PFSS Models for Solar Magnetic Extrapolation

• PFSS models are computational frameworks that extrapolate solar photospheric magnetic fields into the corona by assuming a current-free state and enforcing a radial field at a designated source surface.
• They use spherical harmonic expansions to solve Laplace’s equation under observed boundary conditions, enabling detailed mapping of coronal topology and open magnetic flux.
• PFSS techniques are essential for space weather forecasting and are rigorously validated against in-situ and remote-sensing observations despite their limitations in modeling dynamic current-driven phenomena.

A Potential Field Source Surface (PFSS) model is a widely adopted computational and conceptual framework for extrapolating solar photospheric magnetic field measurements into the solar corona under two primary constraints: the coronal magnetic field is assumed current-free (potential), i.e., $\nabla\times\mathbf{B}=0$, and the field is forced to be purely radial above a chosen spherical surface of radius $R_{ss}$—the so-called “source surface.” This model provides the most fundamental approach to estimating global coronal magnetic topology and open solar flux, undergirding operational space-weather forecasting and the interpretation of solar–heliospheric connectivity.

1. Mathematical Formulation and Solution Structure

In a PFSS model, the coronal domain is defined as the spherical shell $R_\odot \le r \le R_{ss}$, where $R_\odot$ is the solar radius. The coronal field is represented by a scalar potential,

$\mathbf{B} = -\nabla\Psi,$

with $\Psi$ satisfying Laplace’s equation,

$\nabla^2\Psi = 0.$

Boundary conditions:

• Inner boundary (photosphere): Observed synoptic magnetograms provide $$B_r(R_\odot, \theta, \phi) = B_{\mathrm{obs}}(\theta, \phi)$$.
• Outer boundary (source surface): The field is forced to be radial, so $$B_\theta(R_{ss}, \theta, \phi) = B_\phi(R_{ss}, \theta, \phi) = 0$$.

The unique solution is obtained by expanding $\Psi(r, \theta, \phi)$ in Schmidt-normalized spherical harmonics: $$\Psi(r, \theta, \phi) = \sum_{\ell=0}^{\ell_{\max}}\sum_{m=-\ell}^{\ell} \left[a_{\ell m} r^\ell + b_{\ell m} r^{-(\ell+1)}\right] Y_{\ell m}(\theta, \phi),$$ with $b_{\ell m} = -a_{\ell m} R_{ss}^{2\ell+1}$. The coefficients $a_{\ell m}$ are fixed by projecting the photospheric data onto $Y_{\ell m}$. In practice, $\ell_{\max}$ ranges from tens to over one hundred, depending on data and computational constraints (Tähtinen et al., 2024, Petrie et al., 2010, Freed et al., 2014).

2. Model Parameters and Boundary Choices

The critical free parameter in the PFSS model is the source-surface radius $R_{ss}$, typically set empirically to $2.5\,R_\odot$ but varied in the literature from $1.3$ to $3.5\,R_\odot$ depending on solar cycle phase, validation metric, and desired application (Tähtinen et al., 2024, Benavitz et al., 2024, Badman et al., 2019, Huang et al., 2024).

Key points:

• Lower $R_{ss}$: Larger open-field regions, more equatorward extension of coronal holes, greater open flux. May better match in-situ magnetic field reversals and compact connectivity structures observed by Parker Solar Probe (Badman et al., 2019, Song, 2023).
• Higher $R_{ss}$: Restricts open flux to higher latitudes, flattens the heliospheric current sheet and streamer belt, better matches large-scale coronal streamer observations at minimum (Benavitz et al., 2024, Badman et al., 2022).
• Optimal $R_{ss}$ varies: For example, $R_{ss}\approx 2.4$–$2.5\,R_\odot$ provides best agreement with open flux proxies over solar cycles 21–24 (Tähtinen et al., 2024), while eclipse image benchmarking yields $R_{ss}\approx 1.3\,R_\odot$ at maximum and $R_{ss}\approx 3.0\,R_\odot$ at minimum (Benavitz et al., 2024). Adjusting $R_{ss}$ per magnetogram or as a function of photospheric flux has been shown to improve open-flux area matching to MHD reference models (Huang et al., 2024).

3. Calculation of Open Solar Flux and Topological Diagnostics

Once the potential field is fully specified, the central physically relevant quantity is the unsigned open magnetic flux,

$$\Phi_{\mathrm{open}} = \int_0^{2\pi}\int_0^{\pi} |B_r(R_{ss}, \theta, \phi)| R_{ss}^2 \sin\theta\, d\theta\,d\phi,$$

which quantifies the flux transported by the solar wind into the heliosphere (Tähtinen et al., 2024, Arden et al., 2016, Badman et al., 2019). Mapping of closed and open field line footpoints enables direct computation of coronal hole boundaries, streamer belt neutral lines, and quasi-separatrix layers. Metrics such as field-line expansion factors, squashing factors ($Q$), and direct comparison with remote sensing (EUV, white light) are employed to assess topological realism (Baker et al., 2023, Badman et al., 2022, Benavitz et al., 2024).

4. Validation, Physical Interpretation, and Observational Benchmarks

Validation against in-situ and remote data:

• Eclipse images and EUV coronal holes: Rolling Hough Transform-based angle discrepancy metrics show systematic improvements if $R_{ss}$ is cycled with solar activity, but significant angular deviations ($\langle\Delta\theta\rangle>10^\circ$) persist, especially at streamer bases (Benavitz et al., 2024).
• Open flux at 1 AU and PSP perihelion: PFSS open flux best matches empirical and MHD values when $R_{ss}$ is tuned, but a fixed value cannot reconcile open-field area, streamer position, and sector boundaries simultaneously (Huang et al., 2024, Badman et al., 2022, Song, 2023).
• Coronal null-point populations: Systematic PFSS surveys reproduce the gross global distribution and statistics of topological nulls, but discrepancies exist in latitudinal balance and survival due to model idealizations (Freed et al., 2014).

Physical caveats:

• Current-free assumption: PFSS neglects all coronal currents, so cannot model free magnetic energy, twisting structures, or dynamic phenomena (e.g., helmet-streamer transients, interchange reconnection).
• Radial field enforcement at $R_{ss}$: The true open/closed interface is neither perfectly spherical nor static (Kruse et al., 2020, Benavitz et al., 2024).
• Photospheric input limitations: Magnetogram inaccuracies, zero-point errors, and coarse spatial resolution (particularly at the poles) all affect coronal solutions (Song, 2023, Tähtinen et al., 2024).

5. Numerical Techniques and Code Validation

Implementation approaches include truncated spherical-harmonic solvers (Tähtinen et al., 2024), finite-difference Laplace solvers (Song, 2023, Kruse et al., 2020), and mixed formulations for flexibility in grid geometry and non-spherical source surfaces (Kruse et al., 2020). Analytic single-harmonic PFSS solutions serve as standard test cases for code validation, with field-line tracing accuracy and unsigned flux errors typically controlled at the percent level if adequate grid and step size are used (Stansby et al., 2022).

Numerical Method	Typical Use	Accuracy/Performance Note
Spherical harmonics	Standard global PFSS, moderate-high $\ell_{\max}$	Efficient, analytic for sphere
Finite-difference (FD)	Arbitrary grid, non-spherical boundaries	Flexible but requires validation
Mixed (FD + spectral)	High resolution, specialized geometries	Best for ellipsoidal/region-specific surfaces

For spherical PFSS, solvers such as pfsspy enable high-fidelity, rapid extrapolation on large magnetogram sets (Badman et al., 2022, Stansby et al., 2022). Finite-difference approaches generalize to ellipsoidal or otherwise deformed source surfaces (Kruse et al., 2020).

6. Role in Space Weather, Solar Wind, and Coronal Hole Science

PFSS extrapolations provide essential context for interpreting spacecraft data (e.g., Parker Solar Probe, STEREO, SOHO), predicting solar wind source regions, and constraining CME propagation and geoeffectiveness. Predictions of coronal hole topology, open-flux corridors (S-web), and sector-boundary transitions are widely employed in operational and ensemble forecasting frameworks (Heinemann et al., 16 Jan 2026, Baker et al., 2023, Ledvina et al., 2023).

However, the inability to simultaneously optimize coronal-hole coverage, streamer-belt alignment, and sector-crossing accuracy with a single $R_{ss}$ or boundary map demonstrates a core limitation: PFSS is an efficient, foundational model but not a complete physical representation. Physics-rich MHD models (e.g., MAS, AWSoM) and non-potential extensions (e.g., WSA, CSSS, NLFFF) address these gaps but at greatly increased computational and observational demands (Song, 2023, Badman et al., 2022, Kruse et al., 2020).

7. Practical Recommendations and Future Developments

• Tune $R_{ss}$ dynamically: Cycle-dependent adjustment or region-specific $R_{ss}$ values improve open-flux and footpoint mapping (Benavitz et al., 2024, Huang et al., 2024, Badman et al., 2019, Ledvina et al., 2023).
• Consider non-spherical source surfaces: Elliptical or deformed $R_{ss}$ better capture streamer warps and polar/equatorial asymmetries (Kruse et al., 2020).
• Validate consistently: Adopt analytic single-mode tests and cross-compare with independent solar wind, EUV, and white-light signatures (Stansby et al., 2022, Benavitz et al., 2024).
• Integrate with advanced models: For critical applications (CME magnetic connectivity, non-potential active regions), PFSS should be complemented with MHD or NLFFF, at least as a boundary or initialization (Petrie et al., 2010, Ledvina et al., 2023).
• Maintain awareness of regime limits: PFSS excels for global, low-$\beta$, large-scale coronal topology and solar cycle studies; it is insufficient for fine-structure, current-sheet, or rapidly evolving events.

Potential Field Source Surface models thus remain an indispensable element of the modern heliophysics toolkit, combining analytic tractability, computational speed, and satisfactory topological fidelity for many global applications. Their limitations and optimal parameterizations are now well characterized, guiding their use in the era of high-cadence, multi-instrument coronal and heliospheric observations (Tähtinen et al., 2024, Benavitz et al., 2024, Song, 2023).