Stark Broadening of Co II Lines in Stellar Atmospheres

: Data for Stark full widths at half maximum for 46 Co II multiplets were calculated using a modiﬁed semiempirical method. In order to show the applicability and usefulness of this set of data for research into white dwarf and A type star atmospheres, the obtained results were used to investigate the signiﬁcance of the Stark broadening mechanism for Co II lines in the atmospheres of these objects. We examined the inﬂuence of surface gravity (log g), e ﬀ ective temperature and the wavelength of the spectral line on the importance of the inclusion of Stark broadening contribution in the proﬁles of the considered Co II spectral lines, for plasma conditions in atmospheric layers corresponding to di ﬀ erent optical depths.


Introduction
The importance of Co II spectral lines, weak or strong (weak lines of Co II could help in the better adjustment of cobalt abundance measured on the basis of existing strong Co I lines), for the cobalt abundance determination in the spectra of A to F type stars, has been discussed elsewhere [1]. For this reason, Stark full widths at half maximum for 46 Co II multiplets have been calculated [2,3] to be helpful for astrophysical purposes. Calculation for all 46 multiplets were done using the modified semiempirical method (MSE) [4]. Stark broadening of spectral lines is the dominant broadening mechanism in the cases of high-temperature and dense plasma which can be found in hot star atmospheres. It is noticed that disregarding the Stark broadening effect in the process of spectral line synthesis can produce a worse fit of synthetic with observed spectral lines (see, for example, [5]), or can cause errors in abundance determination, especially for A-type stars ( [6], for example).
In this paper, the applicability and usefulness of an electron-impact broadening dataset for Co II lines for investigations of white dwarf and A-type star atmospheres is analyzed. Stark broadening of the lines in the spectra of hot and dense celestial objects such as white dwarfs (WD), because of specific conditions of high electron density and high temperature in their atmospheres, usually dominates on Doppler broadening. Consequently, particular attention has been payed to hydrogen-rich (DA) and helium-rich (DB) types of WD, trying to figure out if a change in the physical conditions in their atmospheres, such as effective temperature or surface gravity, affects the relationship between thermal Doppler and electron-impact broadening for particular spectral lines.
Data 2020, 5, 74 3 of 9 a normal system of configurations, 3d 7 ( M L)nl, which is well known for nl = 4 s and 4p, and according to observations those transitions are expected to be in pure LS coupling [28]. The predicted accuracy of the MSE method is around ±50 percent, but even in the cases of emitters with complex spectra, for example Xe II and Kr II, this method often gives better agreement with experiments, with relative error less than ±30 percent [30,31]. Of course, the used model also has some error bars, but our qualitative conclusions are confirmed with calculations using three different papers with model atmospheres for DA and DB white dwarfs and for A type stars. A high precision can not be achieved since we used the published models and included Stark broadening of spectral lines a posteriori. However, the presence of Stark broadening influence electron density and temperature and, consequently, on parameters of the model of atmosphere and for the best precision the Stark broadening data should be introduced a priori, during the calculation of model atmosphere.
For the purpose of this work, we chose four lines from the list of 46 Co II spectral lines for which Stark widths have already been calculated and published elsewhere [2,3], and we investigated if atmospheric layers with possible domination of Stark broadening over the thermal Doppler broadening for each of these four lines exist (Figures 1-6) To show this, different models of atmosphere of A-type star and DA and DB WD were used. Stark and Doppler broadening were presented as a function of optical depth or temperature of atmospheric layers. For investigation of this dependence, which is shown in Figure 3a,b, the Kurucz model of A spectral type of a star was used with the logarithm of surface gravity log g = 4.5 and effective temperature T eff = 10,000 K [32]. In the case of DA and DB dwarfs, the results of similar investigations are presented in Figures 1 and 2, using the model atmospheres from Wickramasinghe [33]. For the presentation of this dependence according to different T eff or log g for DB stars, appropriate model atmospheres from Koester were used [34].
Data 2020, 5, x FOR PEER REVIEW 3 of 9 = 4 s and 4p, and according to observations those transitions are expected to be in pure LS coupling [28]. The predicted accuracy of the MSE method is around ±50 percent, but even in the cases of emitters with complex spectra, for example Xe II and Kr II, this method often gives better agreement with experiments, with relative error less than ±30 percent [30,31]. Of course, the used model also has some error bars, but our qualitative conclusions are confirmed with calculations using three different papers with model atmospheres for DA and DB white dwarfs and for A type stars. A high precision can not be achieved since we used the published models and included Stark broadening of spectral lines a posteriori. However, the presence of Stark broadening influence electron density and temperature and, consequently, on parameters of the model of atmosphere and for the best precision the Stark broadening data should be introduced a priori, during the calculation of model atmosphere.
For the purpose of this work, we chose four lines from the list of 46 Co II spectral lines for which Stark widths have already been calculated and published elsewhere [2,3], and we investigated if atmospheric layers with possible domination of Stark broadening over the thermal Doppler broadening for each of these four lines exist (Figures 1-6) To show this, different models of atmosphere of A-type star and DA and DB WD were used. Stark and Doppler broadening were presented as a function of optical depth or temperature of atmospheric layers. For investigation of this dependence, which is shown in Figure 3a,b, the Kurucz model of A spectral type of a star was used with the logarithm of surface gravity log g = 4.5 and effective temperature Teff = 10,000 K [32]. In the case of DA and DB dwarfs, the results of similar investigations are presented in Figures 1 and 2, using the model atmospheres from Wickramasinghe [33]. For the presentation of this dependence according to different Teff or log g for DB stars, appropriate model atmospheres from Koester were used [34].   Figure 1a, but for the model atmosphere of a helium-rich (DB) white dwarf [33], with same model parameters, Teff = 15,000 K and log g = 8. (b) Same as Figure 2a, but as a function of atmospheric layer temperature instead of optical depth.

Results and Discussion
In Figures 1-3, the comparisons of Stark widths and Doppler widths of λ2533.2, λ2709, λ9519 and λ9969 Co II spectral lines as a function of the optical depth in the white dwarf and A-star atmospheres are presented, to show in which layers of stellar atmosphere Doppler broadening caused by thermal motion of particles is dominated by Stark broadening caused by impacts of Co II ions with electrons. In astrophysics, optical depth is a measure of the extinction coefficient or absorptivity, integrated from zero towards deeper layers up to a specific depth in stellar atmosphere. So, it is local characteristic as with electron temperature and it increases from zero towards deeper layers. Because it varies with wavelength, it is usually given for a standard wavelength of 5150 Å or as the Rosseland mean optical depth averaged over frequencies. Since we use published models of stellar atmospheres, we use optical depth as provided by authors of the models.
The first two lines, λ2533.2 and λ2709, from multiplets ( 4 P)4s 3 P-( 4 P)4p 3 D o and ( 4 F)4p 3 G o -( 4 F)5s 3 F, respectively, are in the ultraviolet part of the spectrum, while the last two lines considered by us, λ9519 and λ9969, from multiplets ( 4 F)5s 5 F-( 4 F)5p 5 G o and ( 4 F)5s 3 F-( 4 F)5p 5 F o , respectively, are in the infrared part of the spectrum. In Figure 1a,b, this analysis is done for DA WD model atmospheres [33] with parameters Teff = 15,000 K and log g = 8. Stark and Doppler broadening as a function of optical depth τ in the atmosphere at 5150 Å are shown in Figure 1a, and as a function of layer temperature in Figure 1b. The same comparisons but for DB white dwarf atmosphere model with the same parameters are shown in Figure 2a,b. In Figure 3a,b, we can see the behaviors in the   Figure 1a, but for the model atmosphere of a helium-rich (DB) white dwarf [33], with same model parameters, Teff = 15,000 K and log g = 8. (b) Same as Figure 2a, but as a function of atmospheric layer temperature instead of optical depth.

Results and Discussion
In Figures 1-3, the comparisons of Stark widths and Doppler widths of λ2533.2, λ2709, λ9519 and λ9969 Co II spectral lines as a function of the optical depth in the white dwarf and A-star atmospheres are presented, to show in which layers of stellar atmosphere Doppler broadening caused by thermal motion of particles is dominated by Stark broadening caused by impacts of Co II ions with electrons. In astrophysics, optical depth is a measure of the extinction coefficient or absorptivity, integrated from zero towards deeper layers up to a specific depth in stellar atmosphere. So, it is local characteristic as with electron temperature and it increases from zero towards deeper layers. Because it varies with wavelength, it is usually given for a standard wavelength of 5150 Å or as the Rosseland mean optical depth averaged over frequencies. Since we use published models of stellar atmospheres, we use optical depth as provided by authors of the models.
The first two lines, λ2533.2 and λ2709, from multiplets ( 4 P)4s 3 P-( 4 P)4p 3 D o and ( 4 F)4p 3 G o -( 4 F)5s 3 F, respectively, are in the ultraviolet part of the spectrum, while the last two lines considered by us, λ9519 and λ9969, from multiplets ( 4 F)5s 5 F-( 4 F)5p 5 G o and ( 4 F)5s 3 F-( 4 F)5p 5 F o , respectively, are in the infrared part of the spectrum. In Figure 1a,b, this analysis is done for DA WD model atmospheres [33] with parameters Teff = 15,000 K and log g = 8. Stark and Doppler broadening as a function of optical depth τ in the atmosphere at 5150 Å are shown in Figure 1a, and as a function of layer temperature in Figure 1b. The same comparisons but for DB white dwarf atmosphere model with the same parameters are shown in Figure 2a,b. In Figure 3a,b, we can see the behaviors in the function of logarithm of Rosseland optical depth and temperature in the stellar atmospheres for an A-type model atmosphere [32] with parameters log g = 4.5 and Teff = 10,000 K. Stark width in comparison with Doppler width increases as wavelength increases, because if a wavelength is larger than the corresponding atomic energy levels are closer and because of that, the perturbation of the emitter/absorber is larger and the emitted spectral line is broader. We notice also that Stark widths are proportional to λ 2 , while Doppler widths are proportional to λ [3]. For the last line, λ9959, the point where Stark width reaches Doppler width is deeper in the atmosphere than for the previous line, λ9519, because the Stark width values for this line are smaller since the corresponding atomic energy levels are further away than in the previous case and the perturbation is smaller.
We can see that for the hydrogen-rich (DA) type of WD, Stark broadening starts to be more significant than the Doppler broadening already in the atmospheric layers with relatively smaller optical depth, for spectral lines λ9519 and λ9969 near to τ ≈ 0.5. Electron-impact broadening for the line λ2709 becomes more significant for layers after τ ≈ 10, while for line λ2533.2 Doppler broadening is dominant for all considered values of optical depth. For the helium-rich (DB) type of WD, domination of Stark broadening for all four lines over the Doppler broadening starts before optical depth τ ≈ 1, where most of spectral lines are formed, so we can expect that electron-impact broadening for all three lines should be more important than thermal broadening in DB dwarf spectra. Difference between the importance of Stark broadening in comparison with Doppler broadening between DA and DB type of WD is in favor of DB type, because a helium-rich (DB) dwarf can generate more free electrons than the hydrogen-rich (DA) dwarf with the same density, causing higher perturber density [3]. This advantage in the domination of Stark width over Doppler width in the DB type in comparison with the DA type is also obvious from Figure 4, where these widths are presented as a function of optical depth.
From the same analysis for an A-type stellar atmosphere, we can see that for the spectral lines λ9519 and λ9969, Stark broadening also becomes the most significant broadening mechanism, but after reaching the deeper layers of the atmosphere, around optical depth of τ ≈ 50 and τ ≈ 70 respectively, e.g., for temperatures of atmospheric layers around 20,000 and 25,000 K, respectively. For the other two lines, Doppler broadening remains dominant even for layers with larger optical depth, e.g., higher temperatures of considered atmospheric layers.
It is obvious from Figures 1-4 that Stark broadening has larger impact on Co II spectral lines in the infrared spectral range, and this impact will be larger for DB dwarfs than on the rest of the considered objects. So, we decided to investigate how effective temperature and surface gravity of DB WD affect the relationship between Stark and Doppler widths for a particular spectral line.  layers of the DB atmosphere, with temperatures which are more and more smaller than the effective temperature, because in these layers temperature becomes high enough to ionize helium more efficiently, so that electron density is higher. For example the difference between the effective temperature and temperature where Stark and Doppler broadening are approximately equal increases from the model with Teff = 14,000 K where it is several thousand kelvins to the model with Teff = 30,000 K, where it is larger than 10,000 K. Finally, in Figure 6, this comparison for the same line is shown for four model atmospheres of DB white dwarfs [34] with effective temperatures Teff of 12,000 and 30,000 K, with two different values of log g for each temperature. We can see that electron-impact broadening becomes more important in DB white dwarf atmosphere than thermal broadening with the increase in surface gravity.   Finally, in Figure 6, this comparison for the same line is shown for four model atmospheres of DB white dwarfs [34] with effective temperatures Teff of 12,000 and 30,000 K, with two different values of log g for each temperature. We can see that electron-impact broadening becomes more important in DB white dwarf atmosphere than thermal broadening with the increase in surface gravity.

Results and Discussion
In Figures 1-3, the comparisons of Stark widths and Doppler widths of λ2533.2, λ2709, λ9519 and λ9969 Co II spectral lines as a function of the optical depth in the white dwarf and A-star atmospheres are presented, to show in which layers of stellar atmosphere Doppler broadening caused by thermal motion of particles is dominated by Stark broadening caused by impacts of Co II ions with electrons.
In astrophysics, optical depth is a measure of the extinction coefficient or absorptivity, integrated from zero towards deeper layers up to a specific depth in stellar atmosphere. So, it is local characteristic as with electron temperature and it increases from zero towards deeper layers. Because it varies with wavelength, it is usually given for a standard wavelength of 5150 Å or as the Rosseland mean optical depth averaged over frequencies. Since we use published models of stellar atmospheres, we use optical depth as provided by authors of the models.
The first two lines, λ2533.2 and λ2709, from multiplets ( 4 P)4s 3 P-( 4 P)4p 3 D o and ( 4 F)4p 3 G o -( 4 F)5s 3 F, respectively, are in the ultraviolet part of the spectrum, while the last two lines considered by us, λ9519 and λ9969, from multiplets ( 4 F)5s 5 F-( 4 F)5p 5 G o and ( 4 F)5s 3 F-( 4 F)5p 5 F o , respectively, are in the infrared part of the spectrum. In Figure 1a,b, this analysis is done for DA WD model atmospheres [33] with parameters T eff = 15,000 K and log g = 8. Stark and Doppler broadening as a function of optical depth τ in the atmosphere at 5150 Å are shown in Figure 1a, and as a function of layer temperature in Figure 1b. The same comparisons but for DB white dwarf atmosphere model with the same parameters are shown in Figure 2a,b. In Figure 3a,b, we can see the behaviors in the function of logarithm of Rosseland optical depth and temperature in the stellar atmospheres for an A-type model atmosphere [32] with parameters log g = 4.5 and T eff = 10,000 K. Stark width in comparison with Doppler width increases as wavelength increases, because if a wavelength is larger than the corresponding atomic energy levels are closer and because of that, the perturbation of the emitter/absorber is larger and the emitted spectral line is broader. We notice also that Stark widths are proportional to λ 2 , while Doppler widths are proportional to λ [3]. For the last line, λ9959, the point where Stark width reaches Doppler width is deeper in the atmosphere than for the previous line, λ9519, because the Stark width values for this line are smaller since the corresponding atomic energy levels are further away than in the previous case and the perturbation is smaller.
We can see that for the hydrogen-rich (DA) type of WD, Stark broadening starts to be more significant than the Doppler broadening already in the atmospheric layers with relatively smaller optical depth, for spectral lines λ9519 and λ9969 near to τ ≈ 0.5. Electron-impact broadening for the line λ2709 becomes more significant for layers after τ ≈ 10, while for line λ2533.2 Doppler broadening is dominant for all considered values of optical depth. For the helium-rich (DB) type of WD, domination of Stark broadening for all four lines over the Doppler broadening starts before optical depth τ ≈ 1, where most of spectral lines are formed, so we can expect that electron-impact broadening for all three lines should be more important than thermal broadening in DB dwarf spectra. Difference between the importance of Stark broadening in comparison with Doppler broadening between DA and DB type of WD is in favor of DB type, because a helium-rich (DB) dwarf can generate more free electrons than the hydrogen-rich (DA) dwarf with the same density, causing higher perturber density [3]. This advantage in the domination of Stark width over Doppler width in the DB type in comparison with the DA type is also obvious from Figure 4, where these widths are presented as a function of optical depth.
From the same analysis for an A-type stellar atmosphere, we can see that for the spectral lines λ9519 and λ9969, Stark broadening also becomes the most significant broadening mechanism, but after reaching the deeper layers of the atmosphere, around optical depth of τ ≈ 50 and τ ≈ 70 respectively, e.g., for temperatures of atmospheric layers around 20,000 and 25,000 K, respectively. For the other two lines, Doppler broadening remains dominant even for layers with larger optical depth, e.g., higher temperatures of considered atmospheric layers.
It is obvious from Figures 1-4 that Stark broadening has larger impact on Co II spectral lines in the infrared spectral range, and this impact will be larger for DB dwarfs than on the rest of the considered objects. So, we decided to investigate how effective temperature and surface gravity of DB WD affect the relationship between Stark and Doppler widths for a particular spectral line.
In Figure 5, comparison of Stark and Doppler broadening of Co II line λ9969 in white dwarf atmospheres is presented as a function of layer temperature for five different models [34] of DB white dwarf atmospheres with effective temperatures from 14,000 to 30,000 K with a step of 4000 K, and log g = 8. The effective temperature is approximately taken as the temperature of the surface of the star. As effective temperature increases, the Stark broadening becomes more prominent in layers of the DB atmosphere, with temperatures which are more and more smaller than the effective temperature, because in these layers temperature becomes high enough to ionize helium more efficiently, so that electron density is higher. For example the difference between the effective temperature and temperature where Stark and Doppler broadening are approximately equal increases from the model with T eff = 14,000 K where it is several thousand kelvins to the model with T eff = 30,000 K, where it is larger than 10,000 K.
Finally, in Figure 6, this comparison for the same line is shown for four model atmospheres of DB white dwarfs [34] with effective temperatures T eff of 12,000 and 30,000 K, with two different values of Data 2020, 5, 74 7 of 9 log g for each temperature. We can see that electron-impact broadening becomes more important in DB white dwarf atmosphere than thermal broadening with the increase in surface gravity.

Conclusions
In this work, the usefulness and applicability of calculated set of data with Stark widths of 46 Co II lines for the investigations of spectra from atmospheres of stellar type A and hydrogen-rich (DA) and helium-rich (DB) white dwarfs are investigated. One can conclude that Stark broadening is very important for white dwarfs and for the same plasma conditions, its influence is larger for the DB than for the DA type. For A-type stars, Stark broadening may be non-negligible in comparison with thermal Doppler width, especially for higher wavelengths in the red part of the spectrum. Additionally, the influence of Stark broadening increases with the increase in the effective temperature and surface gravity analyzing as an example the DB type of WD.
We hope that the calculated set of 46 Co II Stark widths and these results will be useful for their use for hot star and WD spectroscopy, and also contribute to more accurate cobalt abundance determination. There are no other experimental or theoretical data for Stark broadening of Co II spectral lines analyzed here. As follows from our work, measurements of Stark broadening of Co II spectral lines will be of interest not only for comparison with the results obtained here but also for analysis and synthesis of stellar Co II spectral lines. This set of data, previously published as a hard copy in Ref. [3], is available online here in computer readable form. It will be implemented later and in the STARK-B database [35][36][37][38], which is also a part of the Virtual Atomic and Molecular Data Center (VAMDC) [39,40] and may be accessed through its portal [41].

Conflicts of Interest:
The authors declare no conflict of interest.