Broadening of spectral lines in the Gaia-ESO survey

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Broadening of spectral lines in the

Gaia-ESO survey

Kristoffer Bengtsson

Supervisor: Paul Barklem Subject Reader: Ulrike Heiter

Uppsala University, Department of Physics and Astronomy, Astronomy and Space physics Bachelor’s degree project, 15 credits

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Abstract

Analyzing stellar spectra plays a big role in understanding the evolution of our galaxy. Having good data for spectral line properties is very important when analyzing these spectra. One part of the Gaia-ESO public spectroscopic survey (GES) is to gather data for spectral line properties from stellar spectra. The scope of this project is to study one of these properties, the spectral line width caused by collisional broadening by hydrogen. Collisional broadening by hydrogen occurs when a hydrogen atom collides with a particle. The goal of this project is to successfully calculate the collisional spectral line broadening of iron lines where new data is missing from the GES using modern quantum mechanical calculations. These calculations are done using the ABO theory, which is more advanced than previously established theory. A table of Fe-I (Neutral iron) spectral lines without collisional broadening data in the GES has been provided. Using the ABO theory and the accompanying ABO cross section calculator code, estimates of collisional broadening by hydrogen have been calculated for these lines. The new calculations predict that the line width of the spectral lines are typically twice as large compared to older estimates calculated using simpler theory. This new data can be expected to improve stellar spectrum analysis in the Gaia-ESO survey spectra.

Sammanfattning

Analys av stjärnspektran spelar en stor roll i vår förståelse av vintergatans utveckling. Att ha bra data för spektrallinjers egenskaper är oerhört viktigt vid analys av dessa spektran. En del i Gaia-ESO public spectroscopic survey (GES) är att samla in data för dessa spektrallinjers egenskaper ur stjärnspektran. Omfattningen av detta projekt innefattar att titta närmare på en av dessa egenskaper, spektrallinjebreddning orsakad av kollisionsbreddning av väte. Kollisionsbreddning av väte uppstår när en väteatom kolliderar med en annan partikel. Målet med projektet är att med framgång beräkna kollisionsbreddningen av spektrallinjer från järn där ny data saknas ur GES genom att använda moderna kvantmekaniska beräkningar. Dessa beräkningar är gjorda med den så kallade ABO-teorin, vilken är mer avancerad än tidigare etablerade teorier. En tabell med Fe-I (neutralt järn) spektrallinjer utan kollisionsbreddningsdata i GES har tillhandahållits. Med hjälp av ABO-teorin och den medföljande ABO-tvärsnittsräknar-koden har beräkningar av kollisions-breddning med väte utförts för dessa linjer. De nya beräkningarna förutser att spektrallinjernas bredd blir typiskt två gånger så stor jämfört med de äldre beräkningarna gjorda med enklare teori. Denna nya data kan förväntas att förbättra analysen av stjärnspektrum ur Gaia-ESO kartläggnin-gen.

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1 Introduction

1.1 The Gaia-ESO Survey

The Gaia-ESO survey is a public spectroscopic survey conducting research on stars from all the major components of the Milky Way. The survey is systematically targeting 105 _{stars in} com-ponents ranging from halo to star forming regions in order to build a complete, homogeneous overview of stars in our galaxy. Data is gathered on kinematics and elemental abundances in order to gain a deeper understanding on star- and galactic evolution (Gilmore et al.2012). Spectroscopic analysis is one of the most important tools of astronomy when observing the different components of the universe. Being able to accurately measure the electromagnetic spectrum of celestial objects reveals information about many of their properties. This is especially important when recording stellar spectra, as they contain information on e.g. chemical composition, temperature and dis-tance. A part of this includes calculating the width of certain metal lines, which is the scope of this project. In particular, these metal lines originate from solar-like stars. Gathering abundance data on solar-like stars gives valuable information on how these kinds of stars have evolved and will evolve, how their surroundings affect this process and so on.

The goal of this project is to use spectral line data gathered by a working group within the ESO survey and calculate collisional broadening by hydrogen atoms for lines not in the Gaia-ESO line list that currently do not have modern collisional data. The data set used consists of lines from neutral iron for which no ABO data exists. The calculations will be done using the ABO theory (Barklem, Anstee, and O’Mara1998). The results from these calculations will be compared to existing data in the Vienna Atomic Line Database (VALD). The work on VALD has been done by Piskunov et al. (1995), Kupka et al. (1999), Ryabchikova et al. (1999) and Heiter et al. (2008). This existing data has been calculated by Kurucz (2013) using the Unsöld theory (Unsöld1955), but with a detailed calculation of the interaction constant C6, rather than the approximate formula given in Unsöld’s original formulation. See Kurucz (1981) for a more in-depth explanation.

2 Background

2.1 Spectral Broadening of metals in Solar-like Stars

In a stellar atmosphere, elements exist in their gaseous state due to the high temperature. When atoms interact with photons (which are abundant in a stellar atmosphere) they will sometimes reach an excited state by absorbing a photon and elevating an electron to a higher energy state. When this electron then de-excites, a new photon is released with a specific energy (and therefore wavelength) corresponding to the energy difference between the two states the electron transitions between. However, the uncertainty principle relates the uncertainty of the energy of this excited state with its lifetime. A shorter lifetime will lead to a higher uncertainty in the energy and vice versa. Since the energy of the emitted photon matches that of the energy difference between the two states, the uncertainty carries over. Thus, the emitted photons will vary in wavelength giving rise to a widened spectral line when recording a spectrum.

Figure 1: Example of a Lorentzian profile

This widening process is called natural broadening and gives rise to the typical Lorentzian profile of spec-tral lines. In figure 1 a Lorentzian profile is shown, with the FWHM (Full-Width Half-Maximum) marked out. The FWHM is the width of the profile at half the peak value of the profile. The HWHM (Half-Width Half-Maximum) is half the value of the FWHM.

Another source of spectral line broadening is pres-sure broadening. This broadening occurs when a par-ticle emitting radiation is affected by the presence of other nearby particles. In the atmosphere of solar-like stars, the temperature exceeds 5000 Kelvin. This means that the velocity of particles is sufficiently high and the interaction time sufficiently short compared to the life span of an excited state that an impact approxi-mation can be used. The impact approxiapproxi-mation means

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that it is assumed that the collision is fast, rather than

prolonged. These collisions give rise to broadening of spectral lines. This is called collisional broadening or impact pressure broadening. In collisional broadening, a perturber (e.g. a hydrogen atom) collides with a particle (e.g. an iron atom) during the emission and shortens the life-span of the excited state. However, since the lines that are of interest in this report are absorption lines, it’s the collisions during the absorption process (i.e. excitation by photon) that are relevant. As an example, consider an iron atom going through the process of absorbing a photon when it collides with a hydrogen atom. By doing this, the uncertainty in the emitted energy increases due to the uncertainty principle. Since the wavelength of the emitted photon is proportional to the emitted energy, the spectral line will be broadened accordingly. The effect from collisional broadening is also described by a Lorentzian profile.

2.2 The ABO Theory

The ABO theory has been well developed over the years. It is useful since it more accurately than ever before calculates the broadening of spectral lines per hydrogen atom that interacts with the emitter, giving valuable information about abundances in stellar atmospheres. In order to determine the collisional broadening by hydrogen, the ABO theory first needs the wave function of the optical electron. It describes this using different parameters: species, wavelength, energy levels lower and upper state of emitter, series limits of lower and upper state and lastly the orbital angular momentum quantum number of the upper and lower state.

The theory then calculates the broadening cross section σ and the velocity parameter α which is then used in an equation that calculates the width per perturber. A cross section can be seen as the area where the interacting particles must meet each other in order for any interaction to occur. Thus, the broadening cross section describes the area in which the particles must meet in order for collisional broadening to occur. The probability of the collisional broadening process to occur is represented by the broadening cross section.

For the calculations in this thesis the ABO cross section calculator code (Barklem2016) was used, which first uses data on level energies and series limits to calculate the effective principal quantum numbers (denoted n*) of the upper and lower state(s). The effective principal quantum number is a description of a state if the atom is assumed to be hydrogenic, i.e. behaving like a hydrogen atom. This is done using the following formula:

n∗= r

109678.8

Elimit− Enl

(2.1) Where Elimit is the series limit of the atom and Enl is the energy of the optical electron. Thus,

Elimit− Enl is the binding energy of the optical electron. Both energies are given in wavenumber

units, cm−1. The number 109678.8 is the Rydberg constant for hydrogen, also in cm−1. See section 2.3 for a more in depth explanation of the series limit.

The next step is using these effective principal quantum numbers to determine the cross sec-tion(s) and velocity parameter(s). This is done using pre-existing tables compiled by Barklem and O’Mara (1997), Anstee and O’Mara (1995) and Barklem, O’Mara, and Ross (1998) where the cross section and velocity parameter are tabulated against the principal quantum numbers of the upper and lower states. Cross sections were calculated at different velocities and were then fitted to a function, see equation 2.2. The table for the velocity parameter was compiled using regression.

σ(v) = σ(v0)  v

v0 −α

(2.2) The cross section and velocity parameter is extracted from the tables using interpolation. Lastly, the width of the spectral line is calculated. In the impact approximation which this method utilizes, the width is proportional to the rate of broadening collisions. This is calculated with:

w = N < σv >

where w is the width of the spectral line, N is the hydrogen number density, v is the velocity and σ is the broadening cross section. The width of the spectral line is defined to be the HWHM (Half-Width Half-Maximum) of the Lorentzian profile of the spectral line. The notation <> means the velocity distribution (assumed to be Maxwellian) is averaged.

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In order to calculate this, the mean velocity v of the perturber is calculated assuming that velocity distribution is maxwellian:

v = 8kT πµ 1/2 (2.3) µ = m1m2 m1+ m2 (2.4) Where µ is the reduced mass of the two interacting atoms, k is the Boltzmann constant and T is the temperature. When all of this has been acquired, the method calculates the line width per unit hydrogen atom density using the following formula, where Γ is the gamma function:

w N = 4 π α₂ Γ 4 − α 2 v0σ(v0)  v v0 1−α (2.5)

2.3 The series limit

Figure 2: Schematic view of the series limit of a configuration

When a photon excites an atom, the electron will make a transition to a new state due to the increase in energy. The series limit represents the maximum level an electron can transition to without becoming ionized. If the atom is in the ground state, the series limit Elimit will be

the same as the ionization limit Eion.. If more

than one electron is in an excited state, the series limit is higher than the usual ionization limit (i.e. the ground state of Fe II), as seen in figure 2. Here, the iron atom starts in an ex-cited state (corresponding to the energy shift upwards of Enl) and the series limit increases

from the regular ionization limit. Determining this shift in the series limit requires knowledge of the states electron configuration. If the

par-ent configuration of the state is the ground state of singly ionized iron, the series limit remains the same as the ionization limit. If the parent configuration corresponds to an excited state of singly ionized iron, the energy of this state is added to the ionization limit to obtain the series limit.

3 Method / Data processing

3.1 Acquiring and structuring the data

The data needed for each line for the ABO method consists of the following parameters in this exact order:

1. Species (i.e. what is the perturbed atom, e.g. Fe or Mg)

2. Ionization stage (integer, where 1=neutral, 2= singly ionized and so on) 3. Wavelength (in Angstroms)

4. Level energy, lower (in cm−1) 5. Series limit, lower level (in cm−1) 6. Level energy, upper (in cm−1) 7. Series limit, upper level (in cm−1)

8. Orbital angular momentum quantum number l of the optical electron 9. Orbital angular momentum quantum number l of the optical electron

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The following parameters are not needed by the ABO theory, but they can be used to uniquely identify the states and the transition and are therefore also used.

1. Total angular momentum quantum number J, lower state 2. Total angular momentum quantum number J, upper state

Out of these 11 parameters, 7 were provided initially. the remaining parameters that had to be determined were orbital angular momenta quantum numbers and series limits for both upper and lower states. The series limits were more complicated to determine and is covered in a later section. For each line, the angular momentum quantum numbers had to be determined from the electron configurations provided for the upper and lower state. In this model, only the optical electron is considered, so the orbital angular momentum of the inner electrons is disregarded. Only the shielding effect of the inner electrons on the optical electrons is considered. The optical electron will only see a core with a charge +1, since the core of iron has a charge +26 and there are 25 inner electrons (only neutral iron is considered) with each having a charge of -1.

3.2 Correcting the series limit

In order to properly express the series limit of electron configurations of iron, the energy of the excited core state of Fe II has to be added to the ionization energy in the case that the excited state consists of a state with two excited electrons. These levels can be found in the Atomic Spectra Database provided by NIST (Kramida, Ralchenko, Yu, and Reader2018), under Levels Data.

The first step to finding the proper energy level for a configuration is looking at the com-plete configuration of a transition. The transition of neutral iron with wavelength λ = 5476.5642 Angstrom is a p-s transition with the following electron configurations:

p : 3p6.3d7.(4F ).4p y5D s : 3p6.3d6.(5D).4s.(4D).5s g 5D

Looking at the configuration for the initial p-state, removing the 4p-electron and thusly ionizing the iron, the new term needs to be calculated. Before ionization, the system has a spin of (5−1)/2 = 2, whereas the 4-p electron has spin 1/2 on account of being a fermion. Since the electron is positioned in the p-shell, it also has orbital angular momentum equal to 1. Thus, removing it leaves the configuration with the following possible values for spin and orbital angular momentum:

L = 1(P ), 2(D), 3(F ) S = 3/2 → (2S + 1) = 4 , S = 5/2 → 2S + 1 = 6

In this particular case, the final spin and orbital angular momentum is known, since (4F ) denotes

the spin and orbital angular momentum of the parent configuration to the left. However, this process is required for some configurations, and at the very least provides a reality-check. Looking at the possibilities, orbital angular momentum equal to 3 (term symbol F) matches, as does spin equal to 3/2 (2S+1 = 4). The remaining parent state after ionization therefore becomes:

3d74F

Returning to the ASD, this level has the term energy 2000.3065cm−1 above the ground state of Fe II. Adding this to the ionization energy of Fe I (which is equal to 63737.704 cm−1), the series limit becomes 65738.0105cm−1.

Ionizing the s-state, we lose the 5s electron and through the same process as for the p-state, we acquire the following state

3d6.(5D).4s4D

Here, the term of the parent state (4_{D) is already known. (Going through the process, a conclusion} can be reached that the term could also be6_{D. This is the case for other lines.) Again, returning} to the ASD, this level has the term energy 7904.1885 cm−1. This brings the total series limit up to 71641.8925cm−1.

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3.3 Using the ABO method

When the orbital angular momentum quantum numbers and the series limits were added to the data, everything was first stored in an excel document for easier management. When it was completed and double checked, it was converted to a plain text document so the ABO cross section calculator code (Barklem2016) could properly read it. The document was then put through the abo cross code according to the instructions provided in a readme file, which then produced two output files. These two files were in the .short and .long format, also explained in detail in the readme. For this project, the .long file was used due to it’s more extensive result. Its contents are given in Appendix B.

3.4 Analysing the data

In the output of the code, both the calculated data and the older Kurucz data are given for each line. In order to properly analyze the newly calculated data, a short MatLab script was written and used to plot the data. This process also divides the transitions by type (e.g. s-p, p-d) and plots them separately by color.

The ABO data produced (and the Kurucz data) are logarithmic. For plotting, they were converted to ratios in the following way. The output produced is log₁₀(Γ

N) where Γ is equal to 2w

and w is the HWHM of the Lorentzian profile, as mentioned earlier. This means Γ is equal to the FWHM (Full-Width Half Maximum) of the Lorentzian profile. The following formula was used:

DataP oint = 10ABOData−KuruczData

This comparison was then plotted against the energy of the lower level, upper level and the energy difference of the levels in three different plots.

4 Calculations

In order to show how the ABO method works, the following section will cover two examples of how the calculations are done using the theory. The first example will cover a line from neutral magnesium. Magnesium lines are not covered in the rest of this project, but this line provides a simpler example where doubly excited electrons are not present. The second example covers a line from neutral iron which involves doubly excited electrons, where the correction of the series limit also comes into play. the result for the iron line will be compared to a previously published value using the method.

4.1 Used Equations

n∗= r 109678.8 Elimit− Enl (4.1) v = 8kT πµ 1/2 (4.2) w N = 4 π α₂ Γ 4 − α 2 v0σ(v0)  v v0 1−α (4.3) µ = m1m2 m1+ m2 (4.4)

4.2 Mg I: p-d transition, λ =5528.418 Å

For the p-d transition of neutral magnesium with wavelength 5528.418 Å, the following level energies and ionization limit can be found using the atomic spectra database run by the National Institute of Standards & Technology (NIST):

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Since Magnesium is a lighter metal, having more than one electron in an excited state is rare. In this example, both parent configurations correspond to the ground state of Mg II. Thus, the regular ionization limit is used for both the lower and upper levels when calculating the effective principcal quantum number. Using the above energy values, n∗ for the p- and d-state can be calculated using equation 4.1:

n∗low= r 109678.8 61671.05 − 35051.264 ≈ 2.0298, n ∗ upp= r 109678.8 61671.05 − 53134.642≈ 3.5845 Using table 1 and 2 from Barklem and O’Mara (1997) the cross section σ and velocity parameter

α can be extracted. When using the proper method (which the ABO-cross code utilizes),

inter-polation is used and the principal quantum numbers are not rounded. However, for this examples the numbers were rounded to two digits to mainly show the method itself, not produce the most accurate result. Using n∗_low≈ 2.0 and n∗

upp ≈ 3.6, the following values are extracted:

σ = 1490 a.u., α = 0.313

The reduced mass of the Magnesium-hydrogen system is calculated using equation 4.4:

µ = 1u · 24u

1u + 24u= 0.96u

Using the assumption that the temperature is T = 5000 K, the mean velocity is calculated using equation 4.2: v = 8kT πµ 1/2 = 8 · 5000 · k π · 0.96u 1/2 = 1.05 · 106cm/s

Now the broadening per hydrogen atom can be calculated using equation 4.3, using v0= 106_cm/s as initial velocity: w N = 4 π 0.3132 Γ 4 − 0.313 2 ·106·1490·2.8·10−17 1.05 · 10 6 106 1−0.313 ≈ 4.228·10−8rad s−1cm3

This line lacks a published value in Barklem and O’Mara (1997), but the next line will show that the method produces values in line with other published values.

4.3 Fe I: s-p transition, λ = 5269.54 Å

For the s-p transition of neutral iron with wavelength 5269.54 Å, the following level energies and ionization limit for neutral iron can be found using the atomic spectra database run by the National Institute of Standards & Technology (NIST):

Elow = 6928.68cm−1 Eupp = 25899.989cm−1 Eion= 63737.704cm−1

Since this transition involves iron, it isn’t uncommon for more than one electron to be excited (essentially, one of the inner electrons is also in an excited state) which will shift the series limit upwards. To properly express this limit, the electron configurations for this transition have to be found. After this, their ionized states can be found and their energy levels added to the regular ionization limit. All of this information can also be gathered from the Atomic Spectra Database from NIST using the known wavelength for the transition. See section 3.2 for a step-by-step on how the limits are found, here they will just be written out.

s : 3d7.(4F ).4s a 5F Elimit,lower= 2000.3065cm−1

p : 3d6.(5D).4s.4p.(3P 0) z 5D0 Elimit,upper= 0cm−1

Using these energy values, n∗ for both the s-state and p-state can be calculated using equation 4.1:

n∗_low= r 109678.8 63737.704 + 2000.3065 − 6928.68 = 1.39, n ∗ upp= r 109678.8 63737.704 − 25899.989 = 1.702

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With the effective principal quantum numbers calculated, the rest of the necessary paramaters can be acquired. Using table 1 and 2 from Anstee and O’Mara (1995) cross section σ and velocity pa-rameter α is extracted using n∗_low ≈ 1.4 and n∗

high≈ 1.7. Normally this is done using interpolation

in order to reach a higher precision, but simply rounding shows the idea behind the theory without producing terrible results.

σ = 242 a.u., α = 0.250

The reduced mass of the system is simple to calculate using equation 4.4, and comes out to the following for a hydrogen-iron system, where u is the atomic mass unit:

µ = 1u · 56u

1u + 56u= 0.9824 u

Using the same assumption as in Anstee and O’Mara (1995) for the temperature (T = 5000K), v

can be calculated using equation 4.2:

v = 8kT πµ 1/2 = 8k · 5000 πµ 1/2 ≈ 1.0380 · 106 _cm/s

Where k is the Boltzmann constant. Now the width per hydrogen atom can be calculated, assuming the initial velocity v0= 106 cm/s, and using equation 4.3:

w N = 4 π 0.2502 Γ 4 − 0.250 2 ·106_{·242·2.8·10}−17 1.0378 · 106 106 1−0.250 ≈ 0.68465·10−8_{rad s}−1_cm3 This agrees well with the published value by Anstee and O’Mara (1995) of 0.678 · 10−8rad s−1cm3. As stated above, the published value is calculated without rounding of the effective principal quantum number. Despite this, the produced value using rounding is only roughly 1 percent larger than the published value.

5 Results

5.1 Tables

The table of output data produced by the ABO cross code consists of of the following columns: 1. Species (Element and ionization stage)

2. Wavelength (in Angstroms) 3. Level energy, lower (in cm−1) 4. Series limit, lower level (in cm−1) 5. Level energy, upper (in cm−1) 6. Series limit, upper level (in cm−1)

7. Change in orbital angular momentum quantum number (i.e. transition type, llower→ lupper)

8. Change in total angular momentum quantum number J (Jlower→ Jupper)

9. Principal quantum number n*, lower state 10. Principal quantum number n*, upper state

11. Broadening cross section, σ (in (Bohr radius)2= a2₀, atomic units) 12. Velocity parameter, α

13. Data calculated by the ABO theory for a temperature of 10000 K (log₁₀(ΓABO

N ), cgs units)

14. Existing data, Kurucz for a temperature of 10000 K (log₁₀(ΓKurucz

N ), cgs units)

Due to the large size of this table, it has been placed in appendix B. Below follows an excerpt of the table to show its structure.

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Spec Wavel Elow Elim,low Eupp Elim,upp L J n∗low n

∗

upp σ α ABO Kurucz

FeI 4757.578 26519.4 84154.778 47619.9 89490.705 0->1 1.0->2.0 1.379 1.618 221.92 0.255 -7.79 -7.81

5.2 Graphs

5.2.1 ABO data / Kurucz ratio against Elow

Figure 3: Ratio between ABO data and Kurucz data for a temperature of 10000 K In figure 3 the data calculated using the ABO cross code has been compared to the provided Kurucz data, and it has been plotted against the energy level of the lower state. As seen in the figure, the p-s and p-d transitions have higher ratios between the data sets, compared to the s-p transition. this is explained by the fact that the Kurucz data has been calculated using the Unsöld theory, which does not factor in the Orbital angular momentum quantum number l. They are also slightly grouped energy wise.

5.2.2 ABO data / Kurucz ratio against Eupper

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In figure 4 the data calculated using the ABO cross code has been compared to the provided kurucz data, and it has been plotted against the energy level of the upper state. From the graph it is obvious that the energy state correlates stronger with the transition. Both the p-s and p-d transitions are grouped much tighter compared to when plotting against the energy of the lower state.

5.2.3 ABO data / Kurucz ratio against ∆E = Eupper− Elow

Figure 5: Ratio between ABO data and Kurucz data for a temperature of 10000 K In figure 5 the data calculated using the ABO cross code has been compared to the provided kurucz data, and it has been plotted against the difference in energy between the upper and lower states. This difference is also proportional to the wavelength of the emitted photon. Here, no strong correlation can be found for the different transitions other than the ratios. The ratios are as mentioned earlier explained by the simplicity of the Unsöld theory.

5.3 Lines not calculated

Out of the 103 iron lines acquired, only 82 were calculated by the ABO-cross code. The remaining 21 lines were found to be out-of-bounds of the tables used by the program when calculating cross-sections. Their effective principal quantum numbers (n*) were found to be either too high or too low to be usable, rendering the lines unaccounted for in the rest of the results.

6 Concluding remarks

The predicted line widths produced by the ABO cross code are, except for a few lines, larger compared to pre-existing data. This is likely explained by the fact that the Unsöld theory (used for the Kurucz data) doesn’t take the orbital angular momentum quantum number l into account. As seen in the comparison, most p-s and p-d transitions have a factor of two larger line width, while most of the s-p transitions have a factor of 0.8-1.4 larger line width. This disparity is likely explained by the fact that the orbital angular momentum quantum number plays a different role in the transitions above s-p, since it is the most basic transition. The reason that a few lines have a smaller predicted line width using the ABO theory could be explained by the fact that simpler theory didn’t account for the series limit properly. A higher series limit means that the effective principal quantum number is lowered, which could lead to a smaller line width.

Continuing to use this method to further calculate data for both the Gaia-ESO project and other areas of application is recommended. In this work, the focus has been on spectral lines of iron, but thanks to the design of the ABO method this is easily usable for other metals and elements. Initially, other metal lines were included in the scope of this project, but were cut due to

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time constraints. The sample of lines is also rather small, and the wavelength range is narrow. Due to this one cannot draw general conclusions from this sample, but it is a good start. Continuing to do calculations in larger spectral ranges and for a larger number lines would help with drawing more general conclusions about the reliability of the produced results.

Another interesting aspect that was also cut due to time constraints were investigations of line strength in cool stars. Since cooler stars have longer lifetimes, they are better tracers of galactic evolution. Knowing which spectral lines are strong in cool stars would thus improve abundance analysis further for the Gaia-ESO public spectroscopic survey, since their primary focus is increased knowledge of the galactic evolution of the Milky Way.

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References

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Barklem, P (2016). abo-cross: April 2016 release. doi: 10 . 5281 / zenodo . 50216_{. url:} https : //doi.org/10.5281/zenodo.50216.

Barklem, P. S., S. D. Anstee, and B. J. O’Mara (1998). “Line Broadening Cross Sections for the Broadening of Transitions of Neutral Atoms by Collisions with Neutral Hydrogen”. In:

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Barklem, P. S. and B. J. O’Mara (1997). “The Broadening of p-d and d-p transitions by collisions with neutral hydrogen atoms”. In: Monthly Notices of the Royal Astronomical Society 290.1, pp. 102–106.

Barklem, P. S., B. J. O’Mara, and J. E. Ross (1998). “The Broadening of d-f and f-d Transitions by Collisions with Neutral Hydrogen Atoms”. In: Monthly Notices of the Royal Astronomical

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Kupka, F. et al. (1999). In: A&AS 138, p. 119. Kurucz, R. L. (1981). In: SAO Special Report 390.

— (2013). Kurucz on-line database of observed and predicted atomic transitions. Available on: http://kurucz.harvard.edu/.

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Ryabchikova, T. A. et al. (1999). In: Phys. Scr. T 83, p. 162.

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Appendix

A

MatLab Code

1 ps = ’ Fe_ps . x l s x ’; 2 pd = ’ Fe_pd . x l s x ’; 3 sp = ’ Fe_sp . x l s x ’; 4 % S−P 5 Elowsp = x l s r e a d ( sp ,’D:D ’) ; 6 Ehighsp = x l s r e a d ( sp ,’F : F ’) ; 7 E d i f f s p = Ehighsp − Elowsp ; 8 9 mydatasp = x l s r e a d ( sp , ’N:N ’) ; 10 u n s o l d s p = x l s r e a d ( sp , ’R : R ’) ; 11 d a t a s p = 1 0 . ^ ( mydatasp−u n s o l d s p ) ; 12 13 %P−S 14 ElowPS = x l s r e a d ( ps ,’D:D ’) ; 15 EhighPS = x l s r e a d ( ps , ’F : F ’) ; 16 E d i f f P S = EhighPS − ElowPS ; 17 18 mydataPS = x l s r e a d ( ps , ’N:N ’) ; 19 unsoldPS = x l s r e a d ( ps ,’R : R ’) ; 20 dataPS = 1 0 . ^ ( mydataPS−unsoldPS ) ; 21 22 %P−D 23 ElowPD = x l s r e a d ( pd ,’D:D ’) ; 24 EhighPD = x l s r e a d ( pd ,’F : F ’) ; 25 EdiffPD = EhighPD − ElowPD ; 26 27 mydataPD = x l s r e a d ( pd ,’N:N ’) ; 28 unsoldPD = x l s r e a d ( pd , ’R : R ’) ; 29 dataPD = 1 0 . ^ ( mydataPD−unsoldPD ) ; 30 31 %P l o t s 32 c l o s e a l l 33 A = f i g u r e;

34 p l o t( Elowsp , dat asp ,’+b ’, ElowPS , dataPS , ’ ∗ r ’, ElowPD , dataPD , ’mx ’ , . . . 35 ’ m a r k e r s ’, 1 4 ,’ l i n e w i d t h ’ , 1 . 0 1 ) ;

36 s e t(A, ’ P a p e r P o s i t i o n M o d e ’, ’ a u t o ’) ; 37 s e t(gca, ’ F o n t S i z e ’, 2 0 ) ;

38 t i t l e(’ R a t i o between ABO d a t a and Kurucz data , p l o t t e d a g a i n s t l o w e r

e n e r g y l e v e l ’,’ F o n t S i z e ’, 2 8 ) ,

39 x l a b e l(’ Energy [ cm−1] ’,’ F o n t S i z e ’, 2 8 ) , y l a b e l(’ABO d a t a / Kurucz

d a t a ’,’ F o n t S i z e ’, 2 8 )

40 l e g e n d(’ s−p ’,’ p−s ’,’ p−d ’, ’ F o n t S i z e ’, 2 0 ) 41 B = f i g u r e;

42 p l o t( Ehighsp , dat asp , ’+b ’, EhighPS , dataPS ,’ ∗ r ’, EhighPD , dataPD , ’mx ’,’

m a r k e r s ’, 1 4 ,’ l i n e w i d t h ’ , 1 . 0 1 ) ;

43 s e t(B , ’ P a p e r P o s i t i o n M o d e ’, ’ a u t o ’) ; 44 s e t(gca, ’ F o n t S i z e ’, 2 0 ) ;

45 t i t l e(’ R a t i o between ABO d a t a and Kurucz data , p l o t t e d a g a i n s t h i g h e r

e n e r g y l e v e l ’,’ F o n t S i z e ’, 2 8 ) ,

d a t a ’,’ F o n t S i z e ’, 2 8 )

47 l e g e n d(’ s−p ’,’ p−s ’,’ p−d ’, ’ F o n t S i z e ’, 2 0 ) 48 C = f i g u r e;

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m a r k e r s ’, 1 4 ,’ l i n e w i d t h ’ , 1 . 0 1 ) ;

50 s e t(C, ’ P a p e r P o s i t i o n M o d e ’, ’ a u t o ’) ; 51 s e t(gca, ’ F o n t S i z e ’, 2 0 ) ;

52 t i t l e(’ R a t i o between ABO d a t a and Kurucz data , p l o t t e d a g a i n s t e n e r g y

d i f f e r e n c e o f l e v e l s ’,’ F o n t S i z e ’, 2 8 ) ,

d a t a ’,’ F o n t S i z e ’, 2 8 )

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B

Results table from ABO method

∗

FeI 4757.578 26519.4 84154.778 47619.9 89490.705 0->1 1.0->2.0 1.379 1.618 221.92 0.255 -7.79 -7.81 FeI 4772.803 12611.7 65738.011 33647.4 65738.011 0->1 3.0->3.0 1.437 1.849 284.45 0.254 -7.682 -7.79 FeI 4789.651 28730.7 84154.778 49685.4 90767.246 0->1 2.0->3.0 1.407 1.634 230.51 0.258 -7.774 -7.8 FeI 4802.875 29929.5 93877.03 50827.5 90767.246 0->1 4.0->4.0 1.31 1.657 217.97 0.248 -7.797 -7.8 FeI 4811.033 24875.1 81857.424 45748.8 84742.334 0->1 1.0->1.0 1.387 1.677 234.94 0.252 -7.765 -7.79 FeI 5028.126 28941.3 83873.251 48907.8 89574.581 0->1 5.0->4.0 1.413 1.642 233.48 0.258 -7.769 -7.8 FeI 5036.922 24437.7 81857.424 44371.8 84742.334 0->1 2.0->2.0 1.382 1.648 227.98 0.254 -7.778 -7.82 FeI 5320.036 29500.2 94740.815 48373.2 86639.041 0->1 3.0->2.0 1.297 1.693 223.94 0.247 -7.785 -7.7 FeI 5326.142 28941.3 83873.251 47790 89082.083 0->1 5.0->4.0 1.413 1.63 231.22 0.259 -7.773 -7.82 FeI 5358.113 26738.1 84154.778 45473.4 84742.334 0->1 2.0->2.0 1.382 1.671 232.72 0.252 -7.769 -7.8 FeI 5466.987 28941.3 83873.251 47312.1 84780.446 0->1 5.0->4.0 1.413 1.711 246.43 0.249 -7.744 -7.82 FeI 5534.659 29484 94740.815 47619.9 89574.581 0->1 2.0->2.0 1.296 1.617 208.43 0.25 -7.817 -7.81 FeI 5622.945 29484 94740.815 47336.4 76933.841 0->1 2.0->2.0 1.296 1.925 302.1 0.279 -7.661 -7.75 FeI 6005.542 20962.8 86129.119 37681.2 65738.011 0->1 3.0->2.0 1.297 1.977 327.23 0.288 -7.628 -7.75 FeI 6187.398 22939.2 76933.841 39163.5 65738.011 0->1 2.0->1.0 1.425 2.032 349.86 0.249 -7.591 -7.73 FeI 6221.672 6957.9 65738.011 23093.1 63737.704 0->1 5.0->4.0 1.366 1.643 223.86 0.253 -7.786 -7.83 FeI 6318.018 19869.3 63737.704 35761.5 65738.011 0->1 4.0->3.0 1.581 1.913 329.92 0.245 -7.616 -7.79 FeI 6353.836 7411.5 65738.011 23206.5 63737.704 0->1 4.0->3.0 1.371 1.645 225.28 0.253 -7.783 -7.83 FeI 6464.662 7759.8 65738.011 23295.6 63737.704 0->1 3.0->2.0 1.375 1.647 226.41 0.253 -7.781 -7.83 FeI 6483.944 12028.5 65738.011 27515.7 63737.704 0->1 4.0->3.0 1.429 1.74 255.16 0.244 -7.728 -7.8 FeI 6625.021 8189.1 65738.011 23344.2 63737.704 0->1 1.0->1.0 1.381 1.648 227.59 0.254 -7.779 -7.83 FeI 8868.43 24445.8 89082.083 35761.5 65738.011 0->1 3.0->3.0 1.303 1.913 296.55 0.274 -7.668 -7.79 FeI 4885.43 31444.2 71731.893 51993.9 71641.893 1->0 4.0->3.0 1.65 2.363 766.99 0.233 -7.248 -7.53 FeI 5429.503 33947.1 65738.011 52439.4 71641.893 1->0 2.0->1.0 1.857 2.39 767.11 0.243 -7.249 -7.53 FeI 5476.564 33234.3 65738.011 51572.7 71641.893 1->0 4.0->4.0 1.837 2.338 737.85 0.244 -7.266 -7.53 FeI 5573.102 33947.1 65738.011 51961.5 71731.893 1->0 2.0->2.0 1.857 2.355 738.31 0.239 -7.265 -7.57 FeI 5600.224 34506 71731.893 52439.4 71641.893 1->0 1.0->1.0 1.716 2.39 783.1 0.24 -7.24 -7.53 FeI 5678.379 31460.4 71731.893 49134.6 65738.011 1->0 3.0->2.0 1.65 2.57 1039.39 0.234 -7.116 -7.54 FeI 5715.091 34692.3 65738.011 52261.2 71731.893 1->0 2.0->1.0 1.88 2.373 743.55 0.232 -7.261 -7.57 FeI 5976.777 31938.3 71731.893 48737.7 65738.011 1->0 3.0->3.0 1.66 2.54 988.97 0.231 -7.137 -7.54 FeI 6008.556 31460.4 71731.893 48162.6 65738.011 1->0 3.0->4.0 1.65 2.498 907.74 0.227 -7.173 -7.54 FeI 6232.64 29597.4 63737.704 45700.2 63737.704 1->0 2.0->1.0 1.792 2.466 854.82 0.237 -7.201 -7.54 FeI 6408.017 29856.6 63737.704 45530.1 63737.704 1->0 1.0->2.0 1.799 2.454 843.55 0.241 -7.208 -7.54 FeI 6704.48 34165.8 65738.011 49134.6 65738.011 1->0 1.0->2.0 1.864 2.57 1002.35 0.252 -7.135 -7.54 FeI 6713.046 37316.7 63737.704 52269.3 71641.893 1->0 2.0->2.0 2.037 2.379 786.6 0.266 -7.243 -7.53 FeI 8527.852 40662 63737.704 52439.4 71641.893 1->0 0.0->1.0 2.18 2.39 672.55 0.281 -7.314 -7.53 FeI 8598.829 35534.7 65738.011 47206.8 65738.011 1->0 5.0->5.0 1.906 2.433 813.52 0.231 -7.222 -7.55 FeI 8763.966 37681.2 65738.011 49134.6 65738.011 1->0 2.0->2.0 1.977 2.57 1001.69 0.252 -7.135 -7.54 FeI 8784.44 40143.6 63737.704 51572.7 71641.893 1->0 3.0->4.0 2.156 2.338 631.92 0.289 -7.342 -7.53 FeI 8793.342 37324.8 65738.011 48737.7 65738.011 1->0 3.0->3.0 1.965 2.54 960.04 0.25 -7.153 -7.54 FeI 4872.907 34182 65738.011 54788.4 65738.011 1->2 4.0->4.0 1.864 3.165 1044.32 0.279 -7.122 -7.5 FeI 4952.64 34092.9 71731.893 54367.2 65738.011 1->2 2.0->1.0 1.707 3.106 983.67 0.271 -7.147 -7.52 FeI 4967.897 33947.1 65738.011 54156.6 65738.011 1->2 2.0->1.0 1.857 3.077 937.99 0.278 -7.169 -7.52 FeI 5072.672 34182 65738.011 53978.4 65738.011 1->2 4.0->3.0 1.864 3.054 907.39 0.279 -7.183 -7.51 FeI 5195.472 34182 65738.011 53508.6 65738.011 1->2 4.0->5.0 1.864 2.995 834.25 0.278 -7.219 -7.51 FeI 5196.059 34473.6 65738.011 53800.2 65738.011 1->2 3.0->2.0 1.873 3.031 878.52 0.279 -7.197 -7.51 FeI 5429.842 36231.3 65738.011 54723.6 65738.011 1->2 3.0->3.0 1.928 3.156 1022.65 0.281 -7.132 -7.51 FeI 5455.454 34992 65738.011 53395.2 65738.011 1->2 6.0->6.0 1.889 2.981 817.78 0.278 -7.228 -7.34 FeI 5521.28 35923.5 65738.011 54108 65738.011 1->2 4.0->5.0 1.918 3.071 924.46 0.279 -7.175 -7.51 FeI 5538.516 34165.8 65738.011 52293.6 63737.704 1->2 1.0->2.0 1.864 3.096 960.91 0.279 -7.158 -7.48 FeI 5559.639 40402.8 63737.704 58465.8 71641.893 1->2 2.0->2.0 2.168 2.885 677.07 0.263 -7.307 -7.3 FeI 5577.025 40767.3 63737.704 58773.6 71641.893 1->2 4.0->4.0 2.185 2.919 713.96 0.263 -7.284 -7.39 FeI 5594.655 36846.9 65738.011 54788.4 65738.011 1->2 4.0->4.0 1.948 3.165 1029.83 0.283 -7.129 -7.5 FeI 5595.06 41018.4 63737.704 58959.9 71641.893 1->2 3.0->3.0 2.197 2.941 741.65 0.266 -7.268 -7.42 FeI 5597.061 40427.1 63737.704 58368.6 71641.893 1->2 5.0->4.0 2.169 2.875 666.77 0.264 -7.314 -7.3 FeI 5646.684 34506 71731.893 52293.6 63737.704 1->2 1.0->2.0 1.716 3.096 970.32 0.272 -7.153 -7.48 FeI 5650.705 41196.6 63737.704 58959.9 71641.893 1->2 2.0->3.0 2.206 2.941 741.72 0.266 -7.268 -7.42 FeI 5659.575 41196.6 63737.704 58935.6 71731.893 1->2 2.0->3.0 2.206 2.928 724.13 0.265 -7.279 -7.36 FeI 5667.518 33841.8 65738.011 51556.5 63737.704 1->2 5.0->4.0 1.854 3.001 840.97 0.278 -7.216 -7.53

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∗

FeI 5704.733 40767.3 63737.704 58368.6 71641.893 1->2 4.0->4.0 2.185 2.875 664.98 0.265 -7.315 -7.3 FeI 5752.032 36846.9 65738.011 54294.3 65738.011 1->2 4.0->4.0 1.948 3.096 948.23 0.279 -7.164 -7.51 FeI 5769.323 37324.8 65738.011 54723.6 65738.011 1->2 3.0->3.0 1.965 3.156 1015.45 0.282 -7.135 -7.51 FeI 5848.127 37324.8 65738.011 54488.7 65738.011 1->2 3.0->2.0 1.965 3.122 975.93 0.28 -7.152 -7.52 FeI 5859.586 36846.9 65738.011 53978.4 65738.011 1->2 4.0->3.0 1.948 3.054 902.1 0.277 -7.185 -7.51 FeI 5862.356 36846.9 65738.011 53970.3 65738.011 1->2 4.0->5.0 1.948 3.053 900.94 0.277 -7.186 -7.52 FeI 5947.513 37316.7 63737.704 54197.1 65738.011 1->2 2.0->2.0 2.037 3.083 916.22 0.281 -7.179 -7.52 FeI 5983.68 36846.9 65738.011 53622 65738.011 1->2 4.0->4.0 1.948 3.009 850.4 0.275 -7.211 -7.51 FeI 5984.815 38337.3 65738.011 55112.4 65738.011 1->2 3.0->2.0 2.001 3.213 1072.65 0.289 -7.112 -7.42 FeI 5987.065 38847.6 65738.011 55614.6 65738.011 1->2 2.0->1.0 2.02 3.292 1143.73 0.296 -7.086 -7.42 FeI 6007.96 37681.2 65738.011 54391.5 65738.011 1->2 2.0->3.0 1.977 3.109 957.85 0.279 -7.16 -7.51 FeI 6078.491 38847.6 65738.011 55363.5 65738.011 1->2 2.0->3.0 2.02 3.251 1105.56 0.293 -7.1 -7.42 FeI 6290.965 38337.3 65738.011 54294.3 65738.011 1->2 3.0->2.0 2.001 3.096 938.28 0.279 -7.169 -7.51 FeI 6293.924 39163.5 65738.011 55112.4 65738.011 1->2 1.0->2.0 2.032 3.213 1063.18 0.29 -7.116 -7.42 FeI 6411.106 38337.3 65738.011 54002.7 65738.011 1->2 3.0->4.0 2.001 3.057 897.31 0.275 -7.187 -7.51 FeI 6705.101 37316.7 63737.704 52293.6 63737.704 1->2 2.0->2.0 2.037 3.096 930.21 0.282 -7.173 -7.48 FeI 6707.431 37324.8 65738.011 52293.6 63737.704 1->2 3.0->2.0 1.965 3.096 945.25 0.278 -7.165 -7.48 FeI 6726.666 37316.7 63737.704 52245 63737.704 1->2 2.0->1.0 2.037 3.089 923.12 0.282 -7.176 -7.52 FeI 8571.804 40581 63737.704 52293.6 63737.704 1->2 1.0->2.0 2.176 3.096 898.46 0.288 -7.189 -7.48 FeI 8592.951 40143.6 63737.704 51823.8 63737.704 1->2 3.0->3.0 2.156 3.034 849.87 0.28 -7.212 -7.5 FeI 8613.939 40402.8 63737.704 52058.7 63737.704 1->2 2.0->3.0 2.168 3.064 874.76 0.284 -7.2 -7.47 FeI 8616.28 39795.3 63737.704 51451.2 63737.704 1->2 4.0->5.0 2.14 2.988 802.15 0.271 -7.235 -7.52 FeI 8846.74 40581 63737.704 51929.1 63737.704 1->2 1.0->2.0 2.176 3.048 860.6 0.282 -7.207 -7.5

Table 1: Results produced by the ABO method for a temperature of 10000 K, ordered by transition type

Broadening of spectral lines in the Gaia-ESO survey