Issue 
A&A
Volume 538, February 2012



Article Number  A10  
Number of page(s)  16  
Section  Interstellar and circumstellar matter  
DOI  https://doi.org/10.1051/00046361/201117682  
Published online  26 January 2012 
Discmass distribution in stardisc encounters^{⋆}
^{1} MaxPlanckInstitut für Radioastronomie (MPIfR), Auf dem Hügel 69, 53121 Bonn, Germany
email: mstein@mpifrbonn.mpg.de
^{2} I. Physikalisches Institut, Universität zu Köln, Zülpicher Str. 77, 50937 Köln, Germany
^{3} Astronomisches RechenInstitut (ARI), Zentrum für Astronomie der Universität Heidelberg (ZAH) Mönchhofstr. 1214, 69120 Heidelberg, Germany
^{4} MaxPlanckInstitut für Astronomie (MPIA), Königstuhl 17, 69117 Heidelberg, Germany
^{5} National Astronomical Observatories of China (NAOC), Chinese Academy of Sciences (CAS), 20A Datun Lu, Chaoyang District, Beijing 100012, PR China
Received: 11 July 2011
Accepted: 10 November 2011
Aims. Investigations of stellar encounters in cluster environments have demonstrated their potential influence on the mass and angular momentum of protoplanetary discs around young stars. We investigated how far the initial surface density in the disc surrounding a young star influences the outcome of an encounter.
Methods. The numerical method applied here allows us to determine the mass and angular momentum losses in an encounter for any initial discmass distribution. On the basis of a powerlaw ansatz for the surface density, Σ(r) ∝ r^{ − p}, we perform a parameter study of stardisc encounters with different initial discmass distributions using Nbody simulations.
Results. We demonstrate that the shape of the discmass distribution has a significant impact on the quantity of the discmass and angular momentum losses in stardisc encounters. In particular, the results are most sensitive to how the outer parts of the disc are perturbed by highmass stars. In contrast, discpenetrating encounters lead more or less independently of the discmass distribution always to large losses. However, maximum losses are generally obtained for initially flat distributed disc material. Based on a parameter study, a fit formula is derived, describing how the relative mass and angular momentum loss depend on the initial discmass distribution index p. Encounters generally lead to a steepening of the density profile of the disc. The resulting profiles can have a r^{2}dependence or an even steeper one that is independent of the initial distribution of the disc material.
Conclusions. From observations, the initial density distribution in discs remains unconstrained, hence the strong dependence on the initial density distribution that we find here might require a revision of the effect of encounters in young stellar clusters. The steep surface density distributions induced by some encounters might be a prerequisite to the formation of planetary systems similar to our own Solar System.
Key words: methods: numerical / protoplanetary disks / circumstellar matter
Appendices A and B are available in electronic form at http://www.aanda.org
© ESO, 2012
1. Introduction
There is increasing observational evidence that most, if not all, stars are initially surrounded by a circumstellar disc. For example, Lada et al. (2000) found by examining the Lband excess of young stars in the Trapezium cluster that a fraction of 80 − 85% of all stars are surrounded by discs. With time, these protoplanetary discs become depleted of gas and dust and eventually disappear (Haisch et al. 2001; Hillenbrand 2002; SiciliaAguilar et al. 2006; Hernández et al. 2007; Currie et al. 2008; Hernández et al. 2008; Mamajek 2009; Massi et al. 2010). It is currently unclear which of a variety of physical mechanisms dominates this evolutionary processes, among them internal processes such as viscous torques (e.g. Shu et al. 1987), turbulent effects (Klahr & Bodenheimer 2003), and magnetic fields (Balbus & Hawley 2002), while photoevaporation (Scally & Clarke 2001; Clarke et al. 2001; Matsuyama et al. 2003; Johnstone et al. 2004; Alexander et al. 2005, 2006; Ercolano et al. 2008; Drake et al. 2009; Gorti & Hollenbach 2009) and tidal torques (Heller 1993; Clarke & Pringle 1993; Ostriker 1994; Heller 1995; Hall et al. 1996; Hall 1997; Larwood 1997; Boffin et al. 1998; Pfalzner 2004; Pfalzner et al. 2005a; Moeckel & Bally 2006; Kley et al. 2008) are candidates for external disc destruction processes.
The focus of the present numerical investigation is the effect of gravitational stardisc interactions on the discmass distribution and, therefore, the mass and angular momentum of the discs. A stardisc encounter^{1} can cause matter to become unbound, be captured by the perturbing star or pushed inwards and potentially be accreted by the central star. The extent to which this happens depends on the periastron distance, the mass ratio of the stars, the eccentricity and, moreover, the initial (preencounter) mass distribution in the disc. To date, limitations in observations of protoplanetary discs have prevented the discovery of how consistently the surface density of lowmass discs develops. This means that owing to observational constraints a unique predetermined initial state for the discmass distribution before an encounter does not exist. A wide variety of surface densities have been derived by fitting resolved millimetre continuum or line emission data with parametric disc structure models (e.g. Mundy et al. 1996; Lay et al. 1997) or in combination with broadband spectral energy distributions (SEDs) (Wilner et al. 2000; Testi et al. 2001; Akeson et al. 2002; Kitamura et al. 2002; Andrews & Williams 2007b). While those studies have profoundly shaped our knowledge of disc structures, they have all fundamentally been limited by the low angular resolution of available data.
The standard fitting method underlies the assumption that the surface density Σ has a simple powerlaw dependence of the form (1)out to some cutoff radius (e.g. Andrews & Williams 2007a). Estimates based on numerical powerlaw models fitted to observational data lead to distribution indices p ranging roughly from p = 0 to p = 2. Recent studies have even found anomalous results, e.g. Isella et al. (2009) who measured distribution indices of p < 0.
Analytical approaches have also proposed different discmass distribution indices. The most widely used model is that of a steadystate viscous accretion disc with a surface density distribution index of p = 1 (e.g. Hartmann et al. 1998). However, simulations of the evolution of protostellar discs that form selfconsistently from the collapse of a molecular cloud core have uncovered a surface distribution index of p = 1.5 (Lin & Pringle 1990; Hueso & Guillot 2005; Vorobyov & Basu 2007), while studies that include magnetised disc material have found a flatter discmass distribution of p = 0.75 (Shu et al. 2007). Taking this huge variety of distributions into account, one has to consider different initial discmass distributions to evaluate their effect in stardisc encounters.
Most previous numerical studies of stardisc encounters have used only a single density distribution, mainly focusing on the case of a theoretically motivated r^{1} discmass distribution (Hall et al. 1996; Hall 1997; Pfalzner 2004; Olczak et al. 2006; Moeckel & Bally 2006; Pfalzner & Olczak 2007). Stardisc encounters with different initial discmass distributions have only been considered in a very limited way. Heller (1995) performed numerical simulations of two different mass distributions (p = 0 and p = 1.5) concentrating on parabolic encounters with equal mass stars, while Hall (1997) investigated initial surface distributions of p = 0 and p = 1 for close and penetrating encounters of unity mass ratio. A study of a wide parameter range focusing on multiple initial discmass distributions still needs to be performed.
Nevertheless, numerical studies of stardisc encounters only allow a sectional view of the processes since it is impossible to simulate each combination of encounter parameters. Earlier analytical studies by Ostriker (1994) did not suffer from this shortcoming. In her study, a first order approximation of the angular momentum loss dependent on the initial discmass distribution is given. However, the validity of her results is limited to large periastron radii (for example r_{peri}/r_{disc} > 3 for M_{2}/M_{1} = 1), where the angular momentum loss is usually well below 10%. Close or even penetrating encounters cannot be interpreted by this linear perturbation theory (Ostriker 1994; Pfalzner et al. 2005b) making numerical studies indispensable in this regime.
In this work, the effects of stardisc encounters are investigated for a large parameter space considering most configurations that can be expected in a typical young cluster. The investigated mass distributions cover the entire range of the so far observed discmass distributions.
Section 2 shortly describes the numerical methods and parameter range used in this study, while Sect. 3 presents the results of the simulations including a fit formula for the mass and angular momentum loss depending on the initial discmass distribution index p. A summary and discussion is given in Sect. 4.
2. Methods
The encounter between a discsurrounded star with a secondary star is modelled using a code that is based on the numerical method described in Pfalzner (2003). In our simulations, only one star is initially surrounded by a disc. However, previous investigations have shown that stardisc encounters can be generalised to discdisc encounters as long as there is no significant mass exchange between the discs (Pfalzner et al. 2005a). In the case of a mass exchange, the discs can be to some extent replenished, such that for very close encounters the mass loss determined in this study would be slightly overestimated.
A summary of the disc properties can be found in Table 1. The disc is represented by 10 000 pseudoparticles, distributed according to a given particle distribution ∝ r^{ − b}. Choosing this relatively small number of simulation particles is motivated by the aim to cover a large encounter parameter space. However, performing test simulations with 50 000 particles shows that the lower resolution is sufficient for measuring the properties investigated here.
The simulation particles move initially on Keplerian orbits around the central star. The disc extends from an inner gap of 10 AU to 100 AU. The inner cutoff avoids additional complex calculations of direct stardisc interactions and saves computation time. Any pseudoparticle that reaches a sphere of 1 AU around the central star is removed from the simulation and stated in a commonly used simplified approach as having been accreted (Bate et al. 2002; Vorobyov & Basu 2005; Pfalzner et al. 2008). The vertical density distribution in the disc follows (2)where ρ_{0} is the unperturbed midplane particle density on the equatorial plane with ρ_{0} ∝ r^{ − (b + 1)}, H(r) is the vertical halfthickness of the disc (see Pringle 1981), which is represented by H(r) = 0.05 r according to a temperature profile of T = T_{0}·r^{1} with T = 20 K at the inner edge of the disc.
Initial conditions for all simulations.
The temporal development during the encounter is modelled using a fifthorder RungeKutta CashKarp integrator with an adaptive timestep size control for the numerical calculations. Longrange interactions of the gravitational forces between the disc particles and the perturbing star are calculated using a hierarchical tree method (Barnes & Hut 1986).
The total simulation time of t_{end} ≈ 3000 yrs for each encounter corresponds to three orbital periods of the outermost particles in the investigated standard disc with size r_{disc} = 100 AU around a 1 M_{⊙} star. This time span was found to be adequate for the calculation of all relevant quantities.
The discs obtained at the end of the simulations continue to develop in the sense that taking into account viscous forces the eccentric orbits of perturbed disc pseudoparticles would recircularise on timescales probably in excess of 10^{5} yr. However, bound and unbound material can be clearly distinguished shortly after the encounter.
In this present study, we are interested in protoplanetary discs, which are usually of low mass compared to protostellar discs (Bate 2011). Here lowmass discs mean m_{disc} ≪ 0.1M_{1}, where M_{1} is the mass of the disc surrounded star. The actual adopted discmass value is 10^{4}M_{1}. However, we tested another case of m_{disc} = 10^{3}M_{1} and found no differences in the result when including selfgravity in the simulations. In addition, we performed test simulations including pressure and viscous forces using a SPH code (for a description of the code see Pfalzner 2003), but only found negligible differences (<3%) in the results. Therefore, in most of our simulations selfgravitation and viscous forces were neglected in favour of higher computational performance (Pfalzner 2003, 2004).
The actual values of mass and angular momentum loss induced by the encounters were obtained by averaging over several simulations with different seeds for the initial particle distribution. The errors in the mass and angular momentum losses were determined as the maximum differences between the simulation results. They typically lie in the range of 2–3%.
In addition to these obvious performance benefits, the possibility to neglect the selfgravitation, pressure, and viscous forces has allowed us to optimise the computation time by treating the disc particles as pseudoparticles without a fixed mass during the simulation. The standard method for generating the initial discmass distribution in the simulations is to assign each pseudoparticle the same mass (Boffin et al. 1998; Pfalzner 2004; Olczak et al. 2006; Pfalzner & Olczak 2007). This approach involves the performance of a whole suite of simulations for each variation in the initial discmass distribution. In contrast, the different discmass distributions are realised by using a fixed pseudoparticle distribution and assigning their masses after the simulation process according to the desired density distribution in the disc. The implementation of this flexible numerical scheme allows us to use one suite of simulations for any initial discmass distribution. We note that the postprocessing of the particle mass is only possible because of the restriction to lowmass discs, where selfgravitation and viscous forces can be neglected.
The particle masses are assigned to each pseudoparticle in the diagnostic step to calculate the encounterinduced losses and final mass distributions, which both depend on the initial distribution. For the example of a powerlaw discmass distribution with index p, the mass of a particle, m_{i}, depends on its radial position in the disc r_{i}, the total disc mass m_{disc}, the number of pseudoparticles n_{part}, their radial positions r_{j}, and the underlying powerlaw particle distribution of index b. Since the number of pseudoparticles is limited, the discrete particle masses m_{i} are described by (3)The resulting independence of the mass and the particle distributions allows us to improve the resolution of our simulations by placing most of the pseudoparticles initially where the interaction between the stars and disc is strongest.
In most of our simulations, we adopt a constant particle distribution (b = 0). This is done because the effects on mass and angular momentum are largest at the outskirts of the disc. Using a constant pseudoparticle density (b = 0) throughout the disc ensures a higher resolution in the outer parts, even for steep mass density profiles, than for the standard approach. In contrast, if one were interested in processes that mainly concern the inner parts of the disc, such as accretion, it would be more accurate to use not a constant particle distribution but one that guarantees a high resolution close to the star.
We tested the method of a posteriori mapping of the particle masses against representative test calculations for different initial pseudoparticle distributions and against the standard method. The differences in the results were negligible, thus justifying the generalisation of our results for a constant particle distribution (b = 0).
Apart from the density distribution, the outcome of an encounter depends on the relative periastron distance r_{peri}/r_{disc} and the mass ratio of the involved stars M_{2}/M_{1}, where M_{2} is the perturber mass. The parameter space that we investigated – for a summary see Table 1 – is chosen in such a way that it spans the entire range of encounters likely to occur in a typical young cluster, such as the Orion Nebula Cluster. Studies have shown that the ONC with its high central density might be typical of young clusters with a mass > 1000 M_{⊙} (Pfalzner 2009). The lower limit to the perturber mass ratio was chosen to be M_{2}/M_{1} = 0.1 as for smaller mass ratios the influence of the perturber becomes insignificant. The upper limit, M_{2}/M_{1} = 500, is determined by the maximum possible mass ratio in the ONC, which corresponds to the hydrogen burning limit, the lowest mass for which is 0.08 M_{⊙}, and 40 M_{⊙}, the approximate mass of the θ^{1}COri system, which contains the most massive star in the ONC. The limit where perturbations at large distances become negligible depends on the perturber mass. The inner edge of the disc marks the lower value of r_{peri}/r_{disc} = 0.1.
3. Results
Although in principle any mass distribution of circumstellar discs can be studied, four exemplary discmass distributions are investigated here in more detail. To cover the entire range of numerically and observationally determined discmass distributions, we consider a constant mass distribution (p = 0), representing the lower boundary of expected distributions, and a p = 7/4 distribution, providing an upper limit. Additionally, a p = 1 distribution is investigated for comparison to previous stardisc encounter results and a p = 1/2 distribution, which is in the range of analytical results for lowmass discs (Shakura & Sunyaev 1973; Pringle 1981) and similar to results found in recent investigations of magnetised material (see Sect. 1).
The present investigation focuses on coplanar, prograde encounters on parabolic orbits (e = 1) (see example in Fig. 2a). Previous studies have shown that even for clusters as dense as the inner ONC region, most encounters in star clusters are expected to be close to parabolic (Larson 1990; Ostriker 1994; Olczak et al. 2010). These parabolic encounters provide the strongest impact of the perturber on the disc, since for higher eccentricities the perturber only interacts briefly with the stardisc system and is, therefore, unable to influence the disc significantly. The limitation to a certain orientation is more restricting, as inclined and retrograde encounters can lead to lower losses in mass (Heller 1993; Hall et al. 1996; Clarke & Pringle 1993; Pfalzner et al. 2005b). However, Pfalzner et al. (2005b) showed that up to an inclination of 45° the mass loss of the disc is only slightly smaller than for a coplanar orbit. This means that our results have to be regarded as an upper limit for encounters at a different inclination.
3.1. Surface density distribution
To achieve a clearer understanding of the redistribution of the disc material during a stardisc encounter that depends on the initial discmass distribution, we first focus on the evolution of the surface density profiles. In Fig. 1, we present how the evolution of the surface density in stardisc encounters depends on the periastron distance for both a constant initial discmass distribution (p = 0) and one with a steep initial distribution (p = 7/4). It shows the mass distributions before (solid line) and after a penetrating (r_{peri}/r_{disc} ≤ 1, dasheddotted line), grazing (r_{peri}/r_{disc} = 1, doubledotted line), and distant encounter (r_{peri}/r_{disc} = 3, dashed line). Here, the relative perturber mass was chosen to be M_{2}/M_{1} = 1.
Fig. 1 Initial (solid line) and final surface densities in case of a) initially constant distributed disc material of p = 0 and b) a steep distribution of p = 7/4. In both plots nonpenetrating (r_{peri}/r_{disc} = 3, dashed line), grazing (r_{peri}/r_{disc} = 1, doubledotted line), and penetrating encounters (r_{peri}/r_{disc} = 0.1, dasheddotted line) are plotted for a perturbing star of equal mass (M_{2}/M_{1} = 1). 
As known from previous investigations (Hall 1997; Larwood 1997), an encounter reduces the density in the outer parts of the disc by transporting some part of the outer disc material inwards and some part migrating outwards. The latter might become both captured by the perturber and unbound if the encounter is strong enough. These effects become more pronounced for small r_{peri}/r_{disc} and/or large M_{2}/M_{1}. However, the actual amounts depend on the initial discmass distribution. In an equalmass distant encounter (r_{peri}/r_{disc} = 3), most of the perturbed disc material is pushed inside the disc towards smaller disc radii. In this case, 12.5% of the total disc mass migrates inwards for an initial constant discmass distribution (p = 0), whereas it is only 6% for an initial p = 7/4.
The effects of different initial discmass distributions becomes even more obvious for closer encounters such as grazing (doubledotted lines in Fig. 1) or penetrating flybys (dasheddotted lines in Fig. 1). Here, the migration process and also the differences between the initial distributions are dominated by material moving outwards. In the case of a grazing encounter and an initially constant discmass distribution, about 60% of the disc mass becomes unbound. However, in addition about 10% of the total disc mass can be found outside the initial disc radius of 100 AU that is still bound to the central star. In contrast, for the steep p = 7/4mass distributions it only becomes ~30% of the disc mass unbound but again ~10% is bound but situated outside 100 AU. We conclude that the outer disc material is generally separated from the disc for grazing encounters, while material initially located inside the disc (r/r_{disc} ≤ 0.7) is redistributed but remains bound to the central star. Hence, prominent differences in encounterinduced disc losses for the investigated discmass distributions are expected for strong perturbations of the outer disc parts. In penetrating encounters, part of the disc material is pushed further inside the disc resulting in a higher surface density in the inner disc regions. In extreme cases, the disc loss can increase to >90% so that the final disc structure can no longer be regarded as a disc.
In nearly all cases, a steepening of the density profile is the general effect of an encounter. The degree of steepening depends on the mass ratio, periastron distance, and the initial discmass distribution. Figure 3 shows that even initially flat distributed discmaterial (p = 0) can be redistributed into a surface density profile steeper than p ≈ 2.
How is the vertical structure of the disc influenced by an encounter? As only coplanar encounters are investigated here, the answer is: surprisingly little. Figure 2b shows the vertical particle profile of a strongly perturbed disc after a penetrating encounter of M_{2}/M_{1} = 1. The main effect of the encounter is a decrease in the number of particles in the outer disc regions. In these parts of the disc, the resolution is relatively low, but owing to our constant particle distribution (b = 0) it is significantly higher than in previous investigations using r^{1} particle distributions in their simulations.
Fig. 2 The dashed line in a) shows the boundaries of the initial disc while the solid line indicates the trajectory of a grazing perturber (r_{peri}/r_{disc} = 1) of equal mass (M_{2}/M_{1} = 1). Material that resides within the disc after the perturbation is marked as black squares, while material that is in the end either bound to the perturbing star, unbound, or accreted is shown as grey dots. We note that the simulations were performed in three dimensions as can be seen in b). 
Fig. 3 Initial (p = 0, solid line) and final surface density after an encounter of M_{2}/M_{1} = 5.0 and r_{peri}/r_{disc} = 0.7 (dotted line). The dashed straight line represents a slope of p = 5/2. 
In a few special cases where material is pushed moderately inwards, even partially negative distribution indices as low as p = −1 can be the end product of an encounter (Fig. 4). The effect is most prominent in the inner part of perturbed discs with initially constant distributed disc material. Nonetheless, the outer parts of the disc always have a distribution with positive index, p > 0. This could perhaps explain the negative indices observed for some discs. Observations remain limited to measuring only part of the radial extension of the entire disc. If this were to be the range where negative indices prevail, one would wrongly extrapolate negative indices for the whole disc, while the overall index of the disc would remain positive.
Fig. 4 Initial (p = 0, solid line) and final surface density after an encounter of M_{2}/M_{1} = 1.0 and r_{peri}/r_{disc} = 2.0 (dotted line). The dashed straight line represents a slope of p = −1. 
Fig. 5 The relative discmass loss of a p = 0 (solid line), p = 1/2 (dasheddotted line), p = 1 (dotted line), and p = 7/4 (doubledotted line) discmass distribution including all particles bound more tightly to the central star than to the perturber and excluding unbound and accreted particles. The data is plotted for a) different periastron distances and an equalmass perturber, and b) different perturber mass ratios and r_{peri}/r_{disc} = 1. The vertical dashed line indicates the initial outer disc radius. 
3.2. Relative mass loss
In addition to changing the shape of the discmass distributions, stardisc encounters can remove material from the disc. Figure 5 shows the relative discmass loss Δm_{rel} = (m_{t = 0} − m_{tend})/m_{t = 0} for the four initial discmass distribution indices p = 0, p = 1/2, p = 1, and p = 7/4. Two representative cases of orbital parameters are singled out: Fig. 5a shows the dependence on the periastron distance for an equalmass encounter, whereas Fig. 5b depicts the dependence on the mass ratio for grazing encounters (r_{peri}/r_{disc} = 1).
Qualitatively, the dependence of the relative discmass loss on the periastron distance agrees with previous results (Pfalzner et al. 2005b; Olczak et al. 2006). However, the absolute values change considerably for the different initial discmass distributions. The effect is largest for nearly grazing encounters where the outer disc regions are mainly affected. As expected from the respective fraction of material in the outer regions, maximal mass losses occur for initially constant discmass distributions, while minimal losses occur for the r^{ − 7/4}distribution. The largest difference in mass loss between the investigated discmass distributions in Fig. 5a occurs for r_{peri}/r_{disc} = 0.9, whereas the r^{ − 7/4}distribution only has a mass loss of 33%, and the constant mass distribution has a mass loss of 64%.
The situation is somewhat different for very close penetrating encounters (r_{peri}/r_{disc} ≤ 0.3), where the discs are so strongly perturbed that the resulting structure can hardly be described as a disc. In this case the discmass loss seems relatively independent of the initial mass distribution (see Fig. 5a). At the other end of the parameter space – i.e. at large relative periastron distances – the mass loss becomes too small (Δm_{rel} ≤ 10%) to infer any dependence on the initial distribution.
Similarly, the dependence on the initial mass distribution is less pronounced for M_{2}/M_{1} < 0.3 and M_{2}/M_{1} > 90 (see Fig. 5b for a grazing encounter). Hence, generally weak perturbations  whether distant or of low mass ratio M_{2}/M_{1} – are incapable of significantly influencing the discs, while in the case of strong perturbations nearly the entire disc material is removed independently of the investigated discmass distributions. In both cases, the mass loss does not depend strongly on the discmass distribution. In contrast, encounters of intermediate strength are most sensitive to the discmass distribution. For the case shown in Fig. 5b, we find maximum differences of up to 35% for M_{2}/M_{1} = 3.
For highmass ratios M_{2}/M_{1} > 20 and certain nonpenetrating periastron distances, which restrict the gravitational stardisc interactions to the outer disc parts, differences in mass loss as high as 40% can be inferred for the different initial discmass distributions (see Appendix A).
3.3. Relative angular momentum loss
The different initial discmass distributions do not only influence the discmass loss but also the angular momentum loss. Fig. 6a shows the relative angular momentum loss ΔJ_{rel} = (J_{t = 0} − J_{tend})/J_{t = 0} as a function of the encounter distance for equalmass encounters and the four different discmass distributions used in this work. As expected from previous results (e.g. Pfalzner & Olczak 2007), the general trends in relative angular momentum loss are quite similar to that of the mass loss (compare Figs. 5 and 6) with angular momentum losses being slightly higher than the discmass losses.
While the mass losses for encounters of intermediate strength with r_{peri}/r_{disc} = 0.9 and M_{2}/M_{1} = 1 are 64% for the constant mass distribution compared to 33% for the 7/4mass distribution as shown before, the corresponding angular momentum losses are 75% and 50% , respectively. Material migrating inwards or becoming unbound owing to an encounter leads to an angular momentum loss, while the fraction of the disc mass that is pushed beyond the initial discradius but remains bound to the central star increases the total angular momentum of the disc. In total, the dependence on the initial discmass distributions is less pronounced for the relative angular momentum loss than for the discmass loss.
Fig. 6 The relative angular momentum loss of a p = 0 (solid line), p = 1/2 (dasheddotted line), p = 1 (dotted line), and p = 7/4 (doubledotted line) discmass distribution including all particles bound more tightly to the central star than to the perturber and excluding unbound and accreted particles. The data is plotted for a) different periastron distances and an equalmass perturber, and b) different perturber mass ratios and r_{peri}/r_{disc} = 1. The vertical dashed line indicates the initial outer disc radius. 
However, although the differences are lower in maximum the influence of different initial discmass distributions on the angular momentum loss covers a significantly large parameter range (see also Appendix B).
Since mass and angular momentum losses are generally influenced by perturbations of the outer disc parts, an initially flat particle distribution that has a high resolution in the outer disc regions can help us to achieve a higher accuracy in determining disc losses. Nevertheless, disc losses obtained with initially steep particle distributions (p = 1) found in previous studies (Olczak et al. 2006; Pfalzner & Olczak 2007) are accurately reproduced within the error range in the present study.
3.4. Adapting a fit formula dependent on the initial discmass distribution
The numerical results for the mass and angular momentum loss in this study cover a wide parameter range but, however, present only a discrete classification of the relative losses for the different initial discmass distributions. Analytical approaches are a possible option to avoid this disadvantage but are only valid for a very limited parameter space of distant encounters (Ostriker 1994; D’Onghia et al. 2010). To obtain a general estimate of the effect of arbitrary initial discmass distributions on the encounterinduced mass and angular momentum loss of protoplanetary discs, we present a fit formula that is valid for any initial discmass distribution, given that it can be expressed by a power law of the form Σ(r) ∝ r^{ − p}. For this purpose, we extended the fit function for the relative mass loss of Olczak et al. (2006, see Eq. (4) therein) and for the relative angular momentum loss of Pfalzner & Olczak (2007) (see Eq. (1) therein), which are valid for a r^{1}distribution, towards arbitrary surface density distribution indices 0 ≤ p ≤ 7/4 where r_{p} = r_{peri}/r_{disc} is the relative periastron distance.
For most of the parameter space, the adopted functions fit the data well within the margin of error and extend Ostriker’s analytical function for the angular momentum loss considerably. Larger deviations of the fit functions from the simulated losses occur only for high encounter mass ratios of M_{2}/M_{1} > 20, which was also the case for the established fit functions of Pfalzner & Olczak (2007) and Olczak et al. (2006). In the case of discpenetrating orbits and lowmass ratios, the disc losses can be affected by Lindblad and Corotation resonances that are located in the inner disc regions and cause moderate deviations of the fit function from the expected disc losses.
The extended fit functions provide a significant improvement to previous analytical and numerical results. They cover an extensive parameter space of the most reasonable initial discmass distributions and the most relevant orbital parameter ranges expected for interactions in star clusters of any age.
4. Discussion and conclusion
Most stars are believed to form in clusters (Lada & Lada 2003) and probably undergo at least one encounter closer than 1000 AU during the lifetime of their disc (~10^{6} yrs) (Scally & Clarke 2001). Hence, encounters are likely to have a significant effect on the disc structure. The scope of this study has been to determine the role of the initial mass distribution in the disc in this context. A full parameter study of stardisc encounters, similar to those expected to occur in young dense star clusters such as the Orion Nebula Cluster, has been performed with different discmass distributions of the form Σ(r) ∝ r^{ − p}, p ∈ [0,1/2,1,7/4] , which cover the whole range of observed mass distributions in discs. The main results can be summarised as follows:

1.
The relative discmass loss among the different initial density distributions differs by up to 40% for the same type of encounter. The largest differences are associated with strong perturbations of the outer disc edge, i.e. a grazing encounter in the case of equal stellar masses.

2.
Although larger amounts of relative angular momentum than disc mass are lost due to a stardisc encounter, the dependence on the initial discmass distribution is less pronounced. Nevertheless, half of the parameter range for which the angular momentum loss occurs correspond to differences among the investigated distributions of more than 15%.

3.
The discmass and angular momentum losses caused by a parabolic encounter can be fitted by a function that depends on the perturber mass ratio, the relative periastron distance, and the index of the initial discmass distribution.

4.
A steepening of the surface density slope is a general effect of an encounter for any initial discmass distribution and can result in distribution indices of p > 2 even in the case of initially flat distributions.
Another consequence of such an encounter is a change in the surface density of the disc on short timescales. In this context, the potentially most significant result of this study is that close encounters can provoke density profiles steeper than Σ(r) ∝ r^{2} independently of the initial discmass distributions. Since these density profiles are claimed to be the prerequisite for forming a planetary system similar to the Solar System, encounters in the early history of the Solar System might have provided these conditions (Adams 2010). Desch (2007) analytically inferred an initial surface density profile of the solar nebula protoplanetary disc of roughly Σ(r) ∝ r^{2.2} to form the present planets of the Solar System. As a possible way of forming this steep discmass density profile, he considered photoevaporation by an external massive star (see also Mitchell & Stewart 2010). Kuchner (2004) found similar results for extrasolar planetary systems suggesting large discmass distribution indices of p = 2.0 ± 0.5.
Our results emphasise that these steep density profiles do not have to exist ab initio or be formed by photoevaporation processes but that even discs that initially have evenly distributed material can fulfil the requirements for the formation of a Solar System type planetary system in the inner disc regions after a close encounter. Additional evidence of such an encounter in the case of the early Solar System was given by distant Solar System objects on highly eccentric orbits, such as the transneptunian object Sedna, and also the sharp outer edge of the Solar System at ~50 AU from the Sun (Ida et al. 1999; Allen et al. 2001; Morbidelli & Levison 2004; Kenyon & Bromley 2004).
On the other extreme, encounters cannot lead only to profile steepening but also profile flattening. Even the unexpected surface distribution profiles of p < 0 observed by Isella et al. (2009) can be explained by the influence of a perturbing star. Nonpenetrating encounters of a stardisc system with an initially flat distribution of disc material can easily lead to these density profiles of the inner disc regions. However, photoevaporation as another significant external impact on protoplanetary discs would have to be considered as an additional driver in these considerations.
Consequently, if all young stars were to start out with the same disc density structure, the influence of the cluster environment by means of encounters in the very early and dense phases of cluster evolution could account for the observed multitude of discmass density profiles.
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Online material
Appendix A: Relative mass loss
Relative discmass loss assuming an initial discmass distribution of p = 0.
Relative discmass loss assuming an initial discmass distribution of p = 1/2.
Relative discmass loss assuming an initial discmass distribution of p = 1.
Relative discmass loss assuming an initial discmass distribution of p = 7/4.
Appendix B: Relative angular momentum loss
Relative angular momentum loss assuming an initial discmass distribution of p = 0.
Relative angular momentum loss assuming an initial discmass distribution of p = 1/2.
Relative angular momentum loss assuming an initial discmass distribution of p = 1.
Relative angular momentum loss assuming an initial discmass distribution of p = 7/4.
All Tables
Relative angular momentum loss assuming an initial discmass distribution of p = 0.
Relative angular momentum loss assuming an initial discmass distribution of p = 1/2.
Relative angular momentum loss assuming an initial discmass distribution of p = 1.
Relative angular momentum loss assuming an initial discmass distribution of p = 7/4.
All Figures
Fig. 1 Initial (solid line) and final surface densities in case of a) initially constant distributed disc material of p = 0 and b) a steep distribution of p = 7/4. In both plots nonpenetrating (r_{peri}/r_{disc} = 3, dashed line), grazing (r_{peri}/r_{disc} = 1, doubledotted line), and penetrating encounters (r_{peri}/r_{disc} = 0.1, dasheddotted line) are plotted for a perturbing star of equal mass (M_{2}/M_{1} = 1). 

In the text 
Fig. 2 The dashed line in a) shows the boundaries of the initial disc while the solid line indicates the trajectory of a grazing perturber (r_{peri}/r_{disc} = 1) of equal mass (M_{2}/M_{1} = 1). Material that resides within the disc after the perturbation is marked as black squares, while material that is in the end either bound to the perturbing star, unbound, or accreted is shown as grey dots. We note that the simulations were performed in three dimensions as can be seen in b). 

In the text 
Fig. 3 Initial (p = 0, solid line) and final surface density after an encounter of M_{2}/M_{1} = 5.0 and r_{peri}/r_{disc} = 0.7 (dotted line). The dashed straight line represents a slope of p = 5/2. 

In the text 
Fig. 4 Initial (p = 0, solid line) and final surface density after an encounter of M_{2}/M_{1} = 1.0 and r_{peri}/r_{disc} = 2.0 (dotted line). The dashed straight line represents a slope of p = −1. 

In the text 
Fig. 5 The relative discmass loss of a p = 0 (solid line), p = 1/2 (dasheddotted line), p = 1 (dotted line), and p = 7/4 (doubledotted line) discmass distribution including all particles bound more tightly to the central star than to the perturber and excluding unbound and accreted particles. The data is plotted for a) different periastron distances and an equalmass perturber, and b) different perturber mass ratios and r_{peri}/r_{disc} = 1. The vertical dashed line indicates the initial outer disc radius. 

In the text 
Fig. 6 The relative angular momentum loss of a p = 0 (solid line), p = 1/2 (dasheddotted line), p = 1 (dotted line), and p = 7/4 (doubledotted line) discmass distribution including all particles bound more tightly to the central star than to the perturber and excluding unbound and accreted particles. The data is plotted for a) different periastron distances and an equalmass perturber, and b) different perturber mass ratios and r_{peri}/r_{disc} = 1. The vertical dashed line indicates the initial outer disc radius. 

In the text 
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