Astron. Astrophys. 340, 508-520 (1998)

4. Results

We shall use case A to describe the morphological evolution of externally illuminated disks. Fig. 3 shows several evolutionary times of this simulation. The snapshots are displayed from top to bottom and left to right. The first frame (Fig. 3a) shows the approaching ionization front shortly after the external radiation source above the disk was turned on. At this early stage of the evolution the structure of the disk is still unaffected and material continues to fall onto the disk. Initially of type weak R, the ionization front changes to type strong D where it encounters the disk and the disk's accretion shock (unfortunately the I-front and its associated shock front are poorly resolved in our calculation; c.f. Fig. 3b). In fact the ionization front recedes somewhat at  yr as heated dense disk material begins to expand upwards ¹, is compressed by gas still infalling, and subsequently recombines. This is merely a transitory effect, however, and the ionization front eventually returns to the surface of the central portions of the disk as it continues to wrap around the disk's perimeter.

Fig. 3a-h. Evolution of star-disk system I under the influence of a weak external EUV radiation field (Case A ). The grey scale shows the density structure. The black lines are contours of constant degree of ionization and given for The velocities are represented by arrows. The normalization is given at the upper right corner.

Fig. 4a-h. Evolution of star-disk system II under the influence of an external EUV radiation field (Case B ). The grey scale shows the density structure. The black lines are contours of constant degree of ionization and given for The velocities are represented by arrows. The normalization is given at the upper right corner.

Once part of the I-front enters the "shadow" region below the disk ( yr; c.f. Fig. 3c) it slows down significantly. This part of the I-front is principally driven by EUV radiation resulting from ground state recombinations - the ionizing flux is consequently at a lower level than the previous direct illumination. Also, some material has been swept up; this part of the I-front has now become D type. Because the swept up material on the neutral side of the ionization front was either part of the molecular clump collapsing onto the star-disk system or detached disk material, it possesses some angular momentum. This material cannot easily be pushed to the rotation axis, due to centrifugal effects. Also, the stellar wind can freely expand below the disk, but cannot expand upwards, being contained at the top by the now stationary strong D type ionization front. The result is a tube-like tail, shown in greater detail in Fig. 5, the mass of which is of order . Here the white density contour lines elucidate the density distribution within the neutral tail and disk and reveal the disturbed outer parts of the disk. Most of the mass loss through photoevaporation occurs at the disk's surface, where the ionization front encounters the densest regions. The gas temperature in the dense outer parts of the tail (which rotate with approximate keplerian velocity) is comparable to the dust temperature (100 K - 200 K), whereas the inner part of the tail reaches temperatures of K. In the subsequent evolution (Figs. 3d, 3e and 3f) the tail becomes Rayleigh-Taylor unstable - it expands (centrifugal bounce) before it breaks up into fragments which leave the computational domain with the evaporating flow. The ionization front eventually begins to close immediately behind the disk (Figs. 3g and 3h). Now the ionized gas close to the axis in the disk's shadow is the lower angular momentum material from the inner regions of the disk and the zero angular momentum material from the stellar wind; it doesn't encounter the same centrifugal barrier as the tail material did. The asymmetrical structure of the disk is displayed in greater detail in Fig. 6. The dense inner parts of the disk of model I with a 8.4 central star remain relatively undisturbed by the external radiation field. This is also confirmed when comparing the initial and final radius of the disk which are almost the same.

Fig. 5. Case A 519 yr after the onset of the external radiation field. Density, velocity and ionization structure are displayed as described in Fig. 3. In addition, white contour lines are given for the density within the disk and the cometary tail which vary from to spaced by .

Fig. 6. Density, velocity and ionization structure of model I at the end of the simulation (case A ). Symbols and lines have the same meaning as in Fig. 5.

The evolution of model II for cases B, C and D is much more violent. This is obviously due to the higher photon flux and the less massive (0.58 ) central star. Also, due to the fact that the effects of angular momentum transfer were included in the Yorke & Bodenheimer (1998) models, a small fraction of the disk material in the outer regions gains angular momentum, leading to very extended disks, the outer regions of which are only weakly gravitationally bound. In Fig. 4 the structure at eight evolutionary times of case B are shown. Fig. 4a shows the undisturbed system just before the ionization front reaches the disk. Approximately later (Fig. 4b) the ionization front has swept material "behind" the disk (as seen from the illuminating source). In contrast to case A the tail has a mass of order - a significant fraction of the disk mass. The disk itself is strongly compressed and distorted at its outer edge. As the material behind the disk is photoionized it eventually leaves the immediate vicinity. The weakly bound outer edge of the disk becomes more distorted and more extended away from the illuminating star, displaying a wing-like appearance in our 2D representation. After about the neutral disk material attains its maximum extent (Fig. 4c); several (about 10% of the disk mass) has been forced into these disk wings . Far from the disk's central star these wings are no longer gravitationally bound to the star-disk system (Fig. 4e). They break up (Fig. 4f) and photoevaporate (Fig. 4g). After yr (Fig. 4h) the ionization front completely envelopes the disk - much later than for case A. Fig. 7 and Fig. 8 show with greater detail the density structure within the compact wings (Fig. 4d) and the wing fragments (Fig. 4f).

Fig. 7. Model II in case B yr after turning on the external radiation field. Here the white contour lines for the density vary from to in increments of .

Fig. 8. Model II in case B after yr. Again the white contour lines for the density vary from to in increments of .

The evolution of the disk for cases C and D is qualitatively similar to case B. We continued each simulation until the ionization front completely encloses the disk. The final disk structure for these three cases is shown in Fig. 9 for . The results are displayed with decreasing distance from the ionizing source from top to bottom. Note the decreasing radial extent of the disk and the increasing disturbance of the densest regions with decreasing distance. The temperature of the ionized gas increases from  K to  K above the disk and from  K to K below the disk for case B to D. The resulting slightly higher velocities and the higher densities of the ionized gas surrounding the neutral disks indicate an increasing mass loss rate with decreasing distance.

Fig. 9. Density, velocity and ionization structure of model II at the end of case B , C and D . The results are displayed from top to bottom, respectively.

Typical properties of the models at the end of the simulations are summarized in Table 2. The final appearance of the disk and the duration of a "cometary" phase depend on both the incident photon flux of the illuminating source and the original configuration of the star-disk model. Fig. 10 shows the dependence of the photoevaporation rate on the distance d between the illuminating star and the disk. The crosses mark the photoevaporation rates of the disks shown in Fig. 9, which are determined as described in Papers I and II. The power law fit in Fig. 10 (solid line) corresponds to

Fig. 10. Final photoevaporation rate of case B, C and D versus distance d between the illuminating star and the disk (crosses ) and scaled to the disk radius of case D (squares ). The lines are the results of power-law fits and labeled with the corresponding power-law exponent.

For comparison we show the dependence when is normalized by (boxes and dashed line; see discussion in Sect. 6).

© European Southern Observatory (ESO) 1998

Online publication: November 9, 1998
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