Para conocer la distribución de velocidades de rotación de las estrellas Be, se utiliza la suposición estándar que los ejes de rotación se distribuyen uniformemente. En este trabajo usamos ángulos  proyectados de polarimetría de casi 500 estrellas Be y realizamos un análisis estadístico. Además llevamos a cabo una simulación Monte Carlo cuyos resultados explican las características observadas de los ángulos, confirmando que provienen de una distribución uniforme, pero que sus ángulos proyectados poseen una estructura bimodal. Esta última característica podría cambiar las conclusiones acerca de la distribución de velocidades de rotación de estas estrellas.In order to know the distribution of rotational velocities of Be stars, a standard assumption is used, namely that the rotational axes are uniformed distributed. We use here polarimetric projected angles data from almost 500 Be stars and we make a statistical analysis. Furthermore we perform a Monte-Carlo simulation that explains the observed features of the angles, confirming that it is indeed uniform distributed, but the projected angles possess a bimodal structure. This last features may change the conclusion about the rotational velocity distribution of these stars.Fil: Curé, M.. Universidad de Valparaíso; ChileFil: Christen, A.. Universidad de Chile; ChileFil: Rivinius, Th.. European Southern Observatory; ChileFil: Rial, Diego Fernando. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "luis A. Santaló"; Argentina. Universidad de Valparaíso; Chil

Christen, A.

Curé, M.

Rial, Diego Fernando

Rivinius, Th.

Revista Mexicana de Astronomía y Astrofísica

CONICET Digital

© 2010: Instituto de Astronomía, UNAM - The Interferometric View on Hot StarsEd. Th. Rivinius & M. CureRevMexAA (Serie de Conferencias), 38, 123–124 (2010)SPATIAL DISTRIBUTION OF STELLAR ROTATIONAL AXESFROM BE STARSM. Cure´,1 A. Christen,2 Th. Rivinius,3 and D. F. Rial1,4RESUMENPara conocer la distribucio´n de velocidades de rotacio´n de las estrellas Be, se utiliza la suposicio´n esta´ndar quelos ejes de rotacio´n se distribuyen uniformemente. En este trabajo usamos a´ngulos proyectados de polarimetr´ıade casi 500 estrellas Be y realizamos un ana´lisis estad´ıstico. Adema´s llevamos a cabo una simulacio´n Monte-Carlo cuyos resultados explican las caracter´ısticas observadas de los a´ngulos, confirmando que provienen deuna distribucio´n uniforme, pero que sus a´ngulos proyectados poseen una estructura bimodal. Esta u´ltimacaracter´ıstica podr´ıa cambiar las conclusiones acerca de la distribucio´n de velocidades de rotacio´n de estasestrellas.ABSTRACTIn order to know the distribution of rotational velocities of Be stars, a standard assumption is used, namelythat the rotational axes are uniformed distributed. We use here polarimetric projected angles data from almost500 Be stars and we make a statistical analysis. Furthermore we perform a Monte-Carlo simulation thatexplains the observed features of the angles, confirming that it is indeed uniform distributed, but the projectedangles possess a bimodal structure. This last features may change the conclusion about the rotational velocitydistribution of these stars.Key Words: methods: numerical — methods: statistical — stars: emission-line, Be — stars: rotation1. INTRODUCTIONThe theory of radiation driven winds from hotstars with the inclusion of the centrifugal term dueto the stellar rotation has been revisited by Cure´(2004). He found that for the cases when the ro-tational speed is∼> 0.7Ω(= vrot/vbreakup) there is anew wind solution, that is denser and faster than thestandard solution (Friend & Abbott 1986). Thusis important to know about the mean rotationalspeed of Be stars. The process to calculate thismean rotational speed is from the measurement ofvrot sin(i), where i is the inclination angle of thestar and vrot their rotational speed. After compilinga relevant sample, follows the standard assumptionthat the stellar axes are uniform distributed (see e.g.,Chauville et al. 2001) and the numerical algorithmdescribed by Lucy (1974) is used.In this work we have used the 493 intrinsic po-larization angles (p) from a sample of 497 Be starscompiled by Yudin (2001). Four stars with zero mea-sured polarization have been excluded.1Departamento de F´ısica y Astronomı´a, Universidad deValpara´ıso, Chile.2Departamento de Ingenier´ıa Matematica, Universidad deChile, Santiago, Chile.3European Southern Observatory, Vitacura, Santiago,Chile.4CONICET, Argentina.Interferometric studies so far confirm our first as-sumption that the polarization angle in Be stars isperpendicular to the disk. Our second assumption isthat the disk is plane-parallell to the stellar equator,is supported by the stability of the polarization an-gle in Be stars, as a non-aligned disk would precess.Consequently, the intrinsic polarization angle of a Bestar gives the projection of the stellar rotational axison the celestial sphere.2. STATISTICAL ANALYSIS OF THE SAMPLEAs a first step we made an histogram of the datawith different bins concluding that the angles p areuniform distributed. Furthermore, from the the-ory of an uniform distribution function, the mean(µ) and standard deviation (σ) are µ = 90 andσ =√1802/12 = 51.96, while from Yudin’s databasewe get µ = 86.6 and σ = 52.5 confirming our previ-ous conclusion. But considering that statistical tech-niques such as histograms or the statistical parame-ters of an uniform distribution are simple techniques,we decide to apply more advances ones, such as theKolmogorov-Smirnov (KS) test. We tested thereforeif our sample do really comes from an uniform distri-bution function and the KS test rejected our hypoth-esis, i.e., the sample do not comes from an uniformdistribution. The next statistical tool we used was123© 2010: Instituto de Astronomía, UNAM - The Interferometric View on Hot StarsEd. Th. Rivinius & M. Cure124 CURE´ ET AL.0 50 100 1500.00300.00350.00400.00450.00500.00550.0060Fig. 1. Kernel density estimation from the Yudin’s Sam-ple of 493 Be Stars.0 50 100 1500.0030.0040.0050.006Fig. 2. Kernel density estimation from a MonteCarlosimulation with 500 Stars.the Kernel Density Estimation (hereafter KDE, see,e.g., Royer et al. 2007, and references therein).This kernel density estimation shows clearly thatis not an uniform distribution, but has a bi-modalstructure. Abt et al.(2001) measured the projectedrotational velocities of 1092 northern B stars listedin the Bright Star Catalogue and calibrated themagainst the Slettebak et al.(1975) system. Theyfound a bi-modal distribution of rotational veloci-ties (see their Figure 1) that strongly resembles ourFigure 1. This similarity, in both figures, opens thequestion if the procedure for obtain the mean ro-tational speed from a sample of vrot sin(i) do reallytake in account the bi-modal structure of the pro-jected angles.Thus, in order to answer this question, we per-formed a Monte-Carlo simulation in order to obtaina sample of projected angles.3. MONTE-CARLO SIMULATIONSThe following assumptions for our simulationwere used: (i) the orientation of rotational axes com-ing from an uniform distribution; (ii) the location ofstars is uniform distributed over the galaxy; (iii) thegalaxy structure is a flat disk, modelled with x, y, zCartesian coordinates with the ratio x, y, z = 10 :10 : 1.Figure 2 and Figure 3 show KDE from Monte-Carlo simulations with 500 and 10000 stars respec-0 50 100 1500.0020.0030.0040.0050.006Fig. 3. Kernel density estimation from a MonteCarlosimulation with 10000 Stars.tively. Both figures are similar to the KDE fromYudin’s sample.From these simulations we can conclude that thedistribution of projected polarization angles, p, is bi-modal. We have also calculated KDEs for the caseswhere there exists a privileged direction of rotationalaxes, but the results of the corresponding KDE isdifferent from the observed (Yudin’s) one.Other simulations that we performed were car-ried out assuming that some percentage of the axesare from an uniform distribution and the rest arefrom a particular preferred direction. The resultsof these calculations showed that the percentage ofstars coming from a uniform distribution should beat least of 85% in order that the obtainned KDE issimilar to the observed one.4. CONCLUSIONSWe have used some advanced statistical tech-niques in order to estimate how the rotational axesfrom Be stars are distributed. From our numeri-cal simulations we have confirmed that indeed theseaxes are uniform distributed, but the correspondingdistribution of the projected angles has a bi-modalstructure due to the shape of the galaxy. These re-sults opens the question if the procedure of obtain-ning the mean rotational speed of Be Stars do reallytake into account the projected distribution.REFERENCESAbt, H. A., Levato, H., & Grosso, M. 2002, ApJ, 573,359Chauville, J., Zorec, J., Ballereau, D., Morrell, N.,Cidale, L., & Garc´ıa, A. 2001, A&A, 378, 861Cure´, M. 2004, ApJ, 619, 929Friend, D., & Abbott, D. C. 1986, ApJ, 311, 701Lucy, L. B. 1974 AJ, 79, 745Royer, F., Zorec, J. & Go´mez, A. E. 2007 A&A, 463, 671Slettebak, A., Collins, G. W., II, Boyce, P. B., White,N. M., & Parkinson, T. D. 1975, ApJS, 29, 137Yudin, R. V. 2001, A&A, 368, 912

Spatial distribution of stellar rotational axes from Be stars

LAReferencia - Red Federada de Repositorios Institucionales de Publicaciones Científicas Latinoamericanas

© 2010: Instituto de Astronomía, UNAM - The Interferometric View on Hot StarsEd. Th. Rivinius & M. CureRevMexAA (Serie de Conferencias), 38, 123–124 (2010)SPATIAL DISTRIBUTION OF STELLAR ROTATIONAL AXESFROM BE STARSM. Curé,1 A. Christen,2 Th. Rivinius,3 and D. F. Rial1,4RESUMENPara conocer la distribución de velocidades de rotación de las estrellas Be, se utiliza la suposición estándar quelos ejes de rotación se distribuyen uniformemente. En este trabajo usamos ángulos proyectados de polarimetŕıade casi 500 estrellas Be y realizamos un análisis estad́ıstico. Además llevamos a cabo una simulación Monte-Carlo cuyos resultados explican las caracteŕısticas observadas de los ángulos, confirmando que provienen deuna distribución uniforme, pero que sus ángulos proyectados poseen una estructura bimodal. Esta últimacaracteŕıstica podŕıa cambiar las conclusiones acerca de la distribución de velocidades de rotación de estasestrellas.ABSTRACTIn order to know the distribution of rotational velocities of Be stars, a standard assumption is used, namelythat the rotational axes are uniformed distributed. We use here polarimetric projected angles data from almost500 Be stars and we make a statistical analysis. Furthermore we perform a Monte-Carlo simulation thatexplains the observed features of the angles, confirming that it is indeed uniform distributed, but the projectedangles possess a bimodal structure. This last features may change the conclusion about the rotational velocitydistribution of these stars.Key Words: methods: numerical — methods: statistical — stars: emission-line, Be — stars: rotation1. INTRODUCTIONThe theory of radiation driven winds from hotstars with the inclusion of the centrifugal term dueto the stellar rotation has been revisited by Curé(2004). He found that for the cases when the ro-tational speed is∼> 0.7Ω(= vrot/vbreakup) there is anew wind solution, that is denser and faster than thestandard solution (Friend & Abbott 1986). Thusis important to know about the mean rotationalspeed of Be stars. The process to calculate thismean rotational speed is from the measurement ofvrot sin(i), where i is the inclination angle of thestar and vrot their rotational speed. After compilinga relevant sample, follows the standard assumptionthat the stellar axes are uniform distributed (see e.g.,Chauville et al. 2001) and the numerical algorithmdescribed by Lucy (1974) is used.In this work we have used the 493 intrinsic po-larization angles (p) from a sample of 497 Be starscompiled by Yudin (2001). Four stars with zero mea-sured polarization have been excluded.1Departamento de F́ısica y Astronomı́a, Universidad deValparáıso, Chile.2Departamento de Ingenieŕıa Matematica, Universidad deChile, Santiago, Chile.3European Southern Observatory, Vitacura, Santiago,Chile.4CONICET, Argentina.Interferometric studies so far confirm our first as-sumption that the polarization angle in Be stars isperpendicular to the disk. Our second assumption isthat the disk is plane-parallell to the stellar equator,is supported by the stability of the polarization an-gle in Be stars, as a non-aligned disk would precess.Consequently, the intrinsic polarization angle of a Bestar gives the projection of the stellar rotational axison the celestial sphere.2. STATISTICAL ANALYSIS OF THE SAMPLEAs a first step we made an histogram of the datawith different bins concluding that the angles p areuniform distributed. Furthermore, from the the-ory of an uniform distribution function, the mean(µ) and standard deviation (σ) are µ = 90 andσ =√1802/12 = 51.96, while from Yudin’s databasewe get µ = 86.6 and σ = 52.5 confirming our previ-ous conclusion. But considering that statistical tech-niques such as histograms or the statistical parame-ters of an uniform distribution are simple techniques,we decide to apply more advances ones, such as theKolmogorov-Smirnov (KS) test. We tested thereforeif our sample do really comes from an uniform distri-bution function and the KS test rejected our hypoth-esis, i.e., the sample do not comes from an uniformdistribution. The next statistical tool we used was123© 2010: Instituto de Astronomía, UNAM - The Interferometric View on Hot StarsEd. Th. Rivinius & M. Cure124 CURÉ ET AL.0 50 100 1500.00300.00350.00400.00450.00500.00550.0060Fig. 1. Kernel density estimation from the Yudin’s Sam-ple of 493 Be Stars.0 50 100 1500.0030.0040.0050.006Fig. 2. Kernel density estimation from a MonteCarlosimulation with 500 Stars.the Kernel Density Estimation (hereafter KDE, see,e.g., Royer et al. 2007, and references therein).This kernel density estimation shows clearly thatis not an uniform distribution, but has a bi-modalstructure. Abt et al.(2001) measured the projectedrotational velocities of 1092 northern B stars listedin the Bright Star Catalogue and calibrated themagainst the Slettebak et al.(1975) system. Theyfound a bi-modal distribution of rotational veloci-ties (see their Figure 1) that strongly resembles ourFigure 1. This similarity, in both figures, opens thequestion if the procedure for obtain the mean ro-tational speed from a sample of vrot sin(i) do reallytake in account the bi-modal structure of the pro-jected angles.Thus, in order to answer this question, we per-formed a Monte-Carlo simulation in order to obtaina sample of projected angles.3. MONTE-CARLO SIMULATIONSThe following assumptions for our simulationwere used: (i) the orientation of rotational axes com-ing from an uniform distribution; (ii) the location ofstars is uniform distributed over the galaxy; (iii) thegalaxy structure is a flat disk, modelled with x, y, zCartesian coordinates with the ratio x, y, z = 10 :10 : 1.Figure 2 and Figure 3 show KDE from Monte-Carlo simulations with 500 and 10000 stars respec-0 50 100 1500.0020.0030.0040.0050.006Fig. 3. Kernel density estimation from a MonteCarlosimulation with 10000 Stars.tively. Both figures are similar to the KDE fromYudin’s sample.From these simulations we can conclude that thedistribution of projected polarization angles, p, is bi-modal. We have also calculated KDEs for the caseswhere there exists a privileged direction of rotationalaxes, but the results of the corresponding KDE isdifferent from the observed (Yudin’s) one.Other simulations that we performed were car-ried out assuming that some percentage of the axesare from an uniform distribution and the rest arefrom a particular preferred direction. The resultsof these calculations showed that the percentage ofstars coming from a uniform distribution should beat least of 85% in order that the obtainned KDE issimilar to the observed one.4. CONCLUSIONSWe have used some advanced statistical tech-niques in order to estimate how the rotational axesfrom Be stars are distributed. From our numeri-cal simulations we have confirmed that indeed theseaxes are uniform distributed, but the correspondingdistribution of the projected angles has a bi-modalstructure due to the shape of the galaxy. These re-sults opens the question if the procedure of obtain-ning the mean rotational speed of Be Stars do reallytake into account the projected distribution.REFERENCESAbt, H. A., Levato, H., & Grosso, M. 2002, ApJ, 573,359Chauville, J., Zorec, J., Ballereau, D., Morrell, N.,Cidale, L., & Garćıa, A. 2001, A&A, 378, 861Curé, M. 2004, ApJ, 619, 929Friend, D., & Abbott, D. C. 1986, ApJ, 311, 701Lucy, L. B. 1974 AJ, 79, 745Royer, F., Zorec, J. & Gómez, A. E. 2007 A&A, 463, 671Slettebak, A., Collins, G. W., II, Boyce, P. B., White,N. M., & Parkinson, T. D. 1975, ApJS, 29, 137Yudin, R. V. 2001, A&A, 368, 912

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Spatial distribution of stellar rotational axes from Be stars

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