The Solar Surface Di erential Rotation from Disk-Integrated Chromospheric Fluxes Robert A. Donahue Harvard-Smithsonian Center for Astrophysics 60 Garden Street, Cambridge MA, 02138, USA. (e-mail:
[email protected]) and Steven L. Keil USAF PL/GSSP National Solar Observatory, Sunspot NM, 88349, USA. (e-mail:
[email protected]) Received UNKNOWN;
accepted UNKNOWN
Submitted to: Solar Physics
{2{
ABSTRACT Disk-integrated solar chromospheric Ca II K-line (3933.68 A) uxes have been measured almost daily at Sacramento Peak Observatory since 1977. Using observing windows selected to mimic seasonal windows for chromospheric measurements of lower main-sequence stars such as those observed by Mount Wilson Observatory's HK Project, we have measured the solar rotation from the modulation of the Ca II K-line ux. We track the change of rotation period from the decline of Cycle 21 through the maximum of Cycle 22. This variation in rotation period is shown to behave as expected from the migration of active regions in latitude according to Maunder's \butter y diagram", including an abrupt change in rotation period at the transition from Cycle 21 to Cycle 22. These results indicate the successful detection of solar surface di erential rotation from disk-integrated observations. We argue that the success of our study compared to previous investigations of the solar surface di erential rotation from disk-integrated uxes lies primarily with the choice of the length of the time-series window. Our selection of 200 days is shorter than in previous studies whose windows are typically on the order of one year. The 200-day window is long enough to permit an accurate determination of the rotation period, yet short enough to avoid complications arising from active region evolution. Thus, measurements of the varitaion of rotation period in lower mainsequence stars, especially those that appear to be correlated with long-term changes in chromospheric activity (i.e., cycles), are probably evidence for stellar surface di erential rotation. Subject headings: stars: rotation { stars: activity { stars: chromospheres
{3{
1. Introduction The solar surface di erential rotation (SDR), visible in Maunder's \butter y diagram" (Maunder, 1913; Yallop and Hohenkerk, 1980) was rst detected independently by Carrington (1858, 1863) and Sporer (1861) from observations of the 11-yr sunspot cycle (Schwabe, 1843; Wolf, 1856). All stars with convective zones are expected to di erentially rotate, and the successful measurement of stellar di erential rotation will provide empirical constraints to models of the stellar dynamo, which is frequently posited as the cause of the solar magnetic activity (Parker, 1955; Rosner and Weiss, 1992). A broader perspective which includes information from lower main-sequence stars has not been generally forthcoming because direct con rmation of surface di erential rotation is not yet possible. While the solar SDR is well observed (e.g., Snodgrass, 1983), measurements of SDR for the Sun-as-a-star using di erent activity diagnostics would be helpful in estimating the feasibility of detecting stellar SDR. Several previous attempts have been made using 2.8 GHz (10.7 cm) radio ux (LaBonte, 1984) and Ca II plage data (LaBonte, 1982; Gilliland and Fisher, 1985) with negative or inconclusive results, suggesting that other factors, particularly active region (AR) evolution, make it dicult to accurately monitor changes in the rotation period which could indicate the presence of SDR. However, initial results for a few lower main-sequence stars suggest that some of these stars may indeed show behavior at least consistent with the presence of SDR: Baliunas et al. (1985) using three years of chromospheric Ca II H and K uxes listed 12 stars that appear to be potential SDR candidates. One of these stars, HD 114710 ( Comae), was studied further by Donahue and Baliunas (1992) who found not only a possible link between rotation period and the star's 17-yr activity cycle, but with a dependence opposite the Sun's: i.e., rotation period increases over the course of an activity cycle. Therefore, it appears that further analysis of the disk-integrated solar activity is necessary to determine if it is at all possible to measure
{4{ the solar SDR, and if not, to better understand why the Sun's known behavior is not measurable under the conditions which lower main-sequence stars are observed. We use the solar Ca II K line index which has been measured nearly continuously at Sacramento Peak Observatory as an activity index similar to the Mount Wilson Observatory (MWO) S index (Vaughn, Preston, and Wilson, 1978). We will attempt to mimic the MWO chromospheric time series as closely as possible in order to provide the best approximation to the conditions under which stellar SDR might be observed (including the presence of seasonal observing windows). Therefore, the successful detection of solar SDR such a time series will indicate the possibility that long-term measurements of lower main-sequence stars (i.e., over several years and through one or more activity cycles) may be successful in measuring stellar SDR.
2. Observations The observations have been described in detail by Keil and Worden (1984). Brie y, all observations were made with the Coelostat and horizontal Littrow spectrograph of the Evans (Big Dome) facility at Sacramento Peak. The solar image is focused by a cylindrical lens on the entrance slit of the spectrograph, which forms a one-dimensional image that is slightly smaller in height than the spectrograph slit. Excess light is blocked out by using a 12 mm decker slit. Scattered light is reduced rst from using a mask to block light re ected from the back of the spectrograph lens, and from using the Littrow in double-pass mode. The observed spectrum has a wavelength range of 3898 A to 3954 A with a stepsize of 5.5 m A. The instrumental noise and amount of scattered light are measured by closing the entrance slit and intermediate slits, respectively. For a single day's observation, typically 50 to 150 scans are averaged, depending on the weather conditions. The composite spectrum taken on 1 April 1982 is used as an arbitrarily-chosen reference pro le.
{5{ Each averaged spectrum was subjected to a high-frequency lter to remove some residual noise, and were wavelength-calibrated using crosscorrelation. Flux calibration was done in the manner of White and Livingston (1978), who de ne the average ux within a 0.528 A-wide window that is 1.187 A redward from the core of the K line to be equal to 16.2% of the continuum value at K. This calibration erroneously assumes that the wings of the Ca II K line do not vary with changes in activity: however, over the activity cycle, the variation in the wings will be 1%, which is much smaller than the variation in the K-line emission component. The K-line emission index is then measured as de ned by Wilson (1968): the equivalent width of a 1 A band centered on the K line core. In Figure 1 we plot a portion of a representative spectrum showing the center of the K line and the ux calibration bandpass. In anticipation of applying the results of this investigation to the time series of lower main-sequence stars, the Ca II K-line uxes were converted to the MWO S index, employing the solar 10.7 cm ux as an intermediary (Baliunas et al., 1995). This was necessary because the SPO K-line observations and Mount Wilson S observations do not overlap suciently to permit a direct calibration. Wilson (1978) observed the full Moon as a proxy for the Sun from 1966{1971 and 1974{1977. The HK Project re-initiated monthly observations of the Moon in 1993 in order to provide direct overlap with other solar activity indicies. The resulting time series is shown in Figure 2, where hS i = 0:179, higher than the mean activity index of 0.171 reported by Wilson (1978) whose data only covered Cycle 20, and not the more active Cycles 21 and 22.
3. Analysis The time series was split into 200-day intervals in order to mimic the seasonal windows which are inherent in stellar chromospheric data. Over the entire record, nineteen
{6{ \observing seasons" with at least thirty observations exist. For each of these seasons, a periodogram was calculated (Scargle, 1982) using the prescription of Horne and Baliunas (1986). This technique takes into account that the data are unevenly sampled and provides an estimate for the signi cance of the tallest peak in the periodogram, the false-alarm probability (FAP). The FAP is the likelihood that a peak of given height or higher could occur, assuming the data are purely Gaussian noise. In reality, there are several sources of noise in the data due to di erent processes occurring in the solar atmosphere, primarily AR evolution (discussed below) and none of them are Gaussian in origin. Therefore, we adopt a somewhat conservative value for the FAP, 0:1%, so that the e ects not included in the formal computation of the FAP do not result in over-con dence in the periods observed. After a periodogram was calculated if a signi cant peak was found, it was recorded and then ltered from the data to check for residual secondary peaks. This follows the technique of Baliunas et al. (1985) to look for multiple periods within an observing season. Filtering the data alters the determination of the FAP for secondary peaks because the variance of the data is changed, and because the removed sinusoidal variation at the computed frequency is assumed to be correct. However, the formal values of the period and the height of the secondary peak will still be an indication of the secondary FAP, provided that the initial period determination is correct (Baliunas et al., 1985). All periods measured were converted to sidereal rotation periods, in order to be directly comparable to lower main-sequence star rotation periods (Vaughan et al., 1981; Baliunas et al., 1983).
3.1. Solar Surface Di erential Rotation Table I shows the results of the search for seasonal rotation periods. The date range of the observations and number of observations (cols. 1 and 2), mean S value (col. 3),
{7{ determined period, P , and the uncertainty in the determination of the period (Kovacs, 1981), P (cols. 4 and 5), semi-amplitude (in S units, col. 6), and the FAP in percent (col. 7) are listed for each period determined. A total of nine rotation periods were found in eight observing seasons out of 19 samples of data: a success rate of 42%. Since ARs are almost always present on the Sun, this somewhat low success rate indicates that the e ects of AR evolution and the presence of multiple ARs on the surface at the same time probably do have an e ect upon the time series, making it suciently dicult to reliably and accurately determine a rotation from disk-integrated measurements. Near cycle minimum, ARs may not last long enough within the observing window to show repeated variations that would permit a determination of the rotation period. Figure 3 shows the behavior of rotation period with time. As hoped, a variation in rotation period occurs with time and solar activity cycle phase. During the decline of Cycle 21 (1984{1986), the rotation period decreases (i.e., faster angular rotation, !) coincident with ARs moving closer to the equator. At the beginning of Cycle 22 (circa 1987), the observed rotation period jumps to 28 days as ARs begin forming at mid-latitudes, and through the rise of Cycle 22 the trend of decreasing P is well-marked. The double period in mid-1990 is likely the result of multiple ARs making the time series more complicated. Simulations by Donahue (1993) indicate that multiple ARs at nearly the same latitude but forming at di erent longitudes can produce this sort of behavior.
3.2. E ects In uencing Measured Rotation Periods Several processes unrelated to rotation can a ect the rotational signal in disk-integrated data. For example, the presence of ARs at several longitudes can in uence the observed period by 10% (Donahue, 1993). Images of the Sun frequently show this to be the case, especially at or near cycle maximum. In the worst case, this will completely destroy the
{8{ coherence of the rotational signal and consequently no rotation period will be measured. This e ect undoubtedly occurs in several cases outside of cycle minimum where no rotation period was observed (Fig. 3). Active region growth and decay also plays a role in the ability to accurately measure the rotation period. ARs on the Sun typically have lifetimes of 1 ? 2 rotations (Allen, 1973), and AR complexes at \active longitudes" last up to approximately 180 days (Bogart, 1982; Gaizauskas et al., 1983). A technique for estimating the relative contribution of AR evolution to other mechanisms of variability was developed by Dobson et al. (1990). They introduce the concept of \pooled variance", i.e., the mean variance at a timescale sampled at a series of characteristic timescales 2 =
2
3
N 1 NX 4 1 X ( h S i ? Si )2 5 ; Nbin j =1 Nj ? 1 i=1 bin
j
(1)
where there are Nbin individual determinations of variance. The total variance of the time series is assumed to be the sum of individual contributions due to astrophysical mechanisms such as rotational modulation, activity cycles, ares, extra-cyclic activity (Baliunas and Jastrow, 1990), AR evolution, etc., as well as non-stellar contributors such as photon noise and other instrumental noise. Therefore, it is possible to compare the relative contribution of the rotational modulation, which should peak near 25 days to the change in pooled variance at timescales expected for AR evolution, < 200 days. Figure 4 shows the pooled variance diagram for the Sun. Surprisingly, there does not appear to be any large increase in contributed variance between the rotation period and 1 yr. This might suggest that most of the contribution of AR growth and decay occurs near the rotation period. The mean rotational semi-amplitude, hai = 0:0055 (in S units) can be converted to the expected contribution to the total variance: a2 = hai2=2, which can then be compared to the total variance observed at the rotational timescale. The ratio of a2 to 2=25d is 0.69, suggesting that the amplitude of rotation accounts for most of the variance observed, but
{9{ includes the possibility that some contribution from AR evolution exists. Why then have previous investigations (LaBonte, 1982, 1984; Harvey, 1984; Gilliland and Fisher, 1985) failed to detect the solar SDR? The answer may lie in the choice of the observing window. For the most part, the previous studies used a window length of 1 yr to 400 days, up to double the length used in this investigation. For = 1 yr, the pooled variance ratio drops to 0.49, suggesting that rotational modulation is no longer the primary contributor to the time series. The longer window exceeds the lifetime of most solar long-lived AR complexs (Bogart, 1982; Gaizauskas et al., 1983), and it is likely that more than one AR complex develops in the data at di erent longitudes, complicating the determination of a rotation period. Since severe phase shifts will occur in the data when the longitude of the dominant active region changes, techniques such as periodogram analysis will not be able to reliably determine a rotation period. These phase shifts will not introduce signi cant additional variance to the time series, and therefore will not be immediately visible in plots such as Figure 4. Conversely, the 200-day window is long enough to cover several rotations and yet appears to somewhat minimize the e ects of AR evolution. The existence of active longitudes within the 200-day window help retain some phase constancy in the sinusoidal rotation modulation, despite any net change in the size of the predominant ARs marking out rotation. Even so, with the available time series, we were only able to detect rotation in 42% of the observing windows.
4. Discussion Table I and Figure 3 clearly show a successful detection of the solar SDR from disk-integrated long-term measurements of chromospheric activity. Armed with this result we can now turn our attention to similar long-term measurements of chromospheric data, such as those undertaken by Mount Wilson Observatory's HK Project. Near-nightly
{ 10 { measurements of almost 100 lower main-sequence stars have continued since 1980, and for many of these stars, observations suitable for detecting rotation exist over the length of an activity cycle (Wilson, 1978; Baliunas et al., 1995). The results in this paper represent the foundation for interpreting seasonal rotation period variations and their correlation with chromospheric activity levels for these stars. Successfully detecting and measuring stellar SDR will signi cantly expand the situations for which stellar dynamo models can be tested. In addition, the character of those variations will show to what degree the solar SDR con guration is unique among lower main-sequence stars. We would like to thank the sta of Sacramento Peak Observatory and Mount Wilson Observatory, and in particular, Tim Henry for the diligence and care used in reducing the solar and stellar chromospheric time series used in this research. RAD received support from the Pre-doctoral Fellowship Program of the Smithsonian Instituion.
{ 11 {
REFERENCES Allen, C. W.: 1973, Astrophysical Quantities, 3rd. Edition, Athlone Press, London, x89. Baliunas, S. L., Vaughan, A. H., Hartmann, L., Middlekoop, F., Mihalas, D., Noyes, R. W., Preston, G. W., Frazer, J., and Lanning, H.: 1983, Astrophys. J. 275, 752. Baliunas, S. L., Horne, J. H., Porter, A., Duncan, D. K., Frazer, J., Lanning, H., Misch, A., Mueller, J., Noyes, R. W., Soyumer, D., Vaughan, A. H., and Woodard, L.: 1985, Astrophys. J. 294, 310. Baliunas, S. L. and Jastrow, R.: 1990, Nature 348, 520. Baliunas, S. L., Donahue, R.A., Soon, W.H., Horne, J.H., Frazer, J., Woodard-Eklund, L., Bradford, M., Rao, L.M., Wilson, O.C., Zhang, Q., et al.: 1995, Astrophys. J. 438, 269. Bogart, R. S.: 1982, Solar Phys. 76, 155. Carrington, R. C.: 1858, Monthly Notices Roy. Astron. Soc. 19, 1. Carrington, R. C.: 1863, Observations of the Spots on the Sun from November 9, 1853 to March 24, 1861 made at Redhill, Williams and Norgate, London. Dobson, A. K., Donahue, R. A., Radick, R. R., and Kadlec, K. L.: 1990, G. Wallerstein (ed.), in The Sixth Cambridge Symposium on Cool Stars, Stellar Systems and the Sun, PASP Conference Series, No. 9, 132. Donahue, R. A.: 1993, Ph.D. Thesis, New Mexico State University. Donahue, R. A., and Baliunas, S. L.: 1992, Astrophys. J. (Letters) 393, L63. Gaizauskas, V., Harvey, K. L., Harvey, J. W., and Zwaan, C.: 1983, Astrophys. J. 265, 1056. Gilliland, R. L. and Fisher R.: 1985, Publ. Astron. Soc. Paci c 97, 285.
{ 12 { Harvey, J. W.: 1984, in Solar Irradiance Variations on Active Region Timescales, NASA CP2310, 197. Horne, J. H., and Baliunas, S. L.: 1986, Astrophys. J. 302, 757. Keil, S. L., and Worden, S. P.: 1984, Astrophys. J. 276, 766. Kovacs, G.: 1981, Astrophys. Space Sci. 78, 175. LaBonte, B. J.: 1982, Astrophys. J. 260, 647. LaBonte, B. J.: 1984, Astrophys. J. 276, 335. Maunder, E. W.: 1913, Monthly Notices Roy. Astron. Soc. 74, 112. Parker, E. N.: 1955, Astrophys. J. 122, 293. Rosner, R., and Weiss, N. O.: 1992, in K. Harvey (ed.), The Solar Cycle, PASP Conference Series, No. 27, 511, Scargle, J. D.: 1982, Astrophys. J. 263, 875. Schwabe, H.: 1843, Astr. Nach. 20, No. 205. Snodgrass, H. B.: 1983, Astrophys. J. 270, 288. Sporer, G. F. W.: 1861, Astron. Nachr. 55, No. 1315. Vaughan, A. H., Preston, G. W., and Wilson, O. C.: 1978, PASP 90, 267. Vaughan, A. H., et al.: 1981, Astrophys. J. 250, 276. White, O. R., and Livingston, W. C. 1978, Astrophys. J. 226, 679. Wilson, O. C.: 1968, Astrophys. J. 153, 221. Wilson, O. C.: 1978, Astrophys. J. 226, 379. Wolf, R.: 1856, Astron. Mitt. Zurich No. 14.
{ 13 { Yallop, B. D., and Hohenkerk, C. Y.: 1980, Solar Phys. 68, 303.
This manuscript was prepared with the AAS LATEX macros v3.0.
{ 14 {
0.20
0.15 Iλ/Icont
K2v 0.10
K2r
K1r
K1v K3
0.05 3932.5 3933.0 3933.5 3934.0 3934.5 3935.0 Wavelength
Fig. 1.| A representative K line pro le showing the emission features and the 0.528 A ux calibration passband de ned by White and Livingston (1981) and Keil and Worden (1984).
{ 15 {
0.210 0.200
S
0.190 0.180 0.170 0.160
1980
1985 Year
1990
Fig. 2.| Disk-integrated K-line uxes from Sacramento Peak Observatory converted to the Mount Wilson Observatory S index (Baliunas et al., 1995).
{ 16 {
Fig. 3.| Observed (sidereal) solar rotation period, P , from 200-day windows as a function of time. The expected solar SDR pattern is seen, with an abrupt change in period occurring in the transition between Cycles 21 and 22. Open symbols indicate secondary periods, and errorbars are the uncertainty in the determination of the period.
{ 17 {
Fig. 4.| Pooled variance as a function of timescale, . The solar variance pro le shows an increase near the timescale of rotation ( 25 days) with only a small increase in variance through timescales of 1 yr. The mean rotational amplitude only accounts for approximately 70% of the variance observed, suggesting that AR evolution does in some way contribute to the total variance of the data at timescales from 20{400 days.
{ 18 {
Table I: Observed Rotation Periods Date Range Nobs hS i Prot P Semi-amplitude FAP (JD - 2,440,000) (days) (S -units) (%) 5602{5799 61 0.1777 26:1 0:3 0.0044 0.0048 6401{6598 84 0.1677 24:5 0:2 0.0025 2:8 10?5 6800{6998 102 0.1693 28:5 0:5 0.0017 0.089 7406{7598 107 0.1908 26:1 0:3 0.0057 7:8 10?4 7600{7799 101 0.1939 27:7 0:2 0.0099 6:4 10?12 7800{7994 85 0.1925 25:7 0:2 0.0084 7:1 10?8 8000{8199 109 0.1920 22:8 0:2 0.0054 6:7 10?7 25:0 0:3 0.0035 2:1 10?4 8201{8314 57 0.1955 25:1 0:5 0.0083 0.022
