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Volume 546, October 2012
Article Number L1
Number of page(s) 6
Section Letters
Published online 26 September 2012

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Appendix A: Properties of the sample

After the selection procedure detailed in Sect. 2.1, we list the final PPD sample in Table A.1. To assess the heterogeneity of the sample, we show the wavelengths of the radius measurements as a function of ambient stellar density in Fig. A.1. The vast majority of sources (75%) were measured in a narrow wavelength range below 3 μm (i.e. near infrared wavelengths) with a spread of 0.25 dex and centred at 1.32 μm. The remaining sources were measured at millimetre wavelengths. For the ONC sample, 80% of all sources were measured at 0.66 μm with very little scatter overall. As shown in Fig. A.1, the 16 sources of the total sample measured at millimetre wavelengths all have ambient surface stellar density < 200 pc-2. Therefore the distribution at densities above this value can be considered to be homogeneous.

The regions from which the YSOs are taken to estimate the ambient density are summarized in Table A.2. Using these samples, the ambient surface density of stars around each PPD is estimated as (Casertano & Hut 1985): (A.1)where N is the rank of the Nth nearest neighbour, and dN is the projected angular distance to that neighbour. We use N = 20, which is higher than the commonly-used value of N = 7 (cf. Bressert et al. 2010) and is chosen to improve the statistics of the density estimates. An additional effect of using a higher value of N is a slight decrease of the density estimates. This should be kept in mind when comparing our densities to those in other work.

Table A.1

Distribution of PPD sources over the host star type.

thumbnail Fig. A.1

Wavelengths λ of the radius measurements for the final sample of PPDs versus ambient surface stellar density. Symbols have the same meaning as in the upper panel of Fig. 1.

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Table A.2

Star-forming regions used in this study, listing the names of the regions, their numbers of objects Nobj, and distances D.

Appendix B: A simple model for PPD truncations

In this Appendix, we derive the upper limit to the radii of PPDs due to dynamical encounters. Where appropriate, we emphasize that the derivation is conservative, such that the obtained truncation radius is indeed an upper limit.

Olczak et al. (2006) performed numerical simulations of disc perturbations and provided an expression for the relative disc mass loss ΔM/M due to encounters with other stars (their Eq. (4)). If we assume that the mass loss occurs by stripping the outer disc layers and adopt the disc surface density profile of Σd ∝ r-1 used in their work, then Δr/r = ΔM/M. The expression for Δr/r from Olczak et al. (2006) is consistent to within a factor of three with the scenario in which a disc is always truncated to the equipotential (Lagrangian) point between both stars. If the disc was already smaller than that radius, it is left relatively unperturbed. For the rough estimate made here, it thus suffices to write for the upper limit to the disc radius (B.1)where rp is the pericentre radius at which the perturber passes, m1 is the mass of the perturbed system and m2 is the mass of the perturber. The approximation of Eq. (B.1) follows Eq. (4) of Olczak et al. (2006) with reasonable accuracy for initial disc radii up to a few 103 AU (consistent with the parameter space in Fig. 1), encounter distances rp > 0.002 pc (i.e. rp/rd > 0.2) and mass ratios m2/m1 > 1. We have verified that these conditions are satisfied for the encounters that are expected to determine the disc truncation (see below). Following Binney & Tremaine (1987), the impact parameter b and the encounter radius due to gravitational focusing rp are related as (B.2)where v is the relative velocity of the encounter. This equation is inverted to derive rp for each encounter.

The truncation radius rd of Eq. (B.1) depends on the variable set { b,v,m1,m2 } , for which we specify probability distribution functions (PDFs). For the masses, we use a Salpeter (1955) type initial mass function in the range 0.1 Mmmax, where mmax depends on age due to stellar evolution. For ages τ < 4 Myr we assume mmax = 100 M, while at later ages it is set by the Marigo et al. (2008) stellar evolution models at solar metallicity. The mass function is: (B.3)which is normalized such that . Assuming a Maxwellian velocity distribution, the total number of encounters per unit velocity dv and unit impact parameter db follows from the encounter rate d2N/dbdv as (Binney & Tremaine 1987) (B.4)where ν is the local number density of stars, τ is the age of the region, and σ is the velocity dispersion. The relative velocity ranges from v = 0 − ∞ and the impact parameter from b = 0 − bmax (see below). As in Eq. (B.3), we have normalized  such that , by writing and defining as the total number of encounters at age τ. The factor fdis ≈ 0.3 represents the fraction of encounters that leads to disc mass loss according to Eq. (B.1). This accounts for the fact that encounters with pericentres at inclination angles θ > 45° with respect to the disc plane cause only weak mass loss and retrograde encounters leave the disc almost unperturbed (Pfalzner et al. 2005).

Given a sequence of encounters, the truncation of the PPD is set by the most disruptive encounter (Scally & Clarke 2001; although see Olczak et al. 2006), i.e. rp,min = f(bmin,vmin,m2,max). If we assume that { bmin,vmin,m2,max }  are uncorrelated, implying that the region is not mass-segregated, the PDF of the most disruptive encounter becomes (B.5)where p { b,v,m }  represent the PDFs for the lowest b, lowest v and highest m2, respectively. Following the method of Maschberger & Clarke (2008, Eq. (A5)), these three PDFs are defined as (B.6)\arraycolsep1.75ptwhere are the distribution functions for b and v, with Φb(b) ∝ ντb and Φv(v) ∝ exp( − v2/4σ2)v3/σ3, again normalized to unity in both cases. In Eqs. (B.6), bmin, vmin and m2,max indicate variable limits, and bmax and m2,MIN indicate fixed limits. The fixed limit bmax represents the maximum impact parameter, which is given by the typical interstellar separation bmax = (48/πν)1/3 (the factor was chosen for consistency with Scally & Clarke 2001). It should be noted that while this is a physically motivated choice, it only weakly influences the result since the most likely most disruptive encounter will typically be at bmin ≪ bmax. Assuming an age of τ = 1 Myr, for surface densities of stars Σ ≤ 105 pc-2 we find that pb always peaks at impact parameters bmin > 0.002 pc (i.e. rp/rd ≳ 0.2), whereas for Σ ≥ 100 pc-2 the most likely most disruptive encounter always has m2 ≥ 0.5 M, which after averaging over the mass function to account for the distribution of m1 gives m2/m1 > 2. This validates the use of the approximation in Eq. (B.1).

By combining Eqs. (B.3) and (B.5), the total PDF is (B.7)It should be noted that we did not include the mass of the perturbed object m1 in the PDF of the most likely most disruptive encounter (Eq. (B.5)), but instead average over the mass PDF itself. The reason is that the stars in Fig. 1 span a range of masses, and a “typical” relation between the truncation radius and ambient density is preferable.

Combining the previous equations gives a theoretical estimate for the typical truncation radius rtr as a function of the ambient density, velocity dispersion and age: (B.8)where V indicates the complete phase space, i.e. 0.1 Mmmax in mass, 0–∞ in velocity and 0–bmax in impact parameter. This expression provides the expected radius after the “most likely most disruptive encounter”, averaged over the stellar mass function to account for the unknown mass of the perturbed system.

Appendix C: Evolution of the disc mass function

To calculate the evolution of the disc mass function (DMF), we assume that the initial disc mass md,i is related to the host stellar mass m1 as (C.1)where fd is a constant. We adopt fd = 0.03, which is in good agreement with observations (Andrews & Williams 2005) and sufficiently accurate for the order-of-magnitude estimate made in Sect. 4. Using a Salpeter (1955) stellar initial mass function (cf. Eq. (B.3)), the initial DMF is (C.2)for each host stellar mass, we calculate the characteristics of the most likely most disruptive encounter as in Appendix B, using quantities that are appropriate for NGC 3603 (i.e. τ = 3 Myr, σ = 4.5 km s-1, and R = 1.45 pc). Given a certain encounter, the disc mass loss is calculated using the expression from Olczak et al. (2006, Eq. (4)), which provides Δ ≡ Δmd/md as a function of the host stellar mass m1, the mass of the perturber m2, the pericentre distance rp and the disc radius rd. To account for the dependence of Δ on the radius, it is calculated for all radii from the observed sample at ambient densities 10-1 < Σ/pc-2 < 102 (see Fig. 1), including those with rd < 50 AU since the corresponding regions are all nearby and hence the detection limit is less stringent. At these densities the encounter rate is so low that the observed disc radii can be interpreted as “initial” radii. The obtained values of Δ are then averaged to remove the dependence on rd, and integrated Φtot (see Eq. (B.7)) in the same way as rtr in Eq. (B.8). This provides the expected relative mass loss as a function of host stellar mass ⟨ Δ ⟩ ,1 and hence the final disc mass is approximately (C.3)The final DMF is then given by (C.4)

© ESO, 2012

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