Photometric modelling of starspots – II. The fortran code spotpic

A fortran code which computes synthetic light and colour curves of active, spotted stars has been developed. The main feature of this code is that it can simultaneously model the V light curve and the ( V − R )_c, ( V − I )_c, ( V − K ) colour data. It also uses new effective temperature–colour and Barnes–Evans-like calibrations, temperature and gravity-dependent limb darkening coefficients and different effective surface gravities for the spotted and unspotted photosphere. The code allows for two-component spots, i.e. spots with umbral and penumbral components. Various problematic spot configurations were investigated and we conclude that, in order to be able to differentiate spots with various thermal structures (umbrae, penumbrae, faculae) or polar spots from equatorial bands, the modelling of the infrared colours, especially ( V − I )_c and ( V − K ), is needed.     

Spot Modelling of ζ Andromedae

The photometric light modulation of ζ Andromedae originates from the distorted geometry of the primary, and additionally, from spots of which parameters (temperature, size, location) are variable in time. We present spot modelling results for six two-colour light curves which show that spots preferably appear on the stellar surface towards the companion star and opposite to it, where the distortion also causes dimming. Therefore, simple fitting of the measured data for the ellipticity effect does not yield correct result. Instead, ellipticity calculated from exact stellar parameters should be removed from the data to get reliable spotted light curves.     

Light‐curve inversions with truncated least‐squares principal components: Tests and application to HD 291095 = V1355 Orionis

We present a new inversion code that reconstructs the stellar surface spot configuration from the light curve of a rotating star. Our code employs a method that uses the truncated least‐squares estimation of the inverse problem's objects principal components. We use spot filling factors as the unknown objects. Various test cases that represent a rapidly‐rotating K subgiant are used for the forward problem. Tests are then performed to recover the artificial input map and include data errors and input‐parameter errors. We demonstrate the robustness of the solution to false input parameters like photospheric temperature, spot temperature, gravity, inclination, unspotted brightness and different spot distributions and we also demonstrate the insensitivity of the solution to spot latitude. Tests with spots peppered over the entire stellar surface or with phase gaps do not produce fake active longitudes. The code is then applied to ten years of V and I ‐band light curve data of the spotted sub‐giant HD291095. A total of 22 light curves is presented. We find that for most of the time its spots were grouped around two active longitudes separated on average by 180°. Switches of the dominant active region between these two longitudes likely occurred about every 3.15±0.23 years while the amplitude modulation of the brightness occurred with a possible period of 3.0±0.15 years. For the first time, we found evidence that the times of the activity flips coincide with times of minimum light as well as minimum photometric amplitude, i.e. maximum spottedness. From a comparison with simultaneous Doppler images we conclude that the activity flips likely take place near the rotational pole of the star. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

An analysis of active close binaries (CB) based on photometric measurements

This is the first in a series of papers devoted to the actual problematics in the determination of orbital and physical parameters of active CB on the basis of the interpretation of photometric observations. One solves the problem in two stages: by obtaining a synthetic light curve in the case when the parameters of the corresponding CB model are givena priori (direct problem) and by determining the parameters of the given model for which the best fit between the synthetic light curve and the observations is achieved (inverse problem (see Djura?evi?, 1991). In the first article of the series one presents the basis of the model developed for the synthesis of asymmetric, deformed, light curves of active CB with spots on their components. The modelling of the CB systems is based on the principles orginated in the Wilson and Devinney (1971; hereafter referred to as WD) model for the synthesis of a light curve generalised to include also the case of a nonsynchronous rotation of the components. The shapes of the components correspond to the equipotentials in the Roche model so that the critical Roche limits can be filled up to an arbitrary degree. In a spherical-coordinate system the surfaces of the components are divided into a large number of elementary cells whose intensity and angular radiation distribution are determined by the star temperature. limb darkening, gravitational darkening, and by the effect of reflection in the system. The active regions are approximated with circular spots. The presence of spots (dark or hot) enables to explain the asymmetry and depressions on the light curves of active CB. The model enables to be also interpreted the light curves of classic CB (without spots).     

Measurement of the limb darkening coefficient of stars

This paper introduces a new method of describing the limb darkening phenomenon of stellar atmospheres in the 1st and 2nd order approximations. The limb darkening coefficients are given by direct measurements of relevant physical quantities or measurements of flux and a supplementary quantity, namely, the star's surface temperature. The comparison either with solar observations or with the method of numerical simulation of the atmosphere shows that the new method can very accurately describe the star's limb darkening and determine its relevant coefficients. The accuracy of the new direct method is assessed using 176 observational data points of the Sun, and the mean fractional differences is found to be 0.38% for the 1st-, and 0.26% for the 2nd-order approximations. The mean differences of the indirect method are close to, and slightly larger than these values. When compared with numerical models of the atmosphere, the differences are in the range from under 1% to under 3%.     

Microlensing of close binary stars

The gravity due to a multiple-mass system has a remarkable gravitational effect: the extreme magnification of background light sources along extended so-called caustic lines. This property has been the channel for some remarkable astrophysical discoveries over the past decade, including the detection and characterization of extrasolar planets, the routine analysis of limb darkening, and, in one case, limits set on the apparent shape of a star several kiloparsec distant. In this paper, we investigate the properties of the microlensing of close binary star systems. We show that in some cases it is possible to detect flux from the Roche lobes of close binary stars. Such observations could constrain models of close binary stellar systems.     

Cepheid Limb Darkening Models for the VLTI

We present a new method to compute wavelength- and phase-dependent limb darkening corrections for classical Cepheids. These corrections are derived from hydrodynamic simulations and radiative transfer modeling with a full set of atomic and molecular opacities. Comparison with hydrostatic models having the same stellar parameters show a larger limb darkening for most phases in our models, and temporal variations related to the hydrodynamics of the stellar pulsation. Weassess the implications of our results with respect to the geometric Baade-Wesselink method, which uses interferometric measurements of Cepheid angular diameters to determine their distances and radii. The relevance of the hydrodynamic effects predicted by our model on the limb darkening of pulsating Cepheids is finally discussed in terms of the predicted capabilities of the VLTI. This revised version was published online in July 2006 with corrections to the Cover Date.     

Hubble Space Telescope observations of SV Cam – II. First derivative light-curve modelling using phoenix and atlas model atmospheres

The variation of the specific intensity across the stellar disc is an essential input parameter in surface brightness reconstruction techniques such as Doppler imaging, where the relative intensity contributions of different surface elements are important in detecting star-spots. We use phoenix and atlas model atmospheres to model light curves derived from high precision (signal-to-noise ratio ≃ 5000) Hubble Space Telescope ( HST ) data of the eclipsing binary SV Cam (F9V+K4V), where the variation of specific intensity across the stellar disc will determine the contact points of the binary system light curve. For the first time, we use χ² comparison fits to the first derivative profiles to determine the best-fitting model atmosphere. We show the wavelength dependence of the limb darkening and that the first derivative profile is sensitive to the limb-darkening profile very close to the limb of the primary star. It is concluded that there is only a marginal difference (<1σ) between the χ² comparison fits of the two model atmospheres to the HST light curve at all wavelengths. The usefulness of the second derivative of the light curve for measuring the sharpness of the primary's limb is investigated, but we find that the data are too noisy to permit a quantitative analysis.     

Automated analysis of eclipsing binary light curves – I. EBAS – a new Eclipsing Binary Automated Solver with ebop

We present a new algorithm, Eclipsing Binary Automated Solver (EBAS), to analyse light curves of eclipsing binaries. The algorithm is designed to analyse large numbers of light curves, and is therefore based on the relatively fast ebop code. To facilitate the search for the best solution, EBAS uses two parameter transformations. Instead of the radii of the two stellar components, EBAS uses the sum of radii and their ratio, while the inclination is transformed into the impact parameter. To replace human visual assessment, we introduce a new 'alarm' goodness-of-fit statistic that takes into account correlation between neighbouring residuals. We perform extensive tests and simulations that show that our algorithm converges well, finds a good set of parameters and provides reasonable error estimation.     

The role of stellar envelope convection in gravity-darkening and its relation to observational data

We study the role of convection in the surface heat flow of low mass stellar envelopes with the aid of Paczynski’s public domain program program GOB. This role is important, for example, for the analysis of light curves of close binary stars. We have considered atmospheric models for a range of masses similar to the components of contact or near-contact binaries between 0.4 and 1.1M_⊙ and examined the effects of changing luminosity, surface temperature and mixing length for stars of given mass. Our presentation includes checks of the modeling against published standard stellar envelopes. A slight program modification allowed gravity darkening coefficients for selected models to be calculated directly.If the procedure proposed by Lucy is followed, similar values of the index β (∼0.06–0.1) are obtained for a fairly wide range of masses, luminosities and effective temperatures of cool stars. There also appears no strong dependence on the mixing-length parameter α. There are, however, physical differences between the conditions that apply to this derivation of the index and those of a photosphere distorted by rotation and tides, but having net dynamical stability. Thus, the dependency of T_e on g was argued to come from the ratio of partial derivatives of the adiabatic constant K for the layer where convection starts; but a subconvective equipotential surface, where K would be constant, cannot coincide with such a layer. The adopted procedure is therefore inappropriate, and would involve different heat transfer regimes in different radial directions (say, polar or equatorial). A corresponding evaluation for the layer in which the convective flux becomes maximal shows less sensitivity to T_e. The corresponding index is then much closer to, or even greater than, the von Zeipel value.Other arguments are also considered, including those of Anderson and Shu [Anderson, L., Shu, F.H., 1977. ApJ 214, 798] about the independence of the convective flux to local gravity, as well as the role of superphotospheric circulation effects, which could reduce the gravity darkening index to less than its subphotospheric value.Observational evaluations of gravity darkening for close binary systems, in general, are still inconclusive, due to the strong correlation between β and other parameters characterizing the light curve shape, particularly in the scale photometric ‘ellipticity’ effects. It is possible that very precise light curves that may be obtained in the future will allow this situation to be improved. The general surface distribution of emergent flux in the far infra-red of Jupiter may be a pointer in the direction of future studies of the subject.     

Variation of the period and light curves of the solar-type contact binary EQ Tauri

We present two new sets of complete light curves of EQ Tauri (EQ Tau) observed in 2000 October and 2004 December. These were analysed, together with the light curves obtained by Yang & Liu in 2001 December, with the 2003 version of the Wilson–Devinney code. In the three observing seasons, the light curves show a noticeable variation in the time-scale of years. The more massive component of EQ Tau is a solar-type star (G2) with a very deep convective envelope, which rotates about 80 times as fast as the Sun. Therefore, the change can be explained by dark-spot activity on the common convective envelope. The assumed unperturbed part of the light curve and the radial velocities published by Rucinski et al. were used to determine the basic parameters of the system, which were kept fixed for spot modelling in the three sets of light curves. The results reveal that the total spotted area on the more massive component covers 18, 3 and 20 per cent of the photospheric surface in the three observing seasons, respectively. Polar spots and high-latitude spots are found. The analysis of the orbital period has demonstrated that it undergoes cyclical oscillation, which is due to either a tertiary component or periodic magnetic activity in the more massive component.     

Spotted star mapping by light curve inversion: Tests and application to HD 12545

A code for mapping the surfaces of spotted stars is developed. The concept of the code is to analyze rotational-modulated light curves. We simulate the process of reconstruction for the star surface and the results of simulation are presented. The reconstruction atrifacts caused by the ill-posed nature of the problem are deduced. The surface of the spotted component of system HD 12545 is mapped using the procedure.     

The standard systems of stellar rotation measurements (II)

The limb darkening effect on the measurement of stellar rotation is discussed in this paper. It is shown that this effect plays an important role in the measurement of V_e sin i. In the extreme case with the limb darkening coefficient of 1.0, it may cause a difference up to 17%. As a sequel to paper [1], this work presents the following new explanation for the systematic differences between Slettebak's new and old systems. The main causes of the systematic differences are: (1) The old system made an inadequate twice repeated correction for the limb darkening. (2) Owing to the historical reason, the old system used too large limb darkening coefficients.     

Time‐series photometric spot modelling – VI. A new computer code and its application to 23 years of photometry of the active giant IM Pegasi

We present and apply a new computer program named SpotModeL to analyze single and multiple bandpass photometric data of spotted stars. It is based on the standard analytical formulae from Budding and Dorren. The program determines the position, size, and temperature of up to three spots by minimizing the fit residuals with the help of the Marquardt‐Levenberg non‐linear least‐squares algorithm. We also expand this procedure to full time‐series analysis of differential data, just as real observations would deliver. If multi‐bandpass data are available, all bandpasses can be treated simultaneously and thus the spot temperature is solved for implicitly. The program may be downloaded and used by anyone. In this paper, we apply our code to an ≈23 year long photometric dataset of the spotted RS CVn giant IM Peg. We extracted and modelled 33 individual light curves, additionally, we fitted the entire V dataset in one run. The resulting spot parameters reflect the long term light variability and reveal two active longitudes on the substellar point and on the antipode. The radius and longitude of the dominant spot show variations with 29.8 and 10.4 years period, respectively. Our multicolour data suggests that the spot temperature is increasing with the brightening of the star. The average spot temperature from V, I_C is 3550 ± 150 K or approximately 900 K below the effective temperature of the star.     

Results of the photometry of the spotted dM1‐2e star EY Draconis

We have observed EY Draconis with the 60‐cm telescope of Konkoly Observatory in Budapest for 64 nights. In the first observing season the star produced a stable light curve for more than 60 stellar rotations, however, the light curves observed in the next season and the spot modelling show clear evidence of the evolution of the spotted stellar surface. The changes of the maximum brightness level suggests the existence of a longer period of about 300 days, which seems to be confirmed by the ROTSE archival data. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Absolute parameters of the Algol binary V505 Sgr from infrared JK light curves

We present the first light curves of V505 Sgr in the infrared (IR) J and K bands. The light curves are analysed with a code based on Roche geometry and stellar model atmosphere fluxes in order to determine a new set of stellar and orbital parameters. From the visual–IR photometry we find no evidence of IR excess in the system. We study the effect of the non-synchronous rotation of the primary star in the light and radial velocity curves. The distance of the system is estimated as  112 ± 4 pc  , in close agreement with the Hipparcos parallax.     

Detection of stellar spots from the observations of caustic-crossing binary-lens gravitational microlensing events

Recently, Heyrovský & Sasselov investigated the sensitivity of single-lens gravitational microlensing event light curves to spots and found that, during source transit, spots can cause deviations in amplification larger than 2 per cent, and thus be detectable. In this paper, we explore the feasibility of spot detection from the observations of binary-lens microlensing events instead of single-lens events. For this we investigate the sensitivity of binary-lens event light curves to spots and compare it with that of single-lens events. From this investigation, we find that during caustic crossings the fractional amplification deviations of light curves from those of spotless source events are equivalent to those of single-lens events, implying that spots can also be detected with a similar photometric precision to that required for spot detection by observing single-lens events. We discuss the relative advantages of observing binary-lens events over the observations of single-lens events in detecting stellar spots.     

Photometric modelling of starspots – I. A Barnes–Evans-like surface brightness–colour relation using (Ic−K)

In the first part of this work, the empirical correlation of stellar surface brightness F _V with ( I _c− K ) broad-band colour is investigated by using a sample of stars cooler than the Sun. A bilinear correlation is found to represent well the brightness of G, K and M giant stars. The change in slope occurs at ( I _c− K )∼2.1 or at about the transition from K to M spectral types. The same relationship is also investigated for dwarf stars and found to be distinctly different from that of the giants. The dwarf star correlation differs by an average of −0.4 in ( I _c− K ) or by a maximum in F _V of ∼−0.1, positioning it below that of the giants, with both trends tending towards convergence for the hotter stars in our sample. The flux distribution derived from the F _V −( I _c− K ) relationship for the giant stars, together with that derived from an F _V −( V − K ) relationship and the blackbody flux distribution, is then utilized to compute synthetic light V and colour ( V − R )_c, ( V − I )_c and ( V − K ) curves of cool spotted stars. We investigate the effects on the amplitudes of the curves by using these F _V –colour relations and by assuming the effective gravity of the spots to be lower than the gravity of the unspotted photosphere. We find that the amplitudes produced by using the F _V −( I _c− K ) relationship are larger than those produced by the other two brightness correlations, meaning smaller and/or warmer spots.     

Light‐curve solutions of 20 ecipsing Kepler binaries,most of them with pronounced spot and flare activity

We carried out light curve solutions of the Kepler light curves of twenty detached eclipsing binaries with circular orbits and determined the orbital inclinations, temperatures. relative radii and luminosities of their components. We studied the quality of the solutions with respect to the adopted limb‐darkening law and its coefficients. The detailed tracing of the numerous and uninterrupted data of our targets gave us an unique possibility to detect and learn their spot and flare activity. We established that the out‐of‐eclipse variability of the most targets gradually changes from small‐amplitude two‐waved type to big‐amplitude one‐waved type and vice versa, i.e. their spot activity cycles pass through phase of two almost diametrically opposite spots and phase of big polar cool spot. We found that the low‐temperature targets show flare activity of UV Cet‐type with amplitudes of 0.002–0.22 mag and duration of up to several hours. (© 2016 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solar limb brightening at the extreme limb from photoelectric eclipse observations

Photoelectric observations of the light intensity from the solar crescent before and after totality were made during the eclipses of 7 March, 1970 and 26 February, 1979. Effective wavelengths were determined by interference filters of 20 nm bandwidth. To obtain the limb darkening function, the resulting intensity curves were analyzed by an extension of the method of Julius in which we take into account the actual lunar limb profile. The limb darkening function at 433 nm was obtained for the region 0.937 < sin < 0.9999. Our results are in good agreement with existing center-to-limb measurements in the region of overlap which extends to sin = 0.99. Additional data were obtained at 600 nm for 0.994 < sin < 0.9999. The curves at both wavelengths show a distinctive limb brightening effect at sin = 0.999.