The measurements of the cosmic positron flux with the Alpha Magnetic Spectrometer (AMS) on the International Space Station (ISS) [4,5] and earlier measurements [6] have generated widespread interest and discussions of the observed excess of high-energy positrons. The explanations of these results included three classes of models: annihilation of dark matter particles [1], acceleration of positrons to high energies in astrophysical objects [2], such as pulsars, and production of high-energy positrons in the interactions of cosmic-ray nuclei with interstellar gas [3]. Models describing these phenomena can be compared to data only when time-dependent effects in the heliosphere are well understood.

The fluxes of interstellar charged cosmic rays are thought to be stable on the timescale of decades [7][8][9][10]. Timedependent structures in the energy spectra of galactic cosmic rays are expected from the solar modulation [11] only when they enter the heliosphere. Solar modulation involves convective, diffusive, particle drift, and adiabatic energy change processes. Only particle drift induces a dependence of solar modulation on the particle charge sign [12]. Since positrons and electrons differ only in charge sign, positrons and protons share the same charge sign with different masses, and helium provides different information on both charge and mass, their simultaneous measurement over an |1-year solar cycle offers a unique way to study charge-sign-and mass-dependent solar modulation effects at different timescales.

Previously, AMS has reported the time dependence per Bartels rotation (BR: 27 days) of positron fluxes and separately electron fluxes over six years [13]. AMS has recently reported short-term variations on the scale of days to months and long-term variations on the scale of years in the daily cosmic-ray electron [14], proton [15], and helium [16] fluxes over 11 years. This Letter reports the first daily positron flux measurement. In the past, PAMELA has measured three-month average positron-to-electron flux ratio variation over nine years [17].

In this Letter, we present the daily positron fluxes spanning 11 years over a ngidity 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 [18], and the first part of solar cycle 25. Therefore, both the chargesign-and mass-dependent effects at different solar conditions are studied by comparing the daily positron, daily electron [14], and daily proton [15] fluxes measured simultaneously over an 11-year period. These data provide unique and accurate input to the understanding of the transport processes of charged cosmic rays inside the heliosphere.

Detector.-The layout and description of the AMS detector are presented in Refs. [19,20] and shown in Fig. Sl in Supplemental Material [21]. The key elements used in this measurement are the permanent magnet [22],

the silicon tracker [23][24][25], the transition radiation detector (TRD) [26], the four planes of time of flight (TOF) scintillation counters [27], and the electromagnetic calorimeter (ECAL) [28,29]. Further information on the AMS layout, performance, trigger, and the Monte Carlo simulation [30] is detailed in Supplemental Material [21].

Event selection. -AMS has collected 1.9 x 10!! cosmicray events. In the rigidity range from 1.00 to 41.9 GV, we select positron samples using the combined information of TRD, TOF, inner tracker, and ECAL. The details of the event selection, including the geomagnetic cutoff [31][32][33] and backgrounds, are contained in Supplemental

Material [21] and in Refs. [5,19]. After selection and background subtraction, we obtained 3.4 x 10° positrons.

Data analysis-The daily isotropic flux in the ith rigidity bin (R;, R; + AR;) and jth day is given by "ALL + &) Je! TIAR;

where N! is the number of events corrected for small background (~1%) and bin-to-bin migration using the unfolding procedure described in Ref. [34], A' is the effective acceptance calculated from the Monte Carlo simulation including geometric acceptance, event selection efficiencies, and interactions of positrons in the AMS materials, 6/ is the small correction to the acceptance due to the difference in selection efficiencies between data and Monte Carlo simulation, é/ is the trigger efficiency, and T! is the daily collection time (see Supplemental

Material [21] for details). The positron flux is measured in 12 rigidity bins from 1.00 to 41.9 GV. The binning is similar to that in our electron [14] and proton [15] daily flux measurements.

The small corrections 6/ 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 [5]. The 6) are found to have a small rigidity dependence smoothly varying from -5% at 1 GV, to --1% from 2 to 6 GV, to -5% at 41.9 GV.

The trigger efficiency ¢/ is 100% above 3 GV, decreasing to 83% at 1 GV [19], and is stable over time within errors.

Extensive studies were made of both the time-dependent and time-independent systematic errors. These errors include the uncertainties in background subtraction, the trigger efficiency, the geomagnetic cutoff, the small correction to the acceptance calculation (6/), the unfolding, and the absolute energy scale.

The uncertainty associated with the proton background subtraction includes two parts: the event selection and the statistical fluctuation of the TRD estimator Aypp used to differentiate e+ from p [5]. These two errors are found to be independent and are added in quadrature. The systematic error due to proton background subtraction is found to be < 0.5% of the flux over the entire rigidity range.

The amount of charge confusion is well reproduced by the Monte Carlo simulation [5]. The associated systematic error on the fluxes is negligible (< 0.1%) over the entire rigidity range.

The time-dependent systematic error on the positron fluxes associated with the trigger efficiency measurement is <1% below 3 GV and negligible above 3 GV.

The geomagnetic cutoff is calculated as described in Supplemental Material [21], and the resulting systematic error on the fluxes is less than 2% at 1 GV and negligible (< 0.4%) above 2 GV. |

The systematic error from the correction 6; on the fluxes is time dependent and amounts to < 1.5% over the entire rigidity range.

The systematic error associated with the unfolding includes time-dependent and time-independent errors. The time-independent error is estimated to be 1% of the flux at 1.00 GV and decreases to < 0.2% above 10 GV [5].

The daily flux spectral shape variation leads to an additional time-dependent uncertainty in the unfolding procedure, which is < 1.0% at 1 GV and negligible (< 0.2%) above 5 GV.

The uncertainty on the absolute energy scale [29] is 2.7% at 1 GV, decreasing to 2.0% in the range 2-41.9 GV and is found to be stable at the level of 0.2% for all energies. The energy scale error is treated as an uncertainty of the bin boundaries.

The time-dependent contributions to the systematic error from the background subtraction, the trigger efficiency, the event selection efficiencies, and the unfolding are evaluated independently each day and are found to be uncorrelated. They are added in quadrature to derive a time-dependent systematic error, which is 1.5% at 1 GV and ~1% above 2 GV for all days.

The daily total systematic error is obtained by adding in quadrature the individual contributions of the time-independent systematic errors discussed above and the timedependent systematic errors. At 1 GV, it is less than 3%, and above 3 GV, it is ~1.5% for all days.

Most importantly, independent analyses were performed on the same data sample by three different study groups. The results of those analyses are consistent with this Letter.

Results.-The daily positron fluxes, including statistical errors, time-dependent systematic errors, and _ total systematic errors are tabulated in Tables S1-S3268 in Supplemental Material [21] and in a machine-readable form [35] as functions of the rigidity at the top of the AMS detector. These data are in agreement with our earlier 27-day results [13] in the overlapping time period. AMS in the same rigidity range and time period [14,15]. In these and subsequent figures, the error bars on the fluxes are the quadratic sum of the statistical and time-dependent systematic errors. As seen, ®,+ exhibits short-term variations on the scale of days to months and long-term variations on the scale of years. The time evolution of ®,; and ®,-is presented in Fig. S2 in Supplemental Material [21] for four rigidity bins from 1.00 to 41.9 GV. ®,: and ®,-are shown averaged over 3 days. At low rigidities, below ~8.5 GV, ®,+ and ®,present a different behavior over time. In 2011-2014, ®,. decreases more slowly with time than ®,-. Then, from 2014 to 2017, both fluxes start rising, but ®,+ rises faster than ®,-. From 2017 to 2020, ®,+ rises more slowly than decreases with increasing rigidity, becoming negligible in the rigidity range [22.8-41.9] GV; see Fig. S2(d).

The comparison of the time evolution of 3-day averaged ®,: and ®, in the entire period is shown in Fig. S3 in Supplemental Material [21] for the same four rigidity bins from 1.00 to 41.9 GV. As seen, both fluxes present a similar behavior over time, and at low rigidity [Figs. S3(a) and S3(b)] ®,+ exhibits a larger variation than ®,. At higher rigidities [Fig. S3(c)], the difference in their respective time evolution decreases and becomes negligible in the rigidity range [22.8-41.9] GV [Fig. S3(d)].

Short-term variations in ®,: are shown in Fig. S4 in Supplemental Material [21] in the rigidity range from 1.00 to 2.97 GV, together with ®,-and ®,, measured from January 1, 2016 to January 1, 2017. ®,:, ®,-, and ®, are shown averaged over 3 days. As seen, ®,+ shows time variations that are different from those observed in ®,-. On the contrary, ®,+ and ®, exhibit similar time variations.

These results show that the time evolution of ®,+ is similar to ®,, and distinctly different from ®,-in short term and long term, indicating a clear charge-sign dependence in the solar modulation for positrons and electrons.

To study the recurrent variations in the daily ®,:, a wavelet time-frequency technique [36] was used to locate the time intervals where the periodic structures emerge. The details on the wavelet analysis are described in Supplemental Material [21]. ®,+ for the rigidity interval from 1.00 to 2.97 GV in each year (2011-2021 defined in Table SA in Supplemental Material [21]), together with their time-averaged power spectra and 95% confidence levels, are shown in Figs. S5-S15 in Supplemental Material [21]. Significant values of the normalized power around 27 days are observed in the second half of 2015, the first half of 2016, the first half of 2017, and the first half of 2018. The analysis of ®, presented in Ref. [15] also showed significant 27-day periodicity in these four time intervals.

The long-term variations on the scale of years are related to the 11-and 22-year cycles of the solar magnetic field [11]. To investigate the difference in the modulation of ®,:, B,-and ®,, Fig. 2 shows ®,-and ®, as functions of ®,+ in the rigidity range from 1.00 to 1.71 GV. For Figs. 2(a 151002- 52014-2015 and one after. Around 2017, the hysteresis curve changes such that in 2018-2020 it is nearly parallel to that in 201 1-2013. Similar behavior is observed in the ®,to ®, correlation (see Fig. 3 in Ref. [14]). On the contrary, as seen from Fig. 2(d), there is a nearly linear correlation between ®,+ and ®, in the entire time period. Figure 2 also shows that the three fluxes ®,+, ®,-, and ®, peak in 2020, after which the fluxes start to trace their earlier behavior (2018)(2019)(2020) backwards.

The significance of the hysteresis between ®,+ and ®,has been evaluated following an analysis similar to that described in Ref. [14] (see Figs. S16 and S17 in Supplemental Material [21] for details). The significance is greater than 10o at the rigidity bin [1.00-1.71] GV and greater than 5o for each rigidity bin below 8.48 GV.

To probe structures in the hysteresis, the moving averages of the ®,+ and ®,-are calculated with a finer time window, and the result is shown in Fig. 3 for the rigidity range from 1.00 to 1.71 GV. Figure 3 The moving average of ®,+ and ®,-with a time window of 2 BRs and a step of 1 day is shown in Fig. 3(b). The detailed behavior around dips IV and V is shown in Fig. S18 in Supplemental Material [21].

To analyze the significance of the structures in the positron-electron hysteresis, we study the difference of @®,-at the same ®,+, one in the first half and one in the second half of each region, IV and V (see Fig. S18 and the description in Supplemental Material [21] for details). The significance at the rigidity interval [1.00-1.71] GV for region IV is > 10o [see Fig. S18(c)] and for region V is 40 [see Fig. S18(d)].

The structures in the observed hysteresis in 2015 and 2017 between ®,+ and ®,-are similar to those observed between ®,-and ®, [14] and are likely caused by two series of interplanetary coronal mass ejections [37]. The clear deviation, regions IV and V in Fig. 3 [see also [21]) shows the daily ©, and ®, as a function of time over the entire period for the rigidity range from 1.00 to 1.71 GV. ®, versus ®,+, calculated with a moving average of 2 BRs and a step of | day, is shown in Figs. S19(b)-S19(d) in Supplemental Material [21]. As seen, a nearly linear correlation between positron and proton fluxes is observed, and no significant structures are found.

To compare the daily time variations of ®,+ and ®,, we fit a linear relation between the relative variations of the fluxes for the ith rigidity bin (R;, R; + AR;) as

where k' is the slope of the linear dependence for that bin and (®'.) and (®',) are the positron and proton fluxes in the ith rigidity bin averaged over the entire period, respectively.

Examples of fits of the daily positron and the daily proton fluxes to Eq. ( 2) are shown in Fig. S20 in Supplemental Material [21] for six consecutive rigidity (a) k parameter values obtained from the linear fits to the relative variation of the positron and proton fluxes as function of rigidity [see Eq. ( 2)] and (b) significance, in units of o, of the deviation of the parameter k from unity as a function of rigidity. As seen, k gradually increases with rigidity and is significantly (> 50) greater than unity in the rigidity range from 1.00 to 7.09 GY, indicating that the positron flux is more modulated than the proton flux in this rigidity range. bins from 1.00 to 5.90 GV. Figure 4(a) shows the results of the k' as a function of rigidity. As seen, k' gradually increases with rigidity from 1.055 + 0.004 at the rigidity bin [1.00-1.33] GV to 1.20+0.03 at the rigidity bin [5.90-7.09] GV. As shown in Fig. 4(b), k! is greater than unity with a significance greater than 5o for rigidities from 1.00 to 7.09 GV, indicating that the positron flux is more modulated than the proton flux in this rigidity range.

At a given rigidity below 7 GV, AMS observed that helium, which has a lower velocity than protons, is modulated more than protons [16]. In this Letter, we observe that, remarkably, positrons, which have a higher velocity, are also modulated more than protons. The contradiction in velocity dependence cannot be explained only by differences in the diffusive processes, since these are commonly accepted to be proportional to the velocity. Our simultaneous results on the velocity dependence of positrons, protons, and helium require a comprehensive model to consider other important effects, such as convection, adiabatic energy changes, and the shape of the flux rigidity dependence outside the heliosphere [38].

In conclusion, we presented the precision measurements of daily cosmic positron fluxes spanning 11 years over a rigidity range from 1.00 to 41.9 GV based on 3.4 x 10° positrons. The positron fluxes exhibit variations on multiple timescales. In the 11-year period, the positron fluxes show distinctly different time variations from the electron fluxes at short and long timescales. A hysteresis between the electron flux and the positron flux is observed with a significance greater than 5o at rigidities below 8.5 GV, and significant structures in the electron-positron hysteresis are observed corresponding to sharp variations of both fluxes. On the contrary, positron and proton fluxes show nearly identical time variation. Remarkably, positron fluxes are modulated more than proton fluxes with a significance greater than 5o for rigidities below 7 GV. These continuous daily positron fluxes, together with AMS daily electron, proton, and helium fluxes over an 11-year solar cycle, provide unique input to the understanding of both the charge-sign and mass dependencies of cosmic rays in the heliosphere.

We are grateful for important physics discussions with Igor Moskalenko and Subir Sarkar. 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 12 years. We are grateful for the support of Glen