Statistical Magnitude Analysis and Distance Determination of the Nearby F8V Stars

Transcription

1 Enginring, Tchnology & Applid Scinc Rsarch ol. 4, No. 4, 4, Statistical Magnitud Analysis and Distanc Dtrmination of th Narby F8 Stars Hany R. Dwidar Astronomy, Mtorology and Spac Scinc Dpt. Faculty of Scinc, Cairo Univrsity Giza, Egypt Adl Sharaf Dpartmnt of Astronomy Faculty of Scinc, King Abdul Aziz Univrsity Jddah, Saudi Arabia Abstract Th prsnt papr is of thr folds. First, to provid som basic dscriptiv statistics paramtrs for th apparnt and absolut magnituds of th narby stars of spctral typ F8 stars. Scond, to stablish th frquncy functions (M ) and (m) of th absolut and apparnt magnituds for ths stars. Third, to comput th distanc r of ths stars as a systm assuming that thy scattr around a man absolut magnitud in a Gaussian distribution. Th accuracy of th numrical rsults is satisfactory thus, th prcntag rror btwn r and th man valu is lss than.7%. Kywords distanc dtrminations; spctral typs; frquncy functions; statistical magnitud analysis I. INTRODUCTION An F-typ main squnc star (F ), commonly calld a yllow-whit dwarf, is a main squnc (hydrogn-burning) star of spctral typ F and luminosity class. Ths stars hav from. to.4 tims th mass of th Sun and surfac tmpraturs btwn 6, and 7,6 K []. A star of spctral typ F has a spctrum that shows strong absorption lins of ionizd calcium, mor prominnt than th hydrogn lins. Modratly strong lins du to iron and othr havir lmnts ar also vidnt. Th lins of nutral atoms for th stars of spctral typ F ar wak compard to thos of ionizd atoms. Th hydrogn lins ar strongr, attaining thir maximum intnsitis in A-typ stars. Thraftr, ths absorption lins gradually fad as th hydrogn bcoms ionizd []. Main squnc F stars, of which Procyon is an xampl, hav a surfac tmpratur of 5,8 to 6,9 C, a mass of. to.6 solar masss, and a luminosity of to 6 tims that of th Sun. Rlativly narby, lat-typ mmbrs of this catgory ar gnrally includd in th list of targt stars for xtra solar plant sarchs and targtd NASA SETI programs. F-typ supr giants, of which Canopus and Polaris (th Pol Star) ar xampls, hav a mass up to solar masss and luminosity up to 3, tims that of th Sun. Th importanc of th distanc of F8 stars drivs from that if its moving stllar vrtx and its distanc of ach mmbr ar known, thn on can asily dtrmin th vlocity of th clustr, th position of its cntr [3, 4] and also th distribution of th clustr s mmbrs about this cntr [5, 6]. Modrn obsrvational astronomy has bn charactrizd by an normous growth in acquisition stimulatd by th advnt of nw tchnologis in tlscops, dtctors and computations. This nw astronomical data giv ris to innumrabl statistical problms [7]. In th prsnt papr w considr F8 stars and prform th following: providing som basic dscriptiv statistics paramtrs for th apparnt and absolut magnituds of ths stars stablishing th frquncy functions ( M ) and ( m) of th absolut and apparnt magnituds for ths stars computing th distanc r of ths stars as a systm assuming that thy scattr around a man absolut magnitud in a Gaussian distribution. Th accuracy of th numrical rsults is satisfactory thus, th prcntag rror btwn r and th man valu is lss than.7%. II. BASIC MATERIALS A. Data Th sourc of data is (/7 A) narby stars [8] of (I/39) Hipparcos and Tycho Catalogus [9, ]. In Tabl I of Appndix A, th data for som F8 stars ar listd. Th columns of th tabl hav th following maning: Column : Nam with th following acronyms : o G []. o GJ []. o Wo [3]. o NN nwly addd stars (numbr addd at CDS) Column : mag (unit mag): Apparnt magnitud Column 3: B-(unit mag): Color Column 4: U-B(unit mag): Color Column 5: plx (unit mas milli-scond of arc): Parallax Column 6: M (unit mag): Absolut visual magnitud B. Th distanc quation Th following assumptions will b usd to driv th distanc quation [4]: Dwidar and Sharaf: Statistical Magnitud Analysis and Distanc Dtrmination of th Narby F8 Stars

2 Enginring, Tchnology & Applid Scinc Rsarch ol. 4, No. 4, 4, All th mmbrs in a givn cosmic group ar at th sam distanc, r parscs. Th frquncy function for th absolut magnituds of th mmbrs of th cosmic group is ( MM ) / ( M) () That is th mmbrs scattr around a man absolut magnitud M in a Gaussian distribution with disprsion. Th man apparnt magnitud m of th mmbrs of th cosmic group and th limiting apparnt magnitud m ar rlatd through th quantity whr m m According to th abov assumptions, th distanc r and th frquncy functions (m) of th apparnt magnitud ar givn rspctivly by: ( mm y)/5 r () ( m55log rm ) / ( m) (3) whr y is th solution of th transcndntal quation: y / y ( y) y rf( ), (4) and, rf (z) is th rror function dfind by th intgral z t rf(z) dt, (5) w call (4) th distanc quation. Th rlation btwn M and is givn by [3] M M.38 (6) whr M is th man absolut magnituds of th stllar group. C. Empirical rlation of M vrsus ( B ) An mpirical rlation btwn th absolut visual magnitud M and ( B ) is givn in [4] in a tabular form (Tabl II of Appndix A). From this mpirical rlation on can dduc an intrpolating polynomial btwn ( B ) and M in th form [5]: ( B) q M (7) whr q's ar listd in Tabl III of Appndix A. III. NUMERICAL APPLICATION A. Procdur. comput th parallax p in scond of arc from p plx /. comput for ach star of th group ( B ) from its givn valu of M using (7) 3. comput th absorption A for ach star of th group from: A 3 ( B) ( B ) 4. comput th avrag A of A 5. corrct th apparnt magnituds for absorption from mag=mag- 6. comput th avrag valu M of th absolut magnituds M 7. comput th avrag valu r of th individual distancs r ( r / p ) 8. comput th mdian r mdian of th individual distancs 9. slct start and nd valus of,say and s rspctivly. In th prsnt application,.5 and s. comput th optimum valu of as follows: a. For = s (.) prform th following calculations: M M.38σ comput th distanc r (in parsc) of th stllar group from th valus of mag, M and as obtaind from th statistical mthod using th algorithm of [4]. comput th prcntag rror f in th man from: f ( rr) / r comput th prcntag rror g in th mdian from g ( rrmdian ) / rmdian b. comput th valu that givs th minimum of th f's valus c. comput th valu that givs th minimum of th g's valus d. comput th optimum valu of as =( + )/.. comput som basic dscriptiv statistics paramtrs for th apparnt and absolut magnituds of th stllar group.. comput th masurs of th cntral tndncy for th individual distancs r / p of th stllar group, of ths masurs ar : Th Man distanc r (in parsc) of r Th Mod of r Th Mdian of r Th Harmonic Man of r Th Gomtric Man of r Dwidar and Sharaf: Statistical Magnitud Analysis and Distanc Dtrmination of th Narby F8 Stars

3 Enginring, Tchnology & Applid Scinc Rsarch ol. 4, No. 4, 4, From th optimum valus of M and comput th frquncy functions ( M ) and ( m) of th absolut and apparnt magnituds [() and (3)] of th group. B. Numrical Rsults Numrical rsults ar obtaind using MATHEMATICA 7 and ar listd in Appndix B throughout th following points: Statistics of th apparnt and absolut magnituds of th F8 group Graphical rprsntations: a. Histograms for th apparnt and absolut magnituds of th stllar group b. Graphical rprsntations for th prcntag rrors for th man and mdian Optimum choic of Distanc analysis Th frquncy functions ( M ) and ( m) of th absolut and apparnt magnituds for th F8 group I. CONCLUSION Th aim of th prsnt papr is thr fold. First, to provid som basic dscriptiv statistics paramtrs for th apparnt and absolut magnituds of th narby stars of spctral typ F8 stars. Scond, to stablish th frquncy functions ( M ) and ( m) of th absolut and apparnt magnituds for ths stars. Third, to comput th distanc r of ths stars as a systm assuming that thy scattr around a man absolut magnitud in a Gaussian distribution. Th accuracy of th numrical rsults is satisfactory thus, th prcntag rror btwn r and th man valu is lss than.67%. This prcntag rror indicats th applicability of th usd mthod for distanc dtrmination in th rangs of th considrd stllar group. REFERENCES [] G. J. Habts, J. W. Hintz, "Empirical bolomtric corrctions for th main-squnc", Astronomy and Astrophysics, ol. 46, pp , 98 [] R. O. Gray, R. F. Garrison, "Th Early F-Typ Stars: Rfind Classification, Confrontation with Strömgrn Photomtry, and th Effcts of Rotation", Astrophysical Journal, ol. 69, pp. 3-3, 989 [3] H. I. Abdl-Rahman, I. A. Issa, M. A. Sharaf, M. I. Nouh, A. Bakry, A. I. Osman, A. S. Saad, F. Y. Kamal, Essam, "Statistical Gaussian Distribution Function as a Distanc Indicator to Stllar Groups", Journal of Koran Astronomical Socity, ol. 4, No. 4, pp. 7-79, 9 [4] M. A. Sharaf, I. A. Issa, A. Saad, "A mthod for th dtrmination of cosmic distancs", Nw Astronomy, ol. 8, No., pp. 5-, 3 [5] S. Cassisi, R. d Santis, A. M. Pirsimoni, "Th distanc to Galactic globular clustrs through RR Lyra pulsational proprtis", Monthly Notics of th Royal Astronomical Socity, ol. 36, No., pp , [6] B. W. Carny, B. C. Chaboyr, D. Dunca, T. Grand, D. W. Latham, A. Laydn, A. Mcwilliam, A. Saradini, M. Shao, "Anchoring th Population II Distanc Scal: Accurat Ags for Globular Clustrs and Fild Halo Stars", in Amrican Astronomical Socity Mting Abstracts, ol. 33, [7] E. D. Figlson, G. J. Babu (ds), Statistical Challngs in Modrn Astronomy, Springr-rlag, 99 [8] W. Glis, H. Jahriss, Prliminary rsion of th Third Catalogu of Narby Stars, izir Onlin Data Catalog, ol. 57, 995 [9] ESA, Th Hipparcos and Tycho Catalogus (ESA 997), izir Onlin Data Catalog, ol. 39, 997 [] J. Kovalvsky, "Hipparcos and Tycho Catalogus," in Encyclopdia of Astronomy and Astrophysics, P. Murdin, Ed.: Encyclopdia of Astronomy and Astrophysics, [] W. Glis, Catalogu of Narby Stars. Edition 969, roffntlichungn ds Astronomischn Rchn-Instituts Hidlbrg, ol., 969 [] W. Glis, H. Jahriss, "Narby Star Data Publishd ", Astronomy and Astrophysics, ol. 38, pp , 979 [3] R. van dr Rit Woolly, Catalogu of stars within twnty-fiv parscs of th Sun, Hrstmoncux, Royal Grnwich Obsrvatory, 97 [4] D. Mihalas, J. Bnny, Galactic Astronomy,scond dition. USA: Frman and Company, 98 [5] M. A. Sharaf, "Computations of th cosmic distanc quation", Applid Mathmatics and Computation, ol. 74, No., pp , 6 A. Appndix A: Input data TABLE I.. APPENDIX DATA FOR SOME F8 NEARBY STARS Nam mag B- U-B plx Mv mag mag mag mas mag Gl Gl Gl Gl NN NN G Gl NN NN Gl Gl Gl Wo Gl NN Gl Gl Gl Gl Gl Gl NN Gl NN NN Gl Gl Gl Gl Gl Gl Gl Gl Wo NN Dwidar and Sharaf: Statistical Magnitud Analysis and Distanc Dtrmination of th Narby F8 Stars

4 Enginring, Tchnology & Applid Scinc Rsarch ol. 4, No. 4, 4, TABLE II. EMPIRICAL RELATION BETWEEN ( B ) M ERSUS M ( B ) M TABLE III. THE q 'S COEFFICIENTS OF (7) q q B. Appndix B: Output data ( B ) B.: Statistics of th apparnt and absolut magnituds of th F8 stars Th avrag: For Apparnt Magnituds= For Absolut Magnituds= Th mdian (cntral valu): For Apparnt Magnituds=5.65 For Absolut Magnituds=4.8 Th mod: For Apparnt Magnituds=5.79 For Absolut Magnituds={3.69,4.,4.4,4.53} Th gomtric man: For Apparnt Magnituds=5.576 For Absolut Magnituds=3.99 Th harmonic man: For Apparnt Magnituds=5.584 For Absolut Magnituds= Th root man squar: For Apparnt Magnituds= For Absolut Magnituds=4. Th Quartils: For Apparnt Magnituds={5.65,5.65,6.3} For Absolut Magnituds={3.795,4.8,4.35} Th rang: For Apparnt Magnituds=4.8 For Absolut Magnituds=4.75 Th unbiasd stimat of th varianc: For Apparnt Magnituds= For Absolut Magnituds= Th maximum liklihood stimat of th varianc: For Apparnt Magnituds= For Absolut Magnituds=.57 Th unbiasd stimat of th varianc of th sampl man: For Apparnt Magnituds=.9844 For Absolut Magnituds=.4635 Th unbiasd stimat of th standard dviation: For Apparnt Magnituds=.8453 For Absolut Magnituds= Th maximum liklihood stimat of th standard dviation : For Apparnt Magnituds= For Absolut Magnituds=.757 Th unbiasd stimat of th standard rror(standard dviation)of th sampl man: For Apparnt Magnituds=.4869 For Absolut Magnituds=.975 Th man absolut dviation: For Apparnt Magnituds=.647 For Absolut Magnituds=.4453 Th mdian absolut dviation: For Apparnt Magnituds=.55 For Absolut Magnituds=.55 Th intrquartil rang: For Apparnt Magnituds=.965 For Absolut Magnituds=.5 Th Quartils: For Apparnt Magnituds={5.65,5.65,6.3} For Absolut Magnituds={3.795,4.8,4.35} Th cofficint of skwnss: For Apparnt Magnituds=.6793 For Absolut Magnituds=.63 Parson's first cofficint of skwnss: For Apparnt Magnituds= For Absolut Magnituds={.4856,-.7498, ,-.9867} Parson's scond cofficint of skwnss: For Apparnt Magnituds=-.569 For Absolut Magnituds=-.688 Th quartil cofficint of skwnss: For Apparnt Magnituds=-.435 For Absolut Magnituds= Th kurtosis cofficint: For Apparnt Magnituds=4.445 For Absolut Magnituds=.79 Th kurtosis xcss: For Apparnt Magnituds=.445 For Absolut Magnituds=7.79 Th confidnc intrval of th population man basd on th normal distribution: For Apparnt Magnituds={ ,5.965} For Absolut Magnituds={3.834,4.8655} Th confidnc intrval for th population varianc basd on th chisquar distribution : For Apparnt Magnituds={.46996,.557} For Absolut Magnituds={ , } B. :Graphical Rprsntations Th Histograms of th Apparnt and Absolut Magnituds of th stllar Group B.3: Optimum Choic of Graphical rprsntations of th prcntag rrors for th Man and Mdian Dwidar and Sharaf: Statistical Magnitud Analysis and Distanc Dtrmination of th Narby F8 Stars

5 Enginring, Tchnology & Applid Scinc Rsarch ol. 4, No. 4, 4, Th minimum valu of th prcntag rror in th Man is.553 and occurs at σ=. Th minimum valu of th prcntag rror in th Mdian is.8854 occurs at σ=. Th optimum valu of σ is. B.4: Distanc Analysis Th valu of m is Th valu of m is 7.68 Th valu of is. Th valu of M is Th valu of is.5935 Th valu of y is.493 Th accuracy of th computd valu of y is Th distanc r of th stllar group as obtaind from th statistical mthod is.5357 parsc Th Mdian of th individual distancs is.33 parsc Th Mod of th individual distancs is {8.88,.,5.,5.64} parsc Th gomtric man of th individual distancs is.76 parsc Th harmonic man of th individual distancs is parsc Th man valu of th individual distancs is.6796 parsc Th prcntag rror btwn r and ˆr is B.5: Frquncy functions of th absolut and apparnt magnituds for th F8stars ( M ) ( M ) (-7.5 m) ( m) Dwidar and Sharaf: Statistical Magnitud Analysis and Distanc Dtrmination of th Narby F8 Stars