Temporal Structures in Electron Spectra and Charge Sign Effects in Galactic Cosmic Rays

We present the precision measurements of 11 years of daily cosmic electron fluxes in the rigidity interval from 1.00 to 41.9 GV based on 2 . 0 × 10 8 electrons collected with the Alpha Magnetic Spectrometer (AMS) aboard the International Space Station. The electron fluxes exhibit variations on multiple timescales. Recurrent electron flux variations with periods of 27 days, 13.5 days, and 9 days are observed. We find that the electron fluxes show distinctly different time variations from the proton fluxes. Remarkably, a hysteresis between the electron flux and the proton flux is observed with a significance of greater than 6 σ at rigidities below 8.5 GV. Furthermore, significant structures in the electron-proton hysteresis are observed corresponding to sharp structures in both fluxes. This continuous daily electron data provide unique input to the understanding of the charge sign dependence of cosmic rays over an 11-year solar cycle

Introduction.-Cosmicrays are dominated by positively charged particles and nuclei: protons, helium, etc. Electrons are the most abundant negatively charged particles, but cosmic electrons are rare.The precision study of cosmic electrons requires a magnetic spectrometer in space to separate electrons from positrons and the overwhelming number of positively charged protons and nuclei.
Since installation on the International Space Station on May 20, 2011, AMS has continuously collected and analyzed electron events daily.Most of these events (around 99%) are in the low rigidity range below Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license.Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
41.9 GV.The high rigidity 1% of the spectrum up to 2 TeV provides unexpected results, which will be presented in a future publication.
The fluxes of charged cosmic rays outside the heliosphere are thought to be stable on the timescale of decades [1][2][3][4].Time-dependent variations in the fluxes of galactic cosmic rays measured inside the heliosphere are only expected from the solar modulation [5].Solar modulation involves convective, diffusive, particle drift, and adiabatic energy loss processes [6].Only particle drift induces a dependence of solar modulation on the particle charge sign [7].The systematic measurement of the electron flux and the proton flux offers a unique way to study charge-signdependent solar modulation effects.
Cosmic electrons are primary cosmic rays [8].Their time structure is of particular importance as electrons have been widely used to search for new phenomena in primary cosmic rays, such as the existence of nearby pulsars [9], supernovae remnants [10], or dark matter annihilation [11,12].Models describing these phenomena can only be compared to data when time-dependent effects in the heliosphere are well understood [13][14][15][16].A comprehensive model of the time-dependent solar modulation will have far-reaching consequences for the understanding of the newly observed unexpected features in cosmic-ray fluxes, such as the complex energy dependence of the positron spectrum [17] and of the electron spectrum [18], as well as for other domains of astrophysics, such as the modeling of galactic cosmic-ray propagation [19], the estimate of the galactic cosmic-ray pressure, an important ingredient for models of galaxy formation [20], the interpretation of possible anisotropies in the cosmic-ray arrival directions at the Earth [21], and the understanding of cosmic-ray spectra outside the solar system [22].
Previous experiments measured the time variation of the combined (electron þ positron) flux [23][24][25][26], the electron flux variation averaged over six-and three-month periods [27], or averaged over two days for a total of two months [28].AMS has reported the time dependence of the electron fluxes and the positron fluxes per Bartels rotations (BR: 27 days) over six years [29].In addition, AMS has observed short-term structures in the cosmic-ray proton flux [30] and helium flux [31].
In this Letter, we present the daily electron fluxes based on 2.0 × 10 8 events spanning 11 years over a rigidity range from 1.00 to 41.9 GV.These data cover the major portion of solar cycle 24, which includes the polarity reversal of the solar magnetic field in the year 2013 [32], and the beginning of solar cycle 25.Therefore, the charge-sign-dependent effects are studied at different solar conditions by comparing the daily electron and daily proton [30] fluxes measured simultaneously over an 11-year period.These data provide unique and accurate input to modeling of the transport processes of charged cosmic rays inside the heliosphere.
Event selection.-AMShas collected 1.9 × 10 11 cosmicray events.In the rigidity range from 1.00 to 41.9 GV, we select electron samples using the combined information of the TRD, TOF, and inner tracker.The details of the event selection and backgrounds are contained in Refs.[17,18,[44][45][46] and in the SM [34].After selection, we obtained 2.0 × 10 8 electrons.
Data analysis.-Thedaily isotropic flux in the ith absolute rigidity bin (R i ; R i þ ΔR i ) and jth day is given by where N j i is the number of events corrected for background and bin-to-bin migration; A j i is the effective acceptance calculated from the Monte Carlo simulation, including geometric acceptance, event selection efficiencies, and interactions of electrons in the AMS materials; δ j i is the small correction to the acceptance due to the difference in the event selection efficiencies between data and Monte Carlo simulation; ϵ j i is the trigger efficiency; and T j i is the daily collection time.See the SM [34], Figs.S2  and S3, for more details.In this Letter, the electron flux is measured in ten rigidity bins from 1.00 to 41.9 GV.
The background contribution from antiprotons and light mesons in the data sample is estimated using a template fit to the distribution of TRD estimator Λ TRD [8].The background contribution from charge confusion positrons is estimated to be negligible [8].
Bin-to-bin migration of events is corrected using the unfolding procedures described in Ref. [47].
The small corrections δ j i are estimated by comparing the efficiencies in data and Monte Carlo simulation of every selection cut using information from the detectors unrelated to that cut.The δ j i are found to have a small rigidity dependence: from −5% at 1 GV, decreasing to −2.4% at 10 GV, and becoming constant at −2.8% above 30 GV.
There are extensive studies of the systematic errors.These errors include the uncertainties in the templates definition, the trigger efficiency, the geomagnetic cutoff, the acceptance calculation, the rigidity resolution function, the unfolding, and the absolute rigidity scale.
The uncertainty associated with the Λ TRD templates definition includes two parts: the event selection and the PHYSICAL REVIEW LETTERS 130, 161001 (2023) statistical fluctuations [18].These two errors are added in quadrature.The time-dependent systematic error due to the templates definition amounts to less than 0.5% of the flux below 41.9 GV.
The time-dependent systematic error on the electron fluxes associated with the trigger efficiency measurement is less than 1% over the entire rigidity range and for all days.
The geomagnetic cutoff is calculated as described in the SM [34], and the resulting systematic error on the fluxes is less than 2% at 1 GV and negligible (less than 0.4%) above 2 GV.
The correction δ j i is stable with time within its error, and the associated time-dependent systematic error on the fluxes is less than 1.5% over the entire rigidity range for all days.
The time-independent rigidity resolution function for electrons has a pronounced Gaussian core and non-Gaussian tails.The systematic error on the fluxes due to the rigidity resolution function is obtained by repeating the unfolding procedure while independently varying the width of the Gaussian core by 5% and the amplitude of the non-Gaussian tails by 10% [47].The resulting systematic error on the fluxes is 2% at 1 GV and less than 1% above 2 GV.
The daily variation of the spectral shape leads to an additional uncertainty in the unfolding procedure.The resulting time-dependent systematic error is less than 1% at 1 GV and is negligible (less than 0.2%) above 5 GV for all days.
There are two contributions to the systematic uncertainty on the rigidity scale [47].The first is due to residual tracker misalignment.This error is estimated by comparing the E=p ratio for electrons and positrons, where E is the energy measured with the electromagnetic calorimeter and p is the momentum measured with the tracker.It is found to be 1=30 TV [48].The error is negligible (less than 0.2%) below 41.9 GV.The second systematic error on the rigidity scale arises from the magnetic field map measurement and its temperature corrections.The total time-independent error on the fluxes due to uncertainty on the rigidity scale has been calculated to be less than 0.5% over the rigidity range below 41.9 GV.
The contributions to the systematic error from the trigger efficiency, the reconstruction efficiencies, and the unfolding are evaluated independently each day and are added in quadrature to derive a time-dependent systematic error, which is less than 2% at 1 GV and about 1% above 3 GV for all days.
The daily total systematic error is obtained by adding in quadrature the individual contributions of the timeindependent systematic errors discussed above and the time-dependent systematic errors.At 1 GV, it is less than 3%, and above 3 GV, it is about 1.5% for all days.
Most importantly, several independent analyses were performed on the same data sample by different study groups.The results of those analyses are consistent with this Letter.
Results.-The daily electron fluxes (Φ e − ) including statistical errors, time-dependent systematic errors, and total systematic errors are tabulated in Tables S1-S3300 of the SM [34,49] as functions of the rigidity at the top of the AMS detector.These daily data are in agreement with our earlier 27-day results [29] in the overlapping time period but with improved accuracy.The daily proton flux (Φ p ) data from May 2011 to November 2019 are taken from Ref. [30].The new Φ p data up to November 2021 will be published separately.
Figure 1 shows Φ e − and Φ p for four rigidity bins from 1.00 to 11.0 GV; see also Fig. S6 of the SM [34] for Φ e − in circular format.In this and subsequent figures, the error bars on the fluxes are the quadratic sum of the statistical and time-dependent systematic errors.As seen, Φ e − exhibits both short-term variations on the scale of days to months and long-term variations on the scale of years, and the relative magnitude of these variations decreases with increasing rigidity.The time-dependent behavior of the Φ e − and Φ p is distinctly different, and the differences decrease with increasing rigidity.From 2011 to 2014, Φ e − decreases faster with time than Φ p .From 2015 to mid-2017, Φ e − increase more slowly than Φ p below about 4 GV [Figs.1(a Short-term flux variations can be either recurrent or nonrecurrent.The nonrecurrent variations are mainly caused by transient disturbances in the interplanetary magnetic field [26,28,30,31,50,51].The comparison of the nonrecurrent variation of daily Φ e − and Φ p for three short time intervals is shown in Fig. S7 of the SM [34].As seen, during lower solar activity (left and right columns of Fig. S7), a difference between the short-term evolution of electrons and protons is observed, while during the solar maximum (middle column of Fig. S7), the difference vanishes.For instance, in Figs.S7(b) and S7(j), the slope of the recovery after the dip is different between electrons and protons.These observations indicate a charge-sign dependence in nonrecurrent solar modulation.
Recurrent variations with a period of 27 days and its harmonics are related to solar rotation [52][53][54][55][56][57][58][59][60].To study the recurrent variations in Φ e − , a wavelet time-frequency technique [61] was used to locate the time intervals where the periodic structures emerge.The details on the wavelet analysis are described in the SM [34].All the power spectra in the subsequent figures are drawn with normalized power defined in the SM [34].The Φ e − for four rigidity intervals from 1.00 to 11.0 GV in each year (2011-2021 defined in Table SA of the SM [34]), together with their time-averaged power spectra and 95% confidence levels, are shown in Figs.S8-S18 of the SM [34].
The peak values of the normalized power around 27 days, 13.5 days, and 9 days as a function of rigidity for each year are shown in Figs.S19-S21 of the SM [34], respectively.As indicated by the shaded areas of Fig. S19, the 27-day periodicity is most prominent in the second half of 2011, the second half of 2015, the first half of 2016, and the first half of 2017.As seen in Fig. S20, the 13.5-day periodicity is most prominent in the second half of 2011, the second half of 2015, and the second half of 2016.As seen in Fig. S21, the 9-day periodicity is most prominent in the second half of 2015, the first half of 2016, and the second half of 2016.
The rigidity dependence of the strength of all three periodicities varies in different time intervals, but it does not always decrease with increasing rigidity.These observations do not support the paradigm that, over the AMS rigidity range, the strength of the 27-day (and 13.5-and 9-day) periodicities steadily decreases with increasing rigidity of cosmic rays [62].
Figure 2 shows the normalized power as a function of rigidity and period for Φ e − and Φ p during two time intervals when the 27-day periodicity is most prominent (second half of 2011 and first half of 2017).As seen, the rigidity dependence behavior of the normalized power of electrons and protons is different in these two time intervals.In particular, in the second half of 2011 [Figs.2(a) and 2(b)], the strength of the 27-day period of electrons is greater than that of protons, while in the first half of 2017 [Figs.2(c) and 2(d)], the strength of the 27-day period of electrons is less than that of protons.Figures S22-S24 show the comparison of the peak values of the normalized power including the 95% C.L. between Φ e − and Φ p around 27, 13.5, and 9 days, respectively.As seen, the rigidity dependence of the electron periodicities is different from that of protons [30].
The long-term variations on the scale of years are related to the 11-and 22-year cycles of the solar magnetic field [5].To further investigate the difference in the modulation of Φ e − and Φ p , Fig. 3 shows Φ e − as a function of Φ p for four rigidity intervals from 1.00 to 11.0 GV.For Figs. 3(a peak in 2020, after which the hysteresis curve starts to trace the earlier behavior (2018-2020) backwards.This is consistent with the differences in electron and proton modulation being symmetric with respect to the minimum solar modulation.To assess the significance of this hysteresis, as detailed in the SM [34] (see also Figs.S25 and S26), we study, at different solar conditions, the values of Φ p at the same Φ e − .As seen, the hysteresis is observed with a significance of 47σ at [1.00-1.71]GV, greater than 6σ below 8.48 GV, and 4.1σ at [8.48-11.0]GV.Different methods have been used by several independent analysis groups to quantify the significance of the hysteresis, and they show similar results [63].
To probe structures in the hysteresis, the moving averages of the Φ e − and Φ p are calculated with a finer time window.
The results for the rigidity interval of [1.00-1.71]GV are shown in Fig. 4. Figure 4(a) shows the daily Φ e − and Φ p as a function of time for the 11-year period.The arrows I, II, and III indicate the location of sharp dips in the proton and electron fluxes, and the colored bands IVand V mark the time intervals around the dips in 2015 and 2017.The moving average of Φ e − and Φ p with the time window of 2 BRs and a step of 1 day for this rigidity interval is shown in Fig. 4(b).The detailed behavior around the dips IV and V is shown in Fig. S27.To assess the significance of these structures in hysteresis, we study the difference of Φ e − at the same Φ p ; see SM [34] for details.The significance of the hysteresis structure at [1.00-1.71]GV corresponding to the large dip in 2015 is 15.9σ (IV) and to the large dip in 2017 is 7.0σ (V).The analysis at [1.71-2.97]GV is presented in Fig. S28.The significance of the hysteresis structure corresponding to the large dip in 2015 is 14.6σ and to the large dip in 2017 is 5.3σ.
The structures in the hysteresis in 2015 and 2017 are likely caused by a series of interplanetary coronal mass ejections [64].The clear deviation from the long-term trend implies a charge-sign-dependent modulation during those solar transients on the timescale of several Bartels rotations.
In conclusion, we presented the precision measurements of 11 years of daily cosmic electron fluxes in the rigidity interval from 1.00 to 41.9 GV based on 2.0 × 10 8 electrons.The electron fluxes exhibit variations on multiple timescales.In the 11-year period, the electron fluxes show distinctly different time variations from the proton fluxes.Recurrent electron flux variations with periods of 27 days, 13.5 days, and 9 days are observed.The strength of all three periods of electron fluxes shows different rigidity and time dependence compared to protons.Remarkably, a hysteresis between the electron flux and the proton flux is observed with a significance greater than 6σ at rigidities below 8.5 GV.Furthermore, significant structures in the electronproton hysteresis are observed, corresponding to sharp variations in the fluxes.These continuous daily electron data provide unique input to the understanding of the charge sign dependence of cosmic rays over an 11-year solar cycle.
We are grateful for important physics discussions with Igor Moskalenko.We thank former NASA Administrator Daniel S. Goldin for his dedication to the legacy of the ISS as a scientific laboratory and his decision for NASA to fly AMS as a DOE payload.We also acknowledge the continuous support of the NASA leadership, particularly Kathryn Lueders, and of the JSC and MSFC flight control teams that have allowed AMS to operate optimally on the ISS for over 11 years.We are grateful for the support of Glen Crawford of the DOE including resources from the National Energy Research Scientific Computing Center under Contract No. DE-AC02-05CH11231.We gratefully acknowledge strong support from CERN, including Fabiola Gianotti, and the CERN IT department, including ) and 1(b)].From mid-2020 to 2021, Φ e − decreases faster than Φ p .

2 FIG. 1 .FIG. 2 .FIG. 3 .
FIG. 1. Eleven-year daily AMS electron fluxes Φ e − and daily proton fluxes Φ p in units of [m −2 sr −1 s −1 GV −1 ] for four rigidity bins from 1.00 to 11.0 GV from May 20, 2011 to November 2, 2021.Days with solar energetic particle events are removed from Φ p for the lowest rigidity bins shown.The gaps in the fluxes are due to detector studies and upgrades.Note that Φ p is multiplied by different scale factors as indicated.The scale factor of Φ p is chosen such that Φ e − and Φ p for each rigidity bin are at the same magnitude, on average, during 2014 and 2015.The vertical dashed lines indicate the three time intervals studied in Fig. S7 of the SM [34].As seen, Φ e − exhibits large variations with time, and the relative magnitude of these variations [(a)-(d)] decreases with increasing rigidity.The time-dependent behavior of the Φ e − and Φ p are distinctly different.From 2011 to 2014, Φ e − decreases faster with time than Φ p .From 2015 to mid-2017, Φ e − increases more slowly than Φ p below about 4 GV (a),(b).From mid-2020 to 2021, Φ e − decreases faster than Φ p .

FIG. 4 .
FIG. 4. (a) Daily electron fluxes Φ e − (red, left axis) and daily proton fluxes Φ p (green, right axis) as a function of time for the rigidity interval of 1.00 to 1.71 GV.The arrows I, II, and III indicate the location of sharp dips in the proton and electron fluxes, and the colored bands IV and V mark the time intervals around the dips in 2015 and 2017.(b) Φ e − versus Φ p both calculated with a moving average of 2 BRs and a step of 1 day.The locations of I, II, and III correspond to the flux dips in (a).To assess the significance of the structures in the hysteresis, during dips IV and V (indicated by white boxes), two pairs (white squares and triangles) of nonoverlapping intervals with the same Φ p but different Φ e − are chosen.As shown in Fig. S25 of the SM [34], the large dips in 2015 (IV) and 2017 (V) correspond to additional structures in the overall hysteresis with significance of 15.9σ and 7.0σ, respectively.Fluxes are in units of [m −2 sr −1 s −1 GV −1 ].