Roughly speaking, the second kind error probability optimally tends to 0 with the rate βn [Tn ] ≈ e−nBe (r | ~ρ k ~σ) when the ﬁrst error probability is required to tend to 0 with αn [Tn ] ≈ e−nr or faster.
An Information-Spectrum Approach to Classical and Quantum Hypothesis Testing for Simple Hypotheses
We begin with a detailed analysis of the mass distributions obtained for purely thermal support and the PN07 mass scale corresponding to NBE ≈ 1.2 × 105 .
Numerical and semi-analytic core mass distributions in supersonic isothermal turbulence
The mass distributions for NBE = 103 appear to be better converged, at least in the case of solenoidal forcing.
Numerical and semi-analytic core mass distributions in supersonic isothermal turbulence
For NBE = 103, the scatter around the tails of the mass distributions is too large for the MLE method to be applicable.
Numerical and semi-analytic core mass distributions in supersonic isothermal turbulence
For the solenoidal simulation with turbulent support and NBE = 103 we do not give a value for the power-law exponent because the error bars are simply too large.
Numerical and semi-analytic core mass distributions in supersonic isothermal turbulence
The large error bars are the result of the much smaller number of cores found in each snapshot compared to NBE = 1.2 × 105 .
Numerical and semi-analytic core mass distributions in supersonic isothermal turbulence
CMFs of compressive (solid) and solenoidal (dot–dashed) forcing for different values of NBE with/without turbulent support (see Sect. 2.2 and 3) and a grid resolution of 10243.
Numerical and semi-analytic core mass distributions in supersonic isothermal turbulence
These distributions are independent of the parameter NBE because of the normalization by the total number of cores, Ntot .
Numerical and semi-analytic core mass distributions in supersonic isothermal turbulence
Since the cores are suﬃciently resolved both for solenoidal and for compressive forcing for λ0 J/L = 0.2 (see Table 4), the most likely explanation is that NBE is too small and, consequently, the high-mass cores are not within the asymptotic regime, for which the Padoan-Nordlund theory applies.
Numerical and semi-analytic core mass distributions in supersonic isothermal turbulence
NBE ≈ 1.2 × 105), the clump-ﬁnding distribution is shifted towards higher masses for compressively driven turbulence because the smallest cores cannot be resolved (see Sect. 5.1).
Numerical and semi-analytic core mass distributions in supersonic isothermal turbulence
For hydrodynamic simulations without explicit treatment of self-gravity, the CMF depends on the choice of the global mass scale, which determines the number of Bonnor-Ebert masses, NBE, with respect to the mean density in the computational domain.
Numerical and semi-analytic core mass distributions in supersonic isothermal turbulence
NBE is chosen too large, a signiﬁcant fraction of cores will be numerically unresolved and the resulting CMF will be shifted towards higher masses.
Numerical and semi-analytic core mass distributions in supersonic isothermal turbulence
In our analysis, these constraints are well satisﬁed for two cases: Firstly, solenoidal forcing with NBE ≈ 1.2 × 105 and, secondly, compressive forcing with NBE = 103 .
Numerical and semi-analytic core mass distributions in supersonic isothermal turbulence
The lower value of NBE is consistent with Larson-type relations. 2.
Numerical and semi-analytic core mass distributions in supersonic isothermal turbulence
However, the results for purely thermal support are at odds with the theory of Padoan & Nordlund (2002), which asymptotically applies to NBE → ∞ and predicts power-law tails for the CMF.
Numerical and semi-analytic core mass distributions in supersonic isothermal turbulence
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