Discrete Chi-square Method for Detecting Many Signals

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.Peer reviewe

Sunspot cycles are connected to the Earth and Jupiter

The sunspot number record covers over three centuries.These numbers measure the activity of the Sun. This activity follows the solar cycle of about eleven years. In the dynamo-theory, the interaction between differential rotation and convection produces the solar magnetic field. On the surface of Sun, this field concentrates to the sunspots. The dynamo-theory predicts that the period, the amplitude and the phase of the solar cycle are stochastic. Here we show that the solar cycle is deterministic, and connected to the orbital motions of the Earth and Jupiter. This planetary-influence theory allows us to model the whole sunspot record, as well as the near past and the near future of sunspot numbers. We may never be able to predict the exact times of exceptionally strong solar flares, like the catastrophic Carrington event in September 1859, but we can estimate when such events are more probable. Our results also indicate that during the next decades the Sun will no longer help us to cope with the climate change. The inability to find predictability in some phenomenon does not prove that this phenomenon itself is stochastic.Comment: 40 pages, 15 figures, 21 table

Say Hello to Algol's New Companion Candidates

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.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 (beta 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 1.4 sigma 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.Peer reviewe

Spot activity of LQ Hydra from photometry between 1988 and 2011

Aims. We investigate the spot activity of the young magnetically active main sequence star LQ Hya. Our aims are to identify possible active longitudes, estimate the differential rotation, and study long and short term changes in the activity. Methods. Our analysis is based on 24 years of Johnson V-band photometry of LQ Hya obtained with the T3 0.4 m Automated Telescope at the Fairborn Observatory. We use the previously published continuous period search (CPS) method to model the evolution of the light curve of LQ Hya. The CPS fits a Fourier series model to short overlapping subsets of data. This enables us to monitor the evolution of the light curve and thus the spot configuration of the star with a higher time resolution. Results. We find seasonal variability in the mean level and amplitude of the light curve of LQ Hya. The variability of the light curve amplitude seems not to be cyclic, but the long-term variations in the mean magnitude may be indicative of an approximately 13 year cycle. However, because of the limited length of the observed time series, it is not yet possible to determine whether this structure really represents an activity cycle. Based on fluctuations of the light curve period, we estimate the differential rotation of the star to be small, and the star is potentially very close to a rigid rotator. We search for active longitudes from the inferred epochs of the light curve minima. We find that on time scales up to six months there are typically one or two relatively stable active areas on the star with limited phase migration. On the other hand, on time scales longer than one year, no stable active longitudes have been present except for the period between 2003 and 2009 and possibly also some time before 1995. Neither do we find any signs of flip-flops with a regular period. The mean time scale of change of the light curve during the observation period is determined to be of the same order of magnitude as the estimated convective turnover time for the star

Spot activity of the RS Canum Venaticorum star σ Geminorum

Aims. We model the photometry of RS CVn star σ Geminorum to obtain new information on the changes of the surface starspot distribution, that is, activity cycles, differential rotation, and active longitudes. Methods. We used the previously published continuous period search (CPS) method to analyse V-band differential photometry obtained between the years 1987 and 2010 with the T3 0.4 m Automated Telescope at the Fairborn Observatory. The CPS method divides data into short subsets and then models the light-curves with Fourier-models of variable orders and provides estimates of the mean magnitude, amplitude, period, and light-curve minima. These light-curve parameters are then analysed for signs of activity cycles, differential rotation and active longitudes. Results. We confirm the presence of two previously found stable active longitudes, synchronised with the orbital period Porb = 1960, and found eight events where the active longitudes are disrupted. The epochs of the primary light-curve minima rotate with a shorter period Pmin,1 = 1947 than the orbital motion. If the variations in the photometric rotation period were to be caused by differential rotation, this would give a differential rotation coefficient of α ≥ 0.103. Conclusions. The presence of two slightly different periods of active regions may indicate a superposition of two dynamo modes, one stationary in the orbital frame and the other one propagating in the azimuthal direction. Our estimate of the differential rotation is much higher than previous results. However, simulations show that this may be caused by insufficient sampling in our data

Time series analysis of V815 Herculis photometry between 1984 and 1998

As a case study of the solar-stellar connection, we have analysed a prolonged time series of BV photometry of the chromospherically active binary V815 Her . The surface differential rotation in the rapidly rotating G5{v} primary caused changes of 4.6% in the seasonal photometric rotation periods. This would imply a differential rotation coefficient of k = 0.184, if the rotation of the starspots follows the solar law of differential rotation and the activity is confined within the same latitudinal range as in the Sun, having k = 0.189 and the spectral-type of G2{v}. Our analysis of the primary and secondary minima of the seasonal light curves indicated that the regions of stronger activity have concentrated on one active longitude, which has maintained a constant rotation period of 1.d79244 for about 14 years. No regular activity cycle was detected in the mean brightness changes of V815 Her

Multiperiodicity, modulations and flip-flops in variable star light curves II. Analysis of II Pegasus photometry during 1979–2010

Aims. According to previously published Doppler images of the magnetically active primary giant component of the RS CVn binary II Peg, the surface of the star was dominated by one single active longitude that was clearly drifting in the rotational frame of the binary system during 1994-2002; later imaging for 2004–2010, however, showed decreased and chaotic spot activity, with no signs of the drift pattern. Here we set out to investigate from a more extensive photometric dataset whether this drift is a persistent phenomenon, in which case it could be caused either by an azimuthal dynamo wave or be an indication that the binary system’s orbital synchronization is still incomplete. On a differentially rotating stellar surface, spot structures preferentially on a certain latitude band could also cause such a drift, the disruption of which could arise from the change of the preferred spot latitude. Methods. We analyzed the datasets using the carrier fit (CF) method, which is especially suitable for analyzing time series in which a fast clocking frequency (such as the rotation of the star) is modulated with a slower process (such as the stellar activity cycle). Results. We combined all collected photometric data into one single data set and analyzed it with the CF method. We confirm the previously published results that the spot activity has been dominated by one primary spotted region almost through the entire data set and also confirm a persistent, nearly linear drift. Disruptions of the linear trend and complicated phase behavior are also seen, but the period analysis reveals a rather stable periodicity with Pspot = 671054 ± 000005. After removing the linear trend from the data, we identified several abrupt phase jumps, three of which are analyzed in more detail with the CF method. These phase jumps closely resemble what is called a flip-flop event, but the new spot configurations do not persist for longer than a few months in most cases. Conclusions. There is some evidence that the regular drift without phase jumps is related to the high state, while the complex phase behavior and disrupted drift pattern are related to the low state of magnetic activity. The most natural explanation of the drift is weak anti-solar (pole rotating faster than the equator) differential rotation with a coefficient k ≈ 0.002 combined with the preferred latitude of the spot structure

Multiperiodicity, modulations and flip-flops in variable star light curves II. Analysis of II Peg photometry during 1979-2010

According to earlier Doppler images of the magnetically active primary giant component of the RS CVn binary II Peg, the surface of the star was dominated by one single active longitude that was clearly drifting in the rotational frame of the binary system during 1994-2002; later imaging for 2004-2010, however, showed decreased and chaotic spot activity, with no signs of the drift pattern. Here we set out to investigate from a more extensive photometric dataset whether such a drift is a persistent phenomenon, in which case it could be due to either an azimuthal dynamo wave or an indication of the binary system orbital synchronization still being incomplete. We analyse the datasets using the Carrier Fit method (hereafter CF), especially suitable for analyzing time series in which a fast clocking frequency (such as the rotation of the star) is modulated with a slower process (such as the stellar activity cycle). We combine all collected photometric data into one single data set, and analyze it with the CF method. As a result, we confirm the earlier results of the spot activity having been dominated by one primary spotted region almost through the entire data set, and the existence of a persistent, nearly linear drift. Disruptions of the linear trend and complicated phase behavior are also seen, but the period analysis reveals a rather stable periodicity with P(spot)=6.71054d plus/minus 0.00005d. After the linear trend is removed from the data, we identify several abrupt phase jumps, three of which are analyzed in more detail with the CF method. These phase jumps closely resemble what is called flip-flop event, but the new spot configurations do not, in most cases, persist for longer than a few months.Comment: 9 pages, 7 figures, submitted to Astronomy and Astrophysic

Long-term spot monitoring of the young solar analogue V889 Herculis

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