A fast and reliable method to measure stellar differential rotation from photometric data

Co-rotating spots at different latitudes on the stellar surface generate periodic photometric variability and can be useful proxies to detect Differential Rotation (DR). DR is a major ingredient of the solar dynamo but observations of stellar DR are rather sparse. In view of the Kepler space telescope we are interested in the detection of DR using photometric information of the star, and to develop a fast method to determine stellar DR from photometric data. We ran a large Monte-Carlo simulation of differentially rotating spotted stars with very different properties to investigate the detectability of DR. For different noise levels the resulting light curves are prewhitened using Lomb-Scargle periodograms to derive parameters for a global sine fit to detect periodicities. We show under what conditions DR can successfully be detected from photometric data, and in which cases the light curve provides insufficient or even misleading information on the stellar rotation law. In our simulations, the most significant period P1_{out} could be detected in 96.2% of all light curves. Detection of a second period close to P1_{out} is the signature of DR in our model. For the noise-free case, in 64.2% of all stars such a period was found. Calculating the measured latitudinal shear of two distinct spots \alpha_{out}, and comparing it to the known original spot rotation rates shows that the real value is on average 3.2% lower. Comparing the total equator-to-pole shear $\alpha$ to $\alpha_{out}$ we find that $\alpha$ is underestimated by 8.8%, esp. the detection of DR for stars with $\alpha$ < 6% is challenging. Finally, we apply our method to four differentially rotating Kepler stars and find close agreement with results from detailed modeling. Our method is capable of measuring stellar rotation periods and detecting DR with relatively high accuracy and is suitable for large data sets.Comment: accepted by A&

Rotation and differential rotation of active Kepler stars

We present rotation periods for thousands of active stars in the Kepler field derived from Q3 data. In most cases a second period close to the rotation period was detected, which we interpreted as surface differential rotation (DR). Active stars were selected from the whole sample using the range of the variability amplitude. To detect different periods in the light curves we used the Lomb-Scargle periodogram in a pre-whitening approach to achieve parameters for a global sine fit. The most dominant periods from the fit were ascribed to different surface rotation periods, but spot evolution could also play a role. Due to the large number of stars the period errors were estimated in a statistical way. We thus cannot exclude the existence of false positives among our periods. In our sample of 40.661 active stars we found 24.124 rotation periods $P_1$ between 0.5-45 days. The distribution of stars with 0.5 < B-V < 1.0 and ages derived from angular momentum evolution that are younger than 300 Myr is consistent with a constant star-formation rate. A second period $P_2$ within $\pm30$% of the rotation period $P_1$ was found in 18.619 stars (77.2%). Attributing these two periods to DR we found that the relative shear $\alpha=\Delta\Omega/\Omega$ increases with rotation period, and slightly decreases with effective temperature. The absolute shear $\Delta\Omega$ slightly increases between $T_{eff}=3500-6000$ K. Above 6000 K $\Delta\Omega$ shows much larger scatter. We found weak dependence of $\Delta\Omega$ on rotation period. Latitudinal differential rotation measured for the first time in more than 18.000 stars provides a comprehensive picture of stellar surface shear, consistent with major predictions from mean-field theory. To what extent our observations are prone to false positives and selection bias is not fully explored, and needs to be addressed using more Kepler data.Comment: 19 pages, 18 figures, accepted by A&A. A table containing all periods, KIC number, etc. can be found here: http://www.astro.physik.uni-goettingen.de/~reinhold/period_table.te

Lernstrategien für das Arbeiten mit dynamischen Werkzeugen am Beispiel Dynamischer Geometriesysteme (DGS)

Das Lernen mit Computerunterstützung ist Gegenstand intensiver Forschungsbemühungen in der Mediendidaktik (als einen Zweig der pädagogischen Psychologie). Zugleich interessieren sich die Fachdidaktiken für das computergestützte Lernen aus ihrer jeweiligen fachspezifischen Perspektive. Forschungsstände und Theorierahmen sind hierbei vergleichsweise wenig vernetzt (eine der wenigen Ausnahmen im deutschsprachigen Raum ist Hischer, 2002). Der vorliegende Beitrag soll einige Brückenschläge zwischen Forschung zum Lernen mit computergestützten Lernumgebungen zwischen Mathematikdidaktik und Mediendidaktik andeuten. Diese Überlegungen explorieren die Situation ausgehend vom Bereich des Lernens mit Dynamischen Geometriesystemen

Stellar rotation periods from K2 Campaigns 0-18 -- Evidence for rotation period bimodality and simultaneous variability decrease

Rotation period measurements of stars observed with the Kepler mission have revealed a lack of stars at intermediate rotation periods, accompanied by a decrease of photometric variability. Whether this so-called dearth region is a peculiarity of stars in the Kepler field, or reflects a general manifestation of stellar magnetic activity, is still under debate. Our goal is to measure stellar rotation periods and photometric variabilities for tens of thousands of K2 stars, located in different fields along the ecliptic plane, to shed light on the relation between stellar rotation and photometric variability. We use Lomb-Scargle periodograms, auto-correlation and wavelet functions to determine consistent rotation periods. Stellar brightness variability is assessed by computing the variability range from the light curve. We further apply Gaussian mixture models to search for bimodality in the rotation period distribution. Combining measurements from all K2 campaigns, we detect rotation periods in 29,860 stars. For effective temperatures below 6000K, the variability range shows a local minimum at different periods, consistent with an isochrone age of 750 Myr. Additionally, the K2 rotation period distribution shows evidence for bimodality, although the dearth region is less pronounced compared to the Kepler field. The period at the dip of the bimodal distribution shows good agreement with the period at the local variability minimum. We conclude that the period bimodality is present in different fields of the sky, and is hence a general manifestation of stellar magnetic activity. The reduced variability in the dearth region is interpreted as a cancelation between dark spots and bright faculae. Our results strongly advocate that the role of faculae has been underestimated so far, suggesting a more complex dependence of the brightness variability on the rotation period.Comment: 14 pages, 13+3 figure

Connectedness between G10 currencies: Searching for the causal structure

This paper presents a new approach for modelling the connectedness between asset returns. We adapt the measure of Diebold and Yilmaz, which is based on the forecast error variance decomposition of a VAR model. However, their connectedness measure hinges on critical assumptions with regard to the variance–covariance matrix of the error terms. We propose to use a more agnostic empirical approach, based on a machine learning algorithm, to identify the contemporaneous structure. In a Monte Carlo study, we compare the different connectedness measures and discuss their advantages and disadvantages. In an empirical application we analyse the connectedness between the G10 currencies. Our results suggest that the US dollar as well as the Norwegian krone are the most independent currencies in our sample. By contrast, the Swiss franc and New Zealand dollar have a negligible impact on other currencies. Moreover, a cluster analysis suggests that the currencies can be divided into three groups, which we classify as: commodity currencies, European currencies and safe haven/carry trade financing currencies

New rotation period measurements of 67,163 Kepler stars

The Kepler space telescope leaves a legacy of tens of thousands of stellar rotation period measurements. While many of these stars show strong periodicity, there exists an even bigger fraction of stars with irregular variability for which rotation periods are unknown. As a consequence, many stellar activity studies might be strongly biased toward the behavior of more active stars with measured rotation periods. To at least partially lift this bias, we apply a new method based on the Gradient of the Power Spectrum (GPS). The maximum of the gradient corresponds to the position of the inflection point (IP). It was shown previously that the stellar rotation period $P_{rot}$ is linked to the inflection point period $P_{IP}$ by the simple equation $P_{rot} = P_{IP}/\alpha$, where $\alpha$ is a calibration factor. The GPS method is superior to classical methods (such as auto-correlation functions (ACF)) because it does not require a repeatable variability pattern in the time series. From the initial sample of 142,168 stars with effective temperature $T_{eff}\leq6500K$ and surface gravity $log g\geq4.0$ in the Kepler archive, we could measure rotation periods for 67,163 stars by combining the GPS and the ACF method. We further report the first determination of a rotation period for 20,397 stars. The GPS periods show good agreement with previous period measurements using classical methods, where these are available. Furthermore, we show that the scaling factor $\alpha$ increases for very cool stars with effective temperatures below 4000K, which we interpret as spots located at higher latitudes. We conclude that new techniques (such as the GPS method) must be applied to detect rotation periods of stars with small and more irregular variabilities. Ignoring these stars will distort the overall picture of stellar activity and, in particular, solar-stellar comparison studies.Comment: 15 pages, 18 figures, accepted for publication in A&

The Sun is less active than other solar-like stars

Magnetic activity of the Sun and other stars causes their brightness to vary. We investigate how typical the Sun's variability is compared to other solar-like stars, i.e. those with near-solar effective temperatures and rotation periods. By combining four years of photometric observations from the Kepler space telescope with astrometric data from the Gaia spacecraft, we measure photometric variabilities of 369 solar-like stars. Most of the solar-like stars with well-determined rotation periods show higher variability than the Sun and are therefore considerably more active. These stars appear nearly identical to the Sun, except for their higher variability. Their existence raises the question of whether the Sun can also experience epochs of such high variability.Comment: Accepted for publication in Science. 3 (main) + 10 (supplementary) figure

Solar-type Stars Observed by LAMOST and Kepler

Obtaining measurements of chromospheric and photometric activity of stars with near-solar fundamental parameters and rotation periods is important for a better understanding of solar-stellar connection. We select a sample of 2603 stars with near-solar fundamental parameters from the Large Sky Area Multi-Object Fiber Spectroscopic Telescope (LAMOST)-Kepler field and use LAMOST spectra to measure their chromospheric activity and Kepler light curves to measure their photospheric activity (i.e., the amplitude of the photometric variability). While the rotation periods of 1556 of these stars could not be measured due to the low amplitude of the photometric variability and highly irregular temporal profile of light curves, 254 stars were further identified as having near-solar rotation periods. We show that stars with near-solar rotation periods have chromospheric activities that are systematically higher than stars with undetected rotation periods. Furthermore, while the solar level of photospheric and chromospheric activity appears to be typical for stars with undetected rotation periods, the Sun appears to be less active than most stars with near-solar rotation periods (both in terms of photospheric and chromospheric activity).Comment: 7 pages, 6 figure