The magnetic fields and stellar winds of the mature late F-stars: β Virginis and θ Draconis Free

ABSTRACT

1 INTRODUCTION

As part of our study of magnetism in F-type stars, we present surface magnetic maps for β Vir (F9V) and θ Dra, (F8IV) using Zeeman Doppler Imaging (ZDI), and provide a magnetohydrodynamic (MHD) model of their magnetospheres. ZDI (Semel 1989; Donati & Brown 1997; Donati et al. 2006b) is a powerful technique for studying stellar magnetism since it reconstructs both the location and orientation of surface magnetic features (Kochukhov 2016), and is suitable for both fast and slow rotators (Petit et al. 2008). This contrasts with brightness mapping using Doppler imaging (Deutsch 1958; Vogt & Penrod 1983; Vogt et al. 1987) which is best used for fast rotators with vsin i ≳ 20 km s⁻¹ (Vogt 1988; Rice 2002). ZDI allows the strength, complexity, and axisymmetric nature of the large-scale magnetic field to be determined, including its poloidal, and toroidal components.

The large-scale magnetic field may be described in terms of the poloidal and toroidal components (Elsasser 1946; Chandrasekhar 1961), as well as the axisymmetric/non-axisymmetric and dipole/quadrupole/octupole and higher multipoles (Deutsch 1955; Piskunov & Kochukhov 2002; Donati et al. 2006b). For axisymmetric fields, the poloidal magnetic fields lie in the meridional plane, while toroidal fields have field lines which circle the axis (Elsasser 1956). Both components are expected to be present in stars over the long-term in order to provide magnetic stability, since purely toroidal or purely poloidal fields have been shown to be unstable (Prendergast 1956; Tayler 1973; Wright 1973). Describing the surface magnetic fields in terms of poloidal and toroidal components has been used in dynamo equations (Parker 1955; Backus 1958; Roberts & Stix 1971) and is useful in explaining how stars regenerate, and sustain, a dynamo magnetic field via the αΩ-effect (Parker 1979; Krause & Rädler 1980; Choudhuri et al. 1995; Moffatt & Dormy 2019).

F-star magnetic fields exist in shallow outer convection zones and may have different dynamo properties to solar-type stars (Giampapa & Rosner 1984). The observable effects of magnetism in F-stars differs from later spectral types, and includes strong differential rotation (Barnes et al. 2005a; Reinhold & Reiners 2013; Reinhold & Gizon 2015), weaker X-ray emission (Stern, Schmitt & Kahabka 1995; Lisse et al. 2017; Freund et al. 2020), weaker magnetic fields (Marsden et al. 2014), shorter magnetic cycles (Lopes et al. 2015; Mittag et al. 2019), and inefficiency in producing toroidal magnetic fields (Kitchatinov & Olemskoy 2011). F-stars also span a range of masses where rapid rotational braking occurs (Schatzman 1962; Skumanich 1972; Guedel, Guinan & Skinner 1997; Vidotto et al. 2014c) due to wind-driven angular momentum loss as a consequence of the onset of dynamo magnetic fields (Krause & Rädler 1980; Brandenburg & Subramanian 2005; Moffatt & Dormy 2019) in the outer convection zone. This magnetic braking results in a decrease in rotation from early to late F-stars (Gray 1988; Nielsen et al. 2013) coinciding with an increase in depth of the outer convection zone (Mullan 1972). The properties of stellar winds and the link with magnetic activity is poorly understood in cool stars (Johnstone et al. 2015b), therefore modelling of the stellar magnetosphere (Lammer & Khodachenko 2014; Alvarado-Gómez et al. 2020) is important for understanding how wind properties evolve on the main sequence, and has implications for the habitability of exoplanets (See et al. 2014; Johnstone et al. 2015a). Due to lifespans as short as one billion years, F-stars are considered the hottest stars which allow the possibility of exobiology to develop (Sato et al. 2014).

Relatively little information has been published on the nature of the vector magnetic field in F-stars. Mengel (2005) studied the young, fast rotating F8V star HR 1817 and found a weak complex radial magnetic field with a large rotational shear, which was different from that seen in G and K solar-type stars. The moderately rotating, and hot Jupiter hosting star, τ Boo (F7V) has been studied by several authors with the first magnetic map produced by Catala et al. (2007) who found a predominately poloidal field more complex than that of the Sun, and a high differential rotation rate. Mengel et al. (2016) found a very rapid 120 d chromospheric cycle that was broadly solar-type in contrast to other F-type stars. A chromospheric cycle of 120 d and magnetic cycle of 240 d was found by Jeffers et al. (2018) with the polarity flip of the large-scale magnetic field in phase with the chromospheric cycle. Morgenthaler et al. (2011) found a fast magnetic cycle of ∼3 yr on the slow rotating F9V star HD 78366 and a strong polar radial field and equatorial rotation period of 11.4 d. Fares (2013) presented a magnetic map for the moderately rotating star HD 179949 (F8V) which showed 90 per cent of the magnetic energy in the poloidal component, mainly radial. Waite et al. (2015) showed the young moderately rotating F8V star (HD 35296) contained a complex large-scale magnetic field with significant toroidal component and high level of differential rotation. A short magnetic cycle of 1.06 yr for the F7V star HD75332 was found by Brown et al. (2021), but only a single magnetic field reversal was detected from 12 magnetic map reconstructions over a 12-yr observation period. The authors speculate they may have missed the majority of polarity reversals due to the low frequency of observations. Further studies are needed for F-stars because exactly how the magnetic field topology changes across this spectral class remains to be determined.

In our previous paper (Seach et al. 2020), hereafter called paper I, we determined the longitudinal magnetic field of β Vir (+1.7 ± 0.5 G) and θ Dra (+7.4 ± 2.9 G); however, we were not able to determine if the field was a dynamo, or weak fossil field, from a single night’s observation. In this paper, we use ZDI to reconstruct the surface magnetic field of β Vir and θ Dra, and determine equatorial rotation period, differential rotation, and inclination of the rotational axis. We also present a MHD model of their magnetospheres. Maps of the large-scale magnetic field of F-type stars are important for understanding magnetism and convection in stars with thin outer convection zones, and provide input for models of the coronal magnetic field, and stellar winds. Since F-stars span a range of stellar masses where the transition from fossil to dynamo fields is expected to occur (Durney & Latour 1977; Seach et al. 2020), mapping the large-scale magnetic field will offer insight into the magnetic topology of stars with shallow outer convection zones, and constrain dynamo models.

2 OBSERVATIONS AND DATA REDUCTION

2.1 Target selection

β Vir and θ Dra were identified as magnetic stars in our earlier magnetic snapshot survey of F-type stars (paper I) and the original observations contained a sufficient number of spectra to allow this follow up paper reconstructing the large-scale surface magnetic field using ZDI (Section 3.2). Data were obtained as part of the BRITEpol survey (Neiner et al. 2017), which is a systematic spectropolarimetric survey of stars brighter than magnitude V = 4, which compliments the optical two-colour photometry collected by the BRITE-Constellation Nanosatellite Space Mission (Handler et al. 2017).

β Vir (HIP 57757) is a bright (V = 3.60) F9V star (Morgan & Keenan 1973), with an age of 3.1 ± 0.8 Gyr (Casagrande et al. 2011), metal rich ([Fe/H] = 0.24) (Jofré et al. 2018), and T_eff 6083 ± 58 K (Jofré et al. 2018). It is a slow rotator with vsin i = 6.1 ± 2.6 km s⁻¹ (Glebocki & Gnacinski 2005). β Vir is close to the end of its hydrogen burning phase, or just evolved off the main sequence (Eggenberger & Carrier 2006; North et al. 2009), and is located in a region of the HR diagram where the evolutionary tracks form a loop (Heiter et al. 2015). While β Vir has appeared in planet search programs (Wittenmyer et al. 2006) no exoplanet companion has been found. β Vir is a frequently studied star in the solar neighbourhood (Gehren 1978); however, the nature of its magnetic field is not well known.

θ Dra (HIP 78527) is a bright (V = 4.00), spectroscopic binary consisting of an F8IV primary (Pounds et al. 1991) and a late type secondary (Duquennoy & Mayor 1991; Mazeh et al. 2002) with a mass of 0.62 M_⊙ (Tokovinin 2014) placing it later than spectral type K2. A possible third component was suggested by Mayor & Mazeh (1987) through analysis of radial velocity variations. θ Dra rotates unusually fast for a late F-type star with vsin i = 28.1 ± 0.1 km s⁻¹ (Glebocki & Gnacinski 2005). It is a mature F-star with age = 2.1 ± 0.2 Gyr (Casagrande et al. 2011), based on Padova isochrones (Bertelli et al. 2008, 2009), a T_eff = 6392 ± 63 K (Ramírez et al. 2012), and has been identified as an X-ray source by the Einstein observatory (Maggio et al. 1987).

2.2 Observations

Observations of β Vir and θ Dra were obtained between 2017 February 14 and June 18 using the NARVAL spectropolarimeter at the Telescope Bernard Lyot (TBL) at the Observatoire du Pic du Midi, France (Aurière 2003). NARVAL consists of a cross-dispersed, bench mounted, echelle spectrograph, with a wavelength coverage in a single exposure of 370–1050 nm with small gaps at 922.4–923.4, 960.8–963.6, and 1002.6–1007.4 nm. The full spectrum spans 40 grating orders from #22 in the red to #61 in the blue, with a resolving power of ≈68 000 when using the spectropolarimetric mode. The polarimeter consists of a quarter-wave and two half-wave Fresnel rhombs coupled to a Wollaston prism, which is composed of two perpendicular calcite prisms which produce achromatic polarization (Donati et al. 2006a). One stellar observation consists of four sub-exposures which are combined to produce the Stokes I, combined constructively to produce the V spectrum, and combined destructively to give the null spectrum where the Zeeman signature cancels out, and only non-magnetic effects remains in the profile (Semel, Donati & Rees 1993).

We obtained 47 polarized spectra of β Vir over 108 d between 2017 February 14 and April 24 spanning 7.5 rotation cycles. We obtained 105 polarized spectra of θ Dra over 73 d between 2017 April 06 and June 18, spanning 25.3 rotation cycles. On many nights we obtained multiple spectra for θ Dra, therefore individual LSD profiles were combined to produce nightly averages with higher SNR which were used as inputs for the ZDI code. The averaged LSD profiles were obtained over a time period covering less than 2 per cent of the stellar rotational period, and was considered a small enough time frame to avoid rotational smearing of the results. Using mean nightly LSD profiles did not change the magnetic map compared to using individual LSD profiles. A journal of observations for β Vir are shown in Table 1. A journal of nightly averaged observations for θ dra are shown in Table 2, and the full 105 observations are shown in Table A2.

Figure A1.

Stokes I maximum entropy fit for θ Dra. The black circles represent the observed signatures, while the red lines represent the modelled lines. The radial velocity range of the individual line profiles are shifted to the same range, and line profiles are separated vertically to allow better visibility. The 1.0σ error bars are shown on the left. The rotation cycle is shown on the right with the zero rotation cycle set to the mid JD of observations for each map and assumes P_rot = 9.3 d for β Vir and P_rot = 2.88 d for θ Dra. All figures are set to the same scale.

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Figure A2.

Stokes V maximum entropy fit for β Vir and θ Dra. The black lines represent the observed Zeeman signatures, while the red lines represent the modelled lines. The radial velocity range of the individual line profiles are shifted to the same range, and line profiles are separated vertically to allow better visibility. The 1.0σ error bars are shown on the left. The rotation cycle is shown on the right with the zero rotation cycle set to the mid JD of observations for each map and assumes P_rot = 9.3 d for β Vir and P_rot = 2.88 d for θ Dra. All figures are set to the same scale.

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2.3 Spectropolarimetric analysis

We reconstruct surface magnetic maps using circularly polarized spectra (Stokes V) and enhance the SNR by the line addition technique of least squares deconvolution (LSD; Donati et al. 1997; Kochukhov, Makaganiuk & Piskunov 2010). LSD analysis combines thousands of individual spectral lines into a single line with mean parameters and is useful in boosting the SNR in our F-stars that are expected to show weak magnetic fields (Marsden et al. 2014). Each observation consists of 4 x 223 s sub-exposures for β Vir and 4 x 320 s sub-exposures for θ Dra. The individual exposures are combined to produce an intensity profile (Stokes I), combined constructively to produce a circularly polarized profile (Stokes V), and combined destructively to give a null spectrum. The null spectrum acts as a check to ensure that spurious signals have not contaminated the spectrum. Examples of LSD profiles for β Vir and θ Dra are shown in Fig. 1. See paper I for more details on our spectropolarimetry and extraction of spectra.

Figure 1.

Example of LSD profiles (single spectra) for β Vir on 03 April 2017 (top), and θ Dra on 07 April 2017 (bottom). The upper line (red) shows the Stokes V profile (y-axis expanded by 1000 times and shifted up by 0.2 to allow better visibility). The middle line (yellow) shows the null profile (expanded by 1000 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.

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We determine the mean longitudinal magnetic field (B_l) for individual Stokes V and Null spectra as described in paper I. B_l is the line-of-sight magnetic field component averaged over the stellar disc (Babcock 1947; Schwarzschild 1950). The local magnetic field strength is larger than B_l that suffers from cancellation effects where regions of opposite polarity cancel out, resulting in as little as 5 per cent of the field being detected (Reiners & Basri 2009). Nevertheless, B_l has been a widely used measure of stellar magnetic field strength for many decades due to its relative ease of measurement and because it is measured from the strongest of the Stokes parameters (Stokes V). We use an additional method to determine the presence of a stellar magnetic field, the false alarm probability (FAP), which is a statistical method to determine if a magnetic signature is present in the Stokes V LSD profile or a spurious signal in the Null profile (Donati et al. 1992, 1997).

2.4 Radial velocity

The radial velocity (RV) is a basic stellar parameter (Lindegren & Dravins 2003) and is determined from the Stokes I profile for β Vir and θ Dra using the centre of gravity method (Uitenbroek 2003). The centre of gravity method finds the centroid of the area under an intensity line profile (Rees & Semel 1979) and is considered more accurate than applying a Gaussian fit to the intensity profile, especially for faster rotating stars (Dravins 1975). We use the RV as an input into the ZDI code (Section 3.2) and our determined values are shown in Tables 1, 2, and A2.

3 LARGE-SCALE MAGNETIC FIELD TOPOLOGY

3.1 Brightness map

Doppler Imaging (DI) is a tomographical technique used for mapping bright and dark regions on the surface of a star associated with magnetic activity (Vogt & Penrod 1983; Vogt et al. 1987; Rice et al. 1989; Piskunov & Kochukhov 2002; Rice 2002). We use DI to produce a brightness map for θ Dra to identify regions of enhanced magnetic activity. We are unable to create a brightness map for β Vir, since it rotates too slowly to enable DI to be used. The input parameters for DI are the same as derived for ZDI described in (Section 3.2). The DI code we use (zdipy) is the same as described in Folsom et al. (2018), which inverts the Stokes I LSD profiles using the maximum entropy fitting routine of Skilling & Bryan (1984) to produce a regularized fit to the data which maximizes entropy and minimizes χ². The output of DI produces a map of the stellar surface which we set to show both bright and dark areas compared to the quiet photosphere corresponding to starspots and plages. The rotation cycle for each LSD profile is determined from ZDI using the zero rotation phase as the first observation date for each map. The Stokes I maximum entropy fits for θ Dra from DI is shown in Fig. A1, where we plot the observed signatures and modelled lines. The plot allows a visualization of the agreement between observations and model prediction.

3.2 Magnetic maps

We use ZDI to reconstruct the large-scale magnetic field topology of β Vir and θ Dra using the same ZDI code described above for DI, except for magnetic mapping we use a time-series of Stokes V profiles rather than Stokes I profiles. While there are a large number of possible solutions to the inverse problem of reconstructing a magnetic map from Stokes V LSD profiles, a unique map is achieved by imposing a number of criteria. One criteria is to minimize the sum of χ² of the model fit of the observations. Secondly, we choose the maximum entropy method (Jaynes 1957a, b) that maximizes the Shannon entropy of Hobson & Lasenby (1998), derived from Skilling & Bryan (1984). The Stokes V data are used to reconstruct the magnetic maps of the radial azimuthal and meridional fields (Piskunov & Kochukhov 2002), using the spherical harmonics expansion of the surface magnetic field (Donati et al. 2006b) that allows the strength of the various field modes determined. The magnetic maps are represented using spherical harmonics. Each map comprises an orthogonal sum of components representing simple oscillations on the sphere. The expansion is parametrized by the degree ℓ, ranging from 1 to ℓ_max, and the order m, ranging from 0 to ℓ. The degree (ℓ) governs the total number of oscillations over the sphere for the spherical harmonics expansion (Ismail-Zadeh & Tackley 2010), so that higher values of ℓ correspond to smaller-scale features on the reconstructed magnetic field. ℓ_max is the maximum degree of spherical harmonics expansion used in the magnetic map reconstruction and ℓ_max thus determines the resolution of the magnetogram (Gerth & Yu 2004); however, the resolution is limited in low vsin i stars (Morin et al. 2010). The order (m) can be considered the number of zeros of the spherical harmonic component in the longitudinal direction (Ismail-Zadeh & Tackley 2010). It also determines if the field is axisymmetric (m = 0) or non-axisymmetric (m > 0) (Saikia et al. 2016).

The quantities derived from ZDI are the mean unsigned magnetic field (B_mean), maximal field strength over stellar surface (B_max), the fraction of the large-scale magnetic energy reconstructed in the poloidal and toroidal field components, the fraction of the magnetic energy in the dipolar (ℓ = 1), quadrupolar (ℓ = 2), octopolar (ℓ = 3), and higher (ℓ > 3) components. The fraction of the magnetic energy in the axisymmetric field component (m = 0), as well as the axisymmetry of the poloidal and toroidal fields are determined which gives an indication of the degree of alignment of the magnetic field with the rotation axis (Lehmann et al. 2019). We calculate the maximum degree of spherical harmonic expansion (ℓ_max) by running ZDI over a range of ℓ values and find the value at the point where no more energy is contained in the higher multipoles. For β Vir, we use ℓ_max = 10, and for θ Dra ℓ_max = 20. For θ Dra, we use the brightness map as an input into the ZDI code to account for suppression of the Zeeman signal in dark spots (Gregory et al. 2010; Johnstone et al. 2010; Morgenthaler et al. 2012; Kochukhov et al. 2017).

Using the stellar parameters estimated by the method above, we then determine optimal values and uncertainties for vsin i, inclination, P_rot, and dΩ using the entropy landscape method (Petit et al. 2002). For each stellar parameter, we use ZDI to perform a search across a range of stellar parameter values at a fixed χ² of 1.0, and find the map with the corresponding maximum entropy. This entropy value is then used as an input into a further search using a fixed entropy and variable χ², producing an output of χ² versus stellar parameter. We then fit a Gaussian curve to the results and choose the optimal value as the minimum of the Gaussian curve, and the formal 1σ uncertainties are found from the Gaussian curve χ² statistics. The optimal stellar parameter derived from both the fixed χ² search and fixed entropy search are similar, which adds to the robustness of our results. The values for P_rot and dΩ are further estimated from a grid search described in more detail in Section 3.3. The vsin i for β Vir was taken from Glebocki & Gnacinski (2005), since χ² minimization did not produce a well-defined minimum.

Limitations of ZDI relating to missing information due to axial tilt and only reconstructing the large-scale magnetic field are disused in more detail elsewhere; see for example Gregory et al. (2010), Johnstone et al. (2010), Morgenthaler et al. (2012), and Kochukhov et al. (2017). Despite some limitations, ZDI remains a powerful technique for reconstructing the surface magnetic field in both cool and hot stars (Kochukhov 2016).

3.3 Differential rotation

Differential rotation (dΩ) is one of the key ingredients for creating a large-scale stellar dynamo, whereby non-uniform rotation generates a toroidal field from a poloidal field and helical turbulence recreates the poloidal field (Parker 1955; Steenbeck & Krause 1969; Gilman 1980). A detailed analysis of dΩ is essential for understanding the magnetic activity of the stars (Donahue et al. 1996). While dΩ can be directly measured on the surface of the Sun through sunspot tracing (Kambry & Nishikawa 1990; Beck 2000), different methods must be used to determine stellar surface dΩ due to the inability to directly resolve the stellar surface.

We also provide an estimate of uncertainty in dΩ by producing a variation ellipse which encompasses all the values and 1σ uncertainties generated by individually varying stellar parameters and determining the corresponding dΩ and Ω_eq pairs. The parameters varied for β Vir are vsin i (± 2.6 km s⁻¹), inclination angle (± 2 deg), and χ² aim (± 0.05), for θ Dra vsin i (± 0.2 km s⁻¹), inclination angle (± 2 deg), and χ² aim (± 0.05). The individual error bars indicate the 1σ errors of a Gaussian fit to the χ² landscape.

4 CHROMOSPHERIC ACTIVITY INDICATORS

4.1 S-Index

We investigate presence of chromospheric cycles by applying a generalized Lomb-Scargle (GLS) periodogram (Zechmeister & Kürster 2009) to a time series of our S-index measurements for β Vir and θ Dra. The results are plotted as power versus frequency where the height of the peak is a measure of the goodness of fit of a sine curve to the data. We also calculate the FAP (Baluev 2008; Zechmeister & Kürster 2009) that quantifies the probability of the maximum peak being produced by a signal without a periodic component (VanderPlas 2018). The most likely period is chosen as the peak with the smallest FAP (Baliunas et al. 1995), although it is possible for more than one significant peak to be present which may indicate multiple magnetic cycles (VanderPlas 2018). A detection is considered excellent when FAP is ≤10⁻⁹, good 10⁻⁹ to ≤10⁻⁵, fair 10⁻⁵ to ≤10⁻², and poor 10⁻² to ≤10⁻¹ (Baliunas et al. 1995). The GLS periodogram takes into account measurement errors by introducing weighted sums (Gilliland & Baliunas 1987; Irwin et al. 1989), and a floating mean (Cumming et al. 1999), which allows for statistical fluctuation in the mean. The GLS periodogram is considered more suitable for small numbers of observations, with uneven sampling, and cases where the period is greater than the length of observations (Zechmeister & Kürster 2009). Discussion of our S-index and magnetic cycle determinations are found in (Section 6.5).

4.2 H α Index

5 STELLAR WIND MODELLING

The stellar wind models for β Vir and θ Dra are created using the Alfvén Wave Solar Model (Sokolov et al. 2013; van der Holst et al. 2014) of the Space Weather Modelling Framework (Tóth et al. 2005, 2012) to solve the two-temperature MHD equations along with equation describing the Alfvén wave propagation, reflection, and dissipation along magnetic field lines. We model the wind in two partially overlapping three-dimensional regions. The inner region uses a spherical grid that is irregular in the radial direction; far away from the star the radial spacing is logarithmic, while more radial grid points are added near the star using the methodology of Oran et al. (2013). The outer region uses a Cartesian grid. The wind model covers the regions between the chromosphere and the inner astrosphere, and includes a physical model of the transition region in which the wind is heated to coronal temperatures by Alfvén wave energy emanating from deeper stellar layers, resulting in a Poynting flux |$\Pi _{\small {A}}\propto |\boldsymbol{\mathbf {B}}|$| at the inner model boundary. Other heat exchange and cooling terms are also included as they are necessary (Roussev et al. 2003) to reproduce the slow-fast wind bimodality. The radiative cooling is given by Q_rad = N_iN_eΛ(T_e) where the rate of cooling curve Λ(T_e) is calculated using the CHIANTI data base (Landi et al. 2013) and Solar elemental abundances.

The radial component of the boundary magnetic field is fixed to the local magnetogram value in Figs 2 and  3. The surface parallel components of the magnetic field varies freely as the MHD solution evolves towards a steady state. The outgoing Alfvén wave energy density at the inner boundary is |$(\Pi _{\small {A}}/B)\sqrt{\mu _0 \rho }$|⁠. The model and boundary conditions are similar to the ones in Evensberget et al. (2021); i.e. Solar values are used for the chromospheric base temperature and density, the Poynting flux-to-field ratio, the turbulence transverse correlation length and the Coloumb logarithm (see Table 3). The last three of these parameters control the supply of Alfvén wave energy, the Alfvén wave dissipation rate, and the rate of ion-electron heat exchange respectively. These values have been chosen to reproduce observed Solar conditions, see Sokolov et al. (2013), van der Holst et al. (2014). The simulation is stepped forward until a steady state is reached, in which the magnetic and hydrodynamic forces are in balance.

Figure 2.

Reconstructed magnetic maps using ZDI for β Vir (F9V) at three epochs in 2017. Figure on left uses observations from February 14 to 22. Middle: March 1–29. Right: April 3–24. The rotational phases of the three maps are aligned with zero rotation phase set to MJD = 2457799.62139 (February map), MJD = 2457827.55489 (March map), and MJD = 2457854.5174 (April map). Tick marks at bottom of figure represent phase of individual observations. The colour bar at bottom indicates the magnetic field strength in Gauss.

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Figure 3.

Reconstructed magnetic maps using ZDI for θ Dra at two epochs in 2017. The map at left spans observations from April 6 to 24. The map at right spans observations from 15 May to 18 June. The rotational phases of the two maps are aligned with the zero phase set to MJD = 2457861.4817 (April map) and MJD = 2457910.4710 (May–June map). Tick marks at bottom of figure represent phase of individual observations. The colour bar at bottom indicates the magnetic field strength in Gauss.

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6 RESULTS

6.1 Magnetic maps for β Vir

The large-scale magnetic field topology for β Vir at three epochs derived using ZDI is shown in Fig. 2. The stellar parameters used to construct the map are vsin i = 6.1 km s⁻¹, inclination = 36 deg, and dΩ = 0.19 rad d⁻¹. The spherical harmonics expansion is limited to ℓ_max = 10, since no further magnetic energy in higher multipoles was found by increasing ℓ further, and the chosen value of χ² for β Vir and θ Dra is 0.95. The data sets comprise 9 LSD profiles between 2017 February 14–22, 8 LSD profiles between 2017 March 1–29, and 30 LSD profiles between 2017 April 3–24. We divide observations into three subsets in order to reduce the effects of magnetic field evolution on the final maps, while providing adequate phase coverage to produce a reliable magnetic map. The April map uses more LSD profiles due to the more intensive sampling for that epoch and splitting the data into two maps using a smaller number of LSD profiles does not provide any significant changes in the resulting maps with increased magnetic field complexity still visible in the April β Vir map.

The radial field for β Vir shows a relatively simple structure in February and March 2017, while slightly stronger and more complex in April 2017. The increase in magnetic field strength and complexity of the April 2017 map is not an artefact of the larger number of LSD profiles used in the magnetic map reconstruction but appears to be a real change in the surface magnetic field. We verified this by creating a magnetic map using a subset of 8 LSD profiles, and showed that the April map contained the same maximum and mean magnetic field as the reconstruction using 30 LSD profiles. The simple radial field topology may partly be due to a low vsin i of 6.1 km s⁻¹ where the resolution of a map produced by ZDI is dependent on the number of resolved surface elements (Morin et al. 2010). At all three epochs the radial magnetic field is very dipolar, with a single region of positive field in the Northern hemisphere and negative field in the Southern hemisphere. The magnetic field remains very dipolar despite the moderate differential rotation, a feature which was also seen in the F7V star τ Boo (Donati et al. 2008). The azimuthal field is predominantly negative at all epochs with only a small positive region visible near the north pole. The meridional field is mostly positive with a small region of negative field near the north pole.

The complexity of the magnetic field for β Vir is quantified by calculating the fractional strengths of the magnetic field components (Table 5). Most of the magnetic energy for β Vir is stored in the poloidal field (82–98 per cent) while between 2 and 18 per cent of the magnetic energy is toroidal. The high percentage of poloidal field in β Vir is consistent with the finding of Petit et al. (2008) and Jouve, Brown & Brun (2010) who find stars with rotation periods slower than ∼12 d contains predominantly poloidal magnetic fields. The dominant term in the spherical harmonics expansion for β Vir in the three magnetic maps is the dipole (ℓ = 1) with 73–91 per cent of the magnetic energy. The second largest spherical harmonics term is the quadrupole (ℓ = 2) with 8–18 per cent of the magnetic energy. The octupolar and higher components contain less than 7 per cent of the magnetic energy. The poloidal field is predominantly axisymmetric (80–97 per cent) while the toroidal field is also mainly axisymmetric (85–98 per cent). The uncertainties in magnetic quantities 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. The parameter ranges used are inclination (± 2 deg), vsin i (± 2.6 km s⁻¹), P_rot (± 0.4 d), dΩ (± 0.09 rad d⁻¹), and target χ² (± 0.05) (see Table 5).

6.2 Brightness maps for θ Dra

The reconstructed brightness maps for θ Dra are shown in Fig. 4. The maps are comprised of 14 LSD profiles between 6–24 April 2017 and 8 LSD profiles between 15 May and 18 June 2017. The fast rotation of θ Dra (vsin i = 28.1 km s⁻¹) allows us to use Doppler Imaging to produce brightness maps for two epochs. The brightness maps show both bright and dark areas on the stellar surface compared to the average photospheric brightness at the same latitude (Petit et al. 2017). The dark areas correspond to cooler regions on the surface of the star which are likely associated with starspots (Berdyugina 2005; Strassmeier 2009) and are regions of enhanced magnetic activity which suppresses convection, creating cooler, darker regions on the stellar surface (Chugainov 1966; Vogt 1975, 1983). The brighter regions are thought to correspond to plages which are formed in the upper layers of the stellar atmosphere and are regions of enhanced magnetic activity (Linsky 1983).

Figure 4.

Brightness map for θ Dra at two epochs in 2017. Observations of the top Fig span from 2017 April 6–24. Bottom: Observations span from 2017 May 15 to June 18. The rotational phases of the two maps are aligned with the zero phase set to MJD = 2457861.4817 (April map) and MJD = 2457910.4710 (May–June map). The colour bar at bottom indicates the pixel brightness on a linear scale, where a value of 1.0 represents the quiet photosphere, values less than one are cool spots and values greater than one are bright spots.

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The April 6–24 map for θ Dra shows the presence of both lighter and darker areas that are smaller and more numerous compared with the 15 May–18 June map. This may be due to the larger number of profiles used in constructing the April 6–24 map, which results in better resolution of the surface features. We see a large dark patch at the north polar region on both maps. Similar dark polar regions have been found on other F-stars; see for example one component of σ2 CrB, F6V (Strassmeier & Rice 2003), AF Lep, F8V (Marsden et al. 2006a), and τ Boo, F7V (Fares et al. 2009). The presence of a polar spot on θ Dra and other rapid rotators has been described by Schüssler & Solanki (1992) as being the result of a dominance of Coriolis force over buoyancy in the dynamics of magnetic flux tubes. This results in rising flux tubes following a path nearly parallel to the axis of rotation and erupting at higher latitudes. Our brightness maps for θ Dra are used an input for the ZDI reconstruction of the magnetic maps.

6.3 Magnetic maps for θ Dra

The reconstructed magnetic maps for θ Dra are shown in Fig. 3. The maps are comprised of 14 LSD profiles between 2017 April 6 and 24 and 8 LSD profiles between 2017 May 15 and June 18. The stellar parameters used to construct the map are vsin i = 28.1 km s⁻¹, inclination = 37 deg, and dΩ = 0.18 rad d⁻¹. The spherical harmonic expansion is limited to ℓ_max = 20, since no further detail is obtained with higher degrees. The large polar spot on the brightness maps (Fig. 4) corresponds to regions of enhanced radial field in the magnetic maps (Fig. 3). The magnetic map for θ Dra in 2017 April has a stronger average magnetic field (2.5 ± 0.2 G) and larger S-index (0.2085 ± 0.0019) compared to 2017 May–June map (B_mean = 1.9 ± 0.2 G, S-index = 0.2029 ± 0.0023) (Fig. 3 and Table 4) which also corresponds to the brightness map with the more complex structure. These observations are consistent with θ Dra being more magnetically active in 2017 April compared with 2017 May and June, which indicates θ Dra contains a dynamo magnetic field due to its changing strength over time.

At both epochs the magnetic maps for θ Dra show a strong region of positive radial field at the north polar region, while the mid-northern latitudes shown a mix of positive and negative regions. The azimuthal field is predominantly negative at both epochs with only a small positive regions near the north pole, and a larger region of positive field at mid northern latitudes in April 2017. The meridional field at both epochs contain a mix of positive and negative areas. The complexity of the magnetic field for θ Dra as described by fraction of the large-scale magnetic energy reconstructed in various components is shown in Table 5. The largest terms in the spherical harmonics expansion for θ Dra are dipole (ℓ = 1) with 19–20 per cent of the magnetic energy. The second largest spherical harmonics term is the quadrupole (ℓ = 2) with 14–18 per cent of the magnetic energy, followed by octupolar (ℓ = 3) with 11–16 per cent of magnetic energy. θ Dra has a large amount of magnetic energy stored in the higher multipoles (ℓ > 3) with values ranging from 47 to 55 per cent, which is consistent with a complex dynamo magnetic field.

The amount of energy contained in higher multipoles for θ Dra compared to β Vir is most likely determined by its faster rotation and therefore the number of resolved surface elements (Morin et al. 2010). An increase in surface resolution results in an ability to reconstruct smaller scale magnetic features and therefore higher multipoles. The uncertainties in magnetic quantities 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. The parameter ranges used are inclination (± 2 deg), vsin i (± 0.2 km s⁻¹), P_rot (± 0.02 d), dΩ (± 0.02 rad d⁻¹), and target χ² (± 0.05) (see Table 5).

6.4 Differential rotation

Figure 5.

Figures at left: Surface differential rotation contour plot for β Vir 2017 April 3–24 (top) and θ Dra 6–2017 April 24 (bottom) with 1σ, and 2σ confidence contours marked with black lines. From the χ² landscape we determine the rotation period at the 1σ level of β Vir as 9.1 ± 0.4 d and θ Dra 2.88 ± 0.02. The dΩ for β Vir is 0.23 ± 0.09 rad d⁻¹ and θ Dra 0.18 ± 0.03 rad d⁻¹. Figures at right: Surface differential rotation using χ² minimization landscape of Stokes V profiles for β Vir (top) and θ Dra (bottom). The variation ellipse is generated by individually varying stellar parameters and determining the corresponding Ω_eq and dΩ pairs. The parameters varied for β Vir are vsin i (± 2.6 km s⁻¹), inclination angle (± 2 deg), and χ² aim (± 0.05), for θ Dra vsin i (± 0.2 km s⁻¹), inclination angle (± 2 deg), and χ² aim (± 0.05). The individual error bars indicate the 1σ errors of a Gaussian fit to the χ² landscape.

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6.5 Chromospheric emission

6.5.1 S-index

The S-index determinations for β Vir over the 83 d observation period are shown in Table 1. The S-index varies from 0.1593 to 0.1658 that indicates the star is only showing weak chromospheric activity. The S-index range during our observation period is within the range of values for β Vir published in the literature of 0.138–0.186 (Pace 2013). A plot of the S-index versus rotational phase for β Vir (Fig. 6) shows a possible difference between the three epochs of observations, and the map with the highest S-index (April 3–24) corresponds to the map with the largest mean magnetic field strength (Fig. 7). The S-index determinations for θ Dra are shown as nightly means in Table 2, while the full 105 observations are shown in Table A2. The S-index variation over the observation period of 73 d is from 0.1990 to 0.2144 which indicates the star is showing strong chromospheric activity. Our S-index values for θ Dra are within the published range of 0.164–0.271 by Pace (2013) and indicates the star displays a large range of chromospheric activity. A plot of the S-index versus rotational phase for θ Dra (Fig. 6) shows a difference between the two epochs of observations, and the map with the highest S-index (April 6–24) corresponds with the largest mean magnetic field strength (Fig. 7). The higher S-index for θ Dra compared to β Vir is consistent with observations (Pallavicini et al. 1981) that a faster rotating star is more likely to produce stronger magnetic activity.

Figure 6.

Top: S-index versus rotational phase for β Vir (top). Blue circle indicates data from 2017 February 14 to 22. Red squares indicates data from 2017 March 1 to 29 . Yellow triangle indicates data from 2017 April 3 to 24. The data hint at a possible difference between the three epochs; however, there is no significant difference between the mean S-index of each epoch (see Fig. 7). Bottom: Nightly averaged S-index versus rotational phase for θ Dra. Blue circles indicate data from 2017 April 6 to 24. Pink squares represent data for 2017 May 15 to June 18. The map with the strongest mean magnetic field (see Fig. 7) has a significantly higher S-index.

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Figure 7.

Mean S-index versus mean magnetic field strength (from ZDI) for β Vir (top) and θ Dra (bottom). There is no significant difference between the mean S-index of the 3 maps for β Vir, while there is a significantly larger mean magnetic field strength for the April map. For θ Dra there is a significantly larger mean S-index for the April map that coincides with stronger magnetic field strength (Table 5).

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6.5.2 Chromospheric activity cycles

Results of our GLS periodogram search for chromospheric activity cycles for β Vir and θ Dra are shown in Table 3 and Fig. 8. Since nightly observations may lead to a window function which can create a signal in the periodogram (VanderPlas 2018), we need to look for its presence in our data. We calculate a window function for both stars by applying a Lomb–Scargle periodogram to the data while setting the S-index values to unity. For β Vir, we find a window function with a period of 688 d. Since the window function is much larger than the observation baseline, it does not have a significant effect on the chromospheric cycle period determination for β Vir. For θ Dra, we find a window function with a period of 729 d which is a possibly an harmonic of the highest peak found in the power versus frequency plot that shows the highest peak corresponding to a period of 183 d. We can possibly discount this peak as being an artefact of the nightly observations; however, in Fig. 8, we plot a sine curve with a period of 183 d since this appears to fit the S-index data better than a period of 43 d. On the power versus frequency plots there are no peaks coinciding with the P_rot for β Vir of 9.2 d, (f = 0.109), and 2.88 d (f = 0.347) for θ Dra; therefore, we did not remove any rotational signal from the periodograms.

Figure 8.

Chromospheric cycle determinations for β Vir 2017 (top) and θ Dra 2017 (bottom) applying a GLS periodogram a time-series of S-index measurements. The figures at left show the power versus frequency and FAP, with the highest peaks marked with a red circle, and text describes the associated period and frequency. The dashed horizontal line represents a FAP of <10⁻³ for β Vir and <10⁻⁹ for θ Dra. The figures at right show individual S-index measurements shown with blue circles. The red dashed curves indicate the fitted cycle using a GLS periodogram, and assumes a strictly periodic sine curve with a period of 83 d for β Vir. For θ Dra both 183 d and 43 d sine curves are plotted. The figures at right have the zero day set to the first date of S-index measurements for each star.

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We estimate a chromospheric cycle of 83 ± 14 d for β Vir by applying a GLS periodogram to 49 S-index values over a period of 68 d from 06 January 2017 to 24 April 2017. A FAP of <10⁻³ indicates the detection is considered fair. For θ Dra, we apply a GLS periodogram to 105 S-index values over a period of 73 d from 2017 April 06 to 2017 June 18. We see two peaks with a FAP ≤ 10⁻⁹. The peak corresponding to a cycle length of 43 ± 2 d is chosen as the possible chromospheric cycle length since the highest peak is discounted as an observational artefact as described above. Since we have a short baseline of observation of S-index for β Vir of 69 d (0.8 chromospheric cycles), and for θ Dra 73 d, which is 0.4 or 1.7 cycles (depending on which cycle length is chosen), we can only consider our determination of chromospheric activity cycles for the two stars as a lower limit. A longer baseline of S-index measurements is needed to verify these short cycles.

Our chromospheric activity cycles compared with other F-stars are shown in Fig. 9 and Table A1. The chromospheric cycle periods of 83 ± 14 d for β Vir and 43 ± 2 d for θ Dra are potentially shorter than cycles for other F-stars with published values ranging from 109 d for CoRoT 102780281 (F8V) determined by photometry (Lopes et al. 2015), and 120 d for τ Boo from chromospheric emission (Mengel et al. 2016), to a maximum of 21 yr for KIC 5955122 (F9V) determined by photometry (Bonanno et al. 2014). The older Mt Wilson survey data (Baliunas & Vaughan 1985; Baliunas et al. 1995) may have missed short cycles in some F-stars. The previous biasing of data towards longer chromospheric cycles may be hinted at from Fig. 9, where generally longer cycle lengths are visible in the older data.

Figure 9.

Chromospheric cycle determined from S-index for β Vir and θ Dra versus T_eff compared with 30 F-stars from the literature. Data from Mt Wilson survey papers (Baliunas & Vaughan 1985; Baliunas et al. 1995) are shown in light blue, which may have missed short cycles for F-stars. β Vir data is shown as yellow triangle (period 83 ± 14 d) and θ Dra data are shown as pink square (43 ± 2 d). The dashed grey trend line and orange 95 per cent confidence intervals are shown. No significant trend observed. The data are shown in Table A1.

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6.5.3 H α index

Along with emissions in the core of Ca ii H and K lines and Ca ii triplet lines λ8498, 8542, 8662, the H α-index is one of the best indicators of stellar activity in solar-type stars (Gray & Corbally 2009). In Fig. 10, we show a plot of S-index and H α-index for both β Vir and θ Dra. We see a weak correlation between the two measures of chromospheric activity for β Vir with a Pearson correlation coefficient of r = 0.31, while there is a stronger correlation for θ Dra with correlation coefficient of r = 0.63. The correlation between S-index and H α-index for β Vir and θ Dra is consistent with the trend found in G and K-type stars by Zarro (1983), where the H α absorption increases with Ca ii K strength. We find θ Dra has a larger H α index compared to β Vir and this trend has been observed in K, G, and F stars by Cram & Giampapa (1987), where the strength of the photospheric absorption line increases with increasing T_eff. The increased scatter observed in the S-index for β Vir compared to θ Dra may be due to the low values measured for β Vir which range from 0.159 to 0.166 and may contain a lower relative SNR.

Figure 10.

Individual S-index versus individual H α index for β Vir (top) and θ Dra (bottom). The dashed grey trend line and orange 95 per cent confidence intervals are shown. Significant trend observed for both plots with a Pearson correlation coefficient of r = 0.31 for β Vir and r = 0.63 for θ Dra. These plots indicate the correlation between S-index versus individual H α index holds for late F-stars.

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6.6 B_l versus rotational phase

The nightly average of B_l versus rotational phase for β Vir is shown in Fig. 11. The figure indicates there is no significant change in B_l between the three maps. There is a large scatter in B_l during 2017 April, which is not accounted for by a magnetic field evolution over the two rotational cycles of observations. This was checked by separating the B_l observations into different rotational cycles and observing that the scatter in B_l remained. The scatter in B_l for β Vir in April 2017 is possibly due to the more complex nature of the magnetic field geometry and the location of the fields on the visible surface during an observation may lead to more or less cancellation effects. The nightly average of B_l versus rotational phase for θ Dra is shown in Fig. 11, which also indicates there is no significant difference in B_l between the two maps.

Figure 11.

B_l versus rotational phase for β Vir (top) and θ Dra (bottom). Symbols are the same as Fig. 6. There is no significant difference between the means within the data set for each star. The B_l observations for θ Dra with a mean B_l of −13.1 ± 5.1G at rotation phase 0.811 was cropped from the figure to allow better visibility of data. This value with large errors was caused by cloud interference resulting in an V mag extinction ∼3.

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A possible reason no difference in B_l is observed between the two epochs for θ Dra, while the mean S-index for magnetic maps show a significant difference, is that B_l suffers from cancellation effects from regions of opposite polarity, which does not occur for the S-index. We also find that unlike B_l, the calculation of B_mean using ZDI does show a significant difference between the two maps. This difference may be explained by the reduced amount of flux cancellation that occurs when calculation B_mean using ZDI in faster rotating stars (Lehmann et al. 2019) where θ Dra is a reasonably fast rotator with vsin i = 28.1 km s⁻¹. Another possible reason for the lack of visible change in B_l between the two θ Dra maps is that the error bars are a similar magnitude to B_l.

6.7 Stellar wind modelling

The coronal magnetic field structure when the wind model has reached as steady state is shown in Fig. 12. The mostly dipolar surface magnetic field of β Vir (see Fig. 2) produces a region of closed magnetic field lines surrounding the surface B_r = 0 line, and two regions of open field lines around the north and south magnetic poles. The alignment of the magnetic dipole appears to vary somewhat between the February 14–22 observations and the other two. The region of closed field lines appears smaller for the Febuary 14–22 observation, where the magnetic field is at its weakest. The more complex surface magnetic field geometry of θ Dra (see Fig. 3) gives rise to a similarly complex coronal field; no clear dipolar structure is visible in the coronal field. At both epochs, θ Dra exhibits a large, uniform region of open field lines in the lower hemisphere; this is likely caused by the ZDI regularization in unobservable regions of the star past ∼140^○.

Figure 12.

Coronal field structure and radial field strength for the wind model. The open field lines are truncated at two stellar radii. The stellar surface and the field lanes are coloured according to the local radial magnetic field strength. Note that the colour scale is linear from −1 G to 1 G and logarithmic outside of this range. β Vir, which occupies the three leftmost panels, exhibits a typical dipolar coronal field at all three epochs. A strengthening of the surface field may be observed between the first epoch February 14–22 and the second epoch March 1–29. The increase in field strength is accompanied by an increase in the size of the regions of closed magnetic field lines. θ Dra exhibits similar magnetic field strengths at both epochs. The change in polarity around the south pole region of the star can be seen between April 6–24 and May 15–Jun 18.

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The wind accelerates as it flows away from the star, and eventually the wind speed exceeds the Alfvén wave speed |$v_{\small {A}}= B / \sqrt{\mu _0 \rho }$| (Alfvén 1942). The surface where this first occurs is shown in Fig. 13. The wind states, as it crosses this Alfvén surface determines the stellar angular momentum loss, as the surface functions as an effective lever arm (Schatzman ) increasing the torque effected by the wind upon the star. The xz plane and the Alfvén surface are coloured by the local wind speed. A two-lobed Alfvén surface develops for β Vir at all three epochs; this is typical for a dipole-dominated magnetic field. θ Dra exhibits an irregular Alfvén surface; this is associated with a more complex magnetic field. Fig. 13 also shows the current sheet (Schatten 1972), which we characterize by the radial magnetic field B_r changing sign. The inner current sheet of β Vir appears quite flat but with slightly varying inclinations (see Table 6). The current sheet of θ Dra has multiple kinks and undulations, reflecting the complex magnetic geometry.

Figure 13.

Alfvén surface and inner current sheet. The z-axis coincides with the stellar axis of rotation, and we show the inner current sheet edge-on to emphasize the two-lobed structure of the Alfvén surface. The Alfén surface and the plane of sky (i.e. the xz plane) is coloured according to the wind radial velocity u_r. The current sheet is truncated at 30R_⋆. β Vir exhibits a two-lobed Alfvén surface at all three epochs. An increase of the average Alfvén radius and of the current sheet inclination is seen between February 14 and 22 and the two other epochs, see the aggregate quantities in Table 6. The complicated magnetic field geometry of θ Dra gives rise to a multi-lobed Alfvén surface at both epochs. A decrease in the average Alfvén radius is observed between April 6–24 and May 15–June 18.

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The wind pressure in the stellar equatorial plane is shown in Fig. 14. The non-uniformity of the coronal field creates regions of varying wind pressure as the wind density and acceleration is affected by local differences in the magnetic field. The stellar rotation shapes these regions into spirals. The dipolar coronal field of β Vir creates two spiral co-rotating interaction regions (CIRs) of locally overdense wind, while the θ Dra cases are more complex, showing up to four spiral CIRs. The faster rotation of θ Dra creates a more tightly wound spiral in comparison to β Vir.

Figure 14.

Wind pressure in the star’s equatorial plane from the corona to the inner astrosphere. The dotted circle corresponds to an orbital distance of 1 au. The dipolar magnetic field of β Vir gives rise to two spiral ‘arms’ of overdense wind. These co-rotating interaction regions (CIRs) form when regions of fast, tenuous wind collide with regions of slower, denser wind. The complex field of θ Dra gives rise to multiple CIRs; for the April 6–24 epoch these CIRs persist past 1 au, while the CIRs appear to coalesce into a single spiral at the May 15–June 18 epoch.

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The wind models permit the calculation of a range of wind-related parameters; we give some of these parameters in Table 6. The surface magnetic flux Φ₀ is the total magnetic flux emerging from the stellar surface, and the open magnetic flux fraction Φ_open is the fraction of Φ₀ that extends past the Alfvén surface. This parameter has been studied by Vidotto et al. (2014a) and Réville et al. (2015a, b) as a proxy for wind mass loss and angular momentum loss. A related parameter is the open surface fraction S_open, which quantifies the fraction of the stellar surface from which open magnetic field lines emanate. The current sheet inclination |$i_{B_r}$| gives a quantitative indication of the tilt of the current sheet and Alfvén surface lobes relative to the stellar axis of rotation. The axisymmetric magnetic flux fraction modulates the amount of galactic cosmic rays in the vicinity of the star (see Wang, Sheeley Jr & Rouillard 2006; Vidotto et al. 2014b). The average Alfvén surface |$R_{\small {A}}$| and the torque-averaged Alfvén surface |$|\boldsymbol{\boldsymbol {r}}_{\small {A}} \times \boldsymbol{\boldsymbol {\hat{z}}}|$| are used in lower-dimensional models and empirical scaling laws, e.g. Weber & Davis (1967), Mestel (1984), Kawaler (1988), and Finley & Matt (2018). A more detailed description of the calculation of these parameters is described in Evensberget et al. (2021).

The wind mass-loss |$\dot{M}$| and the wind angular momentum loss is calculated by integrating the flux across the Alfvén surface. The Alfvén surface is chosen for its uniqueness, but the quantities could also be calculated across a spherical surface giving similar results. The |$\dot{M}$| values of β Vir increase from 1–4 times the average solar wind mass loss rate of |$\dot{M}_\odot = 1 \times 10^{9}\, \mathrm{kg\, s}^{-1}$| (e.g. Wood 2004; Mishra et al. 2019), while the angular momentum loss |$\dot{J}$| increases from about 1–5 times the average solar values of |$2 \times 10^{23}\, \mathrm{N\, m}$| (Finley et al. 2019) between the first observation and the last two observations. The biggest change occurs between February 14–22 and March 1–29. θ Dra is does not exhibit as much variation as β Vir; the |$\dot{M}$| values only change by 25 per cent, while the |$\dot{J}$| changes by a factor of about 2. θ Dra’s large radius of 2.67R_⊙ and rapid rotation of 2.88 d means that this star is not quite Sun-like. The numerical model used has mainly been validated for the solar case so some caution should be taken in interpreting θ Dra’s |$\dot{M}$| values which range from 20 to 25 |$\dot{\rm M}_\odot$| and |$\dot{J}$| values which range from 90 to 240 |$\dot{\rm J}_\odot$|⁠, although it can be expected that this large, rapidly rotation star should have high values for both |$\dot{M}$| and |$\dot{J}$|⁠. Compared to the 86 G sub-giant of Ó Fionnagáin et al. (2021) our mass-loss values are lower, and our angular momentum loss values are significantly lower. This is however to be expected due to the sensitivity of |$\dot{J}$| to the magnetic field strength.

We also include the wind pressure for an Earth-like planet at 1 au and the magnetospheric stand-off distance for an Earth-like planet; these quantities permit comparison to the local Earth space weather. The wind pressure is dominated by the wind radial velocity, |$P_\text{w} \approx \rho u_r^2$|⁠. The magnetospheric stand-off distance is calculated from a pressure balance argument similar to Vidotto et al. (2009). The wind pressure and stand-off distance at β Vir is comparable to Earth conditions, where the wind pressure varies from ∼0.1 nPa to ∼10  nPa (King & Papitashvili 2005). The conditions at θ Dra are characterized by more powerful winds; however, they are still comparable to the strong Solar wind conditions when the Sun is at Solar maximum.

7 DISCUSSION

A detailed understanding of surface magnetism and space weather surrounding mature late F-stars has been lacking in the literature, and our study provides an insight into magnetism in stars with thin outer convection zones. The reconstructed surface magnetic maps for β Vir and θ Dra allow us to compare dynamo magnetic fields in fast and slowly rotating late F-stars. The large-scale magnetic topology for the slowly rotating β Vir is relatively simple, and is mostly poloidal and dipolar. The surface field shows some changes between the three epochs and is slightly more complex in the 2017 April 3–24 map, compared with the two earlier maps, which is possibly partly due to a more complete rotational phase coverage of the map. The relatively simple magnetic structure is most likely a dynamo magnetic field due to the changing topology and varying amount of magnetic energy contained in the spherical harmonic modes between epochs. Both the maximum and mean magnetic field strength is significantly larger in the April 3–24 map compared to the two earlier maps, which is also consistent with a dynamo magnetic field present for β Vir.

The magnetic maps for the faster rotating θ Dra show more complex magnetic field topology than β Vir, which may be partly due to a greater surface resolution produced by ZDI for faster rotating stars, and also may be due to faster rotation which drives a stronger and more complex dynamo (Noyes et al. 1984; Pizzolato et al. 2003). θ Dra also has a greater mean magnetic field strength than β Vir due to a higher vsin i, where a strong correlation exists between vsin i and magnetic activity for late F-stars (Wolff et al. 1986) and other cool stars (Noyes et al. 1984). θ Dra has more magnetic energy stored in the toroidal field compared to the slower rotating β Vir, which is consistent with finding by Petit et al. (2008) and Jouve et al. (2010) where the proportion of toroidal energy increases with rotation rate.