A method for the determination of cosmic distances

New Astronomy 8 (2003) 15–21 www.elsevier.com / locate / newast

A method for the determination of cosmic distances M.A. Sharaf a , *, I.A. Issa b , A.S. Saad b a b

Department of Astronomy, Faculty of Science, King Abdul Aziz University, Jeddah, Saudi Arabia Department of Astronomy, National Research Institute of Astronomy and Geophysics, Cairo, Egypt

Received 23 April 2002; received in revised form 1 September 2002; accepted 15 September 2002 Communicated by G.F. Gilmore

Abstract In this paper, a general method is developed for the determination of cosmic distances. The method is based on the assumption that the members of a cosmic group scatter around a mean absolute magnitude in a Gaussian distribution. The basic formula of the method is obtained as a least-squares fit of the solution of a highly transcendental equation as a function of a given apparent magnitude parameter a. For each range of a, the precision criteria of this formula, and also the absolute relative uncertainty in the derived distance, are very satisfactory. Some illustrative examples of the usages of the method are included.  2002 Elsevier Science B.V. All rights reserved. PACS: 98.62.Py Keywords: Distance scales; Methods: statistical; Galaxies: star clusters; Stars: fundamental parameters

1. Introduction One of the most crucial pieces of information needed in astronomy is the distance to the stars or cosmic groups. For example (Robinson, 1985), if the distance d (in parsec) of a star is known as well as its proper motion m (in second of arc per year) then one can calculate its tangential velocity Vt to the line of sight (in km. per second). Also, having measured the distances to the globular cluster, we can study their distribution in the galaxy (Cassisi et al., 2001; Duncan et al., 2001). In moving stellar clusters (e.g. the open clusters Hyades and Pleiades) if the equatorial coordinates of the vertex and the distance of each member are known, then one can easily de-

termine the velocity of the cluster and also the position of its center (Sharaf et al., 2000), thus the distribution of the cluster’s members about this center can be obtained. On the other hand, the determination of distances within our galaxy allows us to calibrate the distance indicators (Shanks, 1997; Tanvir, 1997; Brochkhadze and Kogoshvili, 1999) use to estimate distances outside it. Moreover determining distances would also help astronomers in their quest to understand the size and the age of the universe (Willick and Batra, 2001; Mazumdar and Narasimha, 1999), since it would provide an independent estimation of the size of the first steps on the cosmic distance ladder. Consequently it contributes to the theories about the origin of the universe. Modern observational astronomy has been characterized by an enormous growth in data acquisition, stimulated by the advent of new technologies in

*Corresponding author. E-mail address: sharaf [email protected] (M.A. Sharaf). ] 1384-1076 / 02 / $ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII: S1384-1076( 02 )00198-7

M. A. Sharaf et al. / New Astronomy 8 (2003) 15–21

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telescopes, detectors and computations. This new astronomical data gives rise to innumerable statistical problems (Feigelson and Babu, 1992). Due to the important role of distances as mentioned above in brief, and coping with the recent tendency of reunion between statistical methods and astronomy, the present paper is devoted to developing a general method for distance determination of cosmic groups (e.g. open clusters, globular clusters or clusters of galaxies, etc). The method depends on the assumption that the members scatter around a mean absolute magnitude in a Gaussian distribution. The basic formula of the method is obtained as a least-squares fit of the solution of a highly transcendental equation as a function of a given apparent magnitude parameter a. For each range of a, the precision criteria of this formula and also the absolute relative uncertainty in the derived distance are very satisfactory. Numerical results of the method are in good agreement with the standard values.

M 5 m 1 5 2 5log r.

(2)

The frequency function C (m) of the apparent magnitudes is: ≠(r,M) C (m) 5 ]]F [M(m,r)]. ≠(r,m)

(3)

Since r is constant for all the members of a group, then we get from Eqs. (1) and (2) the following: 1 2(m1525log r 2M 0 ) 2 / 2 s 2 C (m) 5 ]] . ]e Πs 2p

(4)

Let us consider all members brighter than a given ] apparent magnitude m 1 . Then the mean magnitude m is given as:

E mC(m) dm ] m 5 ]]]], E C(m) dm m1

2` m1

(5)

2`

which upon using Eq. (4) we obtain:

2.1. Assumptions • We assume that all the members in a given cosmic group are at the same distance, r parsecs. • We assume with many authors, e.g. Mihalas and Binney, 1981 and Scheffler and Elsasser, 1988 that the frequency function of the absolute magnitudes of the members of a given spectral type is 1 2(M 2M 0 ) 2 / 2 s 2 F (M) 5 ]] . ]e sŒ2p

E me dm ] m 5 ]]]]]]]]]. E e dm m1

2. Basic formulations

(1)

That is the members scatter around a mean absolute magnitude M0 in a Gaussian distribution with dispersion s. Typical values for M0 and s are taken from Mihalas and Binney (1981) and listed in Table A.1 of Appendix A. It should be noted that the dispersion s is due to errors in spectral classification as well as to the actual cosmic dispersion in absolute magnitude.

2.2. Basic equations The relation between the apparent magnitude m and the absolute magnitude M of a cosmic object at a distance r parsec is:

2` m1

2(m 1525log r 2M 0 ) 2 / 2 s 2

(6)

2(m1525log r 2M 0 ) 2 / 2 s 2

2`

It can be shown that the distance r could be determined from: r 5 10 11(m 1 2M 0 2 s y) / 5 ,

(7)

where y is the solution of the transcendental equation: ] 21 p y 2 ] 1 1 erf ] L( y) 5 y 1 e 2y / 2 2a ] Π2 2

Hœ F

S DG J

5 0,

(8)

] (m 1 2m) a 5 ]]], s

(9)

and erf(z) is the error function defined by the series 2 erf(z) 5 ] Œ] p

O

` n 50

(21)n ]]]z 2n11 . n!(2n 1 1)

(10)

If there are A magnitudes of interstellar absorption then Eq. (7) must be written as: r 5 10 11(m 1 2M 0 2 s y 2 A) / 5 .

(11)

M. A. Sharaf et al. / New Astronomy 8 (2003) 15–21

The distance modulus is then given as: m 2 M 5 m 1 2 M0 2 s y 2 A.

(12)

17

where s˜ u is the standard error of u. Finally, it should be noted that if the precision is measured by the probable error e, then e 5 0.6745s˜

2.3. Basic definitions and properties of leastsquares

(18)

2.4. The solution of Eq. (8) Let f be represented by the general linear expression of the form o ni 51 c i fi (x) where f ’s are linearly independent functions of x. Let c be the vector of exact values of the c’s coefficients and cˆ the leastsquares estimators of c obtained from the solution of the normal equations of the form Gcˆ 5b. The coefficient matrix G(n3n) is symmetric positive definite, that is, all its eigenvalues yi ; i 5 1, . . . ,n are positive. Let E(z) denote the expectation of z and s˜ 2 the variance of the fit, defined by:

s˜ 2 5 qn /(N 2 n),

(13)

where: T

T

T

qn 5 (f 2 F cˆ ) (f 2 F ˆc ),

(14)

N is the number of observations, f is a vector with elements fk and F(n3N) has elements Fik 5 Fi (x k ). The transpose of a vector or a matrix is indicated by the superscript ‘T’. According to the least-squares criterion, it could be shown that Kopal and Sharaf (1980) 1. The estimators ˆc given by the least-squares method gives the minimum of qn . 2. The estimators ˆc of the coefficients c, obtained by the method of least-squares are unbiased; i.e, E(cˆ )5c. 3. The variance–covariance matrix Var(cˆ ) of the unbiased estimators ˆc is given by Var(cˆ ) 5 s˜ 2 G 21 ,

(15)

where G 21 is the inverse of G. 4. The average squared distance between ˆc and c is E(L 2 ) 5 s˜ 2

O

n i 51

1 ]. ni

(16)

5. If H(u) is a function of a measured quantity u, then the standard error s˜ H of H is Bevington and Robinson (1992): dH s˜ H 5 ]s˜ u , du

(17)

The solution of Eq. (8) could be obtained by Newton’s iterative method according to the scheme: 2

y i R 2 ( y i ) 1 R( y i ) e 2y i / 2 2 a R 2 ( y i ) y i 11 5 y i 2 ]]]]]]]]]] ; 2 2 R 2 ( y i ) 2 y i R( y i ) e 2y i / 2 2 e 2y i / 2 i 5 0,1,2, . . .

(19)

with a given initial guess y 0 and ] p y R( y) 5 ] 1 1 erf ] . Œ]2 2

œ H

S DJ

(20)

The above procedure is terminated if the following conditions are satisfied:

d #e

and uL( y i 11 )u # 100e,

y i 11 2 y i d 5 ]]] yi

U

U

(21)

if

uy i 11u . 1; d 5u y i 11 2 y iu

if u y i 11u , 1,

(22)

where e is a given tolerance.

3. Approximate relation Clearly, y is the basic quantity for distance determination of a given cosmic group (see Eq. (7)). In what follows, an approximate formula for y as a function of a will be established. To obtain this, two steps are performed: (1) solving Eq. (8) for given ranges of a using the iterative scheme of Subsection 2.4. (2) Find the best fit of y(a ) for each range of a using the method of least-squares described in Subsection 2.3. Applying these two steps with e 5 10 29 (of Eq. (21)), we find for each range of a the relation: y(a ) 5 c 1 1 c 2 a 1 c 3 a 21 .

(23)

In Table B.1 of Appendix B, the coefficients of this relationship together with its error analysis are

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M. A. Sharaf et al. / New Astronomy 8 (2003) 15–21

listed and represented graphically for each range of a. From Eqs. (7) and (17) we obtain for the absolute relative uncertainty in the distance r the expression: e ]r 5 0.560517018s e, (24) r

U U

where s is the dispersion of the Gaussian distribution of Eq. (1) and is given in Table A.1 of Appendix A for each spectral type and e is the probable error of the fit for Eq. (23). In Table C.1, ue r /ru are listed for different spectral types for each range of a.

3.1. Computational algorithm • Purpose: To compute the distance r in parsec and its absolute relative uncertainty ue r /ru of a cosmic group of given spectral type. • Input: m 1 , M0 , ] m, e and A. • Computational sequence: 1. a from Eq. (9). 2. Select Eq. (23) according to the value of a using Table B.1. 3. Determine the distance r in parsec from Eq. (11). 4. Determine the absolute relative uncertainty ue r /ru from Table C.1 for the range to which a belongs. 5. End. 3.2. Applications In what follows are some illustrative examples for the usages of the above algorithm. Extensive applications will be considered later in a separate publication.

3.2.1. Distance of Hyades moving cluster • The following set is an order pairs of the apparent magnitudes and the spectral types of some Hyades stars (Smart, 1939) h(6.42, F2), (5.94, F0), (7.0, F2), (5.59, F0), (5.96, F0), (6.74, F5), (6.71, F0), (7.23, F5), (8., F5), (7.15, F5), (6.66, F2), (6.43, F2), (6.66, F2), (6.96, F5), (6.70, F5), (6.24, F0)j. • The limiting magnitude m 1 is taken as the greatest integer $ the faintest magnitude of the set, so m 1 5 8; ] m 5 6.649; M0 5 3.7; s 5 1.1; a 5 1.228; then using Table B.1 for the range 1 # a # 2 we get y 5 0.903. • Neglecting the absorption we get r 5 45.85pc which is in good agreement with the value 46pc

that adopted recently in Table 22.2 of Cox (2000).

3.2.2. Distance of the open cluster IC2602 • The apparent magnitudes and the spectral types (all of class V) of some stars of this cluster are taken from Table 54 of Acker and Jaschek (1986) as the following order pairs set: h(6.75, B9), (6.82, B9.5), (6.74, B8), (7.18, B9), (7.28, B9.5), (5.36, B6)j. ] 5 6.688; M 5 1.1; • As in Section 3.2.1, m 1 5 8; m 0 s 5 1, a 5 1.312 and y 5 1.038. • Consider the absorption which is 0.04 as given in Table 22.2 of Cox (2000) we get r 5 146.02pc, which again in good agreement with the value 147pc as given in the same table of the reference. 3.2.3. The distance of the globular cluster NGC5139 N(v cen) • The spectral type of this cluster is F5 with m 5 ] m 5 14.53 as given in Table 22.5 of Cox (2000), so m 1 5 15; M0 5 3; s 5 1.1; a 5 0.427; and y 5 2 1.578. • Consider the absorption which is 3.1*0.1 as given in the same table of the reference we get for the distance modulus (Eq. (12)) the value 13.426pc which again in good agreement with the value 13.47pc as given in the same table (column 12) of the reference. In concluding the present paper a general method was developed for distance determination of cosmic groups. The method is based on the assumption that the members of the group scatter around a mean absolute magnitude in a Gaussian distribution. The basic formula of the method is obtained as a leastsquares fit of the solution of a highly transcendental equation as a function of a given apparent magnitude parameter a. For each range of a, the precision criteria of this formula and also the absolute relative uncertainty in the derived distance are very satisfactory. The applications of the method are illustrated for open and globular clusters and the results are in good agreement with the standard values. There is another application of this method, where its basic assumptions (Subsection 2.1) are adequate, and that is to determine distances to clusters of galaxies. Extensive applications of the method will be considered later in a separate publication.

M. A. Sharaf et al. / New Astronomy 8 (2003) 15–21 Table B.1. Continued

Appendix A Table A.1. Parameter M0 and s of Eq. (1) Spectral type

M0

s

B0–B1 B2–B3 B5 B8– A0 A2– A5 dF0–dF5 dG5 dG8–dK3 dK gG gK

24.0 22.0 21.0 10.5 11.5 13.0 15.5 16.0 17.5 12.0 12.4

1.0 1.1 0.9 1.0 1.0 1.1 1.1 1.2 1.3 0.9 0.8

d The range of a is 0.25 # a # 0.5. (1) The solutions and their probable errors: c 1 5 0.25834960.00107616 c 2 5 1.3430260.00147451 c 3 5 2 1.0299860.000190145 (2) The probable error of the fit is e 5 0.000301282 (3) The average squared distance between cˆ and c is Q 5 7.40402 3 10 26 (4) Graph of the raw and the fitted data

Appendix B Table B.1. The relation y 5 c 1 1 c 2 a 1 c 3 a 21 and its error analysis for different ranges of a. d The range of a is 0.15 # a # 0.25. (1) The solutions and their probable errors: c 1 5 0.077677660.000536013 c 2 5 1.6771260.00136059 c 3 5 2 1.0053360.0000518648 (2) The probable error of the fit is e 5 0.0000528336 (3) The average squared distance between cˆ and c is Q 5 4.70644 3 10 26 (4) Graph of the raw and the fitted data

d The range of a is 0.5 # a # 1. (1) The solutions and their probable errors: c 1 5 0.63018660.00106202 c 2 5 0.97850460.000727492 c 3 5 2 1.1262460.000375388 (2) The probable error of the fit is e 5 0.000418306 (3) The average squared distance between cˆ and c is Q 5 3.9522 3 10 26 (4) Graph of the raw and the fitted data

19

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M. A. Sharaf et al. / New Astronomy 8 (2003) 15–21

Table B.1. Continued

Table B.1. Continued

d The range of a is 1 # a # 2. (1) The solutions and their probable errors: c 1 5 0.91920560.000118717 c 2 5 0.82479960.0000406587 c 3 5 2 1.2633660.0000839352 (2) The probable error of the fit is e 5 0.0000659564 (3) The average squared distance between cˆ and c is Q 5 5.00978 3 10 28 (4) Graph of the raw and the fitted data

d The range of a is 3 # a # 4. (1) The solutions and their probable errors: c 1 5 0.12412460.000488389 c 2 5 0.98394160.000070106 c 3 5 2 0.2407160.000845897 (2) The probable error of the fit is e 5 0.0000475766 (3) The average squared distance between cˆ and c is Q 5 2.10788 3 10 26 (4) Graph of the raw and the fitted data

d The range of a is 2 # a # 3. (1) The solutions and their probable errors: c 1 5 0.58368160.00101148 c 2 5 0.90491360.000204224 c 3 5 2 0.9109260.00123875 (2) The probable error of the fit is e 5 0.000195018 (3) The average squared distance between cˆ and c is Q 5 5.71334 3 10 26 (4) Graph of the raw and the fitted data

d The range of a is a $ 4. (1) The solutions and their probable errors: c 1 5 0.00035176960.0000173868 c 2 5 0.99997661.3009 3 10 26 c 3 5 2 0.0012509560.0000549304 (2) The probable error of the fit is e 5 8.20891 3 10 26 (3) The average squared distance between cˆ and c is 29 Q 5 7.30046 3 10 (4) Graph of the raw and the fitted data

M. A. Sharaf et al. / New Astronomy 8 (2003) 15–21

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Appendix C Table C.1. Absolute relative uncertainty ( 3 10 7 ) in the distance r for different spectral types and for different ranges of a Spectral type

0.15 # a # 0.25 0.25 # a # 0.5

0.5 # a # 1

1#a #2

2#a #3

3#a #4

a $4

B0–B1

243.308

1387.45

1926.37

303.739

898.092

219.099

35.7032

B2–B3

267.638

1526.20

2119.01

334.113

987.901

241.008

39.2735

B5

218.977

1248.71

1733.73

273.365

808.282

197.189

32.1329

B8– A0

243.308 243.308 267.638 267.638 291.969 316.30 218.977 194.646

1387.45 1387.45 1526.20 1526.20 1664.94 1803.69 1248.71 1109.96

1926.37 1926.37 2119.01 2119.01 2311.64 2504.28 1733.73 1541.09

303.739 303.739 334.113 334.113 364.487 394.861 273.365 242.991

898.092 898.092 987.901 987.901 1077.71 1167.52 808.282 718.473

219.099 219.099 241.008 241.008 262.918 284.828 197.189 175.279

35.7032 35.7032 39.2735 39.2735 42.8439 46.4142 32.1329 28.5626

A2– A5 dF0–dF5 dG5 dG8–dK3 dK gG gK

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