### 4. AGE SEQUENCES IN BRIGHT RIMS

A description of such pressure equilibrium and the resulting acceleration is given by Bertoldi & McKee (1990). In their equation 5.10, they write the ratio of the self-gravitational acceleration inside the head to the acceleration of the whole head by the rocket effect as

 (4.1)

 (4.2)

for a magnetic head. Here, Mcl is the cloud head mass, MJeans = 1.18 cl4 / (G3 Pcl)1/2 and M = 0.12 / G3/2 are the critical (or maximum) cloud masses for stability without and with a magnetic field, respectively; is the total magnetic flux in the cloud. The pressure at the ionized surface of the cloud enters the expression for MJeans and is

 (4.3) (4.4)

for dimensionless parameter ψ,

 (4.5)

In these expressions, the recombination rate to all but the ground state is , the incident ionizing flux is FII in photons cm-2 s-1, the ionizing luminosity is S49 in photons s-1, the cloud radius is Rcl,pc in pc, the distance to the ionizing source is Rpc, in pc, the velocity dispersion in the cloud is cl, and the velocity dispersion in the HII region is II.

These equations suggest that if the cloud head requires gross instability for a star to form, i.e., Mcl > MJeans or Mcl > M (or, Mcl > MJeans + M in McKee (1989), then the internal gravitational acceleration in the head is always greater than the rocket acceleration. Thus the stars that form in the head should follow the head along as it accelerates away from the HII region. Why are the stars "left behind" in this case?

Another consideration is that the side of the dense core facing the HII region could be continuously peeled away by the ionization. The speed of this peeling is determined by the incident flux. After pressure equilibrium, a D-type ionization front enters the compressed neutral gas; "D" stands for density-bounded, i.e., the ionizing radiation is stopped by gas absorption (Spitzer 1978). The speed of such a front into the dense gas is cl2 / [2II]. Because II >> cl, this D-front speed is always much less than cl. The orbit speed of a newly formed star inside the head is of order cl, however, for a self-gravitating head. Thus the speed at which the ionized side of the pillar gets peeled away is always much less then the embedded stellar speed. Thus, stars should not be exposed by ionization either.

Evidently, for both the rocket effect and ablation by ionization, stars forming in an unstable head should should continuously fall back into the head center faster than the surface of the head moves away from the source of ionization. This makes the exposure of young stars and their age gradients difficult to understand. It could explain, however, why age gradients are seldom obvious - the triggered stars scatter around the head by gravitational forces.

One solution to this problem is that the head is stable on a large scale with cl equal to some turbulent speed that is larger than the sound speed, or perhaps with magnetic support, and yet inside of the head, there are local dense clumps that are unstable in the sense that their masses are larger than the thermal Jeans mass after the magnetic field has diffused out. In these cases, it might be possible that Mcore > MJeans,thermal for localized star formation while at the same time ggrav < grocket for exposure of the star after it forms.

Another model of triggered star formation is that there is a pre-existing pillar-like shape with multiple clumps aligned to the HII region. Then the compression front moves along the pillar from clump to clump, triggering gravitational instabilities as it goes. The exposure of the stars would follow the erosion of each clump, compressed one after another.

Of course it is possible that the stars near the head were not triggered. A key observation for triggering will be the velocities of the young stars near the head in comparison to the head velocity. If the stars are moving much slower than the head, then they could have formed before the compression and rocket-like acceleration.

Simulations by Dale et al. (2007b) of triggered star formation in numerous dense clumps of a pre-existing molecular cloud indicate the difficulty in distinguishing between stars that formed previously and were exposed by clump ionization or motion, and stars that were triggered by the ionization pressure. In this study, the additional amount of star formation that was from the triggering alone was only ~ 30%.