called M-type, and will result in a shock being driven into the clump (RDI stage), which compresses the gas until pressure equilibrium between the ionisation front and the cloud interior is attained. Under these circumstances, the gas just ahead of the ionisation front is compressed initially by the preceding shock wave so that an approximately D -critical ionisation front is able to form (i.e. ) behind the shock front which continues to compress the cloud ahead of it. The structure of the cloud during this implosion stage, moving from the cloud surface towards its interior, consists of a hot, photoevaporating, ionised region, an approximately D -critical ionisation front, a dense, neutral post-shock region, a shock front, and then a pre-shocked neutral region composed of the undisturbed gas in its original state. Provided that the implosion does not induce the cloud to collapse, the post-implosion cloud then undergoes a period of slow evolution as the ionisation front slowly propagates into the cloud.

The high mass stars that provide the ionising radiation are formed within Giant Molecular Clouds (GMCs). The cloud complexes are observed to be composed of clumps, with typical densities 10³ cm^-3, and radii 1 pc, which are embedded in a warm, tenuous interclump medium (ICM ). The ignition of an O or B star causes an R -type ionisation front to propagate out rapidly through the ICM to the Strömgren radius:

where n _i = particle number density of the ICM and = the hydrogen recombination coefficient. The heating of the gas by the ionising flux leads to temperatures of T _i 10⁴ K, and sound speeds of c _i 11.4 km s^-1. When the R -type front reaches a dense clump, it stalls and a shock is driven into the clump (RDI) until it is sufficiently compressed that a steady D -type ionisation front may be maintained. It is this general scenario that we have in mind when making the following numerical estimates for conditions in the fingers of the Eagle nebula.

It is necessary first of all to compare the pressure at the surface of the gas columns with that in their interior, to determine whether or not shocks are being driven into the heads of the fingers. If the pressures were approximately equal, it suggests that the fingers had already been compressed by an ionising-shock front (IS-front). Conditions in the cloud interior would allow the formation of a steady D -type ionisation front at the surface, which will slowly eat into the cloud. If, on the other hand, the external pressure is found to greatly exceed the internal pressure, we will interpret this to mean that an IS-front is currently being driven into the columns.

The line widths of the C¹⁸O spectra are = 2 km s^-1. The pressure inside the cloud is composed of both a turbulent and a thermal contribution, so that

where T is the temperature, is the density, µ is the mean molecular weight, and is the gas constant. Assuming low temperature cores, then the thermal component of the line-width is negligible and = ² / (8 ln 2) represents the square of the inferred turbulent velocity in the cloud. Both the observations and the thermal and chemical modelling, indicate that the temperature in the interior of the cloud is 15-20 K and the average number density n (H ₂) 2 10⁴ cm^-3 However the density in the clumps is estimated to be somewhat larger, with n(H ₂) 2.2 10⁵ cm^-3 (these values are similar to those estimated by Pound 1998). We take the upper value of the internal density to be n (H ₂) 2 10⁵ cm^-3, giving an estimate for the total internal pressure of P _int / = 3.5 10⁷ cm^-3 K. The thermal pressure at the tip of the , estimated by Hester et al. (1996), is P _i / k = 6 10⁷ cm^-3 K, which is almost double the internal pressure. We therefore conclude that equilibrium does not exist between the cloud surface and the interior, and that an IS-front is currently propagating into the column (Megeath & Wilson 1997 find similar values for the pressure difference ( a factor of 2) in the sample of objects they observed). We further note that the conservation conditions across a D -critical ionisation front predict that the pressure just ahead of the front is twice the thermal pressure just behind the front (i.e. P _n = 2 P _i, where the subscripts are n for `neutral' and Ifor `ionised'. This arises because ionised material leaves the front at a velocity v _i = c _i.

If pressure equilibrium has in fact been attained between the ionisation front and the cloud interior, and the additional internal pressure required is provided by a magnetic field, then the field required is:

Work presented by Myers & Goodman (1988) indicates that there is usually an approximate equipartition between the magnetic and kinetic energies in molecular clouds, so that the turbulent line widths should be Alfvénic. If a magnetic field is responsible for providing an internal pressure inside the fingers which is capable of balancing the pressure at the ionisation font, then it is apparent that equipartition does not hold, since the Alfvén speed corresponding to a field of 5.4 10^-4 G is v _A = 1.9 km s^-1. Alfvénic motions would then lead to an observational linewidth of:

which is about twice as large as that observed from the C¹⁸O lines which trace the material in the core.

The pressure of the large scale magnetic field is anisotropic, and provides a stress that is in a direction perpendicular to the field direction. Alfvén waves, on the other hand, are able to provide an isotropic pressure, which is given by:

where B is the perturbation to the mean magnetic field associated with the Alfvén wave. The fluid velocity perturbation associated with this travelling wave is (from McKee & Zweibel 1995):

We therefore have the relation:

where B is the mean magnetic field strength, and is the Alfvén speed associated with the mean field. For , the mean field dominates over the random field component, and internal stresses provided by the magnetic field are strongly anisotropic. If this were the case in the Eagle nebula, then the morphology of the fingers should reflect this internal, anisotropic pressure. The observations of the fingers, and of in particular, indicate a substantial degree of cylindrical symmetry. The direction of elongation of the fingers points towards the ionising sources, and is therefore most probably caused by the ionising radiation rather than by an ordered magnetic field. We therefore conclude that a large-scale ordered magnetic field does not produce an internal pressure that can balance the pressure of the ionisation front. A magnetic field in which the disordered component was comparable to the ordered component would yield internal motions since:

This equipartition of kinetic and magnetic energy is observed in numerical simulations of MHD turbulence (e.g. Gammie & Ostriker 1996). As described above, these motions would lead to linewidths 4.4 km s^-1, which are not seen in the data. We conclude that there is insufficient internal pressure in the fingers of the Eagle nebula to balance the pressure at the ionisation front, and that an ionisation-shock front is currently being driven into the cloud.

The presence of a shock front currently propagating into the columns indicates that the dense clumps located towards their tips are probably not the result of radiatively driven implosion. When an IS-shock front propagates into a cloud, the shock front precedes the ionisation front (located at the optical surface of the cloud) by a small distance. This would preclude it from traversing the top r 0.2 pc of , and forming the clump there. Instead, it seems likely that the dense clumps located towards the tips of the fingers are part of a larger, dense structure that pre-existed the expansion of the HII region, but have now come into stark contrast with their local environment due to the photoionisation of the surrounding, lower density material. The pre-existence of this dense structure, and its associated shadowing effect, have probably contributed to the formation and appearance of the pillars, and the fact that they point towards the external O-stars.

The shock propagation velocity may be derived from the usual shock discontinuity jump conditions, leading to the equation

where and are the pressure and density of the shocked, neutral material and P _n and are the pressure and density of the pre-shocked neutral material. We have no direct knowledge of what to expect for , since we only know the pressure and do not have an independent estimate of the density and sound speed in this region. If we assume that 2 / then the previous equation may be expressed as:

where 1 2, so that the maximum error that we incur in our estimate of V _s, as a result of guessing the value of , is a factor of . We take and = 2 10⁵ m (H ₂), where m (H ₂) is the mass of a hydrogen molecule, which then leads to a shock velocity of V _s 1.3 km s^-1. The time-scale for this shock to propagate through the top 0.2 pc of the finger tip is then = 0.2 pc / V _s 1.5 10⁵ years, which is comparable to shock crossing times derived by Bertoldi (1989) for the dense clumps located in the Rosette nebula that appear to be undergoing implosion.

These shock crossing times are considerably shorter that the estimated ages of the O-stars in the Rosette and Eagle nebula (i.e. 1 Myr), indicating that the structures being observed now (cometary globules or elephant trunks) have only been exposed to the ionising radiation of the nearby stars for a relatively short time. An upper limit for the time taken for an R-type ionisation front to reach the elephant trunks during the initial expansion of the Strömgren sphere is given by = (2 pc / 2 c _i) = 8.8 10⁴ years (i.e. a relatively short time after the switch-on of the O stars). The implication of this rapid time for the initial expansion of the HII region is that previously there must have been intervening dense structures shielding the pillars from the ionising flux of the stars, which have only recently been eroded away, exposing the pillars to the UV radiation.

© European Southern Observatory (ESO) 1999

Online publication: December 22, 1998
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