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Appendix A: Approximations of a Bonnor-Ebert sphere

Here, the mass, radius, and mean mass surface density of spheres with an analytical profile given by Eq. (1) are compared with the correct values of Bonnor-Ebert spheres. They are shown in Fig. A.1. Characteristic parameters are listed in Table A.1.

Fig. A.1 Mass, radius, and mean mass surface density of spherical clouds with fixed bounding pressure p(zsph) and temperature as a function of the pressure ratio pc/p(zsph). The values of a Bonnor-Ebert sphere are compared with the corresponding values of spherical clouds with smooth density profiles as given in Eq. (1) with n = 2 and n = 3 (dashed-dotted and dashed line, respectively). The dotted vertical and horizontal lines mark the values at maximum radius (open circle) and at critical stability (filled circle) of Bonnor-Ebert spheres. The mass, radius, and mean mass surface density of a Bonnor-Ebert sphere in the limit of infinite overpressure are shown as gray horizontal lines.

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Appendix A.1: Mass of Bonnor-Ebert spheres

The mass of a Bonnor-Ebert sphere is given by (Fischera & Dopita 2008) (A.1)where the size θ_cl is determined by pressure equilibrium p_ce^−ω(θ_cl) = p(θ_cl) = p_ext.

In Fig. A.1 the mass of Bonnor-Ebert spheres is compared with the masses of spheres using the approximation n = 3 and n = 2, which are given by (A.2)and (A.3)At given external pressure and temperature, the mass of a Bonnor-Ebert sphere increases with increasing overpressure up to the critical value where the sphere has the highest mass. For higher overpressure the mass is lower than the critical value and fluctuates to an asymptotic value at infinite overpressure given by (A.4)

Appendix A.1.1: Critical stability

Bonnor-Ebert spheres have a well-known critically stable configuration related to the response of the gas pressure at cloud radius in relation to compression. As long as the compression leads to a pressure increase (dp(r_cl)/dr_cl< 0), a cloud is considered stable, otherwise as unstable. Critical stability is given for (A.5)As the derivative of the radius with respect to q is nonzero at the pressure maximum, the critical condition is (A.6)For Bonnor-Ebert spheres, a critically stable sphere of the first pressure maximum, which is also the global maximum, is characterized through an overpressure of 14.04. When we apply the same criterion to spheres with an analytical profile as given by Eq. (1) with n = 3, we find from Eq. (A.2) for fixed mass and fixed K a critical overpressure , close to the critical value of a Bonnor-Ebert sphere. The physical parameters corresponding to the critical value of a sphere with n = 3 are listed in Table A.1.

As pointed out in the paper of Fischera & Martin (2012b), the critical mass of a Bonnor-Ebert sphere is also the highest possible mass for fixed K and external pressure p_ext , but varying pressure ratio q. The same applies for a sphere with n = 3. A cylinder, by comparison, has no critical stability with regard to compression. The highest mass line density is related to infinite overpressure. The situation for spheres with a n = 2-profile would be the same as for cylinders.

Appendix A.2: Radius of Bonnor-Ebert spheres

The radius of the Bonnor-Ebert sphere is given by (A.7)For a cloud with a truncated analytical density profile we find from Eq. (1) and the inner radius where A can be expressed through A² = 4πGp_ext/K²/q,(A.8)For a given q and p_ext the radius increases proportionally to K. Clouds with the same q and K will be smaller in higher pressure regions. Clouds with n> 2 have the greatest extension at an overpressure (A.9)Figure A.1 shows that the radius of the Bonnor-Ebert sphere behaves similarly as a function of overpressure as does the mass. However, the cloud with the greatest extension is subcritical with an overpressure of 4.990. The greatest extension of a sphere with n = 3 corresponds to an overpressure (A.10)which is close to the corresponding value of Bonnor-Ebert spheres. For a higher overpressure, the radius of a Bonnor-Ebert sphere is generally smaller than the radius at critical stability, but at a given q it is larger than a sphere with an analytical density profile with n = 3. In the limit of high overpressure the radius of the Bonnor-Ebert sphere fluctuates asymptotically toward (A.11)which is identical with the radius of a sphere with n = 2 in the limit of infinite overpressure.

Appendix A.3: Mean mass surface density

It has been shown (Fischera & Martin 2012b) that for overpressures where the gas pressure at the cloud boundary of a sphere with fixed mass has local maxima for varying overpressure the mean mass surface density of Bonnor-Ebert spheres is simply given by (A.12)The mean value applies, for example, to a critically stable sphere and for a sphere with infinite overpressure (Fig. A.1). For a sphere with infinite overpressure this can be easily verified by replacing the density ratio e^−ω(θ) by the asymptotic density profile at large scales θ, which is shown in Sect. 2.1.1 to be 2 /θ².

Appendix B: Probability distribution function of the local density

In this section we provide for the sake of completeness the PDF of the local density of spheres and cylinders with a truncated density profile as given by Eq. (1).

The logarithmic PDF is given by (B.1)where V_cl is the cloud volume and where V(r) is the cloud volume within radius r.

For a truncated density profile with a pressure ratio p_ext/p_c = q we obtain for spheres (B.2)In the limit of high overpressure this becomes for (ρ/ρ_c)^{2 /n} ≪ 1 a power law (B.3)or ρP_sph(ρ) ∝ ρ^{−3 /n}, as expected for simple radial density profiles ρ ∝ r⁻ⁿ (Kritsuk et al. 2011; Federrath et al. 2011).

For cylinders the PDF of the local density is given by (B.4)The PDF for a given q over the full range is a power law. For isothermal self-gravitating cylinders we obtain .

Appendix B.1: Local density PDF of Bonnor-Ebert spheres

Fig. B.1 Local density PDF of critical and supercritical Bonnor-Ebert spheres for three different pressure ratios, given in units of the overpressure of a critically stable Bonnor-Ebert sphere. They are directly compared with the PDFs of spheres with density profiles as given in Eq. (1) with n = 3 and n = 2. Their corresponding power-law asymptotes in the limit of infinite overpressure are shown as gray lines.

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Replacing in Eq. (B.1) the density by the solution of the Lane-Emden equation ρ = ρ_ce^−ω(θ) and the radius by the unit-free size θ = rA provides for the local density PDF of a Bonnor-Ebert sphere (B.5)The PDFs for critically stable and supercritical Bonnor-Ebert spheres (here defined as a self-gravitating sphere with an overpressure p_c/p_ext> 14.04) are shown in Fig. B.1. We showed in Sect. 2.1 that the inner part up to p_c/p(r) < ~ 100 of the density profile of self-gravitating spheres can be approximated by an analytical profile as given by Eq. (1) with n = 3. The shape of the PDF of the Bonnor-Ebert spheres at high density values is consequently determined by this profile. For example, the PDF of Bonnor-Ebert spheres at densities close to the central density value is approximately a power law with ρ^-1. For supercritical Bonnor-Ebert spheres, the PDF of density values ρ ≪ ρ_c asypmtotically approaches the PDF of a sphere with an analytical density profile with n = 2. The PDF becomes close to a power law ρ^-1.5.

The values are close to the values obtained with collapse models. Kritsuk et al. (2011) studied the theoretical density distribution in star-forming interestellar clouds. The collapse model provided a PDF that can be approximated over a wide range by a power law ρ^-1.695. At highest densities the PDF showed a flatter shape approximated by a power law ρ^-1 and therefore the same as for Bonnor-Ebert spheres. However, the physical explanation given are rotationally supported cores.

Appendix C: Estimating the maxima of P_BE(Σ) and ΣP_BE(Σ)

The maxima positions for linear and logarithmic PDFs of Bonnor-Ebert spheres were derived by estimating the zero points of the first derivatives. The highest positions of the linear and logarithmic PDF fulfill the conditions

where the linear PDF of the Bonnor-Ebert sphere is given by (C.3)

Appendix C.1: Linear PDF

Inserting the expression C.3 into the condition Eq. (C.1) for the maxima, we obtain (C.4)The condition becomes (C.5)where the expression in brackets is referred to as g(θ_⊥). From the derivative of the mass surface density Eq. (19), it follows that (C.6)where the integration along the radius θ has been changed to the integration along the depths at impact radius θ_⊥. The change of g(θ_⊥) by an infinitely small increase dθ_⊥ is obtained by expanding the upper limit of the integral and the functions depending on θ_⊥ into Taylor series. To first order, we find (C.7)where (C.8)For the derivative we obtain (C.9)

Appendix C.2: Logarithmic PDF

The condition for the highest positions of the logarithmic PDF is derived in a similar manner. Equation (C.2) provides (C.10)

Appendix D: Approximation of the P_BE(Σ)

In this appendix we show for the sake of completeness that for the approximations of Bonnor-Ebert spheres expression 20 of the PDF is identical to Eqs. (21) and (22). Replacing in Eq. (20) the density profile e^−ω(θ) = ρ(θ) /ρ_c by the analytical density profile as given in Eq. (1) and the potential by ω = −ln [ ρ(θ) /ρ_c ] , we obtain (D.1)where we have changed the integration along the radius θ to the integration along the depth . By introducing the integration variable (D.2)and by replacing the projected radius and the cloud radius by θ_⊥ = xθ_cl and the expression (D.1) can be transformed into (D.3)where y_n = (1 − q^{2 /n})(1 − x²) and where the upper limit is given by . The exponent of the integrand can be reduced by using (simplified version of Eqs. (6)−(57) of Joos & Richter 1978) (D.4)which provides (D.5)After applying Eq. (8) for the unit-free mass surface density, we obtain (D.6)which is identical with Eqs. (21) and (22).

Appendix E: Asymptotes at high and low mass surface densities for q-averaged PDFs

In this appendix we provide the asymptotes at high and low mass surface densities of the mean PDF for a considered distribution of the pressure ratios q of an ensemble of isothermal, self-gravitating, and pressurized spheres or cylinders as shown in Figs. 9 and 10. For the sake of simplicity, for Bonnor-Ebert spheres the analytical approximations for low and high overpressure are considered. The external pressure p_ext is considered the same for all clouds in the sample.

For convenience, we introduce the unit-free radius defined by (E.1)The mean PDF as defined by Eq. (52) at a given mass surface density Σ_n is then given by the integral (E.2)where P(q^-1) is the distribution function of the overpressures q^-1 given by Eq. (53), P_cl(Σ_n,q) the conditional PDF under the pressure ratio q, and where is a normalization constant given by (E.3)where κ = 2 for spheres and κ = 1 for cylinders. The lowest overpressure is related to the central mass surface density with Σ_n(0) = Σ_n.

Appendix E.1: Sphere

Appendix E.1.1: Asymptote at high mass surface densities

As shown in Paper I and in Sect. 2.3.3, the PDF of spheres with high overpressure asymptotically approaches within the limit of high mass surface densities the power laws given by Eq. (38). For the product we derive(E.4)Because the probabilities at high mass surface densities are related to spheres with high overpressure (Sect. 2.3), we can simplify the probability distribution Eq. (53) of the overpressure q^-1 through the corresponding power-law asymptote in the limit , so that . For the integral (E.2) we obtain (E.5)where we have replaced the lowest overpressure by the central mass surface density using Eq. (14).

The estimate of the normalization constant (Eq. (E.3)) is straightforward and is(E.6)where (E.7)is the normalized incomplete beta-function with and where (E.8)

Appendix E.1.2: Asymptote at low mass surface densities

In a similar manner as in the previous section, we can derive an asymptote of the mean PDF for low mass surface densities. As discussed in Sect. 2.3.2, the PDF of individual clouds approaches in the limit of low mass surface densities a simple power law given by Eq. (34). We find for the product of the square of the unit-free radius and the PDF the approximation (E.9)The value does not depend on the overpressure q^-1.

In the limit of low mass surface densities we can approximate . The mean asymptotic value of the PDF is then given by (E.10)with and (E.11)The slope of the asymptotic power law for low mass surface densities is the same as for individual clouds.

Appendix E.2: Cylinder

Appendix E.2.1: Asymptote at high mass surface densities

To estimate the mean value of the PDF for high mass surface densities we can apply the approximation Eq. (49). For the product of radius and PDF we obtain (E.12)where we have replaced the central mass surface density by the asymptotic value given by Eq. (50) valid in the limit of high overpressure. The same replacement has been made for the fixed mass surface density Σ₄ in the denominator, which is related to the lowest overpressure . To estimate the integral (E.2) we can make the same simplification as in Appendix E.1.1 by replacing the probability distribution Eq. (53) by the corresponding power law . For the mean PDF of self-gravitating cylinders we obtain as the asymptote at high mass surface densities (E.13)To provide a simple analytical estimate for the normalization constant we can consider the two extreme cases we discussed here. For a distribution in q^-1 with the radius can be simplified to . The estimate of the integral (E.3) is again straightforward and the normalization constant becomes (E.14)where (E.15)For , the distribution of overpressure is a simple power law and the integral can be solved without further simplification. The normalization constant becomes (E.16)

Appendix E.2.2: Asymptote at low mass surface densities

In the limit of low mass surface densities it follows from Eqs. (34) and (41) with (42) that the asymptote of the PDF in the limit of low mass surface densities is given by (E.17)For low mass surface densities the mean PDF is dominated by x ≈ 1. For the product of radius and PDF we obtain (E.18)For large the contribution to the mean PDF is related to high overpressures so that we can simplify the expression by removing the square root using . The asymptote of the mean PDF at low mass surface densities becomes (E.19)where (E.20)For the limit , where the probability distribution P(q^-1) becomes a simple power law, we find (E.21)

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