11institutetext: Space Radio-Diagnostics Research Centre, University of Warmia
and Mazury, R. Prawocheńskiego 9, 10-719 Olsztyn, Poland
11email<EMAIL_ADDRESS>22institutetext: Leibniz-Institut für
Astrophysik Potsdam (AIP), An der Sternwarte 16, 14482 Potsdam, Germany
33institutetext: Institute of Astronomy and National Astronomical Observatory,
Bulgarian Academy of Sciences, Sofia 1784, Bulgaria 44institutetext: ASTRON –
the Netherlands Institute for Radio Astronomy, Oude Hoogeveensedijk 4, 7991 PD
Dwingeloo, the Netherlands 55institutetext: Astronomy & Astrophysics Section,
Dublin Institute for Advanced Studies, Dublin 2, D02 XF86, Ireland
66institutetext: CAS Key Laboratory of Geospace Environment, School of Earth
and Space Sciences, University of Science and Technology of China, Hefei,
Anhui 230026, China 77institutetext: Solar–Terrestrial Centre of
Excellence—SIDC, Royal Observatory of Belgium, 1180 Brussels, Belgium
88institutetext: Centre for mathematical Plasma Astrophysics (CmPA), KU
Leuven, Celestijnenlaan 200B, 3001 Leuven, Belgium 99institutetext: Department
of Physics, University of Helsinki, P.O. Box 64, Helsinki, Finland
1010institutetext: RAL Space, United Kingdom Research and Innovation (UKRI) –
Science and Technology Facilities Council (STFC) – Rutherford Appleton
Laboratory (RAL), Harwell Campus, Oxfordshire OX11 0QX, UK 1111institutetext:
Astronomical Institute, University of Wrocław, Kopernika 11, 51-622 Wrocław,
Poland

# Interferometric imaging of the type IIIb and U radio bursts observed with
LOFAR on 22 August 2017

Bartosz Dabrowski 11 Katarzyna Mikuła 11 Paweł Flisek 11 Christian Vocks 22
PeiJin Zhang 33445566 Jasmina Magdalenić 7788 Alexander Warmuth 22 Diana E.
Morosan 99 Adam Froń 11 Richard A. Fallows 44 Mario M. Bisi 1010 Andrzej
Krankowski 11 Gottfried Mann 22 Leszek Błaszkiewicz 11 Eoin P. Carley 55 Peter
T. Gallagher 55 Pietro Zucca 44 Paweł Rudawy 1111 Marcin Hajduk 11 Kacper
Kotulak 11 Tomasz Sidorowicz 11

(Received 2021; accepted)

###### Abstract

Context. The Sun is the source of different types of radio bursts that are
associated with solar flares, for example. Among the most frequently observed
phenomena are type III solar bursts. Their radio images at low frequencies
(below 100 MHz) are relatively poorly studied due to the limitations of legacy
radio telescopes.

Aims. We study the general characteristics of types IIIb and U with stria
structure solar radio bursts in the frequency range of 20 – 80 MHz, in
particular the source size and evolution in different altitudes, as well as
the velocity and energy of electron beams responsible for their generation.

Methods. In this work types IIIb and U with stria structure radio bursts are
analyzed using data from the LOFAR telescope including dynamic spectra and
imaging observations, as well as data taken in the X-ray range (GOES and
RHESSI satellites) and in the extreme ultraviolet (SDO satellite).

Results. In this study we determined the source size limited by the actual
shape of the contour at particular frequencies of type IIIb and U solar bursts
in a relatively wide frequency band from 20 to 80 MHz. Two of the bursts seem
to appear at roughly the same place in the studied active region and their
source sizes are similar. It is different in the case of another burst, which
seems to be related to another part of the magnetic field structure in this
active region. The velocities of the electron beams responsible for the
generation of the three bursts studied here were also found to be different.

###### Key Words.:

Sun: radio radiation – Sun: X-ray radiation – Sun: UV radiation – Sun:
activity – methods: observational

##  1 Introduction

Type III solar radio bursts are among the most frequently observed radio
events, first described in a series of publications by Wild & McCready (Wild &
McCready, 1950; Wild, 1950a, b). They occur at frequencies in the range from
GHz down to kHz, with individual bursts lasting for several seconds, depending
on the observing frequency. However, they sometimes appear in groups lasting
from a few minutes to over a day in the case of a storm (Benz, 2002). The
brightness temperature of these bursts usually lies between $10^{8}$ and
$10^{12}$ K, sometimes reaching values of $10^{15}$ K (Dulk, 1985). Ginzburg &
Zhelezniakov (1958) were the first to present a type III burst model based on
plasma emission. Type III bursts are believed to be the radio signature of
electrons beams moving through the corona and into interplanetary space along
open magnetic field lines (Lin, 1974). The electron beams are generally
considered to originate from magnetic reconnection or shocks (Dulk et al.,
2000). It is commonly believed that in type III bursts, following their
acceleration, faster electrons outpace the slower ones, resulting in a bump-
on-tail instability in their velocity distribution. This leads to the
generation of Langmuir plasma waves (Ginzburg & Zhelezniakov, 1958). The waves
are later converted into radio waves at the local plasma frequency and its
second or even higher harmonics (Bastian et al., 1998). The plasma frequency
depends on the electron density (e.g., Warmuth & Mann, 2005), which in general
decreases with altitude in the solar corona. Therefore, the plasma frequency
also decreases with height.

Type III radio bursts are characterized by a high frequency drift rate. On the
dynamic spectra, they resemble almost vertical structures. Their slope
indicates fast moving electron beams accelerated in the corona. The drift rate
on the metric wavelengths has been reported by various authors (see Table 1).
It allows us to estimate the velocity of the electrons beams responsible for
the type III bursts (e.g., Mann et al., 1999). On the other hand, the electron
density can be obtained from the Newkirk (1961) radial electron density model
of the solar corona (hereafter called the Newkirk model).

The standard velocities of type III radio bursts are about 0.1c and 0.6c
(Benz, 2002), but Melnik et al. (2021) observed this burst type with URAN-2
and GURT in the frequency band 8 – 80 MHz, and showed that the velocities are
in the range 0.17 – 0.2c (harmonic emission was assumed). On the other hand,
Morosan et al. (2014) observed type III bursts with LOFAR in the 30 – 90 MHz
band and obtained velocities in the range of 0.09 – 0.2c (harmonic emission
for all the type III bursts was assumed).

Table 1: Drift rates for type III solar radio bursts. Drift rate | Freq. Range | Reference
---|---|---
[MHz/s] | [MHz] |
$-0.01f^{1.84}$ | 0.075 – 550 | Alvarez & Haddock (1973)
$-0.007354f^{1.76}$ | 0.04 – 85 | Mann et al. (1999)
$-0.0672f^{1.23}$ | 10 – 80 | Zhang et al. (2018)

The variation in the source sizes of type III bursts with frequency depends on
such several effects, such as (1) divergence of the magnetic field lines as a
function of height in the corona; (2) spatial width of the electron beam; (3)
refractive propagation effects; and (4) scattering (e.g., Saint-Hilaire et
al., 2013; Zhang et al., 2021). A study of source sizes with the Nançay
Radioheliograph in the frequency range of $\sim$150 – 450 MHz performed by
Saint-Hilaire et al. (2013) shows that they are smaller at higher frequencies
than at lower ones, and that their sizes decrease as $f^{-3.3}$. On the other
hand, Dulk & Suzuki (1980) found that the sizes of the source at the
frequencies of 80 MHz and 43 MHz are 11 arcmin and 20 arcmin, respectively.
Sometimes type III bursts show fine spectral substructures. In particular,
bursts classified as type IIIb are characterized by many narrowband bursts
with slow drift, known as stria bursts (de La Noe & Boischot, 1972; Abranin et
al., 1979; Mugundhan et al., 2017; Sharykin et al., 2018; Melnik et al.,
2018). The bandwidth of the striae in the 20 – 80 MHz range is usually about
20 – 100 kHz (de La Noe, 1975; Sharykin et al., 2018; Chen et al., 2018) and
the stria drift rates vary from 0 to 0.3 MHz/s (Sharykin et al., 2018). The
duration of the separate striae is short, typically around 1 s (de La Noe,
1975; Chen et al., 2018). The most typical scenario for the formation of
striae is plasma density fluctuations along the electron beam path (Takakura &
Yousef, 1975). These small-amplitude perturbations can modulate the generation
of Langmuir waves significantly (Kontar, 2001), and hence produce fine stria-
like structures in the dynamic radio spectra.

Bursts classified as type U were first reported by Maxwell & Swarup (1958).
They got their name thanks to their shape in the dynamic spectrum, which
resembles the inverted letter U. They constitute a variant of type III bursts;
they are a manifestation of the signatures of electron beams traveling along
closed magnetic loops. Type U bursts are much rarer than type III bursts. The
rising and the descending branches of the type U bursts have spatially
separated sources (Aurass & Klein, 1997). Their polarization is generally low,
below 10%. In the meter waves they were studied by Aurass & Klein (1997),
Dorovskyy et al. (2015), and Reid & Kontar (2017).

Several papers have recently been published, estimating the emission source
sizes of type IIIb bursts observed with the LOFAR telescope. Kontar et al.
(2017) obtained 400 arcmin2 for 32.5 MHz. Its source was located at $x\approx
280\textrm{ arcsec}$ and $y\approx 390\textrm{ arcsec}$ from the solar center.
On the other hand, Sharykin et al. (2018) found that source size for single
stria equals 320 arcmin2 (located around 290 arcsec, 320 arcsec) and 145
arcmin2 (around $-30$ arcsec, 495 arcsec) for 30.11 and 41.77 MHz,
respectively. It should be noted that in both publications, the authors
obtained their results with the tied-array beam forming method. In turn, the
source size obtained by Zhang et al. (2020) is about 100 arcmin2 (its location
was $-333$ arcsec, $-165$ arcsec from the solar center) for 26.56 MHz, and by
Murphy et al. (2021) is 150.6 arcmin2 ($-1312$ arcsec, $-1064$ arcsec) for
34.76 MHz. In both papers the interferometric observation method was used. It
is also worth mentioning that Dulk & Suzuki (1980) estimated the effective
source sizes (full widths at $1/e$ brightness) of the type III bursts observed
at 43 MHz around 314 arcmin2. On the other hand, Chen et al. (2020)
investigated how the scattering of radio waves on random density fluctuations
affects the time profile evolution, sizes, and positions of the observed type
III–IIIb solar radio bursts emitted via the plasma emission mechanism. They
found that at 32.5 MHz the simulated source size for the fundamental emission
ranged from around 60 to 100 arcmin2 during the 2.5 s of the decay phase,
which they found to be much smaller than typical observed source sizes from
$\sim$300 arcmin2 to $\sim$440 arcmin2.

In this article we study the general characteristics of two type IIIb solar
radio bursts and one U burst with stria structures present (typical of type
IIIb bursts) observed on 22 August 2017 with the LOw-Frequency ARray (LOFAR;
van Haarlem et al. 2013) telescope in the frequency range of 20 – 80 MHz,
including dynamic spectra and imaging observations. This gives the spatial
information of radio bursts and relates it to the height, size, velocity, and
energy of the electron beams responsible for their generation. In order to
better understand the physical properties and evolution of the solar radio
bursts, we used the data in X-rays from the GOES and RHESSI satellites and in
the extreme ultraviolet (EUV) from the Solar Dynamics Observatory (SDO).

Figure 1: The solar dynamic spectra of the analyzed solar radio bursts. a)
Part of the solar dynamic spectrum with the series of the type III solar radio
bursts recorded on 22 August 2017 between 09:00:00 UT and 13:59:59 UT by the
LOFAR remote station RS210 in LBA band. b) Solar dynamic spectrum of the three
investigated bursts, described as #1, #2, and #3. c) Solar dynamic spectrum of
burst #2 with clearly visible fundamental (F) and harmonic (H) components. d)
Solar dynamic spectrum of burst #3.

##  2 Observations and data analysis

LOFAR is a large radio interferometer located in Europe, currently consisting
of 52 stations, 38 of which are located in the Netherlands and 14 in different
countries across Europe: France, Germany, Ireland, Latvia, Poland, Sweden, and
the United Kingdom. The 38 LOFAR stations in the Netherlands include the 24
core stations (6 of which create the “Superterp”) and 14 remote stations. For
the whole International LOFAR Telescope (ILT) the largest baselines reach
about 2000 kilometers. Each station of LOFAR observes with two types of
antennas, the Low Band Antennas (LBA), and the High Band Antennas (HBA),
operating in the ranges 10 – 90 MHz and 110 – 240 MHz, respectively.

Here we used part of the LOFAR spectral and imaging data from the observation
project LC8_013, “Interferometric Observations of the Active Regions in Radio
Domain Before and After the Total Solar Eclipse on 21 August 2017.” For this
research we incorporated 35 LOFAR stations (23 core and 12 remote stations),
including the dynamic spectra and imaging observations at frequencies of 20 –
80 MHz (“LBA Outer” configurations). These 35 stations have a maximum baseline
of about 58.4 km, which allows a radio image with a theoretical spatial
resolution of about 10.6 arcsec at 80 MHz (van Haarlem et al., 2013).
Unfortunately, due to coronal scattering, this resolution is limited to about
1 arcmin (Mercier et al., 2015). We used Taurus A as a calibrator to derive
the stations’ gains. In order to obtain an interferometric image of the bursts
we sampled the dynamic spectrum in 5 MHz intervals, which provided us with
sufficient accuracy. The temporal and frequency resolutions of the dynamic
spectra are 0.0104 s and 0.0154 MHz, respectively. The dynamic spectra were
prepared using the LOFAR Solar Imaging Pipeline developed at the Leibniz-
Institut für Astrophysik Potsdam (Breitling et al., 2015).

The interferometric data were pre-processed with the Default Pre-Processing
Pipeline (DPPP) (van Diepen et al., 2018). The calibration of the gain
solution is obtained from the simultaneous observation of the calibrator. The
gain solution is then applied to the interferometry of the Sun to derive the
calibrated amplitude and phase. After the pre-processing, the imaging is
performed with WSClean (Offringa et al., 2014; Offringa & Smirnov, 2017). The
integration time for each frame of imaging is 1.0 second, representing the
time resolution of the imaging. After deconvolution and the CLEAN process
performed with WSClean, we obtained the flux intensity distribution map in
astronomical coordinates. Then we used the LOFAR-Sun toolbox (Zhang et al.,
2020) to convert the coordinates and units and align them to solar
observations made with other instruments (e.g., SDO).

Using dynamic spectra it is possible to determine the parameters of type III
bursts, such as drift rates, frequency range, and time span of each burst. In
order to do this computation, the raw LOFAR dynamic spectrum data were first
cleaned using a median filter. For each burst we determined approximate burst
boundaries in order to obtain a proper accuracy and diminish the risk of
introducing other unassociated events into the analysis. We found the maximum
intensity value in each frequency channel and established the time of its
occurrence. Collected maxima determine the center-line of the analyzed burst.
In order to obtain a smoother center-line, we used a moving average based in
the frequency and time domains. A logarithmic function was fitted to the
averaged maxima. The obtained values were used to calculate drift rates of the
bursts with a 1 MHz interval. The error of the average drift rate is computed
as the standard deviation of the drift rate at the frequencies that were
determined earlier. On the basis of the obtained drift rates we determined the
velocity and energy of electrons responsible for generation of the analyzed
bursts (e.g., Dabrowski et al., 2021). The reason for using the maxima of the
bursts rather than their leading edges was the extremely irregular shape of
the edges.

In order to obtain an interferometric image of the burst we sampled the
dynamic spectrum every 5 MHz. For each selected frequency channel we
determined the maximum intensity of the burst and time of its occurrence. In
the following step the contour of the radio emission area is set to half of
the maximum intensity. Additionally, for each contour a centroid is
calculated. The centroids of the bursts at each frequency were then used to
estimate the height at which the bursts appeared and the velocity of the
electrons beams responsible for their generation (according to the Newkirk
model of electron density in solar corona). Moreover, it was possible to
estimate the size of the radio emission area at a given height on the basis of
the sizes of the contours. Furthermore, we used soft X-ray fluxes from GOES,
hard X-ray spectroscopic and imaging data from RHESSI (Lin et al., 2002), and
observations in the extreme ultraviolet (EUV) from the Atmospheric Imaging
Assembly (AIA) on board the Solar Dynamics Observatory (SDO) (Lemen et al.,
2012), using the 171 Å channel (which corresponds to the quiet corona and
upper transition region of the solar atmosphere).

Figure 2: Comparison of radio flux with X-ray signals from GOES and RHESSI of
the radio event observed on 22 August 2017. Panel (a) shows GOES fluxes,
including a C1.0 solar flare with a peak at 11:13 UT. Panel (b) shows the
RHESSI count rates in five wide energy bins. Panel (c) presents the radio
total power flux from LOFAR 20 – 80 MHz (LBA) band with a single strong peak
at 11:11:46 UT.

In our observations of the Sun we identified a type III radio event observed
with LOFAR on 22 August 2017 between 09:00:00 UT and 13:59:59 UT, which
included a series of type III solar radio bursts (Fig. 1a). We were unable to
observe the beginning and end of the type III storm because of the limited
observing time. Particularly interesting are the bursts that occurred around
the single strong peak in the total radio flux at 11:11:46 UT, which appears
as a rapid and short-lived increase by a factor of about 30 (but with no
precursor activity), accompanied by a weak flare observed by GOES and RHESSI
at about 11:13 UT (Fig. 2). Here we distinguished three solar radio bursts
described as bursts #1, #2, and #3 (Fig. 1b). Burst #1 (type U) occurred at
11:11:47 UT in the 65 – 80 MHz band; burst #2 (type IIIb, Fig. 1c) was
composed of two components, the fundamental (F), occurring at 11:12:00 UT in
the 29 – 77 MHz band, and the harmonic (H), which is weaker than the
fundamental, occurring at 11:12:03 in the 73 – 79 MHz frequency band; and
burst #3 (type IIIb, Fig. 1d) at 11:12:20 UT in the 28 – 74 MHz band.

The studied radio bursts were associated with a GOES C1.0 solar flare
originating from active region NOAA AR 12671 located at 5.73∘ N, 25.84∘ W. The
flare started at 11:09 UT, reached a maximum at 11:13 UT, and ended at 11:16
UT. It was also registered by RHESSI in the hard X-ray range (Fig. 2) and
observed in the EUV by the AIA in the 94 Å and 171 Å channels. The active
region AR 12671 had a beta-gamma configuration of its photospheric magnetic
field on 22 August 2017, and on that day was a source of many B and C class
flares. The sudden brightening observed at the western edge of the NOAA AR
12671, 4.58∘ N, 32.6∘ W is consistent with X-ray fluxes registered by GOES and
RHESSI.

Figure 3: Image of the Sun received in the AIA 171 Å channel by SDO with
superimposed color contours showing bursts #1 (a), #2 (b), and #3 (c) at a
range of frequencies. In (a) the active region AR 12671 is indicated by a
rectangle. Table 2: Physical parameters of bursts #1, #2, and #3 determined
from the dynamic spectra and the centroids. Columns F and H are respectively
the fundamental and harmonic components of burst #2. The table also includes
velocities determined on the basis of the drift rate obtained by various
authors (see Table 1).

| Burst #1 | Burst #2 | Burst #3
---|---|---|---
|  | F | H |
DYNAMIC SPECTRUM |  |  |  |
Time [UT] | 11:11:47 | 11:12:00 | 11:12:03 | 11:12:20
Freq. band [MHz] | 80 – 65 | 77 – 29 | 79 – 73 | 74 – 28
Height [R⊙] | 1.3 – 1.4 | 1.3 – 1.8 | 1.3 – 1.4 | 1.3 – 1.8
Central freq. [MHz] | 72.5 | 53.1 | 75.7 | 50.8
Drift rate [MHz/s] | $-2.6\pm 0.2$ | $-4.3\pm 2.1$ |  $-4.9$**$*$We were able to estimate only two measuring points for the harmonic burst, which makes the drift rate error unobtainable. | $-9.0\pm 4.8$
Energy [keV] | $0.24\pm 0.04$ | $1.76\pm 1.73$ | 1.72 | $9.06\pm 9.92$
| Velocity [c]
This study | $0.03\pm 0.002$ | $0.08\pm 0.04$ | 0.08 | $0.19\pm 0.10$
Alvarez & Haddock (1973) | 0.31 | 0.29 | 0.48 | 0.28
Mann et al. (1999) | 0.16 | 0.15 | 0.25 | 0.15
Zhang et al. (2018) | 0.15 | 0.17 | 0.23 | 0.17
CENTROIDS |  |  |  |
Freq. band [MHz] | 75 – 65 | 75 – 45 | – | 75 – 35
Height [R⊙] | 1.3 – 1.4 | 1.3 – 1.6 | – | 1.3 – 1.7
Velocity [c] | – | $0.06\pm 1.08\cdot 10^{-4}$ | – | $0.09\pm 3.2\cdot 10^{-5}$
Energy [keV] | – | $0.92\pm 0.003$ | – | $2.26\pm 0.002$

111

Table 3: Source sizes of bursts #1, #2, and #3 at different heights. | Burst #1 | Burst #2 | Burst #3
---|---|---|---
| Height
---
[R⊙]
| Source size
---
[arcmin2]
| Source size
---
[arcmin2]
| Source size
---
[arcmin2]
1.69 (35 MHz) | – | – | $205.9\pm 1.3$
1.61 (40 MHz) | – | – | $132.9\pm 1.2$
1.56 (45 MHz) | – | $46.2\pm 0.9$ | $102.4\pm 1.3$
1.51 (50 MHz) | – | – | $53.1\pm 1.2$
1.46 (55 MHz) | – | – | $56.2\pm 1.3$
1.43 (60 MHz) | – | $50.2\pm 1.0$ | $39.9\pm 1.7$
1.39 (65 MHz) | $33.9\pm 1.6$ | $46.7\pm 1.7$ | $33.3\pm 2.0$
1.37 (70 MHz) | – | $48.3\pm 5.8$ | $42.8\pm 2.6$
1.34 (75 MHz) | $49.6\pm 11.8$ | $36.9\pm 2.2$ | $30.9\pm 5.6$
Figure 4: Change in the size of the emission area with the height for the
investigated bursts (see Table. 3). The horizontal error bars represent the
beam size at FWHM, which varies with frequency. Figure 5: Radial motion of
bursts #2 (left panel) and #3 (right panel) obtained from centroids. The
colors correspond to different frequencies from 35 to 75 MHz. The velocities
were calculated from the slopes of the fits and are given in the upper left
corners. The vertical error bars in the two plots represent the extent of the
FWHM of the beams, which varies with frequency. The horizontal error bars are
due to averaging the data in time steps of 1 s. Figure 6: Map from SDO of the
analyzed active region, and obtained signals from LOFAR and SDO. The left
panel shows the active region AR 12671 during the solar flare registered in
AIA 171 Å by SDO with a superimposed grid and the radio contours at 75 MHz
from the radio bursts #1, #2, and #3. In the right upper corner of the figure
there is a magnification of box 17 with non-thermal hard X-ray footpoints
(dark blue contour) in the 18 – 40 keV range, as derived from RHESSI
observations. The right panels show EUV signals from boxes 17 and 23 and their
comparison with LOFAR total radio flux. The maxima of the EUV signal are
indicated by the red dotted lines.

##  3 Results

The analyzed type III radio bursts are outlined in Table 2, which contains
their physical parameters determined from the dynamic spectra and the
centroids, as well as velocities determined on the basis of the drift rate
obtained by various authors. Burst #1 was connected with the maximum increase
in radio emission (Fig. 2c). Its time-flux profiles look peculiar. In the 65 –
75 MHz band there is a simple profile with a single very clear maximum peak.
Above 75 MHz the time-flux profiles appear as double local maxima very close
to each other. These time-flux profiles suggest that it may be a type U burst
where stria structures (typical for type IIIb bursts) are present. This type
of burst is confirmed by the fact that the source region shows the usual
bipolar structure associated with a flaring loop (which is discussed in detail
below). Burst #2 has two components, fundamental (F) and harmonic (H). The
mean ratio between them is about 1.7 for a given time (Fig. 1c). This is
consistent with other observational values of 1.5 – 1.8 (Suzuki & Dulk, 1985).
The F component of burst #2 at half the frequency of the H component (around
35 – 40 MHz) is registered a few seconds later. There is a possibility that
the harmonic emission is affected by the active region (Fig. 3). The
explanation is that the conditions for producing harmonic emission are not met
immediately, but some time later. The type IIIb burst #3 has a very peculiar
kinked structure.

The single strong peak, observed in the total radio flux at 11:11:46 UT, was
connected with burst #1 and accompanied by a C1.0 flare (Fig. 2). A comparison
of the radio flux with the X-ray flux (in different energy ranges) shows that
the maximum peak in the radio range was observed 18 to 60 seconds earlier than
in the X-ray (Fig. 2). It is worth noting that in the energy range 12 – 25
keV, which is relevant for type III electrons, we observe a jump (at 11:10:52
UT) to an enhanced level about 54 seconds before the radio flux peak (at
11:11:46 UT), although a maximum is reached later. This agrees with the
general flare model that X-ray emission occurs before radio emission from the
upper corona (e.g., Benz, 2017).

For the three analyzed bursts we received the radio images. In Fig. 3 they are
depicted as contours with their centroids superimposed on the image of the Sun
obtained in the AIA 171 Å channel by SDO. In the case of burst #1, observed in
the band 65 – 80 MHz, the radio images were taken only for 65 and 75 MHz just
before the maximum phase in the given frequency because the maximum phase of
the solar radio burst is too bright for the remote baseline in the observation
of the calibrator to form interference correlation, which makes it impossible
to calibrate the corresponding solar observation. We were also unable to
obtain radio images for 70 and 80 MHz, due to radio interference. In the case
of burst #2, observed in the band around 29 – 79 MHz we received images for
45, 60, 65, 70, and 75 MHz. It was not possible to obtain radio images for 30,
35, 40, 50, 55, and 80 MHz. This is due to the fact that in the lower and
upper range of the observed radio band the relative intensity of this burst is
quite low and its structure is irregular. In addition, there is radio
interference at low frequencies. The radio images for burst #3 (observed in 28
– 74 MHz band) were found from 35 up to 75 MHz. It was not possible to obtain
a radio image for 30 MHz because in the lower range of the observed radio band
the relative intensity of this burst is quite low and its structure is
irregular.

From the radio images we found the sizes of the radio emission sources for
three of the analyzed bursts (Fig. 4 and Table 3). For burst #1 they could be
determined only for two altitudes of 1.34 R⊙ (75 MHz) and 1.39 R⊙ (65 MHz),
corresponding to the values of 49.6 arcmin2 and 33.9 arcmin2, respectively. In
the case of burst #2 the sizes of the radio emission sources at different
altitudes are similar, ranging from 36.9 arcmin2 (for a height of 1.34 R⊙) to
50.2 arcmin2 for a height of 1.43 R⊙. Burst #3 is a nice example of burst
evolution through the corona. From the radio images we estimate that the burst
starts at a distance of 1.34 R⊙ (75 MHz) and ends at 1.69 R⊙ (35 MHz). The
source of the burst moves outward with decreasing frequency, as expected with
increasing coronal height, and gets larger. We observe a significant increase
in the size of the source of the emission between the heights of 1.51 R⊙ and
1.69 R⊙, where it was 53.1 arcmin2 and 205.9 arcmin2, respectively. Here, at a
distance of 0.18 R⊙, the size of the source increases nearly four times, while
below 1.51 R⊙ the size of the source did not change that significantly.
Between 1.34 R⊙ and 1.51 R⊙, corresponding to a distance of 0.17 R⊙, the size
of the source changed from 30.9 arcmin2 to 53.1 arcmin2. The analysis shows
that for a fixed height of 1.34 R⊙ (75 MHz) the size of the emission area
equals 49.6, 36.9, and 30.9 arcmin2 for burst #1, #2, and #3, respectively.

Determining the velocities of the electron beams responsible for the emission
of the investigated radio bursts and their energy was performed in two ways.
The first used the dynamic spectra and the second used the centroids of the
emission sources obtained from interferometric radio maps.

Assuming the Newkirk coronal density model, we found from the dynamic spectra
that the velocity of the electron beams responsible for generating the three
analyzed bursts was between 0.03c and 0.19c. This corresponds to an energy of
0.24 keV and 9.06 keV, respectively (see Table 2). On the other hand, the
velocities calculated based on the slope of the fit were estimated for the
centroid points for burst #2 (only for the F component) to be equal to 0.06c
and for burst #3 0.09c (Fig. 5). This corresponds to an energy of 0.92 keV and
2.26 keV, respectively (see Table 2). It should also be noted that the burst
#3 velocity is significantly different from that obtained from the dynamic
spectrum because of the large projection effect visible in Fig. 3c.
Unfortunately, it was not possible to determine the velocity from centroids
for burst #1 (because radio images were determined only for 65 MHz and 75 MHz)
and for the H component of burst #2 (due to the rather complex structure of
the source).

To identify which events in the active region AR 12671 were responsible for
the observed radio bursts, we use observations of the solar flare registered
on the SDO AIA 171 Å image. On Fig. 6 we superimposed a grid of 24 boxes, each
$100\times 100$ arcsec in size. Then we calculated the signal from each box,
which was summed, and in the next step divided by the exposure time of each
SDO image. The obtained signals were enhanced in boxes 17 and 23, where the
C1.0 class solar flare occurred. The comparison of the EUV signal with the
total radio flux in the frequency range 20 – 80 MHz shows a good relationship.
The identification of the source location is independently confirmed by RHESSI
X-ray imaging observations, which show a pair of non-thermal footpoints in the
flaring area seen by AIA (see hard X-ray footpoints in the 18 – 40 keV range
in the upper right corner of Fig. 6).

Burst #1 is likely connected with emission from boxes 17 and 23, the maxima of
which took place at 11:11:45 UT and 11:12:09 UT, respectively. On the other
hand, the maximum of the EUV emission in box 23 appears 22 s later in
comparison with the radio flux (Fig. 6). At the beginning, the solar flare was
observed in box 17 and then the plasma moved along the magnetic field lines to
the analyzed box 23. It should be noted that the RHESSI observations confirm
that burst #1 may be a type U burst (Fig. 6). Nevertheless, the analysis of
its velocity was carried out in the same way as for bursts #2 and #3. Burst #2
is probably connected with emission from boxes 17 and 23 (Fig. 6). The
propagation path of burst #3 seems to be more radial than the other two
bursts, indicating that the electron beam is propagating along a different set
of magnetic field lines. The propagation direction of the beam is clearly
shifted toward Earth, making the projection effects stronger. It was not
possible to determine the exact location of burst #3 due to the complex
structure of this active region and numerous phenomena occurring in it (Fig.
6). Based on the plasma motions along the magnetic field lines we can assume
the burst is also associated with the solar flare. We have to take into
account the temporal resolution of the SDO observations (around 24 – 36 s),
which is lower than that of LOFAR, here reduced to 1 s.

##  4 Discussion and conclusions

In this work we presented a study of the general characteristics of two type
IIIb solar radio bursts and one U burst with stria structures present (typical
for type IIIb bursts), the most interesting among a series of bursts observed
on 22 August 2017 with LOFAR in the frequency range of 20 – 80 MHz, including
dynamic spectra and imaging observations. They are particularly interesting
because they occurred around the maximum emission of this event in the radio
range. Additionally, we also inspected the data recorded by instruments on
board GOES and RHESSI (in X-ray) and SDO (in EUV), which complement the
observations in the radio field.

In this study we determined the source size limited by the actual shape of the
contour at particular frequencies of type IIIb and U solar bursts in a
relatively wide frequency band from 20 to 80 MHz. The works by other authors
(Kontar et al., 2017; Sharykin et al., 2018; Zhang et al., 2020; Murphy et
al., 2021) have focused on the determination of the emission size at a
frequency of around 30 MHz with a 2D elliptical Gaussian function fit, which
is only an approximation of their size (e.g., Kontar et al., 2019). Therefore,
comparison of our results with those obtained by other authors is not
straightforward. In the analyzed case it was possible only for the 35 and 40
MHz frequencies and in our work, using the same method, we obtained 205.9
arcmin2 and 132.9 arcmin2 (both for burst #3), respectively. At 35 MHz the
source size we observed was about 1.4 times bigger than that obtained by
Murphy et al. (2021) (on the same frequency); both results were obtained using
the same observation method (interferometric mode). A comparison of our
results with those obtained by Kontar et al. (2017) and Sharykin et al. (2018)
for the frequency of 32.5 and 30.11 MHz (and therefore very similar to ours)
with the tied-array beam forming method shows that the source sizes we
obtained with the interferometric method are 1.9 and 1.6 times smaller,
respectively. In turn, for 40 MHz it has a similar value to those obtained by
Sharykin et al. (2018) (for 41.77 MHz), despite the different observation
methods (tied-array beam and interferometric in our results). Dulk & Suzuki
(1980) estimated that the effective source sizes of the type III bursts
observed at 43 MHz was about 314 arcmin2. It should be noted that the source
size was obtained with the use of the Culgoora radioheliograph. At 45 MHz, the
source size we observed was about three and seven times smaller (for burst #3
and #2, respectively) than that obtained by Dulk & Suzuki (1980).

In the case of burst #3 the size of the emission source increases with
altitude (Fig. 4). This is well-known and can be explained by coronal
scattering of radio waves and the cross-sectional size of the electron beam as
it moves along magnetic field lines (Zhang et al., 2020). For burst #2 this is
different; the source sizes at different altitudes are similar. Since a big
difference in scattering is not to be expected, which suggests that the
magnetic field configuration is less divergent than for burst #3.

The reason why it was difficult to compare the obtained emission source sizes
of the analyzed bursts with the results received by other authors was the lack
of observations carried out at low frequencies with sufficient spatial
resolution. On the other hand, in several studies (Kontar et al. 2019 and
references therein), the angular source sizes of type III bursts were
determined. On that basis, Kontar et al. (2019) found the relation of the
angular source size of type III emission (determined on the FWHM) versus
frequency in the band 0.05 to 500 MHz. It turns out that the source size
increases with the frequency and is described by the following equation:
size$=(11.78\pm 0.06)\cdot f^{-0.98\pm 0.05}$ (where the source size is given
in degrees and the frequency in MHz). Assuming that the source is circular,
from this relation (which in some cases is far from the truth) we can estimate
its size. In the case of the analyzed radio bursts observed in the 35 – 75 MHz
band, the sizes of the emission areas are 369 arcmin2 and 83 arcmin2,
respectively. Comparison of the source sizes obtained for the bursts analyzed
here at 65 and 75 MHz (only at these two frequencies was it possible to
determine the sizes of the emission for all three bursts) with the sizes
determined from the dependence described above shows that at 65 MHz the size
of the sources we obtained is from 2.3 (for burst #2) to approximately 3.2
(for bursts #1 and #3) times smaller. Furthermore, for 75 MHz our source size
is 1.7 to 2.7 times smaller (for all of the bursts).

Coronal scattering might affect the apparent positions of the sources.
According to the method proposed by Chrysaphi et al. (2018) we estimate that
the source observed at 35 MHz and 75 MHz (corresponding to the bandwidth in
which burst #3 was observed) to be radially shifted by about 0.6
$\textup{R}_{\odot}$ and 0.2 $\textup{R}_{\odot}$, respectively, from its true
location. For this reason the apparent positions of sources observed at 35 MHz
and 75 MHz is $R_{35}=1.69+0.6=2.29\leavevmode\nobreak\ \textup{R}_{\odot}$
and $R_{75}=1.34+0.2=1.54\leavevmode\nobreak\ \textup{R}_{\odot}$. The sources
at this position correspond to the corona described by the $4.3\times$ and
$3.0\times$ Newkirk density model, respectively. The source observed at 35 MHz
is shifted more than the source at 75 MHz. If the apparent sources are
interpreted as true sources, an increase in density due to the radial shifting
is often observed.

Bursts #1 and #2F (at 75 MHz, see Fig. 6) seem to appear at roughly the same
place in active region AR 12671. The bursts are quite different. Burst #1 is a
U-burst, so the electrons are injected into a closed loop, while burst #2
extends to lower frequencies, so the electrons in this burst have been
injected into open magnetic field lines. This means that a small variation in
source location leads to completely different burst characteristics. It also
turned out that the mean values of their source sizes are similar. It is
different in the case of burst #3, which seems to be related to another part
of the magnetic field structure in this active region.

Estimated velocities (and energy) of bursts #1, #2, and #3 generally stick to
the lower boundary of standard type III radio burst velocities of 0.1 – 0.6c
(Benz, 2002). Burst #1 can be classified as a U burst in which stria
structures are present. The low electron velocity (equal to 0.03c) of this
burst is due to the magnetic field line having a significant horizontal
component. Then the projection of the electron velocity on the radial
direction, which determines the drift rate, can be quite low. This certainly
applies to a U burst as the magnetic loop along which the electrons are moving
becomes horizontal at some point. The velocities of bursts #2 and #3 are equal
to 0.1 and around 0.1 – 0.2, respectively, and are comparable with the results
obtained by Morosan et al. (2014), who received 0.09 – 0.2c (harmonic emission
for all the type III bursts was assumed) in the 30 – 90 MHz LOFAR band. We can
conclude that the different values of velocity of the electron beam
responsible for the generation of analyzed bursts suggest their propagation in
the solar corona with different geometric configuration of the magnetic field.
We also observe different energies of the electron beams responsible for their
formation. It ranges from around 0.2 to over 9.0 keV and is different for each
burst.

Comparison of radio and X-ray fluxes shows that the maximum peak in the radio
range was observed up to about 1 min earlier than in the X-ray, but about 1
min before the maximum we also observed a jump in the energy range 12 – 25
keV. This agrees with the standard solar flare model.

The radio images of the three bursts observed at different frequencies allowed
the determination of their height in the solar corona and estimation of the
size of their emission. In turn, the images in EUV from SDO (i.e., from the
AIA 171 Å channel) gave us information about the processes taking place in the
upper transition region. Thanks to imaging at three wavelengths (EUV, X-ray,
and radio) it was possible to better understand the observed event.

###### Acknowledgements.

We acknowledge the National Science Centre, Poland and the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) for granting “LOFAR
observations of the solar corona during Parker Solar Probe perihelion
passages” in the Beethoven Classic 3 funding initiative under project numbers
2018/31/G/ST9/01341 and VO 2123/1-1, respectively. UWM would like to thank the
Ministry of Education and Science of Poland for granting funds for the Polish
contribution to the International LOFAR Telescope (decision number 2021/WK/2)
and for maintenance of the LOFAR PL-612 Baldy (decision number
59/E-383/SPUB/SP/2019.1). Diana E. Morosan acknowledges the Academy of Finland
project ‘RadioCME’ (grant number 333859). The authors thank Derek McKay (Aalto
University Metsähovi Radio Observatory, Finland) and KSP team for support in
solar LOFAR research. This paper is based on data obtained with the
International LOFAR Telescope (ILT) under project code LC8_013. LOFAR (van
Haarlem et al., 2013) is the Low Frequency Array designed and constructed by
ASTRON. It has observing, data processing, and data storage facilities in
several countries that are owned by various parties (each with their own
funding sources), and that are collectively operated by the ILT foundation
under a joint scientific policy. The ILT resources have benefited from the
following recent major funding sources: CNRS-INSU, Observatoire de Paris and
Université d’Orléans, France; BMBF, MIWF-NRW, MPG, Germany; Science Foundation
Ireland (SFI), Department of Business, Enterprise and Innovation (DBEI),
Ireland; NWO, The Netherlands; The Science and Technology Facilities Council,
UK; Ministry of Science and Higher Education, Poland. Thanks to Nicolina
Chrysaphi for sharing the codes (v2.1p) for estimation of scattering effects
and Tomasz Mrozek (Space Research Centre, PAS, Solar Physics Division, Poland)
for help with RHESSI software. We acknowledge the referee for comments and
suggestions which helped us in bringing out the results more clearly.

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# Relativistic frequency shifts in Cr, Ti, Fe, Ni, Ca, Na, and V to search for
variation in the fine structure constant

V. A. Dzuba, V. V. Flambaum  School of Physics, University of New South Wales,
Sydney 2052, Australia  M. T. Murphy, D. A. Berke  Centre for Astrophysics and
Supercomputing, Swinburne University of Technology, Hawthorn, Victoria 3122,
Australia

###### Abstract

Sensitivity of the frequencies of twenty two atomic transitions to the
variation of the fine structure constant $\alpha$ is calculated. The findings
are to be used in search for possible variation of $\alpha$ across our Galaxy
using high quality spectroscopic data for stars similar to our Sun.

##  I Introduction

Theories unifying particle physics and cosmology predict possible variation of
the fundamental constants in space and time. Values of these constants may
depend on a cosmological scalar field representing dark energy or dark matter
Uzan (2011); Olive and Pospelov (2002); Stadnik and Flambaum (2015); Eichhorn
et al. (2018); Davoudiasl and Giardino (2019).

Many searches for variations in the low-energy value of the fine-structure
constant, $\alpha\equiv e^{2}/\hbar c$, rely on understanding the sensitivity
of atomic transtion frequencies on $\alpha$ (e.g.Dzuba et al. (1999); Dzuba
and Flambaum (2009)). Indeed, these were recently calculated for 13
transitions seen in infrared stellar spectra and used to probe for differences
between $\alpha$ in giant stars near the Galactic Centre and the laboratory
value Hees et al. (2020). This is also required for a new probe of
$\alpha$-variation across our Galaxy, using stars very similar to our Sun,
called ‘solar twins’ Murphy et al. (2022); Berke et al. (2022a, b). This star-
to-star comparison largely avoids systematic errors in the usual star-to-
laboratory approach, particularly effective shifts and asymmetries in
absorption lines from several physical processes as they arise in stellar
atmospheres.

The solar twins approach seeks to take advantage of the richness of optical
stellar spectra, with potentially hundreds of strong (but unsaturated)
absorption lines available, in principle, for comparison between stars to
measure any differences in $\alpha$. However, in practice, there are many
reasons to carefully select only transitions which are likely to be most
reliable Berke et al. (2022b). In particular, using closely-separated pairs of
transitions, which have very similar strengths in solar twin spectra,
suppresses many systematic errors of instrumental and astrophysical origin. To
use the thousands of high-quality archival stellar spectra available from the
High-Accuracy Radial velocity Planetary Searcher (HARPS) spectrograph Mayor et
al. (2003) on the European Southern Observatory’s (ESO) 3.6-m telescope at La
Silla Observatory (Chile), 164 different transitions were selected, which form
229 close pairs Berke et al. (2022b). However, the sensitivity of these
transitions to variations in $\alpha$ have not been calculated before. Here we
calculate sensitivies for 22 of these transitions, forming 17 pairs, which
have been used to compare $\alpha$ between 18 solar twins in Murphy et al.
(2022); Berke et al. (2022a).

##  II Calculations

The dependence of atomic transition frequencies on the fine structure constant
$\alpha$ was studied before for a number of systems (see, e.g. Dzuba et al.
(1999); Dzuba and Flambaum (2009)). In particular, Ti and Fe where considered
in Refs. Berengut et al. (2004); Dzuba and Flambaum (2008). However, in
present work we use atomic transition which were never considered before. To
perform the calculations we use similar approaches as before accompanied with
a new method for atoms with complicated electronic structure (many electrons
in open shells). For atomic transitions it is convenient to use atomic units
$\hbar=m_{e}=|e|=1$. In these units dependence of electron energies and
transition frequencies on $\alpha$ appears due to the relativistic
corrections. It is convenient to present frequencies of atomic transitions in
a form

$\omega(x)=\omega_{0}+qx,$ (1)

where $x=(\alpha/\alpha_{0})^{2}-1$, $\omega_{0}$ and $\alpha_{0}$ are the
present values of the transition frequency and the fine structure constant,
$q$ is sensitivity coefficient which can be found by varying the value of
$\alpha$ in computer codes

$q=\frac{\omega(\Delta x)-\omega(-\Delta x)}{2\Delta x}.$ (2)

We use small value of $\Delta x$ ($\Delta x=0.01$) to avoid non-linear
effects, e.g. those which are caused by level crossing Dzuba et al. (2002).

To calculate transition frequencies, we use a range of methods depending on
the number of electrons above closed shells. For atoms with one external
electron (e.g., Na I) we use the correlation potential method Dzuba et al.
(1987). In this method the calculations start from the relativistic Hartree-
Fock (RHF) procedure for the closed-shell core, with one external electron
removed. Then, states of external electron are calculated by solving RHF-like
equations which also include core-valence correlations.

$(\hat{H}_{0}+\hat{\Sigma}_{1}-\epsilon_{v})\psi_{v}=0.$ (3)

Here $\hat{H}_{0}$ is the RHF Hamiltonian, $\hat{\Sigma}_{1}$ is a single-
electron non-local operator (correlation potential) which describes
correlation interaction of an external electron with the core. We calculate
$\hat{\Sigma}_{1}$ in the lowest, second order of the many-body perturbation
theory using the B-spline technique Johnson and Sapirstein (1986). Index $v$
in (3) numerates states of the valence electron. Solutions of (3) are usually
called Bruekner orbitals (BO). Transition frequencies are found as a
difference between two BO energies, $\omega_{vw}=\epsilon_{v}-\epsilon_{w}$.
Calculated energies of the states of Na I which are used in the analysis are
compared with experimental data in Table 1.

Table 1:  Energy levels used in the analysis, comparison of the theory with experiment (cm-1). $Z$ | Atom/ | State | Energy
---|---|---|---
| Ion |  |  | Expt. NIS  | Theory
11 | Na I | $2p^{6}3p$ |  2P${}^{\rm o}_{1/2}$ | 16956.170 | 16862
|  | $2p^{6}3p$ |  2P${}^{\rm o}_{3/2}$ | 16973.366 | 16881
|  | $2p^{6}5s$ |  2S1/2 | 33200.673 | 33057
20 | Ca I | $3p^{6}4s4p$ |  3P${}^{\rm o}_{1}$ | 15210.063 | 15544
|  | $3p^{6}3d4s$ |  3D1 | 20335.360 | 19568
|  | $3p^{6}4s5s$ |  3S1 | 31539.495 | 31816
|  | $3p^{6}4s5p$ |  3P${}^{\rm o}_{0}$ | 36547.688 | 36766
22 | Ti I |  $3d^{3}(^{2}$H$)4s$ | a 3H5 | 18141.265 | 21534
|  |  $3d^{3}(^{2}$H$)4s$ | a 3H6 | 18192.570 | 21369
|  |  $3d^{3}(^{2}$H$)4p$ | x 3H${}^{\rm o}_{5}$ | 39152.103 | 41034
|  |  $3d^{3}(^{2}$H$)4p$ | x 3H${}^{\rm o}_{6}$ | 39198.323 | 41050
| Ti II |  $3d^{2}(^{1}$D$)4s$ | a 2D3/2 | 8710.567 | 10577
|  |  $3d^{2}(^{1}$G$)4s$ | b 2G7/2 | 15265.700 | 17936
|  |  $3d^{2}(^{3}$F$)4p$ | z 4G${}^{\rm o}_{5/2}$ | 29544.454 | 29005
|  |  $3d^{2}(^{3}$F$)4p$ | z 2G${}^{\rm o}_{7/2}$ | 34543.380 | 33930
24 | Cr I | $3d^{4}4s^{2}$ | a 5D2 | 7927.441 | 7114
|  | $3d^{4}4s^{2}$ | a 5D3 | 8095.184 | 7161
|  |  $3d^{4}(^{5}$D$)4s4p(^{3}$P${}^{\rm o})$ | y 5P${}^{\rm o}_{1}$ | 29420.864 | 29204
|  |  $3d^{4}(^{5}$D$)4s4p(^{3}$P${}^{\rm o})$ | y 5P${}^{\rm o}_{2}$ | 29584.571 | 29284
23 | V I |  $3d^{4}(^{5}$D$)4s$ | a 6D9/2 | 2424.809 | 2753
|  |  $3d^{3}(^{4}$F$)4s4p(^{3}$P${}^{\rm o})$ | z 6D${}^{\rm o}_{9/2}$ | 18438.044 | 15575
26 | Fe I |  $3d^{7}(^{4}$F$)4s$ | a 3F2 | 12968.554 | 11964
| even |  $3d^{7}(^{4}$P$)4s$ | a 5P3 | 17550.181 | 23061
|  |  $3d^{7}(^{4}$P$)4s$ | a 5P1 | 17927.382 | 22558
|  | $3d^{6}4s^{2}$ | a 3P1 | 19552.478 | 24345
|  | $3d^{6}4s^{2}$ | a 3H4 | 19788.251 | 22233
|  | $3d^{6}4s^{2}$ | b 3F3 | 20874.482 | 25230
|  | $3d^{6}4s^{2}$ | b 3F2 | 21038.987 | 25746
| Fe I |  $3d^{7}(^{4}$F$)4p$ | y 5D${}^{\rm o}_{2}$ | 33801.572 | 33476
| odd |  $3d^{6}(^{5}$D$)4s4p(^{3}$P${}^{\rm o})$ | z 3P${}^{\rm o}_{2}$ | 33946.933 | 33572
|  |  $3d^{7}(^{4}$F$)4p$ | y 5D${}^{\rm o}_{0}$ | 34121.603 | 34050
|  |  $3d^{7}(^{4}$F$)4p$ | z 3G${}^{\rm o}_{3}$ | 36079.372 | 35648
|  |  $3d^{6}(^{5}$D$)4s4p(^{1}$P${}^{\rm o})$ | y 5P${}^{\rm o}_{2}$ | 37157.566 | 39130
|  |  $3d^{7}(^{4}$F$)4p$ | y 3F${}^{\rm o}_{3}$ | 37162.746 | 37388
|  |  $3d^{7}(^{4}$F$)4p$ | y 3D${}^{\rm o}_{1}$ | 38995.736 | 39402
28 | Ni I |  $3d^{8}(^{1}$D$)4s^{2}$ |  1D2 | 13521.347 | 15768
|  |  $3d^{9}(^{2}$D$)4p$ |  1P${}^{\rm o}_{1}$ | 32982.260 | 30508
|  |  $3d^{9}(^{2}$D$)4p$ |  3D${}^{\rm o}_{3}$ | 29668.893 | 28002

For atoms with two external electrons (e.g., Ca I) we use a combination of the
configuration interaction and many-body perturbation theory (CI+MBPT, Dzuba et
al. (1996)). Calculations are performed in the $V^{N-2}$ approximation Dzuba
(2005). The self-consistent Hartree-Fock procedure is done for the double
ionised ion. Single-electron basis states for valence electrons are calculated
in the field of the ion using the B-spline technique similar to how it is done
for a single-valence-electron atoms. The effective CI Hamiltonian for two
valence electrons has the form

$\hat{H}^{\rm
CI}=\hat{h}_{1}(r_{1})+\hat{h}_{1}(r_{2})+\hat{h}_{2}(r_{1},r_{2}).$ (4)

Here $\hat{h}_{1}=\hat{H}_{0}+\hat{\Sigma}_{1}$ is a single-electron part of
the Hamiltonian. It includes the RHF operator $\hat{H}_{0}$ and the
correlation potential $\hat{\Sigma}_{1}$, the same as in (3). The two-electron
part of the CI Hamiltonian is the sum of the Coulomb interaction and
correlation operator $\Sigma_{2}$,
$\hat{h}_{2}(r_{1},r_{2})=e^{2}/|r_{1}-r_{2}|+\hat{\Sigma}_{2}(r_{1},r_{2})$.
The later can be understood as screening of the Coulomb interaction between
valence electrons by core electrons (see Ref. Dzuba et al. (1996) for
details). Calculated energies of the states of Ca I which are used in the
analysis are compared with experimental data in Table 1.

Table 2:  Electric dipole transitions used in the analysis and their coefficients of sensitivity to the variation of the fine structure constant. Atom/ | Lower state | $J$ | Energy | Upper state | $J$ | Energy | $q$
---|---|---|---|---|---|---|---
Ion |  |  |  | [cm-1] |  |  |  | [cm-1] | [cm-1]
Na I | $2p^{6}3p$ |  2Po | 1/2 | 16956.170 | $2p^{6}5s$ |  2S | 1/2 | 33200.673 | 7(1)
Na I | $2p^{6}3p$ |  2Po | 3/2 | 16973.366 | $2p^{6}5s$ |  2S | 1/2 | 33200.673 | -11(1)
Ca I | $3p^{6}4s4p$ |  3Po | 1 | 15210.063 | $3p^{6}4s5s$ |  3S | 1 | 31539.495 | -69(7)
Ca I | $3p^{6}3d4s$ |  3D | 1 | 20335.360 | $3p^{6}4s5p$ |  3Po | 0 | 36547.688 | -475(24)
Ti I |  $3d^{3}(^{2}$H$)4s$ | a 3H | 5 | 18141.265 |  $3d^{3}(^{2}$H$)4p$ | x 3Ho | 5 | 39152.103 | 440(60)
Ti I |  $3d^{3}(^{2}$H$)4s$ | a 3H | 6 | 18192.570 |  $3d^{3}(^{2}$H$)4p$ | x 3Ho | 6 | 39198.323 | 460(60)
Ti II |  $3d^{2}(^{1}$D$)4s$ | a 2D | 3/2 | 8710.567 |  $3d^{2}(^{3}$F$)4p$ | z 4Go | 5/2 | 29544.454 | 90(90)
Ti II |  $3d^{2}(^{1}$G$)4s$ | b 2G | 7/2 | 15265.700 |  $3d^{2}(^{3}$F$)4p$ | z 2Go | 7/2 | 34543.380 | 340(150)
Cr I | $3d^{4}4s^{2}$ | a 5D | 2 | 7927.441 |  $3d^{4}(^{5}$D$)4s4p(^{3}$P${}^{\rm o})$ | y 5Po | 1 | 29420.864 | 490(60)
Cr I | $3d^{4}4s^{2}$ | a 5D | 3 | 8095.184 |  $3d^{4}(^{5}$D$)4s4p(^{3}$P${}^{\rm o})$ | y 5Po | 2 | 29584.571 | 490(60)
V I |  $3d^{4}(^{5}$D$)4s$ | a 6D | 9/2 | 2424.809 |  $3d^{3}(^{4}$F$)4s4p(^{3}$P${}^{\rm o})$ | z 6Do | 9/2 | 18438.044 | -600(60)
Fe I |  $3d^{7}(^{4}$F$)4s$ | a 3F | 2 | 12968.554 |  $3d^{7}(^{4}$F$)4p$ | y 5Do | 2 | 33801.572 | -830(400)
Fe I | $3d^{6}4s^{2}$ | a 3P | 1 | 19552.478 |  $3d^{7}(^{4}$F$)4p$ | y 3Do | 1 | 38995.736 | 2840(300)
Fe I |  $3d^{7}(^{4}$P$)4s$ | a 5P | 1 | 17927.382 |  $3d^{6}(^{5}$D$)4s4p(^{1}$P${}^{\rm o})$ | y 5Po | 2 | 37157.566 | -1500(900)
Fe I | $3d^{6}4s^{2}$ | a 3H | 4 | 19788.251 |  $3d^{7}(^{4}$F$)4p$ | z 3Go | 3 | 36079.372 | 2710(500)
Fe I | $3d^{6}4s^{2}$ | b 3F | 3 | 20874.482 |  $3d^{7}(^{4}$F$)4p$ | y 3Fo | 3 | 37162.746 | 3010(500)
Fe I |  $3d^{7}(^{4}$P$)4s$ | a 5P | 3 | 17550.181 |  $3d^{7}(^{4}$F$)4p$ | y 5Do | 2 | 33801.572 | 340(400)
Fe I |  $3d^{7}(^{4}$P$)4s$ | a 5P | 1 | 17927.382 |  $3d^{7}(^{4}$F$)4p$ | y 5Do | 0 | 34121.603 | 430(500)
Fe I | $3d^{6}4s^{2}$ | b 3F | 2 | 21038.987 |  $3d^{7}(^{4}$F$)4p$ | y 3Fo | 3 | 37162.746 | 2870(500)
Fe I |  $3d^{7}(^{4}$P$)4s$ | a 5P | 1 | 17927.382 |  $3d^{6}(^{5}$D$)4s4p(^{3}$P${}^{\rm o})$ | z 3Po | 2 | 33946.933 | -1300(900)
Ni I |  $3d^{8}(^{1}$D$)4s^{2}$ |  1D | 2 | 13521.347 |  $3d^{9}(^{2}$D$)4p$ |  1Po | 1 | 32982.260 | 4950(500)
Ni I |  $3d^{8}(^{1}$D$)4s^{2}$ |  1D | 2 | 13521.347 |  $3d^{9}(^{2}$D$)4p$ |  3D | 3 | 29668.893 | 5500(550)

For atoms with more than two external electrons (Ti I, Ti II, Cr I, V I, Fe I,
Ni I) we use a combination of the configuration interaction method with the
perturbation theory (the CIPT method Dzuba et al. (2017)). All electrons above
the Ar-like core are considered as external electrons and treated with
configuration interaction (CI) approach. The CI Hamiltonian has a form

$\hat{H}^{\rm
CI}=\sum_{i}^{N_{v}}\hat{H}_{0}(r_{i})+\sum_{i<j}^{N_{v}}e^{2}/r_{ij},$ (5)

where $\hat{H}_{0}$ is the RHF operator, second term presents Coulomb
interaction between external electrons, the summation goes over all external
(valence) electrons $N_{v}$. Correlation operators $\Sigma_{1}$ and
$\Sigma_{2}$ are not included for reasons explained below. The number of
valence electrons varies from $N_{v}=4$ for Ti to $N_{v}=10$ for Ni. With the
number of external electrons being that large, the number of lines in the CI
matrix may go up to $\sim 10^{8}$ in the standard CI approach. This would make
the calculations impossible. Therefore, we use the CIPT Dzuba et al. (2017)
approach, in which the off-diagonal matrix elements between high-energy states
are neglected. Indeed, it is easy to show that these matrix elements formally
appear in a higher order of the perturbation theory and are suppressed by
large energy denominators. This allows to reduce the size of the effective CI
matrix by many orders of magnitude. The high-energy states are still included
in the correction to the CI matrix elements between low-energy states,

$\langle a|\hat{H}^{\rm eff}|b\rangle=\langle a|\hat{H}^{\rm
CI}|b\rangle+\sum_{n}\frac{\langle a|\hat{H}^{\rm CI}|n\rangle\langle
n|\hat{H}^{\rm CI}|b\rangle}{E-E_{n}}.$ (6)

Here $\hat{H}^{\rm eff}$ is the effective CI operator, $|a\rangle$ and
$|b\rangle$ are many-electron single-determinant low-energy basis states,
$|n\rangle$ are high-energy many-electron basis states, $E_{n}$ is the energy
of the high-energy state, $E_{n}=\langle n|\hat{H}^{\rm CI}|n\rangle$, $E$ is
the energy of the state of interest. Since the later energy is not known, the
CIPT equations, ($H^{\rm eff}-EI)X=0$, are solved iteratively, using the
energy found on the previous iteration to calculate second term in (6), (see
Ref. Dzuba et al. (2017) for details).

Many-electron basis states are constructed using a set of single-electron
basis states calculated in the $V^{N-1}$ approximation using the B-spline
technique Johnson and Sapirstein (1986). The $V^{N-1}$ approximation means
that the initial RHF procedure is done for a system with one external electron
removed. In all considered systems we remove one $4s$ electron from the ground
state and perform the initial RHF calculations for the [Ar]$3d^{n}4s$
configuration (the [Ar]$3d^{3}$ configuration for Ti). The basis states above
the core are calculated in the field of frozen core. These states are linear
combinations of B-splines which are the eigenstates of the RHF Hamiltonian
with the $V^{N-1}$ potential. In the CIPT method it is important to have good
initial approximation so that high states can be treated perturbatively.
Therefore, we cannot include correlations between valence electrons and
electrons in the Ar-like core via the $\Sigma_{1}$ and $\Sigma_{2}$ operators
as it is done for a two-electron system, Eq. (4). These operators are well
defined only in the The $V^{N-M}$ approximation Dzuba (2005), where $M$ is the
number of valence electrons. In any other starting potential one has to
include subtraction diagrams Dzuba et al. (1996) and even three-body diagrams
Berengut et al. (2006), which is significant complication to the method. On
the other hand, in the considered systems, the correlations are dominated by
correlations between valence electrons which are treated pretty accurately in
the CI technique. The effect of neglecting the core-valence correlations is
smaller than the effect of neglecting the off-diagonal matrix elements between
the high-energy states.

Calculated energies of the states of Ti I, Ti II, Cr I, V I, Fe I and Ni I,
which are used in the analysis are compared with experimental data in Table 1.
The accuracy is lower than for systems with simple electron structure, like Na
I and Ca II. However, it is sufficient for the purposes of the present work.

The pairs of transitions, which are used in the analysis, are presented in
Table 2 together with the calculated sensitivity coefficients $q$. Error bars
for the $q$-coefficients are estimated by varying computational parameters,
such as the value of $\Delta x$ in Eq. (2), the number of configurations
included into the low-energy basis, etc. In most of cases the uncertainty is
significantly smaller that the value of $q$.

###### Acknowledgements.

V.A.D. and V.V.F. acknowledge the Australian Research Council for support
through grants DP190100974 and DP200100150, and M.T.M. acknowledges the
Council’s support through Future Fellowship grant FT180100194.

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