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Erratum
This article is an erratum for: [this article]

Issue
A&A
Volume 620, December 2018
Article Number C1
Number of page(s) 4
Section Astrophysical processes
DOI https://doi.org/10.1051/0004-6361/201526104e
Published online 30 November 2018

1. Introduction

A missing normalization coefficient in Eqs. (11) and (12) of de Avillez & Breitschwerdt (2015) affected the results shown in Figs. 3 and 4 and discussed in Sect. 3 of that earlier publication. Here, we present the corrected version of the equations and the corresponding results discussed in Sect. 3 and shown in Figs. 3 and 4 of that paper. In addition we provide supplementary material in the form of tables covering a larger parameter space than the one previously used. These tables are available at the CDS.

2. Temperature-averaged and total Gaunt factor

During the interaction of an electron with the Coulomb field of an ion of ionic state z and atomic number Z the amount of free-free energy emitted per unit time and unit volume is given by

(1)

with z denoting the ionic state of an ion with atomic number Z and number density nZ, z, ne is the electron number density, h is the Planck constant, c is the speed of light, e is the electron charge in Coulombs, me is the electron mass, kB is the Boltzmann constant, T is the temperature, and ⟨gff(γ2, u)⟩ is the temperature averaged Gaunt factor given by (see, e.g., de Avillez & Breitschwerdt 2017)

(2)

where f is the electron distribution function, u = hν/kBT (ν is the frequency of the emitted photon), γ2 = z2Ry/kBT (Ry is the Rydberg constant), and Ne is a normalization coefficient defined such ⟨gf f(γ2, u)⟩ = 1 when gf f(γ2, u)=1. For the κ and Maxwell-Boltzmann (MB) electron distribution functions Ne is given by

(3)

with

(4)

and

(5)

which becomes eu when κ → ∞ (Fig. 1). In these expressions

(6)

Therefore, Eq. (11) in de Avillez & Breitschwerdt (2015) must be rewritten as

(7)

where

The total free-free power associated with an ion (Z, z) given by

(8)

with

(9)

The integral in the RHS of (8) is the total free-free Gaunt factor. When κ → ∞ the Integral in (9) tends to eu and total Gaunt factor for the MB distribution of electrons as defined by Karzas & Latter (1961) is recovered

(10)

The top panel of Fig. 2 and the panels in Fig. 3 display the variation with γ2 of the temperature-averaged Gaunt factor, ⟨gf f(γ2, u)⟩, for different values of u ∈ [10−4, 104] for the MB and κ = 2, ...,1200 distributions. The bottom panel of Fig. 2 highlights in a magnified image of the region γ2 ∈ [10−2, 106] the distribution with γ2 of ⟨gf f(γ2, u)⟩ for different values of u.

It is clear that as κ increases the temperature-averaged κ distributed Gaunt factors approach those calculated with the MB distribution. However, the speed (with κ variation) of this approach depends on the values of u. For u < 10 this approach is faster than for u ≥ 10 (compare top panel of Fig. 2 and the two panels in Fig. 3). For values as high as κ = 1200 there is still no overlap between the κ and Maxwellian values of ⟨gf f(γ2, u)⟩ for u > 102 and γ2 < 10 (Fig. 3). This is a consequence of the slow approach to eu of the integral for u ≥ 10 in comparison to u < 10 (Fig. 1).

thumbnail Fig. 1.

Variation of Integral (5) with κ as function of u. The solid black line represents eu which is overlapped by the integral as κ → ∞.

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thumbnail Fig. 2.

Temperature-averaged Gaunt factors calculated for κ = 2, 3, 5 and MB (black line in both panels) distributions of electrons for the range 10−5γ2 ≤ 1010 (top panel) and a magnification of the region γ2 ∈ [10−2, 106] and ⟨gf f(γ2, u)⟩ ∈ [0.9, 1.5] (bottom panel).

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thumbnail Fig. 3.

Temperature-averaged Gaunt factors calculated for κ = 10, 15, 25, and 50 (left panel), 100, 500, 1000, and 1200 (right panel) and MB (black line in both panels) distributions of electrons for the range 10−5γ2 ≤ 1010. We note the slow approach ⟨gf f(γ2, u)⟩ to the Maxwellian values for u > 10 and γ2 < 10 for large values of the κ parameter.

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Figure 4 displays the total free-free Gaunt factor, ⟨gf f(γ2)⟩, calculated for the MB and κ = 2, 3, 5, 10, 15, and 25 distributions. For larger κ the ⟨gf f(γ2)⟩ almost overlap with the Maxwellian value as shown in the magnification of the regions ⟨gf f(γ2)⟩ ∈ [ − 1.0, 0.8] and ⟨gf f(γ2)⟩ ∈ [1.38, 1.45] (top panel of Fig. 5), and ⟨gf f(γ2)⟩ ∈ [6, 10] and ⟨gf f(γ2)⟩ ∈ [0.995, 1.005] (bottom panel of same Figure). It turns out that even for κ parameters as high as 1000 there is still a slight difference between ⟨gf f(γ2)⟩ calculated for the κ and MB distributions.

thumbnail Fig. 4.

Total free-free Gaunt factor calculated for κ(2, 3, 5, 10, 15, and 25) and MB distributions.

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thumbnail Fig. 5.

Magnification of the total free-free Gaunt factor in two regions in γ2 and in the ⟨gf f(γ2)⟩ profile shown in Fig. 4 but for κ = 50, 100, 500, and 1000. The solid black line refers to the Maxwellian value.

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3. Tables

Supplementary material is available at the CDS with a set of tables referring to the temperature-averaged Gaunt factor versus γ2 and different u, and a table for the total Gaunt factor versus γ2 for results obtained with the κ = 2, 3, 5, 10, 15, 25, 50, 100, and 500 and MB electron distributions. The parameter space comprises γ2 ∈ [10−5, 1010] and u ∈ [10−12, 1011].

Acknowledgments

This research was supported by the project Enabling Green E-science for the SKA Research Infrastructure (ENGAGE SKA), reference POCI-01-0145-FEDER-022217, funded by COMPETE 2020 & FCT, Portugal.

References


© ESO 2018

All Figures

thumbnail Fig. 1.

Variation of Integral (5) with κ as function of u. The solid black line represents eu which is overlapped by the integral as κ → ∞.

Open with DEXTER
In the text
thumbnail Fig. 2.

Temperature-averaged Gaunt factors calculated for κ = 2, 3, 5 and MB (black line in both panels) distributions of electrons for the range 10−5γ2 ≤ 1010 (top panel) and a magnification of the region γ2 ∈ [10−2, 106] and ⟨gf f(γ2, u)⟩ ∈ [0.9, 1.5] (bottom panel).

Open with DEXTER
In the text
thumbnail Fig. 3.

Temperature-averaged Gaunt factors calculated for κ = 10, 15, 25, and 50 (left panel), 100, 500, 1000, and 1200 (right panel) and MB (black line in both panels) distributions of electrons for the range 10−5γ2 ≤ 1010. We note the slow approach ⟨gf f(γ2, u)⟩ to the Maxwellian values for u > 10 and γ2 < 10 for large values of the κ parameter.

Open with DEXTER
In the text
thumbnail Fig. 4.

Total free-free Gaunt factor calculated for κ(2, 3, 5, 10, 15, and 25) and MB distributions.

Open with DEXTER
In the text
thumbnail Fig. 5.

Magnification of the total free-free Gaunt factor in two regions in γ2 and in the ⟨gf f(γ2)⟩ profile shown in Fig. 4 but for κ = 50, 100, 500, and 1000. The solid black line refers to the Maxwellian value.

Open with DEXTER
In the text

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