# The magnetic fields of β Coronae Borealis and the early F-star σ Bootis

J. M. Seach, S. C. Marsden, B. D. Carter, C. Neiner, C. P. Folsom

► To cite this version:

J. M. Seach, S. C. Marsden, B. D. Carter, C. Neiner, C. P. Folsom. The magnetic fields of β Coronae Borealis and the early F-star σ Bootis. Monthly Notices of the Royal Astronomical Society, 2022, 513, pp.4278-4294. 10.1093/mnras/stac1173. insu-03713296

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# The magnetic fields of β Coronae Borealis and the early F-star σ Bootis

J. M. Seach<sup>1,★</sup> S. C. Marsden<sup>1,†</sup> B. D. Carter<sup>1,†</sup> C. Neiner<sup>1,‡</sup> and C. P. Folsom<sup>3</sup>

Accepted 2022 April 25. Received 2022 April 25; in original form 2022 February 25

## ABSTRACT

The study of magnetism in stars close to the transition from fossil to dynamo magnetic fields is important for understanding the nature of the stellar dynamo and dynamics of the outer atmosphere. We present surface magnetic maps for two stars that are located on opposite sides of the suspected transition zone: the chemically peculiar late A-star β Coronae Borealis (A9SrEuCr) and the early F-star σ Bootis (F3V). The large-scale magnetic field reconstructed at six epochs for β Coronae Borealis shows a complex fossil magnetic field, which is highly poloidal, and contains almost half the magnetic energy in higher multipoles ($ℓ > 1$). In contrast, the single epoch magnetic map for σ Bootis contains a simple surface magnetic topology that is mostly poloidal, and predominantly dipolar, and is consistent with observations of other mature late F-stars.

**Key words:** stars: activity – stars: early-type – stars: magnetic field – stars: solar-type.

## 1 INTRODUCTION

We present the third paper in our series on magnetic fields in F-type stars. In the first paper, we published a magnetic snapshot survey of F-type stars (Seach et al. 2020, hereafter Paper 1), and in the second paper we reconstructed surface magnetic maps and created wind models for two late F-type stars β Vir (F9V) and θ Dra (F8IV) (Seach et al. 2022, hereafter Paper 2). In this paper, we present surface magnetic maps for the chemically peculiar star β Coronae Borealis (β CrB) and the F3V star σ Bootis (σ Boo). β CrB is a chemically peculiar star, classified as A9SrEuCr by Renson & Manfroid (2009), which is located on the hot side of the transition between fossil and dynamo magnetic fields. F-type stars are important targets for stellar magnetic studies since they span a range of masses where the transition from fossil to dynamo magnetic fields is believed to occur (Schatzman 1962; Mullan 1972; Schmitt et al. 1985; Wolff, Boesgaard & Simon 1986; Walter & Schrijver 1987; Seach et al. 2022).

Fossil fields (Cowling 1945; Spitzer 1958; Mestel 1967; Dudorov & Khaibrakhmanov 2015) are found in hot stars with little or no outer convection zone. The fields are thought to be left over from the star's formation where it undergoes gravitational collapse from a molecular cloud (Mestel & Spitzer 1956; Mestel 1966), from stellar mergers (Schneider et al. 2019), or from pre-main-sequence dynamos (Moss 2001). Fossil fields are frozen (Cowling 1957; Alfven, Alfvén & Fälthammar 1963) into the stellar atmosphere and essentially remain unchanged over the lifetime of the star. Classically, fossil fields have strengths ranging from a few hundred gauss up to 34 kG (Babcock 1960) and have been found in about 10 per cent of hot stars (Fossati et al. 2015; Grunhut et al. 2016; Sikora et al. 2019). More recently, another class of magnetic fields have been discovered in Ap and

Fp stars with sub-gauss magnetic strengths and are possibly fossil in origin (Lignières et al. 2009; Petit et al. 2010; Neiner, Wade & Sikora 2017). Like the stronger fossil fields, these ultra-weak fields are possibly unchanged over long periods of time (Neiner et al. 2017).

Dynamo magnetic fields are located in the outer convection zone of cool stars where they are most efficiently generated by the action of turbulent convection and stellar rotation (Parker 1955, 1987). The presence of a deepening outer convection zone around early to mid-F-stars corresponds to the onset of dynamo-generated magnetic fields. There is an extensive literature on dynamo magnetic fields; e.g. see reviews by Moffatt (1978), Parker (1979), Krause & Radler (1980), Zeldovich, Ruzmaikin & Sokolov (1983), Roberts (1992), Brandenburg & Subramanian (2005), Mestel (2012), and Moffatt & Dormy (2019). Dynamo-generated magnetic fields are seen on the Sun and other solar-type stars where they are responsible for magnetic features like cool surface spots (Berdyugina 2005; Strassmeier 2009), flares (Pettersen 1989), stellar winds, and space weather phenomenon (Bothmer & Daglis 2007; Hanslmeier 2007; Singh et al. 2021). Increasing interest is developing in the field of exoplanet research (Perryman 2018) where the host star magnetic field has an important influence in the exoplanetary environment (Lammer & Khodachenko 2014; Linsky 2019). Dynamo magnetic fields are distinguished from fossil fields by their dynamic and changing nature with short decay times (Pevtsov, Canfield & Metcalf 1994; Burnette, Canfield & Pevtsov 2004).

β CrB is a well-known magnetic star (Babcock 1949, 1958; Stibbs 1950; Preston & Sturch 1967; Wolff & Wolff 1970) with a fossil field that has not been previously mapped using Zeeman Doppler imaging (ZDI). The star is suitable for magnetic studies due to its brightness and sharp spectral lines (Glagolevskij & Gerth 2003). Wolff & Wolff (1970) noted that the magnetic field of β CrB cannot be represented as a simple dipole and contains a geometry where one pole is stronger than the other. Stift (1975) explained that the shape of the longitudinal magnetic field ($B_1$) is not consistent with axisymmetric morphology,

<footnote n="1"><sup>1</sup>Centre for Astrophysics, University of Southern Queensland, Toowoomba 4350, Australia</footnote>

<footnote n="2"><sup>2</sup>LESIA, Paris Observatory, PSL University, CNRS, Sorbonne University, Université de Paris, 5 place Jules Janssen, F-92195 Meudon, France</footnote>

<footnote n="3"><sup>3</sup>Tartu Observatory, University of Tartu, Observatoorium 1, Tõravere 61602, Estonia</footnote>

<footnote>* E-mail: john.seach@usq.edu.au</footnote>
and this was interpreted by Oetken (1977) as corresponding to a superposition of a dipole and a quadrupole magnetic field with their axes perpendicular to the rotation axis.

The lack of magnetic field symmetry around the rotation axis for β CrB is further discussed by Mathys (1993), and Mathys & Hubrig (1997) indicated that magnetic field is unusual and appears quite complex. A study of magnetic curves by Bagnulo et al. (2000) indicated that the magnetic topology of β CrB is characterized by a multipole expansion of a dipole and an arbitrarily oriented quadrupole. We include β CrB in this paper since it is one of the coolest magnetic Ap stars that is located close to the transition between fossil and dynamo magnetic fields. It provides a contrast with σ Boo that possibly is located on the cool side of the magnetic transition.

σ Boo was identified in our magnetic snapshot survey of F-type stars (Paper 1) as the hottest star in our sample having a possible dynamo magnetic field. Dynamo magnetic fields have been mapped on the surface of several late F-stars; see e.g. HR 1817, F7/8 (Marsden et al. 2006; Järvinen et al. 2015), τ Boo, F7V (Donati et al. 2008; Fares et al. 2009; Jeffers et al. 2018), HD 75332, F7V (Brown et al. 2021), HD 179949, F8V (Fares et al. 2012), θ Dra, F8IV, and β Vir, F9V (Paper 2). Mapping the surface magnetic field of σ Boo (F3V) is important for an understanding of magnetism in early F-stars close to the hot limit of dynamo magnetic fields and helps constrain dynamo models that provide insights into magnetism in stars with shallow outer convection zones.

## 2 OBSERVATIONS

### 2.1 Target selection

β CrB (HIP 75695) is a chemically peculiar (Adelman 1973), spectroscopic binary with component V magnitudes of 3.68 and 5.20 separated by 0.30 arcsec (Mason et al. 2001). The primary component has $T_{\text{eff}} = 7980 \pm 180$ K, and the secondary component has $T_{\text{eff}} = 6750 \pm 230$ K (Bruntt et al. 2010). The star was included in our earlier paper (Paper 1) and identified as a good target for magnetic mapping due to the availability of polarized spectra over multiple epochs.

σ Boo (HIP 71284), $T_{\text{eff}} = 6720 \pm 144$ K (Gaia collaboration 2018), is a $1.6 \pm 0.3$ Gyr old (Casagrande et al. 2011) triple star system with component V magnitudes 4.50, 10.66, and 12.08, respectively (Mason et al. 2001). σ Boo is classified as a standard F2V star in the MK classification system (Johnson & Morgan 1953), while Samus et al. (2017) classify it as F3V. The star was detected as magnetic in our earlier magnetic snapshot survey (Paper 1). There has been increasing interest in σ Boo as a host F-type star used in exoplanet weather modelling (Segura et al. 2003; Godolt et al. 2019; Rushby, Shields & Joshi 2019; Schwieterman et al. 2019; Palubski, Shields & Deitrick 2020; Kaltenegger & Lin 2021).

### 2.2 Observations

Polarized spectra of β CrB were obtained from the PolarBase data base (Petit et al. 2014), which is a publicly available data base containing stellar data collected with the ESPaDOnS (Donati et al. 2006a) and NARVAL (Aurière 2003) high-resolution ($R = 65\,000$) spectropolarimeters. The β CrB data consist of 143 spectra from 2015 April 13 to 2018 July 28 (Table 1) obtained with the NARVAL spectropolarimeter at the Telescope Bernard Lyot (TBL) at the Observatoire du Pic du Midi, France. Observations of β CrB in 2015–2017 were obtained using both circularly polarized (Stokes V)

and linear (Stokes UQ) polarized spectra, while observations in 2018 were only obtained in circular polarization. In this paper, we present magnetic maps using circular polarization only, whereas the linear polarization observations are left for future work.

We obtained polarized spectra for σ Boo using the ESPaDOnS spectropolarimeter at the Canada–France–Hawaii Telescope (CFHT) observing in Stokes V. The observations consist of 105 polarized spectra over 11 nights from 2021 May 18 to 28, corresponding to an estimated 1.2 rotational cycles (Tables 2 and A1).

## 3 MAGNETIC FIELD MEASUREMENTS

### 3.1 Magnetic Zeeman signatures

The line addition technique of least squares deconvolution (LSD; Donati et al. 1997, 2006b) is used to enhance the Stokes V signal, by combining thousands of individual lines into a single line with mean parameters. LSD profiles are created using line masks from VALD3 (Ryabchikova et al. 2015) where the masks contain the wavelength, depths, and effective Landé factors to be used for the LSD code. We normalized the LSD profiles with the values taken from the means of a typical line mask (Folsom et al. 2016). The number of lines used in LSD analysis varies from 17488 to 17652 for β CrB (Table 1) and 17646 to 17654 for σ Boo (Tables 2 and A1). For σ Boo, we added the nightly LSD profiles into a single mean profile that further enhances the SNR and enables a magnetic detection from an otherwise weak magnetic field. The time between the first and last exposure used in each mean LSD profile is less than 2 h which is less than 1 per cent of the estimated rotation period. This was short enough time-span to avoid smearing of the profiles.

We use the statistical criteria of FAP to determine whether the Stokes V profile has a magnetic signal, or whether the Null profile has a spurious signal (Donati, Semel & Rees 1992; Donati et al. 1997). The probability function of Abramowitz & Stegun (1964) is combined with Chi-square ($\chi^2$) statistics inside and outside the spectral lines to give the probability of the Stokes V data differing from a zero magnetic field. The FAP is calculated to determine for the presence of a magnetic field and is defined as being definite if the FAP is smaller than $10^{-5}$, marginal if FAP between $10^{-3}$ and $10^{-5}$, and not detected if FAP is greater than $10^{-3}$. The FAP is used to check for the presence of a magnetic field in addition to a measurement of the longitudinal magnetic field which is discussed in Section 3.2.

In Fig. 1, we compare two LSD profiles for β CrB at similar phases obtained in different years. The figure displays an LSD profile from 2015 April 23 at rotational phase 0.546, and a profile from 2017 March 8 at rotational phase 0.542. Zeeman signatures are visible in the Stokes V profiles which align with the Stokes I profile. The null spectrum does not show a magnetic signature and acts as a check to ensure that spurious signals have not contaminated the spectrum. The plots show almost identical Stokes V profiles that indicates a stable magnetic field over the 2 yr of observations. The longitudinal magnetic field strength (see Section 3.2) for the 2015 April profile 24 is $-501.9 \pm 6.2$ G, while the value for 2017 March 8 is $-496.7 \pm 4.7$ G. Magnetic signatures are visible in all 143 LSD profiles for β CrB and the FAP indicates a magnetic detection in all profiles.

In Fig. 2, we display an example of the mean Stokes V LSD profile for σ Boo on 2021 May 18. Due to the weak magnetic field, a Zeeman signature is not visible in individual LSD profiles, but is visible in the mean nightly profile due to the enhanced SNR. The FAP indicates a definite magnetic a magnetic detection in one of the mean LSD
<table><caption>**Table 1.** Journal of observations for β CrB in 2015–2018. Zero rotation cycle is set to our first observation on 2015 April 13. The table displays the date and time of the observation, zero rotation cycle with respect to the first observation on 2015 April 13, Stokes I profile, radial velocity (RV), longitudinal magnetic field ($B_1$), null value ($N_1$), and the number of lines used in LSD.</caption><thead><tr><th>Date<br>(UT)</th><th>Time<br>(UT)</th><th>Mid-HJD<br>2457000+</th><th>Rot.<br>cycle</th><th>SNR<br>Stokes I</th><th>RV<br>(km s<sup>-1</sup>)</th><th>B<sub>1</sub><br>(G)</th><th>N<sub>1</sub><br>(G)</th><th>LSD lines<br>used</th></tr></thead><tbody><tr><td colspan="9"><strong>2015</strong></td></tr><tr><td>April 13</td><td>23:26:15</td><td>126.47656</td><td>0.000</td><td>264</td><td>-18.45</td><td>608.9 ± 5.9</td><td>-1.6 ± 1.9</td><td>17 614</td></tr><tr><td>April 14</td><td>21:58:41</td><td>127.41575</td><td>0.051</td><td>294</td><td>-18.39</td><td>684.5 ± 7.6</td><td>5.4 ± 5.4</td><td>17 610</td></tr><tr><td>April 21</td><td>02:18:00</td><td>133.59583</td><td>0.385</td><td>280</td><td>-18.36</td><td>-99.8 ± 4.4</td><td>1.4 ± 2.0</td><td>17 619</td></tr><tr><td>April 24</td><td>01:43:00</td><td>136.57153</td><td>0.546</td><td>270</td><td>-18.44</td><td>-501.9 ± 6.2</td><td>-1.3 ± 2.0</td><td>17 616</td></tr><tr><td>April 30</td><td>01:12:28</td><td>142.55032</td><td>0.870</td><td>267</td><td>-18.42</td><td>242.2 ± 4.8</td><td>0.2 ± 1.8</td><td>17 613</td></tr><tr><td>May 12</td><td>00:05:35</td><td>154.50388</td><td>1.516</td><td>277</td><td>-18.38</td><td>-476.7 ± 6.3</td><td>5.0 ± 2.9</td><td>17 615</td></tr><tr><td>May 13</td><td>00:14:40</td><td>155.51019</td><td>1.571</td><td>271</td><td>-18.31</td><td>-518.5 ± 6.5</td><td>-4.0 ± 2.6</td><td>17 616</td></tr><tr><td>May 16</td><td>23:52:13</td><td>159.49459</td><td>1.786</td><td>279</td><td>-18.36</td><td>-65.9 ± 5.0</td><td>-0.4 ± 2.1</td><td>17 620</td></tr><tr><td>May 18</td><td>02:47:31</td><td>160.61633</td><td>1.847</td><td>285</td><td>-18.36</td><td>160.5 ± 5.0</td><td>1.9 ± 2.6</td><td>17 621</td></tr><tr><td>May 26</td><td>21:49:46</td><td>169.40956</td><td>2.322</td><td>272</td><td>-18.11</td><td>127.1 ± 4.1</td><td>-1.4 ± 1.9</td><td>17 621</td></tr><tr><td>May 27</td><td>21:36:39</td><td>170.40045</td><td>2.376</td><td>277</td><td>-18.17</td><td>-79.0 ± 4.4</td><td>0.2 ± 2.1</td><td>17 621</td></tr><tr><td>May 28</td><td>20:29:47</td><td>171.35402</td><td>2.428</td><td>278</td><td>-18.21</td><td>-267.4 ± 5.0</td><td>1.2 ± 2.2</td><td>17 617</td></tr><tr><td>May 31</td><td>02:37:08</td><td>173.60912</td><td>2.550</td><td>269</td><td>-18.24</td><td>-513.3 ± 6.2</td><td>-1.9 ± 2.0</td><td>17 615</td></tr><tr><td colspan="9"><strong>2016</strong></td></tr><tr><td>August 6</td><td>20:15:33</td><td>607.34413</td><td>26.011</td><td>293</td><td>-17.38</td><td>620.1 ± 5.4</td><td>0.6 ± 1.6</td><td>17 616</td></tr><tr><td>August 7</td><td>20:19:40</td><td>608.34699</td><td>26.066</td><td>293</td><td>-17.31</td><td>685.2 ± 5.6</td><td>2.0 ± 1.5</td><td>17 617</td></tr><tr><td>August 8</td><td>20:10:53</td><td>609.34089</td><td>26.119</td><td>291</td><td>-17.28</td><td>676.5 ± 5.5</td><td>1.3 ± 1.6</td><td>17 617</td></tr><tr><td>August 10</td><td>20:10:22</td><td>611.34053</td><td>26.228</td><td>299</td><td>-17.27</td><td>419.5 ± 4.7</td><td>-1.9 ± 1.9</td><td>17 617</td></tr><tr><td>August 11</td><td>20:04:07</td><td>612.33619</td><td>26.281</td><td>295</td><td>-17.26</td><td>224.4 ± 4.2</td><td>0.7 ± 1.9</td><td>17 616</td></tr><tr><td>August 12</td><td>20:10:37</td><td>613.34071</td><td>26.336</td><td>300</td><td>-17.29</td><td>14.2 ± 4.4</td><td>-1.0 ± 2.6</td><td>17 616</td></tr><tr><td>August 18</td><td>19:53:21</td><td>619.32872</td><td>26.660</td><td>292</td><td>-17.42</td><td>-427.6 ± 6.0</td><td>-2.3 ± 2.8</td><td>17 618</td></tr><tr><td>August 21</td><td>19:55:18</td><td>622.33007</td><td>26.822</td><td>290</td><td>-17.40</td><td>57.3 ± 4.4</td><td>0.2 ± 2.0</td><td>17 616</td></tr><tr><td>August 22</td><td>21:17:33</td><td>623.38719</td><td>26.879</td><td>292</td><td>-17.39</td><td>265.4 ± 4.9</td><td>-1.5 ± 2.8</td><td>17 615</td></tr><tr><td>August 23</td><td>21:15:09</td><td>624.37858</td><td>26.933</td><td>290</td><td>-17.37</td><td>440.0 ± 5.0</td><td>2.3 ± 2.2</td><td>17 615</td></tr><tr><td>August 24</td><td>21:11:05</td><td>625.38270</td><td>26.987</td><td>292</td><td>-17.35</td><td>578.1 ± 5.4</td><td>0.1 ± 2.0</td><td>17 617</td></tr><tr><td colspan="9"><strong>2017</strong></td></tr><tr><td>March 2</td><td>01:58:41</td><td>814.58242</td><td>37.221</td><td>300</td><td>-17.22</td><td>440.7 ± 7.5</td><td>-0.4 ± 6.4</td><td>17 619</td></tr><tr><td>March 8</td><td>00:06:05</td><td>820.50422</td><td>37.542</td><td>288</td><td>-17.35</td><td>-496.7 ± 4.7</td><td>-2.3 ± 4.8</td><td>17 621</td></tr><tr><td>March 15</td><td>02:01:08</td><td>827.58412</td><td>37.925</td><td>291</td><td>-17.33</td><td>441.4 ± 4.7</td><td>-1.8 ± 0.8</td><td>17 620</td></tr><tr><td>March 16</td><td>00:15:13</td><td>828.51057</td><td>37.975</td><td>297</td><td>-17.32</td><td>589.7 ± 5.4</td><td>-0.5 ± 0.5</td><td>17 620</td></tr><tr><td>March 17</td><td>04:26:03</td><td>829.68476</td><td>38.038</td><td>297</td><td>-17.29</td><td>697.3 ± 5.5</td><td>-0.1 ± 0.5</td><td>17 620</td></tr><tr><td>March 18</td><td>05:17:31</td><td>830.72050</td><td>38.094</td><td>303</td><td>-17.29</td><td>728.5 ± 5.8</td><td>-0.2 ± 0.5</td><td>17 621</td></tr><tr><td>March 19</td><td>04:03:40</td><td>831.66921</td><td>38.146</td><td>295</td><td>-17.18</td><td>658.5 ± 5.5</td><td>1.0 ± 0.5</td><td>17 622</td></tr><tr><td>March 20</td><td>04:10:28</td><td>832.67394</td><td>38.200</td><td>296</td><td>-17.17</td><td>525.0 ± 4.9</td><td>0.1 ± 0.6</td><td>17 620</td></tr><tr><td>March 29</td><td>02:03:06</td><td>841.58549</td><td>38.682</td><td>286</td><td>-17.39</td><td>-375.1 ± 5.4</td><td>0.6 ± 1.9</td><td>17 619</td></tr><tr><td>March 29</td><td>02:43:57</td><td>841.61385</td><td>38.684</td><td>290</td><td>-17.38</td><td>-385.3 ± 5.2</td><td>0.2 ± 0.5</td><td>17 618</td></tr><tr><td>March 30</td><td>01:28:42</td><td>842.56160</td><td>38.735</td><td>292</td><td>-17.39</td><td>-240.0 ± 5.3</td><td>-4.4 ± 2.8</td><td>17 617</td></tr><tr><td>March 30</td><td>02:16:17</td><td>842.59464</td><td>38.737</td><td>296</td><td>-17.40</td><td>-235.6 ± 4.7</td><td>0.2 ± 0.8</td><td>17 617</td></tr><tr><td>April 3</td><td>01:21:32</td><td>846.55662</td><td>38.951</td><td>316</td><td>-17.36</td><td>513.5 ± 4.7</td><td>0.1 ± 0.8</td><td>17 620</td></tr><tr><td>April 4</td><td>01:00:20</td><td>847.54190</td><td>39.004</td><td>297</td><td>-17.34</td><td>7.2 ± 0.1</td><td>0.0 ± 0.0</td><td>17 620</td></tr><tr><td>April 4</td><td>01:10:46</td><td>847.54914</td><td>39.005</td><td>292</td><td>-17.35</td><td>646.7 ± 5.3</td><td>-0.5 ± 0.4</td><td>17 620</td></tr><tr><td>April 7</td><td>00:55:54</td><td>850.53882</td><td>39.166</td><td>300</td><td>-17.21</td><td>616.4 ± 5.3</td><td>0.3 ± 0.6</td><td>17 622</td></tr><tr><td>April 8</td><td>00:40:43</td><td>851.52828</td><td>39.220</td><td>297</td><td>-17.20</td><td>460.4 ± 4.6</td><td>0.4 ± 0.6</td><td>17 622</td></tr><tr><td>April 9</td><td>02:15:17</td><td>852.59395</td><td>39.278</td><td>336</td><td>-17.21</td><td>227.9 ± 4.4</td><td>4.5 ± 2.6</td><td>17 620</td></tr><tr><td>April 9</td><td>02:25:07</td><td>852.60078</td><td>39.278</td><td>325</td><td>-17.21</td><td>248.0 ± 3.8</td><td>0.0 ± 0.6</td><td>17 621</td></tr><tr><td>April 11</td><td>03:24:33</td><td>854.64205</td><td>39.388</td><td>301</td><td>-17.29</td><td>-144.0 ± 4.4</td><td>0.6 ± 0.7</td><td>17 623</td></tr><tr><td>April 12</td><td>02:29:46</td><td>855.60400</td><td>39.440</td><td>298</td><td>-17.31</td><td>-376.7 ± 4.9</td><td>0.2 ± 0.5</td><td>17 620</td></tr><tr><td>April 13</td><td>03:00:24</td><td>856.62528</td><td>39.496</td><td>298</td><td>-17.32</td><td>-497.8 ± 5.6</td><td>0.3 ± 0.5</td><td>17 623</td></tr><tr><td>April 16</td><td>03:27:12</td><td>859.64389</td><td>39.659</td><td>290</td><td>-17.43</td><td>-439.7 ± 5.5</td><td>-0.3 ± 0.5</td><td>17 623</td></tr><tr><td>April 17</td><td>03:47:30</td><td>860.65799</td><td>39.714</td><td>291</td><td>-17.37</td><td>-308.7 ± 4.9</td><td>-1.1 ± 0.5</td><td>17 620</td></tr><tr><td>April 18</td><td>01:18:22</td><td>861.55442</td><td>39.762</td><td>293</td><td>-17.37</td><td>-153.0 ± 4.4</td><td>-1.1 ± 0.5</td><td>17 620</td></tr><tr><td>April 20</td><td>02:31:18</td><td>863.60507</td><td>39.873</td><td>297</td><td>-17.44</td><td>256.6 ± 4.2</td><td>0.7 ± 0.9</td><td>17 620</td></tr><tr><td>April 21</td><td>01:54:56</td><td>864.57981</td><td>39.926</td><td>296</td><td>-17.37</td><td>451.7 ± 4.6</td><td>-0.6 ± 0.5</td><td>17 621</td></tr><tr><td>April 23</td><td>02:21:19</td><td>866.59814</td><td>40.035</td><td>294</td><td>-17.31</td><td>694.7 ± 5.6</td><td>-0.3 ± 0.5</td><td>17 618</td></tr><tr><td>April 24</td><td>01:48:26</td><td>867.57530</td><td>40.088</td><td>297</td><td>-17.26</td><td>726.7 ± 5.6</td><td>-0.1 ± 0.4</td><td>17 615</td></tr><tr><td colspan="9"><strong>2018</strong></td></tr><tr><td>April 14</td><td>00:00:38</td><td>222.50044</td><td>59.287</td><td>297</td><td>-17.47</td><td>4.3 ± 4.5</td><td>-3.2 ± 3.0</td><td>17 623</td></tr><tr><td>April 18</td><td>03:36:14</td><td>226.65016</td><td>59.511</td><td>285</td><td>-17.54</td><td>-442.7 ± 6.5</td><td>4.7 ± 4.4</td><td>17 623</td></tr><tr><td>April 19</td><td>03:46:04</td><td>227.65699</td><td>59.566</td><td>278</td><td>-17.54</td><td>-471.5 ± 6.4</td><td>2.7 ± 3.7</td><td>17 621</td></tr><tr><td>April 24</td><td>01:24:37</td><td>232.55876</td><td>59.831</td><td>289</td><td>-17.53</td><td>61.4 ± 4.7</td><td>2.9 ± 3.3</td><td>17 617</td></tr></tbody></table>
<table><caption>**Table 1** – continued</caption><thead><tr><th>Date (UT)</th><th>Time (UT)</th><th>Mid-HJD 2457000+</th><th>Rot. cycle</th><th>SNR Stokes I</th><th>RV (km s<sup>-1</sup>)</th><th>B<sub>l</sub> (G)</th><th>N<sub>l</sub> (G)</th><th>LSD lines used</th></tr></thead><tbody><tr><td>April 24</td><td>23:55:42</td><td>233.49701</td><td>59.882</td><td>292</td><td>-17.53</td><td>203.4 ± 4.7</td><td>-4.8 ± 3.6</td><td>17 618</td></tr><tr><td>May 6</td><td>00:27:09</td><td>244.51885</td><td>60.478</td><td>295</td><td>-17.52</td><td>-423.7 ± 5.7</td><td>11.2 ± 2.7</td><td>17 618</td></tr><tr><td>May 11</td><td>00:29:12</td><td>249.52028</td><td>60.748</td><td>298</td><td>-17.57</td><td>-198.5 ± 5.7</td><td>-6.7 ± 2.5</td><td>17 623</td></tr><tr><td>May 17</td><td>01:12:33</td><td>255.55038</td><td>61.075</td><td>289</td><td>-17.42</td><td>703.3 ± 6.2</td><td>0.5 ± 2.4</td><td>17 627</td></tr><tr><td>June 14</td><td>23:45:34</td><td>284.48998</td><td>62.640</td><td>276</td><td>-17.59</td><td>-462.4 ± 5.9</td><td>-1.6 ± 1.9</td><td>17 631</td></tr><tr><td>June 16</td><td>23:13:21</td><td>286.46760</td><td>62.747</td><td>289</td><td>-17.65</td><td>221.7 ± 5.0</td><td>1.2 ± 2.1</td><td>17 629</td></tr><tr><td>June 17</td><td>23:34:11</td><td>287.48207</td><td>62.802</td><td>286</td><td>-17.65</td><td>-32.8 ± 4.5</td><td>1.2 ± 2.1</td><td>17 629</td></tr><tr><td>June 18</td><td>22:55:32</td><td>288.45523</td><td>62.855</td><td>278</td><td>-17.64</td><td>246.5 ± 6.1</td><td>-2.3 ± 4.5</td><td>17 629</td></tr><tr><td>June 19</td><td>21:26:50</td><td>289.39363</td><td>62.905</td><td>288</td><td>-17.62</td><td>343.9 ± 4.8</td><td>-3.8 ± 2.4</td><td>17 629</td></tr><tr><td>June 24</td><td>23:17:23</td><td>294.47041</td><td>63.180</td><td>296</td><td>-17.49</td><td>564.5 ± 5.6</td><td>0.9 ± 2.7</td><td>17 650</td></tr><tr><td>June 25</td><td>22:35:44</td><td>295.44148</td><td>63.232</td><td>279</td><td>-17.49</td><td>413.7 ± 5.4</td><td>-3.5 ± 3.4</td><td>17 652</td></tr><tr><td>June 30</td><td>22:24:02</td><td>300.43336</td><td>63.502</td><td>226</td><td>-17.64</td><td>-236.0 ± 70.0</td><td>-80.3 ± 30.6</td><td>17 488</td></tr><tr><td>July 9</td><td>22:42:15</td><td>309.44601</td><td>63.990</td><td>273</td><td>-17.52</td><td>571.9 ± 5.5</td><td>0.9 ± 1.8</td><td>17 637</td></tr><tr><td>July 22</td><td>23:17:37</td><td>322.47057</td><td>64.694</td><td>271</td><td>-17.63</td><td>-336.8 ± 6.0</td><td>2.5 ± 2.4</td><td>17 642</td></tr><tr><td>July 23</td><td>20:58:50</td><td>323.37419</td><td>64.743</td><td>277</td><td>-17.62</td><td>-182.9 ± 5.9</td><td>-3.9 ± 3.3</td><td>17 641</td></tr><tr><td>July 26</td><td>21:14:16</td><td>326.38491</td><td>64.906</td><td>277</td><td>-17.59</td><td>420.6 ± 5.5</td><td>-3.6 ± 2.8</td><td>17 642</td></tr><tr><td>July 28</td><td>21:03:31</td><td>328.37744</td><td>65.014</td><td>273</td><td>-17.55</td><td>670.0 ± 6.3</td><td>8.8 ± 2.4</td><td>17 642</td></tr></tbody></table>

<table><caption>**Table 2.** Journal of observations for σ Boo in 2021. The table displays the date of the observation, number of LSD profiles combined in the mean profile, zero rotation cycle with respect to the first observation on 2021 May 18, Stokes I profile mean SNR, RV, longitudinal magnetic field (B_l), null value (N_l), and magnetic detection flag for the presence of a magnetic field in the mean LSD profiles, where D is a definite detection, M is marginal, and N is no detection calculated using the false alarm probability (FAP). HJD are the mean values of the nightly observations. Rotation cycle is displayed as the mean time of observation for each night. A journal of the individual observations is shown in the appendix.</caption><thead><tr><th>Date (UT)</th><th>Mid-HJD 2459000+</th><th>No. LSD profiles</th><th>Rot. cycle</th><th>SNR Stokes I</th><th>RV (km s<sup>-1</sup>)</th><th>B<sub>l</sub> (G)</th><th>N<sub>l</sub> (G)</th><th>Det. N, M, D</th></tr></thead><tbody><tr><td>May 18</td><td>352.82771</td><td>12</td><td>0.000</td><td>2468</td><td>0.29 ± 0.01</td><td>-1.0 ± 0.2</td><td>-0.1 ± 0.2</td><td>M</td></tr><tr><td>May 19</td><td>353.80843</td><td>12</td><td>0.117</td><td>2515</td><td>0.34 ± 0.01</td><td>-0.7 ± 0.1</td><td>0.1 ± 0.2</td><td>M</td></tr><tr><td>May 20</td><td>354.82750</td><td>12</td><td>0.238</td><td>2528</td><td>0.34 ± 0.01</td><td>-0.7 ± 0.2</td><td>-0.4 ± 0.2</td><td>D</td></tr><tr><td>May 21</td><td>355.77625</td><td>12</td><td>0.351</td><td>2171</td><td>0.32 ± 0.01</td><td>-1.4 ± 0.6</td><td>-1.0 ± 0.6</td><td>M</td></tr><tr><td>May 22</td><td>356.74181</td><td>3</td><td>0.466</td><td>1048</td><td>0.37 ± 0.01</td><td>-2.5 ± 1.0</td><td>-0.6 ± 1.2</td><td>N</td></tr><tr><td>May 24a</td><td>358.81633</td><td>6</td><td>0.713</td><td>1609</td><td>0.35 ± 0.01</td><td>-1.1 ± 0.3</td><td>0.0 ± 0.3</td><td>N</td></tr><tr><td>May 24b</td><td>358.94385</td><td>6</td><td>0.728</td><td>1640</td><td>0.35 ± 0.01</td><td>-1.1 ± 0.4</td><td>0.1 ± 0.4</td><td>N</td></tr><tr><td>May 25</td><td>359.81721</td><td>6</td><td>0.832</td><td>1607</td><td>0.35 ± 0.01</td><td>-0.6 ± 0.2</td><td>-0.4 ± 0.3</td><td>N</td></tr><tr><td>May 26</td><td>360.92690</td><td>12</td><td>0.964</td><td>2353</td><td>0.30 ± 0.01</td><td>-0.4 ± 0.2</td><td>0.1 ± 0.2</td><td>M</td></tr><tr><td>May 27</td><td>361.77358</td><td>12</td><td>1.065</td><td>2425</td><td>0.36 ± 0.01</td><td>-0.3 ± 0.1</td><td>-0.2 ± 0.1</td><td>N</td></tr><tr><td>May 28</td><td>362.77587</td><td>12</td><td>1.184</td><td>2369</td><td>0.32 ± 0.01</td><td>-0.7 ± 0.2</td><td>-0.4 ± 0.2</td><td>M</td></tr></tbody></table>

profiles, marginal detection in five profiles, and no detection in five profiles (Table 2).

## 3.2 Longitudinal magnetic field

We determine the longitudinal magnetic field (B_l) according to Donati et al. (1997),

$$B_l = -2.14 \times 10^{11} \frac{\int vV(v)dv}{\lambda gc \int [I_c - I(v)]dv}, \quad (1)$$

where B_l is the longitudinal magnetic field in gauss, λ is the mean wavelength of all lines used to calculate the LSD profile in nm, g is the mean value of the Landé factors of all lines used to construct the LSD profile, c is the speed of light (in the same units as v), and I_c is the continuum level of the intensity profile. B_l is defined as the line-of-sight component of the field, which is averaged over the visible stellar disc (Babcock 1947; Schwarzschild 1950), and according to Mathys (1989) and Donati et al. (1997) is measured from the first-order moment of the Stokes V LSD profile. As a first-order approximation B_l is considered to be proportional to the shift in

wavelength between left and right circularly polarized lines (Mathys 1995).

Stars with a fossil magnetic field are expected to be stable over long periods of time and show smooth sinusoidal rotational modulation of B_l that changes polarity over the rotation cycle as the positive and negative magnetic poles become visible. In contrast, dynamo magnetic fields are complex and dynamic and are not expected to show rotational modulation of B_l synchronized with the rotational cycle over longer periods of time due to evolution of the magnetic field.

We apply a generalized Lomb–Scargle periodogram (GLS; Zechmeister & Kürster 2009) to search for a period in the time-series of B_l measurements. We also calculate the FAP (Baluev 2008), which quantifies the probability of the maximum peak being produced by a signal without a periodic component (VanderPlas 2018). A plot of power versus frequency in Fig. 3 shows a peak with an FAP > 10⁻⁹ corresponding to an excellent fit of the sine curve to the data. This peak with maximum power corresponds to sine curve with a period of 18.4748 d, which is consistent with the equatorial rotation period of 18.5 d found by Preston (1967a), 18.497 d (Steinitz 1967),
<figure><img src="image_2.png" /><figcaption>**Figure 1.** Comparison of two LSD profiles for β CrB at similar phases in different years. The LSD profile from 2015 April 24 at rotational phase 0.546 is displayed with the upper red line showing the Stokes V profile (y-axis expanded 4x and shifted up by 0.2 to allow better visibility). The middle green-yellow line shows the null profile (expanded 100x and shifted up by 0.1). The lower blue line shows the Stokes I profile. The profile from 8 March 2017 at rotational phase 0.542 is overplotted in black. The two plots show almost identical Stokes V profiles and a stable magnetic field over the 2 yr of observations, showing the well-known observation that β CrB contains a fossil magnetic field. The zero rotation phase for both figures is aligned to our first observation of β CrB on 2015 April 13.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>**Figure 2.** Mean LSD profile for σ Boo on 2021 May 18. The upper line (red) shows the Stokes V profile (y-axis expanded by 750 times and shifted up by 0.2 to allow better visibility). The middle line (green-yellow) shows the null profile (expanded by 750 times and shifted up by 0.1), which is used as a check for any spurious signals contaminating the spectra. The lower line (blue) shows the Stokes I profile.</figcaption></figure>

18.4868 d (Kurtz 1989; Leroy 1995), and 18.4866 d (Bagnulo et al. 2000). We compute the window function (VanderPlas 2018) that is a check for the presence of periods caused by observational biases in the data. The window function is determined by applying a Lomb-Scargle periodogram to the data with the $B_1$ measurements set to 1. The window function is shown in Fig. 3 and indicates there are no patterns in the timing of observations.

We determine 143 $B_1$ values for β CrB over a 4-yr period between 2015 and 2018 (Table 1). A plot of the rotational modulation of $B_1$ (Fig. 4) shows a sinusoidal curve that results from both positive and negative magnetic poles being visible over the rotation cycle. $B_1$ spans a range of values from +751.6 ± 5.8 G at phase 0.08 to -528.8 ± 6.8 G at phase 0.57 that is consistent with the range found

by previous studies (Wade et al. 2000; Leone & Catanzaro 2001). Fig. 4 shows the rotational modulation $B_1$ for each epoch which remains stable over the 4 yr of observations that is consistent with the behaviour of a fossil field.

We apply a GLS periodogram to the $B_1$ measurements of σ Boo from 2021 May 18 to 28 (Tables 2 and A1), and find a sine period of 7.7 ± 1.0 d with an FAP of approximately 0.1 (Fig. 5). The window function is shown in Fig. 5, and indicates there are no patterns in the timing of observations. A plot of the rotational modulation of 11 mean nightly values of $B_1$ for σ Boo is shown in Fig. 6, where we overlay a sinusoidal curve with a period that corresponds to an estimated stellar rotation period of 8.4 ± 0.2 d determined from ZDI $\chi^2$ minimization (Table 4). The plot of $B_1$ versus rotation phase shows a possible rotational modulation of the magnetic field; however, the large error bars make the determination uncertain.

# 4 MAGNETIC FIELD MODELLING

## 4.1 Longitudinal field modelling

The oblique rotator model (Babcock 1949; Stibbs 1950; Deutsch 1958, 1970; Pyper 1969; Falk & Wehlau 1974) is a useful first step in modelling the longitudinal magnetic field of a hot star where the field is considered to be a simple dipole inclined to the rotation axis. We derive the geometry and strength of the dipole fossil field of β CrB using parameters: inclination of the rotational axis with respect to the observer (i), obliquity of the dipole axis with respect to the rotational axis (β) counted from the positive rotational pole towards the positive magnetic pole, and the polar strength of the surface magnetic field ($B_d$) calculated at the positive magnetic pole. Following the oblique rotator model of Preston (1967b):

$$ \beta = \arctan \left[ \left( \frac{1-r}{1+r} \right) - \cot i \right], \qquad (2) $$

where

$$ r = \frac{B_1(\min)}{B_1(\max)} = \frac{\cos \beta \cos i - \sin \beta \sin i}{\cos \beta \cos i + \sin \beta \sin i}. \qquad (3) $$

We calculate the polar field strength ($B_d$) of following the relation given by Blazère et al. (2020),

$$ B_d = B_{\max} \left( \frac{15 + u}{20(3 - u)} (\cos \beta \cos i + \sin \beta \sin i) \right)^{-1} \qquad (4) $$

where we use the limb-darkening coefficient $u = 0.57$ from Gray (2005), $B_1 \max = +751.6 \pm 5.8$ G and $B_1 \min = -528.8 \pm 6.8$ G calculated from fitting a sinusoidal curve to the data in Fig. 3. The value for $r = 0.70 \pm 0.006$ is calculated using equation (3). The value for $i = 39 \pm 4$ deg is calculated from the mean epoch values determined from ZDI $\chi^2$ minimization, and $\beta = 26 \pm 4$ deg from the mean epoch values calculated using equation (2). We calculate the polar field strength ($B_d$) of β CrB = 2423 ± 51 G. The values are presented in Table 3.

Schwarzschild (1950) showed the maximum value for a longitudinal magnetic field ($B_1$) is 30 per cent of the dipolar field. Our $B_1$ max value of +751.6 ± 5.8 G is 31.0 per cent of our calculated $B_d$ of 2423 ± 51 G, which is consistent with Schwarzschild's prediction.

Landstreet & Mathys (2000) and Bagnulo et al. (2002) suggest the dipolar field obliquity (β) is correlated with stellar rotational period, where low-β geometries are more common in slow rotators. Landstreet & Mathys (2000) indicate that stars with rotation periods of about one month have their magnetic and rotation axes aligned within about 20 deg of each other, while faster rotating stars have
<figure><img src="image_4.png" /></figure>

**Figure 3.** Figure at left: GLS periodogram a time-series of $B_1$ measurements for β CrB spanning 1240 d from 2015 March 7 to 2018 July 28 (Table 1). The figure shows the power versus frequency and FAP, with the highest peaks marked with a red circle, and text describing the associated period and frequency. The FAP quantifies the probability of the maximum peak being produced by a signal without a periodic component. The blue dashed line is FAP $10^{-9}$ level, which indicates an excellent detection for the period determination. The best sine period determined from the GLS periodogram is $18.4748 \pm 0.005$ d. The figure at top right shows the time spacing of the observations. Figure at bottom right shows the window function.

<figure><img src="image_5.png" /></figure>

**Figure 4.** The figure shows the rotational modulation of $B_1$ for β CrB from 73 observations corresponding to magnetic maps from 2015 to 2018. The data from 2015 are shown as blue circle, 2016 red square, 2017 yellow triangle, and 2018 green diamond. The dashed red curve indicates a fitted sine curve to the data with a period of 18.4868 d that corresponds to the rotation period used in ZDI analysis. The rotation phase $\phi$ corresponds to the measured maximum and minimum $B_1$. The figure shows the magnetic field is stable over the 4 yr observation period. The zero day is the first date of $B_1$ measurement on 2018 January 23.

larger values of $\beta$. Since β CrB is a slow rotator with a period of 18.5 d (Section 3.2), we may expect that $\beta$ will be larger 20 deg. We determine $\beta$ values of $29 \pm 3$ deg, using equation (2) and $26 \pm 4$ deg from ZDI $\chi^2$ minimization Section 4.2, which is consistent with the trend from Landstreet & Mathys (2000).

## 4.2 Zeeman Doppler imaging

The surface magnetic fields of β CrB and σ Boo are reconstructed using the inversion technique of ZDI (Semel 1989; Donati & Brown 1997), using a spherical harmonics expansion of the surface magnetic field (Donati et al. 2006b). Recovery of reliable maps depends on modelling Stokes *I* and *V* profiles and using a model local line profile. ZDI involves an iterative procedure that begins with a zero magnetic field, and parameters are adjusted to optimize the fit (parametrized

by reduced $\chi^2$) of the model compared to observations. The local line profile is an approximation to the actual line profile (Piskunov & Rice 1993) and is the function used to calculate the line at one point on the surface of the star. We modelled the local line profile using a Voigt profile (Table 4).

We assume the model star is spherical and divided into approximately equal area surface elements with coordinates of colatitude and longitude. Colatitude increases with distance from the north pole, and longitude increases in the direction of rotation, with the coordinate system aligned with the stellar rotation axis. We use a stellar surface divided into a 4584 point grid, with 120 spatially resolved elements around the equator, and 60 in colatitude similar to Folsom et al. (2018). The minimum $\chi^2$ used in the ZDI code is determined by running the code with increasing values and finding the $\chi^2$ value where there is no increase in magnetic energy as used in Paper 2. Using this method, we adopt a $\chi^2$ of 4.0 for β CrB and 0.95 for σ Boo.

For β CrB, the local line width used are Gaussian width 2.41 km s⁻¹, Lorentzian width 0.89 km s⁻¹, and limb darkening coefficient 0.57. The wavelength 512.6 nm and effective Landé factor 1.195 used for computing Stokes *V* are the normalizing values for the LSD profile. For σ Boo, the local line width used are Gaussian width 2.41 km s⁻¹, Lorentzian width 0.89 km s⁻¹, and limb darkening coefficient 0.70. The wavelength 512.6 nm and effective Landé factor 1.195 used for computing Stokes *V* are the normalizing values for the LSD profile. The profiles for both stars are convolved with a Gaussian instrumental profile of *R* = 65 000.

### 4.2.1 Modelling the LSD stokes profiles

The Stokes *V* maximum entropy fit for β CrB is shown in Fig. 7. The plot shows a good fit between the observed and modelled profiles that adds weight to the reliability of our magnetic map reconstruction. The Stokes *V* LSD profile for σ Boo (Fig. 2) is produced by co-adding the nightly LSD profiles, which further increases the SNR of the weak magnetic field. The Stokes *V* maximum entropy fit for σ Boo is shown in Fig. 8. The plot shows a moderate fit between the observed and modelled profiles.

<figure><img src="image_6.png" /></figure>
<figure><img src="image_7.png" /><figcaption>**Figure 5.** Figure at left: GLS periodogram a time-series of $B_1$ measurements for $\sigma$ Boo spanning 11 d from 2021 May 18 to 28 (Table 2). The figure shows the power versus frequency and FAP, with the highest peak marked with a red circle, and text describing the associated period and frequency. The blue dashed line is FAP $10^{-1}$ level. The best sine period determined from the GLS periodogram is $7.7 \pm 1.0$ d. The figure at top right shows the time spacing of the observations. Figure at bottom right shows the window function.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>**Figure 6.** Figure shows the rotational modulation of $B_1$ for $\sigma$ Boo from 11 observations in 2021 May from the mean LSD profiles from each night. The red dashed line indicates a fitted sine curve to the data with a period of 8.4 d corresponding to an estimated stellar rotation period determined from ZDI.</figcaption></figure>

### 4.2.2 $\beta$ CrB magnetic maps

The large-scale magnetic field topology for $\beta$ CrB at six epochs are shown in Fig. 9. The stellar parameters used to reconstruct the map are determined using a method of $\chi^2$ minimization (Petit, Donati & Collier Cameron 2002) whereby we iteratively change individual parameters and re-run ZDI analysis, to find the parameter with the minimum $\chi^2$ when we fit a Gaussian curve to the $\chi^2$ values. The formal uncertainties in parameter estimation are determined by finding the $1\sigma$ uncertainty around Gaussian fit to the $\chi^2$ minimization. The parameters determined by ZDI analysis of each magnetic map are shown in Table 4.

We adopt the same stellar parameters for all $\beta$ CrB maps displayed in Table 4 that are mean $v \sin i = 9.4$ km s$^{-1}$, mean inclination = 39 deg, and differential rotation $d\Omega = 0.0$ rad d$^{-1}$. The mean and $1\sigma$ errors are derived from the mean values averaged over the six maps. For $P_{\text{rot}}$, we use the value of 18.4868 d from Kurtz (1989) since this value is within the uncertainty of our ZDI determination of $18.59 \pm 0.07$ d. The spherical harmonic expansion is limited to $\ell_{\text{max}} = 20$ since no further detail is obtained with higher degrees. The rotation phase of all maps is aligned to a common zero date, with the

first observation date set to MJD = 2457126.47656 (2015 April 13). The data sets comprise between 8 and 15 LSD profiles (Table 3).

The large-scale magnetic features are similar on all maps and indicate that the field is stable over the 39 month observation period. This is consistent with the fossil field nature of $\beta$ CrB where the field remains frozen into the stellar atmosphere and is expected to remain unchanged over a long period of time. The radial magnetic field shows a large area of negative polarity in the mid-northern latitudes between phases 0.2–0.6 and a region of positive magnetic field in the northern latitudes between phases 0.8–1.0. The azimuthal magnetic field shows a region of negative polarity that approximately matches the position of negative radial field between 0.2–0.6, and a small region of positive field with strengths of about 1000–2000 G at the north pole.

The complexity of the surface magnetic field for $\beta$ CrB is quantified by calculating the fractional strength of the magnetic field components that are displayed in Table 5. The calculated percentages from the Stokes V maps are the average squared magnetic field $\langle B^2 \rangle$, which is proportional to the magnetic energy (Reiners 2012). The uncertainties are derived by varying the stellar parameters over the $1\sigma$ errors for each individual parameter, and taking the extreme values of the results as the variation. The dominant term in the spherical harmonics expansion is the dipole ($\ell = 1$), which ranges from $50_{-0}^{+5}$ to $61_{-9}^{+0}$ per cent over the six epochs from 2015 to 2018. The second largest spherical harmonics term is the quadrupole ($\ell = 2$) with $18_{-1}^{+2}$ to $21_{-4}^{+0}$ per cent of magnetic energy. The octupole ($\ell = 3$) contains between $5_{-0}^{+1}$ and $7_{-2}^{+0}$ per cent of magnetic energy. The higher multipoles ($\ell > 3$) contain a significant fraction of magnetic energy with $12_{-0}^{+13}$ to $27_{-5}^{+0}$ per cent of magnetic energy.

The maximum magnetic field strength of our magnetic map of $\beta$ CrB, $B_{\text{max}} = 4106$ G for the 2017 April 7–24 map, which also has the highest $B_{\text{mean}} = 1192$ G. In contrast, the maximum $B_1$ is only +728.5 G. This large difference between $B_{\text{max}}$ and $B_1$ values indicates a significant transverse magnetic field component may be present in $\beta$ CrB, resulting in magnetic flux cancellation. This flux cancellation is consistent with a complex magnetic field structure on the surface of $\beta$ CrB. The magnetic field configuration is predominately poloidal with values ranging from 73 to 90 per cent (Table 5). The uncertainties are derived by varying the stellar parameters over the $1\sigma$ errors for each individual parameter, and taking the extreme values of the results as
<table><caption>**Table 3.** Stellar parameters for β CrB between 2015 and 2018, and σ Boo in 2021 determined from our ZDI analysis. For β CrB, polar field strength ($B_d$) and obliquity angle (β) are calculated from a dipole model of the star ($\ell_{max} = 1$). $\beta_{ORM}$ is calculated from the oblique rotator model using equation (2). $\beta_{ZDI}$ is calculated using ZDI. $B_d$ is calculated from equation (4) using $\beta$ from equation (2) and inclination from ZDI $\chi^2$ minimization. Maximum dipolar field strength from ZDI output. For σ Boo, the 11 mean LSD profiles constructed from 105 individual profiles.</caption><thead><tr><th>Epoch<br>(UT)</th><th>Obs.<br>rot<br>cycles</th><th>No.<br>LSD<br>profiles</th><th>Incl.<br>(deg)</th><th>v sin i<br>(km s<sup>-1</sup>)</th><th>Eq rot<br>period<br>(d)</th><th>Obliquity<br>angle β<sub>ORM</sub><br>(deg)</th><th>Obliquity<br>angle β<sub>ZDI</sub><br>(deg)</th><th>Polar<br>field str.<br>B<sub>d</sub> (G)</th><th>Dipolar<br>max<br>(G)</th></tr></thead><tbody><tr><td colspan="10"><strong>β CrB</strong></td></tr><tr><td>13 Apr–30 May 2015</td><td>2.5</td><td>13</td><td>35 ± 2</td><td>9.2 ± 0.2</td><td>18.475 ± 0.102</td><td>32 ± 2</td><td>32</td><td>2364</td><td>1389<sup>+52</sup><sub>-30</sub></td></tr><tr><td>6–24 Aug 2016</td><td>1.0</td><td>11</td><td>35 ± 2</td><td>9.8 ± 0.2</td><td>18.450 ± 0.020</td><td>32 ± 2</td><td>22</td><td>2364</td><td>1469<sup>+0</sup><sub>-110</sub></td></tr><tr><td>2 Mar–4 Apr 2017</td><td>0.9</td><td>15</td><td>42 ± 2</td><td>9.1 ± 0.3</td><td>18.465 ± 0.020</td><td>26 ± 2</td><td>21</td><td>2481</td><td>1406<sup>+35</sup><sub>-47</sub></td></tr><tr><td>7–24 Apr 2017</td><td>0.9</td><td>14</td><td>35 ± 2</td><td>9.2 ± 0.2</td><td>18.410 ± 0.021</td><td>32 ± 2</td><td>25</td><td>2401</td><td>1533<sup>+0</sup><sub>-173</sub></td></tr><tr><td>14 Apr–16 May 2018</td><td>1.8</td><td>8</td><td>46 ± 2</td><td>9.6 ± 0.2</td><td>18.625 ± 0.022</td><td>23 ± 2</td><td>28</td><td>2489</td><td>1281<sup>+160</sup><sub>-0</sub></td></tr><tr><td>14 Jun–28 Jul 2018</td><td>2.4</td><td>13</td><td>39 ± 2</td><td>9.4 ± 0.2</td><td>18.525 ± 0.022</td><td>28 ± 2</td><td>25</td><td>2440</td><td>1339<sup>+102</sup><sub>-0</sub></td></tr><tr><td>Mean<br>σ Boo</td><td>–</td><td>–</td><td>39 ± 4</td><td>9.4 ± 0.2</td><td>18.592 ± 0.069</td><td>29 ± 3</td><td>26 ± 4</td><td>2423 ± 51</td><td>1403 ± 82</td></tr><tr><td>18–28 May 2021</td><td>1.2</td><td>11</td><td>15 ± 3</td><td>3.8 ± 0.3</td><td>8.4 ± 0.2</td><td>–</td><td>20</td><td>–</td><td>0.8</td></tr></tbody></table>

<table><caption>**Table 4.** Parameters used to produce the maximum-entropy image reconstructions for β CrB and σ Boo. β CrB $P_{rot}$ from Kurtz (1989).</caption><thead><tr><th>Parameter</th><th>β CrB</th><th>σ Boo</th></tr></thead><tbody><tr><td colspan="3"><strong>ZDI parameters</strong></td></tr><tr><td>Inclination (deg)</td><td>39 ± 4</td><td>15 ± 3</td></tr><tr><td>v sin i (km s<sup>-1</sup>)</td><td>9.4 ± 0.2</td><td>3.8 ± 0.3</td></tr><tr><td>P<sub>rot</sub> (this work) (d)</td><td>18.50 ± 0.07</td><td>8.4 ± 0.2</td></tr><tr><td>P<sub>rot</sub> (literature) (d)</td><td>18.4868</td><td>–</td></tr><tr><td>dΩ (rad d<sup>-1</sup>)</td><td>0.0</td><td>0.0</td></tr><tr><td>χ<sup>2</sup></td><td>4.0</td><td>0.95</td></tr><tr><td>degree ϵ<sub>max</sub></td><td>20</td><td>8</td></tr></tbody></table>

the variation. The parameter ranges used for β CrB are: inclination ± 4 deg, $v \sin i \pm 0.2$ km s$^{-1}$, and $P_{rot} \pm 0.07$ d. The Stokes V LSD fits to the observations are shown in Fig. 7 that indicates an excellent match between the observed Zeeman signatures and modelled line profiles, which adds further evidence of the reliability of our reconstructed magnetic maps.

### 4.2.3 σ Boo magnetic map

The large-scale magnetic field topology for σ Boo in 2021 May is shown in Fig. 10. The stellar parameters used to reconstruct the maps are determined using a method of $\chi^2$ minimization as described in Section 4.2.2 are $v \sin i = 3.8$ km s$^{-1}$, inclination = 15 deg, rotation period = 8.4 d, and $d\Omega = 0.0$ rad d$^{-1}$. The spherical harmonic expansion is limited to $\ell_{max} = 8$ since no further details are observed in the maps with higher degrees. The large-scale magnetic field shows a simple topology, which may partly be due to a low $v \sin i$ of $3.8 \pm 0.3$ km s$^{-1}$ where the resolution is affected by the number of resolution elements (Morin et al. 2010).

The radial field of σ Boo shows a large region of negative polarity in the Northern hemisphere, with a maximum strength of 1.5 G at rotation phase 0.2–0.4. The positive magnetic pole is not visible on the map since it is located near the south polar region in an area that is tilted away from the observer. The azimuthal magnetic field shows mostly weak positive polarity (<0.5 G) that covers most of the surface except of a weak negative patch in the Northern hemisphere between phases 0.5–0.8. The meridional magnetic field shows mostly

weak negative polarity (>$-0.5$ G) that covers most of the surface except of a positive patch at the north pole, and a negative area of approximately $-1.0$ G in the Northern hemisphere between phases 0.7–1.0.

The simplicity of the surface magnetic field of σ Boo is further indicated by the fractional strength of the magnetic field components that are displayed in Table 5. The dominant term in the spherical harmonics expansion is the dipole ($\ell = 1$) which contains $84 \pm 1$ per cent of the magnetic energy. The second largest spherical harmonics term is the quadrupole ($\ell = 2$) with $14 \pm 1$ per cent of the magnetic energy. The octupole ($\ell = 3$) contains between $2^{+0}_{-0}$ per cent of magnetic energy, while the higher order multipoles ($\ell > 3$) contain $2^{+1}_{-2}$ per cent of magnetic energy. The uncertainties are derived by varying the stellar parameters over the $1\sigma$ errors for each individual parameter and taking the extreme values of the results as the variation. The parameter ranges used for σ Boo are: inclination $\pm 3$ deg, $v \sin i \pm 0.3$ km s$^{-1}$, $P_{rot} \pm 0.2$ d.

The maximum magnetic field strength of our magnetic map of σ Boo, $|B_{max}| = 1.5$ G that is consistent with the expected weak magnetic fields for F-type stars (Marsden et al. 2014) and Paper 1. The magnetic field for σ Boo is mainly poloidal which is consistent with the finding of Petit et al. (2008) who found that stars with $P_{rot} < 12$ d are more likely to have increasing proportion of magnetic energy in the poloidal field. The simple magnetic topology of σ Boo is similar to the slow rotating ($v \sin i = 6.1$ km s$^{-1}$) F9V star β Vir (Paper 2) that also has a mainly poloidal field with a weak magnetic field ($B_{max} = 1.6$ G).

## 4.3 Differential rotation

An understanding of differential rotation ($d\Omega$) in hot stars, and the conditions of the atmosphere are incomplete; however, there is evidence from Kepler photometric data that rotational shear is present in about 60 per cent of A-type stars, compared to 75 per cent of F-stars, and 95 per cent of G-stars (Balona & Abedigamba 2016). We do not find evidence of $d\Omega$ in β CrB using the method of $\chi^2$ minimization as described in Section 4.2.2. The lack of detected $d\Omega$ is consistent with the suggestion that hot stars with strong magnetic fields maintain uniform rotation along field lines (Mathis & Zahn 2005; Zahn 2010). The presence of $d\Omega$ in hot stars is also strongly
<figure><img src="image_9.png" /><figcaption>**Figure 7.** Stokes V fit for β CrB corresponding to 2015–2108 magnetic maps. The black lines represent the observed Zeeman signatures, while the red lines represent the modelled line profiles. The 1σ error bars are very small and are shown on the left. The line profiles are arranged vertically by rotation cycle and separated to allow better visibility. The rotation cycle is shown on the right of each profile.</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>**Figure 8.** Stokes V fit for σ Boo corresponding to the 2021 magnetic maps. The description is the same as Fig. 7.</figcaption></figure>

correlated with rotation rate (Balona & Abedigamba 2016), and with β CrB being a slow rotator ($v \sin i = 9.4$ km s⁻¹), we may therefore expect small or no detectable dΩ.

The presence of dΩ is a common feature of solar-type stars (Hall 1972; Gilman 1980; Rüdiger 1989) and is one of the ingredients in the operation of a solar-type dynamo (Parker 1955; Steenbeck & Krause 1969; Gilman 1980). We determine the amplitude of dΩ by applying a solar-type differential rotation law as described in Donati et al. (2000) where equatorial regions have larger angular velocity than polar regions:

$$ \Omega(l) = \Omega_{eq} - d\Omega \sin^2 l, \qquad (5) $$

where l denotes the latitude, Ωeq is the rotation rate at the equator, and dΩ is the rotational shear between the equator and the pole expressed in rad d⁻¹. The presence of dΩ in F-type stars was initially questioned by Gray (1982) who failed to find evidence of it in seven stars from F2-F8 using a Fourier transform of the line profiles. However, using a Fourier transform method, Reiners & Schmitt (2003a) detected dΩ

was present in F-type stars. Further research has shown that strong dΩ may be present in F-type stars compared to cooler stars (Reiners & Schmitt 2003b; Barnes et al. 2005).

We do not find evidence of dΩ in σ Boo using the method of χ² minimization as described in Section 4.2.2. Reiners & Schmitt (2003a) found large dΩ values in comparably slowly rotating stars, therefore we may have expected σ Boo with a $v \sin i = 3.8$ km s⁻¹ to have a high dΩ. Higher rates of dΩ are also expected to be present in earlier spectral types (Barnes et al. 2005). There are several possible reasons we did not find differential rotation in σ Boo. First, the magnetic field in σ Boo may have been too weak to allow magnetic features to be tracked on the surface of the star. A second reason is that rigid rotation may exist for σ Boo, therefore no dΩ was present in the star. A third reason for the lack of detection of dΩ may be due to a change in atmospheric conditions in early F-type stars whereby dΩ is inhibited near the transition to A-type stars (Walter 1983; Ammler-von Eiff & Reiners 2012). A fourth reason may be due to the presence of a varying differential rotation rate caused by a variable stellar dynamo (Marsden, Carter & Donati 2005), and σ Boo may have been observed during a period of low dΩ. The most likely reason for the non-detection of dΩ is the low amplitude of the Stokes V signal relative to the noise, and observations spanning only 1.2 rotation cycles.

## 5 CHROMOSPHERIC ACTIVITY INDICATOR

The S-index introduced by Vaughan, Preston & Wilson (1978) is an indirect measure of stellar magnetic activity in cool stars. It measures the flux in 20 Å wide triangular profiles centred on two resonance lines at 393.3682 nm (K-line) and 396.8492 nm (H-line) compared to the flux in two bands in the continuum on the red and blue side of the H&K line. The onset of chromospheric emission appears abruptly in solar-type stars (Hall 2008), and has been detected in the hot star Altair, A7 IV-V (Ferrero et al. 1995). Observations using the Far Ultraviolet Spectroscopic Explorer have suggested that chromospheres appear in stars as hot as 8250 K (Simon et al. 2002); however, the chromospheric emission is only a few per cent of the solar value. We do not determine S-index for β CrB since the measurement is not used in hot stars due to the absence of significant chromospheres (Hall 2008).

We determine the S-index for σ Boo using the method of Vaughan et al. (1978) and coefficients for the ESPaDOnS spectropolarimeter taken from Marsden et al. (2014). The S-index values for σ Boo are shown in table 6. We plot the rotational modulation of the mean nightly S-index of σ Boo in Fig. 11. The mean nightly S-index values vary from 0.248 ± 0.010 to 0.262 ± 0.005 and possibly show small variations in strength over the 11 d observation period. The figure indicates there is no periodic rotational modulation of the S-index. We confirm the lack of sinusoidal fit to the S-index data by applying a GLS periodogram to the mean S-index values and noted an absence of a detectable period to the 0.1 FAP level (see Fig. A1). Chromospheric emission dominates the flux measured in the S-index of K-type stars. However, Wolff, Heasley & Varsik (1985) suggest in active early F-type stars it is a minor contributor to the S-index. Variability in chromospheric emission may be difficult to detect in F-stars due to possible weakness of chromospheric emission and low contrast relative to the photospheric flux (Baliunas et al. 1995; Knutson, Howard & Isaacson 2010).
<figure><img src="image_11.png" /><figcaption>**Figure 9.** Magnetic maps for $\beta$ CrB at six epochs from 2015–2018 displayed using a Mollweide projection. The magnetic field strength in gauss is indicated in colour bars below the maps. The radial, azimuthal, and meridional components of the reconstructed large-scale magnetic field are plotted from top to bottom. Tick marks at bottom of map indicate observational phases used in magnetic map reconstruction. The rotation phases for all maps are aligned to the same zero date of MJD = 2457126.47656 (2015 April 13). The map displays the visible surface of the star to –39 deg latitude.</figcaption></figure>

# 6 DISCUSSION

## 6.1 $\beta$ CrB

We have reconstructed for the first time surface magnetic maps for $\beta$ CrB using ZDI. Our analysis of $\beta$ CrB provides a rotational modulation of $B_1$ that is consistent with previous studies. We find that the magnetic field is stable over the 4 yr observation period after taking into account the rotation of the magnetic field as a result of the different alignment in rotation and magnetic axes.

Using the simple oblique rotator model for $\beta$ CrB, we find that our $B_1$ max calculation of +751.6 ± 5.8 G is 31.0 per cent of $B_d$ value of 2423 ± 51 G, which is consistent with Schwarzschild’s prediction that $B_1$ should be about a third of the $B_d$ value. Looking into the magnetic field in more detail beyond a simple dipole model, we find a range of maximum strengths of the magnetic field for the maps determined by ZDI from 2660 to 4106 G, which are larger than the $B_1$

values of 902–1120 G. This may be due to some cancellation effects of the $B_1$ where regions of opposite polarity cancel out, while less cancellation effects may occur in ZDI maps (Lehmann et al. 2019). A higher proportion of the magnetic energy is contained in higher multipoles, which is shown in our ZDI analysis that indicates for the six magnetic maps only 54–61 per cent of the magnetic energy is contained in the dipolar magnetic field, while a significant proportion of the energy is contained in the quadrupolar (18–20 per cent) and octupolar (5–8 per cent) components, and higher multipoles ($\ell > 3$) account for 12–27 per cent of magnetic energy. This complex magnetic field for $\beta$ CrB is also displayed in the magnetic maps that show details beyond a simple dipole. There are areas of positive and negative magnetic regions that are more commonly seen in dynamo fields.

Using arguments from Donati et al. (2006b), there are several reasons we conclude the complex magnetic field of $\beta$ CrB is fossil in origin and not a dynamo field. First, $\beta$ CrB is a relative slow rotator
<table><caption>**Table 5.** Fraction of the large-scale magnetic energy reconstructed in the toroidal and poloidal field components; the fraction of the poloidal field in the dipolar (ℓ = 1), quadrupolar (ℓ = 2), and octupolar (ℓ = 3) components; and the fraction of the energy stored in the axisymmetric component (m = 0) for β CrB and σ Boo. The uncertainties are derived by varying the stellar parameters over the 1σ errors for each individual parameter, and taking the extreme values of the results as the variation.</caption><thead><tr><th>Epoch<br>(UT)</th><th>B<sub>mean</sub><br>(G)</th><th>B<sub>max</sub><br>(G)</th><th>Poloidal<br>(per cent tot)</th><th>Toroidal<br>(per cent tot)</th><th>Dipolar<br>(per cent pol)</th><th>Quad.<br>(per cent pol)</th><th>Oct.<br>(per cent pol)</th><th>(ℓ > 3)<br>(per cent pol)</th><th>Axisym.<br>(per cent tot)</th><th>Poloidal<br>(per cent axi)</th><th>Toroidal<br>(per cent axi)</th></tr></thead><tbody><tr><td colspan="12"><strong>β CrB</strong></td></tr><tr><td>13 Apr–30 May 2015</td><td>970<sup>+44</sup><sub>-24</sub></td><td>3462<sup>+147</sup><sub>-87</sub></td><td>88<sup>+1</sup><sub>-1</sub></td><td>12<sup>+1</sup><sub>-1</sub></td><td>54<sup>+2</sup><sub>-2</sub></td><td>18<sup>+2</sup><sub>-1</sub></td><td>5<sup>+1</sup><sub>-0</sub></td><td>23<sup>+2</sup><sub>-2</sub></td><td>9<sup>+2</sup><sub>-2</sub></td><td>10<sup>+2</sup><sub>-2</sub></td><td>3<sup>+0</sup><sub>-0</sub></td></tr><tr><td>6–24 Aug 2016</td><td>1050<sup>+0</sup><sub>-104</sub></td><td>3624<sup>+0</sup><sub>-249</sub></td><td>89<sup>+0</sup><sub>-2</sub></td><td>11<sup>+2</sup><sub>-0</sub></td><td>57<sup>+0</sup><sub>-5</sub></td><td>19<sup>+0</sup><sub>-2</sub></td><td>7<sup>+0</sup><sub>-2</sub></td><td>17<sup>+7</sup><sub>-0</sub></td><td>23<sup>+0</sup><sub>-14</sub></td><td>23<sup>+0</sup><sub>-15</sub></td><td>20<sup>+0</sup><sub>-17</sub></td></tr><tr><td>2 Mar–4 Apr 2017</td><td>1113<sup>+0</sup><sub>-167</sub></td><td>3666<sup>+0</sup><sub>-179</sub></td><td>73<sup>+16</sup><sub>-0</sub></td><td>27<sup>+0</sup><sub>-16</sub></td><td>55<sup>+0</sup><sub>-3</sub></td><td>20<sup>+0</sup><sub>-3</sub></td><td>7<sup>+0</sup><sub>-2</sub></td><td>18<sup>+7</sup><sub>-0</sub></td><td>19<sup>+0</sup><sub>-12</sub></td><td>17<sup>+0</sup><sub>-9</sub></td><td>26<sup>+0</sup><sub>-23</sub></td></tr><tr><td>7–24 Apr 2017</td><td>1192<sup>+0</sup><sub>-246</sub></td><td>4106<sup>+0</sup><sub>-731</sub></td><td>85<sup>+4</sup><sub>-0</sub></td><td>15<sup>+0</sup><sub>-4</sub></td><td>50<sup>+5</sup><sub>-0</sub></td><td>18<sup>+3</sup><sub>-1</sub></td><td>5<sup>+1</sup><sub>-0</sub></td><td>27<sup>+0</sup><sub>-5</sub></td><td>13<sup>+0</sup><sub>-6</sub></td><td>14<sup>+0</sup><sub>-6</sub></td><td>11<sup>+0</sup><sub>-7</sub></td></tr><tr><td>14 Apr–16 May 2018</td><td>902<sup>+112</sup><sub>-0</sub></td><td>2660<sup>+949</sup><sub>-0</sub></td><td>90<sup>+0</sup><sub>-3</sub></td><td>10<sup>+3</sup><sub>-0</sub></td><td>54<sup>+1</sup><sub>-2</sub></td><td>20<sup>+0</sup><sub>-3</sub></td><td>8<sup>+0</sup><sub>-3</sub></td><td>18<sup>+7</sup><sub>-0</sub></td><td>7<sup>+4</sup><sub>-0</sub></td><td>7<sup>+5</sup><sub>-0</sub></td><td>7<sup>+0</sup><sub>-4</sub></td></tr><tr><td>14 Jun–28 Jul 2018</td><td>905<sup>+109</sup><sub>-0</sub></td><td>2906<sup>+703</sup><sub>-0</sub></td><td>87<sup>+2</sup><sub>-0</sub></td><td>12<sup>+0</sup><sub>-2</sub></td><td>61<sup>+0</sup><sub>-9</sub></td><td>21<sup>+0</sup><sub>-4</sub></td><td>6<sup>+0</sup><sub>-1</sub></td><td>12<sup>+13</sup><sub>-0</sub></td><td>21<sup>+0</sup><sub>-14</sub></td><td>20<sup>+0</sup><sub>-12</sub></td><td>30<sup>+0</sup><sub>-27</sub></td></tr><tr><td colspan="12"><strong>σ Boo</strong></td></tr><tr><td>18–27 May 2021</td><td>0.5<sup>+0.0</sup><sub>-0.0</sub></td><td>1.5<sup>+0.0</sup><sub>-0.0</sub></td><td>95<sup>+1</sup><sub>-2</sub></td><td>5<sup>+2</sup><sub>-1</sub></td><td>84<sup>+1</sup><sub>-1</sub></td><td>14<sup>+1</sup><sub>-1</sub></td><td>2<sup>+0</sup><sub>-0</sub></td><td>2<sup>+1</sup><sub>-2</sub></td><td>88<sup>+2</sup><sub>-2</sub></td><td>92<sup>+2</sup><sub>-2</sub></td><td>10<sup>+5</sup><sub>-4</sub></td></tr></tbody></table>

<figure><img src="image_12.png" /><figcaption>**Figure 10.** Magnetic map for σ Boo in 2021 reconstructed from mean LSD profiles. Description is the same as Fig. 9. Zero phase for the map is set to the first date of observations HJD = 2459352.82771. The map displays the visible surface of the star to −15 deg latitude.</figcaption></figure>

(v sin i = 9.4 ± 0.2 km s⁻¹) and dynamo-generated magnetic fields are expected to scale with rotation rate (Noyes et al. 1984). The v sin i is not fast enough to produce a large dynamo magnetic field of the order of several kG in strength. Secondly, we consider the possibility that the field is created by a convective core dynamo (Charbonneau & MacGregor 2001; Brun, Browning & Toomre 2005) that has reached the surface. According to Spruit–Taylor dynamo processes (Spruit 1999, 2002), a core dynamo would only be expected to reach surface strengths of a few gauss in a star rotating as slowly as β CrB. Thirdly, the large-scale magnetic field is stable over at least 3 yr, which contrasts with dynamo fields that are expected to evolve over time.

<figure><img src="image_13.png" /><figcaption>**Figure 11.** Rotational modulation of 11 mean nightly S-index values for σ Boo in 2021 May. The two closely spaced values at rotational phases 0.72 and 0.73 are observation on the same night separated by 2 h. The figure shows a small variation in S-index over time, but no variation due to rotational modulation.</figcaption></figure>

In order to maintain their stability, stellar fossil fields are thought to contain mixed poloidal and toroidal topology of comparable strength in their interior (Prendergast 1956; Tayler 1973; Wright 1973; Braithwaite & Spruit 2004; Braithwaite & Nordlund 2006). However, at the surface, they are expected to be mainly poloidal (Braithwaite 2009). We find that the surface magnetic field of β CrB is mainly poloidal (approximately 85 per cent) with 15 per cent being toroidal. This agrees with the expectation for a hot star fossil field. We also find that the magnetic field geometry of β CrB is not represented by a simple dipole as described by the oblique rotator model, but is more complex as suggested by Preston (1969), Wolff & Wolff (1970), Mathys & Hubrig (1997), and Bagnulo et al. (2000).

Complex fossil fields are rare, but have been found in several stars previously. τ Sco (B0.2 V) was found by Donati et al. (2006b) to contain an unusually complex surface magnetic field for a massive star that differed significantly from a simple dipole. The magnetic field contained a moderate toroidal component, which is unusual for a fossil field. The B₁ varied over the rotation cycle from a minimum of −51.9 ± 5.3 G to a maximum of 87.8 ± 4.2 G. HD 37776 (B2IV) was found to show a complex, non-axisymmetric magnetic
field that differed from a simple dipole (Kochukhov et al. 2011). A quadrupolar component dominated the magnetic field geometry with higher multipoles present up to $ℓ = 6$. The magnetic field intensity varied over the surface from ≈5 to ≈30 kG. The authors found that there were similar poloidal and toroidal components to the field, which was an unexpected result for a hot star that is not expected to contain such a high toroidal component.

The $\delta$ Scuti star HD 41641 (A5III) was found to contain a complex fossil field (Thomson-Paressant et al. 2021) $B_1$ that varied from -178 ± 112 to 182 ± 273 G, and a simple dipole or dipole + quadrupolar configuration did not fit the data. Our finding of a relatively complex surface magnetic field for $\beta$ CrB is at odds with the fossil field hypothesis where a complex magnetic field topology has previously been considered to be produced mostly by dynamo fields. Our results for $\beta$ CrB are consistent with the presence of a complex fossil field and may add to the evidence that complex fossil fields may be more common than previously thought. Another $\delta$ Scuti star $\beta$ Cas (F2III) has recently been shown to be magnetic (Zwintz et al. 2020); however, this time the complex magnetic field was considered to be produced by a dynamo. $\beta$ Cas is therefore an interesting star for the study of magnetic fields in stars with shallow outer convection zones close to the transition between fossil and dynamo magnetic fields.

## 6.2 $\sigma$ Boo

We have reconstructed surface magnetic maps for $\sigma$ Boo at one epoch in 2021 May using ZDI. The surface magnetic topology is simple in structure, axisymmetric, poloidal (95 per cent), and is dominated by the dipole component (84 per cent) of magnetic energy. Less than 4 per cent of magnetic energy is contained in the higher ($ℓ > 2$) multipoles. The simple magnetic field for $\sigma$ Boo is similar to the late F-star $\beta$ Vir F9V (Paper 2) and is more like the dipolar fields seen in hot stars. Complex surface topology does not appear to be a common feature of F-star dynamo magnetic fields, and may possibly be due to the shallow outer convection zones, which do not allow for complex twisting of the flux tubes on their rise to the surface.

$\sigma$ Boo shows a variation of S-index over time, which is not due to rotational modulation. This is possibly due to magnetic areas on the surface of the star varying over time which may indicate a dynamo magnetic field. However, we are not able to definitely say whether the magnetic field of $\sigma$ Boo is a dynamo field or weak fossil from a single epoch magnetic map. One or more additional epochs are required to characterize the field of $\sigma$ Boo, and evolution of the large-scale magnetic field will indicate a dynamo field, while an unchanging field over an additional epoch will indicate that a weak fossil field is present.

The spectral type where the transition from fossil to dynamo magnetic fields occurs has not been determined; however, Paper 1 provided some evidence that it may occur as early as spectral type F3. This identifies $\sigma$ Boo as an important star for magnetic studies. There is some indirect evidence that the F2IV star HD164615 may contain spots (Abt, Bollinger & Burke 1983; Zerbi et al. 1997), and therefore may be generated by a dynamo; however, this was questioned by Hatzes (1998) who concluded that the most likely explanation was non-radial pulsations.

Searching for dynamo magnetic field in stars earlier than F3 will help identify the transition between fossil and dynamo fields. Observing $\sigma$ Boo at additional epochs will help firmly establish the nature of its magnetic field. Our mapping of a complex fossil field in $\beta$ CrB and a simple dynamo field in $\sigma$ Boo adds evidence that magnetic field complexity alone cannot be used to differentiate

between fossil and dynamo fields. We need to look at a combination of factors in stars to fully categorize the type of surface magnetic field present. This includes $B_1$ modelling, magnetic mapping, and chromospheric emission analysis.

## 7 CONCLUSIONS

Using ZDI, we reconstructed surface magnetic maps for $\beta$ CrB (A9SrEuCr), a hot star with a fossil magnetic field, and $\sigma$ Boo, an F3V star with either a dynamo field or a weak fossil field. The surface magnetic field of $\beta$ CrB shows a relatively complex topology that is not commonly seen in hot stars with a fossil field. This adds further evidence that complex fossil fields might be more common than previously recognized. We also reconstruct surface magnetic maps for $\sigma$ Boo and find that it has a relatively simple surface magnetic field topology. Future studies to search for dynamo magnetic fields in stars earlier than F3 will be valuable in order to learn more about the transition between fossil and dynamo magnetic fields.

## ACKNOWLEDGEMENTS

This research was based on observations obtained at the Canada-France-Hawaii Telescope (CFHT), which is operated by the National Research Council of Canada, the Institut National des Sciences de l'Univers of the Centre National de la Recherche Scientifique of France, and the University of Hawaii. This work is based on observations obtained at the Bernard Lyot Telescope (TBL, Pic du Midi, France) of the Midi-Pyrénées Observatory, which is operated by the Institut National des Sciences de l'Univers of the Centre National de la Recherche Scientifique of France. This research was supported by an Australian Government Research Training Program (RTP). This research has made use of the SIMBAD data base, operated at CDS, Strasbourg, France, and the VizieR catalogue access tool, CDS, Strasbourg, France.

## DATA AVAILABILITY

All data presented here are publicly available through the PolarBase data base (http://polarbase.irap.omp.eu/).

## REFERENCES

* Abramowitz M., Stegun A., 1964, Handbook of Mathematical Functions, 9th edn. Dover Press, New York

* Abt H., Bollinger G., Burke E., Jr, 1983, ApJ, 272, 196

* Adelman S. J., 1973, ApJ, 183, 95

* Alfvén H., Alfvén H., Fältthammar C., 1963, Cosmical Electrodynamics: Fundamental Principles, Clarendon Press, London

* Ammler-von Eiff M., Reiners A., 2012, A&A, 542, A116

* Aurière M., 2003, in Magnetism and Activity of the Sun and Stars, EDP Sciences, France, p. 105

* Babcock H., 1947, ApJ, 105, 105

* Babcock H., 1949, The Observatory, 69, 191

* Babcock H. W., 1958, ApJS, 3, 141

* Babcock H. W., 1960, ApJ, 132, 521

* Bagnulo S., Landolfi M., Mathys G., Landi Degl'Innocenti M., 2000, A&A, 358, 929

* Bagnulo S., Degl'Innocenti M. L., Landolfi M., Mathys G., 2002, A&A, 394, 1023

* Baliunas S. A. et al., 1995, ApJ, 438, 269

* Balona L. A., Abedigamba O. P., 2016, MNRAS, 461, 497

* Baluev R. V., 2008, MNRAS, 385, 1279
Barnes J., Cameron A. C., Donati J.-F., James D., Marsden S., Petit P., 2005, MNRAS, 357, L1

Berdyugina S. V., 2005, Living Rev. Sol. Phys., 2, 8

Blazère A., Petit P., Neiner C., Folsom C., Kochukhov O., Mathis S., Deal M., Landstreet J., 2020, MNRAS, 492, 5794

Bothmer V., Daglis I., 2007, Space Weather: Physics and Effects, Springer Praxis Books. Springer-Verlag, Berlin

Braithwaite J., 2009, MNRAS, 397, 763

Braithwaite J., Nordlund Å., 2006, A&A, 450, 1077

Braithwaite J., Spruit H. C., 2004, Nature, 431, 819

Brandenburg A., Subramanian K., 2005, Phys. Rep., 417, 1

Brown E. et al., 2021, MNRAS, 501, 3981

Brun A. S., Browning M. K., Toomre J., 2005, ApJ, 629, 461

Bruntt H. et al., 2010, A&A, 512, A55

Burnette A. B., Canfield R. C., Pevtsov A. A., 2004, ApJ, 606, 565

Casagrande L., Schönrich R., Asplund M., Cassisi S., Ramírez I., Melendez J., Bensby T., Feltzing S., 2011, A&A, 530, A138

Charbonneau P., MacGregor K. B., 2001, ApJ, 559, 1094

Cowling T., 1945, MNRAS, 105, 166

Cowling T., 1957, Magnetohydrodynamics, Interscience Tracts on Physics and Astronomy. Interscience Publishers, London

Deutsch A. J., 1958, in Lehnert B., ed., Electromagnetic Phenomena in Cosmical Physics, Cambridge University Press, Cambridge, p. 209

Deutsch A. J., 1970, ApJ, 159, 985

Donati J.-F., Brown S., 1997, A&A, 326, 1135

Donati J.-F., Semel M., Rees D., 1992, A&A, 265, 669

Donati J.-F., Semel M., Carter B. D., Rees D., Cameron A. C., 1997, MNRAS, 291, 658

Donati J.-F., Mengel M., Carter B., Marsden S., Collier Cameron A., Wichmann R., 2000, MNRAS, 316, 699

Donati J.-F., Catala C., Landstreet J., Petit P., 2006a, in Casini R., Lites B. W., eds, ASP Conf. Ser. Vol. 358, Solar Polarization 4. Astron. Soc. Pac., San Francisco, p. 362

Donati J.-F. et al., 2006b, MNRAS, 370, 629

Donati J.-F. et al., 2008, MNRAS, 385, 1179

Dudorov A. E., Khaibrakhmanov S. A., 2015, Adv. Space Res., 55, 843

Falk A. E., Wehlau W. H., 1974, ApJ, 192, 409

Fares R. et al., 2009, MNRAS, 398, 1383

Fares R. et al., 2012, MNRAS, 423, 1006

Ferrero R. F., Gouttebroze P., Catalano S., Marilli E., Bruhweiler F., Kondo Y., Van Der Hucht K., Talavera A., 1995, ApJ, 439, 1011

Folsom C. P. et al., 2016, MNRAS, 457, 580

Folsom C. et al., 2018, MNRAS, 474, 4956

Fossati L. et al., 2015, A&A, 582, A45

Gaia collaboration, 2018, A&A, 616, A1

Gilman P. A., 1980, Differential Rotation in Stars with Convection Zones. Springer-Verlag, Berlin, p. 19

Glagolevskij Y., 2003, in Piskunov N. et al. eds, Proc. IAU Symp. Modelling of Stellar Atmospheres. Cambridge University Press, Cambridge, UK, p. A3

Godolt M., Tosi N., Stracke B., Grenfell J. L., Ruedas T., Spohn T., Rauer H., 2019, A&A, 625, A12

Gray D., 1982, ApJ, 258, 201

Gray D., 2005, The Observation and Analysis of Stellar Photospheres. Cambridge Univ. Press, Cambridge

Grunhut J. et al., 2016, MNRAS, 465, 2432

Hall D. S., 1972, PASP, 84, 323

Hall J. C., 2008, Living Rev. Sol. Phys., 5, 1

Hanslmeier A., 2007, The Sun and Space Weather, Astrophysics and Space Science Library. Springer, Netherlands

Hatzes A., 1998, MNRAS, 299, 403

Järvinen S. et al., 2015, A&A, 574, A25

Jeffers S. et al., 2018, MNRAS, 479, 5266

Johnson H., Morgan W., 1953, ApJ, 117, 313

Kaltenegger L., Lin Z., 2021, ApJ, 909, L2

Knutson H. A., Howard A. W., Isaacson H., 2010, ApJ, 720, 1569

Kochukhov O., Lundin A., Romanyuk I., Kudryavtsev D., 2011, ApJ, 726, 24

Krause F., Radler K., 1980, Mean-field Magnetohydrodynamics and Dynamo Theory, Pergamon Press, Oxford, UK

Kurtz D., 1989, MNRAS, 238, 261

Lammer H., Khodachenko M., 2014, Characterizing Stellar and Exoplanetary Environments, Astrophysics and Space Science Library. Springer International Publishing, Switzerland

Landstreet J., Mathys G., 2000, A&A, 359, 213

Lehmann G., Hussain G., Jardine M., Mackay D., Vidotto A., 2019, MNRAS, 483, 5246

Leone F., Catanzaro G., 2001, A&A, 365, 118

Leroy J., 1995, A&AS, 114, 79

Lignières F., Petit P., Böhm T., Aurière M., 2009, A&A, 500, L41

Linsky J., 2019, Host Stars and their Effects on Exoplanet Atmospheres: An Introductory Overview, Lecture Notes in Physics. Springer International Publishing, Switzerland

Marsden S., Carter B., Donati J.-F., 2005, in Favata F. ed., 13th Cambridge Workshop on Cool Stars, Stellar Systems and the Sun. ESA, Noordwijk, Netherlands, p. 799

Marsden S. C., Mengel M. W., Donati F., Carter B. D., Semel M., Petit P., 2006, in Casini R., Lites B., eds Solar Polarization 4. ASP, San Francisco, p. 401

Marsden S. et al., 2014, MNRAS, 444, 3517

Mason B. D., et al., 2001, AJ, 122, 3466

Mathis S., Zahn J.-P., 2005, A&A, 440, 653

Mathys G., 1989, Fundam. Cosm. Phys., 13, 143

Mathys G., 1993, in Dworetsky, M., Castelli, F., Faraggiana R, eds, Peculiar versus normal phenomena in A-type and related stars. Cambridge University Press, Cambridge, UK, p. 232

Mathys G., 1995, A&A, 293, 733

Mathys G., Hubrig S., 1997, A&AS, 124, 475

Mestel L., 1966, MNRAS, 133, 265

Mestel L., 1967, in Magnetic and Related Stars. Mono Book Corporation, California, p. 101

Mestel L., 2012, Stellar Magnetism. Oxford Univ. Press, Oxford

Mestel L., Spitzer L., 1956, MNRAS, 116, 503

Moffatt H., 1978, Magnetic Field Generation in Electrically Conducting Fluids. Cambridge Univ. Press, Cambridge

Moffatt K., Dormy E., 2019, Self-Exciting Fluid Dynamos, Cambridge Texts in Applied Mathematics. Cambridge Univ. Press, Cambridge

Morin J., Donati J.-F., Petit P., Delfosse X., Forveille T., Jardine M., 2010, MNRAS, 407, 2269

Moss D., 2001, in Wickramasinghe D., et al., eds, Magnetic Fields Across the Hertzsprung-Russell Diagram. ASP, San Francisco, p. 305

Mullan D., 1972, ApL, 12, 13

Neiner C., Wade G., Sikora J., 2017, MNRAS, 468, L46

Noyes R., Hartmann L., Baliunas S., Duncan D., Vaughan A., 1984, ApJ, 279, 763

Oetken L., 1977, Astron. Nachr., 298, 197

Palubski I. Z., Shields A. L., Deitrick R., 2020, ApJ, 890, 30

Parker E. N., 1955, ApJ, 122, 293

Parker E., 1979, Cosmical Magnetic Fields: Their Origin and Their Activity. OUP, Oxford, UK

Parker E., 1987, Sol. Phys., 110, 11

Perryman M., 2018, The Exoplanet Handbook. Cambridge Univ. Press, Cambridge

Petit P., Donati J.-F., Collier Cameron A., 2002, MNRAS, 334, 374

Petit P. et al., 2008, MNRAS, 388, 80

Petit P. et al., 2010, A&A, 523, A41

Petit P., Louge T., Théado S., Paletou F., Manset N., Morin J., Marsden S., Jeffers S., 2014, PASP, 126, 469

Pettersen B., 1989, Sol. Phys., 121, 299

Pevtsov A. A., Canfield R. C., Metcalf T. R., 1994, ApJ, 425, L117

Piskunov N., Rice J., 1993, PASP, 105, 1415

Prendergast K. H., 1956, ApJ, 123, 498

Preston G., 1967a, in Cameron R., ed., Studies of Stellar Magnetism - Past, Present, and Future, in The Magnetic and Related Stars. Mono Book Corporation, California, p. 3

Preston G. W., 1967b, ApJ, 150, 547
Preston G. W., 1969, ApJ, 158, 1081

Preston G. W., Sturch C., 1967, in Magnetic and Related Stars. Mono Book Corporation, California, p. 111

Pyper D. M., 1969, ApJS, 18, 347

Reiners A., 2012, Living Rev. Sol. Phys., 9, 1

Reiners A., Schmitt J., 2003a, A&A, 398, 647

Reiners A., Schmitt J., 2003b, A&A, 412, 813

Renson P., Manfroid J., 2009, A&A, 498, 961

Roberts P. H., 1992, Dynamo Theory. Springer-Verlag, Berlin, p. 237

Rüdiger G., 1989, Differential Rotation and Stellar Convection: Sun and Solar-type Stars, Fluid Mechanics of Astrophysics and Geophysics. Gordon and Breach Science Publishers, New York

Rushby A. J., Shields A. L., Joshi M., 2019, ApJ, 887, 29

Ryabchikova T., Piskunov N., Kurucz R., Stempels H., Heiter U., Pakhomov Y., Barklem P. S., 2015, Phys. Scr., 90, 054005

Samus N., Kazarovets E., Durlevich O., Kireeva N., Pastukhova E., 2017, Astron. Rep., 61, 80

Schatzman E., 1962, Ann. Astrophys., 25, 18

Schmitt J., Golub L., Harnden F. Jr, Maxson C., Rosner R., Vaiana G., 1985, ApJ, 290, 307

Schneider F. R., Ohlmann S. T., Podsiadlowski P., Röpke F. K., Balbus S. A., Pakmor R., Springel V., 2019, Nature, 574, 211

Schwarzschild M., 1950, ApJ, 112, 222

Schwieterman E. W., Reinhard C. T., Olson S. L., Ozaki K., Harman C. E., Hong P. K., Lyons T. W., 2019, ApJ, 874, 9

Seach J., Marsden S., Carter B., Neiner C., Folsom C., Mengel M., Oksala M., Buysschaert B., 2020, MNRAS, 494, 5682 (Paper 1)

Seach J., Marsden S., Carter B., Evensberget D., Folsom C., Neiner C., Mengel M., 2022, MNRAS, 509, 5117 (Paper 2)

Segura A., Krelove K., Kasting J. F., Sommerlatt D., Meadows V., Crisp D., Cohen M., Mlawer E., 2003, Astrobiology, 3, 689

Semel M., 1989, A&A, 225, 456

Sikora J., Wade G., Power J., Neiner C., 2019, MNRAS, 483, 2300

Simon T., Ayres T. R., Redfield S., Linsky J. L., 2002, ApJ, 579, 800

Singh A. K., Bhargawa A., Siingh D., Singh R. P., 2021, Geosciences, 11, 286

Spitzer L., 1958, in Lehnert B., ed, Electromagnetic Phenomena in Cosmical Physics. Cambridge University Press, London, p. 169

Spruit H., 1999, A&A, 349, 189

Spruit H., 2002, A&A, 381, 923

Steenbeck M., Krause F., 1969, Astron. Nachr., 291, 49

Steinitz R., 1967, in Cameron R., ed., The Magnetic Oblique Rotator, in The Magnetic and Related Stars. Mono Book Corporation, California, p. 83

Stibbs D., 1950, MNRAS, 110, 395

Stift M., 1975, MNRAS, 172, 133

Strassmeier K. G., 2009, A&AR, 17, 251

Tayler R., 1973, MNRAS, 161, 365

Thomson-Paressant K., Neiner C., Zwintz K., Escorza A., 2021, MNRAS, 500, 1992

VanderPlas J. T., 2018, ApJS, 236, 16

Vaughan A. H., Preston G. W., Wilson O. C., 1978, PASP, 90, 267

Wade G., Donati J.-F., Landstreet J., Shorlin S., 2000, MNRAS, 313, 851

Walter F., 1983, ApJ, 274, 794

Walter F., Schrijver C., 1987, in Araki H., et al., eds, Cool Stars, Stellar Systems, and the Sun. Springer, London, p. 262

Wolff S. C., Wolff R. J., 1970, ApJ, 160, 1049

Wolff S. C., Heasley J., Varsik J., 1985, PASP, 97, 707

Wolff S. C., Boesgaard A. M., Simon T., 1986, ApJ, 310, 360

Wright G., 1973, MNRAS, 162, 339

Zahn J.-P., 2010, in Neiner C., et al. eds, Active OB stars: structure, evolution, mass loss, and critical limits, Cambridge University Press, Cambridge, UK, p. 14

Zechmeister M., Kürster M., 2009, A&A, 496, 577

Zeldovich I. B., Ruzmaikin A. A., Sokolov D. D., 1983, Magnetic Fields in Astrophysics. Gordon & Breach, New York

Zerbi F. et al., 1997, MNRAS, 292, 43

Zwintz K. et al., 2020, A&A, 643, A110

## APPENDIX A: ADDITIONAL DATA

<figure><img src="image_14.png" /><figcaption>**Figure A1.** GLS periodogram a time-series of mean nightly S-index measurements for σ Boo spanning 11 d from 2021 May 18–28 (Table 2). The figure shows the power versus frequency and FAP, with no peak visible to the $10^{-1}$ level.</figcaption></figure>
<table><caption>Table A1. Journal of observations for σ Boo in 2021 for the individual spectra. The columns are the same as Table 2.</caption><thead><tr><th>Date (UT)</th><th>Time (UT)</th><th>Mid-HJD 2459000+</th><th>Rot. cycle</th><th>SNR Stokes I</th><th>RV (km s<sup>-1</sup>)</th><th>B<sub>l</sub> (G)</th><th>N<sub>l</sub> (G)</th><th>S-index</th><th>LSD lines used</th></tr></thead><tbody><tr><td>May 18</td><td>06:46:45</td><td>352.78642</td><td>0.0000</td><td>714</td><td>0.29</td><td>-1.5 ± 1.5</td><td>-0.9 ± 1.5</td><td>0.257</td><td>17 654</td></tr><tr><td>May 18</td><td>06:57:27</td><td>352.79386</td><td>0.0009</td><td>715</td><td>0.29</td><td>-0.4 ± 0.5</td><td>0.0 ± 0.5</td><td>0.256</td><td>17 654</td></tr><tr><td>May 18</td><td>07:08:10</td><td>352.80130</td><td>0.0018</td><td>714</td><td>0.30</td><td>-3.0 ± 1.5</td><td>-0.7 ± 1.5</td><td>0.253</td><td>17 654</td></tr><tr><td>May 18</td><td>07:18:53</td><td>352.80874</td><td>0.0027</td><td>717</td><td>0.29</td><td>-0.6 ± 0.5</td><td>0.1 ± 0.5</td><td>0.262</td><td>17 654</td></tr><tr><td>May 18</td><td>07:29:36</td><td>352.81619</td><td>0.0035</td><td>717</td><td>0.28</td><td>-2.4 ± 1.4</td><td>-1.6 ± 1.4</td><td>0.256</td><td>17 654</td></tr><tr><td>May 18</td><td>07:40:20</td><td>352.82360</td><td>0.0044</td><td>714</td><td>0.29</td><td>-1.4 ± 0.5</td><td>0.4 ± 0.5</td><td>0.258</td><td>17 654</td></tr><tr><td>May 18</td><td>07:52:04</td><td>352.83179</td><td>0.0054</td><td>696</td><td>0.29</td><td>-3.5 ± 1.5</td><td>1.4 ± 1.5</td><td>0.247</td><td>17 654</td></tr><tr><td>May 18</td><td>08:02:47</td><td>352.83923</td><td>0.0063</td><td>713</td><td>0.31</td><td>-3.8 ± 1.2</td><td>-0.3 ± 1.2</td><td>0.262</td><td>17 654</td></tr><tr><td>May 18</td><td>08:13:30</td><td>352.84667</td><td>0.0072</td><td>711</td><td>0.30</td><td>-1.2 ± 0.8</td><td>0.5 ± 0.8</td><td>0.262</td><td>17 654</td></tr><tr><td>May 18</td><td>08:24:13</td><td>352.85411</td><td>0.0081</td><td>715</td><td>0.30</td><td>-1.5 ± 0.6</td><td>0.0 ± 0.6</td><td>0.251</td><td>17 654</td></tr><tr><td>May 18</td><td>08:34:56</td><td>352.86155</td><td>0.0089</td><td>715</td><td>0.30</td><td>-1.2 ± 0.5</td><td>0.4 ± 0.6</td><td>0.255</td><td>17 654</td></tr><tr><td>May 18</td><td>08:45:39</td><td>352.86900</td><td>0.0098</td><td>711</td><td>0.30</td><td>-1.3 ± 0.8</td><td>1.2 ± 0.8</td><td>0.260</td><td>17 654</td></tr><tr><td>May 19</td><td>06:18:56</td><td>353.76710</td><td>0.1167</td><td>719</td><td>0.32</td><td>-3.1 ± 1.4</td><td>0.3 ± 1.4</td><td>0.251</td><td>17 653</td></tr><tr><td>May 19</td><td>06:29:40</td><td>353.77455</td><td>0.1176</td><td>723</td><td>0.32</td><td>-2.2 ± 0.6</td><td>0.3 ± 0.6</td><td>0.263</td><td>17 652</td></tr><tr><td>May 19</td><td>06:40:24</td><td>353.78200</td><td>0.1185</td><td>722</td><td>0.32</td><td>-1.6 ± 0.5</td><td>0.8 ± 0.5</td><td>0.258</td><td>17 652</td></tr><tr><td>May 19</td><td>06:51:08</td><td>353.78946</td><td>0.1194</td><td>726</td><td>0.33</td><td>-1.0 ± 0.8</td><td>-0.6 ± 0.8</td><td>0.255</td><td>17 652</td></tr><tr><td>May 19</td><td>07:01:52</td><td>353.79691</td><td>0.1203</td><td>731</td><td>0.33</td><td>-1.6 ± 0.5</td><td>0.4 ± 0.5</td><td>0.259</td><td>17 652</td></tr><tr><td>May 19</td><td>07:12:37</td><td>353.80437</td><td>0.1212</td><td>731</td><td>0.34</td><td>0.7 ± 0.7</td><td>-0.5 ± 0.7</td><td>0.253</td><td>17 653</td></tr><tr><td>May 19</td><td>07:24:12</td><td>353.81242</td><td>0.1221</td><td>731</td><td>0.34</td><td>0.6 ± 0.5</td><td>0.6 ± 0.5</td><td>0.255</td><td>17 652</td></tr><tr><td>May 19</td><td>07:35:00</td><td>353.81992</td><td>0.1230</td><td>731</td><td>0.33</td><td>-0.6 ± 0.5</td><td>0.1 ± 0.5</td><td>0.261</td><td>17 652</td></tr><tr><td>May 19</td><td>07:45:47</td><td>353.82741</td><td>0.1240</td><td>733</td><td>0.34</td><td>1.3 ± 1.0</td><td>0.4 ± 1.0</td><td>0.254</td><td>17 652</td></tr><tr><td>May 19</td><td>07:56:31</td><td>353.83487</td><td>0.1248</td><td>729</td><td>0.34</td><td>-2.5 ± 1.3</td><td>0.5 ± 1.3</td><td>0.257</td><td>17 652</td></tr><tr><td>May 19</td><td>08:07:15</td><td>353.84232</td><td>0.1257</td><td>728</td><td>0.36</td><td>-0.8 ± 1.5</td><td>0.0 ± 0.5</td><td>0.261</td><td>17 652</td></tr><tr><td>May 19</td><td>08:18:04</td><td>353.84982</td><td>0.1266</td><td>711</td><td>0.36</td><td>-1.1 ± 0.5</td><td>0.1 ± 0.5</td><td>0.218</td><td>17 652</td></tr><tr><td>May 20</td><td>06:46:28</td><td>354.78620</td><td>0.2381</td><td>731</td><td>0.35</td><td>-1.5 ± 0.9</td><td>0.0 ± 0.9</td><td>0.256</td><td>17 653</td></tr><tr><td>May 20</td><td>06:57:13</td><td>354.79367</td><td>0.2390</td><td>724</td><td>0.35</td><td>-0.5 ± 0.6</td><td>0.5 ± 0.6</td><td>0.256</td><td>17 653</td></tr><tr><td>May 20</td><td>07:07:57</td><td>354.80112</td><td>0.2398</td><td>726</td><td>0.34</td><td>2.4 ± 1.6</td><td>-1.4 ± 1.6</td><td>0.257</td><td>17 652</td></tr><tr><td>May 20</td><td>07:18:43</td><td>354.80860</td><td>0.2407</td><td>724</td><td>0.34</td><td>-1.4 ± 1.0</td><td>0.5 ± 1.0</td><td>0.258</td><td>17 652</td></tr><tr><td>May 20</td><td>07:29:26</td><td>354.81604</td><td>0.2416</td><td>727</td><td>0.35</td><td>-1.1 ± 0.7</td><td>0.0 ± 0.7</td><td>0.257</td><td>17 653</td></tr><tr><td>May 20</td><td>07:40:12</td><td>354.82350</td><td>0.2425</td><td>732</td><td>0.34</td><td>-0.5 ± 0.5</td><td>-0.1 ± 0.5</td><td>0.258</td><td>17 653</td></tr><tr><td>May 20</td><td>07:51:43</td><td>354.83152</td><td>0.2435</td><td>728</td><td>0.35</td><td>-1.5 ± 0.6</td><td>-0.2 ± 0.6</td><td>0.259</td><td>17 653</td></tr><tr><td>May 20</td><td>08:02:27</td><td>354.83897</td><td>0.2444</td><td>722</td><td>0.34</td><td>-1.1 ± 0.6</td><td>-1.1 ± 0.6</td><td>0.257</td><td>17 652</td></tr><tr><td>May 20</td><td>08:13:10</td><td>354.84641</td><td>0.2452</td><td>730</td><td>0.33</td><td>-0.9 ± 0.7</td><td>0.0 ± 0.7</td><td>0.258</td><td>17 652</td></tr><tr><td>May 20</td><td>08:23:54</td><td>354.85386</td><td>0.2461</td><td>741</td><td>0.34</td><td>-1.2 ± 0.5</td><td>-0.1 ± 0.5</td><td>0.261</td><td>17 652</td></tr><tr><td>May 20</td><td>08:34:40</td><td>354.86134</td><td>0.2470</td><td>736</td><td>0.33</td><td>-0.9 ± 0.6</td><td>-1.3 ± 0.6</td><td>0.259</td><td>17 652</td></tr><tr><td>May 20</td><td>08:45:23</td><td>354.86878</td><td>0.2479</td><td>737</td><td>0.33</td><td>-2.1 ± 0.6</td><td>-0.8 ± 0.6</td><td>0.259</td><td>17 651</td></tr><tr><td>May 21</td><td>05:30:14</td><td>355.73325</td><td>0.3508</td><td>657</td><td>0.31</td><td>-3.0 ± 1.5</td><td>0.8 ± 1.5</td><td>0.249</td><td>17 651</td></tr><tr><td>May 21</td><td>05:41:00</td><td>355.74073</td><td>0.3517</td><td>655</td><td>0.31</td><td>-2.0 ± 0.8</td><td>-0.3 ± 0.8</td><td>0.249</td><td>17 651</td></tr><tr><td>May 21</td><td>05:51:44</td><td>355.74817</td><td>0.3526</td><td>658</td><td>0.31</td><td>-0.4 ± 0.6</td><td>-0.1 ± 0.6</td><td>0.254</td><td>17 651</td></tr><tr><td>May 21</td><td>06:02:27</td><td>355.75562</td><td>0.3535</td><td>653</td><td>0.30</td><td>-2.7 ± 1.6</td><td>-1.9 ± 1.6</td><td>0.255</td><td>17 652</td></tr><tr><td>May 21</td><td>06:13:10</td><td>355.76306</td><td>0.3544</td><td>646</td><td>0.31</td><td>2.2 ± 1.5</td><td>-1.6 ± 1.5</td><td>0.253</td><td>17 652</td></tr><tr><td>May 21</td><td>06:23:53</td><td>355.77051</td><td>0.3552</td><td>643</td><td>0.32</td><td>-2.8 ± 1.2</td><td>0.1 ± 1.2</td><td>0.256</td><td>17 652</td></tr><tr><td>May 21</td><td>06:35:59</td><td>355.77890</td><td>0.3562</td><td>576</td><td>0.33</td><td>-4.3 ± 2.6</td><td>0.1 ± 2.6</td><td>0.258</td><td>17 652</td></tr><tr><td>May 21</td><td>06:52:02</td><td>355.79005</td><td>0.3576</td><td>582</td><td>0.32</td><td>-3.6 ± 1.7</td><td>2.3 ± 1.7</td><td>0.255</td><td>17 652</td></tr><tr><td>May 21</td><td>07:02:45</td><td>355.79749</td><td>0.3585</td><td>608</td><td>0.32</td><td>-2.5 ± 0.9</td><td>0.9 ± 0.9</td><td>0.258</td><td>17 652</td></tr><tr><td>May 21</td><td>07:13:29</td><td>355.80494</td><td>0.3593</td><td>613</td><td>0.31</td><td>-1.5 ± 1.6</td><td>-0.5 ± 1.6</td><td>0.256</td><td>17 651</td></tr><tr><td>May 21</td><td>07:24:12</td><td>355.81239</td><td>0.3602</td><td>612</td><td>0.33</td><td>-4.5 ± 1.8</td><td>-0.9 ± 1.9</td><td>0.256</td><td>17 651</td></tr><tr><td>May 21</td><td>07:34:55</td><td>355.81983</td><td>0.3611</td><td>610</td><td>0.33</td><td>-3.6 ± 3.3</td><td>-3.4 ± 3.3</td><td>0.253</td><td>17 652</td></tr><tr><td>May 22</td><td>05:31:52</td><td>356.73437</td><td>0.4700</td><td>610</td><td>0.38</td><td>-3.3 ± 2.2</td><td>-2.4 ± 2.3</td><td>0.255</td><td>17 652</td></tr><tr><td>May 22</td><td>05:42:36</td><td>356.74181</td><td>0.4709</td><td>622</td><td>0.36</td><td>-3.4 ± 1.8</td><td>0.9 ± 1.8</td><td>0.235</td><td>17 652</td></tr><tr><td>May 22</td><td>05:53:19</td><td>356.74926</td><td>0.4718</td><td>583</td><td>0.37</td><td>-2.2 ± 1.3</td><td>1.8 ± 1.3</td><td>0.256</td><td>17 652</td></tr><tr><td>May 24</td><td>07:03:08</td><td>358.79770</td><td>0.7156</td><td>618</td><td>0.35</td><td>-1.6 ± 0.9</td><td>-0.6 ± 0.9</td><td>0.260</td><td>17 650</td></tr><tr><td>May 24</td><td>07:13:52</td><td>358.80517</td><td>0.7165</td><td>659</td><td>0.35</td><td>-1.4 ± 0.6</td><td>-0.3 ± 0.6</td><td>0.274</td><td>17 650</td></tr><tr><td>May 24</td><td>07:24:36</td><td>358.81261</td><td>0.7174</td><td>671</td><td>0.34</td><td>-3.0 ± 1.2</td><td>1.1 ± 1.2</td><td>0.257</td><td>17 650</td></tr><tr><td>May 24</td><td>07:35:19</td><td>358.82006</td><td>0.7183</td><td>668</td><td>0.35</td><td>-1.2 ± 0.6</td><td>0.0 ± 0.6</td><td>0.262</td><td>17 650</td></tr><tr><td>May 24</td><td>07:46:02</td><td>358.82750</td><td>0.7192</td><td>666</td><td>0.35</td><td>-4.1 ± 0.6</td><td>-0.5 ± 1.6</td><td>0.264</td><td>17 650</td></tr><tr><td>May 24</td><td>07:56:46</td><td>358.83495</td><td>0.7201</td><td>660</td><td>0.35</td><td>-1.3 ± 0.6</td><td>-0.7 ± 0.6</td><td>0.256</td><td>17 650</td></tr><tr><td>May 24</td><td>10:06:47</td><td>358.92524</td><td>0.7308</td><td>671</td><td>0.35</td><td>1.4 ± 1.1</td><td>0.7 ± 1.1</td><td>0.253</td><td>17 651</td></tr><tr><td>May 24</td><td>10:17:31</td><td>358.93269</td><td>0.7317</td><td>671</td><td>0.34</td><td>-1.5 ± 0.9</td><td>0.6 ± 0.9</td><td>0.257</td><td>17 650</td></tr><tr><td>May 24</td><td>10:28:14</td><td>358.94013</td><td>0.7326</td><td>672</td><td>0.34</td><td>-1.2 ± 0.9</td><td>0.2 ± 0.9</td><td>0.252</td><td>17 650</td></tr><tr><td>May 24</td><td>10:38:57</td><td>358.94757</td><td>0.7335</td><td>673</td><td>0.34</td><td>-1.9 ± 1.1</td><td>-1.5 ± 1.1</td><td>0.267</td><td>17 650</td></tr><tr><td>May 24</td><td>10:49:40</td><td>358.95502</td><td>0.7344</td><td>661</td><td>0.36</td><td>-2.8 ± 1.4</td><td>0.3 ± 1.4</td><td>0.259</td><td>17 650</td></tr></tbody></table>
<table><caption>Table A1 – continued</caption><thead><tr><th>Date (UT)</th><th>Time (UT)</th><th>Mid-HJD 2459000+</th><th>Rot. cycle</th><th>SNR Stokes I</th><th>RV (km s<sup>-1</sup>)</th><th>B<sub>l</sub> (G)</th><th>N<sub>l</sub> (G)</th><th>S-index</th><th>LSD lines used</th></tr></thead><tbody><tr><td>May 24</td><td>11:00:23</td><td>358.96246</td><td>0.7352</td><td>671</td><td>0.36</td><td>0.2 ± 0.6</td><td>0.7 ± 0.6</td><td>0.260</td><td>17 650</td></tr><tr><td>May 25</td><td>07:04:24</td><td>359.79856</td><td>0.8348</td><td>657</td><td>0.34</td><td>1.4 ± 1.6</td><td>-1.4 ± 1.6</td><td>0.251</td><td>17 650</td></tr><tr><td>May 25</td><td>07:15:07</td><td>359.80601</td><td>0.8357</td><td>651</td><td>0.35</td><td>1.4 ± 1.7</td><td>0.3 ± 1.7</td><td>0.255</td><td>17 650</td></tr><tr><td>May 25</td><td>07:25:54</td><td>359.81349</td><td>0.8366</td><td>654</td><td>0.34</td><td>-1.4 ± 0.6</td><td>0.0 ± 0.6</td><td>0.255</td><td>17 650</td></tr><tr><td>May 25</td><td>07:36:38</td><td>359.82095</td><td>0.8374</td><td>659</td><td>0.34</td><td>-1.7 ± 1.3</td><td>-0.1 ± 1.3</td><td>0.245</td><td>17 650</td></tr><tr><td>May 25</td><td>07:47:21</td><td>359.82839</td><td>0.8383</td><td>659</td><td>0.35</td><td>-2.7 ± 1.0</td><td>-0.3 ± 1.0</td><td>0.254</td><td>17 650</td></tr><tr><td>May 25</td><td>07:58:08</td><td>359.83588</td><td>0.8392</td><td>658</td><td>0.36</td><td>1.7 ± 1.5</td><td>0.6 ± 1.5</td><td>0.261</td><td>17 650</td></tr><tr><td>May 26</td><td>09:15:13</td><td>360.88939</td><td>0.9646</td><td>671</td><td>0.30</td><td>1.6 ± 1.3</td><td>-0.1 ± 1.3</td><td>0.251</td><td>17 650</td></tr><tr><td>May 26</td><td>09:25:56</td><td>360.89683</td><td>0.9655</td><td>675</td><td>0.30</td><td>-1.1 ± 0.8</td><td>0.8 ± 0.8</td><td>0.251</td><td>17 650</td></tr><tr><td>May 26</td><td>09:36:39</td><td>360.90428</td><td>0.9664</td><td>673</td><td>0.31</td><td>0.9 ± 0.6</td><td>-0.2 ± 0.6</td><td>0.256</td><td>17 650</td></tr><tr><td>May 26</td><td>09:47:22</td><td>360.91171</td><td>0.9673</td><td>678</td><td>0.30</td><td>-0.5 ± 0.5</td><td>0.1 ± 0.5</td><td>0.254</td><td>17 650</td></tr><tr><td>May 26</td><td>09:58:00</td><td>360.91916</td><td>0.9682</td><td>681</td><td>0.30</td><td>-0.6 ± 0.6</td><td>0.0 ± 0.6</td><td>0.255</td><td>17 650</td></tr><tr><td>May 26</td><td>10:08:48</td><td>360.92660</td><td>0.9691</td><td>682</td><td>0.30</td><td>-1.8 ± 1.4</td><td>-0.8 ± 1.4</td><td>0.254</td><td>17 650</td></tr><tr><td>May 26</td><td>10:20:33</td><td>360.93475</td><td>0.9700</td><td>684</td><td>0.30</td><td>-0.6 ± 0.5</td><td>0.0 ± 0.5</td><td>0.254</td><td>17 649</td></tr><tr><td>May 26</td><td>10:31:16</td><td>360.94220</td><td>0.9709</td><td>687</td><td>0.31</td><td>1.0 ± 1.0</td><td>-0.3 ± 1.0</td><td>0.251</td><td>17 650</td></tr><tr><td>May 26</td><td>10:41:59</td><td>360.94965</td><td>0.9718</td><td>687</td><td>0.31</td><td>2.8 ± 1.4</td><td>-1.3 ± 1.4</td><td>0.255</td><td>17 650</td></tr><tr><td>May 26</td><td>10:52:43</td><td>360.95709</td><td>0.9727</td><td>679</td><td>0.32</td><td>-0.8 ± 0.5</td><td>0.6 ± 0.5</td><td>0.253</td><td>17 650</td></tr><tr><td>May 26</td><td>11:03:26</td><td>360.91916</td><td>0.9736</td><td>675</td><td>0.31</td><td>-2.6 ± 1.5</td><td>-0.1 ± 1.5</td><td>0.255</td><td>17 650</td></tr><tr><td>May 26</td><td>11:14:09</td><td>360.97198</td><td>0.9745</td><td>681</td><td>0.32</td><td>-1.3 ± 0.5</td><td>-0.5 ± 0.5</td><td>0.252</td><td>17 649</td></tr><tr><td>May 27</td><td>05:29:08</td><td>361.73236</td><td>1.0650</td><td>710</td><td>0.36</td><td>-1.7 ± 1.4</td><td>-1.1 ± 1.4</td><td>0.242</td><td>17 649</td></tr><tr><td>May 27</td><td>05:39:52</td><td>361.73982</td><td>1.0659</td><td>713</td><td>0.35</td><td>-0.6 ± 0.5</td><td>-0.6 ± 0.5</td><td>0.254</td><td>17 648</td></tr><tr><td>May 27</td><td>05:50:36</td><td>361.74727</td><td>1.0668</td><td>718</td><td>0.36</td><td>-0.6 ± 0.5</td><td>-0.4 ± 0.5</td><td>0.265</td><td>17 648</td></tr><tr><td>May 27</td><td>06:01:19</td><td>361.75471</td><td>1.0677</td><td>715</td><td>0.36</td><td>1.8 ± 1.3</td><td>-0.5 ± 1.3</td><td>0.256</td><td>17 648</td></tr><tr><td>May 27</td><td>06:12:02</td><td>361.76216</td><td>1.0685</td><td>713</td><td>0.37</td><td>-2.0 ± 1.3</td><td>-0.9 ± 1.4</td><td>0.294</td><td>17 648</td></tr><tr><td>May 27</td><td>06:22:45</td><td>361.76960</td><td>1.0694</td><td>720</td><td>0.35</td><td>-1.5 ± 1.0</td><td>0.0 ± 1.0</td><td>0.266</td><td>17 648</td></tr><tr><td>May 27</td><td>06:34:11</td><td>361.77753</td><td>1.0704</td><td>719</td><td>0.35</td><td>-1.6 ± 1.3</td><td>-0.9 ± 1.3</td><td>0.273</td><td>17 648</td></tr><tr><td>May 27</td><td>06:44:54</td><td>361.78498</td><td>1.0713</td><td>472</td><td>0.35</td><td>-0.5 ± 1.5</td><td>1.2 ± 1.5</td><td>0.401</td><td>17 648</td></tr><tr><td>May 27</td><td>06:55:37</td><td>361.79242</td><td>1.0721</td><td>719</td><td>0.37</td><td>-0.7 ± 0.6</td><td>-0.6 ± 0.6</td><td>0.259</td><td>17 648</td></tr><tr><td>May 27</td><td>07:06:24</td><td>361.79991</td><td>1.0730</td><td>718</td><td>0.36</td><td>-0.5 ± 0.5</td><td>0.0 ± 0.5</td><td>0.258</td><td>17 648</td></tr><tr><td>May 27</td><td>07:17:07</td><td>361.80736</td><td>1.0739</td><td>722</td><td>0.36</td><td>1.3 ± 1.3</td><td>0.2 ± 1.2</td><td>0.245</td><td>17 648</td></tr><tr><td>May 27</td><td>07:27:51</td><td>361.81481</td><td>1.0748</td><td>724</td><td>0.36</td><td>-2.1 ± 0.9</td><td>1.4 ± 0.9</td><td>0.254</td><td>17 648</td></tr><tr><td>May 28</td><td>05:32:29</td><td>362.73466</td><td>1.1843</td><td>704</td><td>0.32</td><td>-1.6 ± 1.2</td><td>-0.9 ± 1.2</td><td>0.258</td><td>17 649</td></tr><tr><td>May 28</td><td>05:43:12</td><td>362.74212</td><td>1.1852</td><td>684</td><td>0.32</td><td>-5.0 ± 1.6</td><td>-1.3 ± 1.6</td><td>0.289</td><td>17 649</td></tr><tr><td>May 28</td><td>05:53:56</td><td>362.74956</td><td>1.1861</td><td>701</td><td>0.32</td><td>-1.0 ± 0.5</td><td>-0.9 ± 0.5</td><td>0.255</td><td>17 650</td></tr><tr><td>May 28</td><td>06:04:40</td><td>362.75701</td><td>1.1870</td><td>704</td><td>0.32</td><td>1.2 ± 0.8</td><td>-0.2 ± 0.8</td><td>0.256</td><td>17 650</td></tr><tr><td>May 28</td><td>06:15:23</td><td>362.76446</td><td>1.1879</td><td>434</td><td>0.31</td><td>2.2 ± 1.0</td><td>-2.5 ± 1.0</td><td>NA</td><td>17 646</td></tr><tr><td>May 28</td><td>06:26:06</td><td>362.77190</td><td>1.1887</td><td>712</td><td>0.33</td><td>-1.5 ± 0.6</td><td>-0.5 ± 0.6</td><td>0.259</td><td>17 650</td></tr><tr><td>May 28</td><td>06:37:28</td><td>362.77979</td><td>1.1897</td><td>709</td><td>0.33</td><td>-1.7 ± 0.6</td><td>-0.5 ± 0.6</td><td>0.256</td><td>17 650</td></tr><tr><td>May 28</td><td>06:48:14</td><td>362.78728</td><td>1.1906</td><td>709</td><td>0.33</td><td>-0.8 ± 0.5</td><td>-0.3 ± 0.5</td><td>0.259</td><td>17 649</td></tr><tr><td>May 28</td><td>06:58:58</td><td>362.79473</td><td>1.1915</td><td>708</td><td>0.33</td><td>-0.2 ± 0.5</td><td>-0.4 ± 0.5</td><td>0.260</td><td>17 649</td></tr><tr><td>May 28</td><td>07:09:42</td><td>362.80218</td><td>1.1924</td><td>704</td><td>0.33</td><td>-1.2 ± 0.5</td><td>-0.1 ± 0.5</td><td>0.246</td><td>17 649</td></tr><tr><td>May 28</td><td>07:20:30</td><td>362.80968</td><td>1.1932</td><td>698</td><td>0.32</td><td>-0.8 ± 0.5</td><td>-0.2 ± 0.5</td><td>0.250</td><td>17 650</td></tr><tr><td>May 28</td><td>07:31:13</td><td>362.81712</td><td>1.1941</td><td>694</td><td>0.32</td><td>0.2 ± 0.5</td><td>-0.3 ± 0.5</td><td>0.241</td><td>17 650</td></tr></tbody></table>

This paper has been typeset from a TEX/LATEX file prepared by the author.