Inverse photometric problem for spotted stars

On the basis of the analytical solution of the direct photometric problem for spotted stars is obtained solution of the corresponding inverse problem. It is given expressions for determining of the location, size, and temperature of the spot and of the inclination of the rotational axis of the spotted star with regard to the line-of-sight.     

New analysis of II Peg 1977 light curve

The solution of the inverse photometric problem for spotted stars obtained on the basis of the analytical solution of the direct problem is applied to II Peg 1977 light curve. The geometric parameters of the spot configuration so obtained are unambigously determined in contrast to the cases when analysis of the light curve is done on the basis of numerical modelling. A comparison between both methods has been carried out.     

Rotation broadening of spectral features from star spots

This paper presents an extension of the method of the characteristic points by Kjurkchieva (1987, 1989). This method for the analysis of light curves of spotted stars is further developed by including spectral information. Expressions describing the variation of the rotation broadening of spectral features arising from a star spot are derived for a known spot configuration. The results are used in the solution of the inverse problem by the method of the characteristic points. It is shown that the rotation broadening of spectral features from star spots can be used to easily determine the angular size and the polar distance of a circular spot. It turns out that the equatorial velocity and radius of the spotted star can be obtained in some cases by the combined analysis of the photometric and spectral data.     

食双星RT CrB测光畸变波的黑子模拟及黑子参数的微分改正解

本文将1983年版的Wilson-Devinney方法程序,扩展到包含黑子参数的微分改正解,并提出黑子纬度的网络测定方法,由此,对RS CVn型食双星RT CrB的食外测光扰动波,进行了黑子模拟解,得出其次子星上两个黑子的位置,大小和温度,其理论光变曲线能较好地重现畸变波的观测特征。作者对黑子模型参数微分改正解的收敛性问题和不定性问题进行了初步讨论。     

A Library of Eclipsing Binary Light-Curves: Fast Initial Parameters Estimation, and on the Uniqueness of the Inverse Problem

In order to improve the light curve solutions we suggest to use a new approach. A library of the Fourier-coefficients (up to 20th term) of (several millions of) light curves was constructed each of the light curves has different input parameters. A light curve solution is made in a very fast way: its discrete Fourier-transform yields not more than some simple number which are compared to the Fourier-coefficients in the library. A chi-squared test can help us to estimate very good initial parameters for the differential correction analysis. The library can also be used to study the uniqeness problem of the light curve solutions. Here we investigated the uniqueness of the spot solution of SV Cam.     

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)     

Dynamics of Liquid Mass Spreading over a Planetary Surface

The behavior of liquid mass poured out on the surface of a rotating planet is investigated. Analysis of the numerical experiments indicates that the spreading pattern of the produced spot depends significantly on the nondimensional number Fric that describes the relationship between such parameters of the problem as the acceleration of gravity, the Coriolis parameter, the coefficient of friction, and the depth scale of the spot. In particular, for Fric < 0.1, the spot is shown to make concentric oscillations as it spreads whose origin is similar to that of Poincare waves.     

A multiple spot model for simultaneous solution of light curves and distorted line profiles of spotted stars

This paper is devoted to the effects of dark starspots on observed light curves and line profiles. A new formulation of the starspot model with multiple spots and arbitrary spot shapes is derived which allows for a numerical solution of light and colour curves and distorted line profiles simultaneously. The effects of differential rotation and the existence of a penumbra have also been considered. A parameter study demonstrates the influences of different spot parameters on the theoretical light curves and rotation profiles. The computer code is applicable for single stars as well as for eclipsing close binaries allowing for ellipticity effect and reflection effect.     

磁光效应与太阳黑子磁场结构

本文在考虑磁光效应条件下,根据对斯托克斯参数转移方程组求得的数值解,计算了单极太阳黑子的线偏振讯号的单色像,并与美国马歇尔空间飞行中心的观测资料进行了对比,结果表明,径向黑子磁场模型给出与观测相似的单色像,而旋涡形模型导致与观测有显著差异的图像。因此可以认为径向模型更接近于实际情况。     

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

The subject of the paper is the problem of stellar differntial rotation in close binaries (CB) ofRS CV n type. The differential-rotation parameters we find on the basis of the migration of the depression in the light curves caused by the spot effect over the orbital phase. For that purpose, a simple model (Bussoet al., 1985) and inverse-problem procedure, based on the Marquardt (1963) algorithm, are used. To verify the obtained solutions, the SIMPLEX algorithm (Torczon, 1991) is applied, suitable for the nonlinear parameter optimisation. This algorithm enables a correct solution of the nonlinear equation system describing the differential rotation. The procedure is applied in the determination of the parameters of differential rotation forCV Cam, VV Mon andSS Boo binaries.     

Photoelectric observations of long-period RS CVn binary CQ aurigae

Long-period RS CVn-type binary CQ Aur was observed in 1987 and 1988.B andV light curves for the 1987 and 1988 epochs have been analyzed for the photometric solution by the modified Wilson and Devinney computing code with adjustable spot parameters. One-spot fit is good enough for the 1987 epoch while a wider distribution of spots in longitude is required for the 1988 epoch. The spot covers approximately 3% of total surface area of the cooler component. Absolute dimensions were deduced by combining Popper's spectroscopic results. The volume of the cooler component is approximately 100 times larger than that of the hotter component. Both components are subgiants but the cooler component evolved more than the hotter component.     

Charge-coupled device observations of V388 Cyg

We present BVR full-light curves of V388 Cyg to provide a complete photometric solution for the first time. The light curves show a high degree of asymmetry (O'Connell effect). The maxima at 0.25 phase (Max I) are 0.023, 0.018 and 0.012 mag higher than those at 0.75 phase (Max II) in B , V and R , respectively. Three possible spot models are applied to fit the asymmetric light curves of V388 Cyg, in order to explain the O'Connell effect. We conclude that the model of the cool spot on the cooler star is the most reasonable model for V388 Cyg. The continuous period variation is confirmed by recently collected times of minima, including one minimum that is determined in this paper. The period decrease rate is estimated as d p /d t =−2.055×10⁻⁷ d yr⁻¹ .     

An Analysis of the Light Curves of the Early-Type Binary BF Aurigae

The UBV observations of the massive binary BF Aur were made at the Ankara University Observatory during 1988, 1989 and 1996. Asymmetry of the light curves, arising from unequal height of successive maxima, indicates that the system is active. By analysing these observations in the framework of the Roche model (including the presence of bright regions on the components) one obtains a semidetached configuration of the system, with the cooler secondary component filling its Roche lobe. The analysis of the light curves yields consistent solutions for mass ratio q = m₂/m₁ somewhat less than one. The influence of the mass transfer on the change of the system-orbital-period is relatively small. The upward parabolic character of the O–C diagram (Zhang et al., 1993) indicates a mass transfer from the less massive secondary to the more massive primary. This inturn requires the less massive secondary to fill its Roche lobe. This is consistent with our solution. Based on these facts we introduced the following working hypothesis. At the place where the gas stream from the secondary falls on the primary, relatively small in size but a high temperature contrast active hot-spot (hs) region is formed. As a result of the heating effect caused by the irradiation of the hot-spot region, on the secondary's side facing the hot spot a bright-spot (bs) region is formed. The bright-spot region is larger in size but with significantly lower temperature than the hot spot. This region can be treated as a ‘reflection cap’. By analysing the light curves in the framework of this working hypothesis the basic parameters of the system and the active regions are estimated. The problem is solved in two stages: by obtaining a synthetic light curve in the case when the parameters of the corresponding Close Binary (CB) Roche model (Djurašević, 1992a) are given a priori (the 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 (the inverse problem) (Djurašević, 1992b). This revised version was published online in July 2006 with corrections to the Cover Date.     

Period variation and spot model for the W UMa type binary TY UMa

We present the full VRI light curves and the times of minima of TY UMa to provide a complete photometric solution and a long-term trend of period variation. The light curves show a high degree of asymmetry (the O'Connell effect). The maxima at 0.25 phase (Max I) are 0.021, 0.015, and 0.020 mag fainter than those at 0.75 phase (Max II) in V , R , and I , respectively. The period of TY UMa has varied in a sinusoidal way, superimposed on the long-term upward parabolic variation. The secularly increasing rate of the period is deduced as 1.83 s per century  ( P˙ / P =5.788×10^-10 d d^-1)  . The period of sinusoidal variation is about 57.4 yr. The spot model has been applied to fit the asymmetric light curves of TY UMa, to explain light variations. By changing only the spot parameters, the model light curves can fit the observed light curves for three epochs. This indicates that the variation of the spot location and size is the main reason for changing the shape of light curves, including two different maxima and the interchanging depths of occultation and transit minima.     

Diffuse reflection and transmission of solar radiation by a finite atmosphere bounded by a reflecting surface

The problem of diffuse reflection and transmission of solar radiation through a planetary atmosphere bounded from below by a reflecting surface is solved. The solution method based on rewriting the solution of the proposed problem in terms of the well known standard problem solution, where the planetary surface does not reflect. The solution of the standard problem can be found elsewhere or as we did by using the maximum entropy method. Numerical results for the angular radiation intensity and for the reflection and transmission coefficients are presented and compared with those obtained by Chandrasekhar's method.     

A new approach to impulsive rendezvous near circular orbit

A new approach is presented for the problem of planar optimal impulsive rendezvous of a spacecraft in an inertial frame near a circular orbit in a Newtonian gravitational field. The total characteristic velocity to be minimized is replaced by a related characteristic-value function and this related optimization problem can be solved in closed form. The solution of this problem is shown to approach the solution of the original problem in the limit as the boundary conditions approach those of a circular orbit. Using a form of primer-vector theory the problem is formulated in a way that leads to relatively easy calculation of the optimal velocity increments. A certain vector that can easily be calculated from the boundary conditions determines the number of impulses required for solution of the optimization problem and also is useful in the computation of these velocity increments. Necessary and sufficient conditions for boundary conditions to require exactly three nonsingular non-degenerate impulses for solution of the related optimal rendezvous problem, and a means of calculating these velocity increments are presented. A simple example of a three-impulse rendezvous problem is solved and the resulting trajectory is depicted. Optimal non-degenerate nonsingular two-impulse rendezvous for the related problem is found to consist of four categories of solutions depending on the four ways the primer vector locus intersects the unit circle. Necessary and sufficient conditions for each category of solutions are presented. The region of the boundary values that admit each category of solutions of the related problem are found, and in each case a closed-form solution of the optimal velocity increments is presented. Similar results are presented for the simpler optimal rendezvous that require only one-impulse. For brevity degenerate and singular solutions are not discussed in detail, but should be presented in a following study. Although this approach is thought to provide simpler computations than existing methods, its main contribution may be in establishing a new approach to the more general problem.     

Solution of the Sitnikov Problem

A new analytic expression for the position of the infinitesimal body in the elliptic Sitnikov problem is presented. This solution is valid for small bounded oscillations in cases of moderate primary eccentricities. We first linearize the problem and obtain solution to this Hill's type equation. After that the lowest order nonlinear force is added to the problem. The final solution to the equation with nonlinear force included is obtained through first the use of a Courant and Snyder transformation followed by the Lindstedt–Poincaré perturbation method and again an application of Courant and Snyder transformation. The solution thus obtained is compared with existing solutions, and satisfactory agreement is found.     

Lambert problem solution in the hill model of motion

The goal of this paper is obtaining a solution of the Lambert problem in the restricted three-body problem described by the Hill equations. This solution is based on the use of pre determinate reference orbits of different types giving the first guess and defining the sought-for transfer type. A mathematical procedure giving the Lambert problem solution is described. This procedure provides step-by-step transformation of the reference orbit to the sought-for transfer orbit. Numerical examples of the procedure application to the transfers in the Sun–Earth system are considered. These examples include transfer between two specified positions in a given time, a periodic orbit design, a halo orbit design, halo-to-halo transfers, LEO-to-halo transfer, analysis of a family of the halo-to-halo transfer orbits. The proposed method of the Lambert problem solution can be used for the two-point boundary value problem solution in any model of motion if a set of typical reference orbits can be found.     

On an analytical solution of the optimum trajectory problem in a gravitational field

Mayer's variational problem for a point with a limited mass flow rate is described by differential equations of the fourteenth order, allowing for a few first integrals. By reducing the equations to closed canonical form, these integrals are analyzed from the viewpoint of finding a possible solution to the problem via quadratures on zero, intermediate, and maximum thrust sections. In addition to confirming well-known cases of total integrability, this approach enabled us to establish that the essential difficulty of the solution of the space problem with intermediate thrust is reduced to finding one integral, and the solution of the problem with maximum thrust requires two integrals in involution. It is shown that these integrals can be applied to find particular solutions.     

月球探测器过渡轨道的短弧定轨方法

论述的短弧定轨,是指在无先验信息情况下又避开多变元迭代的初轨计算方法,它需要相应的动力学问题有一能反映短弧内达到一定精度的近似分析解．探测器进入月球引力作用范围后接近月球时可以处理成相对月球的受摄二体问题,而在地球附近,则可处理成相对地球的受摄二体问题,但在整个过渡段的力模型只能处理成一个受摄的限制性三体问题．而限制性三体问题无分析解,即使在月球引力作用范围外,对于大推力脉冲式的过渡方式,相对地球的变化椭圆轨道的偏心率很大(超过Laplace极限),在考虑月球引力摄动时亦无法构造摄动分析解．就此问题,考虑在地球非球形引力(只包含J2项)和月球引力共同作用下,构造了探测器飞抵月球过渡轨道段的时间幂级数解,在此基础上给出一种受摄二体问题意义下的初轨计算方法,经数值验证,定轨方法有效,可供地面测控系统参考．