u/u/erica/BEPaper: BEsphere.tex

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\newcommand{\crc}{cells$/r_c$}
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\shorttitle{The effect of a massive ambient medium on the collapse of a Bonnor-Ebert sphere: deviation from the canonical outside-in collapse}
\shortauthors{Kaminski et al.}

\title{Untitled - BE Crushing Solution} 
%\title{The effect of a massive ambient medium on the collapse of a Bonnor Ebert sphere: deviation from the canonical outside-in collapse}
%\title{A more physically accurate approach to the Bonnor Ebert sphere collapse problem: taking the ambient medium into account}
%\title{Taking the ambient medium into account when modeling the collapse of a BE sphere: a more physically motivated scenario}
%\title{Collapse of a Bonnor Ebert sphere in a massive ambient medium: deviations from the canonical outside-in collapse}
%\title{Considering the effects of a massive ambient medium on the collapse of a Bonnor Ebert sphere}
%\title{A new solution for the Bonnor Ebert collapse problem: the role of the ambient medium on modifying outside-in collapse}




\author{Erica Kaminski\altaffilmark{1}, Adam Frank\altaffilmark{1}, Jonathan Carroll\altaffilmark{1}}
\altaffiltext{1}{Department of Physics and Astronomy, University of Rochester, \
Rochester, NY 14620 \\Email contact: erica@pas.rochester.edu}


%{\it clumps} 

\begin{abstract}
A series of 3D hydrodynamic simulations were carried out to explore the effects of a massive ambient medium on the collapse of a marginally stable Bonnor-Ebert (BE) sphere. In particular, we sought to discern whether a massive ambient medium would be sufficient to drive the collapse of the sphere, whether this collapse would be triggered by the ram pressure of infalling material gravitationally accelerated by the BE sphere, and if this collapse would differ from the canonical {\it outside-in} collapse previous models have shown occurs for BE spheres. To these aims, various uniformly dense ambient environments were initialized to be in pressure equilibrium with the BE sphere. These ambient densities ranged from the conventional 'light' density ($\rho=0.01\rho(Rbe)$, where Rbe is the BE sphere's truncation radius), to a more physically realisitic ambient density equal to that of the sphere at its outer most edge ($\rho=\rho(Rbe)$). In each of the runs, ram pressure at the sphere's outer edge was found to exceed the critical threshold of external pressure on the BE sphere, although the resulting collapse varied. The collapse features were found to be sensitive to the density in the surrounding ambient medium. In the matched case, the sphere collapsed under a crushing wave. . . In contrast, the sphere in the light ambient medium remained dynamically stable, oscillating about its equilibrium values for $\sim$5 crossing times. These results reveal the importance of taking the ambient medium under consideration when modeling the collapse of a Bonnor-Ebert sphere. 
\end{abstract}

\keywords{a list to choose from}
\section{Introduction}

Various properties of the ``Bonnor-Ebert'' (BE) sphere \citep{bonnor1956, ebert1955}, a hydrostatic sphere in pressure equilibrium with its ambient environment, make it a good candidate for numerical modeling of protostellar collapse. First, as a candidate star forming structure is envisaged as gravitationally bound and unstable, it is easy to imagine a protostar evolving from an initially hydrostatic configuration. Indeed, spherical clumps have been observed in or near hydrostatic equilibrium, such as the Bok Globule B68 (check citation). Second, the stability criterion against gravitational collapse has been worked out analytically. Third, pushing the sphere out of the stability regime with various physical perturbations illuminate collapse characteristics. Such features of the collapse may help advance single star formation theory as well as provide clues to observational astronomers in identifying potential star forming sites.  

While the collapse of a BE sphere has been studied extensively over the years, the literature reveals studies of the BE sphere in precarious and unphysical situations; the BE sphere has largely been modeled as residing in artificially low density ambient mediums, seemingly to isolate the collapse of the sphere from the ambient environment \citep{foster1993, ogino1999, banerjee2004}. However, any actual collapsing cloud would not be discontinuous from the ambient medium. Additionally, ad hoc perturbations such as overall density increase within the simulation box, have been employed to force the sphere into collapse. In these ways, previous models have been physically unrealistic. Despite these certain drawbacks though, it is important to note that modeling the BE sphere in this way happened to reveal an important and unexpected feature of the collapse. 

The collapse of a BE sphere was originally anticipated to be an inside-out process, a characteristic feature of Frank Shu's similarity solution for the highly unstable singular isothermal sphere (SIS) \citep{shu1977}. In this well-known paper, \cite{shu1977} speculated that the collapse of {\it any} hydrostatic isothermal sphere, including Bonnor-Ebert spheres, would approach the SIS through a subsonic adjustment to a $1/r^2$ density distribution. The classic {\it inside-out} collapse of the SIS was then, as Shu proposed, a general feature of collapse, applicable to any hydrostatic isothermal sphere, unstable or not.   

Simulations of stable, flat-topped BE spheres in low density ambient environments, however, proved the contrary. \cite{foster1993} explored the collapse of BE spheres of varying truncation radii embedded in ambient backgrounds of uniform $\rho=0.01\rho(Rbe)$, where Rbe is the BE sphere's truncation radius. Their set ups showed that despite initial perturbation methods, as well as whether the sphere was initially in a stable or unstable hydrostatic regime, the collapse proceeded much differently than that of the SIS. Instead of an {\it inside-out} collapse, the collapse of a BE sphere was {\it outside-in}. Further, the collapse was not subsonic, rather supersonic. This directly refuted Shu's proposal. Studies of the collapse with more advanced fluid dynamic codes continued to provide support for outside-in collapse of the BE sphere {\citep[see e.g.][hereafter BPH04]{banerjee2004}. 

Now, although these early models illuminated unique and unexpected features of the BE collapse, they have been physically unrealistic given the discontinuous jump in density across the BE sphere-ambient boundary. More sophisticated simulatations have taken this under consideration. For instance, in calculating protostellar masses due to infall and dispersal of ambient gas, \cite{myers2008} modeled pre-stellar cores \citep{difrancesco2007, ward-thompson2007} as BE spheres seperated from a uniform ambient medium by a continuous, smooth density transition. \cite{hennebelle2003} looked at the effects of increasing external pressure on initiating inward propagating compression waves that drove stable BE spheres into collapse. 

Thus, there was an important missing link in the studies of collapsing isothermal spheres between those embedded in highly unrealistic environments and those in complex systems. This provided the motivation for the present work. We sought to study how the classic outside-in collapse of a BE sphere would be modified when it was placed in a non-negligible and continuous ambient medium. Further we did not artificially perturb the BE sphere into collapse, rather studied the role the ambient alone had in inducing collapse. 

This paper is broken into the following sections. In Section 2 we will discuss our model and methods. In Section 3 we will present our findings. Namely, we found that the collapse resembled a crushing wave solution (Hannebelle, Whitmore, Di Francesco) early on in the presence of a massive ambient medium. As this wave propagated inward with time, the radial density and velocity profiles asymotically approached the outside-in results. In less massive ambient backgrounds, the sphere attempts to reequilibrate, but ultimately is pushed over the stability limit. Collapse resembles an intermediate state between the crushing wave and outside-in solution. In Section 4 we will present discussion and conclusions of these results.   


%"~" is $\sim$


% \cite{kleinwoods1998}

% \citep[see e.g.][]{truelove1998}


%$\propto n^2\Lambda(T)$

%\S~\ref{problem}


\section{Methods}\label{methods}
\subsection{Bonnor-Ebert sphere definitions}\label{BEdefs}
The equations that specify the parameters of the Bonnor-Ebert sphere are derived from the equations of hydrodynamics. Combining the condition for hydrostatic equilibrium in spherical coordinates, Poisson's equation for gravity, and the isothermal equation of state, 

\begin{equation}\label{HSE}
-\frac{\nabla P}{\rho} - \nabla \phi_{g} = 0
\end{equation}

\begin{equation}\label{Poisson}
-\nabla ^2 \phi_{g} = 4 \pi G \rho
\end{equation}

\begin{equation}\label{isoT}
P = \rho C_{s}^2
\end{equation}
where $\phi_{g}$ is the gravitational potential, P is the thermal pressure, ${\rho}$ is the density, and $C_{s}$ is the isothermal sound speed yields \citep[cf.][]{stahler2005},

\begin{equation}\label{rho(r)}
\rho(r) = \rho_{0} exp(-\phi_{g}/C_{s}^2)
\end{equation}

This is an equation for the radial density function of the hydrostatic isothermal sphere in terms of some central density, $\rho_{0}$, and the gravitational potential. The goal from here is to find $\phi_{g}$ so that the functional form of $\rho (r)$ is determined for a specified $\rho_{0}$. The following equation found by combining \ref{rho(r)} and \ref{Poisson} provides $\phi_{g}$,

\begin{equation}\label{phi}
\frac{1}{r^2} \frac{d}{dr}(r^2 \frac{d \phi_{g}}{dr}) = 4 \pi G \rho_{0} exp(-\phi_{g}/C_{s}^2)
\end{equation}
and hence the density profile of the BE sphere with central density $\rho_{0}$. It is common to cast \ref{phi} into a scale-free form by making the following variable substitutions, 

\begin{equation}\label{psi}
\psi = \frac{\phi_{g}}{C_{s}^2}
\end{equation}

\begin{equation}\label{xi}
\xi = (\frac{4 \pi G \rho_{0}}{C_{s}^2})^{1/2} r
\end{equation}
which leads to the famous {\it Lane-Emden} equation for an isothermal sphere, 

\begin{equation}\label{xi}
\frac{1}{\xi ^2} \frac{d}{d \xi}(\xi ^2 \frac{d \psi}{d \xi}) = exp(- \psi)
\end{equation}

There are two classes of solution to this equation. The first is the asymptotic solution, known as the {\it singular isothermal sphere} (SIS). It represents a sphere of infinite central density given by,

\begin{equation}\label{SIS}
\rho = \frac{C_{s}^2}{2 \pi G} r^{-2}
\end{equation}
This is the solution that was studied by Frank Shu in his analytical derivation of inside-out collapse \citep{shu1977}. Of greater physical relevance is the second solution, or {\it family} of  solutions, the non-singular Bonnor-Ebert spheres. These solutions adhere to the boundary conditions $\psi(0) = 0$ and $\psi'(0)=0$. Each  BE sphere is specified by a different truncation radius $R_{BE}$ and central density $\rho_{0}$, and is bound by an external pressure $P_{ext}$. They are characterized by a flat-topped density profile near their centers with outer densities that decline monotonically with radius (Fig.~\ref{fig_BE}). 


The stability of the BE sphere is determined by the value of $\xi$, also known as the BE sphere's {\it non-dimensional radius}. BE spheres are inherently unstable for configurations with $\xi > 6.451$, which is known as the critical radius, $\xi_{crit}$. The critical values of external pressure and radius of such a marginally stable BE sphere of mass $M_{BE}$ and with isothermal sound speed $C_{s}$ are given by (Spitzer),

\begin{equation}\label{Pcrit}
r_{crit} = 0.41 \frac{G M_{BE}}{C_{s}^2}
\end{equation}

\begin{equation}\label{Rcrit}
P_{crit} = 1.40 \frac{C_{s}^8}{G^3 M_{BE}^2}
\end{equation}

\begin{figure}[htbp]
\centering
\epsscale{.60}\plotone{Methods.eps}
\caption{{\small The density profile of a critical Bonnor-Ebert sphere as a function of $\xi$ in log-log space. The y-axis is in scaled units, normalized to the central density $\rho_{0}$ of the BE sphere. Given the scaled nature of this curve, it represents a family of solutions, each BE sphere given by a different $\rho_{0}$ and truncation radius.}}
\label{fig_BE}
\end{figure}

To initialize the BE sphere in our model we used an approximate analytic solution of the Lane-Emden equation \citep{liu1996}, which provided the density contrast of the BE sphere given its a) dimensional radius, b) non-dimensional radius ({\it$\xi$}), and c) central density \footnotemark[2].

\footnotetext[2]{documentation: https://clover.pas.rochester.edu/trac/astrobear/wiki/u/BonnorEbertModule}

\subsection{Model}
We modeled the collapse of the marginally stable BE sphere ($\xi = 6.5$) using AstroBEAR2.0\footnotemark[3], a highly parallelized, multidimensional, adaptive mesh refinement (AMR) code that solves the equations of hydrodynamics and magnetohydrodynamics \citep{andy2009,carroll2011}. AstroBEAR2.0 has a library of multiphysics tools, from heat conduction and resistivity, to radiative cooling and self-gravity. In the present work, AstroBEAR2.0 solved the Euler equations with self-gravity. The Poisson solver for self-gravity uses HYPRE\footnotemark[4], a software package that solves linear systems on massively parallelized systems. Our methods for self-gravity are described in Appendix. 

\footnotetext[4]{documentation: https://computation.llnl.gov/casc/hypre/software.html}

The problem domain was a cube of side L $\approx 50$ pc, at the center of which we initialized the critical BE sphere. This BE sphere had the same properties as that used in BPH04. Namely, the radius of the sphere was r $\approx 1.62 pc$. The gas in the box was assumed to be atomic hydrogen ($\mu = 1$), and had a ratio of specific heats ($\gamma = 1.0001$). Using the ideal gas equation of state, this set the simulation to be effectively isothermal. Note $\gamma = 1.0$ could not strictly be used, as to do so would introduce a division by zero in the solver. The central density was initialized to be $\rho_{0}$ = 2004 $cm^{-3}$. This led to a temperature within the BE sphere of T $\approx$ 20 K, and an isothermal sound speed of $\approx$ 40,793 cm/s.   

\begin{figure}[htbp]
\centering
\epsscale{0.80}
\plotone{Mesh_Plot.eps}
\caption{{\small Schematic of the mesh with an octant of the Bonnor Ebert sphere located at (0,0,0). While all simulations were initialized with 5 levels of refinement to achieve $\backsim$34 cells per initial clump radius, only the first 3 levels are plotted here for clarity. The mesh was dynamic in that additional levels of refinement were added as needed (see text). The color-bar shows variation in $\rho (n/cm^3)$. Note that the maximum value in this plot is lower than the reported 2004 $cm^{-3}$, given the data from the finest cells are not plotted. The z-axis is plotted for convenience. Each major tick mark on the axes of this cube represent $~$1.56 pc.}}
\label{Mesh}
\end{figure}


Given the symmetry of the problem it was possible to minimize computational costs by only simulating the 1st (positive) octant of the domain. That is, the simulated box actually extended from [0,25pc] in x-, y-, and z- directions, thereby encompassing only the 1st octant of the BE sphere centered at (0,0,0) (Figure 2a). Boundary conditions on the box were set to extrapolating (outflow) on all outside-facing sides, and reflecting on the faces that sliced through the BE sphere. The coarse grid was made up of $16^3$ cells, initialized with 5 levels of AMR within the BE sphere. Each level of AMR increased the effective number of cells in each direction by a factor of 2. Thus, at the start of the simulation the effective resolution was $\triangle x_{min} \approx 0.05$ pc, corresponding to $\approx$ 34 cells/$R_{BE}$. Two additional levels of refinement were triggered during the BE sphere collapse when the ''Truelove condition'' \citep{truelove1997},

\begin{equation}\label{Truelove}
\lambda_{J} < 4 \triangle x
\end{equation}
was violated, where $\lambda_{J}$ is the local Jeans length associated with each cell center, and $\triangle x$ is the width of a cell on any given level. Essentially this is an AMR criterion that maintains adequate resolution of the Jeans length. Once the maximum refinement is reached, a sink particle can form inside one of the finest cells if a sequence of conditions are met that insure gravitational instability. These conditions are the same as those outlined in \cite{fedderrath2010}. The formation of a sink in those runs where collapse occurred determined the end of the simulation. By those points the density was not great enough to demand a switch to an adiabatic EOS (cite); an isothermal approximation was justified given the minimal orders of density increase. Further, as this work was designed to track collapse properties of the BE sphere, the formation of a protostellar-like sink particle was a sufficient time to halt the evolution.

\subsection{Simulations}

We performed a set of 6 simulations, listed in Table~\ref{tab_runs}. In each run the marginally stable ($\xi = 6.5$) BE sphere was initialized to be in pressure equilibrium with an ambient medium of a different uniform density. To test the stability of the BE sphere, we chose an ambient density of $\rho_{amb} = 0.01\rho(R_{BE})$, where $R_{BE}$ is the truncation radius of the BE sphere. This we named the ''sparse'' ambient case. We ran the simulation out for 5 crossing times $t_{sc}$, 

\begin{equation}
\label{eqn_tsc}
t_{sc} = r / C_{s} 
\end{equation}
and checked whether the sphere remained upright. Next we added a 10$\%$ overall density enhancement to the BE sphere in the light ambient case. This was the perturbation BPH04 and others used to initiate outside-in collapse of the BE sphere, and so we named it the ''classical'' case. Lastly, in the spirit of testing new conditions on the BE sphere, we did a series of runs that placed the sphere in ambient environments of varying density. Note these runs did not consist of any applied perturbations. These ambients ranged from the sparse ambient, to the denser ambient given by $\rho_{amb} = \rho(R_{BE})$. The latter run we named the ''matched'' case.  

\begin{table}[htbp]\centering\small
\label{tab_runs}
\begin{tabular}{rrc}
\hline\hline
%Run     Effective Resolution    
Run & Density of Ambient  & Perturbation? \\
\hline
%$cells/r_c$ & $cells\ in\ z\times r$  & & $Approx.$\\
1 - Sparse  &    $\rho_{amb} = 0.01 \rho(R_{BE}) \approx 1.5 cm^{-3}$  &  No        \\
2 - Classic &    $\rho_{amb} = 1/100 \rho(R_{BE}) $  &   Yes - 10\% overall density increase \\
3a - Matched  &    $\rho_{amb} = \rho(R_{BE}) \approx 150 cm^{-3} $ & No      \\
3b  &    $\rho_{amb} \approx 1/3 \rho(R_{BE})  $  & No       \\
3c &  $\rho_{amb} = 1/10 \rho(R_{BE})$   &    No   \\
3d &  $\rho_{amb} \approx 1/31 \rho(R_{BE}) $  &   No       \\
\hline
\end{tabular}
\caption{\small Description of the 6 simulations of varying background densities. The first column lists the name of the run. The second column gives the ambient density of the simulation, and the third column states whether a perturbation was applied to initiate collapse. In only the ''classical'' run was a perturbation applied. This was a $10 \%$ overall density enhancement to the box, similar to \cite{banerjee2004}.}
\end{table}



\section{Results}\label{results}
\subsection{A Test of Stability - Sparse Ambient Medium}\label{Stability}

Our numerical model successfully produced a critical BE sphere in hydrostatic equilibrium. We verified this by placing the BE sphere in a low density and warm (sparse case, recall pressure equilibrium) ambient medium, and finding the sphere to be stable against collapse for five sound crossing times (Eq. \ref{eqn_tsc}). The crossing time is a dynamically relevant time scale for these simulations; given the approximate hydrostatic equilibrium of our simulations it represents the time it takes for thermal sound waves to travel back and forth throughout the sphere. For an unstable sphere, it is easy to imagine collapse being triggered by thermal perturbations traveling on the order of a sound crossing time. Therefore, we expected 5 crossing times to be sufficient in bringing out any inherent thermal or numerical instabilities of our model. This though we found to be null; fluid flows over the course of this simulation were insufficient to drive the sphere into collapse. As can be seen from Figure~\ref{light_case}, fluid motions throughout the sphere were largely subsonic. As the sphere ''breathed'' around its initialized equilibrium values, its density fell and rose, returning close to the initial state by the end of the simulation.


\begin{figure}[htbp]
\centering
\epsscale{1}
\plottwo{rhoLight.eps}{vradLight.eps}
\caption{{\small Density and velocity profiles for the sparse case. Legend is in units of sound crossing time $t_{sc}$, as defined by Eq.~\ref{eqn_tsc}. The sphere's initial outer boundary is at $\xi = 6.5$ on x-axis, as can be traced by the sharp discontinuous jump in density that occurs at the BE sphere-ambient boundary. The density in these plots is a radial average of the density of each cell, averaged over successive spherical shells of dr = $\vartriangle x_{min}$. The same averaging method is used for the radial velocity plots as well. As can be seen in the left plot, the sphere oscillated to lower $\rho(r)$, but returned close to the equilibrium profile by t/$t_{sc}$ = 5. The right plot shows small subsonic radial motions throughout the sphere during this time period. As usual, negative velocities indicate inward motions and positive indicate outward. No collapse was triggered by these fluctuations.}}
\label{light_case}
\end{figure}

\subsection{Classic Collapse}\label{classic}

As another test of our code, we verified that we could reproduce the results of other works that modeled the BE sphere collapse by perturbation. An example of such a perturbation would be the 10$\%$ density enhancement BPH04 prescribed. With other conditions being equal to the sparse case, we added a 10$\%$ density enhancement to the entire simulation box. The effect was clearly to increase the mass within the sphere above that which could be supported by thermal pressure alone as dictated by the Lane-Emden equation. Thus, material began to rain inward, starting in the outer regions of the BE sphere where the gravitational acceleration was greatest (cf. BPH04 for detailed discussion on this process) and moving inward with time (Fig. \ref{BP_case}). While the time states plotted in Figure \ref{BP_case} are in units of $t_{sc}$, they are relative to the formation of a sink particle. That is, the last time state plotted in Figure \ref{BP_case}, as well as in the density and velocity plots for the remaining simulations, is equal to the time at which a sink formed. Thus, by t/$t_{sc}$ = 0.82 in this ''classic'' case, a sink particle had formed. By this time, inward radial velocity had become marginally supersonic ($v_{rad}$$\backsim$2.2$C_{s}$), in agreement with other results for this type of BE collapse \citep{banerjee2004, foster1993} (cite Hunter, Ogin''?''). 

The density profile over this collapse is also in agreement with previous results for this model. Namely, the collapse leads to increased density in the inner flat region that both a) decreases in size over the simulation and b) is trailed by an envelope that approaches a $r^{-2}$ profile. Please note that the maximum density at sink particle formation is numerical in the sense that it is associated with the maximum refinement reached ($\vartriangle$x$~\varpropto$$~\lambda_{J}$$~\varpropto$$~\rho^{-1/2}$). Since each simulation had the same ''maximum level of refinement'' possible, the density that triggered the final refinement level (hence sink particle formation and end of simulation) was the same throughout the different runs. That being said, the profiles for both density and velocity qualitatively match those of outside-in collapse.

\begin{figure}[htbp]
\centering
\plottwo{BPrho.eps}{vradBP.eps}
\caption{{\small Density and velocity profiles for the ''Classic'' case (refer to Table~\ref{tab_runs}). As indicated by the left hand density plot, outside-in collapse was established with the formation of a r$^{-2}$ envelope trailing a flat inner collapsing core. The right hand velocity profile also matches canonical outside-in collapse, marked by an inward radial flow that began at larger radii and moved inward with time. The peak velocity became marginally supersonic approaching about 2.2$C_{s}$ as expected (see text). The center sink particle formed by the last time state plotted in all of the figures. Here $t_{sink}/t_{sc}$=0.82.}}
\label{BP_case}
\end{figure}

\subsection{New models}\label{NewModels}

To look at the properties of the collapse under more realistic conditions, we adjusted the set up of \S~\ref{classic} in two ways. First we matched the density profile across the BE sphere-ambient boundary (while maintaining pressure equilibrium at this edge), in contrast to the steep decrease in density of order 1/100 of the classic BE model. Second, we allowed the simulation to progress naturally by withholding any perturbations to force collapse. We followed this setup under 3 different ambient densities, log-spaced between a matched density profile ($\rho=\rho(R_{BE})$) and the Light case discussed in \S~\ref{Stability} to get a sense of any turn-over in the solution that might occur between these simulations.

\subsubsection{Matched Ambient Medium}
   
The qualitative differences between this set of simulations and those that came before are most striking in the ''Matched'' case. As can be seen in the density plot (Figure 
\ref{Fig_Matched}, left), the Matched solution is marked by a build-up of material on the surface of the BE sphere that eventually overwhelmed the sphere and was driven inward via a compression wave. As the simulation was isothermal, the buildup of density at (r = $R_{BE}$, t = 0.3 $t_{sc}$) puffed up the outer edge of the sphere. In this way, the accumulation of matter was associated with an approximate re-equilibration of the BE sphere. This is evident by the a) increase in BE radius (marked by the kink in the density profile at BE-ambient boundary) and b) concomitant increase in outer density ($\rho(R_{BE})$) that smoothly matched the original BE profile. At t = 0.75$t_{sc}$, the outer radius of the BE sphere returned to its initial value, yet the peak density of the perturbed profile (compression wave) exceeded that of the inner regions. Thus in order for the sphere to have re-adjusted to an equilibrium profile, the time scale for equilibration $<$ the compression wave speed. (Or perhaps an appropriate accretion time scale $<$ sound crossing time?). This was not the case. 

By t = 0.88$t_{sc}$ the compression wave had dominated the thermal motions within the sphere, and pushed inward, crushing what remained of the original BE density profile. That process, marked by ever increasing peak and outer densities and shorter, squeezed radius, evolved until about t = 0.94$t_{sc}$. At that time, the shape of the density profile {\it assumed that of the classic collapse profile}. Namely, there was a nice flat inner core region, trailed by a decreasing power law envelope. Interestingly, over the next 2 time states the collapse then followed as usual; the inner flat region collapsed while the outer regions formed a static envelope (static in the sense that d$\rho$/dt = 0 in the envelope). A sink particle formed in the inner most cell of the simulation by t=0.97$t_{sc}$, a little longer than in the classic case. Also in contrast to the classic case, the extended envelope region had a steeper power law dependance than the classic case (Fig.~\ref{FitEnv}). (Some observational papers report sharper envelopes in more packed clusters...).   

A look at the velocity profiles for this case also supports the turn over in the solution from the earlier compression wave phase to the later classic outside-in phase. During the compression wave phase (t/$t_{sc}$ = 0-0.88), fluid motions originated outside of the sphere and moved inward with the density wave. By about 0.5$t_{sc}$ the flow became supersonic. Material moving fastest was at the tail-end of the compression wave. Supersonic material was thus driven up against the hydrostatic ($v_{rad}$=0) inner regions of the BE sphere. By t/$t_{sc}$=0.94, the velocity profile of the inner region of the disturbed and collapsing BE sphere resembled the classic outside-in results, except now velocities were well into the supersonic regime. By sink formation, the peak velocity inside the sphere corresponded with the edge of the flat core region as in the classic case. However, in contrast, this flow had a peak mach $\approx$ 7, compared to $\backsim$2.2 (cite).

Trends:

-steeping into shock in compression phase

-does dv/dt = 0 in envelope during outside-in phase of Matched case, as it does in classic case

-ahead of the compression wave (where the density is the initial BE profile), the velocity turns to 0

-the velocity is non-zero as soon as the density deviates from eq. 

-is shape bondi flow? is it homogous flow? Paste the two solutions together, Bondi and outside-in to test?

-ratio of peak density to min density = constant in compression wave phase?

\begin{figure}[htbp]
\centering
\plottwo{MatchedRho.eps}{MatchedVrad.eps}
\caption{{\small Density and velocity profiles for the Matched run, $\rho(R_{BE})=\rho_{amb}$. The density profile (left) shows material building up on the BE sphere, that eventually exceeded the adjacent inner cells' densities. At that time, the accumulated mass was no longer supported by thermal pressure and fell inward via a compression wave. By t/$t_{sc}$=0.94 the density profile began to mimic the Classic collapse result. The radial velocity plot (right) shows fluid motions to be tied to those regions of the BE sphere where material was being redistributed. At early times velocity profiles turned to 0 in those regions where new matter had not yet accumulated; inner regions of the BE sphere maintained hydrostatic equilibrium before the compression wave entered. As in the density plot, late time states show the concavity of the velocity profile to change, again resembling the Classic results within $\xi \approx$3 by t/$t_{sc}$=0.94. The flow becomes supersonic early on, and reaches a peak Mach $\approx$7 by the end of the run.}}
\label{Fig_Matched}
\end{figure}


\begin{figure}[htbp]
\centering
%\epsscale{0.5}
\includegraphics[height =2in]{FitEnv.eps}
\caption{{\small Envelopes of the Classic and Matched cases. Envelopes are defined as the trailing decreasing density profile adjacent to the collapsing flat core that extend until the BE sphere-ambient boundary. Given this definition, the envelope region of the Matched case is shorter than in the classic case. The best fit power law function is plotted on top of the envelopes, and the equation is given next to the respective fit. Best fits ($R^2$ = 1.0) show the Matched envelope to have a steeper power dependance. Both envelope profiles are taken from the last time state before a sink particle formed.}}
\label{FitEnv}
\end{figure}


\subsubsection{Intermediate Runs}

\begin{figure}[htbp]
\centering
\plottwo{rhow3.eps}{vradw3.eps}
\caption{{\small Ambient is 1/3 $\rho(Rbe)$}}
\label{w3_case}
\end{figure}
\begin{figure}[htbp]
\centering
\plottwo{rhow10.eps}{vradw10.eps}
\caption{{\small Ambient is 1/10 $\rho(Rbe)$}}
\label{w10_case}
\end{figure}
\begin{figure}[htbp]
\centering
\plottwo{rhow31.eps}{vradw31.eps}
\caption{{\small  Ambient is 1/30 $\rho(Rbe)$}}
\label{w30_case}
\end{figure}


\subsection{Discussion}



%The evolution of an adiabatic shocked clump progresses through several stages. The external shock will sweep across or ``pass over'' the clump radius in a time $t_{sp}=r_c/v_s$, where $r_c$ is th%e radius of the clump and $v_s$ the shock velocity. When the shock hits the clump, a new shock wave is transmitted into the clump. This transmitted shock crosses the clump in a time known as the %{\it cloud-crushing time} $t_{cc}$ (KMC94),

%\begin{equation}\label{eq_tcc}
%  t_{cc} = \frac{r_c}{v_{s,c}}\simeq\frac{\chi^{1/2}r_c}{v_s}
%\end{equation}

%\subsection{Radiative cooling}\label{prob_cooling}
% \cite{dysonwilliams1997, zeldovich2002}

%($10^5$--$10^7$ K)


%\begin{table}[htbp]\centering\small
%\begin{tabular}{rrrr}
%\hline\hline
% Run     Effective Resolution    
%Run & Effective Resolution & $\Delta x/L_{cool}$& $\Delta x/L_{cool,inspec.}$\\
%\hline
%$cells/r_c$ & $cells\ in\ z\times r$  & & $Approx.$\\
%12  &    $ 96\times    24$  &   -- & --       \\
%24  &    $192\times    48$  &    1 & $\sim 1$ \\
%48  &    $384\times    96$  &    2 & 1-2      \\
%96  &    $768\times   192$  &    4 &  3       \\
%192 &  $1,536\times   384$  &    7 &  5       \\
%384 &  $3,072\times   768$  &   15 &  7       \\
%768 &  $6,144\times 1,536$  &   29 & 11       \\
%1536& $12,288\times 3,072$  &   58 & 14       \\
%\hline
%\end{tabular}
%\caption{Resolutions of the 8 simulations, from 24 to 1,536 cells per clump radius $r_c$. The third column gives the number of cells per cooling length $L_{cool}$ from Eq.~\ref{eq_lcoolclump}, and the fourth column via visual inspection (see \S~\ref{analysis}). At \rr{12} the cooling length was unresolved.\label{tab_resolutions}}
%\end{table}


%\begin{figure}[htbp]
%\plotone{diagram_rotated.eps}
%\caption{{\small Important features of the flow are given in a synthetic Schlieren image of run \rr{1536} at $t\sim 0.5 t_{cc}$. The image has been reflected about the axis of symmetry, with the reflection showing the location of AMR refined regions. The bow shock wraps tightly around the clump from the strong cooling. The clump surface is ablated by its interaction with the postshock flow. A slip stream forms behind the clump. Transmitted shocks propagate internally through the clump. The external shock is susceptible to the cooling instability, and a conical reflected shock forms off of the axis of symmetry which reengages the flow.}}
%\label{fig_diagram}
%\end{figure}


%\begin{figure}[htbp]
%\centering
%\plotone{schlieren-montage-small.eps}

% Figure~\ref{fig_schlieren}

%\section{Discussion and Conclusion}\label{discussion}

%``Convergence'' has a clear meaning 



%\rr{100}--\rr{200}

%\acknowledgements


\bibliography{erica}
\end{document}