### 2.2 Indices of solar activity

Solar (as well as other) indices can be divided into physical and synthetic according to the way they are obtained/calculated. Physical indices quantify the directly-measurable values of a real physical observable, such as the radioflux, and thus have clear physical meaning. Physical indices quantify physical features of different aspects of solar activity and their effects. Synthetic indices (the most common being sunspot number) are calculated (or synthesized) using a special algorithm from observed (often not measurable in physical units) data or phenomena. Additionally, solar activity indices can be either direct (i.e., directly relating to the sun) or indirect (relating to indirect effects caused by solar activity), as discussed in subsequent Sections 2.2.1, 2.2.2.

#### 2.2.1 Direct solar indices

The most commonly used index of solar activity is based on sunspot number. Sunspots are dark areas on the solar disc (of size up to tens of thousands of km, lifetime up to half-a-year), characterized by a strong magnetic field, which leads to a lower temperature (about 4000 K compared to 5800 K in the photosphere) and observed darkening.

Sunspot number is a synthetic, rather than a physical, index, but it has still become quite a useful parameter in quantifying the level of solar activity. This index presents the weighted number of individual sunspots and/or sunspot groups, calculated in a prescribed manner from simple visual solar observations. The use of the sunspot number makes it possible to combine together thousands and thousands of regular and fragmentary solar observations made by earlier professional and amateur astronomers. The technique, initially developed by Rudolf Wolf, yielded the longest series of directly and regularly-observed scientific quantities. Therefore, it is common to quantify solar magnetic activity via sunspot numbers. For details see the review on sunspot numbers and solar cycles (Hathaway and Wilson, 2004Hathaway, 2010).

Wolf sunspot number (WSN) series

The concept of the sunspot number was developed by Rudolf Wolf of the Zürich observatory in the middle of the 19th century. The sunspot series, initiated by him, is called the Zürich or Wolf sunspot number (WSN) series. The relative sunspot number Rz is defined as

where is the number of sunspot groups, is the number of individual sunspots in all groups visible on the solar disc and denotes the individual correction factor, which compensates for differences in observational techniques and instruments used by different observers, and is used to normalize different observations to each other.

The value of Rz (see Figure 1a) is calculated for each day using only one observation made by the “primary” observer (judged as the most reliable observer during a given time) for the day. The primary observers were Staudacher (1749 – 1787), Flaugergues (1788 – 1825), Schwabe (1826 – 1847), Wolf (1848 – 1893), Wolfer (1893 – 1928), Brunner (1929 – 1944), Waldmeier (1945 – 1980) and Koeckelenbergh (since 1980). If observations by the primary observer are not available for a certain day, the secondary, tertiary, etc. observers are used (see the hierarchy of observers in Waldmeier, 1961). The use of only one observer for each day aims to make Rz a homogeneous time series. As a drawback, such an approach ignores all other observations available for the day, which constitute a large fraction of the existing information. Moreover, possible errors of the primary observer cannot be caught or estimated. If no sunspot observations are available for some period, the data gap is filled, without note in the final WSN series, using an interpolation between the available data and by employing some proxy data. The observational uncertainties in the monthly Rz can be up to 25% (e.g., Vitinsky et al., 1986). The WSN series is based on observations performed at the Zürich Observatory during 1849 – 1981 using almost the same technique. This part of the series is fairly stable and homogeneous. However, prior to that there have been many gaps in the data that were interpolated. Therefore, the WSN series is a combination of direct observations and interpolations for the period before 1849, leading to possible errors and inhomogeneities as discussed, e.g, by Vitinsky et al. (1986); Wilson (1998); Letfus (1999). The quality of the Wolf series before 1749 is rather poor and hardly reliable (Hoyt et al., 1994Hoyt and Schatten, 1998Hathaway and Wilson, 2004).

Note that the sun has been routinely photographed since 1876 so that full information on daily sunspot activity is available (the Greenwich series) with observational uncertainties being negligible for the last 140 years.

The routine production of the WSN series was terminated in Zürich in 1982. Since then, the sunspot number series is routinely updated as the International sunspot number Ri, provided by the Solar Influences Data Analysis Center in Belgium (Clette et al., 2007). The international sunspot number series is computed using the same definition (Equation 1) as WSN but it has a significant distinction from the WSN; it is based not on a single primary solar observation for each day but instead uses a weighted average of more than 20 approved observers.

In addition to the standard sunspot number Ri, there is also a series of hemispheric sunspot numbers and , which account for spots only in the northern and southern solar hemispheres, respectively (note that ). These series are used to study the N-S asymmetry of solar activity (Temmer et al., 2002).

Group sunspot number (GSN) series

Since the WSN series is of lower quality before the 1850s and is hardly reliable before 1750, there was a need to re-evaluate early sunspot data. This tremendous work has been done by Hoyt and Schatten (1998), who performed an extensive archive search and nearly doubled the amount of original information compared to the Wolf series. They have produced a new series of sunspot activity called the group sunspot numbers (GSN – see Figure 1b), including all available archival records. The daily group sunspot number Rg is defined as follows:

where is the number of sunspot groups recorded by the -th observer, is the observer’s individual correction factor, is the number of observers for the particular day, and 12.08 is a normalization number scaling Rg to Rz values for the period of 1874 – 1976. Rg is more robust than Rz since it is based on more easily identified sunspot groups and does not include the number of individual spots. The GSN series includes not only one “primary” observation, but all available observations, and covers the period since 1610, being, thus, 140 years longer than the original WSN series. It is particularly interesting that the period of the Maunder minimum (1645 – 1715) was surprisingly well covered with daily observations allowing for a detailed analysis of sunspot activity during this grand minimum (see also Section 4.3). Systematic uncertainties of the Rg values are estimated to be about 10% before 1640, less than 5% from 1640 – 1728 and from 1800 – 1849, 15 – 20% from 1728 – 1799, and about 1% since 1849 (Hoyt and Schatten, 1998). The GSN series is more reliable and homogeneous than the WSN series before 1849. The two series are nearly identical after the 1870s (Hoyt and Schatten, 1998Letfus, 1999Hathaway and Wilson, 2004). However, the GSN series still contains some lacunas, uncertainties and possible inhomogeneities (see, e.g., Letfus, 2000Usoskin et al., 2003a).

The search for other lost or missing records of past solar instrumental observations has not ended even since the extensive work by Hoyt and Schatten. Archival searches still give new interesting findings of forgotten sunspot observations, often outside major observatories – see a detailed review book by Vaquero and Vázquez (2009) and original papers by Casas et al. (2006); Vaquero et al. (20052007); Arlt (20082009). Interestingly, not only sunspot counts but also regular drawings, forgotten for centuries, are being restored nowadays in dusty archives.

Other indices

An example of a synthetic index of solar activity is the flare index, representing solar flare activity (e.g., Özgüç et al., 2003Kleczek, 1952). The flare index quantifies daily flare activity in the following manner; it is computed as a product of the flare’s relative importance in the -range and duration , , thus being a rough measure of the total energy emitted by the flare. The daily flare index is produced by Bogazici University (Özgüç et al., 2003) and is available since 1936.

A traditional physical index of solar activity is related to the radioflux of the sun in the wavelength range of 10.7 cm and is called the F10.7 index (e.g., Tapping and Charrois, 1994). This index represents the flux (in solar flux units, 1 sfu = 10–22 Wm–2 Hz–1) of solar radio emission at a centimetric wavelength. This emission is close to the peak of solar radio emission and is produced as a result of the nonradiative heating of coronal plasma over active regions. It is a good quantitative measure of the level of solar activity, which is not directly related to sunspots. Close correlation between the F10.7 index and sunspot number indicates that the latter is a good index of general solar activity, including coronal activity. The solar F10.7 cm record has been measured continuously since 1947.

Another physical index is the coronal index (e.g., Rybanský et al., 2005), which is a measure of the irradiance of the sun as a star in the coronal green line. Computation of the coronal index is based on observations of green corona intensities (Fe XIV emission line at 530.3 nm wavelength) from coronal stations all over the world, the data being transformed to the Lomnický Štit photometric scale. This index is considered a basic optical index of solar activity. A synthesized homogeneous database of the Fe XIV 530.3 nm coronal-emission line intensities has existed since 1943 and covers six solar cycles.

Often sunspot area is considered as a physical index representing solar activity (e.g., Baranyi et al., 2001Balmaceda et al., 2005). This index gives the total area of visible spots on the solar disc in units of millionths of the sun’s visible hemisphere, corrected for apparent distortion due to the curvature of the solar surface. The area of individual groups may vary between tens of millionths (for small groups) up to several thousands of millionths for huge groups. This index has a physical meaning related to the solar magnetic flux emerging at sunspots. Sunspot areas are available since 1874 in the Greenwich series obtained from daily photographic images of the sun. In addition, some fragmentary data of sunspot areas, obtained from solar drawings, are available for earlier periods (Vaquero et al., 2004Arlt, 2008).

An important quantity is solar irradiance, total and spectral. Irradiance variations are physically related to solar magnetic variability (e.g., Solanki et al., 2000), and are often considered manifestations of solar activity, which is of primary importance for solar-terrestrial relations.

Other physical indices include spectral sun-as-star observations, such as the CaII-K index (e.g., Donnelly et al., 1994Foukal, 1996), the space-based MgII core-to-wing ratio as an index of solar UVI (e.g., Donnelly et al., 1994Viereck and Puga, 1999Snow et al., 2005) and many others.

All the above indices are closely correlated to sunspot numbers on the solar-cycle scale, but may depict quite different behavior on short or long timescales.

#### 2.2.2 Indirect indices

Sometimes quantitative measures of solar-variability effects are also considered as indices of solar activity. These are related not to solar activity per se, but rather to its effect on different environments. Accordingly, such indices are called indirect, and can be roughly divided into terrestrial/geomagnetic and heliospheric/interplanetary.

Geomagnetic indices quantify different effects of geomagnetic activity ultimately caused by solar variability, mostly by variations of solar-wind properties and the interplanetary magnetic field. For example, the aa-index, which provides a global index of magnetic activity relative to a quiet-day curve for a pair of antipodal magnetic observatories (in England and Australia), is available from 1868 (Mayaud, 1972). An extension of the geomagnetic series is available from the 1840s using the Helsinki Ak(H) index (Nevanlinna, 2004a,b). A review of the geomagnetic effects of solar activity can be found, e.g., in Pulkkinen (2007). It is noteworthy that geomagnetic indices, in particular low-latitude aurorae (Silverman, 2006), are associated with coronal/interplanetary activity (high-speed solar-wind streams, interplanetary transients, etc.) that may not be directly related to the sunspot-cycle phase and amplitude, and therefore serve only as an approximate index of solar activity. One of the earliest instrumental geomagnetic indices is related to the daily magnetic declination range, the range of diurnal variation of magnetic needle readings at a fixed location, and is available from the 1780s (Nevanlinna, 1995). However, this data exists as several fragmentary sets, which are difficult to combine into a homogeneous data series.

Heliospheric indices are related to features of the solar wind or the interplanetary magnetic field measured (or estimated) in the interplanetary space. For example, the time evolution of the total (or open) solar magnetic flux is extensively debated (e.g., Lockwood et al., 1999Wang et al., 2005Krivova et al., 2007).

A special case of heliospheric indices is related to the galactic cosmic-ray intensity recorded in natural terrestrial archives. Since this indirect proxy is based on data recorded naturally throughout the ages and revealed now, it makes possible the reconstruction of solar activity changes on long timescales, as discussed in Section 3.