Starspot Mapping with Adaptive Parallel Tempering. I. Implementation of Computational Code

Abstract

Starspots are thought to be regions of locally strong magnetic fields, similar to sunspots, and they can generate photometric brightness modulations. To deduce stellar and spot properties, such as spot emergence and decay rates, we implement a computational code for starspot modeling. It is implemented with an adaptive parallel tempering algorithm and an importance sampling algorithm for parameter estimation and model selection in the Bayesian framework. For evaluating the performance of the code, we apply it to synthetic light curves produced with three spots. The light curves are specified in the spot parameters, such as the radii, intensities, latitudes, longitudes, and emergence/decay durations. The spots are circular with specified radii and intensities relative to the photosphere, and the stellar differential rotation coefficient is also included in the light curves. As a result, stellar and spot parameters are uniquely deduced, and the number of spots is correctly determined: the three-spot model is preferable because the model evidence is much greater than that of the two-spot model by orders of magnitude and more than that of the four-spot model by a more modest factor, whereas the light curves are produced to have two or one local minimum during one equatorial rotation period by adjusting the values of longitude. The spot emergence and decay rates can be estimated with error less than an order of magnitude, considering the difference of the number of spots.

Superflares, chromospheric activities and photometric variabilities of solar-type stars from the second-year observation of TESS and spectra of LAMOST

“…Zhang et al 2020a) and observing inclination angles (e.g. Notsu et al 2013;Ikuta et al 2020) are other factors affecting the brightness variation amplitude (R var ). Overall, flare stars have a higher S-index and R var than the Sun, which indicates that the Sun is generally less active than a superflare star.…”

“…Moreover, it is also reasonable to consider inclination angle between the rotation axis and observing direction (e.g. Notsu et al 2013;Ikuta et al 2020) as another kind of asymmetry to affect R var . Besides, we also see from Figure 11(a) that those flare stars are not with R var 10 −1 , while their S-indexes become larger.…”

“…A bunch of simulation works have focused on reconstructing light curves for observed stars by using artificial functions of star spots and faculae (e.g. Basri & Shah 2020;Ikuta et al 2020;Is , ık et al 2020;Shapiro et al 2020). These simulations are all based on morphologically artificial functions to describe the generation and evolution of star spots and faculae.…”

Stellar activity and transits

“…• Analytic models describe the stellar disc as the visible part of a sphere emitting a total unit flux, from which the contribution of darker (or brighter) regions with circular shape is subtracted (or added) to calculate the light curve (Budding 1977;Dorren 1987;Eker 1994;Kipping 2012;Béky, Kipping, and Holman 2014;Ikuta et al 2020). Usually, brightness inhomogeneities are distorted according to their projection at a given stellar longitude and latitude.…”

“…Other parameters can be tuned, such as the inclination of the stellar rotation axis, the stellar rotation period, or the stellar limb darkening coefficients. Such a "forward" model is then implemented in a Markov Chain Monte Carlo (MCMC) algorithm (or its variations, such as Nested Sampling) to find the posterior probability distributions for the model parameters (e.g Kipping 2012;Bruno et al 2016;Ikuta et al 2020). 1 An analytic model requires only a fraction of a second to be calculated and, in principle, there is no limit to the number of stellar activity features that can be modelled.…”