Helium Variation Across Two Solar Cycles Reveals A Speed-Dependent Phase Lag

We study the relationship between solar wind helium to hydrogen abundance ratio ($A_\mathrm{He}$), solar wind speed ($v_\mathrm{SW}$), and sunspot number (SSN) over solar cycles 23 and 24. This is the first full 22-year Hale cycle measured with the Wind spacecraft covering a full cycle of the solar dynamo with two polarity reversals. While previous studies have established a strong correlation between $A_\mathrm{He}$ and SSN, we show that the phase delay between $A_\mathrm{He}$ and SSN is a monotonic increasing function of $v_\mathrm{SW}$. Correcting for this lag, $A_\mathrm{He}$ returns to the same value at a given SSN over all rising and falling phases and across solar wind speeds. We infer that this speed-dependent lag is a consequence of the mechanism that depletes slow wind $A_\mathrm{He}$ from its fast wind value during solar wind formation.


INTRODUCTION
Fully ionized hydrogen or protons (p) and fully ionized helium or alpha particles (He or α) are the two most abundant solar wind ion species. The former comprises ∼ 95% of the solar wind ions and the later ∼ 4%, both by number density. Heavier, minor ions constitute the remaining. The alpha particle abundance (A He = 100 × n α /n p ) strongly correlates with solar activity, as indicated by the sunspot number (SSN) (Aellig et al. 2001;Kasper et al. 2007Kasper et al. , 2012. The cross correlation and slope between A He and SSN varies with solar wind speed (v sw ); is strongest in slow wind; markedly falls off above v sw = 426 km s −1 , where A He takes on a stable value between 4% and 5%; and vanishes in the solar wind for speeds below v 0 = 259 km s −1 (Kasper et al. 2007(Kasper et al. , 2012. This helium vanishing speed is within 1 σ of the minimum observed solar wind speed (Kasper et al. 2007), indicating that helium may be essential to solar wind formation in the corona.
In addition to SSN, many other indicators of solar activity also follow a similar ∼ 11 year cycle (Ramesh & Vasantharaju 2014) that demonstrate a distinct phase-offset with SSN, which has been referred to as a hysteresis-like effect. These offsets range from 30 days (Bachmann & White 1994) to 450 days (Temmer et al. 2003). Goelzer et al. (2013) have shown a similar phase lag in the interplanetary magnetic field's response to changes in SSN.
Using observations from the Wind Faraday cups (FC), we extend the study of A He variation with SSN and v sw by Kasper et al. (2007Kasper et al. ( , 2012 to include more than 23 years. This time period encompasses solar cycles 23 and 24 along with the end of solar cycle 22, thereby covering one Hale cycle. In other words, an idealized sun with a pure dipole magnetic field would have experienced two 1 9 9 5 1 9 9 6 1 9 9 7 1 9 9 8 1 9 9 9 2 0 0 0 2 0 0 1 2 0 0 2 2 0 0 3 2 0 0 4 2 0 0 5 2 0 0 6 2 0 0 7 2 0 0 8 2 0 0 9 2 0 1 0 2 0 1 1 2 0 1 2 2 0 1 3 2 0 1 4 2 0 1 5 2 0 1 6 2 0 1 7 2 0 1 8 polarity reversals and be returning to the configuration it had at the end of cycle 22. In this work, we expand on the results of Kasper et al. (2007Kasper et al. ( , 2012. We show a positive correlation between A He and SSN across multiple solar cycles. In the slowest wind, we find a characteristic A He that is consistent across multiple minima and maxima. Examining this relationship over one Hale cycle, we demonstrate that the phase lag between A He and SSN found by Feldman et al. (1978) is a monotonically increasing function of v sw . This delay is characteristic to a given v sw and, at any one v sw , a cyclic delay is sufficient to correct for this lag. Unexpectedly, A He returns to similar values in both maximum 23 and maximum 24 even though SSN Max,24 < SSN Max,23 . Our results are consistent when using the 13-month smoothed, monthly, and daily sunspot numbers.
The remainder of this Letter is dedicated to analyzing and interpreting this speed-dependent lag. Section 2 describes the observations and FC specifics that are key to this study. Section 3 describes the variation of A He with v sw and SSN over two solar cycles. Section 4 analyzes the delay in response of A He to changes in SSN as a function of v sw . Section 5 presents the relationship between A He and SSN in various v sw quantiles, corrected for the delay of peak cross-correlation coefficient. Here, we show that correcting for the lag in A He 's response to changes in SSN reduces this hysteresis effect to a linear relationship. In Section 6, we use A He 's dependence on SSN to investigate the robustness of the A He , v sw , SSN relationship reported by Kasper et al. (2007). In Section 7, we interpret our results and extend earlier hypotheses regarding two sources of slow solar wind. Finally, Section 8 summarizes these results and discusses future work.

DATA SOURCES
The Wind spacecraft has been in continuous operation since its launch in the fall of 1994. Ogilvie et al. (1995) provide a detailed description of the Solar Wind Experiment (SWE) Faraday cups (FC). Kasper et al. (2006) introduce techniques for optimizing the algorithms that extract physical quantities from FC measurements. Maruca & Kasper (2013) and Alterman et al. (2018) build on these algorithms. These data have resulted high precision solar wind measurements of alpha particles (Kasper et al. 2006;Maruca & Kasper 2013) and multiple proton populations (Alterman et al. 2018). The FC ion distributions are available on CDAweb 1 and SPDF 2 . We follow Alterman et al. (2018) and reprocess the raw measurements to extract two proton populations (core and beam) along with an alpha particle population. The proton core is the population with the larger of the two proton densities. We calculate the solar wind speed as the proton center-of-mass velocity and treat the proton core as the proton density when calculating A He .
Two aspects of FCs are key to this work. First, FCs are energy-per-charge detectors. In the highly supersonic solar wind, alpha particles and protons are well separated by the instrument even when they are comoving (Kasper et al. 2008(Kasper et al. , 2017Alterman et al. 2018), as is commonly the case in slow solar wind. Second, the measurement quality has been stable and accurate throughout the mission (Kasper et al. 2006). These two FC characteristics enable our study of A He variation with a single dataset from one instrument suite covering the 23 years necessary to observe one Hale cycle.  Kasper et al. (2007 in each) and can be considered an update to their results. The solar wind speed measurements from the full mission have been split into 12 quantiles. The fastest and slowest quantile have been discarded due to measurement and statistical considerations. Of those quantiles retained, the lower edge of the slowest is 312 km s −1 and the upper edge of the fastest is 574 km s −1 . Consequently, this study is limited to solar wind typically categorized as slow or slow and intermediate speed. 3 As in prior work, the abundance in each v sw quantile is averaged into 250 day intervals. 1 https://cdaweb.sci.gsfc.nasa.gov/misc/NotesW.html# WI SW-ION-DIST SWE-FARADAY 2 ftp://spdf.gsfc.nasa.gov/pub/data/wind/swe/swe faraday/ 3 To be consistent with prior work (e.g. Kasper et al. (2007Kasper et al. ( , 2012), we will use slow and fast to refer to the different extremes presented here. However, the reader should known that truly fast solar wind is excluded from our study.
The 13-month smoothed sunspot number (SILSO World Data Center 2018; Vanlommel et al. 2005, SSN) is interpolated to the measurement time; averaged into the same 250 day intervals as A He ; and plotted on the secondary y-axis. The legend indicates the middle of the solar wind speed quantile along with its corresponding Spearman rank cross correlation coefficient between A He and SSN. For brevity, we henceforth indicate the Spearman rank cross-correlation coefficient between A He and SSN as ρ(A He , SSN). Fig. 1 indicates that ρ(A He , SSN) peaks at v sw = 355 km s −1 . The present drop in A He reflects that the sun is entering Minimum 25. In contrast to the results of Kasper et al. (2007Kasper et al. ( , 2012, ρ(A He , SSN) > 0.6 indicates a meaningful cross-correlation in all but the fastest reported quantile with v sw = 542 km s −1 and ρ(A He , SSN) ≥ 0.7 is highly significant up to v sw = 426 km s −1 . As Feldman et al. (1978) noted, there is a phase offset between A He and ρ(A He , SSN). Although the cycle 23 SSN amplitude is less than the cycle 24 amplitude, A He unexpectedly returns to comparable values during each maximum.

TIME-LAGGED CROSS CORRELATION
Visual inspection indicates a clear time lag between A He and SSN. To quantify this lag, we calculate ρ(A He , SSN) as a function of delay time applied to SSN from −200 days to +600 days in steps of 40 days-slightly longer than one solar rotation-for each v sw quantile. We smooth these results to reduce the impact of discretization. The delay time is the time for which ρ(A He , SSN) peaks as a function of delay. Panel (a) of Fig. 2 plots the peak cross correlation coefficient as a function of v sw for observed (empty marker) and delayed (filled marker) SSN. Marker colors and symbols match Fig. 1 and are maintained throughout the Letter. Dotted lines connect the markers to aid the eye. To estimate the error in this calculation and its sensitivity to averaging timescale, we repeated it for averaging windows N days = 225 to N days = 275 in steps of 5 days. Because a trend is not apparent, we choose to quantify this variability as the standard deviation across N days and represent it as error bars centered on the N days = 250 averaging window utilized in this Letter.
Several features in Panel (a) of Fig. 2 stand out. First, it emphasizes that delayed ρ(A He , SSN) ≥ 0.7 is highly correlated for all v sw quantiles. Second, observed and delayed ρ(A He , SSN) peak at the same v sw = 355 km s −1 . Third, the change in ρ(A He , SSN) is largest and most visually striking in faster wind. However, smaller changes in slower wind's ρ(A He , SSN) are sta-tistically more significant because they are less likely to be due to random fluctuations.
Panel (b) of Fig. 2 examines τ , the delay of peak ρ(A He , SSN), as a function of v sw . A positive delay indicates that changes in SSN precede changes in A He . The insert at the top of the figure indicates the functional form, fit parameters, and quality metrics. As with Panel (a) of Fig. 2, the error bars indicate the variability of τ in each v sw quantile. Solving the fit equation for τ = 0, or the speed at which A He responds immediately to changes in SSN, results in v i = 200 km s −1 . Nevertheless, it is not unambiguously clear if delay time τ monotonically increases with v sw or there are two distinct delay times. If it is actually the latter, then A He in slow wind responds to changes in SSN with a delay time τ slow = 150 days; faster wind responds after τ fast > 300 days; and v i represents a non-trivial conflation of these two delays. If this is not the case, it may be that τ slow is the shortest delay with which A He responds to changes in SSN. As discussed below, in either case all helium released into the solar wind still lags changes in SSN. Panel (a) clearly shows the hysteresis pattern of A He as a function of SSN. As seen with other indices (e.g. Bachmann & White (1994)), time moves counterclockwise in this plot. 4 As noted by Bachmann & White (1994) for several solar indices, the clustering of data at small SSN indicates that the hysteresis effect is stronger at solar maximum and weaker at solar minimum.

PHASE DELAY
In panel (b), the larger R 2 indicates that this spread of A He about the trend decreases. Note that R 2 cor- responds to the square of the correlation coefficient of A He and SSN derived from a robust fit and not directly from the measurements. Although R is similar to ρ(A He , SSN), they are not trivially equal. That delayed χ 2 ν is markedly closer to unity indicates that a linear model better characterizes A He as a function of delayed rather than observed SSN. Because delayed SSN only reduces the spread of A He about the trend, it is expected that the trends and fit parameters in both cases are similar.
6. ROBUSTNESS OF A He (V ) Kasper et al. (2007) describe the relationship between A He and v sw in slow wind (v sw ≤ 530 km s −1 ) using data from a 2 year interval surrounding solar Minimum 23. They find that A He (v) = 1.63 × 10 −2 (v − v 0 ), where v 0 = 259 ± 12 km s −1 is the speed below which helium vanishes from the solar wind. The robust fits in Fig. 3 allow us to extract A He at zero solar activity for all v sw quantiles. This quantity, A He (SSN = 0), represents low solar activity conditions across this Hale cycle that are appropriate for comparison to the minimum 23 results from Kasper et al. (2007). Fig. 4 plots A He (SSN = 0) in all v sw quantiles for delayed SSN with unfilled markers. As observed SSN does not deviate from delayed SSN in this figure, it is omitted for clarity. The black dashed curve is the fit of A He (v) from Kasper et al. (2007). To better compare this analysis to the work of Kasper et al. (2007), filled markers present the results of repeating this analysis for SSN < 25, a range in SSN representative of solar minimum 23. That A He (SSN = 0) is smaller in this reanalysis using a restricted range of SSN further substantiates that our results are consistent with those of Kasper et al. (2007) even though ours cover multiple solar cycles, a larger range in solar activity conditions, and uses a different analysis technique. Furthermore, the agreement between these two distinct analysis techniques supports the interpretation that helium release is essential to solar wind formation (Kasper et al. 2007). The discrepancy between our fastest quantile with v sw = 542 km s −1 and their trend is expected because (1) it is outside of the speed range they fit and (2) they found thet A He at this and similarly high speeds takes on a stable value between 4% and 5%.    Kasper et al. (2007). That repeating our calculation with a reduced range in SSN shows better agreement with the results of Kasper et al. (2007) indicates that our results, covering the full range of solar activity in cycles 23 and 24, are consistent with their results from the two year interval surrounding minimum 23. and lags SSN by 125 days (Bachmann & White 1994). SXR is most common in Active Regions (AR) (van Driel-Gesztelyi & Green 2015), and lags SSN by 300 days to 450 days (Temmer et al. 2003).
While A He is approximately 8.5% within the sun's convection zone and out to the photosphere (Asplund et al. 2009;Laming 2015), it rarely exceeds 5% in the corona (Laming & Feldman 2003;Mauas et al. 2005). It has long been assumed that A He is initially modified in the photosphere. However, the speed-dependent lag in A He 's response to changes in SSN found here suggests additional processes at higher altitudes further modify helium's abundance. Slow solar wind's 150 day lag tracks lags in transition region and chromosphere structures, while faster wind's 300 day lag is more consistent with higher altitude structures in the corona. How could the transition region or corona modify the helium abundance? Kasper et al. (2007) propose that two mechanisms release fully ionized helium into the slow solar wind, one each in the streamer belt and ARs. ARs have a strong magnetic field that extends from the photosphere into the corona, originate well above the equatorial region, tend to migrate towards the equator as they get older, and have loops that tend to grow with age (van Driel-Gesztelyi & Green 2015). In contrast, the streamer belt has a weaker magnetic field, is composed of loops larger than those typical of ARs, is magnetically closed to the heliosphere, and is typically considered the source of slow solar wind (Eselevich & Eselevich 2006). Stakhiv et al. (2016) identify signatures of these two solar sources in ACE/SWICS composition measurements.
If there are two sources of slow wind, solar wind originating in the streamer belt is more processed than that originating in ARs, where SXR is enhanced. Slower wind A He (v sw < 375 km s −1 ) originates from the streamer belt with a phase delay τ slow = 150 days. It appears more depleted than faster solar wind from ARs that has a phase delay τ fast > 300 days. The magnitude of A He 's reduction from its photospheric value and the speed-dependent delay then reflect the extent to which a given source region is magnetically open to the heliosphere. As the phase delay between A He and SSN is an increasing function of v sw , ARs and the streamer belt may be two extreme cases along the continuum of slow wind helium depletion mechanisms.
For illustrative purposes, one candidate mechanism that may contribute to this processing is the FIP effect. The FIP effect is the empirical observation that solar wind ions are fractionated, or their abundances differ from their photospheric value based on their first ionization potential (Meyer 1991(Meyer , 1993Laming 2015, and references therein). Low FIP elements (FIP < 10 eV) tend to increase or experience an enhancement. This low-FIP enhancement also leads to an apparent depletion in high-FIP elements, as with helium. Under the framework of Rakowski & Laming (2012), time averaged coronal Alfvén waves create a pondermotive force that accelerates ions into the corona and leads to fractionation in coronal loops. The FIP effect is strongest in the upper chromosphere and lower transition region, weakest in regions of strong magnetic field, and stronger in longer loops (Rakowski & Laming 2012). Feldman et al. (2005) found that FIP bias in ARs increases with age.
However, this is just one of several possible mechanisms that could cause this phase lag. Other mechanisms that might impact the speed-dependent phase lag may include interchange reconnection (Fisk 2003) and gravitational settling (Hirshberg 1973;Borrini et al. 1981;Vauclair & Charbonnel 1991). Moreover, these are mechanisms are not mutually exclusive. Schwadron et al. (1999); Laming (2004); Rakowski & Laming (2012) include gravitational settling in their models of the FIP effect. Schwadron et al. (1999) also relies on interchange reconnection to create the magnetic structures necessary for FIP fractionation to occur. As Rakowski & Laming (2012) show, the combination of coronal loop length, differences in gravitational scale height, and the FIP effect can lead to the apparent depletion of A He . Whatever the underlying mechanism, it should also account for the observation that A He returns to a similar value during solar maximum, irrespective of SSN during maximum.

CONCLUSION
Following the methods of Kasper et al. (2007Kasper et al. ( , 2012, we have analyzed the relationship between A He and the 13-month smoothed sunspot number (SSN) by studying their cross correlation coefficient using 250 day averages. We have verified that our results are consistent when using the monthly and daily SSN. Our data covers 23 years, including cycle 23 and 24 along with the tail end of cycle 22. This time period is more than the 22 years of a Hale cycle over which the pure dipole field of an idealized sun would experience two polarity reversals and return to an initial configuration. As shown in Fig. 1, the present decrease in A He clearly demonstrates that we are entering solar Minimum 25. While the significance of the cross correlation coefficient ρ(A He , SSN) decreases with increasing v sw , Fig. 1 shows that ρ(A He , SSN) is meaningful up to v sw = 488 km s −1 and highly significant up to v sw = 426 km s −1 . A subject of future work is investigating why A He returns to a similar value in Maximum 24 even though cycle 24's amplitude is markedly smaller than cycle 23's. Feldman et al. (1978) comment on a phase offset between A He and SSN. Panel (b) of Fig. 2 reveals that (1) the length of this delay is an increasing function of v sw and (2) the v sw quantile most correlated with SSN does not change when SSN is appropriately delayed in each quantile. We have also argued that, although changes in ρ(A He , SSN) are most dramatic in faster v sw quantiles, the probability of smaller changes in slower wind's larger ρ(A He , SSN) is much smaller and therefore more significant.
Panel (b) of Fig. 2 presents the delay applied to SSN necessary to maximize ρ(A He , SSN) as a function of v sw . The delay is a monotonically increasing function of v sw and linear fit to this trend reveals that the speed at which A He responds instantaneously to changes in SSN is v i = 200 km s −1 . Yet the speed of instantaneous response is less than the vanishing speed, v i < v 0 . Therefore any helium released into the solar wind will necessarily response to changes in SSN after some delay. If trend in Panel (b) of Fig. 2 is correct, then the mini-mum delay in A He 's response to SSN is 68 ± 13 days, or approximately two Carrington Rotations. Here, we also note that there may be two distinct phase delays (τ slow and τ fast ) with which A He responds to changes in SSN and the fit quantity v i may be a conflation of the physics related to each phase delay. Under either interpretation, helium released into the solar wind is a delayed response to changes in SSN.
In Section 5, we present robust fits to A He as a function of observed and delayed SSN in each v sw quantile. It visually illustrates that applying a time delay to SSN reduces the spread of A He about its trend. In Section 6, we use helium abundance at zero solar activity derived from these fits to demonstrate that our results using 23 years of data are consistent with the trend found by Kasper et al. (2007) for a two year interval surrounding solar minimum 23.
In Section 7, we discuss how the demonstrated phase delay or hysteresis effect is qualitatively similar to the phase delays between SSN and many regularly observed solar indices (Ramesh & Vasantharaju 2014, and references therein). We note that the two aforementioned phase delays (τ slow and τ fast ) are consistent with Lα and SXR and that this consistency is indicative of two distinct source regions. Slower wind (v sw < 375 km s −1 ) with a lower A He originates in the streamer belt and responds to changes in SSN with characteristic delay time τ slow = 150 days. Faster wind with a larger A He originates in ARs and responds to changes in SSN with characteristic delay time τ fast > 300 days. These different delay times indicate that A He is processed by one or more mechanisms above the photosphere. Assuming that the results of Kasper et al. (2007) apply across the solar cycle and helium universally vanishes from the solar wind when v sw < 259 km s −1 irrespective of solar activity, one possible interpretation is that there is a minimum A He necessary for solar wind formation, the mechanisms that reduces A He to a value less than its photospheric value prevents solar wind release below the vanishing speed v 0 , and-using the fit from Panel (b) of Fig. 2-any helium that enters the solar wind is released after 68 days, approximately two Carrington rotations. If this is the case, helium in the high-speed solar wind may represent the solar wind's "ground state" (Bame et al. 1977;Schwenn 2006) and the observed depletion of A He is the result of source regions departing from states that release fast wind, i. e. those magnetically open to the heliosphere. A rigorous study of the relationship between A He and solar indices other than SSN may better constrain helium variation by source region and is a subject of future work. This work highlights the value of recent and forthcoming advances in heliophysics. Parker Solar Probe (Fox et al. 2016, PSP) launched in August, 2018 and completed its first perihelion in November of that year. Solar Orbiter (Müller et al. 2013, SolO) will launch in 2020. The thermal ion instruments on board (Kasper et al. 2016) provide an unprecedented opportunity to study the solar wind, its formation, and its acceleration. For example, PSP will make measurements near and below the Alfvén critical point, i.e. at distances within which mapping the solar wind to specific sources is significantly simplified in comparison with Wind. Mc-Mullin et al. (2016) anticipate that the Daniel K. Inouye Solar Telescope (DKIST) will begin operations in 2020. DKIST's Cryo-NIRSP instrument will be capable of simultaneously imaging solar helium at various heights in the corona. Combining DKIST measurements with PSP and SolO measurements will enhance our ability to differentiate between the mechanisms releasing helium into the solar wind-e.g. from the streamer belt or ARsand better constrain the delay in helium's response to changes in SSN.