A Remarkably Accurate Predictor of Sunspot Cycle Amplitude

A Remarkably Accurate Predictor of Sunspot Cycle Amplitude | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article A Remarkably Accurate Predictor of Sunspot Cycle Amplitude Peter Foukal This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8340346/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 19 Feb, 2026 Read the published version in Solar Physics → Version 1 posted 7 You are reading this latest preprint version Abstract The slopes of the linear relations between sunspot and white light (WL) facular areas at the onset of sunspot cycles 12–21 correlate well with the amplitudes of those cycles between 1878–1980 (Brown and Evans, 1980 ). We use continuum images from the SOHO Michelson Doppler Imager and SDO Heliospheric Magnetic Imager to show that the relation holds also for cycles 24 and 25. The amplitudes of cycles 12–21 and 24 calculated using this relation agree with the observed amplitudes to within 4% rms. It also enabled us in 2022 to correctly predict a larger cycle 25 than estimated by the International Prediction Panel, 3 years before maximum. The technique offers an objective, physically based predictor of cycle amplitudes 3–4 years ahead of their maxima, given a stable source of continuum full disk photospheric images. Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 1. Introduction Predictions of the amplitude of the next sunspot cycle are one of the most important practical applications of solar research and astronomy. The dynamics of the magnetic cycle is still too uncertain to provide a physical basis for reliable predictions so estimates of the cycle amplitude and timing are derived largely from empirical or semi-empirical models (e.g Schatten, 2005 ; Svalgaard, Cliver and Kamide, 2005 ; McIntosh et al, 2020; see e.g. Pesnell, 2020 ; Petrovay, 2020 for recent reviews). This study builds on the finding (Brown and Evans, 1980 ) that the slopes, dA s /dA f , of the linear relation between sunspot and WL facular areas early in a cycle are closely related to the cycle’s amplitude. This relationship was originally discovered in analysis of the measurements of spots and WL faculae made by the Royal Greenwich Observatory (RGO) between 1874 and 1976. Brown and Evans used it to predict the amplitude of cycle 21 to within 6% of its observed amplitude. The RGO program (Royal Observatory, 1973) ended in 1976 and this relationship was not exploited to predict the amplitudes of subsequent cycles. Emulsion inhomogeneity and plate exposure variations make measurement of WL facular areas difficult on digitizations of the Mt Wilson Observatory continuum spectro - heliograms and Kodaikanal Observatory photoheliograms, the two most complete sets of full disk continuum images available after 1976. More recently, excellent daily full disk continuum images were obtained first from the Michelson Doppler Imager (MDI) on the Solar and Heliospheric Observatory (SOHO) beginning in 2001 (Scherrer et al., 1995 ) and then from the Heliospheric and Magnetic Imager (HMI) on the Solar Dynamics Observatory (SDO) since 2011 (Scherrer et al., 2012 ). These enable us to show that the relationship holds also for cycles 24 and 25. 2. Data and analysis The MDI images were obtained in narrow - band (94 mÅ) quasi-continuum in the wing of the Ni 6768 Å line. The 75 mÅ HMI continuum passband was located in the wing of the Fe I 6173 Å line. Both data sets had been corrected for limb darkening by the P.I. teams. These passbands showed faculae only nearer the limb than a heliocentric angle of approximately 50 degrees, in agreement with their appearance on the continuum images used in the RGO measurements. The resolution, uniformity, and dynamic range of the space-borne imagery (which shows the photospheric granulation) are superior to the definition of the photographic plates used by RGO. A representative HMI image is shown in Fig. 1 . Four MDI images equally spaced within each of the 24 months for the two first years 2009–2010 of cycle 24 and the same using 2020–2021 HMI data for cycle 25, were downloaded and printed. This sampling is adequate given the high quality of the MDI and HMI images, the lifetimes of large WL faculae and spots that determine most of these features’ total area, and the 13-month smoothing of our data. The projected areas of spots and faculae on the 130 mm (MDI) and 190 mm (HMI) diameter printed images were measured by counting squares using a magnifier with a reticle grid of 0.5 mm spacing, and corrected for foreshortening. This is similar to the procedure used by the RGO between 1874–1976. Monthly means of these corrected spot and facular areas were then smoothed with an evenly weighted 13-month running mean, following the method used by Brown and Evans ( 1980 ). 3. Results and errors A plot of these running means of spot (A s ) versus facular (A f ) areas for cycle 24 is shown in Fig. 2 . We see that the spot and facular areas are linearly related (R 2 = 0.88) with a slope of 0.24 in this cycle. A similar plot is shown in Fig. 3 for cycle 25. Again, the relation is linear (R 2 = 0.97) but the slope of 0.61 is greater. Brown and Evans ( 1980 ) showed that such linear relationships were characteristic of approximately the first two years of all the cycles measured by RGO. Figure 4 shows the linear relationship (R 2 = 0.95) between the slopes of the cycles 12–21 derived by Brown and Evans, and the peak 13 - month smoothed sunspot number, R m -max, of those cycles. The slope of 0.24 for cycle 24 is indicated by an inverted triangle at the bottom of the plot. It corresponds to a peak cycle amplitude of approximately 115, which agrees with its 13- month smoothed amplitude of 115 achieved in 2014. The values of R m –max used in this study are on the re-calibrated International Sunspot Number scale (Clette et al., 2016 ). They are about 35% higher than those on the earlier Zurich sunspot number scale used by Brown and Evans ( 1980 ). The slope of 0.61 measured here from HMI data for cycle 25 yields a peak amplitude of R m –max = 185, about 16% higher than the actual R m -max achieved in mid-2025. Györi (2012) reports that the MDI spot and facular areas exceed the HMI values because of the lower angular resolution of the MDI images. His relations for HMI corrected areas relative to MDI values are: A s MDI = (A S HMI − 4)/0.91, and A f MDI = (A f MDI – 260)/0.67. Using these we obtain a slope of 0.44 in Fig. 3 and a lower R m -max ~ 145. That analysis compared MDI data obtained during the decline of cycle 23 with different structures observed by HMI during the rise of cycle 24. We compared instead the areas obtained when the same structures are observed co-temporally during the August – December 2010 overlap period between the MDI and HMI missions. The results shown in Figs. 5 and 6 yield the transformation relations: As MDI = (As HMI − 2)/0.88 (r 2 = 0.93) and A f MDI = (A f HMI – 4)/0.87 (r 2 = 0.62). The comparison of projected rather than corrected areas used in these Figures avoids error from unnecessary foreshortening corrections. Our transformations essentially cancel and make no significant difference to the slope of Fig. 3. This is consistent with the blurring interpretation of the larger MDI areas since the RGO method of area measurement we adopted for consistency introduces similar blurring for the MDI and HMI images. It is also consistent with the report by Jha et al. ( 2019 ) of good agreement between MDI spot areas and those measured from Kodaikanal Solar Observatory photo - heliograms when both are degraded to similar angular resolution. The corrections reported by Györi (2012) apply to comparison of the MDI images with those obtained from HMI at significantly higher angular area resolution so they cannot be applied directly to our measurements. Unfortunately, the low activity level during the short overlap period precludes defining our facular correction accurately enough to decide confidently whether it can account for the 16% difference between calculated and actual R m – max in cycle 25. 4. Discussion and Conclusions We have shown that the linear relations between spot and facular areas, A s and A f , and between the slopes dA s /dA f and cycle amplitudes R m – max, first reported for the onsets of cycles 12-21 by Brown and Evans (1980) hold also for cycles 24 and 25. The mean difference between the calculated and actual amplitudes of cycles 12-24 (see Table I) is below 1 (+/- 4) %.This represents unprecedented accuracy over an unequaled span of 1 ½ centuries. Table I Comparison of calculated and observed 13-month smoothed amplitudes of cycles 12 21 and 24, 25. The 16% over-estimate in our 2022 prediction of the cycle 25 amplitude of R m – max = 185 (Foukal, 2023) was probably caused mainly by the uncertain correction between facular areas measured from the high - resolution HMI images, as compared to those we measured from the MDI and those measured by RGO from ground based photo – heliograms of similar resolution. Despite this uncertainty we were able to correctly predict a cycle 25 amplitude larger than the prediction by the International Prediction Panel at that epoch (e.g. Miesch, 2023) The approximately 3 ½ year advance notice of our cycle 25 prediction is shorter than provided by some of the 50 or more other methods used by the IPP (see e.g. Miesch, 2023 for an overview of cycle 25 predictions). Nor does this method predict the timing of maxima, although a useful relation between R m - max and the cycle rise time (Waldmeier, 1978) can be helpful in this respect. The method described here is empirical but its physical basis can be understood in terms of the evidence on the relative areas of spots and faculae in the Sun and other late type stars (e.g. Foukal, 1994; 1998). Dark spots are produced by magnetic flux tubes whose areas lie at the upper end of the photospheric flux tube size spectrum. The WL faculae lie at the lowest end (e.g. Spruit, 1976). The increase in the slope dA s /dA f between cycles 24 and 25 seen in Figures 2 and 3 therefore corresponds to a shift in the size spectrum of photospheric magnetic fields towards more power at the larger scales, as the cycle amplitude increases. Such a tendency to shift dynamo power towards lower spatial wavenumbers as magnetic flux output increases has been recognized in observations of large spots on younger late-type stars (e.g. Radick, Lockwood and Baliunas, 1990) and in dynamo calculations (e.g. Cattaneo, Cieunh and Hughes, 1990). The relationship studied here between A s and A f does not hold as well between sunspots and bright chromospheric plages (Foukal, 1998) because these cover not only the most slender flux tubes observed as WL faculae, but also pores and small spots. This decreases the sensitivity of the present diagnostic. So daily Ca K plage areas available for cycles 22 and 23 (e.g. Foukal et al., 2009) cannot be used to substitute the missing facular data for those cycles. But extension of the method to cycle 23 using the enhanced contrast continuum images from the Kanzelhöhe Observatory (Pötzi, 2025) deserves investigation. Declarations Author Contribution The work was performed entirely by the author. Acknowledgement Acknowledgements This study was facilitated by availability of the MDI and HMI images obtained by the Stanford University and Lockheed Corporation Solar Groups on Spaceweather.com, and on the Debrecen Observatory site (fenyi.solarobs.hunren.hu). I am grateful to Frédéric Clette and Mark Miesch for information on the SC 25 Prediction Panel, to Leif Svalgaard for drawing my attention to Waldmeier’s rule and to Neil Sheeley and an anonymous referee for comments on an original version of this paper. I thank Tom Williams for drawing my attention to the Spaceweather.com site and Jack Martin for facilitating access to the RGO archive at Cambridge University; I am grateful to the archivist, Dr. Emma Saunders, for her kind assistance. References Brown, G., Evans, D.: 1980, The Use of Solar Faculae in Studies of the Sunspot Cycle. Solar Phys . 66 , 233. Cattaneo, F., Cieunh, T., Hughes, D.: 1990, Buoyancy Driven Instabilities and the Non - Linear Break Up of a Sheared Magnetic Layer. J. Fluid Mech . 219 , 1. Clette, F., Cliver, E., Lefevre, L., Svalgaard, L., Vacquero, J., Leibacher, J.: 2016, Recalibration of the Sunspot Number. Solar Phys . 291 , 2479. Foukal, P.: 1994, Stellar Luminosity Variations and Global Warming. Science , 264 , 238. Foukal, P.: 1998 , What Determines the Relative Areas of Spots and Faculae on Sun – Like Stars. Astrophys. J. , 500 , 958. Foukal, P. et al.: 2009, A Century of Solar Ca II Measurements and Their Implications for Solar UV Driving of Climate. Solar Phys . 255 , 229. Foukal, P.: 2023, A Remarkably Accurate Predictor of Solar Cycle Amplitude. Arxiv.org/submit/5200355. Gyori, L.: 2012, Study of Differences Between Sunspot and Facular Areas from SOHO MDI and SDO HMI Observations. Solar Phys . 280 , 365. Jha, B., Mandal, S., Banerjee, D.: 2019, Solar Phys . 294 , 72. McIntosh, S., Chapman, S., Leamon, R., Egeland, R., Watkins, N.: 2020, Overlapping Magnetic Cycles and the Sunspot Number: Forecasting Sunspot Cycle 25. Solar Phys. 295 , 163. Miesch, 2023, private communication. (https://www.swpc.noaa.gov/products/solar-cycle-progression Pesnell, D.: 2020, Lessons Learned from Predictions of Solar Cycle 24. J. Space Weather Space Climate 10, 60. Pötzi, W., 2025, private communication. Petrovay, K.: 2020, Solar Cycle Prediction. Living Rev. Sol. Phys. 17 (1). Radick, R., Lockwood, W., Baliunas, S.: 1990, Stellar Activity and Brightness Variations: A Glimpse at the Sun’s History. Science , 247 , 39. Royal Observatory Annals , 1973, Vol. 8 , (Photoheliograph Results). Schatten, K.:2005, Fair Space Weather for Cycle 24. Geophys. Res. Letts , 32 , 21. Scherrer, P., et al.: 1995, The Solar Oscillations Investigation - Michelson Doppler Imager. Solar Phys . 162 , 129-188 Scherrer, P., et al.: 2012, The Heliospheric and Magnetic Imager (HMI) Investigation for the Solar Dynamics Observatory (SDO) Solar Phys . 275 , 17-40. Spruit, H.: 1976, Pressure Equilibrium and Energy Balance of Small Photospheric Flux Tubes. Solar Phys . 50 , 269-295. Svalgaard, L., Cliver, F., Kamide, Y.: 2005, Sunspot Cycle 24: Smallest Cycle in 100 Years? Geophys.Res. Letts ., 32 , 1. Waldmaier, M.: 1978, Sunspot Numbers and Sunspot Areas. Astron.Mitt. Zurich Nr 358. Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 19 Feb, 2026 Read the published version in Solar Physics → Version 1 posted Editorial decision: Revision requested 02 Jan, 2026 Reviews received at journal 02 Jan, 2026 Reviewers agreed at journal 16 Dec, 2025 Reviewers invited by journal 16 Dec, 2025 Editor assigned by journal 12 Dec, 2025 Submission checks completed at journal 12 Dec, 2025 First submitted to journal 11 Dec, 2025 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-8340346","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":561464151,"identity":"edd7b7ee-da5e-475c-b8b4-cdfb3071f310","order_by":0,"name":"Peter Foukal","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA4klEQVRIie3PMQrCMBSA4VcCnR50fWLxDC2FShfvIoVOggdQqlDQRZ07eAiPUAimS8G14lJwdaiLOIkNbipRN4f8W0K+JA9Ap/vLjAogA9v6gTBHEmxNH0uSt0y/Ik72LelSaByx6KGX8+3mOo5jaCd5pSJBGjIPyxD9Ior2C8EJbNFXvuIUO9HGmqFfon8AMyOggfuBcLMhE/RSSW5xQ4ZnNcmThpQcHWqIMWPyFfX4wTxh7rrIkeQsyxVvzShyUxXpMmZUJzHqWHMuyusltiwKq1r5sZcdU3X8PdHpdDrdc3cg8Ec9NT0QXQAAAABJRU5ErkJggg==","orcid":"","institution":"","correspondingAuthor":true,"prefix":"","firstName":"Peter","middleName":"","lastName":"Foukal","suffix":""}],"badges":[],"createdAt":"2025-12-11 23:08:20","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8340346/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8340346/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s11207-026-02611-7","type":"published","date":"2026-02-19T15:57:36+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":98761650,"identity":"9c418494-b0c0-4c44-b9e6-6561e4aafb83","added_by":"auto","created_at":"2025-12-22 09:55:51","extension":"docx","order_by":0,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":267872,"visible":true,"origin":"","legend":"","description":"","filename":"RemarkablePredictorSolarPhys.Copy.docx","url":"https://assets-eu.researchsquare.com/files/rs-8340346/v1/8b36340a3b69b5b9642ccf04.docx"},{"id":98761611,"identity":"0ce3498f-976f-41ae-9510-3f657b44aaa4","added_by":"auto","created_at":"2025-12-22 09:55:43","extension":"json","order_by":1,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":2559,"visible":true,"origin":"","legend":"","description":"","filename":"77882467a6ae49c48d7c40a0bf3646cd.json","url":"https://assets-eu.researchsquare.com/files/rs-8340346/v1/5f276491c820c95f3e60131c.json"},{"id":98761603,"identity":"beb4b537-f21c-48e6-82ee-b9b521bc99c3","added_by":"auto","created_at":"2025-12-22 09:55:42","extension":"xml","order_by":2,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":39672,"visible":true,"origin":"","legend":"","description":"","filename":"77882467a6ae49c48d7c40a0bf3646cd1enriched.xml","url":"https://assets-eu.researchsquare.com/files/rs-8340346/v1/18fd77b3bcf63a77884973d5.xml"},{"id":98761630,"identity":"40ba5cb6-595d-4b6c-846d-81e78aa54f7f","added_by":"auto","created_at":"2025-12-22 09:55:44","extension":"eps","order_by":4,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":97999,"visible":true,"origin":"","legend":"","description":"","filename":"drawingimage2.eps","url":"https://assets-eu.researchsquare.com/files/rs-8340346/v1/a80fd9e20479ded554b9b8b6.eps"},{"id":98779372,"identity":"e756ddcc-71b2-43b2-868d-64666d0fff2d","added_by":"auto","created_at":"2025-12-22 12:30:17","extension":"eps","order_by":6,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":1087,"visible":true,"origin":"","legend":"","description":"","filename":"drawingimage4.eps","url":"https://assets-eu.researchsquare.com/files/rs-8340346/v1/59c5348d85315c9395b7cf7a.eps"},{"id":98761640,"identity":"ea84eb88-35fc-46e2-9239-4ed1e065a2d2","added_by":"auto","created_at":"2025-12-22 09:55:50","extension":"eps","order_by":7,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":1059,"visible":true,"origin":"","legend":"","description":"","filename":"drawingimage5.eps","url":"https://assets-eu.researchsquare.com/files/rs-8340346/v1/04e6a20fe29b4d01ad537038.eps"},{"id":98761614,"identity":"a613d8ea-d816-4c5f-8d33-827780731616","added_by":"auto","created_at":"2025-12-22 09:55:43","extension":"eps","order_by":9,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":100741,"visible":true,"origin":"","legend":"","description":"","filename":"drawingimage7.eps","url":"https://assets-eu.researchsquare.com/files/rs-8340346/v1/c343daa36642bf0046c73c50.eps"},{"id":98761649,"identity":"a4b1fb7c-6395-4ee8-b93f-f70c55e29ba1","added_by":"auto","created_at":"2025-12-22 09:55:51","extension":"jpeg","order_by":10,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":72566,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage1.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-8340346/v1/37bfd1f436596e8c5696b055.jpeg"},{"id":98761643,"identity":"6bb31aa0-3622-4034-95bf-9f4f4b7a35fb","added_by":"auto","created_at":"2025-12-22 09:55:51","extension":"png","order_by":11,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":46145,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage1.png","url":"https://assets-eu.researchsquare.com/files/rs-8340346/v1/ea367349ba7ded07fb06525b.png"},{"id":98761606,"identity":"846d26c6-04c7-4a64-929f-56065534ba16","added_by":"auto","created_at":"2025-12-22 09:55:42","extension":"xml","order_by":12,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":37996,"visible":true,"origin":"","legend":"","description":"","filename":"77882467a6ae49c48d7c40a0bf3646cd1structuring.xml","url":"https://assets-eu.researchsquare.com/files/rs-8340346/v1/efe45925c054b2970ce2d72c.xml"},{"id":98761621,"identity":"3157179d-8f8f-410d-a9b1-9bfc498b09a5","added_by":"auto","created_at":"2025-12-22 09:55:43","extension":"html","order_by":13,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":44486,"visible":true,"origin":"","legend":"","description":"","filename":"earlyproof.html","url":"https://assets-eu.researchsquare.com/files/rs-8340346/v1/36ea61b3a1c5f4611ce976bf.html"},{"id":98761655,"identity":"2f7f1307-0214-4676-a57c-23e65a727d50","added_by":"auto","created_at":"2025-12-22 09:55:52","extension":"jpg","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":99735,"visible":true,"origin":"","legend":"\u003cp\u003eClose-up of limb faculae imaged by HMI on 1 November 2011.\u003c/p\u003e","description":"","filename":"1.jpg","url":"https://assets-eu.researchsquare.com/files/rs-8340346/v1/09968b02024f459a887e129d.jpg"},{"id":98777489,"identity":"d69a6311-db9e-446b-a178-ef8f21cd8cb5","added_by":"auto","created_at":"2025-12-22 12:27:40","extension":"jpg","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":23899,"visible":true,"origin":"","legend":"\u003cp\u003eA\u003csub\u003es\u003c/sub\u003e versus A\u003csub\u003ef\u003c/sub\u003e (µh) for 2009 – 2010.\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp; \u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;\u003c/p\u003e","description":"","filename":"2.jpg","url":"https://assets-eu.researchsquare.com/files/rs-8340346/v1/d4180cec5a370cf38c2908ee.jpg"},{"id":98761635,"identity":"ea484609-2214-41d2-b008-8759871a7544","added_by":"auto","created_at":"2025-12-22 09:55:49","extension":"jpg","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":25129,"visible":true,"origin":"","legend":"\u003cp\u003eA\u003csub\u003es\u003c/sub\u003e versus A\u003csub\u003ef\u003c/sub\u003e (µh) for 2020 – 2021.\u003c/p\u003e","description":"","filename":"3.jpg","url":"https://assets-eu.researchsquare.com/files/rs-8340346/v1/0b6f395d45688d378089ccd1.jpg"},{"id":98761628,"identity":"7e3b764c-f63a-41f9-8b2f-182a7ae60ccb","added_by":"auto","created_at":"2025-12-22 09:55:44","extension":"jpg","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":25905,"visible":true,"origin":"","legend":"\u003cp\u003eR\u003csub\u003em\u003c/sub\u003e-max versus slopes dA\u003csub\u003es\u003c/sub\u003e/dA\u003csub\u003ef\u003c/sub\u003e. The dotted line represents a linear fit to the 10 points derived by Brown and Evans (1980), shown as filled circles. The slope values derived here for cycles 24 and 25 are shown as the inverted and upright triangles\u003cstrong\u003e,\u003c/strong\u003e respectively.\u003c/p\u003e","description":"","filename":"4.jpg","url":"https://assets-eu.researchsquare.com/files/rs-8340346/v1/82f92d8aade77861cd4282ba.jpg"},{"id":98761631,"identity":"43d266f5-0405-4f3d-88dc-2082b6cb22fc","added_by":"auto","created_at":"2025-12-22 09:55:44","extension":"jpg","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":26790,"visible":true,"origin":"","legend":"\u003cp\u003eA\u003csub\u003es \u003c/sub\u003eHMI versus MDI (millionths of disk).\u003c/p\u003e","description":"","filename":"5.jpg","url":"https://assets-eu.researchsquare.com/files/rs-8340346/v1/181fa69af9ddc2a85ed0c686.jpg"},{"id":98761645,"identity":"d4bae039-9560-4ea3-a232-c5ebb686dd63","added_by":"auto","created_at":"2025-12-22 09:55:51","extension":"jpg","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":28880,"visible":true,"origin":"","legend":"\u003cp\u003eA\u003csub\u003ef\u003c/sub\u003e HMI versus MDI (millionths of disk).\u003c/p\u003e","description":"","filename":"6.jpg","url":"https://assets-eu.researchsquare.com/files/rs-8340346/v1/9aeb2272bc0d5c588d0f770b.jpg"},{"id":103251157,"identity":"24b49a0b-85b7-49c0-802f-924ca26c8458","added_by":"auto","created_at":"2026-02-23 16:05:24","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":597158,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8340346/v1/e2571251-23a2-4ed0-b9f3-3c42390e37b3.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"\u003cp\u003eA Remarkably Accurate Predictor of Sunspot Cycle Amplitude\u003c/p\u003e","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003ePredictions of the amplitude of the next sunspot cycle are one of the most important practical applications of solar research and astronomy. The dynamics of the magnetic cycle is still too uncertain to provide a physical basis for reliable predictions so estimates of the cycle amplitude and timing are derived largely from empirical or semi-empirical models (e.g Schatten, \u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e2005\u003c/span\u003e; Svalgaard, Cliver and Kamide, \u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e2005\u003c/span\u003e; McIntosh et al, 2020; see e.g. Pesnell, \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e2020\u003c/span\u003e; Petrovay, \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e2020\u003c/span\u003e for recent reviews).\u003c/p\u003e \u003cp\u003eThis study builds on the finding (Brown and Evans, \u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1980\u003c/span\u003e) that the slopes, dA\u003csub\u003es\u003c/sub\u003e/dA\u003csub\u003ef\u003c/sub\u003e, of the linear relation between sunspot and WL facular areas early in a cycle are closely related to the cycle\u0026rsquo;s amplitude. This relationship was originally discovered in analysis of the measurements of spots and WL faculae made by the Royal Greenwich Observatory (RGO) between 1874 and 1976. Brown and Evans used it to predict the amplitude of cycle 21 to within 6% of its observed amplitude.\u003c/p\u003e \u003cp\u003eThe RGO program (Royal Observatory, 1973) ended in 1976 and this relationship was not exploited to predict the amplitudes of subsequent cycles. Emulsion inhomogeneity and plate exposure variations make measurement of WL facular areas difficult on digitizations of the Mt Wilson Observatory continuum spectro - heliograms and Kodaikanal Observatory photoheliograms, the two most complete sets of full disk continuum images available after 1976.\u003c/p\u003e \u003cp\u003eMore recently, excellent daily full disk continuum images were obtained first from the Michelson Doppler Imager (MDI) on the Solar and Heliospheric Observatory (SOHO) beginning in 2001 (Scherrer et al., \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e1995\u003c/span\u003e) and then from the Heliospheric and Magnetic Imager (HMI) on the Solar Dynamics Observatory (SDO) since 2011 (Scherrer et al., \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e2012\u003c/span\u003e). These enable us to show that the relationship holds also for cycles 24 and 25.\u003c/p\u003e"},{"header":"2. Data and analysis","content":"\u003cp\u003eThe MDI images were obtained in narrow - band (94 m\u0026Aring;) quasi-continuum in the wing of the Ni 6768 \u0026Aring; line. The 75 m\u0026Aring; HMI continuum passband was located in the wing of the Fe I 6173 \u0026Aring; line. Both data sets had been corrected for limb darkening by the P.I. teams. These passbands showed faculae only nearer the limb than a heliocentric angle of approximately 50 degrees, in agreement with their appearance on the continuum images used in the RGO measurements. The resolution, uniformity, and dynamic range of the space-borne imagery (which shows the photospheric granulation) are superior to the definition of the photographic plates used by RGO. A representative HMI image is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFour MDI images equally spaced within each of the 24 months for the two first years 2009\u0026ndash;2010 of cycle 24 and the same using 2020\u0026ndash;2021 HMI data for cycle 25, were downloaded and printed. This sampling is adequate given the high quality of the MDI and HMI images, the lifetimes of large WL faculae and spots that determine most of these features\u0026rsquo; total area, and the 13-month smoothing of our data.\u003c/p\u003e \u003cp\u003eThe projected areas of spots and faculae on the 130 mm (MDI) and 190 mm (HMI) diameter printed images were measured by counting squares using a magnifier with a reticle grid of 0.5 mm spacing, and corrected for foreshortening. This is similar to the procedure used by the RGO between 1874\u0026ndash;1976. Monthly means of these corrected spot and facular areas were then smoothed with an evenly weighted 13-month running mean, following the method used by Brown and Evans (\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1980\u003c/span\u003e).\u003c/p\u003e"},{"header":"3. Results and errors","content":"\u003cp\u003eA plot of these running means of spot (A\u003csub\u003es\u003c/sub\u003e) versus facular (A\u003csub\u003ef\u003c/sub\u003e) areas for cycle 24 is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e. We see that the spot and facular areas are linearly related (R\u003csup\u003e2\u003c/sup\u003e\u0026thinsp;=\u0026thinsp;0.88) with a slope of 0.24 in this cycle. A similar plot is shown in Fig.\u0026nbsp;3 for cycle 25. Again, the relation is linear (R\u003csup\u003e2\u003c/sup\u003e\u0026thinsp;=\u0026thinsp;0.97) but the slope of 0.61 is greater. Brown and Evans (\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1980\u003c/span\u003e) showed that such linear relationships were characteristic of approximately the first two years of all the cycles measured by RGO.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e4\u003c/span\u003e shows the linear relationship (R\u003csup\u003e2\u003c/sup\u003e\u0026thinsp;=\u0026thinsp;0.95) between the slopes of the cycles 12\u0026ndash;21 derived by Brown and Evans, and the peak 13 - month smoothed sunspot number, R\u003csub\u003em\u003c/sub\u003e-max, of those cycles. The slope of 0.24 for cycle 24 is indicated by an inverted triangle at the bottom of the plot. It corresponds to a peak cycle amplitude of approximately 115, which agrees with its 13- month smoothed amplitude of 115 achieved in 2014. The values of R\u003csub\u003em\u003c/sub\u003e\u0026ndash;max used in this study are on the re-calibrated International Sunspot Number scale (Clette et al., \u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e2016\u003c/span\u003e). They are about 35% higher than those on the earlier Zurich sunspot number scale used by Brown and Evans (\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1980\u003c/span\u003e).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe slope of 0.61 measured here from HMI data for cycle 25 yields a peak amplitude of R\u003csub\u003em\u003c/sub\u003e\u0026ndash;max\u0026thinsp;=\u0026thinsp;185, about 16% higher than the actual R\u003csub\u003em\u003c/sub\u003e-max achieved in mid-2025.\u003c/p\u003e \u003cp\u003eGy\u0026ouml;ri (2012) reports that the MDI spot and facular areas exceed the HMI values because of the lower angular resolution of the MDI images. His relations for HMI corrected areas relative to MDI values are: A\u003csub\u003es\u003c/sub\u003e MDI = (A\u003csub\u003eS\u003c/sub\u003e HMI \u0026minus;\u0026thinsp;4)/0.91, and A\u003csub\u003ef\u003c/sub\u003e MDI = (A\u003csub\u003ef\u003c/sub\u003e MDI \u0026ndash; 260)/0.67. Using these we obtain a slope of 0.44 in Fig.\u0026nbsp;3 and a lower R\u003csub\u003em\u003c/sub\u003e-max\u0026thinsp;~\u0026thinsp;145.\u003c/p\u003e \u003cp\u003eThat analysis compared MDI data obtained during the decline of cycle 23 with different structures observed by HMI during the rise of cycle 24. We compared instead the areas obtained when the \u003cem\u003esame\u003c/em\u003e structures are observed co-temporally during the August \u0026ndash; December 2010 overlap period between the MDI and HMI missions. The results shown in Figs.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e5\u003c/span\u003e and 6 yield the transformation relations: As MDI = (As HMI \u0026minus;\u0026thinsp;2)/0.88 (r\u003csup\u003e2\u003c/sup\u003e\u0026thinsp;=\u0026thinsp;0.93) and A\u003csub\u003ef\u003c/sub\u003e MDI = (A\u003csub\u003ef\u003c/sub\u003e HMI \u0026ndash; 4)/0.87 (r\u003csup\u003e2\u003c/sup\u003e\u0026thinsp;=\u0026thinsp;0.62). The comparison of projected rather than corrected areas used in these Figures avoids error from unnecessary foreshortening corrections.\u003c/p\u003e \u003cp\u003eOur transformations essentially cancel and make no significant difference to the slope of Fig.\u0026nbsp;3. This is consistent with the blurring interpretation of the larger MDI areas since the RGO method of area measurement we adopted for consistency introduces similar blurring for the MDI and HMI images. It is also consistent with the report by Jha et al. (\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e2019\u003c/span\u003e) of good agreement between MDI spot areas and those measured from Kodaikanal Solar Observatory photo - heliograms when both are degraded to similar angular resolution.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe corrections reported by Gy\u0026ouml;ri (2012) apply to comparison of the MDI images with those obtained from HMI at significantly higher angular area resolution so they cannot be applied directly to our measurements. Unfortunately, the low activity level during the short overlap period precludes defining our facular correction accurately enough to decide confidently whether it can account for the 16% difference between calculated and actual R\u003csub\u003em\u003c/sub\u003e \u0026ndash; max in cycle 25.\u003c/p\u003e"},{"header":"4. Discussion and Conclusions","content":"\u003cp\u003eWe have shown that the linear relations between spot and facular areas, A\u003csub\u003es\u003c/sub\u003e and A\u003csub\u003ef\u003c/sub\u003e, and between the slopes dA\u003csub\u003es\u003c/sub\u003e/dA\u003csub\u003ef\u003c/sub\u003e and cycle amplitudes R\u003csub\u003em\u003c/sub\u003e – max, first reported for the onsets of cycles 12-21 by Brown and Evans (1980) hold also for cycles 24 and 25. The mean difference between the calculated and actual amplitudes of cycles 12-24 (see Table I) is below 1 (+/- 4) %.This represents unprecedented accuracy over an unequaled span of 1 ½ centuries.\u003c/p\u003e\n\u003cp\u003eTable I Comparison of calculated and observed 13-month smoothed amplitudes of cycles 12 21 and 24, 25.\u003c/p\u003e\n\u003cp\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"358\" height=\"378\"\u003e\u003c/p\u003e\n\u003cp\u003eThe 16% over-estimate in our 2022 prediction of the cycle 25 amplitude of R\u003csub\u003em\u003c/sub\u003e – max = 185 (Foukal, 2023) was probably caused mainly by the uncertain correction between facular areas measured from the high - resolution HMI images, as compared to those we measured from the MDI and those measured by RGO from ground based photo – heliograms of similar resolution. Despite this uncertainty \u0026nbsp;we were able to correctly predict a cycle 25 amplitude larger than the prediction by the International Prediction Panel at that epoch (e.g. Miesch, 2023) \u0026nbsp; \u0026nbsp;\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eThe approximately 3 ½ year advance notice of our cycle 25 prediction is shorter than provided by some of the 50 or more other methods used by the IPP (see e.g. Miesch, 2023 for an overview of cycle 25 predictions). Nor does this method predict the timing of maxima, although a useful relation between R\u003csub\u003em\u0026nbsp;\u003c/sub\u003e- max and the cycle rise time (Waldmeier, 1978) can be helpful in this respect.\u003c/p\u003e\n\u003cp\u003eThe method described here is empirical but its physical basis can be understood in terms of the evidence on the relative areas of spots and faculae in the Sun and other late type stars (e.g. Foukal, 1994; 1998). Dark spots are produced by magnetic flux tubes whose areas lie at the upper end of the photospheric flux tube size spectrum. The WL faculae lie at the lowest end (e.g. Spruit, 1976).\u003c/p\u003e\n\u003cp\u003eThe increase in the slope dA\u003csub\u003es\u003c/sub\u003e/dA\u003csub\u003ef \u0026nbsp;\u0026nbsp;\u003c/sub\u003ebetween cycles 24 and 25 seen in Figures 2 and 3 therefore corresponds to a shift in the size spectrum of photospheric magnetic fields towards more power at the larger scales, as the cycle amplitude increases. Such a tendency to shift dynamo power towards lower spatial wavenumbers as magnetic flux output increases has been recognized in observations of large spots on younger late-type stars (e.g. Radick, Lockwood and Baliunas, 1990) and in dynamo calculations (e.g. Cattaneo, Cieunh and Hughes, 1990).\u003c/p\u003e\n\u003cp\u003eThe relationship studied here between A\u003csub\u003es\u003c/sub\u003e and A\u003csub\u003ef\u003c/sub\u003e does not hold as well between sunspots and bright chromospheric plages (Foukal, 1998) because these cover not only the most slender flux tubes observed as WL faculae, but also pores and small spots. This decreases the sensitivity of the present diagnostic. So daily Ca K plage areas available for cycles 22 and 23 (e.g. Foukal et al., 2009) cannot be used to substitute the missing facular data for those cycles. But extension of the method to cycle 23 using the enhanced contrast continuum images from the Kanzelhöhe Observatory (Pötzi, 2025) deserves investigation. \u0026nbsp;\u003c/p\u003e"},{"header":"Declarations","content":"\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eThe work was performed entirely by the author.\u003c/p\u003e\u003ch2\u003eAcknowledgement\u003c/h2\u003e\u003cp\u003e\u003cstrong\u003eAcknowledgements \u0026nbsp; \u0026nbsp; \u0026nbsp;\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThis study was facilitated by availability of the MDI and HMI images obtained by the Stanford University and Lockheed Corporation Solar Groups on Spaceweather.com, and on the Debrecen Observatory site (fenyi.solarobs.hunren.hu). I am grateful to Frédéric Clette and Mark Miesch for information on the SC 25 Prediction Panel, to Leif Svalgaard for drawing my attention to Waldmeier’s rule and to Neil Sheeley and an anonymous referee for comments on an original version of this paper. I thank Tom Williams for drawing my attention to the Spaceweather.com site and Jack Martin for facilitating access to the RGO archive at Cambridge University; I am grateful to the archivist, Dr. Emma Saunders, for her kind assistance.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eBrown, G., Evans, D.: 1980, The Use of Solar Faculae in Studies of the Sunspot Cycle. \u003cem\u003eSolar Phys\u003c/em\u003e. \u003cstrong\u003e66\u003c/strong\u003e, 233.\u003c/li\u003e\n\u003cli\u003eCattaneo, F., Cieunh, T., Hughes, D.: 1990, Buoyancy Driven Instabilities and the Non - Linear Break Up of a Sheared Magnetic Layer. \u003cem\u003eJ. Fluid Mech\u003c/em\u003e. \u003cstrong\u003e219\u003c/strong\u003e, 1.\u003c/li\u003e\n\u003cli\u003eClette, F., Cliver, E., Lefevre, L., Svalgaard, L., Vacquero, J., Leibacher, J.: 2016, Recalibration of the Sunspot Number. \u003cem\u003eSolar Phys\u003c/em\u003e. \u003cstrong\u003e291\u003c/strong\u003e, 2479.\u003c/li\u003e\n\u003cli\u003eFoukal, P.: 1994, Stellar Luminosity Variations and Global Warming. \u003cem\u003eScience\u003c/em\u003e, \u003cstrong\u003e264\u003c/strong\u003e, 238.\u003c/li\u003e\n\u003cli\u003eFoukal, P.: 1998\u003cem\u003e, \u003c/em\u003eWhat Determines the Relative Areas of Spots and Faculae on Sun \u0026ndash; Like Stars. \u003cem\u003eAstrophys. J.\u003c/em\u003e, \u003cstrong\u003e500\u003c/strong\u003e, 958.\u003c/li\u003e\n\u003cli\u003eFoukal, P. et al.: 2009, A Century of Solar Ca II Measurements and Their Implications for Solar UV Driving of Climate. \u003cem\u003eSolar Phys\u003c/em\u003e. \u003cstrong\u003e255\u003c/strong\u003e, 229.\u003c/li\u003e\n\u003cli\u003eFoukal, P.: 2023, A Remarkably Accurate Predictor of Solar Cycle Amplitude. Arxiv.org/submit/5200355.\u003c/li\u003e\n\u003cli\u003eGyori, L.: 2012, Study of Differences Between Sunspot and Facular Areas from SOHO MDI and SDO HMI Observations. \u003cem\u003eSolar Phys\u003c/em\u003e.\u003cstrong\u003e280\u003c/strong\u003e, 365.\u003c/li\u003e\n\u003cli\u003eJha, B., Mandal, S., Banerjee, D.: 2019, \u003cem\u003eSolar Phys\u003c/em\u003e. \u003cstrong\u003e294\u003c/strong\u003e, 72.\u003c/li\u003e\n\u003cli\u003eMcIntosh, S., Chapman, S., Leamon, R., Egeland, R., Watkins, N.: 2020, Overlapping Magnetic Cycles and the Sunspot Number: Forecasting Sunspot Cycle 25. \u003cem\u003eSolar Phys.\u003c/em\u003e \u003cstrong\u003e295\u003c/strong\u003e, 163.\u003c/li\u003e\n\u003cli\u003eMiesch, 2023, private communication. (https://www.swpc.noaa.gov/products/solar-cycle-progression\u003c/li\u003e\n\u003cli\u003ePesnell, D.: 2020, Lessons Learned from Predictions of Solar Cycle 24. \u003cem\u003eJ. Space Weather Space Climate\u003c/em\u003e \u003cstrong\u003e10,\u003c/strong\u003e 60.\u003c/li\u003e\n\u003cli\u003eP\u0026ouml;tzi, W., 2025, private communication.\u003c/li\u003e\n\u003cli\u003ePetrovay, K.: 2020, Solar Cycle Prediction. \u003cem\u003eLiving Rev. Sol. Phys.\u003c/em\u003e \u003cstrong\u003e17\u003c/strong\u003e (1).\u003c/li\u003e\n\u003cli\u003eRadick, R., Lockwood, W., Baliunas, S.: 1990, Stellar Activity and Brightness Variations: A Glimpse at the Sun\u0026rsquo;s History. \u003cem\u003eScience\u003c/em\u003e, \u003cstrong\u003e247\u003c/strong\u003e, 39.\u003c/li\u003e\n\u003cli\u003e\u003cem\u003e Royal Observatory Annals\u003c/em\u003e, 1973, \u003cstrong\u003eVol. 8\u003c/strong\u003e, (Photoheliograph Results).\u003c/li\u003e\n\u003cli\u003eSchatten, K.:2005, Fair Space Weather for Cycle 24. \u003cem\u003eGeophys. Res. Letts\u003c/em\u003e, \u003cstrong\u003e32\u003c/strong\u003e, 21.\u003c/li\u003e\n\u003cli\u003eScherrer, P., et al.: 1995, The Solar Oscillations Investigation - Michelson Doppler Imager. \u003cem\u003eSolar Phys\u003c/em\u003e. \u003cstrong\u003e162\u003c/strong\u003e, 129-188\u003c/li\u003e\n\u003cli\u003eScherrer, P., et al.: 2012, The Heliospheric and Magnetic Imager (HMI) Investigation for the Solar Dynamics Observatory (SDO) \u003cem\u003eSolar Phys\u003c/em\u003e.\u003cstrong\u003e 275\u003c/strong\u003e, 17-40.\u003c/li\u003e\n\u003cli\u003eSpruit, H.: 1976, Pressure Equilibrium and Energy Balance of Small Photospheric Flux Tubes. \u003cem\u003eSolar Phys\u003c/em\u003e. \u003cstrong\u003e50\u003c/strong\u003e, 269-295.\u003c/li\u003e\n\u003cli\u003eSvalgaard, L., Cliver, F., Kamide, Y.: 2005, Sunspot Cycle 24: Smallest Cycle in 100 Years? \u003cem\u003eGeophys.Res. Letts\u003c/em\u003e., \u003cstrong\u003e32\u003c/strong\u003e, 1.\u003c/li\u003e\n\u003cli\u003eWaldmaier, M.: 1978, Sunspot Numbers and Sunspot Areas. \u003cem\u003eAstron.Mitt. Zurich\u003c/em\u003e \u003cstrong\u003eNr\u003c/strong\u003e \u003cstrong\u003e358.\u003c/strong\u003e\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"solar-physics","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"sola","sideBox":"Learn more about [Solar Physics](http://link.springer.com/journal/11207)","snPcode":"11207","submissionUrl":"https://submission.nature.com/new-submission/11207/3","title":"Solar Physics","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"","lastPublishedDoi":"10.21203/rs.3.rs-8340346/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8340346/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThe slopes of the linear relations between sunspot and white light (WL) facular areas at the onset of sunspot cycles 12\u0026ndash;21 correlate well with the amplitudes of those cycles between 1878\u0026ndash;1980 (Brown and Evans, \u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1980\u003c/span\u003e). We use continuum images from the SOHO Michelson Doppler Imager and SDO Heliospheric Magnetic Imager to show that the relation holds also for cycles 24 and 25. The amplitudes of cycles 12\u0026ndash;21 and 24 calculated using this relation agree with the observed amplitudes to within 4% rms. It also enabled us in 2022 to correctly predict a larger cycle 25 than estimated by the International Prediction Panel, 3 years before maximum. The technique offers an objective, physically based predictor of cycle amplitudes 3\u0026ndash;4 years ahead of their maxima, given a stable source of continuum full disk photospheric images.\u003c/p\u003e","manuscriptTitle":"A Remarkably Accurate Predictor of Sunspot Cycle Amplitude","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-12-22 09:53:41","doi":"10.21203/rs.3.rs-8340346/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2026-01-02T20:51:20+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-01-02T05:27:36+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"235975187569511209673035470674261441164","date":"2025-12-17T02:04:55+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2025-12-16T18:03:19+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2025-12-12T06:49:43+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2025-12-12T06:49:06+00:00","index":"","fulltext":""},{"type":"submitted","content":"Solar Physics","date":"2025-12-11T23:03:58+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"solar-physics","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"sola","sideBox":"Learn more about [Solar Physics](http://link.springer.com/journal/11207)","snPcode":"11207","submissionUrl":"https://submission.nature.com/new-submission/11207/3","title":"Solar Physics","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"73e7a1dd-1d87-4e86-bf72-66a54df89bf6","owner":[],"postedDate":"December 22nd, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[],"tags":[],"updatedAt":"2026-02-23T16:01:54+00:00","versionOfRecord":{"articleIdentity":"rs-8340346","link":"https://doi.org/10.1007/s11207-026-02611-7","journal":{"identity":"solar-physics","isVorOnly":false,"title":"Solar Physics"},"publishedOn":"2026-02-19 15:57:36","publishedOnDateReadable":"February 19th, 2026"},"versionCreatedAt":"2025-12-22 09:53:41","video":"","vorDoi":"10.1007/s11207-026-02611-7","vorDoiUrl":"https://doi.org/10.1007/s11207-026-02611-7","workflowStages":[]},"version":"v1","identity":"rs-8340346","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8340346","identity":"rs-8340346","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}