The mainstream dynamo models predict that the sunspot cycle is non-stationary and stochastic. The official Solar Cycle Prediction Panel forecasts only the ongoing sunspot cycle because any forecast beyond one cycle is considered impossible. We analyse the sunspot data using our Discrete Chi-square Method (DCM). This method can detect many periodic signal superimposed on an arbitrary trend. We detect the extremely significant strong 10, 11, and 11.86 years signals. The Discrete Fourier Transform (DFT) cross-check confirms that these signals are certainly real. The interference of the 10 and 11 years signals ca cause the detected long 110 years signal in real data. Long periods or interference can never appear in simulated non-stationary stochastic sunspot data. We detect long periods and interference in the real data. Our deterministic DCM model predictions are more accurate and longer than the official Solar Cycle Prediction Panel forecast. The DCM models can predict the past prolonged activity minima, like the Maunder minimum era, although we do not have data from these periods. We claim that the sunspot data are stationary and multi-periodic, and therefore deterministic. We connect the detected signals to the orbital periods of Mercury, Venus, the Earth and Jupiter. If the planets cause the sunspot cycle, the exoplanets may cause the starspot cycles observed in other chromospherically active stars.

Previous studies have related surface temperature maps, obtained with the Doppler imaging (DI) technique, of LQ Hya with long-term photometry. We compare surface magnetic field maps, obtained with the Zeeman Doppler imaging (ZDI) technique, with contemporaneous photometry, with the aim of quantifying the star's magnetic cycle characteristics. We inverted Stokes IV spectropolarimetry into magnetic field and surface brightness maps using a tomographic inversion code that models high signal-to-noise ratio mean line profiles produced by the least squares deconvolution (LSD) technique. The magnetic field and surface brightness maps reveal similar patterns to previous DI and ZDI studies: non-axisymmetric polar magnetic field structure, void of fields at mid-latitudes, and a complex structure in the equatorial regions. There is a weak but clear tendency of the polar structures to be linked with strong radial field and the equatorial ones with the azimuthal. We find a polarity reversal in the radial field between 2016 and 2017 coincident with an activity minimum seen in the long-term photometry. The inverted field strengths cannot easily be related with the observed spottedness, but we find that they are partially connected with the retrieved field complexity. Comparing to global magnetoconvection models for rapidly rotating young Suns, this field topology and dominance of the poloidal field component could be explained by a turbulent dynamo, where differential rotation does not play a major role (so called alpha^2 Omega or alpha^2 dynamos), and axi- and non-axisymmetric modes are excited simultaneously. The complex equatorial magnetic field structure could arise from the twisted (helical) wreaths often seen in these simulations, while the polar feature would be connected to the mostly poloidal non-axisymmetric component having a smooth spatial structure.

Constant orbital period ephemerides of eclipsing binaries give the computed eclipse epochs (C). These ephemerides based on the old data cannot accurately predict the observed future eclipse epochs (O). Predictability can be improved by removing linear or quadratic trends from the O − C data. Additional companions in an eclipsing binary system cause light-time travel effects that are observed as strictly periodic O − C changes. Recently, Hajdu et al. estimated that the probability of detecting the periods of two new companions from the O − C data is only 0.00005. We apply the new discrete chi-square method to 236 yr of O − C data of the eclipsing binary Algol (β Persei). We detect the tentative signals of at least five companion candidates having periods between 1.863 and 219.0 yr. The weakest one of these five signals does not reveal a “new” companion candidate, because its 680.4 ± 0.4 day signal period differs only 1.4σ from the well-known 679.85 ± 0.04 day orbital period of Algol C. We detect these same signals also from the first 226.2 yr of data, and they give an excellent prediction for the last 9.2 yr of our data. The orbital planes of Algol C and the new companion candidates are probably coplanar because no changes have been observed in Algol’s eclipses. The 2.867 day orbital period has been constant since it was determined by Goodricke.

Regular periodic changes in the observed (O) minus the computed (C) epochs of binary eclipses may reveal the presence of a third or a fourth body. Recently, Hajdu et al. (2019) estimated that the probability for detecting a fourth body from the O-C data is only 0.00005. We apply the new Discrete Chi-square Method (DCM) to 236 years of O-C data of the eclipsing binary Algol ($\beta$ Persei), and detect the probable signatures of six wide orbit stars having periods between 1.9 years (Algol~C) and 119.4 years (Algol~H). Since the $680.7 \pm 0.4$ days period of the weakest detected sixth signal differs $2.1 \sigma$ from the well-known $679.85 \pm 0.04$ days orbital period of Algol~C, the five strongest detected signals could also be real orbital periods. These signals are stable, and they can be used to predict Algol's short-term O-C changes, as well as the direction of long-term changes. The orbital planes of Algol~C and all five new wide orbit star candidates are most probably co-planar, because no changes have been observed in Algol's eclipses after Goodricke (1783) discovered its 2.867 days orbital period. The mass transfer between Algol~A and Algol~B should have increased this orbital period. We show that the orbital period decreased before the year 1900, but since then it has increased.

Unambiguous detection of signals superimposed on unknown trends is difficult for unevenly spaced data. Here, we formulate the Discrete Chi-square Method (DCM) that can determine the best model for many signals superimposed on arbitrary polynomial trends. DCM minimizes the Chi-square for the data in the multi-dimensional tested frequency space. The required number of tested frequency combinations remains manageable, because the method test statistic is symmetric in this tested frequency space. With our known tested constant frequency grid values, the non-linear DCM model becomes linear, and all results become unambiguous. We test DCM with simulated data containing different mixtures of signals and trends. DCM gives unambiguous results, if the signal frequencies are not too close to each other, and none of the signals is too weak. It relies on brute computational force, because all possible free parameter combinations for all reasonable linear models are tested. DCM works like winning a lottery by buying all lottery tickets. Anyone can reproduce all our results with the DCM computer code. All files, variables and other program code related items are printed in magenta colour. Our Appendix gives detailed instructions for using this http URL. We also present one preliminary real use case, where DCM is applied to the observed (O) minus the computed (C) eclipse epochs of a binary star, XZ And. This DCM analysis reveals evidence for the possible presence of a third and a fourth body in this system. One recent study of a very large sample of binary stars indicated that the probability for detecting a fourth body from the O-C data of eclipsing binaries is only about 0.00005. Open J. Astrophys. 3(2020) 4

A third body in an eclipsing binary system causes regular periodic changes in the observed (O) minus the computed (C) eclipse epochs. Fourth bodies are rarely detected from the O-C data. We apply the new Discrete Chi-square method (DCM) to the O-C data of the eclipsing binary XZ Andromedae. These data contain the periodic signatures of at least ten wide orbit stars (WOSs). Their orbital periods are between 1.6 and 91.7 years. since no changes have been observed in the eclipses of XZ And during the past 127 years, the orbits of all these WOSs are most probably co-planar. We give detailed instructions of how the professional and the amateur astronomers can easily repeat all stages of our DCM analysis with an ordinary PC, as well as apply this method to O-C data of other eclipsing binaries.

Context. Starspots are important manifestations of stellar magnetic activity. By studying their behaviour in young solar analogues, we can unravel the properties of their magnetic cycles. This gives crucial information of the underlying dynamo process. Comparisons with the solar cycle enable us to infer knowledge about how the solar dynamo has evolved during the Sun’s lifetime. Aims. Here we study the correlation between photometric brightness variations, spottedness, and mean temperature in V889 Her, a young solar analogue. Our data covers 18 years of spectroscopic and 25 years of photometric observations. Methods. We use Doppler imaging to derive temperature maps from high-resolution spectra. We use the Continuous Period Search method to retrieve mean V-magnitudes from photometric data. Results. Our Doppler imaging maps show a persistent polar spot structure varying in strength. This structure is centred slightly off the rotational pole. The mean temperature derived from the maps shows an overall decreasing trend, as does the photometric mean brightness, until it reaches its minimum around 2017. The filling factor of cool spots, however, shows only a weak tendency to anti-correlate with the decreasing mean brightness. Conclusions. We interpret V889 Her to have entered into a grand maximum in its activity. The clear relation between the mean temperature of the Doppler imaging surface maps and the mean magnitude supports the reliability of the Doppler images. The lack of correlation between the mean magnitude and the spottedness may indicate that bright features in the Doppler images are real.

An ancient Egyptian Calendar of Lucky and Unlucky Days, the Cairo Calendar (CC), assigns luck with the period of 2.850 days. Previous astronomical, astrophysical and statistical analyses of CC support the idea that this was the period of the eclipsing binary Algol three millennia ago. However, next to nothing is known about who recorded Algol's period into CC and especially how. Here, we show that the ancient Egyptian scribes had the possible means and the motives for such astronomical observations. Their principles of describing celestial phenomena as activity of gods reveal why Algol received the title of Horus

For seven decades, the widely held view has been that the formation, the migration and the decay of short-lived starspots explain the constantly changing light curves of chromospherically active stars. Our hypothesis is that these deceptive observed light curves are interference of two real constant period light curves of long-lived starspots. The slow motion of these long-lived starspots with respect to each other causes the observed light curve changes. This hypothesis contradicts the current views of starspots. Therefore, we subject it to eight reproducible tests. Our new period finding method detects the two real light curves of FK Com. Our hypothesis is a total success: all real light curve parameters are directly connected to the long-lived starspots which are also seen in the Doppler images of FK Com.These parameters are spatially and temporally correlated just like in the Sun, including weak solar-like surface differential rotation. As for other chromospherically active stars, all eight reproducible tests also support our hypothesis. It explains many spurious phenomena: the rapid light curve changes, the short starspot life-times, the rapid rotation period changes, the active longitudes, the starspot migration, the period cycles, the amplitude cycles and the minimum epoch cycles. It also explains why the light curves and the Doppler images give contradicting surface differential rotation estimates even for the same individual star, as well as the abrupt 180 degrees shifts of activity (the flip-flop events) and the long-term mean light curves. We argue that the current views of starspots need to be revised.

The starspots on the surface of many chromospherically active binary stars concentrate on long-lived active longitudes separated by 180 degrees. Shifts in activity between these two longitudes, the "flip-flop". events, have been observed in single stars like FK Comae and binary stars like sigma Geminorum. Recently, interferometry has revealed that ellipticity may at least partly explain the flip-flop events in sigma Geminorum. This idea was supported by the double-peaked shape of the long-term mean light curve of this star. Here we show that the long-term mean light curves of 14 chromospherically active binaries follow a general model that explains the connection between orbital motion, changes in starspot distribution, ellipticity, and flip-flop events. Surface differential rotation is probably weak in these stars, because the interference of two constant period waves may explain the observed light curve changes. These two constant periods are the active longitude period(P-act) and the orbital period (P-orb). We also show how to apply the same model to single stars, where only the value of Pact is known. Finally, we present a tentative interference hypothesis about the origin of magnetic fields in all spectral types of stars.

Long‐term photometry is commonly used to monitor chromospheric activity of late–type stars. We study standard Johnson differential V photometry of the RS CVn binary BM Canum Venaticorum (BM CVn) spanning over a quarter of a century. Our main aims are to determine the activity cycles, the rate of surface differential rotation, and the rotation period of the active longitudes of BM CVn. The continuous period search (CPS) algorithm is applied to the photometry. The changes of the mean and amplitude of the light curves are used to search for activity cycles. The rotation period changes give an estimate of the rate of surface differential rotation. The Kuiper method is applied to the epochs of the primary and secondary minima to search for active longitudes. The photometry reveals the presence of a stable mean light curve (MLC) connected to the orbital period P orb =20. d 6252 of this binary. We remove this MLC from the original V magnitudes, which gives us the corrected magnitudes. These two samples of V and data are analyzed separately with CPS. The fraction of unreliable CPS models decreases when the MLC is removed. The same significant activity cycle of approximately 12.5 years is detected in both V and samples. The estimate for the surface differential rotation coefficient, k ⩾0.10, is the same for both samples, but the number of unrealistic period estimates decreases after removing the MLC. The same active longitude period of P al =20. d 511 ± 0. d 005 is detected in the V and magnitudes. This long‐term regularity in the epochs of primary and secondary minima of the light curves is not caused by the MLC. On the contrary, the MLC hampers the detection of active longitudes.

Recently, we presented a general model for the light curves of chromospherically active stars, where the observed light curve is interference of two real constant period light curves of long-lived starspots. In this first paper, we make six specific questions which undermine this argument, because it contradicts the current widely held views about the stellar surface differential rotation and the starspots. Our aim is to answer these six questions. We present evidence that the long-lived starspots of our general model have already been detected in the earlier surface imaging studies. The Lomb-Scargle power spectrum method analysis of the real and the simulated data of FK Com reveals that this method fails to detect the two real constant period light curves of our general model. If our model is valid, this method gives incompatible period, amplitude and minimum epoch estimates telling nothing about the real periods, the real amplitudes and the real minimum epochs of the two real light curves. This would mean that all earlier one-dimensional period analyses of the light curves of chromospherically active stars have given spurious results which have been widely and uncritically accepted since the discovery of the starspots in the year 1947. However, we arrive at a dead end, because we can not solve the real light curves of FK Com. In our second paper, we solve these real light curves with a new two-dimensional period finding method, prove the validity of our general model, and answer all six questions made in this first paper.

{To show that the observed light curves of FK Comae Berenices are interference of a nonstationary and a stationary light curve. These two light curves follow the active longitude period $(P_{mathrm{act}})$ and the rotation period $(P_{mathrm{rot}})$ of FK Com.} {The recently formulated general model for the light curves of chromospherically active binary stars is applied to 14 years of standard Johnson $V$ photometry of FK Com. } {We determine a unique value, $P_{mathrm{rot}}=2.3976 pm 0.0018$ days, for the rotation period of FK Com. The observed light curves can be modelled as interference of two real light curves, $f_1(phi_{mathrm{act}})$ and $f_2(phi_{mathrm{rot}})$, which are nonstationary and stationary in the rotational frame of reference. The observed flip ~events, where the activity abruptly shifts 180 degrees in longitude, are connected to the interaction of these two curves at rotational phases $phi_{mathrm{rot}}=0.0$ and 0.5.} {Our light curve model can remove the veil of interference from the observed light curve, and give unique solutions for the real light curves $f_1(phi_{mathrm{act}})$ and $f_2(phi_{mathrm{rot}})$. The magnetic field evolution in chromospherically active single and binary stars is probably connected to the presence or absence of nonstationary and stationary parts in their observed light curves.}

Aims. We study a sample of 21 young and active solar-type stars with spectral types ranging from late F to mid K and characterize the behaviour of their activity.Methods. We apply the continuous period search (CPS) time series analysis method on Johnson B-and V-band photometry of the sample stars, collected over a period of 16 to 27 years. Using the CPS method, we estimate the surface differential rotation and determine the existence and behaviour of active longitudes and activity cycles on the stars. We supplement the time series results by calculating new log R-HK' = log F-HK'/sigma T-eff(4) emission indices for the stars from high resolution spectroscopy.Results. The measurements of the photometric rotation period variations reveal a positive correlation between the relative differential rotation coefficient and the rotation period as k alpha P-rot(1.36), but do not reveal any dependence of the differential rotation on the effective temperature of the stars. Secondary period searches reveal activity cycles in 18 of the stars and temporary or persistent active longitudes in 11 of them. The activity cycles fall into specific activity branches when examined in the log P-rot/P-cyc vs. log Ro(-1), where Ro(-1) = 2 Omega tau(c), or log P-rot/P-cyc vs. log R-HK' diagram. We find a new split into sub-branches within this diagram, indicating multiple simultaneously present cycle modes. Active longitudes appear to be present only on the more active stars. There is a sharp break at approximately log R-HK' = -4.46 separating the less active stars with long-term axisymmetric spot distributions from the more active ones with non-axisymmetric configurations. In seven out of eleven of our stars with clearly detected long-term non-axisymmetric spot activity the estimated active longitude periods are significantly shorter than the mean photometric rotation periods. This systematic trend can be interpreted either as a sign of the active longitudes being sustained from a deeper level in the stellar interior than the individual spots or as azimuthal dynamo waves exhibiting prograde propagation.

The Ancient Egyptians wrote Calendars of Lucky and Unlucky Days that assigned astronomically influenced prognoses for each day of the year. The best preserved of these calendars is the Cairo Calendar (hereafter CC) dated to 1244-1163 B.C. We have presented evidence that the 2.85 days period in the lucky prognoses of CC is equal to that of the eclipsing binary Algol during this historical era. We wanted to find out the vocabulary that represents Algol in the mythological texts of CC. Here we show that Algol was represented as Horus and thus signified both divinity and kingship. The texts describing the actions of Horus are consistent with the course of events witnessed by any naked eye observer of Algol. These descriptions support our claim that CC is the oldest preserved historical document of the discovery of a variable star. The period of the Moon, 29.6 days, has also been discovered in CC. We show that the actions of Seth were connected to this period, which also strongly regulated the times described as lucky for Heaven and for Earth. Now, for the first time, periodicity is discovered in the descriptions of the days in CC. Unlike many previous attempts to uncover the reasoning behind the myths of individual days, we discover the actual rules in the appearance and behaviour of deities during the whole year.