The exoplanetary magnetosphere extension in Sun-like stars based on the solar wind - solar UV relation

The solar wind is a continuous plasma flow emitted from the upper atmosphere of the Sun, mostly consisting of ions and electrons (e.g. Verscharen19). At 1 AU, it is characterised by a typical speed ranging between 250 km/s and 800 km/s, a density of a few particles per cubic centimeter, and it carries out a magnetic field of the order of a few nanoTeslas (Parks18), with a dependency over the solar activity cycle (Poletto2013). In the last 40 years the solar wind has been investigated with increasing deeper details, both in terms of instrument resolution and spacecraft locations. This refined our understanding of its dynamical properties (Escoubet97; Stone98; Burch16), opening new insights on the the source regions of the fast solar wind, i.e. the lower solar atmosphere layers of the Coronal Holes (e.g. Bryans2020). It is considered as a natural laboratory to investigate several kinds of processes and mechanisms such as turbulence and intermittency, plasma instabilities, waves and structures, small- vs. large-scale dynamics (e.g. Bavassano1998; Bruno16). Solar wind can be described as a multiscale system whose dynamic occurs over a wide range of scales. If we focus on time scales longer than the so-called inertial range (i.e., longer than a few hours), the dynamics of the solar wind is mainly related with solar source mechanisms as active regions, coronal mass ejections, and flares (Tu95). Thus, these large-scale phenomena are the main reason for the variations of the planetary environments, through the interaction with planetary magnetospheres and/or ionospheres. A wide variety of processes are generated, such as geomagnetic storms and substorms, particle precipitation, auroral activity, localized energy transfer processes, impacting the climate and habitability of terrestrial extrasolar planets (e.g. Russell93; Blanc05; Airapetian2020; Galuzzo2021). A clear manifestation of the solar wind-magnetosphere interactions is the observed change of the standoff distance of the nose of the magnetospheric cavity, thus affecting both its size and shape. The standoff distance is defined as the distance at which the solar wind dynamic pressure equals the magnetic pressure of the magnetospheric cavity, i.e.,

Assuming the incompressibility condition for the solar wind, i.e., a constant mass density $\rho=\rho_{0}$, and writing the magnetic field as a dipolar shape, i.e., $B=\frac{M_{E}}{r^{3}}$, being $M_{E}$ the Earth’s dipole moment, we can obtain the standoff distance as

In the last two decades, several efforts have been made to increase our capabilities in forecasting the dynamical behaviour of the solar wind as well as its effects on Earth (Bothmer07). Thus, investigating and characterising the relations between long-term solar activity proxies and in-situ solar wind measurements are of crucial impact for any Space Weather and Space Climate forecasting scheme and could be fundamental for characterising Sun-like stars and their interaction with their own planetary systems (e.g. Airapetian2020).
The solar magnetic activity main periodicity is the 22-year Hale cycle (Hale1925), with the reversal of polarity that results in the well-known 11-year Schwabe solar cycle (Schwabe1844). Periods close to 11 years have been found in most solar wind parameters since the very first observations from satellites (Siscoe1978; King1979; Neugebauer1981). However, a not perfect match of the solar wind long-term behaviour with the shape and phase of the sunspot cycle, stimulated a discussion about the observed lag with geomagnetic indices (see e.g., Hirshberg73; Intriligator1974; Feynman82). As longer time-series of near-Earth solar wind measurements became available, several studies have investigated the periodicity of the solar wind parameters and the relation between the Sunspot Number and solar wind proxies (Petrinec1991; Kohnlein96; ElBorie02; Katsavrias12; Richardson12; Li2016; Li2017; Venzmer2017; Samsonov19), or geomagnetic data such as the aa index (Echer2004; Dmitriev2005; Lockwood2009; Du2011).

However, this is the first study to analyse the near-Earth solar wind measurements in relation to Ca II K index over the last five solar cycles. The emission in the Ca II K resonance line, related to the mean emission of the Sun’s chromosphere, has been proven to be a great proxy of solar activity (see e.g., Judge2006; 2016SoPh..291.2967B; Chatzistergos2019).
Starting from the 1960s, astronomers began looking with particular attention at other "suns" (i.e., stars with physical properties similar to solar ones or Sun-like stars), with the aim to search for stellar cycles in solar analogs and to understand where the Sun stands on a broader context (Wilson68). To carry out this purpose, a long-term observational campaign regarding their emissions in Ca II H & K lines, expressed in terms of the dimensionless S-index (Wilson78; Vaughan78), was conducted at Mount Wilson Observatory (MWO) starting from 1966 (Wilson68) and then continued at Lowell Observatory starting from 1995 (see e.g., Hall98). The data provided by those surveys have enormously improved our knowledge about the long-term chromospheric variations of Sun-like stars, as well as how these variations are connected with changes in brightness (see e.g., Skumanich72; Baliunas95; Radick98).
When we study such type of stars, the Sun represents a sort of Rosetta stone, so that our understanding about how its magnetic activity affects the solar-system planets can be seen as a starting point to assess the way Sun-like stars influence the environment around them. Assessing the magnetospheric compression of planets orbiting Sun-like stars is a fundamental point to evaluate their habitability conditions (Airapetian2020). It is well known that the presence of a large enough magnetosphere is a crucial point to shield the planetary atmosphere from the effects of stellar activity, since sufficiently strong stellar wind may also be able to erode it (see e.g., Lammer2012; Cohen2015; Rodriguez-Mozos2019).
The goal of this work is to extend the model of the Earth’s magnetosphere standoff distance based on the Ca II K index to exoplanets orbiting Sun-like stars. To achieve this goal, we take advantage of chromospheric measurements provided by the MWO campaign, by selecting a sample of Sun-like stars that are in a faculae-dominated activity regime as the Sun (see e.g., Radick98; Shapiro2016). Such selection has been made by considering only Sun-like stars with a Rossby number $R_{0}>1$ which, as pointed out by Reinhold19, corresponds to a stellar age $\gtrsim 2.55$ Gyr and to the transition to faculae-dominated activity.

To study the relation between solar magnetic variability and near-Earth solar wind parameters, on decennial time-scales, we need to use datasets which cover a sufficiently extended time interval. As previously described, the magnetic activity of the Sun can be quantified by using different solar indices (Hathaway2010). For this work we use a physical index which measures the mean properties of the solar chromospheric emission: the Ca II K index. Monthly measurements of this index starting from 1907 and covering more than one century are publicly available from the National Solar Observatory (NSO) ¹¹1https://solis.nso.edu/0/iss/. The Ca II K index dataset contains measurements up to October 2017, but it is possible to use other solar activity proxies linked to chromospheric emission, such as the Mg II index (Viereck01), to extend the analysis almost up to date.
For the solar wind we use data available from the OMNI database, which provides various near-Earth solar wind parameters at different time resolutions (more details about the data are given in Sect. 2.1). In particular, we focus our attention to the hourly-resolution measurements of the plasma ion density $n$ and speed $v$, thus determining the solar wind dynamic pressure $P$, defined as $1/2\,m_{p}\,n\,v^{2}$, where we assume proton mass ($m_{p}$) as the mean ion mass. The OMNI database provides plasma measurements only starting from 1965 and, therefore, covers a shorter time interval with respect to the Ca II K index time series. This places a limit on the length of the time period over which a relation between solar proxies and solar wind parameters can be studied. Despite this, by using the Mg II index to reconstruct Ca II K index to date, the latter and the solar wind parameters time series have an overlapping time period which goes from July 1965 up to April 2021, covering almost entirely the last five solar cycles (SCs 20-24).
The properties of each dataset are described in detail later in this section. Here we concentrate on the first steps in the data treatment for the OMNI solar wind dataset. We first bin our data to obtain monthly means. Therefore, starting from the monthly values of Ca II K index, solar wind speed and dynamic pressure, we follow the approach used by Kohnlein96 and apply a 37-month moving average to look at the long-term behaviour of these quantities. These time series are shown in Figure 1. The size of the time window used for the moving average allows to filter and remove the effects of the solar variability related to the solar rotation, transient phenomena, and any other source of variation below the yearly time-scales.
The availability of the observations in the Ca II H & K lines provided by the HK Project at the Mount Wilson Observatory (Wilson68; Wilson78; Duncan91; Baliunas95) is fundamental to extend the model calibrated on the Sun-Earth system to Sun-like star systems. These measurements are accessible for thousands of stars and constitute a broad and long dataset spanning nearly 30 years in many cases. We take advantage of the availability of such measurements to relate, in stars other than the Sun, the mean chromospheric emission to stellar wind dynamic pressure, and hence to study its impact on the exo-planetary magnetospheres.
In the following subsections a detailed description of the datasets used for this work will be provided.

As a first step in our analysis, we assess the relationship between solar activity and solar wind parameters by computing the Pearson’s correlation coefficient over the whole time extent of the dataset. Our analysis is applied to the 37-month moving average quantities presented in Sect. 2. As shown in the scatter plots of Figure 2, we found an almost zero correlation coefficient (r = -0.01) between Ca II K index and solar wind speed, with a similar result also for the dynamic pressure (r = 0.01). Hence, for the whole time interval, from July 1965 to April 2021, we do not find a significant correlation (the p-value is higher than 0.05) between Ca II K index and the solar wind parameters.

As shown in previous studies, the declining phase of solar cycles are characterised by the presence of high-speed solar wind streams (Gosling76; Luhmann09; Tokumaru10; Richardson12). Thus, it is reasonable to expect a time lag between solar activity and solar wind response. To investigate this hypothesis, we compute the cross-correlation between Ca II K index and solar wind speed, where we assume that the latter has a delayed response to changes in solar activity, which means that we are considering only positive time lags of the solar wind with respect to Ca II K index (the result for negative lags is also shown for completeness). As shown in the top panel of Figure 3, the correlation coefficient between the two quantities peaks, with a value of 0.65, at time lag of $3.2\pm 0.1$ yr. The two time-series as visible in the panel b of Figure 3, where the solar wind speed has been back-shifted by the time lag found, have a quite similar phase when a time delay is considered. This result is quite in agreement to that reported by Li2016, which found that the daily means of the solar wind velocity lag the ones of SSN by about 3 yr (see their fig. 3). A 3-year time shift was also found by Venzmer2017 for the correlation of the yearly averages of the same quantities.
A slightly bigger time lag has been found performing the cross-correlation of Ca II K index with the solar wind dynamic pressure data. In this case, we find a moderate positive correlation with a correlation coefficient of 0.57 for a time lag of $3.6\pm 0.1$ yr, as shown in the panel c of Figure 3. Although this latter value is lower than the one found for the solar wind speed, it is higher than the aforementioned correlation coefficients found in literature for SSN and solar wind velocity (0.2 in Li2016 and 0.45 in Venzmer2017). Further comments on the strength of the correlation can be found in the discussion section.

To assess our results in a stronger framework, we also explored the nonlinear features of shared information between the Ca II K index and solar wind parameters using the mutual information analysis (Shannon48). Given a pair of time series $(x(t_{j}),y(t_{k}))$ the mutual information coefficient ($MI$) is defined as

where $p(x,y)$ is the joint probability of observing the pair of values $(x,y)$, while $p(x)$ and $p(y)$ are the independent distributions. For statistically independent time series $MI=0$, while for correlated time series $MI\geq MI_{th}$, where $MI_{th}$ is a threshold associated with a particular statistical significance level. For the following analysis we adopt a threshold value $MI_{th}$ that corresponds to a $95\%$ statistical significance.

As reported in Figure 4 the mutual information coefficient reaches its maximum for time delays of $\sim 3.2$ yr and $\sim 3.4$ yr for the solar wind speed and dynamic pressure with respect to the Ca II K index, respectively. These values are consistent with those estimated via the cross-correlation analysis (see Figure 3). This seems to suggest that there is a significant non-null probability of observing a relation between the Ca II K index and solar wind parameters. In the following, since the same delay is observed both for the MI and the cross-correlation analysis, we use the results from the latter.

By using the above time lags, we found linear relationships for the Ca II K index with both solar wind speed and dynamic pressure. The relationships between these quantities are shown in the two scatter plots in Figure 5, where the two black lines show the best linear fits to the data points. The corresponding empirical equations are:

We can conclude that, by considering a time lag, the correlation coefficients between Ca II K index and the two solar wind parameters investigated here significantly increase. Similar results were recently reported by Samsonov19 using yearly averaged data for the last five solar cycles. They found that the maximum correlation of both solar wind speed and dynamic pressure with the SSN is reached with a 3-year time lag. In particular, they found a correlation coefficient r = 0.57 between SSN and SW dynamic pressure, in line with our results of r = 0.57 for 3.6-year lag. Moreover, they found r = 0.68 focusing only over the last three solar cycles taking 2-year lag. If we limit our analysis on the last three solar cycles as well, we find the same correlation coefficient (r = 0.68) for 2.4-year lag.

The magnetosphere shields the Earth’s atmosphere from the erosion processes due to solar wind, energetic particles, and radiation. Thanks to the magnetic shielding, the Earth has retained its atmosphere for a long time, but this is not the case of all Solar-system planets. Because of the lack of a strong intrinsic magnetic field, Mars has suffered a significant solar wind-induced atmospheric loss (see e.g., Kass1995; Chassefiere2004; Dong2018), with the result of nowadays very thin atmosphere.
Since the Earth’s magnetopause standoff distance is mainly controlled by the solar activity via the solar wind pressure, it is interesting to see how its average value changes over a solar cycle. Moreover, assessing the extent of a planetary magnetosphere constitutes a point of fundamental importance also in the characterisation of extra-solar planets, as the presence of a large enough magnetic shield protecting the underlying atmosphere is directly related to any habitability evaluation.

In the present article we introduce a convenient relationship to deduce the long-term variability of the solar wind by studying its correlation with the solar magnetic activity. To this scope, we used an index that measures the chromospheric emission in the Ca II K resonance line. The relationships between two solar wind parameters, speed and dynamic pressure, and the Ca II K index covering almost 5 solar cycles have been studied using 37-month averaged data.
Our study, based on cross-correlation techniques and mutual information analysis, shows that the solar wind properties follow the solar activity behaviour with a time lag of 3.2 years for the solar wind speed and 3.6 years for the dynamic pressure. The derived linear relationships between the Ca II K index and the solar wind speed and dynamic pressure are found to be valid for the whole time interval (five solar cycles) covered by this investigation.
In Figure 5 we showed the correlation between Ca II K index and solar wind parameters. Hysteresis cycles that could be linked to the 11-year solar cycles are made evident once the evolution in time is made explicit using the colour code. This is particularly evident in the case of the solar wind pressure, where parallel branches of the hysteresis cycle have the same slope of the fit representing the correlation between the parameters. Other cases of hysteresis cycles involving solar activity proxies are known (see e.g., hyst2018MUPB...73..216B; hyst2019SoPh..294....8R; hyst2019SoPh..294...86S). We plan to study the hysteresis cycles shown in this paper more in details in a future work, exploring the effects of a dynamic lag between the variables as opposed to the constant lag considered in this paper. A possible outcome of this future study is a new relation able to constrain the scatter plots on the fitted relation, providing higher correlation coefficients.
Having a relationship that links the solar wind variations to that of Ca II K index is remarkable both for an historical reconstruction of the solar wind parameters and to even filling gaps for which measurements are not available. In particular, the solar Ca II K index is available since the beginning of the 20th century, but it has been synthetically reconstructed since 1750 in berrilli2020 by using different solar atmospheric models that represent quiet and magnetic regions. Furthermore, the time-shifted relations obtained can be employed for an attempt at short-time predictions into the future (up to 1.7 years for the solar wind speed and 2.1 years for its dynamic pressure). Those predictions could be very useful to assess the mean solar wind conditions from the point of view of human space missions, but also to forecast solar wind parameters during the flight phase of solar focused missions like Parker Solar Probe or Solar Orbiter, as in Venzmer2017. In particular our model predicts a minimum in the solar wind dynamic pressure in mid-2022.
We believe that these results are not only helpful to achieve a better knowledge of the interaction between Sun and Earth, but also for developing new skills to study the space-climate variability of other solar-type stars, in particular those with exoplanets, enabling us to characterise the interactions between planets and their host stars and the wind conditions of exoplanetary environment. Measurements in the Ca II H & K lines are still performed by different ground-base telescopes, even directly expressed in terms of the S-index as at the TIGRE telescope (see e.g., Schmitt2014; Gonzalez-Perez2022). Since our relations have been calibrated on an intermediate age star, as the Sun, the most appropriate targets to which extend them are constituted by faculae dominated G-spectral type stars, characterised by an age $\gtrsim$ 2.6 Gyr and with a Rossby number $R_{0}\gtrsim 1$ (Reinhold19). For the future, we plan to compare the results from our model to that of other independent models, in order to assess the possibility to extend our procedure also to late F-type or early K-type stars.
In analogy to the case of the Sun, by making use of the relations we found, the variation and hence the effects of the stellar wind of Sun-like stars on their planets can be studied by analysing the temporal evolution of the chromospheric measurements already collected for several targets. Given the impossibility to obtain in-situ measurements of the stellar wind, it is difficult to recover the phase lag between stellar UV emission and stellar wind properties, and thus the stellar wind level in a precise moment in time. Nevertheless, having the information on at least a complete UV stellar cycle, our model enables to compute the mean magnetospheric standoff distance for planets nearby Sun-like stars.
It is useful to point out that asteroseismic observations, like those obtained by the successful photometric space missions, i.e., Kepler (Borucki2010) or TESS (Ricker2014), could also provide accurate fundamental parameters, in particular ages with an accuracy better than $15\%$ (see e.g., Chaplin2014; Lebreton2014; DiMauro2017), which can be used in combination with the present method to study the variation of the magnetospheric extension as a star like the Sun evolves, by comparing results for stars with different ages. In addition, asteroseismic observations could also be used to study the variation of the magnetic activity of a Sun-like star by analysing the temporal evolution of oscillation parameters (see e.g., Reda2022; DiMauro2022). In fact, it has been demonstrated by Bonanno2014 for a sample of 19 Sun-like stars, the presence of clear relations between the S-index and some asteroseismic parameters such as the amplitude of the observed acoustic oscillation modes or the ’small frequency separation’, known as age indicator. The targets selected in Table 1 have been observed by TESS in 120 s and 20 s cadence mode and the asteroseismic analysis will be considered in the next future for a comparison with the present results. Clearly, this will open an independent way to estimate the erosion of exoplanetary atmospheres. Thus, we plan for the future to extend the present analysis to a wider set of stars, by exploiting the relationship between solar wind properties and UV emission over the Mount Wilson Observatory (MWO) measurements, which regularly observed the Ca II H & K emission since 1966 (Wilson78) for several stars of different spectral types. Further, in order to complete the phenomenological scenario, we will complement and verify our results by employing the independent procedure based on the asteroseismic method.

RR and PG are PhD students of the PhD course in Astronomy, Astrophysics and Space Science, a joint research program between the University of Rome “Tor Vergata”, the Sapienza University of Rome, and the National Institute of Astrophysics (INAF). The authors thank the reviewer P. Judge (HAO, NCAR) for the thoughtful comments that improved the quality of this paper.

The Time series of the Ca II K index uses SOLIS data obtained by the NSO Integrated Synoptic Program (NISP), downloaded from the SOLIS website (https://solis.nso.edu/0/iss/; Monthly-Averaged ISS/SP/KKL Ca II K 1Å Emission Index Time Series). NISP is managed by the National Solar Observatory, which is operated by the Association of Universities for Research in Astronomy (AURA), Inc. under a cooperative agreement with the National Science Foundation. The Mg II composite is available from the University of Bremen (http://www.iup.uni-bremen.de/UVSAT/Datasets/mgii). The OMNI data are available from Coordinated Data AnalysisWeb (CDAWeb; http://cdaweb.gsfc.nasa.gov). The datasets of the HK Project at the Mount Wilson Observatory are available from the National Solar Observatory (NSO) website (https://nso.edu/data/historical-data/mount-wilson-observatory-hk-project/).