SEMI-ANALYTICAL FORMULAS FOR THE FUNDAMENTAL PARAMETERS OF GALACTIC EARLY B SUPERGIANTS

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1 Serb. Astron. J , UDC DOI: /SAJ Z Original scientific paper SEI-ANALYTICAL FOULAS FO THE FUNDAENTAL PAAETES OF GALACTIC EALY B SUPEGIANTS L. Zaninetti Dipartimento di Fisica Generale, Via Pietro Giuria Torino, Italy E mail: zaninett@ph.unito.it eceived: ay 21, 2009; Accepted: August 7, 2009 SUAY: The publication of new tables of calibration of some fundamental parameters of Galactic B0-B5 supergiants in the two classes I a I b allows to particularize the eight parameters conjecture that model five fundamental parameters. The numerical expressions for visual magnitude, radius, mass, luminosity surface gravity are derived for supergiants in the range of temperature between K K. The availability of accurate tables of calibration allows us to estimate the efficiency of the derived formulas in reproducing the observed values. The average efficiency of the new formulas, expressed in percent, is 94 for the visual magnitude, 81 for the mass, 96 for the radius, 99 for the logarithm of the luminosity 97 for the logarithm of the surface gravity. Key words. Stars: fundamental parameters Hertzsprung-ussell H C- Stars: general 1. INTODUCTION A recent study on the supergiants of spectral type B0-B5, see Searle et al. 2008, provided a fine grid of the fundamental parameters such as the temperature, luminosity, radius, surface gravity mass. In this paper we first review the four fundamental parameters as modeled by the eight parameters conjecture we add to the list the surface gravity, see Section 2. Section 3 contains the numerical expressions for five fundamental parameters as well as the efficiency of such an evaluation. 2. THE FUNDAENTAL FOULAS We briefly review the formulas that characterize the visual magnitude, V, the mass,, the radius,, the luminosity, L, of the stars for each K class as function of the intrinsic, unreddened color index B V 0, see Zaninetti 2008 for details. The first equation models the visual magnitude, V V = 2.5 a L 2.5 b L a T 2.5 b L b T log 10 B V 0 K BV K BC + 10 log 10 + B V 0 K BV T BC [B V 0 K BV ] + T bol,. 1 BV The second equation connects the mass of the star,, with B V 0 69

2 L. ZANINETTI log 10 = a T + b T ln ln 10 1, 2 B V 0 K BV where is the solar mass. The third equation relates the radius, with B V 0 log 10 = 1/2a L + 1/2b L a T + 2 ln T ln /2b L b T ln ln 10 1 B V 0 K BV 2 ln ln 10 1, 3 B V 0 K BV where is the solar radius. The fourth equation connects the luminosity of a star, L, with B V 0 log 10 L L = a L + b L a T 1 +b L b T ln, 4 B V 0 K BV ln 10 where L is the solar luminosity. The eight numerical parameters compared above are reported in Table 1 along with the physical or empirical formula that relates them. Table 1. Synoptic table of the eight coefficients. BC sts for the bolometric correction Coefficient adopted relationship K BV, B V 0 = K BV + /T K BC,T BC BC = T BC T 10 log T + K BC a L, b L logl/l = a L + b L log/ a T, b T log/ = a T + b T logt/t A fifth fundamental parameter is the surface gravity, g, that is defined as g = G 2, 5 where is the mass of the body, its radius G is the Newtonian gravitational constant G = 11 m ohr Taylor kgs 2 By adopting = m = kg, see Cox 2000, we obtain the following expression for the logarithm of the surface gravity log 10 g[cm/s 2 ] = lne 2.3a T 2.3a L 2.3b L a T bt bl bt+4. 6 B V 0 K BV 3. APPLICATION TO THE SUPEGIANTS The eight parameters conjecture may represent an acceptable fit of five fundamental parameters of the stars once the calibration data are available in the considered K class, in our case B0-B5 supergiants. The Table 5 in Searle et al provides the calibration of log 10 L L, log 10, log 10 g[cm/s 2 ] as function of the temperature T in the range < T < Table 2 reports the eight parameters as well as the source where the calibrated data are taken from. The fundamental parameters of the stars are parametrized according to the K class to which they belong, to the intrinsic, unreddened color index B V 0 or the temperature as derived, for example, from spectroscopic arguments. The conversion between temperature B V 0 is obtained through the following two formulas Table 2. The adopted coefficients for B0-B5 supergiants. Coefficient Ia Ib source of data K BV Table 15.7 in Cox 2000 [K] Table 15.7 in Cox 2000 K BC Table 15.7 in Cox 2000 T BC [K] Table 15.7 in Cox 2000 a L Table 5 in Searle et al b L Table 5 in Searle et al a T Table 5 in Searle et al b T Table 5 in Searle et al

3 GALACTIC EALY B SUPEGIANTS B V 0 = T K < T < K, T = B V < B V 0 < From a numerical point of view the visual magnitude, V, is given by V = ln B V B V 0 9 supergiants Ia 0.25 < B V 0 < 0.14, = ln B V supergiants Ib 0.25 < B V 0 < Fig. 2. shows the theoretically determined mass,, for the two classes considered here, as well as the observational data extracted from Table 4 of Searle et al V = ln B V B V 0 10 supergiants Ib 0.25 < B V 0 < Fig. 1. shows the theoretical visual magnitude, V, for the two classes considered here, as well as the observational data extracted from Table 3 of Searle et al Fig. 2. Theoretical mass,, versus B V 0 for Galactic early B supergiants: Ia solid line Ib dotted line. The observed values of mass extracted from Table 4 of Searle et al are also shown: Ia full dots Ib empty diamonds. The radius,, is given by expression Fig. 1. Theoretical visual magnitude, V, versus B V 0 for Galactic early B supergiants: Ia solid line Ib dotted line. The observed values of magnitude extracted from Table 3 of Searle et al are also shown: Ia full dots Ib empty diamonds. The mass can be expessed as = ln B V supergiants Ia 0.25 < B V 0 < 0.14, = ln B V supergiants Ia 0.25 < B V 0 < 0.14, = ln B V supergiants Ib 0.25 < B V 0 < Fig. 3. shows the theoretical radius,, for the two classes considered here, as well as the observational data extracted from Table 4 of Searle et al

4 L. ZANINETTI Fig. 3. Theoretical radius,, versus B V 0 for Galactic early B supergiants: Ia solid line Ib dotted line. The observed values of radius extracted from Table 4 of Searle et al are also shown Ia full dots Ib empty diamonds. The logarithm of the luminosity, log 10 L L, is given by log 10 L L = ln 4012 B V supergiants Ia 0.25 < B V 0 < 0.14, log 10 L L = ln 4012 B V supergiants Ib 0.25 < B V 0 < Fig. 4. shows the theoretically determined logarithm of the luminosity, log 10 L L, for the two classes considered here, as well as the observational data extracted from Table 4 of Searle et al Fig. 4. Theoretical logarithm of the luminosity, log 10 L L, versus B V 0 for Galactic early B supergiants: Ia solid line Ib dotted line. The observed values of luminosity extracted from Table 4 of Searle et al.2008 are also shown: Ia full dots Ib empty diamonds. The logarithm of the surface gravity, log 10 g, is given by log 10 g[cm/s 2 ] = ln B V ln B V supergiants Ia when 0.25 < B V 0 < 0.14, log 10 g[cm/s 2 ] = ln B V ln B V supergiants Ib when 0.25 < B V 0 < Table 3. Efficiency in deriving the fundamental parameters for B0-B5 supergiants Ia Ib Fundamental Parameter ɛ min % ɛ % ɛ max % ɛ min % ɛ % ɛ max % Visual magnitude ass adius Logarithm Luminosity Logarithm surface gravity

5 GALACTIC EALY B SUPEGIANTS Fig. 5. shows the theoretically determined logarithm of the surface gravity for the two classes considered here, as well as the observational points extracted from Table 4 of Searle et al Fig. 5. Theoretical logarithm of the surface gravity, log 10 g[cm/s 2 ], versus B V 0 for Galactic early B supergiants: Ia solid line Ib dotted line. The observed values of surface gravity extracted from Table 4 of Searle et al are also shown: Ia full dots Ib empty diamonds. From the practical point of view, ɛ, a measure of the reliability of our results expressed in percents can also be introduced: ɛ = 1 F obs F num F obs 100, 19 where F obs is one of the fundamental parameters as given by the astronomical observations, F num the analogous fundamental parameter as given by our numerical relationships. The minimum, average maximum efficiency in reproducing the observed fundamental parameters respectively ɛ min, ɛ ɛ max, as given by formula 19, are reported in Table CONCLUSIONS The eight parameters conjecture which allows to model five fundamental parameters of the stars is connected with the availability of calibration data on mass luminosity as a function of the temperature or the intrinsic, unreddened color index B V 0. Tables 3 4 in Searle et al offer both the splitting of early supergiants in two classes a good coverage of the fundamental parameters in the range K < T < K. The five fundamental parameters analyzed here can be presented as functions of B V 0 Table 3 reports the efficiency of the model implemented here. As an example, the two relationships pertaining to the visual magnitude have an average accuracy of 0.34 mag in Ia 0.5 mag in Ib. This paper also presents the logarithm of the surface gravity, log 10 g[cm/s 2 ], as a function B V 0, see formulas 6, 18, 19 Fig. 5. This allows to cast doubt to the fits of T that contain log 10 g[cm/s 2 ] as a parameter, see for example Sekiguchi Fukugita 2000, Kovtyukh et al. 2000, because the reverse is true: log 10 g is a function of T or its observational counterpart B V 0. Acknowledgements I thank Lorenzo Ducci who drew my attention to Table 5 in Searle et al EFEENCES Cox, A. N.: 2000, Allen s astrophysical quantities New York: Springer. Kovtyukh, V. V., Soubiran, C., Belik, S. I., et al.: 2008, Kinematics Physics of Celestial Bodies, 24, 171. ohr, P. J. Taylor, B. N.: 2005, eviews of odern Physics, 77, 1. Searle, S. C., Prinja,. K., assa, D. yans,.: 2008, Astron. Astrophys., 481, 777. Sekiguchi,. Fukugita,.: 2000, Astron. J., 120, Zaninetti, L.: 2008, Serb. Astron. J., 177,

6 L. ZANINETTI POLU-ANALITIQKE FOULE ZA OSNOVNE PAAETE GALAKTIQKIH ANIH B SUPE INOVA L. Zaninetti Dipartimento di Fisica Generale, Via Pietro Giuria Torino, Italy E mail: zaninett@ph.unito.it UDK Originalni nauqni rad Daju se nove vrednosti osam kalibrisanih koeficijenata pomo u kojih se izvode relacije za apsolutnu vizuelnu magnitudu, radijus, masu, luminoznost i povrxinsku gravitaciju zvezda u funkciji efektivne temperature. Numeriqki izrazi ovih parametara se odnose na galaktiqke B0 B5 super inove klase sjaja I a, I b i temperature između K i K. Taqnost izvedenih formula u odnosu na posmatrane vrednosti parametara, u procentima je 94 za magnitudu, 81 za masu, 96 za radijus, 99 za logaritam sjaja i 97 za logaritam povrxinske gravitacije. 74