Intro
Triggered star formation, in which winds generated from supernova blasts or wind-blown bubbles enters a clumpy region and compresses the clumps into dense cores that can violate local Jean's criterion and trigger gravitational collapse, is one of the mechanisms used to explain star forming regions such as Cygnus Loop, and more importantly, the Solar System. The triggering mechanism provides a natural way of forming stars while at the same time injecting SLRI's into the star and its disk. Recent years' literature has seen a noticeable increase in the amount of numerical works dedicated to this topic, as projects lead by Boss (Boss et al 2008) carried out pioneering work on the study of the shock condition of such successful triggering and mixing. In the general cases, the higher the Mach number of the shock, the more difficult it is to trigger collapse. From an intuitive point of view, the shock being too fast will shred the clump material away at a shorter time scale compared to the time scale for collapse. At the mean time, faster shock speed allows better mixing because of the enhanced Rayleigh-Taylor instability growth rate.

Previous studies have been focusing on the precursor of triggered star formation: the condition of which a compressed clump can form a star, instead of forming one in the numerical study. Part of the difficulty lies in the numerical tool to generate collapsing zones that represents stars and follow its subsequent evolution. This leaves a lot of interesting questions unanswered, for instance, what is the mass accretion rate of such a star formed by triggering? How much post-shock material ends up accreted onto the star? Does the formed star have a disk? If so, is the disk stable? Some of these questions, such as those related to the disk formation, are essential to estimate the likelihood of triggering being the mechanism of forming a star.

In this project, we study the shock-induced triggering of a stable Bonnor-Ebert cloud following, for the first time, the long-term evolution of the flow after a star has formed. We will be focusing on the triggering event as well as the post-triggering evolution of the star. We will impose initial rotation as well as internal magnetic field to study their influence on the post-triggering evolution.

Hydro Triggering with Initial Rotation
We setup an initial marginally stable Bonnor-Ebert sphere as the triggering target. The cloud has 1 M_{\odot} and radius of 0.058 pc, with central density 6.3\times10^{-19} g/cc and edge density of 3.6\times 10^{-20} g/cc. The cloud has a uniform interior temperature at 10K. The ambient medium is setup to satisfy the pressure balancing at the cloud edge, with density 3.6\times 10^{-22} g/cc and temperature of 1000 K. We have performed simulations to check the stability of the cloud, and found that the initial cloud breathes at a time scale of about 10 cloud crushing time scales (here after t_{cc}), which is longer than the time span of our simulation (4t_{cc}). t_{cc} is defined as the time for the transmitted shock to pass the cloud, which is estimated to be t_{cc} \approx 276 kyrs. The free-fall time of the cloud is t_{ff} \approx 84 kyrs.

To make a comparison between slow and fast shock cases, we setup the incoming shock at two different Mach numbers: either 1.5 or 3.16. With initial rotation added to the cloud, we identify K = \Omega \times t_{ff} as a parameter characterizing the importance of the rotational energy, where \Omega is the angular velocity. We assume K = 0.1 for all the rotational cases. There are two possible rotation orientations: axis parallel to the shock normal or axis perpendicular to shock normal. First set of runs are set up as follows:

code	Mach	Rotation	K
N	1.5	None	0
N'	3.16	None	0
R1	1.5	Parallel	0.1
R2	1.5	Perpendicular	0.1

Simulation lasts till about 4t_{cc}, which is about 1 million years. The wind should be turned off before that, so we are really looking at an extreme case where the wind lasts for unusually long time.
The simulation box has a resolution of 320 x 192 x 192, with 3 AMR levels of particle refinement.
The following figure shows the at 0.6 million years. (a) non-rotation Mach 1.5(case N), (b) non-rotation Mach 3.16(case N'), © parallel rotation Mach 1.5(case R1), (d) perpendicular rotation Mach 1.5(case R2).





Movies:
Case N:
​http://www.pas.rochester.edu/~shuleli/tsf_paper/norot.gif
Case N':
​http://www.pas.rochester.edu/~shuleli/tsf_paper/mach3final.gif
Case R1:
​http://www.pas.rochester.edu/~shuleli/tsf_paper/parrot.gif
Case R2:
​http://www.pas.rochester.edu/~shuleli/tsf_paper/mach3final.gif ​http://www.pas.rochester.edu/~shuleli/tsf_paper/krumf.gif

​http://www.pas.rochester.edu/~shuleli/tsf_paper/perrot.gif
Generally, the evolution can be divided into three stages: As we can see the two rotating cases develop disks with the axis aligned with the initial rotation. The 3D rendering of cloud material tracer for case R1, is as follows:

We can study important physics quantities by line plots. The first set of line plots are 1. star mass formed by triggering, 2. accretion rate, 3. mixing ratio of the wind material onto the star, 4. bounded mass (basically the mass of the disk for the rotation cases. for the non-rotating cases, the bounded material evaporates very fast after stage I.

Choice of Algorithm
We implement Krumholz's (Krumholz et al 2006) sink particle algorithm apart from Federrath et al 2010. Basic difference between the two algorithms are listed below:

The two algorithms give different results for the non-rotating case when the isothermal approximation \gamma = 1.0001 is used. As shown in the following movie for K = 0.1, parallel rotation for Krumholz algorithm and Federrath algorithm. For the Federrath case, after the bounded material enters the post-shock region, the disk disappears due to an "explosion".


Movies:
parallel rotation, Krumholz:
​http://www.pas.rochester.edu/~shuleli/tsf_paper/parrot.gif
​http://www.pas.rochester.edu/~shuleli/tsf_paper/krumf.gif
parallel rotation, Federrath:
​http://www.pas.rochester.edu/~shuleli/tsf_paper/fedf.gif

This is due to the fact that under this condition, the initial compression can increase the temperature of the bounded core around the star for the Federrath case, since it keeps a hot accretion zone surrounding the central star. When the bounded material starts to move out of the cloud profile, they are exposed to colder incoming wind and expands, which is seen as an "explosion". This can be demonstrated by the following temperature movies:
temperature evolution, Krumholz:
​http://www.pas.rochester.edu/~shuleli/tsf_feed/krumtmp1.gif
temperature evolution, Federrath:
​http://www.pas.rochester.edu/~shuleli/tsf_feed/fedtmp1.gif
This phenomenon disappears if a closer to 1 \gamma is used. Here's a movie for Federrath algorithm with rotation as before, but using \gamma = 1.000001:
​http://www.pas.rochester.edu/~shuleli/tsf_feed/fediso.gif

There is also a difference in asymptotic star mass due to the difference in accretion routines.

MHD Triggering
When added magnetic field to the cloud, the triggering behavior can be very different. Here, we suppose the initial rotation to be zero first, and look at the case where there is a global uniform magnetic field along .

Next, we study if the field is contained within the cloud. We divide the simulations into categories where the contained field is either poloidal or toroidal, and look at the triggering behavior when the poloidal or toroidal axis is parallel or perpendicular to the incoming shock. As shown in our earlier paper about shock interaction with a magnetized clump, it is possible to form multiple compression cores in such a scenario. One natural question is, can we have formation of more than one stars within a single magnetized cloud?