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Appendix A: Solution for the mass surface density profile

Appendix A.1: Case n ≥ 1

In case n> 1 the integral can be expressed through the incomplete and complete beta function (A.1)with the condition a,b> 0 where the normalized incomplete beta function is given by (A.2)The beta function is equal to (A.3)where Γ(x) is the Γ-function given by (A.4)In Appendix D the asymptotic behavior of I_ξ(a,b) with a = (n − 1)/2 and b = 0.5 is discussed. In general the asymptotic behavior is better for higher powers of a.

The mass surface density for given external pressure and overdensity through the cloud center for n> 1 is (A.5)In the limit of high overdensity (q → 0) the central mass surface density becomes the asymptotic value (A.6)

Appendix A.2: Case n < 1

To estimate the profile for n< 1 we can transform the integral to (A.7)The integral can then be calculated using the complete and incomplete beta function as in Eq. (A.1). The mass surface density profile becomes (A.8)In the limit of high overdensity the central mass surface density approaches asymptotically a maximum value given by (A.9)

Appendix A.3: Analytical profiles of the mass surface densities

The mass surface density profiles of the truncated analytical density profile for any natural number n = 1,2,... can be expressed through simple analytical functions. For n = 1, 2, 3, and 4 the profiles are for example given by where y_n = (1 − x²)(1 − q^{2 /n}). The profiles of higher orders can be derived by applying successively the integral transform Eq. (A.7).

The profile n = 4 applies for isothermal self-gravitating pressurized cylinders. For cylinders exist a maximum mass line density given by [M/l] _max = 2K/G where G is the gravitational constant. K is a constant given by K = kT/ (μm_H) where T is the effective temperature, k the Boltzmann constant, μ the mean molecular weight and m_H is the atomic mass of hydrogen. If we replace the pressure ratio with q = (1 − f)² where f = (M/l)/ [M/l] _max ≤ 1 is the normalized mass line density we obtain the expression given in the work of Fischera & Martin (2012a). The profile for n = 3 closely describes the profile of Bonnor-Ebert spheres with overdensities less than ~100 (Fischera 2014).

Appendix A.4: Gaussian approximation (n →∞)

Under certain circumstances the inner region of the profile can be approximated by a Gaussian function as will be shown in the following where the width is related to physical parameters as the overdensity q^-1 and the inner radius r₀.

The density profile can in general be expressed through (A.14)where (A.15)Where (xr_cl/r₀)² ≪ 1 we can linearize the logarithm using ln [1 + (xr_cl/r₀)²] ≈ (xr_cl/r₀)² and we obtain a Gaussian density profile (A.16)where the variance is given by (A.17)The approximation improves with power n. In case of a pressure ratio q the density profile becomes approximately a Gaussian function for all impact parameters if n ≫ − 2lnq/ ln2 or . For large n the variance becomes (A.18)which decreases slowly with overdensity.

In a similar approach we can derive the asymptotic profile of the mass surface density. Considering the same condition for n as above we obtain for example for (A.19)also a Gaussian form where the variance of the mass surface density is given by (A.20)where σ_Σ ≈ σ_ρ = σ for large n.

The central region (x ≪ r₀/r_cl) of the profile is therefore approximately given by a Gaussian function (A.21)The approximation provides the exact central mass surface density.

We want to consider the case of high overdensity and large power n so that the contribution of the incomplete beta function in the central region of the cloud becomes negligible (see Appendix D). In the limit of large n the beta function becomes (A.22)Replacing n through the variance of the density profile we obtain for the asymptotic profile for given overdensity (A.23)

Appendix B: Asymptotes of the PDF

Appendix B.1: Asymptotes in the limit y_n → 0 (low X_n)

In the limit of large impact parameters (x → 1) it follows that y_n → 0 and consequently . The integrand in Eq. (4) is approximately a constant so that the unit-free mass surface density becomes (B.1)From Eq. (11) we find directly the corresponding asymptotic behavior for spheres which is given by (B.2)For cylinders we find from Eq. (15) (B.3)For q^{2 /n} → 0 we obtain the asymptote of the probability function of homogeneous spheres or cylinders.

Appendix B.2: Asymptotes in the limit y_n → 1 (high X_n)

Appendix B.2.1: Asymptotes for n> 1

In the limit of low impact parameter (x → 0) and high overdensity (q ≪ 1) we have y_n → 1 − q^{2 /n} ~ 1. The normalized mass surface density is then approximately given by (B.4)where ζ_n = 0.5 B((n − 1)/2,1/2). We can use this relation to replace 1 − y_n to obtain an expression of the PDF as function of the mass surface density.

For the PDF of spheres we find that at high mass surface densities the PDF approaches asymptotically a power law given by (B.5)For the asymptotic PDF of cylinders we find (B.6)Replacing 1 − y_n in the Eq. (12) with the above expression for the mass surface density provides the asymptotic behavior of the PDF of cylinders at the pole given by (B.7)where is the central mass surface density. A power law is only established for cylinders with sufficiently high overpressure so that (X_n/X_n(0))^{2/(n − 1)} ≪ 1.

Appendix B.2.2: Asymptotes for n = 1

In the limit of large y₁ the mass surface density behaves as (B.8)Replacing (B.9)in Eq. (11) and in Eq. (15) provides the asymptotes (B.10)for and(B.11)For the special case n = 1 the PDF at high mass surface densities is therefore approximately described by a simple exponential function. The asymptotic behavior of the PDF for a cylinder including the region at the pole is (B.12)where X₁(0) = ln(2q^-1).

Appendix B.2.3: Asymptotes for n< 1

As pointed out in Appendix A.2 for n< 1 the mass surface density has a maximum possible value. In the limit y_n → 1 we have (B.13)As can be shown we have P_sph(X_n) → 0 and for X_n → 1/(1 − n).

Appendix B.3: Asymptote of the PDF for high/low n

Appendix B.3.1: Asymptote in the limit n→∞

As we have seen in the previous section in the limit of high n the power law slope at high mass surface densities approaches asymptotically α = 1. The corresponding PDF can be directly obtained from Eq. (B.5). Replacing n − 1 by the standard deviation of the Gaussian approximation we get for spheres with high overdensity in the limit of n → ∞(B.14)The same result is obtained from Eq. (A.23) by deriving the corresponding derivative and using Eq. (8). For the asymptotic PDF of cylinders Eq. (B.6) (B.15)

Appendix B.3.2: Asymptote in the limit n→ 0

In the limit n → 0 it follows from Eq. (A.8) for the mass surface density (B.16)In case of spheres the PDF of the mass surface density becomes (B.17)which is the PDF of homogeneous spheres. Likewise, we find for the asymptotic PDF of cylindrical clouds in the limit n → 0 that (B.18)which is the PDF of a homogeneous cylinder.

Appendix C: Condition for PDF maxima

The maxima position were derived for both linear and logarithmic PDFs of spheres and cylinders. For cylinders the asymptotic PDF as defined in Eq. (15) was considered.

Appendix C.1: For linear values (P(Σ_n))

The condition for maxima of the linear PDF is given by (C.1)This leads to (C.2)in case spheres and to (C.3)for cylinders. For n = 1 and n = 3 the maxima are simple analytical expressions listed in Table 1.

Appendix C.1.1: Approximation for n≫ 1

In the limit n ≫ 1 the condition for maxima of the linear PDF is equal for spheres and cylinders and is given by (C.4)As y_n → 0 for n → ∞ it follows from Eqs. (D.7) and (A.22) that the mass surface density behaves approximately as (C.5)where P(χ²,1) is the PDF of the χ²-distribution and where (C.6)In the limit of n ≫ 1 the condition for maxima becomes a function of ny_n = C where C is a constant. Solving (C.7)provides C ≈ 0.58404. The maxima position is therefore approximately given by

Appendix C.2: For logarithmic values (Σ_n P(Σ_n))

The maxima of the PDF of logarithm values are given by the condition (C.10)This leads to (C.11)in case of spheres and to (C.12)in case of cylinders. For n = 3 the maxima positions are again simple analytical expressions given in Table 1.

Appendix D: Asymptotic behavior of the incomplete beta function

To derive the asymptotic behavior of the incomplete beta function in Eq. (4) for the mass surface density in the limit n ≫ 1 we consider the approximation (Eq. (26.5.20), of Abramowitz & Stegun 1972) (D.1)where P(χ²,ν) is the χ² distribution function of ν events where

In our case we have a = (n − 1)/2 and b = 1/2. The incomplete beta function is then approximately given by (D.4)where (D.5)For given power n the approximation improves with increasing ξ.

In the limit y_n → 0 we can use the replacement (D.6)so that the mass surface density becomes (D.7)In the limit n → ∞ we have (D.8)so that the beta function becomes independent of n.

Appendix D.1: Incomplete beta function for b = 1/2

The power law approximation of the PDF at large mass surface densities for n> 1 as presented in Appendix B are valid for negligible contribution of the incomplete beta function to the mass surface density. We have seen in Fig. 4 that the power law is only a good representation for large mass surface densities and that the approximation of the PDF improves for larger n.

Figure D.1 shows the value of the incomplete beta function as given in Eq. (4) for different assumptions for the powers n and the density ratio q. As we see the value of the incomplete beta function for given q decreases for larger n. In the limit n → ∞ for given q we obtain the asymptotic value of the incomplete beta function given in the previous section. In the limit n → 1 we have (D.9)

Fig. D.1 Incomplete beta function Iξ(a,b) (black contours) for a = (n − 1)/2 and b = 1/2 as function of ξ = 1 − yn. The lines are labeled with the corresponding power n. The line for a power n = 3 is emphasized through a thick line. The gray lines correspond to ξ = q2 /n for fixed density ratio q and varying power n. The lines are labeled with log 10q. The gray dashed lines are obtained using the approximation of the incomplete beta function (Eq. (D.4)). The asymptotic value of the incomplete beta function for given q in the limit n → ∞ is indicated through dotted lines. The filled circle corresponds to a power n = 3 and an overdensity of a critical stable sphere.

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