ISSN 0038-0946, Solar System Research, 2009, Vol. 43, No. 2, pp. 151–157. © Pleiades Publishing, Inc., 2009. Original Russian Text © A.P. Sarychev, E.M. Roshchina, 2009, published in Astronomicheskii Vestnik, 2009, Vol. 43, No. 2, pp. 160–166.
Comparison of Three Solar Activity Indices Based on Sunspot Observations A. P. Sarychev and E. M. Roshchina Sternberg Astronomical Institute, Universitetskii pr. 13, Moscow, 119899 Russia Received March 19, 2008
Abstract—The following sunspot formation indices are analyzed: the relative sunspot number Rz, the normalized sunspot group number Rg, and the total sunspot area A. Six empirical formulas are derived to describe the relations among these indices after 1908. The earlier data exhibit systematic deviations from these formulas, which can be attributed to systematic errors of the indices. The Greenwich data on the sunspot total area A and the sunspot group number in 1874–1880 are found to be doubtful. Erroneous data at the beginning of the Greenwich series must spoil the values of the index Rg in the XVII–XIX centuries. The Hoyt–Schatten series of Rg may be less reliable than the well-known Wolf number series Rz. PACS: 96.60.qd DOI: 10.1134/S0038094609020087
INTRODUCTION It is known that the central star of the Solar System exhibits quasi-periodic activity. This activity is transferred over the entire Solar System by virtue of solar wind and the variable component of electromagnetic emission and modulates the intensity of galactic cosmic rays. The available data on the solar activity far back in the past are based on indirect data of naked eye observation of auroras, comets, and sunspots and on measurements on 14ë and 10Be isotope abundances in tree rings, ices, and bottom sediments. The currently available data were analyzed by Nagovitsyn (2007) and Ogurtsov (2007). Telescopic observations of sunspots have been used to monitor the solar activity level only in the last 300– 400 years. The best known data sequence of this kind is the time series of the relative sunspot number Rz (the socalled Wolf number), dated since 1700 (Vitinskii et al., 1986). The 1610–1995 time series of the observed sunspot group number Rg has recently been published by Hoyt and Schatten (1998). The next data sequence in decreasing duration is the Greenwich series of the total sunspot area A, dated since 1874. Advantages and drawbacks of the series of indices Rz, Rg, and A were discussed by Nagovitsyn (2005) and in references therein. According to that work, the above-mentioned indices represent physically different characteristics of sunspot formation. Since all the indices described a single process, one can expect certain correlations among them. This work is devoted to revealing and analyzing these relations. Empirical relations among the indices can be used, e.g., to evaluate the index Ä before 1874, i.e., before the beginning of measurements of the total sunspot area at the Greenwich Observatory (Nago-
vitsyn, 1997; Vaquero et al., 2004; Sarychev and Roshchina, 2007). RELATION BETWEEN THE INDICES Rz AND Rg Wolf performed visual observations of the Sun regularly since 1849 and determined the relative sunspot number Rz given by the formula R z = k ( 10G + N ), where G is the number of sunspot groups, N is the total number of individual spots, and the factor k reduces the observations by any author to the Wolf scale for which k = 1. The method for the calculation of Rz was changed later on. However, measures were undertaken to preserve the historical scale of this index (Vitinskii et al., 1986). Daily evaluation of Rz from observations have been carried out until recently. In addition, based on archival observations, Wolf retrieved the monthly average index Rz from the past until 1749 and the annual average index Rz up to 1700. The resulting duration of the time series of Rz is over 300 years. It should be kept in mind that this series comprises segments having a fortiori different accuracies. The segment retrieved by Wolf from the data of other observers is the least reliable. The retrieval after 1825 based on regular observations by Schwabe is probably quite reliable. The most reliable data were acquired since 1849 by specialized observations. However, the time series of Rz can be inherently nonuniform, even in this time interval (see Nagovitsyn, 2005 and references therein). The longer-term series of the index Rg has existed since the beginning of telescopic observations of the Sun in 1610. This series has been compiled as follows.
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1.5
1.3
1.1
0.9
0.7
0.5 1825
1875
1925
1975 Year
Fig. 1. Ratio Rz/Rg of the observed annual mean indices (solid curve) and the results of calculations based on Eq. (1) (dashed curve).
The observed sunspot group number N was determined regularly in 1874–1976 at the Greenwich Observatory. The best of white light photographic images of the Sun obtained daily at a few observatories has been selected for this purpose. Using these uniform data, Hoyt and Schatten (1998) set the index scale Rg = 12.08 N, where the factor 12.08 was introduced to make the values of Rg and Rz close to unity. This scale was used for all sunspot group numbers retrieved from available archival observations. The composite Greenwich and archival time series of the index Rg covers the time interval from 1610 to1995. Similar to the Wolf number series, it consists of segments having different accuracies. The least reliable values of Rg correspond to the first half of this series (XVII–XVIII centuries). There were attempts to correct these segments of the series (Usoskin et al., 2003; Vaquero, 2004; Vaquero et al., 2005; 2007). An estimate of the quality of the entire series of Rg has been provided by Nagovitsyn (2005). A comparative analysis of indices Rg and Rz as solar activity indicators has been given by Hathaway et al. (2002), and Usoskin and Kovaltsov (2004). The solid curve in Fig. 1 shows the annual mean ratio Rz/Rg as a function of time. Here, the time axis originates at 1825 and corresponds to the expected interval of fairly reliable data. The ratio Rz/Rg in 1908– 1984 varies according to the Gleisberg secular solar activity cycle. The 23-point moving average (Rg)23of
the annual mean values of Rg are used for a quantitative description of this cycle. The effect of the 11- and 22-year solar cycles are eliminated by virtue of such smoothing, but the effect of longer-term cycles is retained. The least-square method yields the following linear regression for the data in 1908–1984: R z /R g = 0.6443 + 0.005141 ( R g ) 23 .
(1)
It is shown by the dashed curve in Fig. 1. It is seen in this figure that the solid and dashed curves are close for 1908–1984 and noticeably diverge before 1908. The correlation coefficient of the values of Rz/Rg calculated from the observational data and the according to Eq. (1) is r = 0.70, within the interval of coincidence. Figure 1 should be interpreted taking into account the physical meaning of the ratio Rz/Rg. The ratio Rz/Rg is proportional to (10 + η), where η is the mean number of sunspots in a group (Vitinskii et al., 1986). Therefore, variations in η and Rz/Rg should be similar. The right half of Fig. 1 implies than the mean number of sunspots in a group is in phase with respect to the secular activity cycle. At the same time, this is not true for the left half of the figure. The apparent discrepancy can be explained either by a rapid variation in the solar activity near 1908 or by the violation of the internal uniformity of one or both studied activity index series. In our opinion, the latter explanation is more feasible. SOLAR SYSTEM RESEARCH
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Rz/Rz(Rg) 1.9
1.7
1.5
1.3
1.1
0.9
0.7 1825
1875
1925
1975 Year
Fig. 2. Ratio of the observed values of Rz to the value of Rz(Rg), calculated according to Eq. (2). The dashed curve is plotted through the points in the interval 1825–1907. The solid line marks the points from 1908 to 1984.
Equation (1) can be reduced to the following empirical formula for calculating the relative sunspot number Rz from the known values of the index Rg: R z ( R g ) = [ 0.6443 + 0.005141 ( R g ) 23 ]R g .
(2)
The following formula describing the relation between the annual mean values of Rg and Rz in 1908–1995 can be derived in a similar way: R g ( R z ) = [ 1.3522 – 0.004892 ( R z ) 23 ]R z .
(3)
Here, (Rz)23 is the result of the smoothing of the annual mean values of Rz by their 23-point moving average. Let us evaluate the validity of Eqs. (2) and (3) using the time series used for their derivation. Here, the correlation coefficient of the observed and calculated values of the indices is r ≈ 0.995. The expected error of the single calculated annual mean index is no worse than ±7.5%. Thus, a value of either Rz or Rg after 1908 can be retrieved with a fairly high accuracy given the value of the other index. Meanwhile, a weak quasi-periodic trend seen in Fig. 2 remains unaccounted for in Eqs. (2) and (3). The ratio of the observed Rz to the corresponding values calculated using Eq. (2) is shown in this figure. The solid curve is plotted via the points belonging to the interval 1908–1984. The dashed curve connects the observational points dated earlier than 1908. A sinusoid segment in the right half of Fig. 2 fits the trend in the interval 1908–1984. A period of 48.4 ± 3.8 years and relative amplitude of 0.052 ± 0.010 are derived by the least-square method. The obtained period is close to the period of 45 years found by Kuklin (1984) from a SOLAR SYSTEM RESEARCH
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study of the time series of annual mean Wolf numbers in 1700–1979. According to this figure, the sine-trend amplitude is comparable to fluctuations of the fitted data. Thus, Eq. (2) without allowance for the trend is valid in the first approximation since 1908. However, the deviations are inadmissibly large before 1908. These deviations can originate due to systematic errors of one or both time series of indices Rz and Rg before 1908. The same conclusions follow from an analysis of Eq. (3) and the corresponding figure similar to Fig. 2 (not shown in the present work). To point out the index whose time series is erroneous before 1908, we consider the dependences of each index on the total sunspot area index A. RELATIONS OF THE INDICES Rz AND Rg WITH THE INDEX A It was already noted above that solar photographs obtained within the framework of a specialized campaign were used since 1874 to measure the total sunspot area. The summed areas of the observed sunspot are preliminarily corrected for the effect of line-of-sight projection and expressed in ppm of the solar hemisphere. The scale of the index A in 1874–1976 has be established historically using uniform observations at the Greenwich Observatory. The index A in the Greenwich system is being determined from the observations of the US solar observatory network. To reduce the American total sunspot area to the Greenwich system, a multiplier of 1.4 is used. As far as the three sunspot
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formation indices Rg, Rz, and A are concerned, the index A is the most “physical” (Nagovitsyn, 2005), but has the least time period of regular determinations. Empirical dependences A(Rz) and A(Rg) were used by Nagovitsyn (1997), Vaquero et al.(2004), and Sarychev and Roshchina (2007) to retrieve the index A before the beginning of its regular determination at Greenwich. It was noted in the two latter works that the retrieval results depend on the used index (Rz or Rg). A possible explanation for this ambiguity is proposed in this section. It was shown in our previous work (Sarychev and Roshchina, 2007) that the relation between the sunspot area parameter A and sunspot number parameters Rz and Rg can be fitted well by a power law. The relation between A and Rz depends on the secular cycle phase, while the relation between A and Rg is almost independent of this cycle. Hence, the dependence of A on Rg is fitted by a power law, while the dependence of A on Rz, by a power law multiplied by a linear function of (Rz)23 or of (Ä)23 , where (Ä)23 is the 23-point moving average of the annual mean values of Ä. The fit is applied to the data in the time interval 1908–1995 for which, according to the previous section, fairly reliable values of indices Rz and Rg are available. The numerical parameters of the four fitting functions are derived by the leastsquare method: A ( R z ) = [ 11.014 – 0.04361 ( R z ) 23 ]R z
,
R z ( A ) = [ 0.1395 + 5.634 × 10 ( A ) 23 ] A
0.847
1.143
–5
A ( R g ) = 6.517R g
1.194
R g ( A ) = 0.2418 A
,
0.818
(4) ,
(5) (6)
.
(7)
The parameter values in Eqs. (4) and (6) are different from the corresponding values given by Sarychev and Roshchina (2007). This is explained by the different time intervals used to derive the fitting formulas. The correlation coefficient r for the observed and calculated values of the index was evaluated for each formula. The four resulting values of r are within the range 0.984 < r < 0.987. This is indicative of certain relations between index Ä and indices Rz and Rg in 1908–1995. Figure 3 shows the deviation of the observed index Ä from the calculated values A(Rz) and A(Rg), i.e., illustrates the quality of fits (4) and (6). Here, the time intervals 1875–1907 and 1908–1995 are shown by the dashed and solid curves, respectively. Small systematic deviations of the solid curves from unity are seen in the lat interval. A quadratic function can fit these deviations. This parabolic trend can be neglected in the first approximation and Ä can be estimated without allowance for this trend using Eqs. (4) and (6). A parabolic trend similar to the one shown in Fig. 3 is present in the time dependences of Rz(A)/Rz and Rg(A)/Rg after 1908. Therefore, such a trend is formed if the numerator of the ratio is either the observed index Ä or a function of Ä (see Eqs. (5) and (7)). This justified the assumption
that the trend is caused by small gradual changes in the scale of index Ä. Consider the interval 1875–1907 shown by the dashed curve in Figs. 3a and 3b. No patterns similar to the solid curve segments are seen in this case. The dashed curve in Fig. 3b looks like a continuation of the solid one, except for the interval 1875–1879 at the beginning of the plot. The solid and dashed curves in Fig. 3a are matched in a completely different way. In this case, two apparently different time dependences are seen, which do not continue each other, but intersect near 1908. This picture can be interpreted as follows. According to Fig. 3b, variations in indices A and Rz in the interval 1880–1995 are correlated according to empirical formula (4). Figure 3a shows that the extrapolation of formula (4) relating indices Ä and Rg from 1908 to 1880 is not confirmed by the observations. The short segment 1875–1879 in Figs. 3a and 3b exhibits the peculiar behavior of the ratios Ä/A(Rz) and Ä/A(Rg). In what follows, the assumption is justified on the possibly erroneous values of index Ä in the first years of the Greenwich series observations. Meanwhile, Fig. 3 has been analyzed since 1880. Six formulas, (2)–(7), reflect the relations among the sunspot formation indices Rz, Rg, and Ä after 1908. In earlier years, Eqs. (2) and (3) describing the relations among Rz, and Rg were not valid (see Fig. 2). According to the results of the analysis of Fig. 3, the relation between indices Ä and Rg is also violated during this time period, but the relation between Ä and Rz remains valid. This behavior of the three solar activity indices can be attributed to the systematic underestimation of the true values of Rg before 1908. On the other hand, the adopted values of indices Rz and Ä are considered to be fairly correct. The reason for the erroneous series of Rg before 1908 may be the violation of internal uniformity of the initial segment of the Greenwich observational data. Recall that the scale of the Rg series is based on the Greenwich determinations of the sunspot group number (Hoyt and Schatten, 1998). The Rg series was formed by extrapolating the Greenwich scale to the past using a chain of successively overlapping observations by different authors. Thus, the error of matching the last section of the chain with the Greenwich data spoils the entire “pre-Greenwich” segment of the Rg series. Note that the beginning of the Greenwich series is almost coincident with the time period 1878–1882 when moist collodion plates were replaced by dry gelatin- bromide photographic materials (Chibisov, 1951). The radical transformation of the photographic process could bias the Greenwich data scale based on photographic observations. An indirect evidence for such a change can be Fig. 4 showing the ratio between the annual mean valGr ues of R g for the Greenwich observations and the values of Rg for the Hoyt–Schatten series. Similar to Fig. 3b, the strongest deviations are seen in the interval 1875–1879. The Greenwich data should be considered doubtful within this interval. Truncation of these data SOLAR SYSTEM RESEARCH
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(a)
A/A(Rg) 1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4 A/A(Rz) 1.8
(b)
1.6
1.4
1.2
1.0
0.8
0.6
0.4 1875
1895
1915
1935 Year
1955
1975
1995
Fig. 3. Ratio of the observed total sunspot area A to the results of its reconstruction (the points are connected with the solid curve since 1908) based on (a) Eq. (6), using the annual mean values of Rg and (b) Eq. (4), using the annual mean values of Rz. SOLAR SYSTEM RESEARCH
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mates based on Eq. (10) can be obtained if only index Rg is known. Needless to say, we should be confident that the used values of Rg are reliable since 1610. The ratio A/Rg is proportional to the mean area S of the observed sunspot groups. If index Ä is unknown, the physical parameter S can be roughly estimated using only the index Rg. With allowance for the definition of index Rg and Eq. (6), the value of S in the sunspot area units (ppm of the solar hemisphere) is given by
RgGr/Rg 1.3 1.2 1.1 1.0
S = 12.08 A/R g ≈ 80R g . 0.2
0.9 0.8 0.7 1975
1885
1905 Year
1895
Fig. 4. Comparison of index Rg from the Hoyt–Schatten Gr
series with similar values of R g , based only on the Greenwich observations.
makes the Greenwich series only 5% shorter. To correct the segments of the Rg series before 1908, the entire procedure of matching the archival observations with the Greenwich ones should be repeated. DISCUSSION AND CONCLUSIONS Thus, analysis of the observed relations among indices Rz, Rg, and A stirred doubts about the reliability of index Rg in the interval 1610–1907. The preliminary conclusion that index Rg is not reliable is important since the Hoyt–Schatten series of Rg is widely used by many authors. The problem of the reliability of index Rg is also important because this index can be used to estimate the so-called primary sunspot formation indices f0 and í0, where f0 is the number of sunspot groups formed on the entire solar surface per unit time and í0 is the mean lifetime of a sunspot group. According to the monograph by Vitinskii et al. (1986), the primary indices derived from observations are related by the following approximate formulas: Rg ≈ c1 f 0 T 0 ,
A ≈ c2 f 0 T 0 , 2
(8)
where f0 is an unknown constant. Formulas (8) make it possible to express the primary indices in terms of the observed ones: f 0 ≈ c 3 R g / A, 2
T 0 ≈ c 4 A/R g .
(9)
These formulas can be simplified using the empirical formula (6): f 0 ≈ c5 Rg , 0.8
T 0 ≈ c6 Rg . 0.2
(10)
A physical description of sunspot formation based on Eq. (9) requires the values of Ä and Rg. Rougher esti-
(11)
According to Eqs. (9)–(11), index Rg in combination with index A is one of the basic characteristics of sunspot formation. This differs index Rg from the relative sunspot number Rz, missing such a clear physical meaning. The obtained empirical relations among indices Rz, Rg, and A are self-sustained since they allow for retrieving an unobserved index given another one is known. Consider the retrieval of index A as an example (Nagovitsyn, 1997; Vaquero et al., 2004; Sarychev and Roshchina, 2007). The last two works revealed a significant discrepancy of the retrievals A(Rz) and A(Rg) before 1887. This ambiguity of the retrieval can be attributed to a step-like change in the scale of Rg in the end of the XIX century. Since a similarly strong change in the stability of Rz is not found, one must admit that the reconstruction A(Rz) is more reliable than A(Rg). The new solar activity index Rg is more informative from the physical viewpoint in comparison with the well-known index Rz. Moreover, the series of Rg begins in 1610, while the series of Rz only in 1700. These are the obvious advantages of index Rg, as compared to Rz. However, index Rg should be used with caution until concerns on the internal uniformity of its time series before 1908 are not waived. In general, the study of the Rg series is not completed since it was published only ten years ago. In conclusion, we formulate the main results of the present work: 1) Since 1908, the indices Rz, Rg, and A exhibit relations which sometimes are dependent on the secular activity cycle; 2) The solar-activity indices Rz and Rg are proportional with a coefficient linearly dependent on the quantitative index of secular solar activity (see Eqs. (2) and (3)); 3) The relation between indices Rg and A is fitted well by a power law independent of the secular cycle (see Eqs. (6) and (7)); 4) The relation between indices Rz and A is fitted by a product or a power law and a linear function of the quantitative secular cycle parameter (see Eqs. (4) and (5)); and 5) There is evidence that the Greenwich values of A obtained before 1880 and the Hoyt–Schatten series of Rg before 1908 are incorrect. SOLAR SYSTEM RESEARCH
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The indices were analyzed assuming that the physical relations among them were the same during the entire time of telescopic observations of the Sun. Hence, deviations from the empirical relations were related to systematic errors of the indices. The time series of the solar activity indices uploaded at the following web sites were used in this work: 1. The index Rz: http://www.sidc.oma.be/DATA/ yearssn.dat 2. The index Rg: ftp://ftp.ngds.noaa.gov/ STP/SOLAR_DATA/ 3. The index A: http://sciens/msfc.nasa.gov.ssl/ pad/solar/grinwch.htm REFERENCES Chibisov, K.V., Modern Photographic Materials and Prospects for Their Further Improvement, Usp. Nauchn. Fotografii, 1951, vol. 1, p. 5. Hathaway, D.H., Wilson, R.M., and Reichmann E.J., Group Sunspot Numbers: Sunspot Cycle Characteristics, Solar Phys., 2002, vol. 211, p. 357. Hoyt, D.V. and Schatten, K.H., Group Sunspot Numbers: A New Solar Activity Reconstruction, Solar Phys., 1998, vol. 181, p. 491. Kuklin, G.V., Results of Analyses of the 11- and 22-Year Wolf-Number Cycles, Issledovaniya po geomagnetizmu, aeronomii i fizike Solntsa (Studies on Geomagnetism, Aeronomy, and Solar Physics), 1984, no. 68, p. 45. Nagovitsyn, Yu.L., Sunspot Area Index Sequence in the Greenwich System in 1821–1989, Solnechnye dannye. Stat’i I soobshcheniya 1995–1996 (Solar Data. Papers and Reports of 1995–1996), 1997, p. 38. Nagovitsyn, Yu.L., To the Description of Long-Term Variations in the Solar Magnetic Flux: The Sunspot Area
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