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%http://arxiv.org/abs/0911.0293
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\newcommand {\oric} {\mbox{$\theta^1{\rm{C}}\:{\rm{Ori}}$}}
\newcommand {\Msun} {\mbox{M$_{\odot}$}}
\newcommand {\Lsun} {\mbox{L$_{\odot}$}}
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\begin{document}
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\title{Stellar interactions in dense and sparse star clusters}
\titlerunning{Stellar interactions in dense and sparse star clusters}
\author{C. Olczak\inst{1}
\and S. Pfalzner\inst{1}
\and A. Eckart\inst{1,2}}
\institute{I. Physikalisches Institut, Universit\"{a}t zu K\"{o}ln,
Z\"{u}lpicher Str.77, 50937 K\"{o}ln, Germany \\
\email{olczak@ph1.unikoeln.de}
\and MaxPlanckInstitut f\"{u}r Radioastronomie, Auf dem H\"{u}gel
69, 53121 Bonn, Germany}
\date{Received ; accepted}
% _______________________________________________________________________
%
% 
% Abstract
% 
%
% \abstract{}{}{}{}{}
% 5 {} token are mandatory
\abstract
% context heading (optional)
% {} leave it empty if necessary
{Stellar encounters potentially affect the evolution of the
protoplanetary discs in the Orion Nebula Cluster (ONC). However, the
role of encounters
in other cluster environments is less known.}
% aims heading (mandatory)
{We investigate the effect of the encounterinduced discmass loss in
different cluster environments.}
% methods heading (mandatory)
{Starting from an ONClike cluster we vary the cluster size and density
to determine the correlation of collision time scale and discmass loss. We
use the {\textsc{\mbox{nbody6\raise.2ex\hbox{\tiny{++}}}}} code to
model the dynamics of these clusters and analyze the discmass loss due to
encounters.}
% results heading (mandatory)
{We find that the encounter rate depends strongly on the cluster density
but remains rather unaffected by the size of the stellar population. This
dependency translates directly into the effect on the
encounterinduced discmass loss. The essential outcome of the
simulations are: i) Even in
clusters four times sparser than the ONC the effect of encounters is
still apparent. ii) The density of the ONC itself marks a threshold: in less
dense and less massive clusters it is the massive stars that dominate
the encounterinduced discmass loss whereas in denser and more massive
clusters the lowmass stars play the major role for the disc mass
removal.}
% conclusions heading (optional), leave it empty if necessary
{It seems that in the central regions of young dense star clusters 
the common sites of star formation  stellar encounters do affect the
evolution of the protoplanetary discs. With higher cluster density
lowmass stars become more heavily involved in this process. This finding
allows for the extrapolation towards extreme stellar systems: in case
of the Arches cluster one would expect stellar encounters to destroy the
discs of most of the low and highmass stars in several hundred
thousand years, whereas intermediate mass stars are able to retain to
some extant
their discs even under these harsh environmental conditions.}
\keywords{stellar dynamics 
methods: Nbody simulations 
stars: premain sequence, circumstellar matter}
\maketitle
% _______________________________________________________________________
%
% 
% Introduction
% 
\section{Introduction}
To current knowledge, planetary systems form from the accretion discs
around young stars. These young stars are in most cases not isolated but are
part of a cluster \citep[e.g.][]{2003ARA&A..41...57L}. Densities in
these cluster environments vary considerably, spanning a range of
10\,pc$^{3}$
(e.g. $\eta$ Chameleontis) to $10^6$\,pc$^{3}$ (e.g. Arches Cluster).
Though it is known that discs disperse over time
\citep{2001ApJ...553L.153H,2002astro.ph.10520H,2006ApJ...638..897S,2008ApJ...672..558C}
and that in dense clusters ($n$\,$\gtrsim$\,$10^3$\,pc$^{3}$)
the disc frequency seems to be lower in the core
\citep[e.g.][]{2007ApJ...660.1532B}, it is an open question as to how
far interactions with the
surrounding stars influence planet formation in clusters of different
densities.
Two major mechanisms are potentially able to strongly affect the
evolution of protoplanetary discs and planets in a cluster environment:
photoevaporation and gravitational interactions. Photoevaporation causes
the heating and evaporation of disc matter by the intense UV radiation from
massive stars. First models of photoevaporation were developed by
\citet{1998ApJ...499..758J} and \citet{1999ApJ...515..669S} (see also
references in
Hollenbach, Yorke \& Johnstone 2000), and have been much improved in the
past years
\citep{2001MNRAS.328..485C,2003ApJ...582..893M,2004RMxAC..22...38J,2005MNRAS.358..283A,2006MNRAS.369..229A,2008ApJ...688..398E,2009ApJ...690.1539G,2009ApJ...699L..35D}.
Gravitational interactions are another important effect on the
population of stars, discs, and planets in a cluster environment. An
encounter between
a circumstellar disc and a nearby passing star can lead to significant
loss of mass and angular momentum from the disc. While such isolated
encounters
have been studied in a large variety
\citep{1993ApJ...408..337H,1993MNRAS.261..190C,1994ApJ...424..292O,1995ApJ...455..252H,1996MNRAS.278..303H,1997MNRAS.287..148H,2004ApJ...602..356P,2005ApJ...629..526P,2006ApJ...653..437M,2007ApJ...656..275M,2008A&A...487..671K},
only few numerical studies have investigated the effect of stellar
encounters on circumstellar discs in a dense cluster environment directly
\citep{2001MNRAS.325..449S,2006ApJ...641..504A}.
Only recently has it been shown from numerical simulations that stellar
encounters do have an effect on the discs surrounding stars in a young dense
cluster
\citetext{\citealp[][]{2006ApJ...642.1140O,2006A&A...454..811P,2006ApJ...652L.129P,2006ApJ...653..437M,
2007ApJ...661L.183M,2007ApJ...656..275M,2007A&A...462..193P,2007A&A...475..875P};
\citealp[see also the review by][]{2007ARA&A..45..481Z}}. The
massive stars in the centre of such a stellar cluster act as
gravitational foci for the lower mass stars \citep{2006A&A...454..811P}.
Discs are most
affected when the masses of the stars involved in an encounter are
unequal \citep{2006ApJ...642.1140O,2007ApJ...656..275M}, so it is the
massive stars
that dominate the encounterinduced discmass loss in young dense
clusters \citep{2006ApJ...642.1140O}. Observational evidence for this
effect has
been found by \cite{2008A&A...488..191O}.
The numerical results obtained in our previous investigations are based
on a dynamical model of the Orion Nebula Cluster (ONC)  one of the
observationally most intensively studied young star cluster. It was
demonstrated that in the ONC stellar encounters can have a significant
impact on
the evolution of the young stars and their surrounding discs
\citep{2006ApJ...652L.129P,2006ApJ...642.1140O,2006A&A...454..811P,2007A&A...462..193P,2008A&A...488..191O,2008A&A...487L..45P}.
However,
investigating one model star cluster is not sufficient to draw general
conclusions  in fact, one could not answer questions as: How would things
change in a \emph{denser} cluster? Would a higher density inevitably
imply that stellar encounters play a more important role in the star and
planet
formation process? And what would be the situation in more
\emph{massive} clusters? Would the larger number of stars  in
particular \emph{massive}
stars  play a role?
Conclusive answers to these questions demand further numerical
investigations covering a larger parameter space in cluster parameters.
Here we realise
this by modelling scaled versions of the standard ONC model  clusters
with varying stellar numbers and sizes. The focus of this investigation
is on
the encounterinduced discmass loss. Throughout this work we assume
that initially all stars are surrounded by protoplanetary discs. This is
justified by observations that reveal disc fractions of nearly 100\,\%
in very young star clusters
\citep[e.g.][]{2000AJ....120.1396H,2000AJ....120.3162L,2001ApJ...553L.153H,2005astro.ph.11083H}.
In Section~\ref{sec:observations_onc} we briefly
outline the observationally determined basic properties of the ONC, that
serves as a reference model for the other cluster models. The computational
method is described in Section~\ref{sec:numerical_method} and the
construction of the numerical models is outlined in
Section~\ref{sec:numerical_models}. Afterwards we present results from a
numerical approach to this problem in
Section~\ref{sec:numerical_results} and
compare with analytical estimates in
Section~\ref{sec:analytical_results}. The conclusion and discussion mark
the last section of this paper.
% _______________________________________________________________________
%
% 
% Observations or calculations or mathematical derivations
% 
\section{Structure and dynamics of the ONC}
\label{sec:observations_onc}
The ONC is a rich stellar cluster with about 4000 members with masses
$m$\,$\ge$\,0.08\,{\mbox{M$_{\odot}$}} and a radial extension of
$\sim$2.5\,pc
\citep{1998ApJ...492..540H,2000ApJ...540..236H}. Most of the objects are
T~Tauri stars. The mean stellar mass is about
$\bar{m}$\,$\approx$\,0.5\,{\mbox{M$_{\odot}$}} and the halfmass radius
$R_{\rm{hm}}$\,$\approx$\,1\,pc \citep{1998ApJ...492..540H}. Recent studies
of the stellar mass distribution
\citep{2000ApJ...540..236H,2000ApJ...540.1016L,2002ApJ...573..366M,2004ApJ...610.1045S}
reveal no significant
deviation from the generalized IMF of \citet{2002Sci...295...82K},
\begin{equation}
\xi(m)=
\begin{cases}
\, m^{1.3} & , \quad 0.08 \le m/{\mbox{M$_{\odot}$}} <0.50 \,, \\
\, m^{2.3} & , \quad 0.50 \le m/{\mbox{M$_{\odot}$}} <1.00 \,, \\
\, m^{2.3} & , \quad 1.00 \le m/{\mbox{M$_{\odot}$}} <\infty \,.
\end{cases}
\label{eq:kroupa_imf}
\end{equation}
The mean age of the whole cluster has been estimated to be
$t_{\rm{ONC}}$\,$\approx$\,1\,Myr, although a significant age spread of
the individual
stars is evident \citep{1997AJ....113.1733H,2000ApJ...540..255P}.
The density and velocity distribution of the ONC resembles an isothermal
sphere. The central number density $\rho_{\rm{core}}$ in the inner 0.053\,pc
reaches $4.7$\,$\cdot$\,$10^4$\,pc$^{3}$ \citep{2002Msngr.109...28M}
and makes the ONC the densest nearby ($<$\,1\,kpc) young stellar
cluster. The
dense inner part of the ONC, also known as the Trapezium Cluster (TC),
is characterized by $R_{\rm{TC}}$\,$\lesssim$\,0.3\,pc and
$N_{\rm{TC}}$\,$\approx$\,500, or
$\rho_{\rm{TC}}$\,$\approx$\,$4\cdot10^3$\,pc$^{3}$.
In the most detailed study on circumstellar discs in the Trapezium
Cluster, \citet{2000AJ....120.3162L} found a fraction of 8085\,\% discs
among the
stellar population from the $L$band excess. This is in agreement with
an earlier investigation of the complete ONC in which
\citet{1998AJ....116.1816H} report a disc fraction of 5090\,\% (though
relying only on $I_{\mathrm{C}}K$ colors) and justifies the assumption of a
100\,\% primordial disc fraction in the here presented simulations.
In the following we describe the construction of the numerical cluster
models that have been used in our simulations.
% _______________________________________________________________________
%
% 
% Observations or calculations or mathematical derivations
% 
\section{Computational method}
\label{sec:numerical_method}
The basic dynamical model of the ONC used here is described in
\citet{2006ApJ...642.1140O}, with several extensions discussed in
\citet{2007A&A...475..875P}. We summarize the main aspects of our model:
The initial stellar population consists of 4000 members with masses between
$0.08\,\Msun$ and $50\,\Msun$ sampled from the standard Kroupa IMF (see
Eq.~(\ref{eq:kroupa_imf})). The system is initially in virial equilibrium,
%
\begin{equation}
\label{eqn:vir}
Q_{\rm{vir}}=\frac{R_{\rm{hm}}\sigma^2}{2GM} = 0.5 \,,
%
Q_{\rm{vir}}=\frac{R_{\rm{hm}}\left({\mbox{$\sigma_{\rm{3D}}^{\rm{JW}}$}}\right)^2}{2GM}
\approx 1.5 \,,
\end{equation}
%
where $R_{\rm{hm}}$ is the halfmass radius of the cluster, $\sigma$ the
velocity dispersion, $M$ the total mass, and $G$ the gravitational
constant. It is characterized by a radial density profile, $\rho \propto
r^{2}$, with a central density $\sim$4$\cdot10^4$\,pc$^{3}$. We adopt a
Maxwellian velocity distribution as would be expected from theory of
star cluster formation \citep[e.g.][]{2000prpl.conf..151C} and roughly in
agreement with observations of starforming regions
\cite[e.g.][]{1990ApJ...359..344F,1995ApJ...450L..27M}. We use here a
single star model only and
do not include the effect of gas expulsion or stellar evolution. All
simulations have been performed with the direct $N$body code $\nbodypp$
\citep{1999JCoAM.109..407S,2003grav.book.....A}.
For the generation of star cluster models in the present investigation
the initial radial density profile has been modified slightly. To first
order
the isothermal sphere, $\rho(r) \propto r^{2}$, represents the
projected density distribution of the ONC, yet a flattening in the core,
$\Sigma(r)
\propto r^{0.5}$, $0 < r \le R_\text{core}$, $R_\text{core} \approx
0.2$\,pc, is observed \citep{2005MNRAS.358..742S}. Validating the
initial setup
by means of the best reproduction of the \emph{current projected}
density distribution of the ONC after a simulation time of 1\,Myr, the
evaluation of
numerous initial configurations led to the following best estimate of
the \emph{initial threedimensional} density distribution:
%
\begin{align}
\label{eq:density_distribution_initial}
\rho_0(r) & =
\begin{cases}
\rho_0\,(r/R_{\mathrm{core}})^{2.3} & , \quad r \in
(0,R_{\mathrm{core}}] \\
\rho_0\,(r/R_{\mathrm{core}})^{2.0} & , \quad r \in
(R_{\mathrm{core}},R] \\
\qquad 0 & , \quad r \in (R,\infty] \\
\end{cases} \quad ,
\end{align}
%
where $\rho_0 = 3.1 \cdot 10^3\,\text{stars}\,\text{pc}^{3}$,
$R_{\mathrm{core}} = 0.2\,\text{pc}$, and $R = 2.5\,\text{pc}$.
Moreover, the generation of the highmass end of the mass function has
been modified. In the case of the ONC the upper mass was chosen to be
50\,\Msun\ because this value corresponds to the mass of the most
massive stellar system in the ONC. However, stars with larger masses are
expected to
form in more massive clusters
\citep{2005ApJ...620L..43O,2006MNRAS.365.1333W}. Thus in the framework
of this numerical investigation the upper mass
limit has been set to the current accepted fundamental upper mass limit,
$m_{\mathrm{max}} = 150\,\Msun$
\citep{2005Natur.434..192F,2005ApJ...620L..43O,2006MNRAS.365..590K,2006MNRAS.365.1333W,2007ApJ...660.1480M,2007ARA&A..45..481Z}.
The choice of a fixed upper mass limit, though in disagreement with the
wellestablished \emph{nontrivial} correlation of the mass of a star
cluster
and its most massive member
\citep[e.g.][]{2003ASPC..287...65L,2006MNRAS.365.1333W,2008MNRAS.391..711M},
was motivated by the fact that the exact
relation is not known. However, a comparison with the ``sorted sampling
algorithm'' of \citet{2006MNRAS.365.1333W} in
Table~\ref{tab:cluster_parameters} shows that  at least in a
statistical sense  the exact prescription for the generation of the
maximum stellar
mass in a cluster is not as important as it might seem. The values
obtained by random sampling are only slightly larger than those from sorted
sampling.
\begin{table*}
\centering
\begin{tabular}{c*2{*{4}{c}}{*{3}{c}}}
\hline
\hline
family of models & \multicolumn{4}{c}{densityscaled} &
\multicolumn{4}{c}{sizescaled} & \multicolumn{3}{c}{} \\
\hline
\hline
$N$
& model & $R$ & $\rho_{\mathrm{TC}}$ & $\sigma_{\mathrm{3D}}$
& model & $R$ & $\rho_{\mathrm{TC}}$ & $\sigma_{\mathrm{3D}}$
& $m_{\mathrm{max}}^{\mathrm{med}}$ &
$m_{\mathrm{max}}^{\mathrm{sort}}$ & $m_{\mathrm{max}}^{\mathrm{obs}}$ \\
& & [pc] & [$10^3$\,pc$^{3}$] & [\kms]
& & [pc] & [$10^3$\,pc$^{3}$] & [\kms]
& [\Msun] & [\Msun] &
[\Msun] \\
\hline
~~1000 & D0 & ~~2.50 & ~~1.3 & 1.15 & S0 & ~~0.63 &
~~4.8 & 2.42 & ~~36 & ~~32 & $25 \pm 15$ \\
~~2000 & D1 & ~~2.50 & ~~2.7 & 1.64 & S1 & ~~1.25 &
~~5.1 & 2.37 & ~~52 & ~~45 & $25 \pm 15$ \\
~~4000 & D2 & ~~2.50 & ~~5.3 & 2.26 & S2 & ~~2.50 &
~~5.3 & 2.26 & ~~79 & ~~63 & $55 \pm 25$ \\
~~8000 & D3 & ~~2.50 & 10.5 & 3.11 & S3 & ~~5.00 &
~~5.3 & 2.13 & ~~94 & ~~80 & $75 \pm 25$ \\
16000 & D4 & ~~2.50 & 21.1 & 4.34 & S4 & 10.00 &
~~5.3 & 2.11 & 125 & 112 & $95 \pm 35$ \\
32000 & D5 & ~~2.50 & 42.1 & 6.04 & S5 & 20.00 &
~~5.2 & 2.03 & 137 & 126 & $100\pm 35$ \\
\hline
\hline
\end{tabular}
\caption{Averaged initial parameters of the cluster models, divided
among the families of densityscaled and sizescaled models. The first
column
contains the number of stars, $N$, the next eight columns contain
the designation, the cluster size, $R$, the number density in a sphere
of radius 0.3\,pc,
$\rho_{\mathrm{TC}}$ (equivalent to the Trapezium Cluster in the
ONC, see Section~\ref{sec:observations_onc}), and the
threedimensional velocity dispersion, $\sigma_{\mathrm{3D}}$, for
the densityscaled and sizescaled models, respectively. The last three
columns
denote the median maximum stellar mass in each simulation,
$m_{\mathrm{max}}^{\mathrm{med}}$, the median maximum stellar mass for
sorted
sampling, $m_{\mathrm{max}}^{\mathrm{sort}}$, and estimates from
observational data, $m_{\mathrm{max}}^{\mathrm{obs}}$, both taken from
Fig.~7 of
\citet{2006MNRAS.365.1333W}.}
\label{tab:cluster_parameters}
\end{table*}
Stellar encounters in dense clusters can lead to significant transport
of mass and angular momentum in protoplanetary discs
\citep{2006ApJ...642.1140O,2006A&A...454..811P,2007A&A...462..193P}. In
the present investigation we assume that all discs are of low mass, i.e.
a low
mass ratio of disc and central star, $m_{\mathrm{disc}}/m_{\star} \ll
0.1$, in agreement with observations of the ONC
\citep{1998AJ....116..854B,2005ApJ...634..495W,2009ApJ...699L..55M}. We
use Eq.~(1) from \citet{2006A&A...454..811P} to keep track of the
\emph{relative} discmass loss of each stardisc system due to
encounters. Approaches of stars are only considered to be encounters if
the calculated
relative diskmass loss is higher than 3\,\%, corresponding to the
1$\sigma$ error in our simulations of stardisc encounters. In this case the
relative discmass loss is independent of the disc mass and depends only
on the mass ratio of the interacting stars and the orbital parameters
\citep[see][]{2006A&A...454..811P}. Our estimate of the accumulated
discmass loss is an upper limit because the underlying formula is only
valid for
coplanar, prograde encounters, which are the most perturbing. A
simplified prescription assigns stars into one of two distinct groups:
if the
relative discmass loss exceeds 90\,\% of the initial disc mass, stars
are marked as ``discless''; otherwise they are termed ``stardisc
systems''. As
in our previous investigations, the determination of the discmass loss
involves two different models of the initial distribution of disc sizes:
i)~scaled disc sizes $r_{\mathrm{d}}$ with $r_{\mathrm{d}} =
150\,\mathrm{AU} \sqrt{M_1} [\Msun]$, which is equivalent to the
assumption of a fixed
force at the disc boundary, and ii)~equal disc sizes with
$r_{\mathrm{d}} = 150\,\mathrm{AU}$. Whenever results are presented, we
will specify which
of these two distributions has been used.
% _______________________________________________________________________
%
\section{Numerical models}
\label{sec:numerical_models}
For the present study we have set up a variety of scaled versions of the
standard ONC model cluster with varying sizes and stellar numbers. In total,
eleven cluster models have been set up (see
Table~\ref{tab:cluster_parameters}). They form two parametric groups,
the ``densityscaled'' (D0D5) and
the ``sizescaled'' (S0S5) group, both containing six clusters with
stellar numbers of 1000, 2000, 4000, 8000, 16000, and 32000,
respectively. Models
D2 and S2 are identical and correspond to the standard ONC model with
the adopted higher stellar upper mass limit, $m_{\mathrm{max}} =
150\,\Msun$. The other ten cluster models have been set up as scaled
representations of ONClike clusters. As in the case of the numerical
model of
the ONC, for each cluster model a set of simulations has been performed
with varying random configurations of positions, velocities, and masses,
according to the given distributions, to lower the effect of statistical
uncertainties. For the clusters with 1000, 2000, 4000, 8000, 16000, and
32000
particles, a number of 200, 100, 100, 50, 20, and 20 simulations seemed
appropriate to provide sufficiently robust results.
It has to be noted that these ``artificial'' stellar systems are not
just theoretical models but have as well counterparts in the observational
catalogues of star clusters: the young star cluster NGC~2024
\citep[e.g.][]{2000AJ....120.1396H,2003A&A...404..249B,2003AJ....126.1665L}
is well
represented by the 1000 particle model, whereas the 16000 particle model
has its counterpart in the massive cluster NGC~3603
\citep[e.g.][]{1999AJ....117.2902D,2004AJ....128..765S,2006AJ....132..253S}.
%                                    
\subsubsection*{Densityscaled cluster models}
The six densityscaled cluster models (D0D5) have been simulated with
the same initial size as the ONC ($R=2.5$\,pc). Due to the adopted number
density distribution, roughly represented by $\rho(r) = \rho_0 \,
r^{2}$, the density of the models scales as the stellar number,
%
\begin{equation}
\label{eq:relation_number_density_size}
N = \int_0^R \rho(r) \, r^2 \, dr \, d\Omega \propto \rho_0 \, R \,,
\end{equation}
%
though for an exact treatment one would have to consider the steeper
density profile of the core, $\rho_{\mathrm{core}}(r) =
\rho_{\mathrm{core},0} \,
r^{2.3}$,
%
\begin{equation}
N_{\mathrm{core}} = \int_0^{R_{\mathrm{core}}} \rho_{\mathrm{core}}(r)
\, r^2 \, dr \, d\Omega \propto \rho_{\mathrm{core},0} \;
R_{\mathrm{core}}^{0.7} \,.
\label{eq:relation_number_density_size_core}
\end{equation}
%
However, since the core population is not dominant in terms of number,
the clusters are characterised in good approximation by densities that
are 1/4,
1/2, 1, 2, 4, and 8 times the density of the ONC (at any radius),
respectively. These models are used to study the importance of the
density for the
effect of stardisc encounters in a cluster environment.
%                                    
\subsubsection*{Sizescaled cluster models}
Five more cluster models have been simulated with the same initial
density as the ONC but with varying extension. The set of sizescaled
cluster
models (S0S5) is used to study the pure effect of the size of the
stellar population. Due to the
relation~(\ref{eq:relation_number_density_size}),
the initial size of these clusters scales as the stellar number and was
set up with 1/4, 1/2, 1, 2, 4 and 8 times the initial size of the ONC,
respectively.
\bigskip
The initial parameters of the cluster models, for each model averaged
over all configurations, are presented in
Table~\ref{tab:cluster_parameters}. Here the number density in the
Trapezium Cluster, $\rho_{\mathrm{TC}}$, is taken as a reference value
for all
simulations. As expected, the density scales with the number of stars
for the densityscaled models, while it is rather constant for the
sizescaled
models. The velocity dispersion, that satisfies the relation
%
\begin{equation}
\label{eq:relation_velocity_dispersion}
\sigma = \sqrt{ \frac{2GM}{R} } \propto \sqrt{ \frac{N}{R} } \,,
\end{equation}
%
shows the expected scaling of $\sqrt{N}$ for the densityscaled models,
and is again roughly constant for the sizescaled models, as expected
from $N
\propto R$ (Eq.~(\ref{eq:relation_number_density_size})) and the above
relation. The reason for the slight increase of the velocity dispersion with
decreasing stellar number for the sizescaled models is the steeper
density profile in the cluster core, which becomes more dominant in terms of
stellar number with decreasing cluster size. Combining
$N_{\mathrm{core}} \propto R^{0.7}$ from
Eq.~(\ref{eq:relation_number_density_size_core}) and
the above relation gives roughly $\sigma_{\mathrm{core}} \propto
N^{0.3}$ and thus explains the correlation.
% _______________________________________________________________________
%
% 
% Results
% 
\section{Results of numerical simulations}
\label{sec:numerical_results}
In this section the results of the numerical simulations of the two
families of cluster models will be presented. A short discussion of the
characteristic scaling relations and differences of the cluster dynamics
will be followed by a more detailed investigation of the encounterinduced
discmass loss among the two families of densityscaled and sizescaled
cluster models.
% 
\subsection{Cluster dynamics}
\label{sec:numerical_results:cluster_dynamics}
In Fig. \ref{fig:projected_density_distribution_sample_density_scaled}
the evolution of the projected density distribution of the densityscaled
models is shown. The shape of the distributions is in all cases very
similar. The evolved distributions have nearly identical shapes and are
separated by vertical intervals of 0.3 in logspace, which corresponds
to the difference of the initial densities by a factor~2. Only in the
innermost
cluster regions slight deviations between the evolved distributions are
apparent. These are attributed to the poorer random sampling of the initial
particle distribution due to the very steep density profile, $\rho(r)
\propto r^{2.3}$, as is evident from the larger scatter among the initial
profiles. However, after 1\,Myr these deviations are smoothed out to a
large degree.
\begin{figure}
\centering
\includegraphics[width=1.0\linewidth]{figures/projected_density_distribution__1_myr__sample_density_scaled__paper.pdf}
\caption{Projected density profiles of the densityscaled models
compared to observational data. The initial profile (grey lines) and the
profile at
a simulation time of 1\,Myr (black lines) are shown. From bottom to
top in each colour regime the cluster models D0D5 are marked by a
shortdashed, longdashed, solid, dotted, dotlongdashed, and
dotshortdashed line, respectively. The observational data are from a
compilation
of \citet{2002Msngr.109...28M} and \citet{1997AJ....113.1733H}.}
\label{fig:projected_density_distribution_sample_density_scaled}
\end{figure}
Due to the nearly exact qualitative \emph{and} scaled quantitative
evolution of the densityscaled cluster models, it is justified to ascribe
differences of the effects of stardisc encounters on the stellar
population mainly to one parameter, namely the initial density of the
cluster
models. Nevertheless, the influence of the particle number has to be
considered, too.
\begin{figure}
\centering
\includegraphics[width=1.0\linewidth]{figures/density_vs_time__sample_size_scaled__paper.pdf}
\caption{Time evolution of stellar densities $\rho_{\mathrm{TC}}$ (in
a volume of radius $R_{\mathrm{TC}}=0.3\,$pc, see
Section~\ref{sec:observations_onc}) of the sizescaled cluster
models. The cluster models S0S5 are marked by a shortdashed,
longdashed, solid,
dotted, dotlongdashed, and dotshortdashed line, respectively.
The horizontal error bar marks the corresponding observational estimates for
comparison.}
\label{fig:density_vs_time_sample_size_scaled}
\end{figure}
The sizescaled cluster models show a different dynamical evolution
compared to the densityscaled models. The temporal evolution of the
densities
$\rho_{\mathrm{TC}}$ in
Fig.~\ref{fig:density_vs_time_sample_size_scaled} demonstrates that the
clusters evolve on slightly different time scales,
where the density declines faster for the less populated clusters S0 and
S1. However, the densities of the models S1S5 differ not much, and are
consistent with a coeval decline. The evolution of the cluster densities
does not  to first order  depend on the number of particles, probably
with the exception of the model S0. The sizescaled models are thus well
suited to investigate the effect of the number of cluster stars on stardisc
encounters and the corresponding induced discmass loss.
% 
\subsection{Encounter dynamics}
\label{sec:numerical_results:encounter_dynamics}
\begin{figure}
\centering
\includegraphics[width=1.0\linewidth]{figures/disc_destruction_vs_time__sample_density_scaled__paper.pdf}
\caption{Time evolution of the cluster disc fraction of the
densityscaled models for a region of the size of the Trapezium Cluster
($R_{\mathrm{TC}}=0.3\,\mathrm{pc}$). The curves have been smoothed
by Bezier curves to avoid intersecting lines. Here initially equal disc
sizes
have been assumed. From top to bottom the cluster models D0D5 are
marked by a shortdashed, longdashed, solid, dotted, dotlongdashed, and
dotshortdashed line, respectively. The typical error bar is
indicated in the upper right.}
\label{fig:disc_destruction_vs_time__sample_density_scaled}
\end{figure}
We investigate the effect of the cluster density on the
encounterinduced discmass loss via the evolution of the cluster disc
fraction (CDF). The
distributions in
Fig.~\ref{fig:disc_destruction_vs_time__sample_density_scaled} show the
average fraction of stars that are surrounded by disc
material and correspond from top to bottom to clusters with increasing
density. The curves have been smoothed with Bezier curves to provide a
clearer
view. It is evident that the fraction of destroyed discs in the
Trapezium Cluster increases significantly with increasing cluster
density. In
particular, the effect becomes much stronger for the models D3D5 with
28 times the density of the ONC. In the case of the densest model D5,
even up
to 60\,\% of the stars in the Trapezium Cluster could have lost their
discs after 1\,Myr of dynamical evolution. But it is also interesting to
note
that even in a cluster 4 times less dense than the ONC (model D0), still
1015\,\% of the stars could lose their surrounding discs due to
gravitational interactions with cluster members. However, one has to
treat these numbers with care due to the  partially  significant
uncertainties that go into the calculations
\citep[see][]{2006ApJ...642.1140O}. Nonetheless, what is more important
here  and relies only on the
\emph{relative} quantities  is the fact that the distributions in
Fig.~\ref{fig:disc_destruction_vs_time__sample_density_scaled} are not
equidistant
but do show larger differences with increasing density.
The impression that the disc fraction increases at later stages of the
cluster evolution is solely due to the acceleration of perturbed systems
leaving the sampling volume. The escape rate follows the trend of the
disc destruction rate with a time delay that is determined by the
difference of
the crossing time of discless stars and stardisc systems in the
sampling volume. Hence, the decrease of the disc fraction is followed by
an increase
and eventually remains constant some time after disc destruction has
stopped.
The preferred mass range of stars that become disproportionately high
involved in perturbing encounters changes significantly between the
models D2
and D4. From previous investigations
\citep{2006ApJ...642.1140O,2006A&A...454..811P} we already know that in
the model D2, representing an ONClike
cluster, the highmass stars' discs are mostly affected by encounters.
These encounters occur preferentially with lowmass stars due to
gravitational
focusing. However, in the model cluster D4, a four times denser system,
it is the lowmass stars that dominate the discmass loss of the cluster
population.
\begin{figure}
\centering
\includegraphics[width=1.0\linewidth]{figures/encounters_vs_relative_perturber_mass__disc_mass_loss__lowmass__n1000_n4000_n16000__sample_density_scaled__paper.pdf}
\caption{Comparison of the fraction of encounters as a function of the
relative perturber mass, i.e. the mass ratio of perturber and perturbed
star,
of the group of lowmass stars (see
Appendix~\ref{app:star_cluster_models:mass_groups} for the width of the
mass intervals) for the models D0
(dotted), D2 (solid), and D4 (dotlongdashed), respectively. Here
initially equal disc sizes have been assumed.}
\label{fig:encounters_vs_relative_perturber_mass__disc_mass_loss__lowmass__4000_1000_16000__sample_density_scaled}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=1.0\linewidth]{figures/encounters_vs_eccentricity__disc_mass_loss__lowmass__n1000_n4000_n16000__sample_density_scaled__paper.pdf}
\caption{Comparison of the fraction of encounters as a function of
eccentricity of the group of lowmass stars (see
Appendix~\ref{app:star_cluster_models:mass_groups} for the width of
the mass intervals) for the models D0 (dotted), D2 (solid), and D4
(dotlongdashed), respectively. Here initially equal disc sizes
have been assumed.}
\label{fig:encounters_vs_eccentricity__disc_mass_loss__lowmass__4000_1000_16000__sample_density_scaled}
\end{figure}
We demonstrate this transition by binning the fraction of encounters,
normalised to the total number of encounters of each model (so that the
integrated surface is unity in each case). Recall that encounters have
been defined as flybys causing more than 3\,\% diskmass loss (see
Section~\ref{sec:numerical_method}).
Fig.~\ref{fig:encounters_vs_relative_perturber_mass__disc_mass_loss__lowmass__4000_1000_16000__sample_density_scaled}
shows for the group of lowmass stars that the dominance of
perturbations from much more massive stars in the models D0D2,
represented by the peak at
relative perturber masses $\gtrsim 100$, changes dramatically towards
encounters of nearly equalmass stars in the model D4 (with a peak at
relative
perturber masses $\approx 3$). Similarly, the typical geometry of the
encounter orbits underlies a strong transition. As is evident from
Fig.~\ref{fig:encounters_vs_eccentricity__disc_mass_loss__lowmass__4000_1000_16000__sample_density_scaled},
most encounters of lowmass stars in the
models D0D2 are parabolic (with an eccentricity $e \approx 1$). In
contrast, in the model D4 these stars usually experience a strongly
hyperbolic
flyby of a perturber, mostly with eccentricity $e > 10$. Hence, with
increasing cluster density, the dominant mode of stardisc interactions
changes
towards hyperbolic encounters of lowmass stars between each other.
This transition is a consequence of gravitational focusing by highmass
stars becoming significantly less important among the models
D0D5. Gravitational focusing forces a deflection of the interacting
stars onto a less eccentric orbit, effectively increasing the
crosssection above
geometrical:
%
\begin{equation}
\label{eq:cross_section}
\pi{b}^2 = \pi{r_{\mathrm{min}}^2} (1 + \Theta) \,,
\end{equation}
%
where
%
\begin{equation}
\Theta = \frac{v_{\mathrm{min}}^2}{\langle v_{\mathrm{rel}} \rangle^2}
= \frac{\pi}{16}\frac{v_{\mathrm{min}}^2}{\sigma^2}
\end{equation}
%
is the gravitational focusing term or Safronov number, $b$ the impact
parameter, $r_{\mathrm{min}}$ the minimum distance, $v_{\min}$ the
escape speed
at the minimum distance, $\langle v_{\mathrm{rel}} \rangle$ the mean
relative speed, and $\sigma$ the cluster velocity dispersion (one finds
$\langle
v_{\mathrm{rel}} \rangle = 4\sigma / \sqrt{\pi}$ for a Maxwellian
distribution with dispersion $\sigma$). Adopting the typical cluster and
encounter
parameters from Table~\ref{tab:cluster_parameters} and
\ref{tab:encounter_rate_parameter_space}, respectively, and substituting
$v_{\mathrm{min}} =
v_{\mathrm{enc}}$ and $r_{\mathrm{min}} = r_{\mathrm{enc}}$, we find
that gravitational focusing by lowmass stars is negligible ($\Theta \ll
1$) in
all our cluster models. In contrast, highmass stars in the model D0
play a substantial role as gravitational foci ($\Theta \approx 120$),
whereas
their cross section is reduced by more than one order of magnitude in
the model D5 ($\Theta \approx 4$).
In summary, the encounterinduced discmass loss in cluster environments
of different densities shows two important features: i) lowmass stars
lose a
larger fraction of their disc material with increasing cluster density,
and ii) the discs of the most massive stars are (nearly) completely
destroyed,
independent of the density of the cluster environment. The important
finding for i) is that the correlation is not linear, but shows a much
larger
increase for the cluster models with higher densities than model D2,
implying that there exists a critical density $\rho_{\mathrm{crit}}$
that marks
the onset of a much more destructive effect of stardisc encounters.
This critical density seems to be close to the density of the ONC,
$\rho_{\mathrm{crit}} \approx \rho_{\mathrm{ONC}}$.
We find that the evolution of the CDF in the Trapezium Cluster region is
very similar among the sizescaled models and corresponds to the
distribution
of the model D2 in
Fig.~\ref{fig:disc_destruction_vs_time__sample_density_scaled}. The
sizescaled models are obviously equivalent in their
environmental effect on protoplanetary discs despite the slightly
different dynamical evolution. The density of the models S0 and S1
decreases faster
than for the more massive clusters, even up to a factor of 2 in case of
the model S0 (see Section~\ref{sec:numerical_results:cluster_dynamics}).
Thus
one would expect a lower encounter rate in these smaller systems and,
consequently, on average a lower discmass loss. However, this is
obviously not
the case, and is explained by the fact that, similarly to the finding
for the densityscaled models, highmass stars become less important as
gravitational foci for the lowmass stars in clusters with larger
stellar populations. Hence in terms of encounter statistics, the lower
density of
the models S0 and S1 is compensated by the more frequent interactions of
the highmass stars.
% 
\subsection{Validation of the numerical method}
\label{sec:numerical_results:validation}
So far the discmass loss has been calculated from Eq.~(1) of
\citet{2006A&A...454..811P} by treating all encounters as parabolic. To
account for the
lower discmass loss in hyperbolic encounters, we have determined a
function that quantifies the reduction of the discmass loss in
dependence of the
orbital eccentricity:
%
\begin{equation}
\label{eq:fit_function_mass_loss_vs_eccentricity}
\begin{split}
\widehat{\Delta{m}}(e) =& \exp[0.12(e  1)] \, \times \\
& \{0.83  0.015(e  1) + 0.17 \exp[0.1(e  1)]\} \,.
\end{split}
\end{equation}
%
Eq.~(\ref{eq:fit_function_mass_loss_vs_eccentricity}) is a fit function
to the median distribution of the relative discmass loss as a function of
eccentricity, normalised to the parabolic case, for all stardisc
simulations that have been performed.
\begin{figure}
\centering
\includegraphics[width=1.0\linewidth]{figures/encounters_vs_eccentricity__disc_mass_loss__sample_density_scaled_n16000__paper.pdf}
\includegraphics[width=1.0\linewidth]{figures/encounters_vs_eccentricity__disc_mass_loss__ecc_model__sample_density_scaled_n16000__paper.pdf}
\caption{Number of encounters as a function of eccentricity
(logarithmic bins), plotted for three different groups by means of
discmass loss per
encounter. The white surface represents all encounters \citep[i.e.
minimum 3\,\% discmass loss, cf.][]{2006ApJ...642.1140O}, the light
grey those
that removed at least 50\,\% of the discmass, while the dark grey
stands for the most destructive encounters that caused a discmass loss
of at
least 90\,\%. The two plots show the distributions for the model D4.
\emph{Top:} Discmass loss calculated assuming parabolic
encounters. \emph{Bottom:} Discmass loss calculation corrected for
effects of eccentricity by using
Eq.~(\ref{eq:fit_function_mass_loss_vs_eccentricity}).}.
\label{fig:encounters_vs_eccentricity__disc_mass_loss__parabolic_and_ecc_model__sample_density_scaled_n16000}
\end{figure}
As shown in
Fig.~\ref{fig:encounters_vs_eccentricity__disc_mass_loss__parabolic_and_ecc_model__sample_density_scaled_n16000},
the number of hyperbolic
encounters is significantly reduced if the eccentricity is considered
explicitly in the calculation of the discmass loss using the fit function
(\ref{eq:fit_function_mass_loss_vs_eccentricity}). This is a consequence
of our definition of an encounter (see Section~\ref{sec:numerical_method}):
the fraction of perturbations that cause a relative discmass loss above
3\,\% is lower for higher eccentricities, hence the number of eccentric
encounters decreases. However, because the fit function represents the
\emph{median} distribution of all simulated stardisc encounters and the
effect
of strongly perturbing encounters is only weakly dependent on the
eccentricity, the number of strongly perturbing encounters in
Fig.~\ref{fig:encounters_vs_eccentricity__disc_mass_loss__parabolic_and_ecc_model__sample_density_scaled_n16000}b
(light and dark grey surfaces) is
underestimated.
\begin{figure}
\centering
\includegraphics[width=1.0\linewidth]{figures/disc_destruction_vs_time__ecc_model__sample_density_scaled__paper.pdf}
\caption{Time evolution of the cluster disc fraction of the
densityscaled models, not restricted to parabolic encounters, for a
region of the size
of the Trapezium Cluster ($R_{\mathrm{TC}}=0.3\,\mathrm{pc}$). The
curves have been smoothed by Bezier curves to avoid intersecting lines. Here
initially equal disc sizes have been assumed. From top to bottom the
cluster models D0D5 are marked by a shortdashed, longdashed, solid,
dotted, dotlongdashed, and dotshortdashed line, respectively.
The typical error bar is indicated in the upper right.}
\label{fig:disc_destruction_vs_time__ecc_model__sample_density_scaled}
\end{figure}
Hence it is the discmass loss induced by weak hyperbolic interactions
that has been overestimated in the calculations. These events are most
numerous
in the models D3D5 with densities $\rho > \rho_{\mathrm{crit}}$ and
result preferentially from interactions of lowmass stars with roughly
equalmass
perturbers. Consequently, when considering explicitly the reduced
discmass loss in hyperbolic encounters (see
Fig.~\ref{fig:disc_destruction_vs_time__ecc_model__sample_density_scaled}),
the outstanding role of these dense clusters as environments of huge disc
destruction becomes less pronounced (cf.
Fig.~\ref{fig:disc_destruction_vs_time__sample_density_scaled}), though
the encounterinduced discmass loss
is still considerably larger compared to the sparser clusters D0D2.
\bigskip
\begin{figure}
\centering
\includegraphics[width=1.0\linewidth]{figures/disc_mass_loss_vs_mass_bins__disc_size_fixed__basic_onc_model_50_msun_and_150_msun__paper.pdf}
\caption{Average relative discmass loss at 1\,Myr for the Trapezium
Cluster as a function of the stellar mass for initially equal disc
sizes. The
standard ONC model with a stellar upper mass limit of 50\,\Msun\
(dark grey bars) is compared to a simulation of the same model with an
upper mass
limit of 150\,\Msun\ (light grey bars).}
\label{fig:disc_mass_loss_vs_mass_bins__disc_size_fixed__basic_onc_model_50_msun_and_150_msun}
\end{figure}
The effect of the higher upper mass limit on our simulation results
compared to the basic ONC model is investigated in
Fig.~\ref{fig:disc_mass_loss_vs_mass_bins__disc_size_fixed__basic_onc_model_50_msun_and_150_msun}.
It shows the average relative discmass loss as a
function of the stellar mass for the standard ONC model with a stellar
upper mass limit of 50\,\Msun\ (dark grey boxes), and the same model with an
upper mass limit of 150\,\Msun\ (light grey boxes). For masses below
10\,\Msun\ the two distributions are quantitatively in good agreement.
In the range of $\sim$1050\,$\Msun$ the average discmass loss of the
50\,$\Msun$limit model is significantly higher. This is to be expected
because
stars in this mass range can act as additional strong gravitational foci
in the presence of a 50\,$\Msun$ star, while their effect is largely reduced
if a 150\,$\Msun$ star is gravitationally dominating. The most massive
stars in the range $80150\,\Msun$ show, as expected, the largest average
relative discmass loss. That it is somewhat lower than that of the most
massive star of the 50\,$\Msun$limit model is in agreement with the
stronger
gravitational attraction of their disc, leading on average to a reduced
discmass loss per encounter. However, since the highest mass bin is only
populated by 9 stars, any further conclusions about the average relative
discmass loss of the most massive stars would be highly speculative.
% _______________________________________________________________________
%
% 
% Results
% 
\section{Comparison of numerical results and analytical estimates}
\label{sec:analytical_results}
In this section the numerical results will be compared to analytical
estimates. The treatment of encounters involves one important time
scale, the
\emph{collision time} $t_{\mathrm{coll}}$
\citep[][Eq.~8123]{1987gady.book.....B},
%
\begin{equation}
\label{eq:tcoll}
t_\mathrm{coll} = \left[ 16 \sqrt{\pi} \rho \sigma r_\star^2 \left(1 +
\frac{Gm_\star}{2\sigma^2r_\star}\right) \right]^{1} \,.
\end{equation}
%
Here the inverse of the collision time will be introduced as the
encounter rate, $f_{\mathrm{enc}} \equiv t_{\mathrm{coll}}^{1}$. Using
the escape
velocity $v_{\star}$ from the stellar surface,
%
\begin{equation}
\label{eq:escape_velocity_stellar_surface}
v_{\star} = \sqrt{ \frac{ 2 G m_{\star} }{ r_{\star}} } \,,
\end{equation}
%
the encounter rate can be written as
%
\begin{equation}
% \begin{aligned}
\label{eq:encounter_rate_relative}
f_{\mathrm{enc}}
% &= 16 \sqrt{\pi} \rho \sigma r_\star^2 \left( 1 +
\frac{Gm_\star}{2\sigma^2r_\star} \right) \\
% &= 16 \sqrt{\pi} \rho \sigma r_\star^2 \left( 1 + \frac{ v_{\star}^2
}{ 4\sigma^2 } \right) \,,
= 16 \sqrt{\pi} \rho \sigma r_\star^2 \left( 1 + \frac{ v_{\star}^2 }{
4\sigma^2 } \right) \,,
% \end{aligned}
\end{equation}
%
where $G$ denotes the gravitational constant, $m_{\star}$ and
$r_{\star}$ the stellar mass and radius, and $\rho$ and~$\sigma$ the
density and
velocity dispersion of the star cluster. In the following, the stellar
radius~$r_{\star}$ will be replaced by the ``typical interaction radius''
$r_{\mathrm{enc}}$, that means the radius at which the star is subject
to a significant (but still frequent) perturbation that potentially can
remove
some fraction of the discmass. Eq.~(\ref{eq:encounter_rate_relative})
will be evaluated for three different stellar masses, representing
stellar mass
groups of low\mbox{,} intermediate and highmass stars. Appropriate
typical interaction radii have been taken from Table~3 of
\citet{2006ApJ...642.1140O}. The set of masses $m_{\star}$, radii
$r_{\mathrm{enc}}$, and ``encounter escape speeds'' $v_{\mathrm{enc}}$
resulting
from Eq.~(\ref{eq:escape_velocity_stellar_surface}) is shown in
Table~\ref{tab:encounter_rate_parameter_space}. The last column contains the
``gravitational focusing parameter'' $\gamma_{\mathrm{enc}}$, an
approximation parameter defined as the power of ten best representing
$v_{\mathrm{enc}}^2$,
%
\begin{equation}
% \nonumber
\label{eq:gravitational_focusing_parameter}
\gamma_{\mathrm{enc}} \equiv 10^{\displaystyle \, \lfloor
\log{v_{\mathrm{enc}}^2} + 1/2 \rfloor} \,,
\end{equation}
%
where $\lfloor{x}\rfloor$ denotes the floor function\footnote{The floor
function $\lfloor{x}\rfloor$ gives the largest integer less than or equal to
$x$.} of $x$. For clarity, massdependent quantities,
$x_{\mathrm{enc}}$, will be explicitly denoted as functions of mass,
$x_{\mathrm{enc}}(m_{\star})$, in the following.
\begin{table}
\centering
\begin{tabular}{l*4{c}}
\hline
mass groups & $m_{\star}$ & $r_{\mathrm{enc}}$ &
$v_{\mathrm{enc}}$ & $\gamma_{\mathrm{enc}}$ \\
& [$\Msun$] & [AU] & [AU/yr] & \\
\hline
low mass & $0.11~~~~$ & $10^{23~~}$ &
$\sim0.3$ & $10^{1}$ \\
intermediate mass & $~~~110~~$ & $10^{22.5}$ & $\sim
2$ & $1$ \\
high mass & $~10100$ & $10^{2~~~~~}$ &
$\sim 6$ & $10$ \\
\hline
\end{tabular}
\caption{Typical parameters adopted for the calculation of the
encounter rate of cluster stars. The first column denotes the three mass
groups, the
second column contains the adopted mass ranges, $m_{\star}$, while
in the third typical interaction radii, $r_{\mathrm{enc}}$, are listed
for each
mass group. In the last two columns the resulting ``encounter escape
speeds'', $v_{\mathrm{enc}}$, and the ``gravitational focusing
parameter'', $\gamma_{\mathrm{enc}}$, are
noted (see text for definitions of these quantities).}
\label{tab:encounter_rate_parameter_space}
\end{table}
Because the model D2/S2 represents the standard ONC model, which has
been intensively studied, the calculations will be normalised to this
model. All
quantities related to this model will be thus denoted by a ``0'' as
subscript. Adopting the initial velocity dispersion of the model D2/S2,
$\sigma_0
\approx 2.3\,\kms \approx 0.5\,\mathrm{AU/yr}$, using
%
\begin{equation}
4\sigma^2 =
4\sigma_0^2 \left( \frac{ \sigma } { \sigma_0 } \right)^2 \approx
\left( \frac{ \sigma } { \sigma_0 } \right)^2
\frac{\mathrm{AU}^2}{\mathrm{yr}^2} \,,
\nonumber
\end{equation}
%
and the numbers given in Table~\ref{tab:encounter_rate_parameter_space},
Eq.~(\ref{eq:encounter_rate_relative}) can be simplified
to
%
\begin{equation}
\label{eq:encounter_rate_relative_compact}
f_{\mathrm{enc}} = 16 \sqrt{\pi} \rho \sigma
r_{\mathrm{enc}}^2(m_{\star}) \left[ 1 +
\gamma_{\mathrm{enc}}(m_{\star}) \left( \frac{ \sigma_0 } { \sigma }
\right)^2 \right] \,.
\end{equation}
%
An even more compact representation is achieved by considering the
scaling properties of the two families of models: as can be derived from
Eqs.~(\ref{eq:relation_number_density_size}) and
(\ref{eq:relation_velocity_dispersion}), the scaling relations for the
densityscaled models are
$\rho \propto N$ and $\sigma \propto \sqrt{N}$, while $\rho =
\mathrm{const}$ and $\sigma = \mathrm{const}$ is found for the
sizescaled models. Using
these relations, transforming $r_{\mathrm{enc}}(m_{\star})$ to
$\gamma_{\mathrm{enc}}(m_{\star})$ via
Eqs.~(\ref{eq:escape_velocity_stellar_surface})
and (\ref{eq:gravitational_focusing_parameter}), and normalising the
encounter rate to the model D2/S2, $f_{\mathrm{enc}}^{\mathrm{norm}} \equiv
f_{\mathrm{enc}} / f_{\mathrm{enc,0}}$, one obtains
%
\begin{equation}
\label{eq:encounter_rate_normalized}
\begin{split}
% f_{\mathrm{enc}}^{\mathrm{norm}} = & \left(
\frac{r_{\mathrm{enc}}(m_{\star})} {r_{\mathrm{enc},0}(m_{\star})}
\right)^2 \times \\
f_{\mathrm{enc}}^{\mathrm{norm}} = & \left(
\frac{m_{\star}}{m_{\star,0}}
\frac{\gamma_{\mathrm{enc}}(m_{\star,0})}{\gamma_{\mathrm{enc}}(m_{\star})}
\right)^2 \times \\
&
\begin{cases}
\left( \dfrac{\rho}{\rho_0} \right)^{3/2}
\dfrac{1 + \left( \frac{\rho}{\rho_0} \right)^{1}
\gamma_{\mathrm{enc}}(m_{\star})} {1 + \gamma_{\mathrm{enc}}(m_{\star,0})}
& \text{Dmodels} \,, \\[2ex]
%
\phantom{\left( \dfrac{\rho}{\rho_0} \right)^{3/2}}
\dfrac{1 + \gamma_{\mathrm{enc}}(m_{\star})} {1 +
\gamma_{\mathrm{enc}}(m_{\star,0})}
& \text{Smodels} \,.
\end{cases}
\end{split}
\end{equation}
%
The derived relation for the normalised encounter rate predicts very
different scaling relations for the two families of cluster models. Density
scaledmodels are expected to show large variations of the number of
encounters, with a strong dependency on the density for lowmass stars,
i.e. when
$\gamma_{\mathrm{enc}}(m_{\star}) \ll 1$. In contrast, the encounter
rate for the sizescaled models is expected to vary only for the
different mass
groups, but not among different models. For a better overview of the
scaling in terms of numbers,
Table~\ref{tab:encounter_rate_normalized_tabulated}
lists approximated relative encounter rates
$f_{\mathrm{enc}}^{\mathrm{norm}}$, normalised to the lowmass group of
the model
D2/S2. Table~\ref{tab:encounter_rate_normalized_tabulated} demonstrates
that the gravitational focusing parameter plays an important role for the
massive stars. The encounter rates increase dramatically by roughly one
order of magnitude from the low and intermediatemass stars to the
highmass
stars for the model D2/S2. This finding agrees well with the number of
encounters of the ONC model found earlier
\citep[cf.][]{2006A&A...454..811P}.
\begin{table*}
\centering
\begin{tabular}{l*{6}{c}c}
\hline
family of models & \multicolumn{6}{c}{densityscaled} &
sizescaled \\
cluster model & D0 & D1 & D2 & D3 & D4 & D5 &
S0S5 \\
\hline
low mass & 1/5 & 2/5 & 1 & 3 & 7 & 21 & 1 \\
intermediate mass & 1/2 & 1 & 2 & 4 & 9 & 23 & 2 \\
high mass & 5 & 7 & 10 & 15 & 25 & 44 & 10 \\
\hline
\end{tabular}
\caption{Approximate relative encounters rates
$f_{\mathrm{enc}}^{\mathrm{norm}}$ from
Eq.~(\ref{eq:encounter_rate_normalized}) of the
densityscaled and sizescaled models, normalized to the lowmass
group of the model D2/S2.}
\label{tab:encounter_rate_normalized_tabulated}
\end{table*}
In summary, what one would expect from the numerical simulations of the
densityscaled models is a steep increase of the encounter rate $\propto
\rho^{3/2}$ in case of the lowmass stars and a considerably shallower
dependency $\propto \rho^{1/2}$ for the highmass stars. For low densities,
corresponding to low particle numbers, one would expect that the
highmass stars dominate the encounter rate via gravitational focusing,
favouring on
overall scaling $\propto \rho^{1/2}$. In contrast, the sizescaled
models should produce very similar results in terms of encounter rate,
independent
of the specific cluster model.
\begin{figure*}
\centering
\includegraphics[width=0.49\linewidth]{figures/normalized_encounter_frequency__sample_density_scaled__paper.pdf}
\includegraphics[width=0.49\linewidth]{figures/normalized_encounter_frequency__sample_size_scaled__paper.pdf}
\caption{Normalised encounter rate $f_{\mathrm{enc}}^{\mathrm{norm}}$
from the simulations in comparison with the analytical estimate given by
Eq.~(\ref{eq:encounter_rate_normalized}). The filled squares
represent all stars in the Trapezium Cluster region
($R=0.3\,\mathrm{pc}$), the other
open symbols stand for predefined mass groups: highmass, $m \ge
10\,\Msun$ (upward triangles), intermediatemass, $10\,\Msun \ge m \ge
1\,\Msun$
(circles), and lowmass stars, $m \le 1\,\Msun$ (downward
triangles). The lines depict the analytical estimate of the encounter
rate for highmass
(dashed line) and lowmass stars (solid line). The ranges of the
mass groups have been chosen here different from those in previous
figures to
account for the mass regimes of the encounter rate presented in
Table~\ref{tab:encounter_rate_parameter_space}. \emph{Left:} Densityscaled
cluster models. \emph{Right:} Sizescaled cluster models (here the
analytical estimates of the encounter rate for highmass and lowmass
stars are
identical).}
\label{fig:normalized_encounter_frequency}
\end{figure*}
These expectations are in good agreement with the results from the
numerical simulations. We demonstrate this in
Fig.~\ref{fig:normalized_encounter_frequency} via the average encounter
rate of stars of all masses and of the three mass groups, respectively,
normalised in each case to the model D2/S2.
Fig.~\ref{fig:normalized_encounter_frequency}a shows that the encounter
rates of the cluster models D0D2
are scaling roughly as $N^{1/2}$. For higher particle numbers the
distribution becomes more complex. Here the highmass stars show a trend of
\emph{decreasing} encounter rate with particle number. This feature
accounts for the decreasing importance of the highmass stars as
gravitational
foci (for the lower mass stars) and is a consequence of the decreasing
ratio of the mass of the most massive star and the cluster mass.
Accordingly,
the distribution of the encounter rate tends towards the analytical
limit of $N^{3/2}$ for lowmass stars, representing the more frequent
interaction
of lowmass stars with each other in the models D3D5.
Fig.~\ref{fig:normalized_encounter_frequency}b shows that the
sizescaled models are equivalent in their environmental effect on
protoplanetary discs.
In the case of low and intermediate mass stars the presented encounter
rates, normalised to the model S2, are in good agreement with the analytical
estimate, which predicts a constant distribution as a function of
particle number. In contrast, the normalised encounter rate of the
highmass stars
decreases with increasing particle number. This trend shows that the
highmass stars, similarly to the finding for the densityscaled models,
become
less important as gravitational foci for the lowmass stars in clusters
with larger stellar populations.
% _______________________________________________________________________
%
% 
% Conclusions
% 
\section{Conclusion and Discussion}
\label{sec:conclusions}
The influence of different cluster environments on the encounterinduced
discmass loss has been investigated by scaling the size, density and
stellar
number of the basic dynamical model of the ONC. The findings can be
summarized as follows:
%
\begin{enumerate}
\item The discmass loss increases with cluster density but remains
rather unaffected by the size of the stellar population.
\item The density of the ONC itself marks a threshold:
\begin{enumerate}
\item in less dense and less massive clusters it is the massive stars
that dominate the encounterinduced discmass loss via gravitational
focusing of
lowmass stars whereas
\item in denser and more massive clusters the interactions of lowmass
stars with equalmass perturbers play the major role for the removal of disc
mass.
\end{enumerate}
\item In clusters four times sparser than the ONC the effect of
encounters is still apparent.
\end{enumerate}
%
These findings from numerical simulations are well confirmed via
observations. It is widely known that the disc frequency of even very
young clusters
(e.g. NGC~2024) is significantly below 100\,\%
\citep{2001ApJ...553L.153H,2005astro.ph.11083H}. This implies a very
short timescale for the disc
destruction of a part of the cluster population. In fact, stellar
interactions are a potential candidate for a very rapid physical process
and thus
the encounterinduced discmass loss is potentially a vital mechanism
for disc destruction at the earliest stages of cluster evolution.
Observational
evidence for the role of this mechanism has also been provided for the
$\sim$1\,Myr old Orion Nebula Cluster \citep{2008A&A...488..191O}. Moreover,
observations of NGC~2024, ONC, and NGC~3603 confirm also the correlation
of decreasing cluster disc fraction (CDF) with increasing cluster
density and
even provide evidence for a critical density $\rho_{\mathrm{crit}}
\approx \rho_{\mathrm{ONC}}$. A compilation of observational data shows
that these
similar aged ($\sim$1\,Myr) clusters with peak densities of roughly
$10^3\,\mathrm{pc}^{3}$, $10^4\,\mathrm{pc}^{3}$, and
$10^5\,\mathrm{pc}^{3}$,
respectively, show CDFs of $85\,\%$, $80\,\%$, and $40\,\%$
\citep{2000AJ....120.1396H,2000AJ....120.3162L,2004AJ....128..765S}. The
fact that
NGC~2024 is sparser than the ONC but has a similar CDF is in good
agreement with our simulations.
\begin{figure}
\centering
\includegraphics[width=1.0\linewidth]{figures/ONC_range.pdf}
\caption{Cluster density as a function of cluster size for clusters
more massive than $10^3\,\Msun$ and embedded clusters with more than 200
observed members from \cite{2009A&A...498L..37P}. The parameter
space covered by the simulations in this work is indicated by the large pink
cross centered on the ONC.}
\label{fig:cluster_sequences__simulated_parameter_space}
\end{figure}
Our results have several important implications for the general picture
of star and cluster formation. Very recently, \cite{2009A&A...498L..37P} has
shown that there exist two cluster sequences evolving in time along
predefined tracks in the densityradius plane, the ``leaky'' and the
``starburst'' clusters. The simulations performed in the present
investigation cover the parameter space of the ``leaky'' clusters in
their embedded
stage (see Fig.~\ref{fig:cluster_sequences__simulated_parameter_space}).
Comparison with our findings shows that at the earliest evolutionary stage
leaky clusters have densities above the critical density. Hence in leaky
clusters stardisc systems are initially efficiently destroyed via
encounters
that occur preferentially between lowmass stars. The ONC corresponds to
an intermediate stage in the embedded phase of leaky clusters with the
transition towards a preferred discmass loss of highmass stars via
gravitational focusing of lowmass stars. At the final stage of the embedded
phase the encounterinduced discmass loss in leaky clusters ceases.
Gravitational focusing by highmass stars may still affect single
objects yet at
this age is most probably exceeded by other discdestructive processes
like photoevaporation or planet formation. The effects in starburst
clusters
would be similar yet much more pronounced. In case of the Arches cluster
one could expect stellar encounters to destroy the discs of most of the low
and highmass stars in several hundred thousand years. Combining our
results with the finding of \cite{2003ARA&A..41...57L} that most stars
are born
in clusters, it becomes evident that a significant fraction of all stars
must have been affected by stellar encounters at the early evolutionary
stages of their hosting environment.
The application of our results to the dynamics of embedded clusters 
though obtained from simulations that do not contain gas  is justified for
three reasons:
\begin{enumerate}
\item Rather than simulating the \emph{evolution} of leaky clusters
(which would explicitly require the treatment of gas), we use our
cluster models
to map certain \emph{evolutionary stages} of the sequence of leaky
clusters in terms of "dynamical snapshots". The dynamical effects of these
numerical models are then used to estimate the effect of encounters in
the observed clusters at their current dynamical state.
\item The effect of gas in an embedded cluster is to lower the frequency
of close encounters (due to the smoother cluster potential), yet unless the
gas mass is dominating cluster dynamics  as is not the case for the
only partially embedded leaky clusters shown in
Fig.~\ref{fig:cluster_sequences__simulated_parameter_space} (e.g.
NGC~2024)  the effect is minor.
\item Gas expulsion causes the clusters to expand and thus their density
to decrease much faster than in our simulations. Consequently, when mapping
our results to the current dynamical state, we \emph{underestimate}
the initial cluster density and thus the effect of encounters in the early
evolutionary phases. Hence, our results are least accurate for older
clusters, yet well applicable for the young evolutionary stages were
encounters
have the largest effect.
\end{enumerate}
The effect of the encounterinduced discmass loss in the early
evolution of stellar systems has important implications for the
formation and
evolution of planets. From our findings we conclude that the probability
to find planets around intermediate mass stars is highest. While the high
initial densities of leaky clusters imply that planets around lowmass
stars are expected to be less frequent, hardly any are expected in
starburst
clusters. Independent of the cluster environment, planet formation
around highmass stars seems to be completely hindered via interactions
with other
cluster members.
% _______________________________________________________________________
%
% 
% Acknowledgements
% 
\begin{acknowledgements}
We thank the anonymous referee and the editor for careful reading and
very useful comments and suggestions which improved this work. C. Olczak
appreciates fruitful discussions with S. Portegies Zwart concerning
the analytical estimates and scaling relations. We also thank R. Spurzem for
providing the {\textsc{\mbox{nbody6\raise.2ex\hbox{\tiny{++}}}}} code
for the cluster simulations. Simulations were partly performed at the
J\"{u}lich Supercomputing Centre (JSC), Research Centre J\"{u}lich,
Project HKU14. We are grateful for the excellent support by the JSC Dispatch
team.
\end{acknowledgements}
% _______________________________________________________________________
%
% 
% Bibliography
% 
% for the bibliography, at the end
\bibliographystyle{aa}
\bibliography{references}
% _______________________________________________________________________
%
% 
% Appendix
% 
\appendix
% _______________________________________________________________________
%
\section{Determination of boundaries of mass groups}
\label{app:star_cluster_models:mass_groups}
Boundaries of mass groups of low, intermediate and highmass stars
have been determined individually for different sizes of stellar
populations on
the basis of the IMF of \citet[][see also
Eq.~\ref{eq:kroupa_imf}]{2001MNRAS.322..231K}. The derivation involves
the requirement for the three mass
ranges to be equidistant in logarithmic space, weighted by the slope of
the IMF (of each mass range). The weighting accounts for the steepness
of the
slope in the highmass regime which would otherwise cause a very
sparsely populated group of highmass stars.
In the case of a lower mass cutoff at $m_0 = 0.08\,\Msun$, and an upper
mass limit $m_3$, the IMF is characterised by just two different slopes,
$\alpha_1=1.3$ in the range $m_0 \le m < 0.50\,\Msun$, and
$\alpha_2=2.3$ in the range $0.50\,\Msun \le m \le m_3$. Because the
break in the slope of
the IMF at the critical mass $m_c^{\mathrm{br}} = 0.5\,\Msun$ does not
necessarily coincide with one of the boundaries of the mass ranges, the
cases
$m_1 < m_c^{\mathrm{br}}$ and $m_1 \ge m_c^{\mathrm{br}}$ have to be
differentiated. Though from the theoretical point of view the same
differentiation would be required for the higher mass boundary $m_2$,
this is not relevant for the stellar systems in the focus of the present
work. The four mass ranges, $m_k$, $k=0,..,3$, and the two slopes,
$\alpha_k$, $k=1,2$, are then interrelated as follows:
%
\begin{equation}
\nonumber
\begin{split}
(m_1& \ge m_c^{\mathrm{br}}) \wedge (m_2 \ge m_c^{\mathrm{br}}): \\
& \phantom{\equiv~} (\log{m_1}  \log{m_c^{\mathrm{br}}})
\alpha_2^{1} + (\log{m_c^{\mathrm{br}}}  \log{m_0}) \alpha_1^{1} \\
& \equiv (\log{m_2}  \log{m_1})\alpha_2^{1} \\
& \equiv (\log{m_3}  \log{m_2})\alpha_2^{1} \,,\\
\\
(m_1& < m_c^{\mathrm{br}}) \wedge (m_2 \ge m_c^{\mathrm{br}}): \\
& \phantom{\equiv~} (\log{m_1}  \log{m_0}) \alpha_1^{1} \\
& \equiv (\log{m_2}  \log{m_c^{\mathrm{br}}}) \alpha_2^{1} +
(\log{m_c^{\mathrm{br}}}  \log{m_1}) \alpha_1^{1} \\
& \equiv (\log{m_3}  \log{m_2}) \alpha_2^{1} \,.\\
\end{split}
\end{equation}
%
\\
Solving these equations, and substituting
%
\begin{equation}
\nonumber
\begin{aligned}
\alpha_{12}& \equiv \alpha_1 \alpha_2^{1}, \\
\alpha_{21}& \equiv \alpha_2 \alpha_1^{1}, \\
\Gamma_{12}& \equiv 1  \alpha_{12}, \\
\Gamma_{21}& \equiv 1  \alpha_{21},
\end{aligned}
\end{equation}
%
one obtains
%
\begin{equation}
\label{eq:mass_groups_solution}
\begin{split}
(m_1& \ge m_c^{\mathrm{br}}) \wedge (m_2 \ge m_c^{\mathrm{br}}): \\
& \log{m_1} = \tfrac{1}{3} \left[ \log{m_3} + 2 \Gamma_{21}
\log{m_c^{\mathrm{br}}} + 2 \alpha_{21} \log{m_0} \right] \,, \\
& \log{m_2} = \tfrac{1}{3} \left[ 2 \log{m_3} + \Gamma_{21}
\log{m_c^{\mathrm{br}}} + \alpha_{21} \log{m_0} \right] \,, \\
\\
(m_1& < m_c^{\mathrm{br}}) \wedge (m_2 \ge m_c^{\mathrm{br}}): \\
& \log{m_1} = \tfrac{1}{3} \left[ \alpha_{12} \log{m_3} +
\Gamma_{12} \log{m_c^{\mathrm{br}}} + 2 \log{m_0} \right] \,, \\
& \log{m_2} = \tfrac{1}{3} \left[ 2 \log{m_3} +
\Gamma_{21} \log{m_c^{\mathrm{br}}} + \alpha_{21} \log{m_0} \right] \,. \\
\end{split}
\end{equation}
%
\\
The choice of the appropriate solution is determined by the upper mass
limit $m_3$. For this purpose the ``critical maximum mass''
$m_c^{\mathrm{max}}$,
%
\begin{equation}
\nonumber
m_c^{\mathrm{max}} = \log^{1}{[ ( 1 + 2 \alpha_{12} )
\log{m_c^{\mathrm{br}}}  2 \alpha_{12} \log{m_0} ]} \,,
\end{equation}
%
is estimated from Eq.~(\ref{eq:mass_groups_solution}) and $m_1 \equiv
m_c^{\mathrm{br}}$. Consequently, the following relations hold:
%
\begin{equation}
\nonumber
\begin{aligned}
m_3 < m_c^{\mathrm{max}} \quad & \Longrightarrow \quad m_1 \ge
m_c^{\mathrm{br}} \,, \\
m_3 \ge m_c^{\mathrm{max}} \quad & \Longrightarrow \quad m_1 <
m_c^{\mathrm{br}} \,.
\end{aligned}
\end{equation}
%
With the given values of the parameters $m_0$, $\alpha_1$, and
$\alpha_2$ one finds
%
\begin{equation}
\nonumber
m_c^{\mathrm{max}} \approx 3.97\,\Msun \,.
\end{equation}
%
The derived mass boundaries, $m_k$, $k=0,..,3$, for each cluster of the
families of models are presented in Table~\ref{tab:mass_groups}.
%
\begin{table*}
\centering
\begin{tabular}{l*6{c}}
\hline
\hline
& 1000 & 2000 & 4000 & 8000 & 16000 & 32000 \\
\hline
$m_0 [\Msun]$ &
$8.00 \cdot 10^{2}$ & $8.00 \cdot 10^{2}$ & $8.00 \cdot
10^{2}$ & $8.00 \cdot 10^{2}$ & $8.00 \cdot 10^{2}$ & $8.00
\cdot 10^{2}$ \\
$m_1 [\Msun]$ &
$3.30 \cdot 10^{1}$ & $3.54 \cdot 10^{1}$ & $3.82 \cdot
10^{1}$ & $3.95 \cdot 10^{1}$ & $4.15 \cdot 10^{1}$ & $4.24
\cdot 10^{1}$ \\
$m_2 [\Msun]$ &
$2.94$ & $3.78$ &
$4.95$ & $5.58$ & $6.61$ &
$7.14$ \\
$m_3 [\Msun]$ &
$1.47 \cdot 10^{2}$ & $1.47 \cdot 10^{2}$ & $1.48 \cdot
10^{2}$ & $1.48 \cdot 10^{2}$ & $1.50 \cdot 10^{2}$ & $1.50
\cdot 10^{2}$ \\
\hline
\end{tabular}
\caption{Boundaries of the three mass groups of low, intermediate,
and highmass stars of cluster models with 1000,
2000, 4000, 8000, 16000, and 32000 particles.}
\label{tab:mass_groups}
\end{table*}
% density
% n1000 8.00E02 3.30E01 2.94E+00 1.47E+02
% n2000 8.00E02 3.54E01 3.78E+00 1.47E+02
% n4000 8.00E02 3.82E01 4.95E+00 1.48E+02
% n8000 8.00E02 3.95E01 5.58E+00 1.48E+02
% n16000 8.00E02 4.15E01 6.61E+00 1.50E+02
% size
% n1000 8.00E02 3.30E01 2.94E+00 1.47E+02
% n2000 8.00E02 3.54E01 3.78E+00 1.47E+02
% n4000 8.00E02 3.82E01 4.95E+00 1.48E+02
% n8000 8.00E02 3.95E01 5.58E+00 1.48E+02
% n16000 8.00E02 4.17E01 6.72E+00 1.50E+02
% _______________________________________________________________________
%
% 
% End of document
% 
\end{document}
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Dr. Christoph Olczak olczak@ph1.unikoeln.de
currently:
The Kavli Institute for Astronomy and Astrophysics, Peking University
Yi He Yuan Lu 5, Hai Dian Qu
Beijing 100871
P. R. China
University of Cologne
I. Physics Institute
Zuelpicher Str.77
50937 Koeln
Germany
Phone: +49 (0)221 470 6157
