Empirical Model of 10 – 130 MeV Solar Energetic Particle Spectra at 1 AU Based on Coronal Mass Ejection Speed and Direction

Abstract

We present a new empirical model to predict solar energetic particle (SEP) event-integrated and peak intensity spectra between 10 and 130 MeV at 1 AU, based on multi-point spacecraft measurements from the Solar TErrestrial RElations Observatory (STEREO), the Geostationary Operational Environmental Satellites (GOES), and the Payload for Antimatter Matter Exploration and Light-nuclei Astrophysics (PAMELA) satellite experiment. The analyzed data sample includes 32 SEP events occurring between 2010 and 2014, with a statistically significant proton signal at energies in excess of a few tens of MeV, unambiguously recorded at three spacecraft locations. The spatial distributions of SEP intensities are reconstructed by assuming an energy-dependent 2D Gaussian functional form, and accounting for the correlation between the intensity and the speed of the parent coronal mass ejection (CME), and the magnetic-field-line connection angle. The CME measurements used are from the Space Weather Database Of Notifications, Knowledge, Information (DONKI). The model performance, including its extrapolations to lower/higher energies, is tested by comparing with the spectra of 20 SEP events not used to derive the model parameters. Despite the simplicity of the model, the observed and predicted event-integrated and peak intensities at Earth and at the STEREO spacecraft for these events show remarkable agreement, both in the spectral shapes and their absolute values.

Appendix: Estimate of the Magnetic Footpoint Location

The HGS coordinates ($\alpha _{\mathrm{sc}}$, $\beta _{\mathrm{sc}}$) of the footpoint location of the IMF line passing through a given spacecraft can be estimated by assuming a simple 3D Parker spiral model (Parker, 1958). Specifically, the longitude is calculated as

$$ \beta _{\mathrm{sc}} \approx b_{\mathrm{sc}} + \frac{\Omega _{\odot } }{V_{\mathrm{sw}}} \hspace{1mm} \left ( R_{\mathrm{sc}} - R_{0}' \right ) \hspace{1mm} \cos (\alpha _{\mathrm{sc}}), $$

(15)

where $b_{\mathrm{sc}}$ is the longitude of the spacecraft location, $R_{\mathrm{sc}}$ is its radial distance,

$$ R_{0}' = R_{0} \hspace{1mm} \left [ 1 + \log \left ( \frac{R_{\mathrm{sc}}}{R_{0}} \right ) \right ], $$

(16)

with $R_{0}{=}2.5$ R_⊙ the radius of the “source” surface; $V_{\mathrm{sw}}$ is the solar-wind speed [km s⁻¹] – assumed to be constant and purely radial – and $\Omega _{\odot }$ is the differential solar rotation rate at the latitude $\alpha _{\mathrm{sc}}$ of the footpoint location:

$$ \Omega _{\odot } = A - B \hspace{1mm} \sin ^{2}(\alpha _{\mathrm{sc}}) - C \hspace{1mm} \sin ^{4}(\alpha _{\mathrm{sc}}), $$

(17)

with $A{=}2.972{\pm} 0.010$ μrad s⁻¹, $B{=}0.484{\pm} 0.038$ μrad s⁻¹, and $C{=}0.361{\pm} 0.051$ μrad s⁻¹ (Snodgrass and Ulrich, 1990); in particular, the value of $A$ is related to the sidereal rotation period at the Equator (≈24.47 days). The footpoint latitude $\alpha _{\mathrm{sc}}$ is assumed to coincide with the heliographic latitude of the central point of the solar disk as seen by the spacecraft, accounting for the Sun’s rotation axis’ tilt of about 7.25^∘ relative to the Ecliptic plane. In general, at large radial distances the Parker spiral IMF is simplified as an Archimedes spiral by neglecting the logarithmic term ($R_{\mathrm{sc}}-R_{0}'\approx R_{\mathrm{sc}}-R_{0}$).

The uncertainty associated with the footpoint longitude can be calculated as

$$ \delta \beta _{\mathrm{sc}} = \sqrt{(\delta \beta _{\mathrm{sw}})^{2}+( \delta \beta _{\mathrm{tra}})^{2}}, $$

(18)

where

$$ \delta \beta _{\mathrm{sw}} = \frac{(R_{\mathrm{sc}} - R_{0}')\hspace{1mm}\cos (\alpha _{\mathrm{sc}})}{V_{\mathrm{sw}}} \hspace{1mm} \sqrt{ \left [ \left ( \frac{\partial \Omega _{\odot }}{\partial \alpha _{\mathrm{sc}}} - \tan (\alpha _{\mathrm{sc}}) \hspace{1mm} \Omega _{\odot }\right ) \delta \alpha _{\mathrm{sc}} \right ]^{2} + \left [ \frac{\Omega _{\odot }}{V_{\mathrm{sw}}} \hspace{1mm} \delta V_{\mathrm{sw}} \right ]^{2}}, $$

(19)

with $\delta V_{\mathrm{sw}}$ the uncertainty in the solar-wind speed estimate, and $\delta \beta _{\mathrm{tra}}$ accounts for interplanetary transport effects ignored by the model. Similarly, the error on the footpoint latitude takes into account the deviations from the spiral-model approximation out of the equatorial plane and effects related to the differential changes across the solar surface.