COMMISSION 27 OF THE I. A. U.
           INFORMATION BULLETIN ON VARIABLE STARS
                       Number 2425

                                        Konkoly Observatory
                                        Budapest
                                        1 November 1983

                                        HU ISSN 0374-0676



   DETERMINATION OF CEPHEID DISTANCES BY MEANS OF WESENHEIT FUNCTION


  It is well known that the determination of Cepheid distances is complicated 
by the necessity of taking into account the effect of the interstellar
absorption. Here we propose a method for determination of true distances when
only the ratio R of total A_V to selective absorption E_B-V is known:

                  R= A_V / E_B-V

  To solve this problem we use the Wesenheit function W introduced by van
den Bergh (1968) and discussed by Madore in a series of papers, e.g. Madore
(1982). The important property of this function is its independence on 
interstellar absorption:

           W = <V>_0 - R<B-V>_0 = <V> - R<B-V>          (1)

where <V> = <V>_0 + A_V  and  <B-V> = <B-V>_0 + E_B-V  .

A similar function for absolute magnitude M_<V> has been defined by Madore
(1976):

               W_M = M_<V> - R<B-V>_0                  (2)

This expression can be compared with the P-L-C relation for classical 
Cepheids pulsating in fundamental mode:

           M_<V> = a * logP_F + b<B-V>_0 + c           (3)
From Eqs. (2) and (3) we have:

           W_M = a*logP_F + (b-R)<B-V>_0 + c           (4)

  It was stated many times that the numerical value of the coefficient b is
very poorly determined by the observational data. Even the method of calculation, 
least squares or maximum likelihood, may influence significantly the
value of this coefficient, as is shown in the paper by Martin, Warren and
Feast (1979) where quantities from 2.18 to 3.08 have been obtained. The reason
is that b determines the slope of constant P lines on the M<V> - <B-V>_0
plane. These lines are nearly perpendicular to the axis of rather narrow
stripe occupied by classical Cepheids and it is difficult to fix the accurate
value of the slope b.
  Taking into account this fact we propose to assume that b equals R, for
which we accept the value 3.20 according to Seaton (1979):

b = R = 3.20 .

This makes W_M depending only on P_F :

  W_M = a*logP_F + c                    (5)

This way we take advantage of the accidental, numerical similarity of two
dimensionless quantities: R describing the optical property of the interstellar
matter and b appearing in the P-L-C relation, when V and B photometric 
systems are used.
  To illustrate the result of the proposed procedure we have calculated
P-L-C and P-W_M relations by means of the data of the author's paper (Opolski,
1982, Table II) where the values logP_F, M_<V> and <B-V>_0 for 66 Cepheids are
listed. The results achieved by the least squares method are as follows:

  M<V> = -3.8601 log P_F + 2.731<B-V>_0 - 2.782        (6)

s.d.= +-0.202

   W_M = M_<V> -3.20<B-V>_0 = -4.0988 log P_F - 2.897  (7)

s.d.= +-0.205 .

Comparing the standard deviations in these formulae we see that the accuracy
of the three parameter relation for M_<V> with b = 2.731 is practically the
same as that for W_M with two parameters where we constrain the coefficient b
to be equal to 3.20.
  This fact makes it easy to calculate the true distance moduli avoiding
the direct taking into account the interstellar absorption. From Eqs. (1) and
(2) we have:
  
       W - W_M = <V>_0 - M<V> = Mod_W = 5 * log r_W - 5        (8)

where Mod_W is the distance modulus and r_W the true distance in parsecs. Now
taking advantage of the properties of the functions W and W_M, Eqs. (1) and
(5), we get the relations:

  Mod_W = W - W_M = <V> - 3.20<B-V> - a log P_F - c          (9)

  logr_W = 0.2[<V> - 3.20<B-V> - a log P_F - c] + 1          (10)

where only directly observed quantities P_F, V and <B-V> are used.
  The numerical values for a and c in Eqs. (9) and (10) can be taken from
the formula (7). But we can also calculate W_M for calibration Cepheids, 
members of clusters or associations for which Mod are known:

  W_M = W = Mod = <V> - 3.20<B-V> - Mod                     (11)

and then we can fix the P_F - W_M relation. From the published moduli of


Table I. Distance moduli Mod and function W_M for calibration Cepheids
---------------------------------------------------------------------
           log P      <V>       <B-V>    Mod      W_M
-----------------------------------------------------
SU Cas     0.290       5.969m   0.725m   7.76    -4.11
EV Sct     0.490      10.128    1.157   11.29    -4.86
CE Cas b   0.651      10.988    1.140   12.79    -5.45
CF Cas     0.687      11.110    1.227   12.79    -5.61
CE Cas a   0.711      10.919    1.213   12.79    -5.75
UY Per     0.730      11.306    1.591   12.16    -5.94
CV Mon     0.731      10.296    1.365   11.37    -5.44
U Sgr      0.828       6.714    1.142    9.24    -6.18
DL Cas     0.903       8.942    1.216   11.54    -6.49
S Nor      0.989       6.414    0.969   10.02    -6.71
VX Per     1.037       9.300    1.242   12.16    -6.83
SZ Cas     1.134       9.826    1.505   12.16    -7.15
VY Car     1.277       7.446    1.193   11.73    -8.10
RU Sct     1.294       9.500    1.779   11.76    -7.95
RZ Vel     1.310       7.114    1.209   11.39    -8.14
SW Vel     1.371       8.126    1.252   12.25    -8.13
RS Pup     1.617       7.006    1.489   11.56    -9.32
SV Vul     1.654       7.221    1.534   11.77    -9.46
S Vul      1.826       9.00     1.92    13.38   -10.52
-----------------------------------------------------

calibration Cepheids we have selected 19 stars listed in Table I. The moduli 
have
been corrected to one system based on the Hyades modulus Mod = 3.31 according
to Hanson (1978). The P_F - W_M relation achieved by the least squares method is
as follows:

                        W_M = -4.0455 log P_F - 2.797       (12)
s.d.= +-0.158.

This result is very similar to Eq.(7) but has better accuracy and we propose
it as final. So for galactic classical Cepheids pulsating in the fundamental
mode we have distance moduli Mod_W and distances r_W expressed by the formulae:

Mod_W = <V> - 3.20<B-V> + 4.0455 log P_F + 2.797           (13)

log r_W = 0.2<V> - 0.64<B-V> + 0.8091 log P_F + 1.559      (14)

With the aid of these relations it is possible to calculate Mod_W and r_W for
a large number of Cepheids, e.g. contained in the catalogue by Schaltenbrand
and Tammann (1971). Such an investigation is in progress.
An attempt to apply Eq.(14) to galactic Cepheids was made by comparing
log r calculated by means of the standard relation:

            log r = 0.2[<V> - M<V> + 5 - 3.20 E_B-V ]      (15) 

with log r_W resulting from Eq. (14). The data needed for these calculations
have been taken from papers: <V> and <B-V> - Schaltenbrand and Tammann (1971),
M<V> - Opolski (1982), E_B-V - Dean, Warren and Cousins (1978). As the result
we got the mean difference Delta log r = log r - log r_W = +0.0286+-0.0044 and standard
deviation of an individual star +-0.038.
  Application of Eq. (13) to the extragalactic Cepheids in the Magellanic
Clouds with the photometric data taken from Gascoigne (1969) and Madore (1975)
for stars with periods 0.3 < logP_F < 2.2 led to the following results:

    LMC - 33 stars: Mod_W = 19.07, r_W = 65.2 kpc
                           +-0.21       +-6.6

    SMC - 32 stars: Mod_W = 19.79, r_W = 90.8 kpc
                           +-0.18       +-7.8

  The details of these investigations will be published in a separate paper.




                       ANTONI OPOLSKI
                 Wroclaw University Observatory
                       Wroclaw, Poland




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Dean, J.F. Warren, P.R. and Cousins, A.W.J., 1978, M.N.R.A.S., 183, 569. [BIBCODE 1978MNRAS.183..569D ]
Gascoigne, S.C.B., 1969, M.N.R.A.S., 146, 1. [BIBCODE 1969MNRAS.146....1G ]
Hanson, R.B., 1978, IAU Symposium 80, 154, ed. A.G. D. Philip and D.S. Hayes,
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								[CoKon No. 83]
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