u/u/erica/BEPaper: BEsphere.2.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}}
\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 (Not sure if we want to use this angle - will wait til we examine results more carefully), 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, 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 (Myers). 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 (cite), seemingly to isolate the collapse of the sphere from the ambient environment. 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). In his well-known 1977 paper (cite), Shu 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. In 1994, Foster and Chevalier (cite) 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. (NOTE - THE SINK FORMATION IN THE FLAT TOPPED STUDIES MAY MORE CLOSELY RESEMBLE THE COLLAPSE OF THE SIS... PERHAPS THIS IS WHAT HE WAS TALKING ABOUT). Studies of the collapse with more sophisticated fluid dynamic codes provided further support for outside-in collapse of the BE sphere (cite BP).

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 and the artificial perturbations to trigger collapse. If the BE sphere might be considered as an initial structure of star formation, it should be examined in a more physically plausible setting (cite). Such a simulation would more accurately describe astrophysical situations that resemble collapsing hydrostatic structures. Little research has been pursued to this aim. Myers in 2008 did . Hannebelle did . The question of putting the sphere in a quiescent, realistic environment has been overlooked. 


%"~" 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 (cf. Stahler) 

\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 (cite) in his analytical derivation of inside-out collapse. 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 $\rho_{0}$. 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 $\xi > 6.451$ (cite), 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,

\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}


\subsection{Numerics}
\subsection{Simulation parameters}



\section{Results}\label{results}

\begin{figure}[htbp]
\centering
\epsscale{1}
\plottwo{rhoLight.eps}{vradLight.eps}
\caption{{\small Light case - Ambient is 1/100 $\rho (Rbe)$ - 5 crossing times}}
\label{light_case}
\end{figure}

\begin{figure}[htbp]
\centering
\plottwo{BPrho.eps}{vradBP.eps}
\caption{{\small 10$\%$ Overdensity}}
\label{BP_case}
\end{figure}

\begin{figure}[htbp]
\centering
\plottwo{MatchedRho.eps}{MatchedVrad.eps}
\caption{{\small Matched Case}}
\label{Matched_case}
\end{figure}

\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}




%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


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