TRIGGERING COLLAPSE OF THE PRESOLAR DENSE CLOUD CORE AND INJECTING SHORT-LIVED RADIOISOTOPES WITH A SHOCK WAVE. II. VARIED SHOCK WAVE AND CLOUD CORE PARAMETERS

Table 1 summarizes the three standard models (v10, v40, and v70), which are identical to the standard cloud and shock described above, differing only in the assumed shock speeds: 10, 40, and 70 km s⁻¹, respectively. These models are similar to those presented by Boss et al. (2010), except that the present models are for target clouds with twice the mass of the Boss et al. (2010) clouds. Boss et al. (2010) found that shock speeds in the range of 5–70 km s⁻¹ were able to simultaneously trigger collapse and inject significant shock wave material: slower shocks were unable to trigger collapse, while faster shocks shredded the target clouds. Hence we focus here on the same range of shock speeds.

Figure 1 depicts the evolution of model v40, which is typical in all respects of the models where simultaneous triggered collapse and injection occurs. The initial shock wave, moving at 40 km s⁻¹, strikes and compresses the edge of the target cloud, generating R-T fingers, leading to the onset of dynamic collapse along the symmetry axis (R = x = 0) and the formation of a protostar within ∼10⁵ yr. Figure 1 shows that the protostar is accelerated by the shock front to a speed of ∼1 km s⁻¹. While significant, such speeds are small with respect to the typical velocity dispersions seen for T Tauri stars in Taurus and Orion, where dispersions of ∼5 to 10 km s⁻¹ are observed (Walter et al. 2000).

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Figure 2 shows the extent to which gas and dust from the shock wave, the post-shock wind, and the ambient medium, are injected into the collapsing protostar seen in Figure 1(d). The velocity vectors show that while most of this gas is blown downwind by the shock and post-shock wind, some fraction of this material will be accreted by the protostar.

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Boss & Keiser (2012) discussed several of the reasons why estimates of the injection efficiency are difficult to determine. Here we follow Boss & Keiser (2012), and define the injection efficiency f_i to be the fraction of the incident shock wave material that is injected into the collapsing cloud core. Table 1 then summarizes the amount of mass derived from the shock wave, post-shock wind, and the ambient medium that is injected into the densest region of the collapsing protostar, i.e., regions with density greater than 10⁻¹⁶ g cm⁻³, as used by Boss & Keiser (2012). Table 1 shows that the injection efficiencies for shock wave material are of order f_i ∼ 10⁻² for the standard models, as found by Boss & Keiser (2010, 2012). In addition, the amount of injected gas derived from the post-shock wind is typically 10 times higher, while that derived from the ambient medium (i.e., the pre-shock gas) is typically over 10 times lower.

We first consider variations on standard model v40, where the density and radius of the target cloud are changed. Figure 3 shows the result for a cloud that was shredded by a 40 km s⁻¹ shock and failed to collapse—model v40-m1.9-r2, with a target cloud density 10 times lower than for model v40 (initial central density of 1.24 × 10⁻¹⁹ g cm⁻³) and a cloud radius twice as large (0.116 pc), leading to an initial cloud mass of 1.9 M_☉. Given that the shock in this model was identical to that in model v40 (Figures 1 and 2), it is clear that certain shock waves will only trigger the collapse of suitably dense initial cloud cores. In fact, the same result—a shredded cloud without sustained collapse—was obtained when the initial central density was decreased by only a factor of five to 2.48 × 10⁻¹⁹ g cm⁻³ along with the doubled cloud radius, for a cloud mass of 3.7 M_☉. Shredding continued for these clouds when the shock width was reduced by a factor of 10 to w_s = 0.0003 pc and the standard shock gas density (ρ_s = 3.6 × 10⁻²⁰ g cm⁻³) was increased by factors of 10, 100, 200, 300, 400, and 1000. All of these models refused to collapse, even when the computational volume was expanded to 20 blocks in Z = y, the direction of shock wave propagation, to allow more time for collapse to begin before the cloud disappears off the lower edge of the AMR grid.

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However, when the initial central density was decreased by a factor of 3.33 to 3.72 × 10⁻¹⁹ g cm⁻³, along with the doubled cloud radius, raising the initial cloud mass to 5.4 M_☉, dynamic collapse was achieved for the standard 40 km s⁻¹ shock (model v40-m5.4-r2). Table 2 summarizes the injection efficiencies achieved for several of these models with varied shock densities and widths, all with the number of blocks in the R = x direction doubled to 10. It can be seen that the values of f_i are typically ∼10⁻⁴ to ∼10⁻³ for these 5.4 M_☉ cloud models; however, f_i = 0 for model v40-m5.4-r2-w0.1-s400, as this model was compressed to a maximum density of ∼10⁻¹⁶ g cm⁻³, but was not quite driven into sustained collapse.

Table 3 summarizes the models with variations in the initial angular velocity (Ω_i, in rad s⁻¹) about the Z = y axis. All models had the standard shock with v_s = 40 km s⁻¹ and the standard 2.2 M_☉ target cloud, differing only in the initial rate of solid body rotation. The initial rotation rates of 10⁻¹⁶ rad s⁻¹ to 10⁻¹² rad s⁻¹ span the range from essentially no rotation to the most rapid rotation rate possible for collapse to occur for an isolated dense cloud core. Typical observed rotation rates for dense cloud cores span the range of ∼10⁻¹⁴ rad s⁻¹ to ∼10⁻¹³ rad s⁻¹ (Goodman et al. 1993).

The rotation axis is the same as the symmetry axis, as that is the only possible configuration allowed for 2D, axisymmetric clouds. The effect of rotation will then be to flatten a collapsing cloud into the direction perpendicular to the rotation axis (i.e., outward along the x = R axis), forming a rotationally flattened, protostellar disk that will be smacked face-on by the shock front. Such a configuration should lead to enhanced injection efficiencies by the target cloud. Table 3 shows that, compared to the standard, non-rotating model v40, when cloud rotation is included, significantly higher injection efficiencies are obtained, as expected, by as much as a factor of ∼10. Figure 4 shows the results for four models with increasingly higher values of Ω_i: as Ω_i increases, the collapsing protostar is accompanied by a progressively larger region with relatively high gas density (yellow-green regions), compared to the non-rotating model v40 (Figure 2(a)), allowing more shock wave material to be intercepted in cloud gas that is likely to be accreted. Model v40-o13, in fact, can be seen to have formed a structure similar to a large scale (∼1000 AU) protostellar disk (Figure 4(c)). However, when Ω_i is increased to 10⁻¹² rad s⁻¹, as in model v40-o12, the cloud does not collapse, and instead is shredded by the shock wave. Hence the extent to which initial cloud rotation can enhance the injection efficiency is limited in the end by the desire to also achieve high enough densities for dynamic collapse to ensue. In addition, shock fronts which strike clouds where the rotation axis is not aligned with the direction of shock propagation will not be able to inject quite as much shock front matter, lowering the injection efficiencies. The present results for aligned rotation axes and shock propagation directions thus represent upper bounds for the possible enhancements due to rotation. A fully 3D study will be needed to learn about injection efficiencies in the case of non-aligned rotation axes.

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Table 4 summarizes the models with variations in the initial target cloud density distribution, namely a perturbation to the Bonnor–Ebert sphere with a binary perturbation along the symmetry (Z = y) axis, as seen in Figure 5(a). These models have a reference density of 4.1 × 10⁻¹⁹ g cm⁻³, three times lower than the standard target clouds, yielding an initial cloud mass of 1.0 M_☉. These clouds are so low in mass, in fact, that when evolved without any shock wave, they expand and do not collapse. Hence these clouds represent clumpy regions of the ISM that would not collapse on their own, in contrast to the other, more massive target clouds, which eventually collapse on their own, even without being struck by a shock wave.

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Figure 5 shows that in spite of this basic difference, when struck by a suitable shock wave, the 1.0 M_☉ clouds can still be driven into sustained gravitational collapse, in much the same way as the other models. However, when v_s is increased to 20 km s⁻¹ or above, these low density clouds are shredded and do not collapse. Table 4 summarizes these results, and shows that for the shock speeds that do lead to collapse, the injection efficiencies can be significant, i.e., f_i as high as 5.2 × 10⁻² for model v10-m1.0.

Table 5 summarizes the models with variations in the maximum shock (T_s) and post-shock wind (T_ps) temperatures, compared to a standard shock model with T_s = T_ps = 1000 K. All models collapse, except for model v40-T100-s100, where the shock density was too high to result in collapse, given the other parameters. The injection efficiencies varied from 2.7 × 10⁻⁴ to 1.8 × 10⁻² for the T_s = T_ps = 100 K models, differing only in the value of ρ_s. These models demonstrate that injection efficiencies continue to be significant, even for considerably cooler shock and post-shock temperatures than the standard values.

Tables 6, 7, and 8 present the results for variations in the shock densities (ρ_s) and widths (w_s), for shock speeds (v_s) of 10, 40, and 70 km s⁻¹, respectively. Many of these models had extended grids (N_y = 30 or 40) in the shock direction (Z = y) in order to allow the cloud to be followed downstream far enough to determine if collapse would ensue. The models with M_s = 0 were models where collapse did not occur.

Table 6 lists the models with v_s = 20 km s⁻¹, showing that with a wide variety of shock densities and widths, injection efficiencies of ∼10⁻² to ∼10⁻¹ can result if collapse occurs. All of the 40 km s⁻¹ models listed in Table 7 were also described in Boss & Keiser (2010), but with the difference that an improved methodology for estimating the injection efficiencies (Boss & Keiser 2012) is used here, which usually results in higher estimates of f_i. Both estimates are listed in Table 7 for ease of comparison. Finally, Table 8 shows that for 70 km s⁻¹ shocks, injection efficiencies are still significant, with f_i ∼ 10⁻³ to ∼10⁻², but are typically lower; e.g., f_i = 7.4 × 10⁻², 5.0 × 10⁻², and 8.6 × 10⁻³ for models v10-w0.1, v40-w0.1, and v70-w0.1, respectively.

We now turn to a discussion of which models might be the most appropriate representations of observed stellar outflows and shock waves that might be carriers of freshly synthesized SLRIs. SNe resulting from the core collapse of massive stars in the range of ∼20–60 M_☉ and planetary nebulae derived from intermediate-mass (∼5 M_☉) AGB (Huss et al. 2009) or higher mass (∼7–11 M_☉) super-AGB stars (Lugaro et al. 2012) have been proposed as possible sources of many solar system SLRIs. In addition, significant amounts of ²⁶Al are contained in the outflows of massive WR stars, which are the predecessors to many Type II SN (Tatischeff et al. 2010). ²⁶Al is abundant in the ISM: Diehl et al. (2010) detected γ-rays from the Sco-Cen association that implied the presence of ∼1.1 × 10⁻⁴ M_☉ of live ²⁶Al. If this amount of ²⁶Al pollutes a ∼10⁶ M_☉ GMC containing ²⁷Al at solar system abundances (Anders & Grevesse 1989), then the GMC would have a ratio ²⁶Al/²⁷Al ∼3 × 10⁻⁶, well below the inferred initial solar system canonical level of ²⁶Al/²⁷Al ∼ 5 × 10⁻⁵ (MacPherson et al. 2012).

The Cygnus Loop is a ∼10⁴ yr old core collapse (Type II) SN remnant (SNR) with v_s ≈ 170 km s⁻¹ and w_s < 10¹⁵ cm (Blair et al. 1999), a width consistent with our thin shock models, with 0.1 w_s = 9.3 × 10¹⁴ cm. W44 is also a core collapse SNR with v_s = 20–30 km s⁻¹ and a width less than 10¹⁶ cm (Reach et al. 2005), comparable to our standard shock width w_s = 9.3 × 10¹⁵ cm. Hence SNR appear to be most consistent with our models with relatively high speed (70 km s⁻¹) shocks and 0.1 w_s or with models with lower speed (10–40 km s⁻¹) shocks and 1 or 0.1 w_s, respectively.

The W44 SNR is expanding into gas with a number density n ∼ 10² cm⁻³ (Reach et al. 2005). Boss & Keiser (2012) presented results for a 3D version of our 2D model v40-w0.1-s400-20 (Table 7), a model with v_s = 40 km s⁻¹, shock number density n_s ≈ 4 × 10⁶ cm⁻³, and a shock width of ≈10¹⁵ cm. Boss & Keiser (2012) noted that the shock density n_s for an isothermal shock propagating in an ambient gas of density n_am is n_s/n_am = (v_s/c_am)², where c_am is the ambient gas sound speed. For model v40-w0.1-s400-20, with c_am = 0.2 km s⁻¹ and n_am = 10² cm⁻³, as in the case of the W44 SNR, the shock density should be n_s = 4 × 10⁶ cm⁻³, the same as the density in model v40-w0.1-s400-20. Hence shock densities about 400 times higher than in the standard shock model appear to be similar to what is expected to be the case in realistic SNRs (Boss & Keiser 2012).

During the transition from an AGB star to a post-AGB star, low mass stars generate outflows known as planetary nebulae (van Marle & Keppens 2012), with speeds of order 20 km s⁻¹. The planetary nebula A39 is estimated to have a thickness of ∼3 × 10¹⁷ cm (Jacoby et al. 2001), while planetary nebula PFP-1 is estimated to have a thickness of ∼5 × 10¹⁷ cm (Pierce et al. 2004). These thicknesses are considerably larger than the standard shock width w_s = 9.3 × 10¹⁵ cm, but are only a few times thicker than the models with 10 w_s = 9.3 × 10¹⁶ cm. Hence planetary nebulae appear to be most consistent with our models with low speed (∼10 km s⁻¹) shocks and large (10 w_s) shock widths.

3D models of the interaction of a WR fast wind (initially moving at 2 × 10³ km s⁻¹) with a previously emitted red supergiant wind (moving at 10–15 km s⁻¹) led to the formation of R-T fingers and expansion velocities of the interacting region less than 100 km s⁻¹ (van Marle & Keppens 2012). The thickness of the shocked region was typically ∼10¹⁷ cm and the density was ∼4 × 10⁻²² g cm⁻³. These quantities correspond to shock widths and densities of ∼10 w_s and 0.01 ρ_s, respectively. The latter value is considerably lower than any of the values explored in the present study. Foster & Boss (1996) found that such a low-density shock would not have enough momentum to compress a target cloud into collapse. Furthermore, the gas temperature was forced to remain above 10⁴ K everywhere in the van Marle & Keppens (2012) simulations. WR winds are fully ionized and are driven by thermal pressure. For these reasons a direct comparison to the present models is not possible. It is unclear what would happen when a WR wind shell encounters a dense molecular cloud core, but the fact that the gas is fully ionized and unable to cool below 10⁴ K implies that such a hot shock would shred a target cloud to pieces (Foster & Boss 1996).

The dilution factor D is defined as the ratio of the amount of mass injected into the collapsing cloud that was derived from the shock wave (M_s) to the amount of mass derived from the target cloud. The tables list M_s for each of the models. For a final system mass of 1 M_☉, the dilution factor is then simply D = M_s/M_☉. However, because a SN shock starting at ∼2000 km s⁻¹ must be slowed down by snowplowing a large amount of the ISM to reach a speed as low as in these models, we follow Boss & Keiser (2012) in defining a third parameter, β, to be the ratio of shock front mass originating in the SN to the total mass in the shock, including that swept up from the intervening ISM. In this case, the dilution factor becomes D = βM_s. For a SNR to slow down from 2000 km s⁻¹ to 10–70 km s⁻¹, as in our models, β = 0.005–0.035. This means that the shock front must sweep up between ∼30 and ∼200 times its own mass before slowing to the proper speeds. In the case of the W44 SNR, moving at 20–30 km s⁻¹, the shock must have already swept up about 100 times its original mass. For a W44-like shock with n_s = 4 × 10⁶ cm⁻³ and w_s = 10¹⁵ cm, this means that the amount of swept-up intervening gas was ∼99% of the shock's current gas mass, i.e., a column depth (by number) of ∼4 × 10²¹ cm⁻². For the gas that W44 is currently expanding into, with a number density ∼10² cm⁻³ (Reach et al. 2005), this would require passing through ∼10 pc of such intervening gas. The diameter of the W44 SNR is ∼11 pc (Reach et al. 2005), in agreement with this estimate. Such number densities (∼10² cm⁻³) are typical of the average giant molecular cloud densities (Williams et al. 2000).

The dilution factors needed to match inferred meteoritical abundances of SLRIs are uncertain, in part because the inferred initial abundances for some SLRIs are in flux (e.g., for ⁶⁰Fe, as previously noted), but also because estimates of the production rates of SLRIs such as ⁶⁰Fe and ²⁶Al are uncertain by large factors (Tur et al. 2010), as noted by Boss & Keiser (2012). Given these caveats, dilution factors for a SN source range from D ∼ 10⁻⁴ to 10⁻³ (Takigawa et al. 2008) to D = 3 × 10⁻³ (Gaidos et al. 2009). Given the β factor, this means that a SN source requires models with 1 w_s and 10 km s⁻¹ (β = 0.005) to have M_s = 0.02–0.6 M_☉; 0.1 w_s and 40 km s⁻¹ (β = 0.02) to have M_s = 0.005–0.15 M_☉; or 0.1 w_s and 70 km s⁻¹ (β = 0.035) to have M_s = 0.003–0.09 M_☉; in order to cover the range of D ∼ 10⁻⁴–3 × 10⁻³. High shock densities are also indicated for a SN source. The tables show that while none of the v10-series or v70-series models meet these requirements, two models in Table 7 (v40-w0.1-s100 and v40-w0.1-s200) reach M_s = 0.0025 and 0.0039 M_☉, respectively, close to the minimum amount needed. Given that rotation (Table 3) can increase the injection efficiencies by factors as large as 10, it appears likely that a rotating target cloud may be able to meet the required dilution factor for a v_s = 40 km s⁻¹ shock.

For an AGB star, Trigo-Rodríguez et al. (2009) estimate D ∼ 3 × 10⁻³. Since β ∼ 1 for an AGB source, D = M_s/M_☉, and an AGB source would require a model with M_s ∼ 3 × 10⁻³ M_☉, 10 w_s, and v_s ∼ 20 km s⁻¹. The models with v_s = 10 or 40 km s⁻¹ come the closest to the correct shock speed. Table 6 shows that none of the v10-w10-series succeeded in triggering collapse. Table 7 shows that one of the v40-w10 series of models, v40-w10, was able to trigger collapse and inject M_s ∼ 4 × 10⁻⁴ M_☉, about a factor of 10 times too low to support the AGB star scenario. However, the rotating target cloud models (Table 3) show that with significant rotation, M_s can be increased by a factor of 10, so in that case an AGB star remains as a possibility.

The Table 3 models show that target cloud rotation is likely to be a key factor for achieving the dilution factors apparently required for the solar system's SLRIs. Tables 2 and 4 show that target clouds with larger masses than the standard cloud are also more likely to achieve collapse and significant injection. Table 5 shows that the maximum temperatures assumed for the shock and post-shock gas can also have an effect on the outcome.

Table 1 shows that more post-shock wind is likely to be accreted by the collapsing protostar than shock wave matter, by as much as a factor of 10. This is similar to the result found in the 2D models of Boss & Foster (1998). Hence, these models suggest that explaining the evidence for low ⁶⁰Fe/⁵⁶Fe ratios and high ²⁶Al/²⁷Al ratios as being due to their synthesis in different layers of a massive star (Tang & Dauphas 2012) will only work if the inner layers fall back onto the stellar remnant, not if they are ejected and follow behind the leading edge of the SN shock.

Finally, we note that the present estimates strictly apply only to the injection of shock wave gas carrying SLRIs, or dust grains small enough to be carried along with the gas. Large grains can be injected with higher efficiencies, as found by Ouellette et al. (2010). However, such large grains appear to be rare in the ISM, and Boss & Keiser (2010) concluded that the overall effect should not be large. Nevertheless, to the extent that large grains carrying SLRIs are present in shock waves, the present estimates of SLRI injection efficiencies should be considered to be lower bounds.