solar magnetic helicity injected into the heliosphere - IOPscience

The Astrophysical Journal, 705:L48–L52, 2009 November 1  C 2009.

doi:10.1088/0004-637X/705/1/L48

The American Astronomical Society. All rights reserved. Printed in the U.S.A.

SOLAR MAGNETIC HELICITY INJECTED INTO THE HELIOSPHERE: MAGNITUDE, BALANCE, AND PERIODICITIES OVER SOLAR CYCLE 23 M. K. Georgoulis1,2 , D. M. Rust1 , A. A. Pevtsov3 , P. N. Bernasconi1 , and K. M. Kuzanyan4,5 2

1 The Johns Hopkins University Applied Physics Laboratory, Laurel, MD 20723, USA Research Center for Astronomy and Applied Mathematics, Academy of Athens, Athens GR-11527, Greece 3 National Solar Observatory, Sacramento Peak, Sunspot, NM 88349, USA 4 National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, China 5 IZMIRAN, Troitsk, Moscow Region 142190, Russia Received 2009 June 9; accepted 2009 September 22; published 2009 October 12

ABSTRACT Relying purely on solar photospheric magnetic field measurements that cover most of solar cycle 23 (1996–2005), we calculate the total relative magnetic helicity injected into the solar atmosphere, and eventually shed into the heliosphere, over the latest cycle. Large active regions dominate the helicity injection process with ∼5.7×1045 Mx2 of total injected helicity. The net helicity injected is 1% of the above output. Peculiar active-region plasma flows account for ∼80% of this helicity; the remaining ∼20% is due to solar differential rotation. The typical helicity per active-region CME ranges between (1.8–7) × 1042 Mx2 depending on the CME velocity. Accounting for various minor underestimation factors, we estimate a maximum helicity injection of ∼6.6 × 1045 Mx2 for solar cycle 23. Although no significant net helicity exists over both solar hemispheres, we recover the well-known hemispheric helicity preference, which is significantly enhanced by the solar differential rotation. We also find that helicity injection in the solar atmosphere is an inherently disorganized, impulsive, and aperiodic process. Key words: Sun: atmosphere – Sun: coronal mass ejections (CMEs) – Sun: magnetic fields – Sun: photosphere

magnetic field measurements. We further investigate (1) whether this helicity is balanced, and (2) whether periodicities other than the solar cycle exist in the timeseries of helicity buildup in the solar atmosphere. The analysis methods are described in Section 2, with our results in Section 3, and the discussion and our conclusions in Section 4.

1. INTRODUCTION The discovery of magnetic helicity patterns in the Sun (Rust 1994; Pevtsov et al. 1994, 1995) confirmed the importance of helicity in solar activity (Berger & Field 1984; Seehafer 1990) and triggered a strong interest in magnetic helicity and its role in the solar cycle. This interest is justifiable—magnetic helicity signals the existence of magnetic energy beyond the “ground” (current-free) state but, unlike magnetic energy, it cannot be dissipated in magnetic reconnection episodes (Berger 1999). As a result, magnetic helicity emerging with magnetic flux from beneath the photosphere builds up in the corona and must be transported to the heliosphere which, many believe, necessitates coronal mass ejections (CMEs; Low 1994; Rust 1994; Rust & Kumar 1994; Bieber & Rust 1995). It is comforting to believe that magnetic helicity generated by the solar dynamo should be balanced between the solar hemispheres; that is, algebraically summing to zero. A strict theoretical or observational proof is lacking, however. Fast astrophysical dynamos do not depend on the existence of magnetic helicity and, if helical, they may or may not give rise to a net helicity effect (e.g., Gilbert et al. 1988; Hughes et al. 1996; Vishniac & Cho 2001; Brandenburg 2005). Solar observations can determine whether helicity in the two hemispheres is balanced and how much magnetic helicity is injected in the solar atmosphere and eventually escapes in the heliosphere over a typical solar cycle. The latter has been so far inferred semianalytically or by models involving interplanetary measurements at 1 AU, with estimated helicity ranging between ∼2 × 1045 Mx2 (Bieber & Rust 1995) and ∼1046 Mx2 (DeVore 2000). Note that helicity injection in the solar atmosphere may be very different from total solar helicity production, which may be even an order of magnitude larger, with only a small part of helicity ever escaping from the solar interior (Berger & Ruzmaikin 2000). Here we perform the first calculation of the solar magnetic helicity shed into the heliosphere over cycle 23 using only solar

2. THE ANALYSIS 2.1. Previous Work LaBonte et al. (2007) performed the groundwork for this analysis by studying the evolution of 393 solar active regions (ARs) observed by the Michelson Doppler Imager (MDI; Scherrer et al. 1995) onboard the Solar and Heliospheric Observatory (SOHO) between mid-1996 and mid-2005. The data set included many ARs with major flares, along with others hosting only subflares. The studied ARs were the largest in the nine-year period of interest. Each AR was observed for up to 6–7 days, which is the time required to traverse an 82 deg “observation zone” from 41◦ E to 41◦ W, imposed to avoid extreme projection effects in the measured line-of-sight magnetic field. To quantify the impact of projection effects, we repeated this analysis using a 60 deg observation zone (30◦ E to 30◦ W). The difference in the results was marginal (∼5%), so the original 82 deg zone was also adopted for this work. After approximately correcting for projection effects, calculating the heliographic plane, and interpolating the normal magnetic field component on it, LaBonte et al. (2007) calculated the injection rate (dHm /dt) of the relative magnetic helicity Hm in the ARs via the equation (D´emoulin & Berger 2003)  dHm = −2 Bn u · Ap dS, (1) dt S where Bn is the normal field component on the heliographic plane S, Ap is the current-free vector potential (calculated L48

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Figure 1. Relative magnetic helicity injected in the solar atmosphere from the 393 largest ARs of solar cycle 23. Helicity due to the solar differential rotation is included in the left column of plots (a–c) but is removed in the right column of plots (d–f). The rows include helicity injected in both solar hemispheres (a, d), in the northern (b, e), and in the southern (c, f) hemisphere. The reddish line in (a) and (d) indicates the 60 day averaged total solar flux, as calculated by SOHO/MDI over the period of interest (readings in the right ordinate).

following Chae 2001), and u is the flow velocity field in the ARs. Following the above authors, the velocity field was approximated by the local correlation tracking (LCT) velocity (November & Simon 1988). One may integrate (dHm /dt) over a given interval to calculate the  total injected helicity ΔHm over this interval. From ΔHm = (dHm /dt)dt, we have  ΔHm =

Hm(max)

dHm = Hm(max) − Hm(min) ,

(2)

Hm(min)

where Hm(min) , Hm(max) are the extremes of the relative helicity. Evidently, this analysis cannot tell whether the injected helicity ΔHm adds to or subtracts from the helicity budget of an AR because the total helicity of the AR at the start of the observation is unknown. We account for and discuss this effect in Section 3. 2.2. Present Analysis

of an AR. When more than one CMPs fall into the same time bin, we algebraically sum the respective total injected helicities. Figure 1 shows the resulting timeseries. In the left column (Figures 1(a)–(c)), the total injected helicities include the contribution from solar differential rotation. This contribution has been removed in the timeseries in the right column (Figures 1(d)–(f)) as discussed in LaBonte et al. (2007). Figure 1 implies that helicity injection in ARs over a solar cycle is dominated by relatively few impulses, each corresponding to a few ARs, at most. For each helicity timeseries, we calculate the following quantities. 1. The total right-handed helicity for the N+ ARs with righthanded total injected helicities ΔHm(+) =

N+ 

ΔHmi ;

where

ΔHmi > 0.

(3)

i=1

We bin the median observation times for all 393 ARs in intervals of 0.01 yr, or ∼3.65 days. To each median time, we assign the respective total injected helicity ΔHm calculated in the aforementioned 82 deg observation zone. Since this zone is symmetric to the central solar meridian, the median time is approximately the instant of central meridian passage (CMP)

2. The total left-handed helicity for the N− ARs with lefthanded total injected helicities ΔHm(−) =

N−  i=1

ΔHmi ;

where

ΔHmi < 0.

(4)

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Table 1 Results of the Analysis Applied to the Timeseries of Figure 1 Figure 1(a) 1(b) 1(c) 1(d) 1(e) 1(f)

N

Hemisphere

393 197 196 393 197 196

Both Northern Southern Both Northern Southern

ΔHm(+) ΔHm(−) (×1044 Mx2 ) (×1044 Mx2 ) 5.37 0.9 4.47 4.12 1.66 2.46

−5.45 −4.23 −1.22 −4.28 −2.53 −1.75

εHm

Differential Rotation?

−0.0076 −0.65 0.57 −0.019 −0.21 0.17

YES YES YES NO NO NO

Note. Shown are the number N of ARs used in each calculation, the total produced right- and left-handed relative helicity, ΔHm(+) and ΔHm(−) , respectively, and the helicity imbalance εHm .

3. The imbalance of the total injected helicity for the N = N+ + N− ARs εHm =

ΔHm(+) + ΔHm(−) . N i=1 |ΔHmi |

(5)

In case both hemispheres are considered, N = 393. For the northern and southern hemispheres, we have N = 197 and N = 196, respectively. To search for possible periodicities in the timeseries of Figures 1(a) and (d), we calculate their periodograms (Scargle 1982) as implemented by Horne & Baliunas (1986). We also calculate (1) the false alarm probability (fap), and (2) the approximate full width at half-maximum (FWHM) of the primary periodogram peaks. We treat (1/2)FWHM as the uncertainty of the periods associated with these peaks. We investigated periods ranging between ∼3 time bins (10 days) and ∼137 time bins (500 days). 3. RESULTS The results of Equations (3)–(5) for all timeseries of Figure 1 are summarized in Table 1. We immediately notice that regardless of including the helicity due to solar differential rotation the imbalance εHm appears small (between 0.76% and 1.9%) when both hemispheres are considered. Calculating εHm for each hemisphere separately reveals the expected hemispheric helicity preference with left-handed and right-handed excess of helicity in the northern and southern hemispheres, respectively (Pevtsov et al. 1995). Note, however, that this hemispheric preference is much stronger when the differential-rotation helicity is taken into account (Figures 1(a)–(c)). This finding was also reported by LaBonte et al. (2007) who concluded that the hemispheric helicity preference is primarily imposed by the differential-rotation helicity term although this term has smaller amplitude. The helicity injection purely due to flow velocities within ARs is more intense but much more disorganized. Although the imbalances εHm in Figures 1(a) and (d) appear small, we must achieve independent confirmation that helicity is indeed balanced. For this we randomly changed the signs— but not the amplitudes—of the different peaks in Figures 1(a) and (d). We then calculated εHm for each randomized timeseries. Performing this test numerous times (104 randomizations here), we counted the number of cases where the randomized εHm was smaller than the actual εHm . In the case of Figure 1(a), the randomized εHm < 0.0076 in ∼4.2% of the cases. Therefore, the likelihood that global helicity balance due to differential rotation and flows peculiar to ARs does not occur by chance is

Figure 2. Periodograms for the timeseries of (a) Figure 1(a), (b) Figure 1(d), and (c) the SOHO/MDI total magnetic flux. The primary peaks are indicated by ovals. The yellow-shaded zone is centered on one solar rotation period (∼27.3 days) and has a half-width equal to one time bin (3.65 days). The dotted line indicates the fap near-certainty line (fap = 0.99).

>95%. For Figure 1(d) the randomized εHm < 0.019 in ∼9.5% of the cases, so the likelihood that global helicity balance due to peculiar AR plasma flows alone does not occur by chance is >90%. Figure 2 depicts the periodograms of Figures 1(a) in panel (a) and 1(d) in panel (b) in comparison with the periodogram of the total SOHO/MDI flux over the period of interest shown in (c). Clearly, only the total solar flux shows a prominent periodicity around the solar synodic rotation period of 27.3 days. The periodogram of the helicity injection timeseries without differential rotation (Figure 2(b)) roughly follows this peak (23.7 ± 0.12 days), albeit with a substantial fap  0.93. Including differential rotation (Figure 2(a)) the periodogram peaks at 14.3 ± 0.1 days, also with a substantial fap  0.78. Therefore, contrary to the total solar flux, the total helicity injection timeseries for solar cycle 23 does not show any prominent periodicities. We now calculate the total magnetic helicity injected in the solar atmosphere over solar cycle 23. From Table 1 (first row),

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we find ΔHm(+) +|ΔHm(−) |  1.08×1045 Mx2 . This number needs to be corrected by a factor of (360/82) because we observe an 82 deg meridional zone of the solar sphere—assuming, of course, that the helicity injection in the unobserved part occurs at the same rate as in the observed zone. This raises the total helicity injected to ∼4.74 × 1045 Mx2 and should now reflect the upper limit of helicity budgets, rather than the integrated helicity rates, in the studied ARs. To account for the fact that our observations cover most (∼9.03 yr) but not the entire (∼11.3 yr) solar cycle 23, we have fitted the averaged total solar magnetic flux of Figures 1(a) and (d) by a chi-square function that can model non-symmetric cycles, with a prolonged decay phase in our case. Integrating the best-fitted flux over time, we find that the observed period includes ∼91% of the total flux of the cycle. Given that magnetic helicity is proportional to the square of magnetic flux, the total helicity calculated above may be enhanced by a factor of 1/0.912  1.21 to yield a total helicity ∼5.74 × 1045 Mx2 for solar cycle 23. Ignoring the solar differential rotation, the total injected helicity is ∼ 4.46 × 1045 Mx2 . Therefore, differential rotation is responsible for only ∼22% of the total helicity injection in the solar active-region atmosphere. 4. SUMMARY AND DISCUSSION We calculated the total relative magnetic helicity injected into the solar atmosphere, and hence transported to the heliosphere, over the latest solar cycle 23. We further determined with likelihood >95% that the Sun produces no net helicity. Finally, we looked for periodicities in the helicity injection timeseries and found none to be convincing. The heliosphere over cycle 23 was supplied with a total helicity of ∼5.7 × 1045 Mx2 . About 80% of it, or ∼4.5 × 1045 Mx2 , stems from peculiar plasma flows in the studied ARs. The remaining ∼1.2 × 1045 Mx2 comes from solar differential rotation acting on these ARs. Although an upper limit of the helicity injected in the studied ARs, this budget does not account for helicity injection due to (1) ARs that were too small to satisfy the selection criteria of LaBonte et al. (2007), (2) quiet-Sun magnetic structures, and (3) the perceived underestimation of helicity injection by the LCT velocity (Chae 2007). Accounting for these effects, we start with a crude calculation of the helicity contribution of smaller ARs: our studied 393 ARs were a mere ∼14% of the 2811 magnetic structures that were registered as ARs by NOAA over the nine-year period of interest. The median flux of the studied ARs was Φmed ∼ 1022 Mx, while the smallest AR of our sample had a median flux Φmin ∼ 2.8 × 1020 Mx. Assuming that all rejected ARs were smaller than the smallest AR used in this study, their maximum total helicity should be a factor of (2811/393 − 1) × (Φmin /Φmed )2 ∼ 0.005 of the total helicity calculated in Section 3, assuming a characteristic twist in each AR. Therefore, the studied 14% of ARs contribute ∼99.5% of the total helicity injected by ARs. For the quietSun helicity contribution, Equation (1) gives a helicity injection rate ∼4.3 × 1035 Mx2 s−1 and for the entire solar cycle (∼3.56 × 108 s) we obtain ∼1.5 × 1044 Mx2 . For this we have assumed Ap ∼ Bn Δl, where Δl  1.425 × 108 cm (1. 98), u  5 × 104 cm s−1 as a typical quiet-Sun velocity, and mean Bn ranging between 1.39 G (equator) and 0.08 G (poles), as modeled by Scherrer et al. (1977)—see also Boberg et al. (2002). Our calculated quiet-Sun helicity is almost an order of magnitude larger than the 2 × 1043 Mx2 of (mutual)

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helicity calculated by Welsch & Longcope (2003), but still corresponds to only ∼2.5% of the total helicity calculated for the 393 studied ARs. Finally, Chae (2007) reported a systematic but small (∼10%) underestimation of the helicity injection due to the LCT velocity. Overall, our total helicity of ∼5.7 × 1045 Mx2 might be enhanced by a factor of 1.15, at most, to account for all sources of underestimation. This raises the maximum total magnetic helicity shed into the heliosphere during solar cycle 23 to ∼6.6 × 1045 Mx2 . This helicity is ∼3.3 times larger than the estimate of Bieber & Rust (1995) but smaller than the estimate of 1046 Mx2 by DeVore (2000) for solar cycle 21. The difference with DeVore might be either due to the stronger cycle 21 or, simply, because of the factor-of-two error that he assigned to his results. We attempt here a rough estimation of the helicity shed into the heliosphere over a typical solar cycle using the total magnetic flux emerged in the solar atmosphere during, say, the last three cycles (Arge et al. 2002). Total (unsigned) fluxes equal to Φ21 ∼ 8×1025 Mx, Φ22 ∼ 7.5 × 1025 Mx, and Φ23 ∼ 6 × 1025 Mx for cycles 21, 22, and 23, respectively, might give total respective helicities Hm(21) ∼ (Φ21 /Φ23 )2 Hm(23) ∼ 1.2 × 1046 Mx2 (very close to the estimate of DeVore (2000)) and Hm(22) ∼ (Φ22 /Φ23 )2 Hm(23) ∼ 1046 Mx2 for our calculated Hm(23) ∼ 6.6 × 1045 Mx2 . To infer a typical helicity content per active-region CME, we utilize the SOHO/LASCO CME Catalog (Yashiro et al. 2004). For active-region CMEs with velocities VCME  750 km s−1 (Sheeley et al. 1999) and VCME  900 km s−1 (Georgoulis 2008), the catalog lists 1273 CMEs and 662 CMEs, respectively, over our nine-year study period. The helicity injected by ARs over this period is ∼4.7 × 1045 Mx2 . Assuming that LASCO detects only CMEs launched in the earthward solar hemisphere, that most likely overestimates the total number of active-region CMEs, we can enhance the above numbers by a factor of 2 and calculate a lower limit of the CME helicity. This limit is ∼1.8 × 1042 Mx2 and ∼ 3.5 × 1042 Mx2 for VCME  750 km s−1 and VCME  900 km s−1 , respectively. Assuming now that LASCO detects all CMEs, including those launched in the farside, we can find from the above numbers an upper limit of the CME helicity to be ∼ 3.6 × 1042 Mx2 and ∼ 7 × 1042 Mx2 for VCME  750 km s−1 and VCME  900 km s−1 , respectively. Typical CME helicities should be biased toward these upper limits because LASCO observes most large CMEs independently of their origin location. In any case, our estimates are in good agreement with the estimates of 2 × 1042 Mx2 (DeVore 2000; D´emoulin et al. 2002) and ∼5 × 1042 Mx2 (Lepping et al. 1990) per CME. The net solar helicity, if any, is most likely very small. A single AR injecting a right-handed helicity of ∼8 × 1042 Mx2 would be sufficient to set the net helicity to zero. This needed helicity is ∼27% of the total helicity apparently produced by the 2418 ARs that were excluded from this study and ∼5% of the total helicity apparently produced by the quiet Sun. Therefore, we have no sound reason to believe that there was any significant or systematic net helicity over solar cycle 23. The timing of helicity injection in the solar atmosphere shows no credible periodicity, either (Figure 2). The total solar magnetic flux shows a prominent periodicity around the synodic solar rotation period of ∼27.3 days (Figure 2(c)) that is long-known (e.g., Sheeley & DeVore 1986), quasi-stationary, and attributed to persisting solar activity complexes (Henney & Harvey 2002) or active longitudes (Vitinskij 1969). For the helicity injection periodograms, however, we believe— and have verified by Monte Carlo simulations—that the weak

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peak close to the solar rotation period (Figure 2(b)) is an artifact occurring because we observe large ARs, typically surviving for more than a solar rotation (∼4 weeks), for about a week. This ∼27-day peak appears but is even weaker in the periodogram of Figure 2(a) that peaks primarily at ∼14.3 days, a likely harmonic of the ∼27-day peak, probably enhanced by differential rotation. Regardless, the substantial false alarm probabilities un-mistakenly characterize both primary peaks in Figures 2(a) and (b) as insignificant. Indeed, the only compelling periodicity in the timeseries of solar magnetic helicity injection, found to be dominated by the active-region contribution, is the solar cycle itself. This work was borne out of a discussion on magnetic helicity held among the authors in 2008 July at JHU/APL. We thank the anonymous referee for helping us improve the accuracy of the text. Our thanks are also due to the tremendously successful SOHO mission, its MDI magnetograph, and the MDI team at Stanford University that provided us with the means to perform this study. K.K. acknowledges support from the National Natural Science Foundation of China and Russian Fund for Basic Research under grants 08-02-92211 and RFBR grant 07-0200246. National Solar Observatory (NSO) is operated by the Association of Universities for Research in Astronomy, AURA Inc., under cooperative agreement with the National Science Foundation. The JHU/APL authors wish to acknowledge partial support by the NASA Guest Investigator grant NNX01AJ10G. REFERENCES Arge, C. N., Hildner, E., Pizzo, V. J., & Harvey, J. W. 2002, J. Geophys. Res., 107, SSH 16 Berger, M. A. 1999, Plasma Phys. Control. Fusion, 41, B167

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