\section{Introduction} \label{sec:intro}
{For the past} decade, exoplanetary systems are being discovered at a spectacular rate \citep[e.g.][]{mayor04,borucki11}. Driven by these discoveries, there is an increasing interest in the global properties of planetary systems, from the epoch of their formation in protoplanetary discs ({PPDs}) to their long-term stability. While there is a natural focus on {internal processes} that govern the {evolution} of such systems {\citep[e.g.][]{lee03,dullemond05,gorti09,morbidelli09,brasser09,blum10,williams11}}, it is clear that {not all planetary systems form in isolation and} environmental effects should be considered as well. Particularly, theoretical studies show that {external} photoevaporation {\citep{scally01,adams04, adams06, fatuzzo08}} and dynamical interactions {\citep{bonnell01,pfalzner05,olczak06,olczak10,spurzem09,lestrade11,dukes12,parker12,bate12}} can lead to the truncation of {PPDs} and planetary systems.

{While the external photoevaporation of PPDs has been studied observationally \citep{odell93,robberto08,rigliaco09,mann10}, there is no conclusive evidence of dynamical effects \citep{eisner06,olczak08,reche09}.} In part, this {is likely} due to the relatively short lifetimes of {PPDs} \citep[up to $\sim8$~Myr, e.g.][]{haisch01,hernandez08,ercolano11,smith12} compared to the time it takes stellar encounters to have an observable effect on the disc {population \citep[$\sim0.1$--$1$~Gyr, e.g.][]{adams10}. Previous observational studies on the Orion Nebula Cluster {(ONC)} \citep[e.g.][]{eisner06,olczak08} did aim to find traces of dynamical interactions in the population of PPDs, but lacked a sufficient number of sources and/or suffered from uncertainties on the disc mass measurements.}

We address the problem statistically by considering the sizes of {PPDs} as a function of {their} ambient stellar density, using samples of {PPDs} and young stellar objects {(YSOs)} from the {latest infrared surveys}. If stellar encounters truncate {PPDs} by tidally stripping the outskirts of the discs {\citep[e.g.][]{clarke93,heller95,hall96}}, {this should be observable} above some characteristic {ambient stellar} density {\citep[$\sim10^3~{\rm pc}^{-3}$][]{adams10}}, because the {encounter rate} increases with density {\citep[e.g.][Eq.~7-61]{binney87}}. In this Letter, we find model-independent evidence of {a change in the {PPD} size distribution for ambient stellar surface densities $\Sigma\mathrel{\copy\simgreatbox} 10^{3.5}~{\rm pc}^{-2}$} at the $>97\%$ confidence level.




\section{{Protoplanetary} discs and their environment} \label{sec:method}

\subsection{Data selection} \label{sec:data}
To verify whether a relationship exists between the sizes of {PPD}s and their {ambient stellar surface density, $\Sigma$, we} combine existing catalogues of {PPD}s and young stellar objects {in star-forming regions (SFRs) of} the solar neighbourhood. 

The data for the {PPD}s is taken from {circumstellardisks.org} {(Karl Stapelfeldt, NASA/JPL)}. This catalogue gathers resolved {PPD}s that have been confirmed and described in the literature. {If a {PPD} is resolved} in different wavelengths {\citep[probing different parts of the disc, see e.g.][]{lada06}}, the catalogue lists the largest {measured} diameter, implying that the disc radii used in this work are lower limits. {About $75\%$ of the {PPD} radii in our final sample {(see below)} are measured at wavelengths around $1~\mu{\rm m}$, with only $\sim25\%$ of the {PPD}s (all in low-density regions) being observed at mm wavelengths {(see Appendix~\ref{sec:app0})}. The {catalogue contains} an estimate for how well-resolved each disc is by listing the number of diffraction-limited beams that fit within its diameter. We only consider the discs for which this value is greater than unity. {This provides us with $133$ {PPD} sources from which we exclude those {whose host star is} classified as a main sequence star, weak-line T-Tauri star, {or} Class~0 YSO. Sources at distances $>500$~pc {(which covers all our YSOs) are also excluded}. These criteria reduce the sample to a total of $101$ sources.

To estimate the local ambient surface density of each {PPD} source, we use publicly available near-infrared data of nearby SFRs (see Table~\ref{tab:regions} in Appendix~\ref{sec:app0}). The ambient surface density of stars around each {PPD} is estimated as in \citet{casertano85} {-- see Appendix~\ref{sec:app0} for a detailed explanation}. The thus-obtained angular {ambient} surface densities are converted to {physical ambient stellar} surface densities, $\Sigma$, using the distances listed in Table~\ref{tab:regions}. In cases where the listed {PPD} distance differs from the distance to the region that it is a member of, we adjust the {PPD} distance and size. Since a low surface density can be due to incomplete YSO coverage, any discs with $\Sigma<0.1~{\rm pc}^{-2}$ are omitted from our analysis. {Moreover, the minimum {PPD} radius that can be resolved increases with distance and hence introduces a distance-dependent selection bias. To avoid this, we exclude any {PPD}s smaller than the smallest radius ($\sim50$~AU) that is resolved in the most distant region of our sample, which is the Orion Nebula Cluster (ONC). The final sample thus contains $67$ sources (see Table~\ref{tab:sources} in Appendix~\ref{sec:app0})}.}{ Completeness does not affect the densities because the surveys of Table~\ref{tab:regions} are complete down to the hydrogen-burning limit.}

\subsection{A simple theoretical estimate for the truncation of discs} \label{sec:theory}
To interpret the data, we include a rough theoretical estimate for the expected truncation radii of {PPD}s as a function {of $\Sigma$. This is obtained by combining the truncation induced by each particular encounter with the stellar encounter rate}. The derivation is presented in detail in {Appendix~\ref{sec:app1}.

We use the numerical simulations of disc perturbations in clustered environments by \citet{olczak06} to obtain the disc radius as a function of the encounter parameters. {We convert their expressions for the disc mass loss to a radial truncation assuming that it occurs by stripping the outer disc layers to the Lagrangian point between both stars, and adopting a PPD surface density profile $\Sigma_{\rm d}\propto r^{-1}$ \citep{olczak06}}. Under these assumptions, {we write for the upper limit to the disc radius}
\begin{equation}
\label{eq:rd}
r_{\rm d}(r_{\rm p},m_1,m_2)=\frac{r_{\rm p}}{\sqrt{m_2/m_1}+1} ,
\end{equation}
where $r_{\rm p}$ is the pericentre radius at which the perturber passes, $m_1$ the mass of the perturbed system, and $m_2$ the mass of the {perturber.} This approximation is validated in Appendix~\ref{sec:app1}.

The encounter radius $r_{\rm p}$ is obtained from the impact parameter $b$, encounter velocity $v$, and masses $m_1$ and $m_2$ by accounting for gravitational focussing {(see Appendix~\ref{sec:app1})}. The masses are assumed to follow a \citet{salpeter55} type initial mass function in the range 0.1--100~\mbox{M$_\odot$}, and the velocity distribution is taken to be Maxwellian {with a} velocity dispersion of $\sigma=2$~km~s$^{-1}$, as is typical for SFRs \citep{hillenbrand98,covey06}. This enables the derivation of the encounter rate as a function of $b$ and $v$ \citep{binney87}, which for a given age provides the total number of encounters $n$. Because encounters with pericentres at inclination angles $\theta\simgreat45^\circ$ with respect to the disc plane affect the disc only mildly, about $30\%$ of the $n$ encounters lead to the disc truncation described by Eq.~\ref{eq:rd} \citep{pfalzner05}. We use the probability distribution functions (PDFs) for $b$, $v$, and $m_2$ to calculate the PDF of the `most disruptive' encounter, i.e.~the parameter set that gives the smallest disc size, according {to \citet[Eq.~A5]{maschberger08}. We} then integrate the product of the disc radius $r_{\rm d}$, the PDF of the `most disruptive encounter', and the mass PDF of the perturbed object $m_1$ to obtain the expected disc truncation radius $r_{\rm tr}$. {To} compare the theoretical estimate to the observations, we relate the stellar volume density $\nu$ to the surface density $\Sigma$ as $\nu=\Sigma/2R$, where $R\approx 2$~pc is a typical radius for the SFRs in our sample \citep{hillenbrand98,evans09}. 

\section{Results}\label{sec:result}
\subsection{Evidence for environmental effects} \label{sec:theplot}
\begin{figure}
\center\resizebox{9cm}{!}{\includegraphics{theplot.ps}}\\
\caption[]{\label{fig:sizes}
      {{\it Top panel}: Protoplanetary disc (PPD) radius distribution versus ambient stellar density. Squares, triangles, stars, and crosses correspond to ONC sources, T-Tauri stars, Herbig stars, and YSOs, respectively. Dotted lines represent the theoretical truncation for ages of $0.3$ (top) and $1$~Myr (bottom). {\it Bottom panel}: KS test $p_{\rm KS}$-values for the PPD radii at that density to be consistent with the sample at lower densities, using $\{6,7,8\}$ (triangles, squares, diamonds) objects per density bin (see text). The grey thick line shows the $p_{\rm KS}$-value for ONC PPDs only when dividing the subsample in two at the indicated density. The horizontal dashed line and grey area indicate the median $p_{\rm KS}$ and its $1\sigma$ dispersion retrieved from 30,000 Monte Carlo experiments to verify the significance of the results (see text). Dotted lines mark $95\%$ (top) and $99\%$ (bottom) confidence levels.
                 }}
\end{figure}
The upper panel of Fig.~\ref{fig:sizes} {shows the observed {PPD}} radii  {versus $\Sigma$. The} distribution is} relatively insensitive to the ambient density {until $\Sigma\sim10^{3.5}~{\rm pc}^{-2}$, where it} {appears truncated at large radii. This is consistent with the simple theoretical approximation from Sect.~\ref{sec:theory}, which predicts a truncation at these densities for ages between $0.3$ and $1$ Myr} ({see} Fig.~\ref{fig:sizes}). The affected {PPD}s are all {in the ONC. If the truncation} is interpreted as being due to dynamical interactions, the theoretical curves {suggest that youngest sources in the ONC have ages} of $\sim0.6$~Myr, which is reasonably consistent with observations \citep{palla99,dario10,jeffries11}. {However, this is not a unique explanation, since the truncation might also be due to external photoevaporation {by nearby massive stars} \citep[e.g.][]{clarke07}. It is also important to keep in mind that some fraction of the {PPD}s at this projected surface density will actually reside in a region of lower volume density, either behind or in front of the high-density core of the ONC.} The distribution also shows some evidence of a density-independent {upper limit to the {PPD} radius} of $\sim10^3$~AU, which could be related to binarity \citep{artymowicz94,kraus12} or {be intrinsic} \citep{basu98}.

To test the statistical significance of the {change {in} the radius distribution,} we perform a Kolmogrov-Smirnov (KS) test, {which }is theory-independent {and} thus insensitive to model assumptions. {It gives the {probability $p_{\rm KS}$ that two samples were drawn from the same parent distribution}. Starting at the high-density end, we first divide the sample in density bins {of} $\{6,7,8\}$ objects per bin, which {represents a balance between good statistics} and enough bins to resolve the regime where the radius distribution changes}. The KS test is then carried out {comparing radii} in a bin at density $\Sigma_{\rm bin}$ with {those at lower densities (i.e.~$10^{-1}~{\rm pc}^{-2}<\Sigma<\Sigma_{\rm bin}$)}. We do not include bins at $\Sigma_{\rm bin}<10^1~{\rm pc}^{-2}$ to avoid low-number statistics in the reference sample. {The bottom panel of Fig.~\ref{fig:sizes} shows the results of the test}. At intermediate densities, {all} KS tests give high {$p_{\rm KS}$-values, but for densities higher than $\Sigma\simgreat10^{3.5}~{\rm pc}^{-2}$ there is} a pronounced drop. As such, the KS test yields {a detection} of a change in the {PPD} radius distribution {at the $>97\%$ confidence level}. {Note that this does not change when excluding the largest {PPD} in the sample (at $\Sigma\sim200~{\rm pc}^{-2}$). {The result} also holds within the ONC only {(grey} line in Fig.~\ref{fig:sizes}), when} dividing the ONC subsample in two at each density and running a KS test for the radii at both sides of the separation. {To check the result, we performed 30,000 Monte Carlo experiments in which the KS test was applied in the same way to distributions of randomly paired radii and surface densities, i.e. erasing any possible correlation. Figure~\ref{fig:sizes} indicates the resulting median $p_{\rm KS}$ and its $1\sigma$ dispersion, showing that the results obtained for the original sample are unlikely to be due to the adopted statistical method.}

\subsection{Implications for habitable zone occupancy lifetimes}\label{sec:hz}
\begin{figure}
\center\resizebox{8.5cm}{!}{\includegraphics{hz.ps}}\\
\caption[]{\label{fig:hz}
      {Habitable zone lifetime} as a function of stellar mass {for} different ambient stellar densities. The dotted line indicates the total lifetime of the host star, and the diamond indicates Earth. 
}
\end{figure}
If the drop of the $p_{\rm KS}$-value at high densities is indeed caused by dynamical truncation, then the model can be used to give the maximum {time} during which the {habitable zone (HZ)} can host a planet or a {PPD} {(`HZ lifetime')}. {We calculate} $r_{\rm tr}$ as a function of $\Sigma$ and $\tau$ as in Fig.~\ref{fig:sizes}, but without averaging over the host stellar mass to retain the mass dependence. We then compare it to the inner radius of the HZ, $r_{\rm HZ}^{\rm in}$, and determine until which age they overlap, {as a function of mass and ambient density}. The radius $r_{\rm HZ}^{\rm in}$ {depends on} the required radiative equilibrium temperature $T_{\rm eq}$, on the {stellar} luminosity $L$ and on the {properties} of the planet \citep[i.e.~$f$, a proxy for atmospheric thermal circulation, and the Bond albedo $A$, see][]{kasting93,tarter07}:
\begin{equation}
\label{eq:rhzmin}
r_{\rm{HZ}}^{\rm in}=\frac{1}{2} \sqrt{\frac{L f (1-A)}{4 \pi \sigma_{\rm{T}} T_{\rm{eq}}^4}} ,
\end{equation}
where $\sigma_{\rm{T}}$ is the Stefan-Boltzman constant. {Following \cite{borucki11}, we calculate $r_{\rm HZ}^{\rm in}$ for an Earth-like planet using $f=1$, $A=0.3$, and $T_{\rm eq}=307~K$, which is the maximum temperature that allows for the presence of liquid water when accounting for the greenhouse effect.} The luminosity is taken from stellar evolution models at solar metallicity \citep{marigo08}. {The results are} shown in Figure~\ref{fig:hz} for ambient densities of $\Sigma=\{10^2,10^3,10^4,10^5\}~{\rm pc}^{-2}$. The HZ lifetime decreases with density and stellar mass, due to the enhanced encounter rate and the large $r_{\rm HZ}^{\rm in}$, respectively. {These estimates for the HZ lifetime hold both for {PPD}s and planetary systems, {since} dynamical interactions {would have comparable effects in both cases} {\citep[e.g.][]{olczak06,parker12}}.} 


Figure~\ref{fig:hz} shows that, based on the Earth's existence alone, the solar system cannot have formed in a dense ($\Sigma>10^3~{\rm pc}^{-2}$) {environment}, unless the ambient density decreased on a short ($\tau\mathrel{\copy\simlessbox} 100$~Myr) timescale. Conversely, meteoritic evidence indicates that the young solar system must have endured nearby supernovae \citep[e.g.][]{cameron77}, which provides a lower limit to the product {$\Sigma\tau\simgreat10^{3.8}~{\rm Myr}~{\rm pc}^{-2}$} \citep[see the review by][]{adams10}. {A plausible scenario is thus} that the solar system formed in a massive {($\sim10^4~\mbox{M$_\odot$}$, $\Sigma\sim10^3~{\rm pc}^{-2}$)}, but unbound association, which dispersed on a short ($\tau\sim10$~Myr) timescale \citep[see e.g.][although they refer to such a system as a `cluster']{dukes12}. Our results seem to disagree with \citet{eisner06} who derive the disc fraction {in the ONC} and find no evidence of disc truncations. {However, their conclusion may result from low-number statistics, and PPD mass estimates are more uncertain than radius measurements. On the other hand, our results agree with the studies of \citet{bonnell01} and \citet{spurzem09}, and would explain} why no planets have been found in the globular clusters {47~Tuc and} NGC~6397 \citep{gilliland00,nascimbeni12}, where $\tau\Sigma\sim10^8~{\rm Myr}~{\rm pc}^{-2}$ {within the half-mass radius}. This implies such a high number of encounters that it is {improbable} that {\textit{any}} bound planets survived, {most of them likely to have escaped the cluster due to two-body relaxation \citep[e.g.][]{kruijssen09c}}.

{We} note that our theoretical estimates are conservative and provide upper limits to the disc sizes, because (1) we do not account for potentially higher ambient densities in the past \citep[e.g.][]{bastian08b}, (2) {we neglect the presence of massive stars at earlier ages of the SFRs}, (3) we only consider the most disruptive encounters and {ignore} the cumulative effect of weak perturbations. Figure~\ref{fig:hz} thus provides strict upper limits to the HZ lifetimes.


\section{Further observational avenues} \label{sec:outlook}
We {present evidence} for a change in the {PPD} radius distribution at ambient stellar densities of $\Sigma>10^{3.5}~{\rm pc}^{-2}$ {at the $>97\%$ confidence level}, in line with the expected range due to close encounters with other stars {on a $\sim1$~Myr timescale}. These densities are only reached in the  densest parts of the ONC, which {is not only consistent with the detection of reduced {PPD} masses in the centre of the region \citep{mann10}, but also with} {studies concluding} that encounters are not important in the ONC as a whole \citep[e.g.][]{scally01}. Our results demonstrate that {the} stellar environment {can be} an important factor in setting the habitability of planetary systems. {For instance, the existence of unbound, free-floating planets \citep[see e.g.][]{bihain09,sumi11,strigari12} is a natural outcome of our results}. However, a ubiquity of Earth-like planets in the HZ {of stars remains likely} because a large fraction {($\sim90\%$)} of stars forms in unbound associations {(see \citealt{kruijssen12d} for a recent review, and observational references therein)}, of which the density quickly decreases after their formation.

\begin{figure}
\center\resizebox{8.5cm}{!}{\includegraphics{mdf2.ps}}\\
\caption[]{\label{fig:mdf}
      {{Evolution of the disc mass function (DMF) due to dynamical encounters at an age of 3 Myr for densities $\{2.25\times10^4, 2.5\times10^3, 5\times10^2\}~{\rm pc}^{-2}$ \textbf{(solid, dashed, dash-dotted)}, sampling NGC~3603 at \textbf{radius intervals of $\{0"-5",16.5"-43.5",46.5"-73.5"\}$. The dotted curve} shows the initial DMF, and the vertical dotted line indicates the $4\sigma$ sensitivity limit of ALMA Cycle~1 (30~min at 1.3~mm) at the distance of NGC~3603.}
                 }}
\end{figure}
To verify our results, more observations of {PPD}s in clustered environments are desirable. {A} fruitful approach would be to probe {PPD} truncations using disc mass measurements, {which would provide much larger samples of {PPDs. Figure~\ref{fig:mdf} shows the effect of (only) dynamical encounters on the disc mass function (DMF) using the disc mass loss description from \citet{olczak06} (see Appendix~\ref{sec:app2})}. {This} will be easily observable in dense and young stellar clusters with ALMA. While the full ALMA array will be able to directly measure disc sizes, we predict that, already from Early Science Cycle 1, the sensitivity will be sufficient to detect the variation of the DMF caused by the truncation (see Fig.~\ref{fig:mdf}), which would verify the result of Fig.~\ref{fig:sizes}.}


\begin{acknowledgements}
{We are grateful to the anonymous referee for a thoughtful and constructive report.} We thank Cathie Clarke, Michael Meyer, and Christoph Olczak for insightful comments on the manuscript, and David Jewitt, Thomas Maschberger, and Niels Oppermann for helpful discussions. NB is supported by the DFG cluster of excellence `Origin and Structure of the Universe' and HC by the Millennium Science Initiative, Chilean Ministry of Economy, Nucleus P10-022-F. 
\end{acknowledgements}


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