Calculation of molecular line intensity in stellar atmospheres

TL;DR

This paper discusses methods for calculating molecular line intensities in stellar atmospheres, introduces a new spectrum synthesis code for cool stars, and demonstrates its application to various spectral features.

Contribution

It presents a comprehensive description of molecular line calculations and introduces a new, publicly available spectrum synthesis code for cool star analysis.

Findings

Successful application to molecular bands in UV, visible, and near-infrared

Comparison with reference star spectra validates the methods

User-friendly tools facilitate spectrum synthesis for cool stars

Abstract

Molecular line intensity calculations are not a straightforward task. We present a description of the basics for including molecular linesin synthetic spectra, and of the input data needed. We aim both at describing ways in which molecular lines are computed in the context of photospheres of F-G-K stars, and to present a new online available version of our code for spectrum synthesis of cool stars. We apply calculations to molecular bands in the ultraviolet, visible and near-infrared main features, and comparisons with spectra of reference stars are shown. We provide user-friendly tools for the free use of the code PFANT. The code is available at http://trevisanj.github.io/PFANT.

Tables

Table 1. Table 1: Molecular lines included in PFANT and respective sources of data: line lists, dissociation potential D o , electronic oscillator strength f el .

Molecule	System	No. lines	$\lambda \lambda$(nm)	No. $(v^{\prime },v^{\prime \prime })$	Linelists	$D_{\circ }$(eV)	$f_{\mathrm{el}}$ (ref.)
MgH	A2$\Pi -$X2$\Sigma$	1,945	470$-$609	13	(1)	1.34 (14)	0.2793 (17)
C2 Swan	d3$\Pi -$a3$\Pi$	11,254	428$-$677	35	(2)	6.21 (15)	0.033 (18)
CN blue	B2$\Sigma -$X2$\Sigma$	92,851	300$-$600	197	(3)	7.72 (16)	0.0388 (19)
CN red	A2$\Pi -$X2$\Sigma$	23,828	483$-$2715	88	(4)	”	0.00676 (20)
CH-AX	A2$\Delta -$X2$\Pi$	10,137+13,830†	321$-$787	20+31†	(5)	3.46 (14)	0.005257 (21)
CH-BX	B2$\Sigma -$X2$\Pi$	2,016	360$-$683	10	(5)	”	0.0025 (22)
CH-CX	C2$\Sigma -$X2$\Pi$	2,829	269$-$425	12	(5)	”	0.00595 (23)
CO-NIR	X1${\Sigma }^{+}$	7,088	1557$-$5701	63	(6)	11.09 (14)	– (24)
NH blue	A3$\Pi -$X3$\Sigma$	8,599	300$-$600	55	(7)	3.47 (14)	0.008 (22)
OH blue	A2$\Sigma -$X2$\Pi$	6,018	300$-$540	46	(3)	4.39 (14)	0.0008 (22)
OH-NIR	X2$\Pi$	2028	746$-$25294	43	(8,9)	”	– (24)
FeH	A4$\Delta -$X4$\Delta$	2,705	778$-$1634	9	(10)	1.63 (16)	0.001 (25)
TiO $\gamma$	A3$\Phi -$X3$\Delta$	26,007	622$-$879	23	(11,12,13)	6.87 (14)	0.15 (11)
TiO ${\gamma }^{\prime }$	B3$\Pi -$X3$\Delta$	219,367	500$-$915	81	(11,12,13)	”	0.14 (11)
TiO $\alpha$	C3$\Delta -$X3$\Delta$	360,726	388$-$837	79	(11,12,13)	”	0.12 (11)
TiO $\beta$	c1$\Phi -$a1$\Pi$	91,805	431$-$796	63	(11,12,13)	”	0.006 (11)
TiO $\delta$	b1$\Pi -$a1$\Delta$	189,019	622$-$1480	66	(11,12,13)	”	0.05 (11)
TiO $ϵ$	E3$\Pi -$X3$\Delta$	253,755	641$-$1341	61	(11,12,13)	”	0.014 (11)
TiO $\varphi$	b1$\Pi -$d1$\Sigma$	105,082	664$-$1800	65	(11,12,13)	”	0.052 (11)

Equations

$$\begin{multlined}\kappa_{mol}\;=\\ {\pi e^{2}\over m_{\rm e}c^{2}}\;\lambda^{2}_{0J^{{}^{\prime}}J^{{}^{\prime\prime}}}\;f\;N^{{}^{\prime\prime}}_{nvj}\;{H(a,v)\over\sqrt{\pi}\;\Delta\lambda_{D}}\;(1-e^{\;{-hc\over\lambda_{0J^{{}^{\prime}}J^{{}^{\prime\prime}}}kT}})\end{multlined}\kappa_{mol}\;=\\ {\pi e^{2}\over m_{\rm e}c^{2}}\;\lambda^{2}_{0J^{{}^{\prime}}J^{{}^{\prime\prime}}}\;f\;N^{{}^{\prime\prime}}_{nvj}\;{H(a,v)\over\sqrt{\pi}\;\Delta\lambda_{D}}\;(1-e^{\;{-hc\over\lambda_{0J^{{}^{\prime}}J^{{}^{\prime\prime}}}kT}})$$

$$\Delta {\lambda }_{\mathrm{D}}=\frac{{\lambda }_0}{c}\sqrt{\frac{2kT}{M}+v_t^2}$$

$$E_{J^{\prime }{J}^{\prime \prime }}=N_{J^{\prime }}h{\nu }_{J^{\prime }{J}^{\prime \prime }}{A}_{J^{\prime }{J}^{\prime \prime }}$$

$$\begin{multlined}A_{J^{\prime}J^{\prime\prime}}={64\pi^{4}\nu^{3}\over 3hc^{3}d}{\sum_{M^{\prime}M^{\prime\prime}}\mid{\bf R}_{n^{\prime}v^{\prime}\Lambda^{\prime}J^{\prime}p^{\prime},n^{\prime\prime}v^{\prime\prime}\Lambda^{\prime\prime}J^{\prime\prime}p^{\prime\prime}}\mid^{2}}=\\ {64\pi^{4}\nu^{3}\over 3hc^{3}d}\mathcal{S}\end{multlined}A_{J^{\prime}J^{\prime\prime}}={64\pi^{4}\nu^{3}\over 3hc^{3}d}{\sum_{M^{\prime}M^{\prime\prime}}\mid{\bf R}_{n^{\prime}v^{\prime}\Lambda^{\prime}J^{\prime}p^{\prime},n^{\prime\prime}v^{\prime\prime}\Lambda^{\prime\prime}J^{\prime\prime}p^{\prime\prime}}\mid^{2}}=\\ {64\pi^{4}\nu^{3}\over 3hc^{3}d}\mathcal{S}$$

$$R_{ij}={\int }_{-\infty }^{\infty }{\Psi }_i^{\ast }P{\Psi }_{j}dV$$

$$R_{ij}=<{\Psi }_i\mid \mathbf{P}\mid {\Psi }_j>,$$

$$\begin{multlined}\sum_{M^{\prime}M^{\prime\prime}}\mid{{\bf R}_{n^{\prime\prime}v^{\prime\prime}\Lambda^{\prime\prime}p^{\prime\prime}J^{\prime\prime}}^{n^{\prime}v^{\prime}\Lambda^{\prime}p^{\prime}J^{\prime}}\mid^{2}}=\sum_{M^{\prime}M^{\prime\prime}}{\mid{\bf R}_{nv\Lambda pJ}\mid^{2}}=\\ \mathcal{S}_{nv\Lambda pJ}={R^{\rm nv}_{\rm rad}\mathscr{S}_{\Lambda pJ}}\end{multlined}\sum_{M^{\prime}M^{\prime\prime}}\mid{{\bf R}_{n^{\prime\prime}v^{\prime\prime}\Lambda^{\prime\prime}p^{\prime\prime}J^{\prime\prime}}^{n^{\prime}v^{\prime}\Lambda^{\prime}p^{\prime}J^{\prime}}\mid^{2}}=\sum_{M^{\prime}M^{\prime\prime}}{\mid{\bf R}_{nv\Lambda pJ}\mid^{2}}=\\ \mathcal{S}_{nv\Lambda pJ}={R^{\rm nv}_{\rm rad}\mathscr{S}_{\Lambda pJ}}$$

$${\Psi }_{\mathrm{total}}={\Psi }_{\mathrm{electronic}}\times {\Psi }_{\mathrm{nuclear}}$$

$${\mathbf{R}}_{\mathrm{rad}}={\int }_{-\infty }^{\infty }{\Psi }_{\mathrm{e}}^{{}^{\prime }\ast }{\Psi }_{\mathrm{v}}^{{}^{\prime }\ast }({\mathbf{P}}_{\mathrm{e}}+{\mathbf{P}}_{\mathrm{N}}){\Psi }_{\mathrm{e}}^{{}^{\prime \prime }}{\Psi }_{\mathrm{v}}^{{}^{\prime \prime }}dV$$

$$\begin{multlined}{\bf R}_{\rm rad}=\int_{-\infty}^{\infty}\Psi_{\rm v}^{{}^{\prime}*}(\Psi_{\rm e}^{{}^{\prime}*}{\bf P}_{\rm e}\Psi_{\rm e}^{{}^{\prime\prime}}dV_{\rm e})\Psi_{\rm v}^{{}^{\prime\prime}}dr+\\ \int_{-\infty}^{\infty}\Psi_{\rm v}^{{}^{\prime}*}{\bf P}_{N}(\Psi_{\rm e}^{{}^{\prime}*}\Psi_{\rm e}^{{}^{\prime\prime}*}dV_{\rm e})\Psi_{\rm v}^{{}^{\prime\prime}}dr\end{multlined}{\bf R}_{\rm rad}=\int_{-\infty}^{\infty}\Psi_{\rm v}^{{}^{\prime}*}(\Psi_{\rm e}^{{}^{\prime}*}{\bf P}_{\rm e}\Psi_{\rm e}^{{}^{\prime\prime}}dV_{\rm e})\Psi_{\rm v}^{{}^{\prime\prime}}dr+\\ \int_{-\infty}^{\infty}\Psi_{\rm v}^{{}^{\prime}*}{\bf P}_{N}(\Psi_{\rm e}^{{}^{\prime}*}\Psi_{\rm e}^{{}^{\prime\prime}*}dV_{\rm e})\Psi_{\rm v}^{{}^{\prime\prime}}dr$$

$${\mathbf{R}}_{\mathrm{e}}={\int }_{-\infty }^{\infty }{\Psi }_{\mathrm{e}}^{{}^{\prime }\ast }{\mathbf{P}}_{\mathrm{e}}{\Psi }_{\mathrm{e}}^{{}^{\prime \prime }}d{V}_{\mathrm{e}}=<{\Psi }_{\mathrm{e}}^{{}^{\prime }\ast }\mid {\mathbf{P}}_{\mathrm{e}}\mid {\Psi }_{\mathrm{e}}^{{}^{\prime \prime }}>$$

$${\mathbf{R}}_{\mathrm{rad}}={\int }_{-\infty }^{\infty }{\Psi }_{\mathrm{v}}^{{}^{\prime }\ast }{\mathbf{R}}_{\mathrm{e}}{\Psi }_{\mathrm{v}}^{{}^{\prime \prime }}dr$$

$${\mathbf{R}}_{\mathrm{rad}}={\mathbf{R}}_{\mathrm{e}}{\int }_{-\infty }^{\infty }{\Psi }_{\mathrm{v}}^{{}^{\prime }\ast }{\Psi }_{\mathrm{v}}^{{}^{\prime \prime }}dr$$

$$q_{v^{\prime }{v}^{\prime \prime }}={\left|{\int }_{-\infty }^{\infty }{\Psi }_{\mathrm{v}}^{{}^{\prime }\ast }{\Psi }_{\mathrm{v}}^{{}^{\prime \prime }}dr\right|}^2,$$

$$R_{\mathrm{rad}}^{\mathrm{nv}}=R_{\mathrm{rad}}^{\mathrm{ev}}=\mid {\mathbf{R}}_{\mathrm{e}}{\mid }^2{q}_{v^{\prime }{v}^{\prime \prime }}$$

$$\mathcal{S}=\mid {\mathbf{R}}_{\mathrm{e}}{\mid }^2{q}_{v^{\prime }{v}^{\prime \prime }}{\mathcal{S}}_{J^{\prime }{J}^{\prime \prime }}$$

$$\sum {\mathcal{S}}_{J^{\prime }{J}^{\prime \prime }}(J)=(2-{\delta }_{0,{\Lambda }^{\prime }+{\Lambda }^{\prime \prime }})(2S+1)(2J+1),$$

$$\begin{multlined}f=f_{J^{\prime}J^{\prime\prime}}=f_{n^{\prime\prime}v^{\prime\prime}\Lambda^{\prime\prime}p^{\prime\prime}J^{\prime\prime}}^{n^{\prime}v^{\prime}\Lambda^{\prime}p^{\prime}J^{\prime}}=f_{nv{\Lambda}pJ}=\\ {8\pi^{2}m_{\rm e}\nu\over{3he^{2}d}}{\sum_{M^{\prime}M^{\prime\prime}}\mid{\bf R}_{n^{\prime}v^{\prime}\lambda^{\prime}J^{\prime}p^{\prime},n^{\prime\prime}v^{\prime\prime}\lambda^{\prime\prime}J^{\prime\prime}p^{\prime\prime}}\mid}{{}^{2}}=\\ {8\pi^{2}m_{\rm e}\nu_{J^{\prime}J^{\prime\prime}}\over{3he^{2}d}}\mid{\bf R}_{\rm e}\mid^{2}q_{v^{\prime}v^{\prime\prime}}\mathscr{S}_{J^{\prime}J^{\prime\prime}}={8\pi^{2}m_{\rm e}\nu\over{3he^{2}d}}\mathcal{S}\end{multlined}f=f_{J^{\prime}J^{\prime\prime}}=f_{n^{\prime\prime}v^{\prime\prime}\Lambda^{\prime\prime}p^{\prime\prime}J^{\prime\prime}}^{n^{\prime}v^{\prime}\Lambda^{\prime}p^{\prime}J^{\prime}}=f_{nv{\Lambda}pJ}=\\ {8\pi^{2}m_{\rm e}\nu\over{3he^{2}d}}{\sum_{M^{\prime}M^{\prime\prime}}\mid{\bf R}_{n^{\prime}v^{\prime}\lambda^{\prime}J^{\prime}p^{\prime},n^{\prime\prime}v^{\prime\prime}\lambda^{\prime\prime}J^{\prime\prime}p^{\prime\prime}}\mid}{{}^{2}}=\\ {8\pi^{2}m_{\rm e}\nu_{J^{\prime}J^{\prime\prime}}\over{3he^{2}d}}\mid{\bf R}_{\rm e}\mid^{2}q_{v^{\prime}v^{\prime\prime}}\mathscr{S}_{J^{\prime}J^{\prime\prime}}={8\pi^{2}m_{\rm e}\nu\over{3he^{2}d}}\mathcal{S}$$

$$\frac{A}{f}=\frac{8{\pi }^2{\nu }^2{e}^2}{c^3{m}_{\mathrm{e}}}$$

$$f_{v^{\prime }{v}^{\prime \prime }}=\frac{8{\pi }^2{m}_{\mathrm{e}}\nu }{3h{e}^2}{R}_{\mathrm{rad}}^{\mathrm{nv}}=\frac{8{\pi }^2{m}_{\mathrm{e}}{\nu }_{J^{\prime }{J}^{\prime \prime }}}{3h{e}^2}\mid {\mathbf{R}}_{\mathrm{e}}{\mid }^2{q}_{v^{\prime }{v}^{\prime \prime }},$$

$$f=\frac{f_{v^{\prime }{v}^{\prime \prime }}{\mathcal{S}}_{J^{\prime }{J}^{\prime \prime }}}{d}.$$

$$f_{n^{\prime \prime }{v}^{\prime \prime }{\Sigma }^{\prime \prime }{p}^{\prime \prime }{J}^{\prime \prime }}^{n^{\prime }{v}^{\prime }{\Sigma }^{\prime }{p}^{\prime }{J}^{\prime }}=\frac{f_{v^{\prime }{v}^{\prime \prime }}{\mathcal{S}}_{J^{\prime }{J}^{\prime \prime }}}{(2J^{\prime \prime }+1)}$$

$$\begin{multlined}\kappa_{mol}\;=\;{\pi e^{2}\over m_{\rm e}c^{2}}\;\lambda^{2}_{0J^{{}^{\prime}}J^{{}^{\prime\prime}}}\;f_{v^{\prime}v^{\prime\prime}}{\mathscr{S}_{\Lambda{pJ}}\over{(2J^{\prime\prime}+1)}}\;N^{{}^{\prime\prime}}_{nvj}\;\times\;\\ {H(a,v)\over\sqrt{\pi}\;\Delta\lambda_{D}}\;(1-e^{\;{-hc\over\lambda_{0J^{{}^{\prime}}J^{{}^{\prime\prime}}}kT}}),\end{multlined}\kappa_{mol}\;=\;{\pi e^{2}\over m_{\rm e}c^{2}}\;\lambda^{2}_{0J^{{}^{\prime}}J^{{}^{\prime\prime}}}\;f_{v^{\prime}v^{\prime\prime}}{\mathscr{S}_{\Lambda{pJ}}\over{(2J^{\prime\prime}+1)}}\;N^{{}^{\prime\prime}}_{nvj}\;\times\;\\ {H(a,v)\over\sqrt{\pi}\;\Delta\lambda_{D}}\;(1-e^{\;{-hc\over\lambda_{0J^{{}^{\prime}}J^{{}^{\prime\prime}}}kT}}),$$

$$\begin{multlined}{n({\rm A})n({\rm B})\over n({\rm A})}={(\sum_{i}g_{i}e^{-\epsilon_{i}/kT})_{\rm A}(\sum_{i}g_{i}e^{-\epsilon_{i}/kT})_{\rm B}\over(\sum_{i}g_{i}e^{-\epsilon_{i}/kT})_{{\rm A}{\rm B}}}=\\ {Q_{{\rm total}_{\rm A}}Q_{{\rm total}_{\rm B}}\over Q_{{\rm total}_{{\rm A}{\rm B}}}}\end{multlined}{n({\rm A})n({\rm B})\over n({\rm A})}={(\sum_{i}g_{i}e^{-\epsilon_{i}/kT})_{\rm A}(\sum_{i}g_{i}e^{-\epsilon_{i}/kT})_{\rm B}\over(\sum_{i}g_{i}e^{-\epsilon_{i}/kT})_{{\rm A}{\rm B}}}=\\ {Q_{{\rm total}_{\rm A}}Q_{{\rm total}_{\rm B}}\over Q_{{\rm total}_{{\rm A}{\rm B}}}}$$

$$\frac{n(\mathrm{A})n(\mathrm{B})}{n(\mathrm{A})}=\frac{Q_{\mathrm{A}}{Q}_{\mathrm{B}}}{Q_{\mathrm{AB}}}{e}^{-({ϵ}_{0_{\mathrm{A}}}+{ϵ}_{0_{\mathrm{B}}}-{ϵ}_{0_{\mathrm{AB}}})/kT}$$

$$\begin{multlined}{n({\rm A})n({\rm B})\over n({\rm A})}=\left({2\pi mkT\over h^{2}}\right)^{3/2}{V_{A}V_{B}\over V_{\rm AB}}\mu_{\rm AB}^{3/2}\;\times\\ {Q_{\rm int_{A}}Q_{\rm int_{B}}\over Q_{\rm int_{AB}}}e^{-D_{0}/kT},\end{multlined}{n({\rm A})n({\rm B})\over n({\rm A})}=\left({2\pi mkT\over h^{2}}\right)^{3/2}{V_{A}V_{B}\over V_{\rm AB}}\mu_{\rm AB}^{3/2}\;\times\\ {Q_{\rm int_{A}}Q_{\rm int_{B}}\over Q_{\rm int_{AB}}}e^{-D_{0}/kT},$$

$$K_{\mathrm{AB}}^P=\frac{p_{\mathrm{A}}{p}_{\mathrm{B}}}{p_{\mathrm{AB}}}$$

$$\frac{p_{\mathrm{A}}{p}_{\mathrm{B}}}{p_{\mathrm{AB}}}={\left(\frac{2\pi kT}{h^2}{\mu }_{\mathrm{AB}}\right)}^{3/2}\frac{Q_{in{t}_{\mathrm{A}}}{Q}_{in{t}_{\mathrm{B}}}}{Q_{in{t}_{\mathrm{AB}}}}kT{e}^{\frac{-D_0}{kT}}$$

$$\frac{(N_{\mathrm{AB}}{)}_{nvJ}}{N_{\mathrm{AB}}}=\frac{(Q_{in{t}_{\mathrm{AB}}}{)}_{nvJ}}{Q_{in{t}_{\mathrm{AB}}}}$$

$$\frac{(N_{\mathrm{AB}}{)}_{nvJ}}{N_{\mathrm{AB}}}={\left(\frac{2\pi kT}{h^2}{\mu }_{\mathrm{AB}}\right)}^{-3/2}\frac{N_A{N}_B}{Q_{in{t}_A}{Q}_{in{t}_B}}{Q}_{in{t}_{\mathrm{AB}}}{e}^{\frac{D_0}{kT}}$$

Full text

1 Introduction

Molecular bands in the spectra of cool stars need to be considered in the calculation of synthetic spectra of cool stars, first of all because they are present all over the spectra, affecting considerably the continuum, and as well as because there are several strong lines. Bands involving C, N, O in particular, are key indicators of many important aspects of both stellar evolution and chemical evolution. For example, photometric filters placed to measure CH, NH, CN, OH, as used in Piotto et al. (2015), can reveal multiple stellar populations in globular clusters.

The main nitrogen indicator used in the literature are CN bands, among which the strongest ones are in the near-UV, in particular CN(0,0) at 388.3 nm. For the use of CN, one has to previously derive C, available in the optical from the CH G-bandhead at 431 nm or a weak C2(0,0) bandhead at 563.5 nm. In the near-UV there is the unique available strong NH bandhead at 336 nm. The NH band measurement in important samples would allow a direct measurement of nitrogen (e.g. Pasquini et al. 2008). Note on the other hand, that discrepancies have been found in the nitrogen abundances derived from CN and NH bandheads, still unsolved (see Spite et al. 2005).

Oxygen has several strong OH lines occurring in the near-UV, at \lambda$$\mathrel{\hbox{\raise 2.15277pt\hbox{<}\hbox to0.0pt{\hss\lower 2.15277pt\hbox{\sim}}}} 330 nm, and in the near-infrared in the H band. In metal-poor turn-off dwarf stars, the UV OH lines are the only measurable oxygen indicator. Furthermore, molecular bands of CH and CN in the visible and near-infrared are the best way of deriving the isotopic composition of in particular $12$C/$13$C (e.g. Smith et al. 2002), which is an indicator of stellar mixing (e.g. Smiljanic et al. 2009).

Therefore the inclusion of molecular lines in synthetic spectra calculations of stars cooler than \mathrel{\hbox{\raise 2.15277pt\hbox{<}\hbox to0.0pt{\hss\lower 2.15277pt\hbox{\sim}}}}7000 K is of utmost importance. Line lists of molecular bands come however in different formats, and as well the different codes use these lists adapted in different ways. The main difference in using molecular line lists is the use of either the Einstein $A$ coefficients, such as given in Goorvitch et al. (1994) for the near-infrared (NIR) lines of CO X1$\Sigma^{+}$ system, which are a sum of all ingredients, or when this is not available, the different constituents that produce the line. In the present paper we aim to show how to use these different data.

Codes for spectrum synthesis available include about a dozen of them, among which SYNTHE (Kurúcz 1970, 1993a), MOOG (Sneden 1973), the codes BSYN (Edvardsson et al. 1993), and Turbospectrum (Alvarez & Plez 1998), from the Uppsala group, and SME (Valenti & Piskunov 1996; Piskunov & Valenti 2017), among a few others. In the present paper we give details on the updated version of the spectrum synthesis code PFANT, that is now available online.

In Sect. 2 we describe the calculation of molecular lines of diatomic molecules. In Sect. 3 we describe the spectrum synthesis code PFANT and the line lists used. In Sect. 4 a short summary is given.

2 Molecular absorption coefficient

In the 60s and early 70s a few authors provided the basis for the understanding and correct calculation of molecular line formation. There are some inconsistencies in normalization matters among these papers, which made the subject a hot topic at that time. These include Schadee (1964, 1967, 1975, 1978), Tatum (1967), and Whiting & Nicholls (1974). A particularly clear work on normalization factors was given later by all these authors gathered in Whiting et al. (1980). In the following, a description on the calculation of diatomic molecular lines is given, trying to be as straightforward as possible, and preparing for the next Section where these equations are included in the code PFANT presented in this paper.

The opacity of a molecular line as a result of a transition between two rotational levels is given by:

[TABLE]

where $f$ = molecular oscillator strength, $\lambda^{2}_{0J^{\prime}J^{\prime\prime}}$ is the line central wavelength, $J$ = rotational quantum number, $v$ = vibrational quantum number, $n$ = electronic quantum number, $N^{\prime\prime}_{nvj}$ = population of the lower level, and $\Delta\lambda_{\rm D}$ is the Doppler width given by

[TABLE]

where $v_{t}$ = microturbulence velocity. The constants correspond to $c$ = light speed (2.997$\times$108m/s), $m_{\rm e}$ = mass of hydrogen (1.6737236$\times$10*-24g), $e$ = electron charge (1.6022$\times$10-19*C). Finally, $H(a,v)$ = Hjerting’s function for line wing broadening, where $v$ = ${\Delta\lambda\over\Delta\lambda_{\rm D}}$ the damping constant $\Gamma$ = $\lambda^{2}$/4$\pi$c$\Delta\lambda_{\rm D}$, where $a$ = damping parameter (see e.g. Gray 2005).

2.1 Line strength and Einstein coefficients

Einstein (1917) introduced the $A$ and $B$ coefficients to describe spontaneous emission, and induced absorption and emission. The Einstein $A$ coefficient is defined in terms of spontaneous emission from level 2 to level 1 by the probability: $dW=A_{21}dt$. In a radiation field with a radiation density $\rho_{\nu}$ at frequency $\nu$, a transition from 1 to 2 has probability $dW=B_{12}\rho_{\nu}dt$, and also induced emission with the probability $dW=B_{21}{\rho}dt$. In a system with $N_{1}$, $N_{2}$ atoms in levels 1 and 2, the total rate of transitions is: $W_{21}=A_{21}N_{2}$ for spontaneous emission, $W_{12}=B_{12}^{\nu}\rho_{\nu}N_{1}$ for induced absorption, and $W_{21}=B_{21}^{\nu}\rho_{\nu}N_{2}$ for induced emission. He also showed that $B_{21}^{\nu}=(\pi^{2}c^{3}/\-{h}\nu_{21}^{3})A_{21}$ and $B_{12}^{\nu}=(g_{2}/g_{1})B_{21}^{\nu}$, with $g$ = degeneracy factor.

The line intensity resulting from a spontaneous photon emission in a rotational transition, from levels ($J^{\prime}$, $J^{\prime\prime}$) is

[TABLE]

where $J^{\prime}$, $J^{\prime\prime}$ are the rotational quantum numbers of upper and lower levels of the transition, respectively.

The Einstein probability coefficient for a rotational transition $J^{\prime}J^{\prime\prime}$ between two levels ($n^{\prime}v^{\prime}\Lambda^{\prime}J^{\prime}p^{\prime}$, $n^{\prime\prime}v^{\prime\prime}\Lambda^{\prime\prime}J^{\prime\prime}p^{\prime\prime}$) will be:

[TABLE]

where $\mathcal{S}$ = line strength; $\Lambda$ = electronic sublevel of spin; $p$ = parity (due to dedoubling of rotational states); $M$ = magnetic quantum number; $d$ = degeneracy; $R_{ij}$ = elements of the matrix of electric dipole moment = transition moments; ${\mid{\bf R}\mid}^{2}$ = transition probability. Note that each rotational level $J$ has $(2J+1)$ sublevels characterized by a quantum number $M$. Each of the two states of $\Lambda$ doubling is considered a distinct rotational state (according to Whiting & Nichols 1974 and contrariwise to Tatum 1967 and Schadee 1971).

2.2 Transition Moments R

The transition moment R is defined as:

[TABLE]

or, in Dirac’s notation,

[TABLE]

corresponding to transition moments for a transition between two states $(i,j)$ such as $(J^{\prime},J^{\prime\prime})$ or $(v^{\prime},v^{\prime\prime})$ etc., where $\Psi_{i}$, $\Psi_{j}$ are total wave functions of states $i$ and $j$, and P = electric dipole moment.

Hönl & London (1925) have shown that the wave function can be separated in radial and angular parts, such that the transition moment can be written as:

[TABLE]

The superscript “nv” symbolizes the dependence of $R^{\rm nv}_{\rm rad}$ on the electronic and vibrational eigenfunctions.

2.2.1 The Vibronic transition moment R${}^{\rm nv}_{\rm rad}$

The Born-Oppenheimer approximation (1927) allows the wave function of a molecule to be broken into two components, the electronic and the nuclear (vibrational and rotational), given that the electronic wave function varies very slowly with the nuclear coordinates. Therefore

[TABLE]

The dipole moment can thus be broken into two components, electronic and nuclear, i.e., ${\bf P}={\bf P}_{\rm e}+{\bf P}_{\rm N}$. The transition moment becomes then

[TABLE]

and $dV$ is defined by electronic and nuclear coordinates: $dV=dV_{\rm e}dV_{\rm N}$ or, since $\Psi_{\rm v}$ and PN are functions only of the internuclear distance $r$, the formula can be reduced to a radial dependence only, therefore

[TABLE]

The wave functions of different electronic states are orthogonal, therefore the second term of this equation can be canceled. The electronic transition moment can be defined as:

[TABLE]

[TABLE]

where ${\bf R}_{\rm e}$ can be measured in laboratory experiments, or computed theoretically. As a first approximation, we can admit that ${\bf R}_{\rm e}$ = constant, and a mean value can be adopted:

[TABLE]

where the Franck-Condon factor is

[TABLE]

and the band intensity (force de bande) is

[TABLE]

2.2.2 The line strength $\mathcal{S}$

The line strength for rotational lines will be (Whiting et al. 1980):

[TABLE]

with $\mathscr{S}_{J^{\prime}J^{\prime\prime}}$ = $\mathscr{S}_{\Lambda pJ}$ = Hönl-London factor (HLF), giving the relative intensity of the different rotational transitions. Depending on the Hund’s coupling associated with each case, formulae for computing the HLF are tabulated by Kovács (1969).

The normalization of the HLF is as follows, valid for all cases of $\Lambda$-S coupling (Whiting et al. 1980):

[TABLE]

where the sum is carried out for all branches and fixed $J$. The Kronecker delta $\delta_{0,\Lambda^{\prime}+\Lambda^{\prime\prime}}$ equals 1 for $\Sigma-\Sigma$ transitions and 0 otherwise. The term $(2-\delta_{0,\Lambda^{\prime}+\Lambda^{\prime\prime}})(2S+1)$ represents the degeneracy of the electronic state, $(2J+1)$ the degeneracy of each rotational level, and $(2S+1)$ represents the multiplicity of the transition: 1 for singlet, 2 for doublet, etc.

2.2.3 The oscillator strength $f$

The oscillator strength $f$ is defined as:

[TABLE]

Therefore, by combining equations (4) and (18), we get

[TABLE]

Designating $f_{v^{\prime}v^{\prime\prime}}$ = vibronic oscillator strength, we have:

[TABLE]

therefore

[TABLE]

The degeneracy factor $d$ is $(2J^{\prime\prime}+1)$ for a single rotational line:

[TABLE]

The molecular absorption coefficient can then be expressed by:

[TABLE]

2.3 Calculation of the lower level population $N^{{}^{\prime\prime}}_{nvj}$

Consider the equilibrium reaction A + B $\rightleftharpoons$ AB, between two atoms A and B, of the AB molecule, which is expressed by the relation:

[TABLE]

where $g_{i}$ = statistical weight, $n({\rm X})$ = number of particles X in a volume V(X), $Q_{\rm total}=Q_{\rm translational}Q_{\rm internal}=(\sum_{i}g_{i}e^{-\epsilon_{i}/kT})$ = total partition function, and the energies ($\epsilon_{i}$)A, ($\epsilon_{i}$)B are defined with respect to the fundamental state, i.e., $(\epsilon_{i})_{\rm A,B}=[(\epsilon_{0})_{\rm fundamentalstate}+(\epsilon_{i})_{\rm exc})]_{\rm A,B}$. The AB molecule energies $\epsilon_{i}/kT$AB are given relative to the vibrational fundamental state (quantum number $v=0$), of the fundamental electronic state of the molecule: $(\epsilon_{i})_{\rm AB}=[(\epsilon_{0})_{\rm fundamentalstate}(v=0)+(\epsilon_{i})_{\rm exc}]_{\rm AB}$.

By rewriting eq. (24), we have

[TABLE]

where $\epsilon_{0_{A}}+\epsilon_{0_{B}}-\epsilon_{0_{AB}}=D_{0}$, and $D_{0}$ is the dissociation energy of molecule AB.

Noting that $Q_{\rm translational}=({2\pi mkT\over h^{2}})^{3/2}V$, we get:

[TABLE]

where $\mu_{\rm AB}=m_{\rm A}m_{\rm B}/m_{\rm AB}$ is the reduced mass.

On the other hand, the definition of the dissociation pressure constant is:

[TABLE]

where $p_{\rm A}$, $p_{\rm B}$, and $p_{\rm AB}$ are the partial pressures of particles A, B, and AB.

[TABLE]

Considering now the Boltzmann equation for the population of the level $nvJ$, $(N_{\rm AB})_{nvJ}$ being the number of molecules AB in the electronic level $n$, vibrational level $v$, and rotational level $J$, per volume unit, and $N_{\rm AB}$ the total number of molecules AB per volume unit, we have:

[TABLE]

and one obtains:

[TABLE]

2.4 The partition functions Q${}_{int_{\rm AB}}$

Considering that for a molecule the electronic, vibrational, and rotational energies are independent of each other, Qint may be written as the product:

[TABLE]

where:

[TABLE]

and,

(a) gel, $g_{\rm vibr}$, and $g_{rot}$ are the statistical weights of the electronic, vibrational, and rotational levels, respectively. The product of the statistical weight $g_{\rm el}$, $g_{\rm vibr}$, and $g_{\rm rot}$ are given for each case of molecular transition by Tatum (1967);

(b) $T_{\rm e}$ = term value of electronic energy of a multiplet, in first aproximation it is given by

[TABLE]

where $T_{0}$ = value of the term if the spin is negligible; $A$ = constant of spin coupling; $\Lambda$ and $\Sigma$ = respectively, orbital electronic angular momentum and spin angular momentum, projected in the internuclear axis (not to confound with other $A$ and $\Lambda$ defined in Sect. 2.1).

(c) $F(J)$ = term of rotational energy, given by

[TABLE]

where $B_{v}$ = rotational constant, $D_{v}$ = centrifugal distortion constant.

Note: $F(J)$ is given with respect to the lowest level, i.e., $F(J)$ is in reality $F(J)-F(1)$.

(d) $G(v)$ = term of vibrational energy.

A diatomic molecule, by the vibrational movements of its nuclei, can be treated as an anharmonic oscillator with a potential $U$ approximately given by:

[TABLE]

where: $g\ll f$ and $r_{e}$ is the internuclear distance in equilibrium.

Introducing $U$ in the wave equation, one obtains the following energy eigenvalues:

[TABLE]

The values of the anharmonic constants $\omega_{e}$, $\omega_{e}$xe, and $\omega_{e}$ye are tabulated in the literature (for example, in Huber & Herzberg 1979, or in the NIST Chemistry web book at http://webbook.nist.gov/chemistry).

Note: $G(v)$ is given with respect to the zero point energy, i.e., $v$ = 0, so in reality one uses $G_{0}(v)=G(v)-G(0)$.

A discussion on the multiple forms used for partition functions is presented in Popovas (2014).

3 PFANT: calculation of synthetic spectra

The spectra were computed using the code PFANT (http://trevisanj.github.io/PFANT). The first version of this Fortran code, named FANTOM, was developed by Spite (1967) for the calculation of atomic lines. Barbuy (1982) included the calculation of molecular lines, implementing the dissociative equilibrium by Tsuji (1973) and molecular line computations as described in Cayrel et al. (1991). It has been further improved for calculations of large wavelength coverage and inclusion of hydrogen lines as described in Barbuy et al. (2003), where M.N. Perrin had an important rôle, wherefrom comes the P in PFANT. Coelho et al. (2005) unified the code to contain the full line lists from the ultraviolet (Castilho et al. 1999), together with the visible (Barbuy et al. 2003), and near-infrared (Meléndez et al. 2001).

Given a stellar model atmosphere and lists of atomic and molecular lines, the code computes a synthetic spectrum assuming local thermodynamic equilibrium (LTE).

3.1 New changes to the PFANT code

During 2015-2017, the code was entirely upgraded with homogeneization of language to Fortran 2003, optimization for speed, improved error reporting in case of problems with the input data files, and command-line configuration options. The code now has a manual including installation and tutorial sections.

Important existing accessory codes in Fortran were also upgraded and incorporated to form a four-step workflow as follows (Figure 1):

innewmarcs – interpolate a grid of MARCS atmospheric models (Gustafsson et al. 2008), to generate star-specific model (the original code comes from the Meudon research group); 2. 2.

hydro2 – calculate hydrogen lines profiles (Praderie 1967); 3. 3.

pfant – calculate synthetic spectrum; 4. 4.

nulbad – convolve synthetic spectrum with a Gaussian profile to simulate spectral measurement.

The list above describes commands to be typed in a computer terminal. All commands may be invoked with the --help option.

3.2 Input and output data

This section describes the most relevant data files needed or generated during PFANT code execution. The following subsections refer to these files by their default names, which can be changed via command-line, if necessary.

3.2.1 Stellar parameters and abundances

Stellar parameters and chemical abundances are defined in files main.dat and abonds.dat. The code available for download includes reference solar abundances that can be chosen to be those from Asplund et al. (2009), Grevesse et al. (1996), Scott et al. (2015a,b) and Grevesse et al. (2015), or Grevesse et al. plus A(O)=8.76 from Steffen et al. (2015).

3.2.2 Model atmospheres

innewmarcs, the grid interpolation code, allows to call for a chosen grid among the MARCS model atmospheric grids, and interpolates for the stellar parameters needed, creating files named modeles.mod and modeles.opa. If these parameters, i.e., $\mathbf{v}$ = (Teff, log g, [Fe/H]) are outside the grid, the code does not extrapolate, but either stops, or if forced to run (innewmarcs --allow T), projects $\mathbf{v}$ the onto closest grid “wall”, then interpolates.

It is possible to replace the grid among the different options given in MARCS. To create a new grid, it suffices to download the atmospheric models of interest (files “.mod” and “.opa”) from the MARCS website (http://marcs.astro.uu.se), place them in a single folder, and run create-grid.py (a Python script included with the pyfant package, described in Sect. 3.3).

3.2.3 Hydrogen lines profiles

The Balmer, Paschen and Brackett series line data are included in file hmap.dat. For each hydrogen line, the central wavelength, the levels of the transition, excitation potential, and a constant related with the oscillator strength are tabulated. Hydrogen lines profiles are computed through a revised version of the code presented in Praderie (1967) (hydro2), generating separate files for each hydrogen line.

It is well-known that hydrogen lines in red giant stars cannot be well-reproduced. Cayrel et al. (1991) suggested that the bottom of the lines would be due to chromospheric layers, which are not taken into account in stellar atmosphere models. For higher order Balmer lines the overlapping of lines is taken into account, but combined with the Balmer decrement present in the continuum opacities, we get a bump in the continuum at 3650-3662 Å. This particular region should be used with caution.

3.2.4 Continuum opacity

The code PFANT computes the continuum in pure absorption. It takes into account absorption due to H*-, H, H${}_{2}^{+}$, He and He+*, Rayleigh diffusion by H and H2 and Thomson diffusion by electrons. A validation of these continuum calculations was carried out by Trevisan et al. (2011), through comparisons with the Uppsala code BSYN (Edvardsson et al. 1993, and updates since then). All steps of the calculation were carefully compared: optical depths of lines, continuum opacities $\kappa_{\rm c}$, line broadening, and final abundances.

The inclusion of continuum from scattering is taken into account by adding the scattering opacity from the MARCS models. This is illustrated in Figure 2 for the stellar parameters of star HP1-2115 with ($T_{\rm eff}$=4530, ${\rm log}~{}g$=2.0, [Fe/H]=-1.0, $v_{\rm t}$=1.45) (Barbuy et al. 2016), in the near-ultraviolet, where this effect is more pronounced. MARCS opacities are interpolated by innewmarcs using a grid of atmospheric models with opacities included (grid.moo), generating file modeles.opa, which is identical in structure to “.opa” files downloaded from MARCS website. A similar calculation was presented in Figure 1 of Barbuy et al. (2011), based on calculations by B. Plez using the code Turbospectrum.

3.2.5 Atomic line lists

Atomic line lists can be converted from VALD3 (extended format) (Piskunov et al. 1995, Ryabchikova et al. 2015) lists of atomic lines. The VALD atomic line list started with basically the list by Kurúcz (1995, 2017), with progressive implementation of improvements coming from different sources. The default atomic line list in the range 3000 - 10.000 Å is from VALD, where the collisional broadening is obtained by adopting a general formula for the van der Waals broadening (e.g. Gray 2005). Better broadening constant values can be obtained through the code from Anstee & O’Mara (1995), Barklem & O’Mara (1997), Barklem et al. (1998), and Barklem et al. (2000) (this series of papers is referred to hereafter as ABO) for neutral lines.

The list of lines in the near-infrared adopted is from Meléndez & Barbuy (1999). An updated such line list is given by Shetrone et al. (2015).

Other options of line lists are available, in particular our own list that includes damping parameters from ABO for most lines in the visible region, and lists of hyperfine structure for several elements, that can be obtained by contacting the authors.

3.2.6 Molecular line lists

The list of molecules, their system, wavelength coverage, number of vibrational transitions $(v^{\prime},v^{\prime\prime})$, the constants dissociation potential $D_{0}$ (eV) and the electronic oscillator strength $f_{\rm el}$, that is related to the transition moment Re, with corresponding references are given in Table LABEL:listmol. This list of molecular lines is included in the code package, in file molecules.dat.

In the near-ultraviolet and visible, the line lists from Kurúcz (1993b) (http://kurucz.harvard.edu/molecules.html) were adopted (and transformed to our format) for OH, and CN blue. CH lines are from Luque & Crosley (1999). MgH lines are from Balfour & Cartwright (1976). The NH blue lines are adopted from the recent line lists by Fernando et al. (2018). The CN red, and C2 Swan lines are from C2 laboratory line lists by Phillips & Davis (1968), and CN ${\rm A}^{2}\Pi-{\rm X}^{2}\Sigma$ red line lists by Davis & Phillips (1963). TiO line lists are from Alvarez & Plez (1998), Plez (1998), and Jorgensen (1994) – see Schiavon et al. (1999). FeH line lists are from Phillips et al. (1987), and constants are described in Schiavon et al. (1997).

Note that the MgH, CN red, C2 and FeH adopted correspond to laboratory measurements, differently to line lists available in most of the other codes.

In the near-infrared, the CO X${}^{1}\Sigma$ lines are from Goorvitch (1994). The OH X${}^{2}\Pi$ lines were made available by S.P. Davis, that were originally from Abrams et al. (1994) with a few theoretical lines from Goldman et al. (1998) – see Meléndez & Barbuy (1999) and Meléndez et al. (2001).

Next we describe the structure of the PFANT molecular line lists files, which are hierarchically organized as follows:

for each molecular system: constants including the electronic oscillator strength fel, dissociation constant D0 2. 2.

for each $(v^{\prime},v^{\prime\prime})$ transition in each molecular system: $q_{v^{\prime}v^{\prime\prime}}$, $G_{0}(v)$, $B_{v}$, $D_{v}$ 3. 3.

for each molecular line in each $(v^{\prime},v^{\prime\prime})$ transition: $\lambda_{0J^{\prime}J^{\prime\prime}}$, $J^{\prime\prime}$, $(\mathscr{S}_{\rm norm})_{J^{\prime}J^{\prime\prime}}$, where (using Eq. 17)

[TABLE]

so that (sum for all branches, fixed $J$)

[TABLE]

$G_{0}(v)$, $B_{v}$, $D_{v}$ are calculated as follows:

[TABLE]

3.2.7 Extending the molecular line list

Several databases present updated line lists. Some of the most well-known are described below:

Bertrand Plez’s website111http://www.pages-perso-bertrand-plez.unaiv-montp2.fr links to a directory of line lists in various formats. A constant update in line lists such as CH (Masseron et al. 2014), and NH (Fernando et al. 2018) carried out within Turbospectrum, makes it extremely important to be able to implement these new data in other codes (and our own), by converting the formats from each other - see below and Sect. 3.3.1.

The Robert L. Kurúcz website222http://http://kurucz.harvard.edu/ compiles several line lists converted to a standardized format (herein referred to as the “Kurúcz format”).

The High Resolution Transmission (HITRAN) brings molecular line lists, with description updated in Rothman et al. (2013).

As concerns complete line lists, the drawback is that they can reach millions of theoretical lines, with most of them being weak. The situation is similar to the millions of weak atomic lines (Kurúcz 1995). It is not our intent here to cover all these weak lines. This is accomplished for example by Tennyson & Yurchenko (2012), where laboratory and earlier calculations of molecular lines are completed by ab inition quantum mechanical treatment. Their line lists are presented in a standardized format using Einstein coefficients, in the EXOMOL database 333http://www.exomol.com/data/molecules.

The webpage by Peter Bernath444http://bernath.uwaterloo.ca/molecularlists.php contains a database on papers regarding a series of molecules.

The webpage by Jeremy Bailey555http://newt.phys.unsw.edu.au/j̃bailey/vstar_mol.html brings line lists, links to other databases, and software tools.

The Virtual Atomic and Molecular Data Centre Consortium - VAMDC666http://www.vamdc.edu gives access to 36 inter-connected atomic and molecular databases (Dubernet et al. 2016).

Given all the different formats of the line lists available, line list conversion codes are valuable resources, as they allow for validations of synthesis codes and exchanges among research groups. We implemented a tool to convert molecular line lists from the Kurúcz formats (old and new), and also from the Turbospectrum/BSYN format, to the PFANT format, as described in Sect. 3.3.1. Other conversion paths are planned.

3.2.8 Output spectra

The code pfant generates three output files corresponding to the continuum, un-normalized flux, and normalized flux. As usual, the normalized flux is $flux_{\rm norm}(\lambda)=flux_{\rm un{\text{-}}normalized}(\lambda)/flux_{\rm continuum}(\lambda)$.

The code nulbad takes pfant output spectra, convolves it with a Gaussian profile (GP), and generates a file that, by default, contains the FWHM (full width at half maximum, given in Å) of the Gaussian profile in its name. This is a two-column text file with the wavelength (Å) and flux.

3.3 Python interface and tools

With package pyfant (http://trevisanj.github.io/pyfant), one can write scripts that use PFANT spectral synthesis in the core of more complex tasks (e.g. automatic determination of stellar parameters) in the Python language. The pyfant package API (application programming interface) includes resources to run PFANT spectral synthesis in series and in parallel, and to load, filter, transform, save and visualize relevant data files.

In addition, pyfant contains a series of command-line and graphical tools (command-line and graphical interfaces) to perform various tasks, including a scraper of molecular constants from the NIST chemistry web book, an atomic lines converter from VALD3 to PFANT format, a molecular line list converter to the PFANT format, a tool to create grids of MARCS models, an editor for atomic lines, and an editor for molecular lines. A complete list of tools is obtained by running script programs.py in the computer terminal.

3.3.1 Conversion of molecular line lists

A systematic method to convert molecular line lists to the PFANT format was implemented. Conversion can be carried out with very little user interaction, provided that:

• •

Diatomic molecular constants are available in the NIST chemistry web book for the particular molecular systems of choice777NIST is a convenient source because their data can be automatically retrieved, using the tool moldbed.py. However in some cases the NIST database does not contain the most recent data, so it is of interest to consult other databases, such as reported in Sect. 3.2.7.

• •

The input line list format contains sufficient information to determine the branch of each line.

In this section we discuss some particularities with the conversions of the Kurúcz and Turbospectrum line lists. For the Kurúcz format, the branch can be determined from the tabulated $J^{\prime}$, $J^{\prime\prime}$, $spin^{\prime}$, and $spin^{\prime\prime}$ as follows:

• •

“P” if $J^{\prime\prime}>J^{\prime}$;

• •

“Q” if $J^{\prime\prime}=J^{\prime}$;

• •

“R” if $J^{\prime\prime}<J^{\prime}$.

And in more detail if it is “P1”, “P2”, “P3”, etc. is indicated by the spin of the lower state and upper states. If $spin^{\prime\prime}=spin^{\prime}$, the branch is “(P/Q/R)($spin$)”, If $spin^{\prime\prime}\neq spin$, the branch is “(P/Q/R)($spin^{\prime}$)($spin^{\prime\prime}$)”.

Turbospectrum linelists already contain the branch given as a string, e.g. “R23”.

The branch is used to determine the specific formula from Kovács (1969) that will be used to calculate $\mathscr{S}_{J^{\prime}J^{\prime\prime}}$. This choice of formula is a detailed procedure, and depends on:

• •

the multiplicity of the transition (singlet, doublet, triplet);

• •

the sign and value of $\Delta\Lambda=\Lambda^{\prime}-\Lambda^{\prime\prime}$;

• •

the sign of $A$; and

• •

the branch.

The Franck-Condon factors $q_{v^{\prime}v^{\prime\prime}}$ (eq. 16) are computed with the code for transition probabilities of molecular transitions by Jarmain & McCallum (1970) (TRAPRB).The $q_{v^{\prime}v^{\prime\prime}}$ factors are dependent on the system, and the vibrational levels, and have almost no dependence on rotational constants (Singh & de Almeida 1982).

The procedure described above to convert molecular linelists is coded as part of the pyfant API, and is available as a graphical application named convmol.py. This application interfaces with the NIST chemistry web book to get the diatomic molecular constants $A$, $B_{\rm e}$, $\alpha_{\rm e}$, $\beta_{\rm e}$, $\omega_{\rm e}$, $\omega_{\rm e}x_{\rm e}$, and $\omega_{\rm e}y_{\rm e}$ (these can be later tuned if necessary), and interfaces with code TRAPRB to calculate $q_{v^{\prime}v^{\prime\prime}}$.

3.4 Comparison with stellar spectra and code Turbospectrum

A validation of the code PFANT is presented here, with the comparison of the CN blue ${\rm B}^{2}\Sigma-{\rm X}^{2}\Sigma$ ($v^{\prime}=0$,$v^{\prime\prime}=0$) bandhead (Figure 3), the CH ${\rm A}^{2}\Delta-{\rm X}^{2}\Pi$ (0,0) G-bandhead (Figure 4), and the NH A${}^{3}\Pi-$X${}^{3}\Sigma$ (0,0) bandhead (Figure 5) of the spectra resulting from the calculations with PFANT, Turbospectrum, and the observed solar spectrum (Kurúcz et al. 1984).

4 Summary

We present the description of calculation of diatomic molecular lines, given that this task is not straightforward. Molecular line intensities in the literature are given in either of the two formats: the Einstein $A$ constant, related to the oscillator strength (see Sect. 2), or else the inclusion of all the ingredients that build together the oscillator strength is needed.

We present a description of a new updated online version of the code for spectrum synthesis PFANT, together with many accessory tools and documentation. We describe the molecular line lists adopted, recalling that they need constant updating and implementation of new molecules. One example of an important update is the recent line list for NH (Fernando et al. 2018). The package includes codes for conversion of molecular line lists to the PFANT format, in particular so far from Kurúcz’s and Plez’s line lists. The atomic line list is also in constant improvement, both from the VALD3 database, as our own line list (available upon request).

Acknowledgements.

We are grateful to the referee for constructive comments that helped improving the code and the paper. We thank J. Meléndez and R. Smiljanic for useful comments on a previous version of the manuscript. BB and AA acknowledge grants from CAPES - Finance code 001, CNPq and FAPESP. JT acknowledges the TT-5 FAPESP fellowships 2014/25987-5 and 2016/00777-3.