Parâmetros físico-químicos de estrelas nos campos de exoplanetas do satélite corot

A rotação estelar é um dos mais importantes observáveis da evolução estelar. Neste sentido, o satélite CoRoT representa uma oportunidade única de medir os períodos rotacionais para uma amostra de estrelas estatisticamente robusta, oferecendo dados absolutamente necessários para o estudo da rotação e seu papel na evolução estelar. Para conseguir isto, um passo fundamental é a caracterização física e química das estrelas observadas pelo CoRoT, especificamente devido ao fato de que o cálculo de períodos rotacionais confiáveis é um trabalho difícil sem a ajuda dos parâmetros estelares. Desta forma, foi elaborado um importante seguimento observacional das estrelas nos campos do CoRoT do anticentro LRa01 e do centro LRc01, permitindo a correta identificação dos períodos que reflitam a modulação rotacional. Nesta tese de doutorado são apresentados os resultados de tal seguimento. Parâmetros físicos e químicos, tais como temperatura efetiva Teff , gravidade superficial log(g), velocidade de microturbulência Vmic, abundância de ferro [Fe/H], velocidade de rotação projetada Vsin(i), e abundância de lítio A(Li) são apresentados para uma amostra de 116 estrelas dos campos CoRoT. Elas se encontram em diferentes estágios evolutivos, desde a sequência principal (SP) até o ramo das gigantes vermelhas (GV). As observações foram feitas utilizando os espectrógrafos UVES (VLT) e HYDRA (CTIO). Para a derivação de tais parâmetros foram utilizados o programa TurboSpectrum e os modelos de atmosfera de MARCS. Paralelamente, velocidades rotacionais Vsin(i) foram obtidas a partir do ajuste dos perfis observados e sintéticos das linhas de ferro e por meio de uma calibração de função de correlação cruzada (CCF). Períodos rotacionais Prot para 77 estrelas da amostra foram obtidos a partir das curvas de luz do satélite CoRoT. Extensas tabelas destes parâmetros e seus respectivos erros são apresentadas. Foram encontradas diferenças nas distribuições de Teff , [Fe/H] e estágios evolutivos entre os diferentes campos do CoRoT, indicando possíveis efeitos de seleção na amostra, assim como a existência de diferentes populações estelares do disco Galáctico. Por outro lado, o comportamento rotacional e as abundâncias de lítio não apresentam diferenças entre estrelas de parâmetros físicos similares, mas que pertencem a diferentes campos do CoRoT. A partir da análise de temperaturas, foi encontrada uma maior extinção por avermelhamento para estrelas do CoRoT localizadas no campo LRc01, assim como um gradiente deste valor em função da distância. Os resultados mostram que as abundâncias de lítio, as velocidades de rotação e os períodos rotacionais apresentam o mesmo comportamento descrito na literatura. Por outro lado, é apresentada pela primeira vez a relação que existe entre o lítio e o período de rotação em diferentes estágios evolutivos, mostrando, tal como era esperado, que ambas as grandezas possuem uma anticorrelação. Também é apresentada a evolução simultânea da rotação e do lítio, e foram calculadas relações que permitem obter valores médios de A(Li) como função da temperatura efetiva e do período rotacional. Os dados apresentados nesta tese de doutorado representam um importante ponto de partida para serem utilizados como uma amostra de calibração para diferentes programas no contexto da missão do satélite CoRoT, uma vez que a lista de estrelas aqui analisadas são parte das mais brilhantes que compõem o campo Exo do CoRoT

Em busca de um novo indicador Espectroscópico do período de rotação das Estrelas do tipo solar

The study of sunspots consistently contributed to a better understanding of magnetic phenomena of the Sun, as its activity. It was found with the dynamics of sunspots that the Sun has a rotation period of twenty-seven days around your axis. With the help of Project Sun-As-A-Star that solar spectra obtained for more than thirty years we observed oscillations of both the depth of the spectral line and its equivalent width, and analysis of the return information about the characteristics of solar magnetism. It also aims to find patterns of solar magnetic activity cycle and the average period of rotation of the Sun will indicate the spectral lines that are sensitive to magnetic activity and which are not. Sensitive lines how Ti II 5381.0 Å stands as the best indicator of the solar rotation period and also shows different periods of rotation cycles of minimum and maximum magnetic activity. It is the first time we observe clearly distinct rotation periods in the different cycles. The analysis also shows that Ca II 8542.1 Å and HI 6562.0 Å indicate the cycle of magnetic activity of eleven years. Some spectral lines no indicated connection with solar activity, this result can help us search for programs planets using spectroscopic models. Data analysis was performed using the Lomb-Scargle method that makes the time series analysis for unequally spaced data. Observe different rotation periods in the cycles of magnetic activity accounts for a discussion has been debated for many decades. We verified that spectroscopy can also specify the period of stellar rotation, thus being able to generalize the method to other stars

Interação estrela planeta: sobre o magnetismo de planetas gigantes gasosos

In this thesis we analyze the effects that the presence of a near gas giant planet can cause in its host star. It has been argued that the star planet interaction can cause changes in the coronal and chromospheric stellar activity. With this in mind, we analyze a sample of 53 extrasolar planets orbiting F, G and K main sequence stars, among them three super-Earths. In this analysis, we look for evidence of changes in the chromospheric activity due to the proximity of the giant planet. We show that, so far, there is not enough evidence to support such a hypothesis. Making use of the same sample and also taking in account available data for the Solar System, we revisit the so-called magnetic Bode s law. This law proposes the existence of a direct relationship between magnetism and rotation. By using estimations for the stellar and planetary magnetic momentM and the angular momentumL, we construct a Blackett s diagram (logL 􀀀logM). In this diagram is evident that the magnetic Bode s law is valid for both the Solar System and the new planetary systems

Um estudo da abundância de lítio, rotação, atividade cromosférica e magnetismo das estrelas análogas e gêmeas solares

The study physical process that control the stellar evolution is strength influenced by several stellar parameters, like as rotational velocity, convective envelope mass deepening, and magnetic field intensity. In this study we analyzed the interconnection of some stellar parameters, as Lithium abundance A(Li), chromospheric activity and magnetic field intensity as well as the variation of these parameters as a function of age, rotational velocity, and the convective envelope mass deepening for a selected sample of solar analogs and twins stars. In particular, we analyzed the convective envelope mass deepening and the dispersion of lithium abundance for these stars. We also studied the evolution of rotation in subgiants stars, because its belong to the following evolutionary stage of solar analogs, and twins stars. For this analyze, we compute evolutionary models with the TGEC code to derive the evolutionary stage, as well as the convective envelope mass deepening, and derive more precisely the stellar mass, and age for this 118 stars. Our Investigation shows a considerable dispersion of lithium abundance for the solar analogs stars. We also realize that this dispersion is not by the convective zone deep, in this way we observed which the scattering of A(Li) can not be explained by classical theories of mixing in the convective zone. In conclusion we have that are necessary extra-mixing process to explain this decrease of Lithium abundance in solar analogs and twins stars. We analyzed the subgiant stars because this are the subsequent evolutionary stage after the solar analogs and twins stars. For this analysis, we compute the rotational period for 30 subgiants stars observed by Co- RoT satellite. For this task we apply two different methods: Lomb-Scargle algorithm, and the Plavchan Periodogram. We apply the TGEC code we compute models with internal distribution of angular momentum to confront the predict results with the models, and the observational results. With this analyze, we showed which solid body rotation models are incompatible with the physical interpretation of observational results. As a result of our study we still concluded that the magnetic field, convective envelope mass deepening, and internal redistribution of angular momentum are essential to explain the evolution of low-mass stars, and its observational characteristics. Based on population synthesis simulation, we concluded that the solar neighborhood presents a considerable quantity of solar twins when compared with the discovered set nowadays. Altogether we foresee the existence around 400 solar analogs in the solar neighborhood (distance of 100 pc). We also study the angular momentum of solar analogs and twins, in this study we concluded that added angular momentum from a Jupiter type planet, putted in the Jupiter position, is not enough to explain the angular momentum predicted by Kraft law (Kraft 1970)

Um estudo sobre o momentum angular total de estrelas com planetas

Since Michel Mayor and his student Didier Queloz s pioneer announcement, in 1995, of the existence of a planet orbiting the star 51 Peg, up to present date, 695 extrasolar planets orbiting stars of spectral type F, G, K and M have been discovered. A study on the behavior of the total angular momentum of the planetary systems known up to present date becomes relevant when we know that about 98% of the angular momentum of the solar system is associated with the planets, although they represent only 0.15 percent of the mass of the whole system. In this dissertation we study the behavior of stellar angular momentum, orbital angular momentum and total angular momentum in a sample of 282 stars harboring planets, including 40 multiple systems. We observed that planetary systems containing more than one known planet have both higher orbital angular momentum and total angular momentum compared to those who have only one planet. This analysis shows that multiplanet systems tend to have higher momenta, suggesting that the planets in such systems that contribute to the greater portion momenta have been found. Thus, planetary systems with lower values for the momenta represent the best candidates to the discovery of new planets

Um estudo da rotação em estrelas da seqüência principal

Conselho Nacional de Desenvolvimento Científico e Tecnológico

Rotação diferencial em estrelas do tipo solar

Stellar differential rotation is an important key to understand hydromagnetic stellar dynamos, instabilities, and transport processes in stellar interiors as well as for a better treatment of tides in close binary and star-planet systems. The space-borne high-precision photometry with MOST, CoRoT, and Kepler has provided large and homogeneous datasets. This allows, for the first time, the study of differential rotation statistically robust samples covering almost all stages of stellar evolution. In this sense, we introduce a method to measure a lower limit to the amplitude of surface differential rotation from high-precision evenly sampled photometric time series such as those obtained by space-borne telescopes. It is designed for application to main-sequence late-type stars whose optical flux modulation is dominated by starspots. An autocorrelation of the time series is used to select stars that allow an accurate determination of spot rotation periods. A simple two-spot model is applied together with a Bayesian Information Criterion to preliminarily select intervals of the time series showing evidence of differential rotation with starspots of almost constant area. Finally, the significance of the differential rotation detection and a measurement of its amplitude and uncertainty are obtained by an a posteriori Bayesian analysis based on a Monte Carlo Markov Chain (hereafter MCMC) approach. We apply our method to the Sun and eight other stars for which previous spot modelling has been performed to compare our results with previous ones. The selected stars are of spectral type F, G and K. Among the main results of this work, We find that autocorrelation is a simple method for selecting stars with a coherent rotational signal that is a prerequisite to a successful measurement of differential rotation through spot modelling. For a proper MCMC analysis, it is necessary to take into account the strong correlations among different parameters that exists in spot modelling. For the planethosting star Kepler-30, we derive a lower limit to the relative amplitude of the differential rotation. We confirm that the Sun as a star in the optical passband is not suitable for a measurement of the differential rotation owing to the rapid evolution of its photospheric active regions. In general, our method performs well in comparison with more sophisticated procedures used until now in the study of stellar differential rotation

Estudo sistemático de estrelas variáveis na era dos grandes surveys

The interest in the systematic analysis of astronomical time series data, as well as development in astronomical instrumentation and automation over the past two decades has given rise to several questions of how to analyze and synthesize the growing amount of data. These data have led to many discoveries in the areas of modern astronomy asteroseismology, exoplanets and stellar evolution. However, treatment methods and data analysis have failed to follow the development of the instruments themselves, although much effort has been done. In present thesis, we propose new methods of data analysis and two catalogs of the variable stars that allowed the study of rotational modulation and stellar variability. Were analyzed the photometric databases fromtwo distinctmissions: CoRoT (Convection Rotation and planetary Transits) and WFCAM (Wide Field Camera). Furthermore the present work describes several methods for the analysis of photometric data besides propose and refine selection techniques of data using indices of variability. Preliminary results show that variability indices have an efficiency greater than the indices most often used in the literature. An efficient selection of variable stars is essential to improve the efficiency of all subsequent steps. Fromthese analyses were obtained two catalogs; first, fromtheWFCAMdatabase we achieve a catalog with 319 variable stars observed in the photometric bands Y ZJHK. These stars show periods ranging between ∼ 0, 2 to ∼ 560 days whose the variability signatures present RR-Lyrae, Cepheids , LPVs, cataclysmic variables, among many others. Second, from the CoRoT database we selected 4, 206 stars with typical signatures of rotationalmodulation, using a supervised process. These stars show periods ranging between ∼ 0, 33 to ∼ 92 days, amplitude variability between ∼ 0, 001 to ∼ 0, 5 mag, color index (J - H) between ∼ 0, 0 to ∼ 1, 4 mag and spectral type CoRoT FGKM. The WFCAM variable stars catalog is being used to compose a database of light curves to be used as template in an automatic classifier for variable stars observed by the project VVV (Visible and Infrared Survey Telescope for Astronomy) moreover it are a fundamental start point to study different scientific cases. For example, a set of 12 young stars who are in a star formation region and the study of RR Lyrae-whose properties are not well established in the infrared. Based on CoRoT results we were able to show, for the first time, the rotational modulation evolution for an wide homogeneous sample of field stars. The results are inagreement with those expected by the stellar evolution theory. Furthermore, we identified 4 solar-type stars ( with color indices, spectral type, luminosity class and rotation period close to the Sun) besides 400 M-giant stars that we have a special interest to forthcoming studies. From the solar-type stars we can describe the future and past of the Sun while properties of M-stars are not well known. Our results allow concluded that there is a high dependence of the color-period diagram with the reddening in which increase the uncertainties of the age-period realized by previous works using CoRoT data. This thesis provides a large data-set for different scientific works, such as; magnetic activity, cataclysmic variables, brown dwarfs, RR-Lyrae, solar analogous, giant stars, among others. For instance, these data will allow us to study the relationship of magnetic activitywith stellar evolution. Besides these aspects, this thesis presents an improved classification for a significant number of stars in the CoRoT database and introduces a new set of tools that can be used to improve the entire process of the photometric databases analysis

Evolução da atividade cromosférica, abundância de lítio e rotação das estrelas análogas e gêmeas solares

The study of solar-type stars also includes the familiar solar analogs and twins. These objects have been one of the major research subjects in astrophysics nowadays. A direct comparison of solar activity with chromospheric activity indices for a set of stars very similar to the Sun (twins and analogs) provides an excellent opportunity to study the evolution of stellar activity on timescales of the order of the lifetime on the main sequence. This work deals with the relationship between the abundance of lithium, chromospheric activity, X-ray emission and rotation period in terms of stellar ages. We explore the influence of stellar evolution in the global properties of the stars and the aspects linked to its coronal, chromospheric and magnetic activity. Our main objective is to probe the law of decay of each of these parameters based on a sample of stars classified as well-connected as analogs stars and solar twins.

O enigmático problema das gigantes ricas em Lítio e as perspectivas com o satélite Kepler

Lithium (Li) is a chemical element with atomic number 3 and it is among the lightest known elements in the universe. In general, the Lithium is found in the nature under the form of two stable isotopes, the 6Li and 7Li. This last one is the most dominant and responds for about 93% of the Li found in the Universe. Due to its fragileness this element is largely used in the astrophysics, especially in what refers to the understanding of the physical process that has occurred since the Big Bang going through the evolution of the galaxies and stars. In the primordial nucleosynthesis in the Big Bang moment (BBN), the theoretical calculation forecasts a Li production along with all the light elements such as Deuterium and Beryllium. To the Li the BNB theory reviews a primordial abundance of Log log ǫ(Li) =2.72 dex in a logarithmic scale related to the H. The abundance of Li found on the poor metal stars, or pop II stars type, is called as being the abundance of Li primordial and is the measure as being log ǫ(Li) =2.27 dex. In the ISM (Interstellar medium), that reflects the current value, the abundance of Lithium is log ǫ(Li) = 3.2 dex. This value has great importance for our comprehension on the chemical evolution of the galaxy. The process responsible for the increasing of the primordial value present in the Li is not clearly understood until nowadays. In fact there is a real contribution of Li from the giant stars of little mass and this contribution needs to be well streamed if we want to understand our galaxy. The main objection in this logical sequence is the appearing of some giant stars with little mass of G and K spectral types which atmosphere is highly enriched with Li. Such elevated values are exactly the opposite of what could happen with the typical abundance of giant low mass stars, where convective envelops pass through a mass deepening in which all the Li should be diluted and present abundances around log ǫ(Li) ∼1.4 dex following the model of stellar evolution. In the Literature three suggestions are found that try to reconcile the values of the abundance of Li theoretical and observed in these rich in Li giants, but any of them bring conclusive answers. In the present work, we propose a qualitative study of the evolutionary state of the rich in Li stars in the literature along with the recent discovery of the first star rich in Li observed by the Kepler Satellite. The main objective of this work is to promote a solid discussion about the evolutionary state based on the characteristic obtained from the seismic analysis of the object observed by Kepler. We used evolutionary traces and simulation done with the population synthesis code TRILEGAL intending to evaluate as precisely as possible the evolutionary state of the internal structure of these groups of stars. The results indicate a very short characteristic time when compared to the evolutionary scale related to the enrichment of these stars

Efeitos não-gaussianos em astrofísica e cosmologia

The recent astronomical observations indicate that the universe has null spatial curvature, is accelerating and its matter-energy content is composed by circa 30% of matter (baryons + dark matter) and 70% of dark energy, a relativistic component with negative pressure. However, in order to built more realistic models it is necessary to consider the evolution of small density perturbations for explaining the richness of observed structures in the scale of galaxies and clusters of galaxies. The structure formation process was pioneering described by Press and Schechter (PS) in 1974, by means of the galaxy cluster mass function. The PS formalism establishes a Gaussian distribution for the primordial density perturbation field. Besides a serious normalization problem, such an approach does not explain the recent cluster X-ray data, and it is also in disagreement with the most up-to-date computational simulations. In this thesis, we discuss several applications of the nonextensive q-statistics (non-Gaussian), proposed in 1988 by C. Tsallis, with special emphasis in the cosmological process of the large structure formation. Initially, we investigate the statistics of the primordial fluctuation field of the density contrast, since the most recent data from the Wilkinson Microwave Anisotropy Probe (WMAP) indicates a deviation from gaussianity. We assume that such deviations may be described by the nonextensive statistics, because it reduces to the Gaussian distribution in the limit of the free parameter q = 1, thereby allowing a direct comparison with the standard theory. We study its application for a galaxy cluster catalog based on the ROSAT All-Sky Survey (hereafter HIFLUGCS). We conclude that the standard Gaussian model applied to HIFLUGCS does not agree with the most recent data independently obtained by WMAP. Using the nonextensive statistics, we obtain values much more aligned with WMAP results. We also demonstrate that the Burr distribution corrects the normalization problem. The cluster mass function formalism was also investigated in the presence of the dark energy. In this case, constraints over several cosmic parameters was also obtained. The nonextensive statistics was implemented yet in 2 distinct problems: (i) the plasma probe and (ii) in the Bremsstrahlung radiation description (the primary radiation from X-ray clusters); a problem of considerable interest in astrophysics. In another line of development, by using supernova data and the gas mass fraction from galaxy clusters, we discuss a redshift variation of the equation of state parameter, by considering two distinct expansions. An interesting aspect of this work is that the results do not need a prior in the mass parameter, as usually occurs in analyzes involving only supernovae data.Finally, we obtain a new estimate of the Hubble parameter, through a joint analysis involving the Sunyaev-Zeldovich effect (SZE), the X-ray data from galaxy clusters and the baryon acoustic oscillations. We show that the degeneracy of the observational data with respect to the mass parameter is broken when the signature of the baryon acoustic oscillations as given by the Sloan Digital Sky Survey (SDSS) catalog is considered. Our analysis, based on the SZE/X-ray data for a sample of 25 galaxy clusters with triaxial morphology, yields a Hubble parameter in good agreement with the independent studies, provided by the Hubble Space Telescope project and the recent estimates of the WMAP

Sobre a relação entre rotação, atividade crosmosférica e abundância de lítio em estrelas subgigantes

The connection between rotation, CaII emission flux and lithium abundance is analyzed for a sample of subgiant stars, with evolutionary status was determined from the Toulouse-Geneve code and HlPPARCOS trigonometric parallax measurements. We noted that the distribution of rotation and CaII emission flux, as a function of effective temperature, shows a discontinuity located around the same spectral type, F8IV. Stars located blueward of this spectral type, exhibit a large spread of values of rotation and CaII flux, whereas stars redward of F8lV show essentially low ratation anel low CaII flux. The strength of these declines nevertheless, depends on stellar mass. The distribution of lithium abundances also shows a discontinuity, however with behavior a little more complex for subgiants with mass lower than about 1.2 Solar Masses, this decrease is observed later than that in rotation and CaII flux, whereas for masses higher than 1.2 Solar Masses the decrease in lithium abundance is located around the spectral type F8IV. The discrepancy between the location of the discontinuities of rotation and CaII flux and log n(Li) for stars with masses lower than 102 Solar Masses, seems to reflect the sensitivity of these phenomena to the mass of the convective envelope. The drop in rotation, which results mostly from a magnetic braking, requires an increase in the mass of the convective envelope less than that required for the decrease in lithium abundance The location of the discontinuity in log n( Li) in the same region of the discontinuity ties in rotation and CaII flux, for stars with masses higher than 1.2 Solar Masses, may also be explained by the behavior of the deepening of the convective envelope. In contrast to the relationship between rotation and CaII flux the relationship between lithium abundance and rotation shows no dear tendency toward linear behavior. Similarly, the same tendency is observed in the relationship between lithium abundance and CaII flux in spite of these facts, subgiants with high lithium content also have high rotation and high CaII emission flux. We also observed that stars with high lithium content present, in its majority, an undeveloped convective envelope, whereas stars with low lithium content have a developed convective envelope. In the case of the rotation, stars with undeveloped convective envelope, show rotational velocities as much high as low, whereas stars with developed convective envelope only present low rotation

Lítio e rotação nas estrelas análogas solares: análise teórica das observações

The projected rotational velocity together with lithium abundance and the onset of the dilution by the deepening in mass of the convective envelope provide a key tool to investigate the so far poorly understood processes at work in stellar interiors of solar-analog stars. To investigate the link between abundances, convection and rotational velocities in solar-analog G dwarf stars, we study a bona fide sample of 118 selected solar-analog G dwarf stars presenting measured lithium abundances, rotational velocities, and fundamental parameters together with computed evolutionary tracks (Toulouse-Geneva code) for a range of stellar masses around 1 M and metallicity consistent with the solar-analog range. The aim of this work is to build up an evolution of lithium and rotation as a function of stellar age, mass, effective temperature, and convection. We analyze the evolutionary status of the sample of 118 solar-analog G dwarf in the HR diagram based on Hipparcos data and using a grid of stellar models in the effective temperature and mass range of the solar-analog stars. We discuss the deepening (in mass) of the convective envelope and the influence on the Li abundances and projected rotational velocities. We determined the stellar mass and the mass of the convective envelope for a bona fide sample of 118 selected solar-analog G dwarf and checked the evolutionary link between the rotational velocity, lithium abundance, and the deepening of the convective envelope. Fast rotators (vsini 6 km s��1) are also stars with high Li content. Slow rotators present a wide range of values of log n(Li). Our results shed new light on the lithium and rotational behavior in G dwarf stars. We confirmed the presence of a large Li abundance spread among the solar-analog stars and concluded that the solar twins probably share a similar mixing history with the Sun

Atividade cromosférica induzida por planetas extrasolares gigantes

In the present work, we have analyzed the behavior of the chromospheric activity of stars with planets, as a function of different planetary parameters, searching for possible effects of planets on the chromosphere of the hosting star. For this study we have selected a sample of 73 main sequence stars with planets, of spectral types F, G and K. Our analysis shows that among stars with planets presenting semi-major axis smaller than 0.15 AU, a few ones present enhanced CaII emission flux, paralleling recent results found in the literature for coronal X-ray flux. Nevertheless, in contrast to Kashyap et al. (2008), who claim that enhanced X-ray flux in stars with planets is associated to massive close-in planetary companions, we suggest that such an aspect, at least in the context of CaII emission flux, is rather an effect of stellar sample selection. We have also studied the behavior of the CaII emission as a function of orbital parameters such as orbital period and eccentricity, and no clear trend was found, reinforcing our present suggestion that enhanced chromospheric activity in stars with planets is an intrinsic stellar phenomenon

Fundamentação cinética da estatística não gaussiana : efeitos em politrópicas

Considering a non-relativistic ideal gas, the standard foundations of kinetic theory are investigated in the context of non-gaussian statistical mechanics introduced by Kaniadakis. The new formalism is based on the generalization of the Boltzmann H-theorem and the deduction of Maxwells statistical distribution. The calculated power law distribution is parameterized through a parameter measuring the degree of non-gaussianity. In the limit = 0, the theory of gaussian Maxwell-Boltzmann distribution is recovered. Two physical applications of the non-gaussian effects have been considered. The first one, the -Doppler broadening of spectral lines from an excited gas is obtained from analytical expressions. The second one, a mathematical relationship between the entropic index and the stellar polytropic index is shown by using the thermodynamic formulation for self-gravitational systems