ISSN: Title: Statistical Magnitude Analysis and Distance Determination of the Nearby Stars

Transcription

1 ISSN: Title: Statistical agnitude Analysis and Distance Determination of the Nearby Stars By ohamed Adel Sharaf Radwa ohamed Amin Said A. azen Inas K. Elzawawy

2 Greener Journal of Physical Sciences ISSN: ol. 3 (4), pp , ay 13. Research Article Statistical agnitude Analysis and Distance Determination of the Nearby Stars ohamed Adel Sharaf 1, Radwa ohamed Amin 1*, Said A. azen and Inas K. Elzawawy 1 1 Physics division, National research center, Cairo, Egypt. agnetic Semiconductor Lab. Physics Dept. Faculty of Science, Zagazig University, Zagazig, Egypt. *Corresponding Author s hassanselim@yahoo.com ABSTRACT Goals of this paper are: to provide some basic descriptive statistical parameters for the apparent and absolute magnitudes of the nearby stars of spectral type G5. Second, to establish the frequency functions Φ ( ) and Ψ(m) of the absolute and apparent magnitudes for these stars. Third, to compute the distance r of these stars as a system assuming that they scatter around a mean absolute magnitude in a Gaussian distribution. The accuracy of the numerical results is satisfactory. Keywords: Distance determinations, Spectral types, Frequency functions, Statistical magnitude analysis. INTRODUCTION G stars, of which the Sun is one, are yellow with temperatures around 5 to 6 K. Another example of the G class is Alpha Centauri A. The spectra of these stars are solar type spectra, with CaII lines extremely strong, neutral metals prominent, ions weaker, band (CH) strong and H lines are weaker [Robinson,1985]. As for the late spectral type stars, G class is less concentrated in the ilky Way than the early types[e.g. Scheffler and Els& & asser 1988], as an example, the number of the G type stars per square degree up to the limiting magnitude m pg 8. 5 is about.4, the corresponding number of the A type stars at the latitude 5 o. G5 stars of luminosity class, are denoted as G5 (main sequence ), their effective temperature about 56K, bolometric correction -.1 [ihalas and Binney,1981], absolute magnitude +5.1,mass: log( / Sun ) =.3,radius: log( R / RSun ) =. 3, luminosity : log( L / LSun ) =. 1 and mean density (g cm 3 ) : log ρ =. [Zombeck,199] In the present paper, we consider G5 stars and perform the following: First, to provide some basic descriptive statistics parameters for the apparent and absolute magnitudes of these stars. Second, to establish the frequency functions Φ ( ) and Ψ(m) of the absolute and apparent magnitudes for these stars. Third, to compute the distance r of these stars as a system assuming that they scatter around a mean absolute magnitude in a Gaussian distribution. The accuracy of the numerical results is satisfactory in that, the percentage error between r and the mean value is less than.5%

3 Greener Journal of Physical Sciences ISSN: ol. 3 (4), pp , ay 13. BASIC ATERIALS.1 Data The source of data is( /7 A) Near by stars, preliminary 3 rd. version (Gliese+1991) of (I/39) Hipparcos and Tycho Catalogues (ESA 1997) In Table I of Appendix A, the data for some G5 stars are listed. The columns of the table have the following meaning: Column 1: Name with the following acronyms: G1 Gliese: GJ Gliese& Jahreiss,AA&AS,3843(1979) Wo Woolley et al., Roy. Obs.Ann.No.5(197) NN newly added stars (number added at CDS) Column : mag (unit mag): Apparent magnitude Column 3: B- (unit mag): Color Column 4: U-B (unit mag): Color Column 5: plx (unit mas milli-second of arc): Parallax Column 6: (unit mag): Absolute visual magnitude Appendix A: Input data Table I: Data for some G5 near by stars Name mag B- U-B plx v mag mag mag mas mag Gl Wo Wo Gl Gl Gl NN Gl Wo Gl Wo Gl Gl Gl Wo Gl Gl Gl

4 Greener Journal of Physical Sciences ISSN: ol. 3 (4), pp , ay 13. Table 1 Continues NN NN NN Gl Gl Gl NN NN NN Gl Gl Gl NN Gl NN Gl Gl Gl Gl Gl NN Gl Gl Wo Gl Gl Gl Gl Wo Gl Gl Gl Gl Gl Gl Gl Gl GJ Gl Gl Wo

5 Greener Journal of Physical Sciences ISSN: ol. 3 (4), pp , ay 13.. The distance equation The assumptions upon which the distance equation was derived were [Sharaf et al. 3] (1) All the members in a given cosmic group are at the same distance, r parsecs. () The frequency function for the absolute magnitudes of the members is 1 ) / ( σ Φ = e (1) ( ) σ π That is the members scatter around a mean absolute magnitude in a Gaussian distribution with dispersion σ. (3) The mean apparent magnitude m of the members of the cosmic group and thelimiting apparent magnitude m 1 are related through the quantity α where α m m = 1 σ According to the above assumptions, the distance r and the frequency functions Ψ (m) of the apparent magnitude are given respectively by 1+ ( m1 σ y)/5 r = 1 () 1 ( m+ 5 5log r ) / σ Ψ ( m) = e (3) σ π Where y is the solution of the transcendental equation: Λ ( y) = y + e y / π 1 erf ( + y ) 1 α =, (4) And, erf (z) is the error function defined by the integral We call Equation (4) the distance equation. z e t erf(z) = dt (5) π The relation between and σ is given by[ Bok, 1937 ] = σ (6) Where is the mean absolute magnitude of the stellar group

6 Greener Journal of Physical Sciences ISSN: ol. 3 (4), pp , ay Empirical relation between versus ( B ) Table II: Empirical relation between versus ( B ) (B ) ( B ) An empirical relation between the absolute visual magnitude versus ( B ) is given in ihalas and Binney,book [1981] in a tabular form (Table II of Appendix A). From this empirical relation, we deduce an interpolating polynomial between ( B ) and in the form: Where q's are listed in Table III of Appendix A. q = 1 ( B ) = (7) Table III: The q's coefficients of Equation(7) q q

7 Greener Journal of Physical Sciences ISSN: ol. 3 (4), pp , ay 13. NUERICAL APPLICATION 3.1 Procedure 1-Compute the parallax p in seconds of arc from p = plx /1 -Compute for each star in the group ( B ) from its given value of using Equation ((7) 3-Compute the absorption A [( B ) ( B ] = 3 ) 4- Compute the average A of A A for each star of the group from 5-Correct the apparent magnitudes for absorption from mag=mag- A 6-Compute the average value of the absolute magnitudes 7-Compute the average value r of the individual distances r r = 1/ p ) ( 8-Compute the median r median of the individual distances 9-Select start and end values of σ,say σ s and σ e respectively. [In the present application, σ s =.5 andσ e =] 1-Compute the optimum value of σ as follows a-for σ = σ s (.1) σ e perform the following calculations = σ From the values of mag, and σ compute the distance r (in parsec) of the stellar Group as obtained from the statistical method using the algorithm of Sharaf et al. (3). Compute the percentage error f in the mean from f = ( r r) / r 1 Compute the percentage error g in the median from g ( r r median ) / r 1 = median b- Compute the value σ 1 that gives the minimum of the f's values c- Compute the value σ that gives the minimum of the g's values 136

8 Greener Journal of Physical Sciences ISSN: ol. 3 (4), pp , ay 13. d- Compute the optimum value of σ as σ =( σ 1+ σ )/. 11- Compute some basic descriptive statistics parameters for the apparent and absolute magnitudes of the stellar group. 1-Compute the measures of the central tendency for the individual distances of the stellar group, of these measures are: The ean distance r (in parsec) of r The ode of r The edian of r The Harmonic ean of r The Geometric ean of r 13-From the optimum values of and σ compute the frequency functions Φ ( ) and Ψ(m) of the absolute and apparent magnitudes [Equations (1) and (3)] of the group 3. Numerical Results Numerical results are listed in Appendix B covering the following points: Statistics of the apparent and absolute magnitudes of the G5 group Graphical representations: a- Histograms for the apparent and absolute magnitudes of the stellar group b- Graphical representations for the percentage errors for the mean and median Optimum choice of σ Distance analysis The frequency functions Φ( ) and Ψ(m) the G5 group of the absolute and apparent magnitudes for CONCLUSION In concluding, we draw the attention that the aim of the present paper is three folds: First, to provide some basic descriptive statistical parameters for the apparent and absolute magnitudes of the nearby stars of spectral type G5 stars. Second, to establish the frequency functions Φ ( ) and Ψ(m) of the absolute and apparent magnitudes for these stars. Third, to compute the distance r of these stars as a system assuming that they scatter around a mean absolute magnitude in a Gaussian distribution. The accuracy of the numerical results is satisfactory in that, the percentage error between r and the mean value less than.5%. This percentage error indicates the applicability of the used method for distance determination in the ranges of the considered stellar group. The analysis of the present paper will be extended for the stars of early spectral types and will constitute the task to which we shall consider soon

9 Greener Journal of Physical Sciences ISSN: ol. 3 (4), pp , ay 13. REFERENCES Bok, B.J. (1937). The Distribution of Stars in Space, Chicago University Press, Chicago. ihalas,d. and Binney,J. (1981) Galactic Astronomy, Second Edition, Freeman and Company, USA. Robinson, R., (1985). The Cosmological Distance Ladder, W. H. Freeman Scheffler, H. and Els& & asser,h. (1988), Physics of the Galaxy and Interstellar atter, Translated by Armstrong, A. H., Springer-erlag, Berlin Sharaf,.A., Issa, I.A. and Saad, A. (3), A ethod for the Determination of Cosmic Distance New Astronomy,8:15-1. Zombeck,.,. (199) Handbook of Space Astronomy &Astrophysics, second edition, Cambridge University Press, Cambridge, New York. Appendix B: Output data B.1: Statistics of the apparent and absolute magnitudes of the G5 stars The average: For Apparent agnitudes = For Absolute agnitudes= The median (central value): For Apparent agnitudes = 6.68 For Absolute agnitudes= 5.17 The mode: For Apparent agnitudes = {5.86,6.35,6.45,6.48,6.87,6.97} For Absolute agnitudes= {4.9,4.94,5.11,5.14,5.} The geometric mean: For Apparent agnitudes = 6.7 For Absolute agnitudes= The harmonic mean: For Apparent agnitudes = For Absolute agnitudes= The root mean square: For Apparent agnitudes = For Absolute agnitudes= The Quartiles: For Apparent agnitudes = {6.335,6.68,7.38} For Absolute agnitudes= {4.94,5.17,5.36} The range: For Apparent agnitudes = 4.34 For Absolute agnitudes=.59 The unbiased estimate of the variance: For Apparent agnitudes =.7666 For Absolute agnitudes=.1581 The maximum likelihood estimate of the variance: For Apparent agnitudes = For Absolute agnitudes=

10 Greener Journal of Physical Sciences ISSN: ol. 3 (4), pp , ay 13. The unbiased estimate of the variance of the sample mean: For Apparent agnitudes =.194 For Absolute agnitudes= The unbiased estimate of the standard deviation: For Apparent agnitudes = For Absolute agnitudes= The maximum likelihood estimate of the standard deviation : For Apparent agnitudes = For Absolute agnitudes= The unbiased estimate of the standard error(standard deviation)of the sample mean: For Apparent agnitudes = For Absolute agnitudes= The mean absolute deviation: For Apparent agnitudes = For Absolute agnitudes=.6456 The median absolute deviation: For Apparent agnitudes =.5 For Absolute agnitudes=. The inter-quartile range: For Apparent agnitudes = 1.45 For Absolute agnitudes=.4 The Quartiles: For Apparent agnitudes = {6.335,6.68,7.38} For Absolute agnitudes= {4.94,5.17,5.36} The coefficient of skewness: For Apparent agnitudes = For Absolute agnitudes= Pearson's first coefficient of skewness: For Apparent agnitudes = {3.131, ,1.9846,.99939, , } For Absolute agnitudes= {.117,1.864,.5784,.35197,-.1564} Pearson's second coefficient of skewness: For Apparent agnitudes = For Absolute agnitudes=.1574 The quartile coefficient of skewness: For Apparent agnitudes = For Absolute agnitudes= The kurtosis coefficient: For Apparent agnitudes = For Absolute agnitudes= The kurtosis excess: For Apparent agnitudes =.5379 For Absolute agnitudes=

11 Greener Journal of Physical Sciences ISSN: ol. 3 (4), pp , ay 13. The confidence interval of the population mean based on the normal distribution: For Apparent agnitudes = { , } For Absolute agnitudes= {5.8341,5.8993} The confidence interval for the population variance based on the chi-square distribution : For Apparent agnitudes = { ,1.5444} For Absolute agnitudes= {.11768,.3894} B. : Graphical Representations The Histograms of the Apparent and Absolute agnitudes of the stellar Group For Apparent agnitudes ForAbsolute agnitudes B.3: Optimum Choice of σ Graphical representations of the percentage errors for the ean and edian Precentage error for the mean against s s Precentage error for the median against s s The minimum value of the percentage error in the ean is = occurs at σ =.98 The minimum value of the percentage error in the edian is = occurs at σ = 1.4 The optimum value of σ is = 1.1 B. 4: Distance Analysis The value of m 1 is = The value of m is = The value of σ is = 1.1 The value of is = The value of α is = The value of y is =

12 Greener Journal of Physical Sciences ISSN: ol. 3 (4), pp , ay The accuracy of the computed value of y is = The distance r of the stellar group as obtained from the statistical method is = parsec The edian of the individual distances is =.8333 parsec The ode of the individual distances is = {3.558,5.,3.33} parsec The geometric mean of the individual distances is =.65 parsec The harmonic mean of the individual distances is = parsec The mean value of the individual distances is = parsec The percentage error between r and r is = B. 5: Frequency functions of the absolute and apparent magnitudes for The G5stars ( ) Φ ( ) = e Ψ ( m) = e ( m ) 141