2.1. Modeling Inputs

All models require an input lower boundary condition, a measurement of the photospheric magnetic field, which is typically taken from the ground-based Global Oscillation and Network Group (GONG; Harvey et al. Reference Harvey, Hill, Hubbard, Kennedy, Leibacher, Pintar, Gilman, Noyes, Title, Toomre, Ulrich, Bhatnagar, Kennewell, Marquette, Patron, Saa and Yasukawa1996) or the space-based Helioseismic and Magnetic Imager (HMI; Scherrer et al. Reference Scherrer, Schou, Bush, Kosovichev, Bogart, Hoeksema, Liu, Duvall, Zhao, Title, Schrijver, Tarbell and Tomczyk2012) aboard the Solar Dynamics Observatory (SDO; Pesnell et al. Reference Pesnell, Thompson and Chamberlin2012). These measurements provide a synchronic (single point in time) measurement of the photospheric field which can then be extrapolated with a model of your choosing. However, these synchronic measurements only provide a measurement of the Earth-facing hemisphere of the solar photosphere. Modeling the global magnetic field requires an input of the full Sun, these are often known as Carrington or synoptic maps. Full-Sun photospheric magnetic field maps are typically produced one of two ways: by combining a Carrington rotation’s worth of observations of the photospheric field with some merging technique and accounting for line-of-sight effects, or via evolution of the observed magnetic flux to the part of the disk that is not observed. This method is known as flux transport, and models such as the Air Force Data Assimilative Photospheric Flux Transport (ADAPT; Arge et al. 2010) provide well tested pathways to do this.

2.2. Connectivity via Potential Field Extrapolations

The simplest method for extrapolation of the photospheric magnetic field is known as the Potential Field Source Surface (PFSS; Altschuler and Newkirk Reference Altschuler and Newkirk1969) model. It requires two user inputs as boundary conditions: the photospheric magnetic field and a source surface height ( $R_{ss}$ ) where the magnetic field is considered purely radial. Between these two points, the model assumes a magnetostatic or current-free ( $\nabla \times \mathbf{B} = 0$ ) corona, and represents the magnetic field as a gradient of a scalar potential ( $\mathbf{B} = - \nabla \psi$ ). As $\nabla \cdot \mathbf{B} = 0$ , we can write

(1) \begin{equation} \nabla \cdot \mathbf{B} = \nabla \cdot ( - \nabla \psi) = \nabla^2 \psi = 0\end{equation}

giving Laplace’s equation (Equation 1). A solution can then be determined with the two boundary conditions described above. This can be solved in spherical coordinates to provide an estimated extrapolation of the global magnetic field out to the source surface ( $R_{ss}$ ).

To connect spacecraft observations to the extrapolated magnetic field, requires other assumptions about the propagation of the solar wind beyond the source surface. The simplest method is to ballistically propagate the spacecraft trajectory at its (variable) heliocentric distance ( $R_{sc}$ ) down to the source surface height. This method assumes that any point in the heliosphere can be propagated inwards by assuming propagation along a Parker spiral line, with some set speed (Nolte and Roelof Reference Nolte and Roelof1973). While the real solar wind is accelerating as it expands, the errors associated with the constant speed assumption has been shown to be $\leq 10^\circ$ (Nolte and Roelof Reference Nolte and Roelof1973; Badman et al. Reference Badman, Bale, Martínez Oliveros, Panasenco, Velli, Stansby, Buitrago-Casas, Réville, Bonnell, Case, Dudok de Wit, Goetz, Harvey, Kasper, Korreck, Larson, Livi, MacDowall, Malaspina, Pulupa, Stevens and Whittlesey2020; Ervin et al. 2024c). With this ballistic propagation, the spacecraft trajectory is now connected to the model of the magnetic field. All that is left is to trace along modeled field lines to estimate the spacecraft’s connectivity to the photosphere. It is important to note that this method can only trace along field lines that are topologically ‘open’ at the source surface, meaning that we can never show connection to topologically ‘closed’ structures that may open at larger heights in the corona.

This PFSS + ballistic propagation method is an incredibly computationally efficient, and easily usable method for providing an estimation of the magnetic connectivity of in-situ observations to their source. Despite its usability, the PFSS and ballistic propagation combination includes a variety of user introduced, and general sources of error. The choice of which input magnetogram to use (what data source, what observation time, how to extrapolate to the far-side of the Sun), which source surface height, and which speed for the ballistic propagation can lead to uncertainties on the order of 10s of degrees in the modeled footpoints. The choice of PFSS boundary conditions can be optimized by comparing the modeled heliospheric current sheet with the observed crossing of the neutral line, or by comparing the modeled magnetic field strength with what is observed in-situ (Badman et al. Reference Badman, Bale, Martínez Oliveros, Panasenco, Velli, Stansby, Buitrago-Casas, Réville, Bonnell, Case, Dudok de Wit, Goetz, Harvey, Kasper, Korreck, Larson, Livi, MacDowall, Malaspina, Pulupa, Stevens and Whittlesey2020; Ervin et al. 2024c). In addition, the assumptions that go into the model itself are not physically accurate, and the steady-state nature of these models is not able to capture the time-dependent nature of magnetic connectivity in the heliosphere. Despite this, with rigorous uncertainty estimation and the understanding that these are estimates of the spacecraft connectivity, PFSS models provide an incredible tool for understanding the solar wind’s varied sources.

2.3. Connectivity via Magnetohydrodynamic Simulations

While the PFSS model provides a computationally efficient method for extrapolating the global field, it makes nonphysical assumptions about the potential nature of the magnetic field in the lower corona. More physically complex methods to extrapolate the magnetic field exist, and a commonly used method is magnetohydrodynamic (MHD) modeling. These models take an equivalent inner boundary condition as potential field models, a full-Sun map of the photospheric magnetic field. The model then integrates the MHD equations (with varied assumptions and in varied forms) to provide a description of the plasma and magnetic fields. Different models use varied assumptions (e.g. different heating terms) for driving the supersonic solar wind, and provide a method through which coronal heating can be studied.

There are a variety of state-of-the-art codes to extrapolate the photospheric magnetic field by integrating the MHD equations with some set of assumptions. In our work, we use the Predictive Science Inc. (PSI) MHD algorithm outside a Sphere (MAS) code to solve the MHD equations on a nonuniform mesh grid (Riley et al. Reference Riley, Lionello, Caplan, Downs, Linker, Badman and Stevens2021). This code has been shown to well replicate remote and in-situ observations that can be derived from the model. Synthetic observables can be produced to qualitatively study how well the model does at replicating the observed magnetic field and density structure in the observed corona. One can fly a spacecraft through the modeled field to produce a timeseries of modeled in-situ observations, which can then be directly compared to spacecraft measurements. In contrast to the potential field extrapolation, MAS can model the magnetic field throughout the inner heliosphere, and thus ballistic propagation is not necessary for estimating spacecraft connectivity. We show an example comparing the outputs of PFSS and MAS simulations with in-situ measurements of polarity in Figure 1. We note that the PFSS and MAS extrapolations produce similar current sheets. This Figure shows how a comparison of modeled and measured magnetic field polarities can be used a method of model validation, and for inter-model comparison.

Figure 1.

Figure from Ervin et al. (Reference Ervin, Bale, Badman, Bowen, Riley, Paulson, Rivera, Romeo, Sioulas, Larson, Verniero, Dewey and Huang2024a) showing a comparison of model outputs between PFSS and MHD (PSI/MAS) magnetic field extrapolations compared with in-situ measurements from Parker Solar Probe and Solar Orbiter. Panel (a) shows the magnetic field polarity as a function of source surface (2.5 $\mathrm{R}_\odot$ ) longitude. The modeled polarities are pulled directly from the PFSS and MHD coronal field extrapolations. The Parker and Orbiter polarities are projected down from the spacecraft trajectory to 2.5 $\mathrm{R}_\odot$ using ballistic propagation as described in Section 2.2. Panel (b) shows the modeled current sheets from the PFSS (dotted) and MHD (solid) simulations. The background is the global magnetic field at 2.5 $\mathrm{R}_\odot$ from the MHD simulation. The projected Parker Solar Probe trajectory is overlaid and colored by the measured polarity at the spacecraft.

Figure 1 Long description

Panel A: A bar graph shows magnetic field polarity as a function of source surface longitude. The x-axis represents source surface longitude in degrees, ranging from 0 to 360 degrees. The y-axis lists different models and measurements: PFSS, MHD, Parker, and Orbiter. The bars are color-coded to indicate polarity, with blue representing negative polarity and red representing positive polarity. Panel B: A contour plot displays modeled current sheets from PFSS (dotted lines) and MHD (solid lines) simulations. The background is a color map of the global magnetic field at 2.5 solar radii from the MHD simulation, with colors ranging from blue (negative) to red (positive). The x-axis represents Carrington longitude in degrees, and the y-axis represents Carrington latitude in degrees. The projected Parker Solar Probe trajectory is overlaid and colored by the measured polarity at the spacecraft.

2.4. Accessing source region parameters

Estimation of the footpoint location for where in-situ measurements of the solar wind emerged from allows us to study characteristics associated with the source. To do this, we utilize remote observations of the photosphere and corona, specifically magnetograph and EUV observations. We extract values of the remote observations at the pixel coordinate associated with the location of the estimated footpoint. We can do this for each point in our in-situ timeseries where we have an estimated source region location (see Ervin et al. (Reference Ervin, Bale, Badman, Rivera, Romeo, Huang, Riley, Bowen, Lepri and Dewey2024b)). This allows us to produce a ‘timeseries’ of source region parameters. We do this for measurements of the photospheric field, which allows us to understand the magnetic field strength at the solar wind’s estimated source, and differentiate between weak and strong photospheric fields. The same method is used for EUV observations from the Atmospheric Imaging Assembly (AIA; Lemen et al. 2012) on the SDO, giving insight into the temperature and density of the plasma in the corona.

We also look at the magnetic expansion factor ( $f_s$ ) in the same way. This provides a measure of how over-expanded the magnetic field is, relative to purely $1/r^2$ spherical expansion, between the photosphere ( $R_0, \lambda_0, \phi_0$ ) and a second point ( $R_1, \lambda_1, \phi_1$ ).

(2) \begin{equation} f_{s} = \left( \frac{R_0}{R_1} \right)^2 \frac{B_R{R_0, \lambda_0, \phi_0}}{B_R{R_1, \lambda_1, \phi_1}}\end{equation}

This can be done between the photosphere and the source surface ( $f_{ss}$ ) or between the photosphere and the spacecraft location. A large body of work exists that has studied the relation between solar wind speed and expansion factor (e.g. Wang and Sheeley Reference Wang and Sheeley1990). In general, it has been found that slower wind emerges from regions with larger expansion factors. A more rapidly diverging field (larger expansion) has a higher critical point (where the wind becomes supersonic) and thus more heating would occur prior to the transition of the wind from subsonic to supersonic. Heating above the critical point causes the wind speed to increase (Leer and Holzer Reference Leer and Holzer1980), and thus slower wind from rapidly expanding fields follows this relation. However, not all slow wind is associated with large expansion factors, and there may be an additional dependence on Alfvénicity in addition to just wind speed.

3.1. Heavy Ion Ratios

Heavy ion ratios allow us to probe the ionization level of the solar wind plasma. The processes that ionize the solar wind occur low down in the corona as ionization and recombination rates are dependent upon (and proportional to) the electron density (Owocki et al. Reference Owocki, Holzer and Hundhausen1983). As the wind radially expands, there is a dramatic drop in the electron density to a point where the density is too low for ionization and recombination processes to occur. Known as the freeze-in point, after this radial distance (typically 1-2 $R_\odot$ depending on the ion), ionization ratios are frozen-in and undergo little radial evolution (Hundhausen Reference Hundhausen1968). Therefore, in-situ measurements of heavy ion ratios can serve as a proxy for the electron properties at the source. Various works have shown the relation of low charge state ratios with wind emerging from CH structures (von Steiger and Zurbuchen Reference von Steiger and Zurbuchen2011), while higher charge states have been related to CH boundaries (Antiochos et al. Reference Antiochos, Linker, Lionello, Mikić, Titov and Zurbuchen2012), helmet streamers (Zhao et al. Reference Zhao, Zurbuchen and Fisk2009; Zhao and Fisk Reference Zhao and Fisk2011; Zhao et al. Reference Zhao, Landi, Zurbuchen, Fisk and Lepri2014), and active regions (Liewer et al. Reference Liewer, Neugebauer and Zurbuchen2003).

3.2. Elemental Composition

Total elemental composition, specifically ratios between elements, allow us to probe the effects of the first ionization potential (FIP) bias, where abundances of low-FIP elements are enhanced over high-FIP elements. There are various ratios that can be studied to probe this fractionation, the most common of which is Fe/O. Spectroscopic measurements of coronal loops have shown a correlation between FIP bias and loop lifetime, e.g. the longer the loop exists, the larger the Fe/O ratio (Feldman and Marsch Reference Feldman and Marsch1997). Theoretical work by (Laming Reference Laming2015; Laming Reference Laming2017; Laming et al. Reference Laming, Vourlidas, Korendyke, Chua, Cranmer, Ko, Kuroda, Provornikova, Raymond, Raouafi, Strachan, Tun-Beltran, Weberg and Wood2019) has tied the enhancement of low-FIP elements in coronal loops to the generation of the ponderomotive force in the upper chromosphere. As this force is only felt by ions, it will lead to an enhancement in ions over neutrals, leading to elemental fractionation in the upper solar atmosphere. Therefore, plasma that quickly flows out of the lower solar atmosphere along open field lines should show low FIP bias in comparison to plasma that spent time within coronal loops. One can study the in-situ FIP bias by comparing measurements of elemental abundance ratios, to their photospheric values determined from remote spectroscopic observations (e.g. $\mathrm{(Fe/O)} / \mathrm{(Fe/O)}_\mathrm{phot})$ . A larger ratio indicates more FIP enhancement (canonically tied to more time spent in coronal loops).