4 NLTE calculations for Sr8pt10ptII

The NLTE problem for Sr II was first treated on the base of a realistic model atom by Belyakova & Mashonkina (1997). Here we describe briefly the atomic data and new results.

The Sr II model atom contains all levels with $n \leq 12$ and $l \leq 4$. Doublet fine structure is neglected except for the 4d²D and 5p²P$^\circ$ splitting. Thus, 40 bound levels of Sr II and the ground state of Sr III are included in the model atom. The corresponding Grotrian diagram is shown in Fig. 3. The Sr I levels are taken into account only for number conservation because in all stellar atmospheres considered the ratio n(Sr I)/n(Sr II) is smaller than 10^-4 due to the low ionization energy of Sr I: $\chi \rm (Sr~I) = 5.695$ eV.

Figure 4: Departure coefficients bi for some levels of Sr II in the model atmosphere of the Sun. Tick marks indicate the locations of line center optical depth unity for the Sr II lines. The resonance line core forms above $\log\tau_{5000} = -4$.

The energy levels are from Moore (1952) and Lindgard & Nielsen (1977). Sr II transition probabilities from Wiese & Martin (1980) are believed to be the best. If they are not available the data of Kurucz (1994) or Lindgard & Nielsen (1977) are taken giving preference to the first of the two sources. Photoionization cross-sections for ns, np and nd levels have been calculated by the quantum defect method using Peach's (1967) tables. For the remaining levels hydrogenic cross-sections are computed. For electron impact excitation we use the formula of van Regemorter (1962) for allowed transitions and that of Allen (1973) for forbidden ones. Electron impact ionization cross-sections are computed according to Drawin (1961). For hydrogen collisions, we use the formula of Steenbock & Holweger (1984). Since this formula provides only an order of magnitude estimate, the cross-sections were multiplied by appropriate scaling factors in order to produce the best fit to the solar Sr II line profiles.

The Sr II kinetic equilibrium is calculated using the code NONLTE3 (Sakhibullin 1983), which is based on the complete linearization method as described by Auer & Heasley (1976). The advanced method of calculations has been described in detail in our previous work (Mashonkina et al. 1999).

The Sr II term structure is similar to that of Ba II, and the same mechanisms of departures from LTE are responsible for both ions. NLTE effects for Ba II were described in detail earlier (Mashonkina et al. 1999). In Fig. 4 the departure coefficients, b_i, are shown for the solar atmosphere as a function of continuum optical depth $\tau_{5000}$ at $\lambda = 5000$ Å. In the first place, we are interested in the behaviour of the levels contributing to the subsequent line profile synthesis. These are the $\rm 5s$, $\rm 4d$, $\rm 5p$ and $\rm 6s$ levels.

As Sr II is the dominant ionization stage, no process affects the ground state population, and $\rm 5s$ keeps its thermodynamic equilibrium value. The metastable level $\rm 4d$ is separated by 1.8 eV from the ground state and by 1.14 eV from $\rm 5p$, and therefore collisional and radiative transitions $\rm 4d{-}5p$ have stronger effects on the $\rm 4d$ level population compared with collisional coupling of this level to the ground state. The departure coefficients of $\rm 4d$ and $\rm 5p$ begin to deviate from 1 at the depths around $\log\tau_{5000}$ = -1 where photon losses in the weakest line $\lambda$10036 of the multiplet $\rm 4d{-}5p$ start to become important. The $\rm 4d$ overpopulation and $\rm 5p$ underpopulation are amplified in the upper layers which are transparent with respect to the radiation of the two strong lines of that multiplet. The overpopulation outside $\log\tau_{5000} = 0.4$ of all levels above $\rm 6s$ is due to line pumping. Inside $\log\tau_{5000} \simeq -1.5$ the $\rm 5p$-level follows the ground state due to strong radiative and collisional coupling. Several transitions such as $\rm 5p \rightarrow 6d$, $\rm 5p \rightarrow 7s$, $\rm 4d \rightarrow 4f$ are pumped by $J_\nu - B_\nu(T_{\rm e})$ excess radiation in the layers where the line wing optical depth drops below 1.

From this behaviour of departure coefficients we expect that the Sr II resonance lines $\lambda$4077, $\lambda$4215, and the lines $\lambda$10036, $\lambda$10327, $\lambda$10914 of multiplet $\rm 4d{-}5p$ are amplified, whereas $\lambda$4161 arising from $\rm 5p$ is weakened compared with the LTE case. In line formation layers the departure coefficients of the lower levels of $\rm 5s{-}5p$ and $\rm 5p{-}6s$ transitions equal 1, and NLTE effects for the resonance line and $\lambda$4161 are caused by a deviation of the source function $S_{\rm lu}$ from $B_\nu(T_{\rm e})$: $S_{\rm 5s,5p} \simeq b_{\rm 5p}/b_{\rm 5s}\ B_\nu(T_{\rm e}) < B_\nu(T_{\rm e})$ for the resonance lines and $S_{\rm 5p,6s} \simeq b({\rm 6s})/b({\rm 5p})\ B_\nu(T_{\rm e}) > B_\nu(T_{\rm e})$ for $\lambda$4161. For the infrared triplet lines both $b_{\rm l} > 1$ and $b_{\rm u}/b_{\rm l} < 1$ are valid in line formation layers resulting in much larger NLTE effects compared with the resonance lines and $\lambda$4161: for the Sun the NLTE abundance correction $\Delta_{\rm NLTE}$ is between -0.03 and -0.01 dex for $\lambda4215$ and between 0.02 and 0.03 dex for $\lambda4161$ depending on the efficiency of H atom collisions while $\Delta_{\rm NLTE}$ ranges from -0.18 to -0.35 dex for $\lambda10327$.

Figure 5: Synthetic NLTE (continuous line) flux profiles of the Sr II lines compared with the observed spectrum of the Kurucz et al. (1984) solar flux atlas (bold dots). The pure Sr II $\lambda$4215 and $\lambda$4161 NLTE profiles (dotted line) are shown for comparison. For the Sr II $\lambda$10327 the LTE profile is given as dashed line. See text for discussion of the fitting parameters.

A similar behaviour of the departure coefficients resulting in an amplification of the $\lambda4215$ line and a weakening of $\lambda$4161 has been found for all the stars of our sample. The first NLTE calculations for Sr II (Belyakova & Mashonkina 1997) have shown that in very metal-poor atmospheres the Sr II resonance lines are not strengthened but weakened compared with the LTE case. The same phenomenon was found for the Ba II lines, too (see Sect. 3). For the Ba II lines $\Delta_{\rm NLTE}$ changes its sign at [Fe/H] between -1.5 and -1.9 depending on $T_{\rm eff}$ while for the Sr II lines such a transition range is shifted to lower metallicities between -2.1 and -3.0. At [Fe/H] = -2.5 NLTE effects for the Sr II resonance lines depend strongly on $T_{\rm eff}$, $\log g$ and [Fe/H], and neglecting NLTE effects can lead to strontium abundance errors up to 1 dex (Belyakova & Mashonkina 1997).