Limitation of energetic ring current ion spectra

Summers, Danny; Shi, Run

doi:10.1002/2015ja021482

Cited by 5 publications

(10 citation statements)

References 28 publications

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“…We suppose that the EMIC waves generated by the distribution (1) undergo a specified gain G in amplitude during propagation over a convective path length LR E along a magnetic field line, where L denotes magnetic shell and R E is the Earth's radius. Then, as detailed by Summers and Shi (), it can be shown that a limiting energy distribution g ( p ) = g * ( p ) can be achieved where g * is determined by the equation,

\begin{matrix} lefttrue \\ \int_{X_{0}}^{\infty} {([], 1 - {(\frac{1 - italicγω / Ω_{p}}{1 - γ_{0} ω / Ω_{p}})}^{2} \frac{X_{0}}{X})}^{s} \\ \cdot [\frac{s}{γ} - (1 + s) \frac{ω}{Ω_{p}} - \frac{((), 1 + s)}{γ} {((), \frac{ω}{Ω_{p}})}^{2} \frac{((), 1 - italicγω / Ω_{p})}{{(1 - γ_{0} ω / Ω_{p})}^{2}} X_{0}] g^{*} (p) dX, \\ = \frac{1}{{(m_{p} c)}^{2}} \cdot \frac{1}{2 π^{3}} \cdot \frac{G}{m_{p} c^{2}} \cdot \frac{{cB}_{E}}{{eR}_{E}} \cdot \frac{{(1 - γ_{0} ω / {normalΩ}_{p})}^{2}}{L^{4} X_{0}} \end{matrix}

where ω is the (real) wave frequency, X = p 2 ,

γ = \sqrt{1 + X}

, and Ω p = eB 0 /( m p c ) is the proton gyrofrequency where e is the electronic charge. We assume that the background magnetic field strength B 0 equals the equatorial dipole value, B 0 = B E / L 3 .…”

Section: Calculation Of the Limiting Energy Spectrummentioning

confidence: 99%

“…Equation , which is an integral equation for g * , is valid for all frequencies ω over which wave growth occurs. The equation can be readily transformed into a linear Volterra integral equation of the first kind (see Summers & Shi, ) and can be solved by standard numerical techniques, as given, for instance, by Press et al (). Once the solution for g * has been found, the limiting omnidirectional differential proton flux (from ) is then given by

J_{4 π}^{*} ((), E) = 2 π^{3 / 2} \frac{normalΓ ((), s + 1)}{normalΓ ((), s + 3 / 2)} {(m_{p} c)}^{2} X g^{*} ((), p),

where

E = ((), \sqrt{1 + X} - 1) m_{p} c^{2}

.…”

Section: Calculation Of the Limiting Energy Spectrummentioning

confidence: 99%

“…While analytical solutions of equation are not readily found, Summers and Shi () were able to obtain a useful asymptotic solution for g * valid for large X (or, equivalently, large E ) of the form g * ~1/ X 3/2 as X → ∞. The corresponding asymptotic solution for the limiting energy spectrum

J_{4 π}^{*}

given by result is found to be

J_{4 π}^{*} ((), E) = \frac{{italiccB}_{E}}{π^{3 / 2} {eR}_{E}} ((), \frac{s + 1}{s}) \frac{normalΓ ((), s + 1)}{normalΓ ((), s + 3 / 2)} \frac{G}{L^{4} E}, E \to \infty

…”

Section: Calculation Of the Limiting Energy Spectrummentioning

confidence: 99%

“…They showed that the limiting electron differential flux moderated by whistler mode waves is determined by an integral equation that can be solved by standard numerical techniques. Subsequently, Summers and Shi () likewise showed in a relativistic regime that the limiting ion differential flux subject to scattering by EMIC waves in a cold multi‐ion background plasma is also given by an integral equation that must be solved numerically.…”

Section: Introductionmentioning

confidence: 99%

“…In the present study, we test the validity of the formulation of the limiting ion flux by Summers and Shi () by comparing numerical (theoretical) solutions for the limiting flux with experimental proton flux profiles. Specifically, we measure proton differential flux profiles in the region 3 ≤ L ≤ 6 over two chosen magnetic storms using data from the RBSPICE‐B instrument on the Van Allen Probes.…”

Section: Introductionmentioning

confidence: 99%

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Energetic Proton Spectra Measured by the Van Allen Probes

et al. 2017

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We test the hypothesis that pitch angle scattering by electromagnetic ion cyclotron (EMIC) waves can limit ring current proton fluxes. For two chosen magnetic storms, during 17–20 March 2013 and 17–20 March 2015, we measure proton energy spectra in the region 3 ≤ L ≤ 6 using the RBSPICE‐B instrument on the Van Allen Probes. The most intense proton spectra are observed to occur during the recovery periods of the respective storms. Using proton precipitation data from the POES (NOAA and MetOp) spacecraft, we deduce that EMIC wave action was prevalent at the times and L‐shell locations of the most intense proton spectra. We calculate limiting ring current proton energy spectra from recently developed theory. Comparisons between the observed proton energy spectra and the theoretical limiting spectra show reasonable agreement. We conclude that the measurements of the most intense proton spectra are consistent with self‐limiting by EMIC wave scattering.

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\begin{matrix} lefttrue \\ \int_{X_{0}}^{\infty} {([], 1 - {(\frac{1 - italicγω / Ω_{p}}{1 - γ_{0} ω / Ω_{p}})}^{2} \frac{X_{0}}{X})}^{s} \\ \cdot [\frac{s}{γ} - (1 + s) \frac{ω}{Ω_{p}} - \frac{((), 1 + s)}{γ} {((), \frac{ω}{Ω_{p}})}^{2} \frac{((), 1 - italicγω / Ω_{p})}{{(1 - γ_{0} ω / Ω_{p})}^{2}} X_{0}] g^{*} (p) dX, \\ = \frac{1}{{(m_{p} c)}^{2}} \cdot \frac{1}{2 π^{3}} \cdot \frac{G}{m_{p} c^{2}} \cdot \frac{{cB}_{E}}{{eR}_{E}} \cdot \frac{{(1 - γ_{0} ω / {normalΩ}_{p})}^{2}}{L^{4} X_{0}} \end{matrix}

where ω is the (real) wave frequency, X = p 2 ,

γ = \sqrt{1 + X}

Section: Calculation Of the Limiting Energy Spectrummentioning

confidence: 99%

J_{4 π}^{*} ((), E) = 2 π^{3 / 2} \frac{normalΓ ((), s + 1)}{normalΓ ((), s + 3 / 2)} {(m_{p} c)}^{2} X g^{*} ((), p),

where

E = ((), \sqrt{1 + X} - 1) m_{p} c^{2}

.…”

Section: Calculation Of the Limiting Energy Spectrummentioning

confidence: 99%

J_{4 π}^{*}

given by result is found to be

J_{4 π}^{*} ((), E) = \frac{{italiccB}_{E}}{π^{3 / 2} {eR}_{E}} ((), \frac{s + 1}{s}) \frac{normalΓ ((), s + 1)}{normalΓ ((), s + 3 / 2)} \frac{G}{L^{4} E}, E \to \infty

…”

Section: Calculation Of the Limiting Energy Spectrummentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Energetic Proton Spectra Measured by the Van Allen Probes

et al. 2017

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Survey of radiation belt energetic electron pitch angle distributions based on the Van Allen Probes MagEIS measurements

Shi

Summers

et al. 2016

JGR Space Physics

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A statistical survey of electron pitch angle distributions (PADs) is performed based on the pitch angle‐resolved flux observations from the Magnetic Electron Ion Spectrometer (MagEIS) instrument on board the Van Allen Probes during the period from 1 October 2012 to 1 May 2015. By fitting the measured PADs to a sinnα form, where α is the local pitch angle and n is the power law index, we investigate the dependence of PADs on electron kinetic energy, magnetic local time (MLT), the geomagnetic Kp index, and L shell. The difference in electron PADs between the inner and outer belt is distinct. In the outer belt, the common averaged n values are less than 1.5, except for large values of the Kp index and high electron energies. The averaged n values vary considerably with MLT, with a peak in the afternoon sector and an increase with increasing L shell. In the inner belt, the averaged n values are much larger, with a common value greater than 2. The PADs show a slight dependence on MLT, with a weak maximum at noon. A distinct region with steep PADs lies in the outer edge of the inner belt where the electron flux is relatively low. The distance between the inner and outer belt and the intensity of the geomagnetic activity together determine the variation of PADs in the inner belt. Besides being dependent on electron energy, magnetic activity, and L shell, the results show a clear dependence on MLT, with higher n values on the dayside.

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A statistical study of proton pitch angle distributions measured by the Radiation Belt Storm Probes Ion Composition Experiment

Shi

Summers

et al. 2016

JGR Space Physics

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A statistical study of ring current‐energy proton pitch angle distributions (PADs) in Earth's inner magnetosphere is reported here. The data are from the Radiation Belt Storm Probes Ion Composition Experiment (RBSPICE) on board the Van Allen Probe B spacecraft from 1 January 2013 to 15 April 2015. By fitting the data to the functional form sinnα, where α is the proton pitch angle, we examine proton PADs at the energies 50, 100, 180, 328, and 488 keV in the L shell range from L = 2.5 to L = 6. Three PAD types are classified: trapped (90° peaked), butterfly, and isotropic. The proton PAD dependence on the particle energy, magnetic local time (MLT), L shell, and geomagnetic activity are analyzed in detail. The results show a strong dependence of the proton PADs on MLT. On the nightside, the n values outside the plasmapause are clearly lower than those inside the plasmapause. At higher energies and during intense magnetic activity, nightside butterfly PADs can be observed at L shells down to the vicinity of the plasmapause. The averaged n values on the dayside are larger than on the nightside. A maximum of the averaged n values occurs around L = 4.5 in the postnoon sector (12–16 MLT). The averaged n values show a dawn‐dusk asymmetry with lower values on the dawnside at high L shells, which is consistent with previous studies of butterfly PADs. The MLT dependence of the proton PADs becomes more distinct with increasing particle energy. These features suggest that drift shell splitting coupled with a radial flux gradient play an important role in the formation of PADs, particularly at L > ~ 4.5.

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Limitation of energetic ring current ion spectra

Cited by 5 publications

References 28 publications

Energetic Proton Spectra Measured by the Van Allen Probes

Energetic Proton Spectra Measured by the Van Allen Probes

Survey of radiation belt energetic electron pitch angle distributions based on the Van Allen Probes MagEIS measurements

A statistical study of proton pitch angle distributions measured by the Radiation Belt Storm Probes Ion Composition Experiment

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