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NASA NTRS1993

Future operations of Pioneer 10 and 11

The Pioneers 10 and 11 spacecraft were launched on 2 March 1972 and 5 April 1973 and are now 53 and 35 AU from the sun. Pioneer 10 is now the most distant man-made object in our solar system and the in situ measurements of the gas and dust surrounding the sun have been obtained for almost two solar sunspot cycles. Plasma analyzer measurements out to 50 AU show that the mean velocity is about 430 km/sec, the mean density decreases as R exp -2, and that the terminal shock has not been encountered. The magnetic field is drawn out to form Archimedean spirals in the ecliptic place and dipolelike asymmetry in the polar directions as predicted by the Parker model. Galactic cosmic ray measurements of the intensity and radial gradient indicate a 'modulation boundary' between 70 to 100 AU from the sun. All measurements to date indicate that both spacecraft are within the heliosphere and proceeding toward the outer boundary where the modulating effect of solar activity on cosmic ray intensity ceases.

Dyal, P.↗
NASA NTRS1999

The Heliosphere as Resonant Cavity

If a disturbance in the supersonic solar wind reaches the heliospheric shock, a number of events ensue. First, the shock itself responds with inward or outward motion. Secondly, the disturbance propagates outward through the heliosheath as a sound or magnetoacoustic wave; eventually it reaches the heliopause and is (partially) reflected back toward the termination shock. The reflected wave can return to the shock, affecting the shock's motion, and be reflected yet again. The repetition of these processes can produce a 'ringing' in the heliosheath. This suggests that it may be useful to regard the heliosheath as a resonant acoustic cavity with inner and outer boundaries at the termination shock and heliopause, respectively. To evaluate this concept we have developed a simple model of small-amplitude resonant oscillations in an outwardly flowing gas, with appropriate boundary conditions (shock on the interior, tangential discontinuity on the exterior boundary). The fundamental mode of oscillation has a period of order T approx. 2D/C, where C is the speed of sound in the heliosheath and D is the distance between the two boundaries. Typical numerical models of the heliosphere give C approx. 200-500 km/s and D approx. 20 - 100 AU, giving T approx. 0.5 - 2.5 years. Hence we suggest that motions of the heliosheath and termination shock will occur with time scales of the order of a year, and are the consequence of the resonant nature of the heliospheric cavity rather than the history of variation at the Sun and/or in the solar wind. In particular, we suggest that the motion of the termination shock may be unrelated to solar variations over the time scale of the sunspot cycle.

Bames, Aaron↗
NASA NTRS2003

June 1997 ER-2 Flight Measurements

Within our current understanding of the atmospheric ionizing radiation, the ER-2 flight package was designed to provide a complete characterization of the physical fields and evaluate various dosimetric techniques for routine monitoring. A flight plan was developed to sample the full dynamic range of the atmospheric environment especially at altitudes relevant to the development of the High Speed Civil Transport. The flight of the instruments occurred in June of 1997 where predictive models indicated a maximum in the high altitude radiation environment occurring approximately nine months after the minimum in the solar sunspot cycle. The flights originated at Moffett field at the Ames Research Center on ER-2 aircraft designated as 706. The equipment was shipped mid- May 1997 for unpacking and checkout, size fitting, systems functional test, and preflight testing on aircraft power with flight readiness achieved on May 30, 1997. The equipment was qualified on its first engineering flight on June 2, 1997 and the subsequent science gathering flights followed during the period of June 5-15, 1997. Herein we give an account of the flight operations.

Jones, Irby W.↗
NASA NTRS2003

Local Helioseismology: A New Window into the Sun

In recent years the focus of helioseismology has shifted from the classical measurements of global properties using normal mode frequencies to possibly a more difficult problem of measuring the three-dimensional, time-dependent structure of the solar interior, now called local helioseismology. Problems to be studed include the subsurface structure of observed surface features such as sunspots, convective flow patterns, and the magnetic field of the 1 1-year sunspot cycle. A variety of techniques have been developed to attack these problems, including a 'local' normal mode method, a method of measuring travel times between &stant locations, and a method for holographically reconstructing the acoustic signal in the solar interior. Recent results from the field will be reviewed.

Duvall, Thomas↗
NASA NTRS2004

Long-Term Total Solar Irradiance (TSI) Variability Trends: 1984-2004

The incoming total solar irradiance (TSI), typically referred to as the solar constant, is being studied to identify long-term TSI changes, which may trigger global climate changes. The TSI is normalized to the mean earth-sun distance. Studies of spacecraft TSI data sets confirmed the existence of 0.1 %, long-term TSI variability component with a period of 10 years. The component varied directly with solar magnetic activity associated with recent 10-year sunspot cycles. The 0.1 % TSI variability component is clearly present in the spacecraft data sets from the 1984-2004, Earth Radiation Budget Experiment (ERBE) active cavity radiometer (ACR) solar monitor; 1978-1993, Nimbus-7 HF; 1980-1989, Solar Maximum Mission [SMM] ACRIM; 1991-2004, Upper Atmosphere Research Satellite (UARS) ACRIM; 1996-2003, Solar and Heliospheric Observatory (SOHO)/VIRGO, Space Science (ATLAS), 2000-2004, ACRIMSAT; and 2003-2004 SOlar Radiation and Climate Experiment (SORCE) active cavity radiometer (ACR) missions. From October 1984, through March 2004, the ERBS/ERBE solar monitor was used to produce the longest continuous data set of total solar irradiance (TSI) variability measurements. The solar monitor is located on Shuttle Atmospheric Laboratory for Applications and the NASA Earth Radiation Budget Satellite (ERBS). Maximum TSI values occurred during the 1989-1991, and 1998-2002, time frames; while minimum [quiet sun] TSI levels occurred during 1986 and 1996. Recent ERBS measurements indicate that the TSI is decreasing to forecasted, minimum levels by 2006. Using the discontinuous non-operational Nimbus-7, SMM ACRIM, and UARS ACRIM mission TSI data sets, Wilson and Mordvinor (2003) suggested the existence of an additional long-term TSI variability component, 0.05 %, with a period longer than a decade. Analyses of the ERBS/ERBE data set do not support the Wilson and Mordvinor analyses approach because it used the Nimbus-7 data set which exhibited a significant ACR response shift of 0.7 Wm-2

Lee, Robert Benjamin, III↗
NASA NTRS2005

Forecasting the Impact of an 1859-calibre Superstorm on Satellite Resources

We have assembled a database of operational satellites in orbit as of 2004, and have developed a series of simple models to assess the economic impacts to this resource caused by various scenarios of superstorm events possible during the next sunspot cycle between 2010 and 2014. Despite the apparent robustness of our satellite assets against the kinds of storms we have encountered during the satellite era, our models suggest a potential economic loss exceeding $10(exp 11) for satellite replacement and lost profitability caused by a once a century single storm similar to the 1859 superstorm. From a combination of power system and attitude control system (the most vulnerable) failures, we estimate that 80 satellites (LEO, MEO, GEO) may be disabled as a consequence of a superstorm event. Additional consequences may include the failure of many of the GPS, GLONASS and Galileo satellite systems in MEO. Approximately 98 LEO satellites that normally would not have re-entered for many decades, may prematurely de-orbit in ca 2021 as a result of the temporarily increased atmospheric drag caused by the superstorm event occurring in 2012. The $10(exp 11) International Space Station may lose at least 15 kilometers of altitude, placing it in critical need for re-boosting by an amount that is potentially outside the range of typical Space Shuttle operations during the previous solar maximum in ca 2000, and at a time when NASA plans to decommission the Space Shuttle. Several LEO satellites will unexpectedly be placed on orbits that enter the ISS zone of avoidance, requiring some action by ground personnel and ISS astronauts to avoid close encounters. Radiation effects on astronauts have also been considered and could include a range of possibilities from acute radiation sickness for astronauts inside spacecraft, to near-lethal doses during EVAs. The specifics depends very sensitively on the spectral hardness of the accompanying SPE event. Currently, the ability to forecast extreme particle events and coronal mass ejections, or predict their fluences and geo-severity in the 24-hrs prior to the event, appears to be no better than 50/50. If the events of the 1859 superstorm serve as a guide, the scope of a contemporary superstorm will most certainly be an awesome event, but one that the vast majority of our other satellite resources may reasonably be expected to survive.

Odenwald, Sten↗
NASA NTRS2007

SOHO-Ulysses Coordinated Studies During the Two Extended Quadratures and the Alignment of 2007-2008

During SOHO-Sun-Ulysses quadratures the geometry of the configuration makes it possible to sample "in situ" the plasma parcels that are remotely observed in the corona. Although the quadrature position occurs at a well defined instant in time, we typically take data while Ulysses is within +/- 5 degrees of the limb, with the understanding that plasma sampled by Ulysses over this time interval can all be traced to its source in the corona. The relative positions of SOHO and Ulysses in winter 2007 (19 Dec 2006-28 May 2007) are unusual: the SOHO-Sun-Ulysses included angle is always between 85 and 95 degrees - the quadrature lasts for 5 months! This provides an opportunity for extended observations of specific observing objectives. In addition, in summer 2007, Ulysses (at 1.34 AU) is in near-radial alignment with Earth/ACE/Wind and SOHO, allowing us to analyze radial gradients and propagation in the solar wind and inner heliosphere. Our own quadrature campaigns rely heavily on LASCO and UVCS coronal observations: LASCO giving the overall context above 2 solar radii while the UVCS spectrograph acquired data from - 1.5 to, typically, 4-5 solar radii. In the past, coronal parameters have been derived from data acquired by these two experiments and compared with "in situ" data of Ulysses' SWOOPS and SWICS. Data from other experiments like EIT, CDS, SUMER, Sac Peak Fe XIV maps, magnetic field maps from the Wilcox solar magnetograph, MLSO, from MDI, and from the Ulysses magnetograph experiment have been, and will be, used to complement LASCO/UVCS/SWOOPS and SWICS data. We anticipate that observations by ACE/WIND/STEREO/Hinode and other missions will be relevant as well. During the IHY campaigns, Ulysses will be 52-80 degrees south in winter 2007, near sunspot minimum. Hence, our own scientific objective will be to sample high speed wind or regions of transition between slow and fast wind. This might be a very interesting situation - not met in previous quadratures - allowing us to study the variation of element abundances across streamer borders in the corona and,correspondingly, "in situ". Also, observing plasma over prolonged time intervals and at different latitudes, we may examine the latitude variation of the fractionation effect (e.g. Yon Steiger, 2002) and its dependence on time (e.g. Woo, 2004). For instance, if the FIP bias is dictated by the duration of plasma confinement in solar loops, its value in streamers should depend on the streamer age. Analogous variations should be found in the abundances of "in situ" plasma released by the streamer. All Ulysses and SOHO data is in the public domain and accessible so different objectives could be addressed by others. The winter 2008 quadrature will be well into the next sunspot cycle and there might be significant high mid-latitude activity. Ulysses will be up to 80 degrees in the northern hemisphere at this time and able to detect the resulting ICMEs and ejecta, allowing us to study the physical parameters of the propagating CME bubble/core/front, and/or to study the coronal and "in situ" characteristics of the current sheet associated with the event (e.g. Bemporad et al. 2005), so far still vaguely defined.

Suess, S. T.↗
NASA NTRS2011

Voyagers 1 and 2 in a Shrunken and Squashed Heliosphere

We have extended our earlier calculations of the distance to the heliospheric termination shock (HTS), which covered the period from the launch of V1 and V2 in 1977 to 2005, to the period from 2006 to 2011. During this latter period, the solar wind speed, ram pressure, and magnetic field decreased to the lowest levels in recent history, related to the sunspot minimum in 2008-2009. The HTS distance has decreased correspondingly so that V1, which was crossed by the HTS at 94 AU in late 2004, would now, in early 2011, be expected to reach the HTS at a distance of approx 80 AU, when the HTS distance would be expected to be at its minimum. Similarly, V2, which was crossed by the HTS at 84 AU in mid-2007, would, in early 2011, reach the HTS at a distance of only 74 AU. These distances, in early 2011, are approx 15% less than those at which V1 and V2 initially reached the HTS. The distance to the heliopause (HP) is more uncertain, but recent calculations place its equilibrium distance at between 1.4 and 1.6 times the HTS distance. Allowing for an additional 1 year for the HP to reach its equilibrium minimum distance relative to the HTS would mean that, assuming this distance remains a constant fraction larger than the HTS distance, the HP distance would be at its minimum distance of (1.4-1.6) X 80 AU = 112-128 AU in the direction of V1 in early 2012. At this time, V1 will be at a distance of approx 120 AU so that there is a possibility that V1 could cross the HP and enter interstellar space at the time 2012.0 +/- 1 year. If the crossing does not happen during this time period, then it is unlikely that V1 will reach this defining boundary before about 2016 because of the expected outward motion of the HTS and the HP toward their more normal distances of 85-96 and approx 120-140 AU, coincident with the maximum of the new sunspot cycle.

Webber, W. R.↗
NASA NTRS2013

On the Trend of the Annual Mean, Maximum, and Minimum Temperature and the Diurnal Temperature Range in the Armagh Observatory, Northern Ireland, Dataset, 1844 -2012

Examined are the annual averages, 10-year moving averages, decadal averages, and sunspot cycle (SC) length averages of the mean, maximum, and minimum surface air temperatures and the diurnal temperature range (DTR) for the Armagh Observatory, Northern Ireland, during the interval 1844-2012. Strong upward trends are apparent in the Armagh surface-air temperatures (ASAT), while a strong downward trend is apparent in the DTR, especially when the ASAT data are averaged by decade or over individual SC lengths. The long-term decrease in the decadaland SC-averaged annual DTR occurs because the annual minimum temperatures have risen more quickly than the annual maximum temperatures. Estimates are given for the Armagh annual mean, maximum, and minimum temperatures and the DTR for the current decade (2010-2019) and SC24.

Wilson, Robert M.↗
NASA NTRS2022

Space Weather Modeling and Prediction for Intermediate Time-scales

Much of solar activity within a sunspot cycle occurs as bursts, or 'seasons' of strong activity over several months, separated by periods of much less activity. The most important space weather effects occur during these bursts. Previous modeling and forecasting efforts have focused on time-scales of hours-to-days and decades-to-centuries. The recent discovery of Rossby waves in the Sun, together with recently developed global models of solar MHD Rossby waves and their interactions with differential rotation and spot-producing magnetic fields, reveal the opportunity to simulate and predict the occurrence, strength and location of enhanced activity bursts a few weeks up to several months in advance. We now have a golden opportunity to fill in this gap in time-scales of forecasting space weather. This requires a) continuous observations of solar Rossby waves by various techniques; b) development of coupled nonlinear MHD models that simulate both global Rossby waves and the much smaller spatial scale emergence of new active regions; c) application of advanced data assimilation techniques to couple surface observations to update the model-system to integrate forward in time for creating forecasts months ahead. Then it will be possible to build operational prediction models to meet the needs of customers and stakeholders, including support of future NASA missions, regarding what kind and level of space weather to expect a few weeks to several months ahead.

SMD↗
NASA NTRS1998

On the Correlation Between Maximum Amplitude and Smoothed Monthly Mean Sunspot Number during the Rise of the Cycle (from t = 0-48 months Past Sunspot Minimum)

During the rise from sunspot minimum to maximum, the observed value of smoothed monthly mean sunspot number at maximum RM is found to correlate with increasing strength against the current value of smoothed monthly mean sunspot number R(t), where t is the elapsed time in months from minimum. On the basis of the modern era sunspot cycles (i.e., cycles 10-22), the inferred linear correlation is found to be statistically important (i.e., at the 95-percent level of confidence) from about 11 mo past minimum and statistically very important (i.e.. at the 99-percent level of confidence) from about 15 mo past minimum; ignoring cycle 19, the largest cycle of the modern era, the inferred linear correlation is found to be statistically important from cycle onset. On the basis of R(t), estimates of RM can be gauged usually to within about +/- 30 percent during the first 2 yr and to within about +/- 20 percent (or better) after the first 2 yr of a cycle's onset. For cycle 23, because controversy exists regarding the placement of its minimum (i.e., its onset), being either May 1996 or perhaps August 1996 (or shortly thereafter), estimates of its RM are divergent, being lower (more like a mean size cycle) when using the earlier epoch of minimum and higher (above average in size) when using the later-occurring minimum. For smoothed monthly mean sunspot number through October 1997 (t = 17 or 14 mo, respectively), having a provisional value of 32.0. the earlier minimum date projects an RM of 110.3 +/- 33.1, while the later minimum date projects one of 137.2 +/- 41.2. The projection is slowly decreasing in size using the earlier onset date, while it is slowly increasing in size using the later onset date.

Wilson, Robert M.↗
NASA NTRS2010

Predicting the Size and Timing of Sunspot Maximum for Cycle 24

For cycle 24, the minimum value of the 12-month moving average (12-mma) of the AA-geomagnetic index in the vicinity of sunspot minimum (AAm) appears to have occurred in September 2009, measuring about 8.4 nT and following sunspot minimum by 9 months. This is the lowest value of AAm ever recorded, falling below that of 8.9 nT, previously attributed to cycle 14, which also is the smallest maximum amplitude (RM) cycle of the modern era (RM = 64.2). Based on the method of Ohl (the preferential association between RM and AAm for an ongoing cycle), one expects cycle 24 to have RM = 55+/-17 (the +/-1 - sigma prediction interval). Instead, using a variation of Ohl's method, one based on using 2-cycle moving averages (2-cma), one expects cycle 23's 2-cma of RM to be about 115.5+/-8.7 (the +/-1 - sigma prediction interval), inferring an RM of about 62+/-35 for cycle 24. Hence, it seems clear that cycle 24 will be smaller in size than was seen in cycle 23 (RM = 120.8) and, likely, will be comparable in size to that of cycle 14. From the Waldmeier effect (the preferential association between the ascent duration (ASC) and RM for an ongoing cycle), one expects cycle 24 to be a slow-rising cycle (ASC > or equal to 48 months), having RM occurrence after December 2012, unless it turns out to be a statistical outlier.

Wilson, Robert M.↗
NASA NTRS1978

A physical mechanism for the prediction of the sunspot number during solar cycle 21

On physical grounds it is suggested that the sun's polar field strength near a solar minimum is closely related to the following cycle's solar activity. Four methods of estimating the sun's polar magnetic field strength near solar minimum are employed to provide an estimate of cycle 21's yearly mean sunspot number at solar maximum of 140 plus or minus 20. This estimate is considered to be a first order attempt to predict the cycle's activity using one parameter of physical importance.

Schatten, K. H.↗
NASA NTRS1982

Sunspot variation and selected associated phenomena: A look at solar cycle 21 and beyond

Solar sunspot cycles 8 through 21 are reviewed. Mean time intervals are calculated for maximum to maximum, minimum to minimum, minimum to maximum, and maximum to minimum phases for cycles 8 through 20 and 8 through 21. Simple cosine functions with a period of 132 years are compared to, and found to be representative of, the variation of smoothed sunspot numbers at solar maximum and minimum. A comparison of cycles 20 and 21 is given, leading to a projection for activity levels during the Spacelab 2 era (tentatively, November 1984). A prediction is made for cycle 22. Major flares are observed to peak several months subsequent to the solar maximum during cycle 21 and to be at minimum level several months after the solar minimum. Additional remarks are given for flares, gradual rise and fall radio events and 2800 MHz radio emission. Certain solar activity parameters, especially as they relate to the near term Spacelab 2 time frame are estimated.

Wilson, R. M.↗
NASA NTRS2012

The Effects of Solar Variability on Earth's Climate: A Workshop Report

Solar irradiance, the flux of the Sun s output directed toward Earth, is Earth s main energy source.1 The Sun itself varies on several timescales over billions of years its luminosity increases as it evolves on the main sequence toward becoming a red giant; about every 11 years its sunspot activity cycles; and within just minutes flares can erupt and release massive amounts of energy. Most of the fluctuations from tens to thousands of years are associated with changes in the solar magnetic field. The focus of the National Research Council's September 2011 workshop on solar variability and Earth's climate, and of this summary report, is mainly magnetically driven variability and its possible connection with Earth's climate variations in the past 10,000 years. Even small variations in the amount or distribution of energy received at Earth can have a major influence on Earth's climate when they persist for decades. However, no satellite measurements have indicated that solar output and variability have contributed in a significant way to the increase in global mean temperature in the last 50 years. Locally, however, correlations between solar activity and variations in average weather may stand out beyond the global trend; such has been argued to be the case for the El Nino-Southern Oscillation, even in the present day. A key area of inquiry deals with establishing a unified record of the solar output and solar-modified particles that extends from the present to the prescientific past. The workshop focused attention on the need for a better understanding of the links between indices of solar activity such as cosmogenic isotopes and solar irradiance. A number of presentations focused on the timescale of the solar cycle and of the satellite record, and on the problem of extending this record back in time. Highlights included a report of progress on pyroheliometer calibration, leading to greater confidence in the time history and future stability of total solar irradiance (TSI), and surprising results on changes in spectral irradiance over the last solar cycle, which elicited spirited discussion. New perspectives on connections between features of the quiet and active areas of the photosphere and variations in TSI were also presented, emphasizing the importance of developing better understanding in order to extrapolate back in time using activity indices. Workshop participants reviews highlighted difficulties as well as causes for optimism in current understanding of the cosmogenic isotope record and the use of observed variability in Sun-like stars in reconstructing variations in TSI occurring on lower frequencies than the sunspot cycle. The workshop succeeded in bringing together informed, focused presentations on major drivers of the Sun-climate connection. The importance of the solar cycle as a unique quasi-periodic probe of climate responses on a timescale between the seasonal and Milankovitch cycles was recognized in several presentations. The signal need only be detectable, not dominant, for it to play this role of a useful probe. Some workshop participants also found encouraging progress in the top-down perspective, according to which solar variability affects surface climate by first perturbing the stratosphere, which then forces the troposphere and surface. This work is now informing and being informed by research on tropospheric responses to the Antarctic ozone hole and volcanic aerosols. In contrast to the top-down perspective is the bottom-up view that the interaction of solar energy with the ocean and surface leads to changes in dynamics and temperature. During the discussion of how dynamical air-sea coupling in the tropical Pacific and solar variability interact from a bottom-up perspective, several participants remarked on the wealth of open research questions in the dynamics of the climatic response to TSI and spectral variability. The discussion of the paleoclimate record emphasized that the link between solar varbility and Earth s climate is multifaceted and that some components are understood better than others. According to two presenters on paleoclimate, there is a need to study the idiosyncrasies of each key proxy record. Yet they also emphasized that there may be an emerging pattern of paleoclimate change coincident with periods of solar activity and inactivity, but only on long timescales of multiple decades to millennia. Several speakers discussed the effects of particle events and cosmic-ray variability. These are all areas of exciting fundamental research; however, they have not yet led to conclusive evidence for significant related climate effects. The key problem of attribution of climate variability on the timescales of the Little Ice Age and the Maunder Minimum were directly addressed in several presentations. Several workshop participants remarked that the combination of solar, paleoclimatic, and climate modeling research has the potential to dramatically improve the credibility of these attribution studies.

Source record↗
NASA NTRS1978

Using dynamo theory to predict the sunspot number during solar cycle 21

On physical grounds it is suggested that the polar field strength of the sun near a solar minimum is closely related to the solar activity of the following cycle. Four methods of estimating the polar magnetic field strength of the sun near solar minimum are employed to provide an estimate of the yearly mean sunspot number of cycle 21 at solar maximum of 140 + or - 20. This estimate may be considered a first-order attempt to predict the cycle activity using one parameter of physical importance based upon dynamo theory.

Schatten, K. H.↗
NASA NTRS1986

Bimodality of the solar cycle

An alternative characterization of the solar cycle is offered that is consistent with the sunspot data for cycles 1-20 (1775-1976) but suggests a different physical interpretation. For sunspot cycles 1-20, all cycles occurred in strings (two to six cycles in length) during which the period remained longer or shorter than the sample mean period. These strings have coincided with long-term trends of growth or decay in the amplitude of the cycle. In six out of six cases, the period of the cycle has switched from long to short (or the reverse) in coincidence with the turning points in the long-term trend. This suggests that the solar dynamo has two modes with different mean periods. In the short-period mode, the amplitude of the cycle grows; in the long-period mode, the amplitude decays. The transition between modes has occurred at irregular intervals. A persistence of the long-period mode would eventually produce a grand minimum such as the Maunder minimum; a persistence of the short-period mode would produce a grand maximum. Unless the present interval between transitions turns out to be shorter than any previously observed interval, the present cycle (cycle 21) is part of a long-period, decaying trend and will be of longer-than-average duration (more than 133 months).

Rabin, D.↗
NASA NTRS2014

Comparison of the Variations of Sunspot Number, Number of Sunspot Groups, and Sunspot Area, 1875 -2013

Examined are the yearly variations and ratios of sunspot number, the number of sunspot groups, and the total corrected sunspot area for the interval 1875-2013. While yearly sunspot number independently correlates strongly (r = 0.98) with the yearly number of sunspot groups (y = -2 + 11.99x) and the total corrected sunspot area (y = 5 + 0.059x), the strongest correlation (Ry12 = 0.99) is the one based on the bivariate fit of sunspot number against the combined variations of the number of sunspot groups and sunspot area (y = 1 + 5.88x1 + 0.031x2, where y refers to sunspot number, x1 refers to the number of sunspot groups, and x2 refers to the sunspot area). While all cycle minima based on the bivariate fit are concurrent with the observed minimum in sunspot number, cycle maxima are sometimes found to differ. For sunspot cycles 12, 19, 20, and 23, cycle maximum is inferred to have occurred in 1884, 1958, 1970, and 2002, respectively, rather than in 1883, 1957, 1968, and 2000, based on the observed sunspot number. Also, cycle 19's maximum amplitude based on observed sunspot number seems too high in comparison to that found using the bivariate fit. During the 139-year interval 1875-2013, the difference between the observed and predicted sunspot number based on the bivariate fit is <1 standard error of estimate (se) (<6.4) for 111 years, between 1 and <2 se (6.4 to <12.8) for 28 years, and ≥2 se (≥12.8) for only 4 years, these years being 1957 (16.6), 1978 (-15.8), 1980 (23), and 1982 (-16.3). For sunspot cycle 24, the difference between observed and predicted values has been only -0.7 and 3.2 (≤0.5 se).

Wilson, Robert M.↗