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\IBVShead{5354}{17 December 2002}

\IBVStitle{The Absolute Magnitude (M$_{v}$) of Type {\large \it ab} RR Lyrae Stars}


\IBVSauth{KINMAN, T. D.}

\IBVSinsto{Kitt Peak National Observatory, NOAO,$^{\dagger}$\IBVSfoot{The National Optical Astronomy Observatories are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Founda
tion} P.O.Box 26732, Tucson, Arizona 85726, USA; \\ email: kinman@noao.edu}

\IBVStyp{ RR }
\IBVSkey{photometry}
\IBVSabs{The relation of Kovacs & Walker giving M_v for type}
\IBVSabs{ab RR Lyrae stars in terms of their period and Fourier}
\IBVSabs{coefficients is calibrated using the new HST parallax of RR Lyrae }
\IBVSabs{itself (Benedict et al.). The M_v from this calibrated }
\IBVSabs{relation are found for a sample of nearby stars using data given by }
\IBVSabs{Jurcsik & Kovacs. These M_v agree well with those obtained }
\IBVSabs{from a linear relation in [Fe/H] by Cacciari, using data from Layden }
\IBVSabs{et al. A similar comparison is made for a third simpler }
\IBVSabs{empirical relation for M_v that only involves the }
\IBVSabs{period of the star.}

\begintext

RR Lyrae stars are commonly used as tracers of the halo population in both 
galactic and extragalactic work. Such studies require a knowledge of their   
absolute magnitude M$_{v}$. If their metallicity is known, it has been 
customary to use an empirical linear relation in terms of [Fe/H]:  
\begin{equation} 
         M_{v} = A [Fe/H] + B                               
\end{equation} 
Chaboyer (1999) gives 0.23 and 0.93 while Cacciari (2002) gives 0.23 and 0.92 
for the coefficients A and B, respectively. These values are consistent with 
the widely-used 
LMC modulus of 18.50$\pm$0.10 (Freedman {\it et al.}, 2001), van der Marel 
{\it et al.}~(2002)). If we assume a mean RR Lyrae metallicity of $-$1.6, these
relations 
give an $\langle M_{v}\rangle$ of +0.55 or +0.56 with an uncertainty of about $\pm$0.10 mag.; this is in the middle of the range of recent direct determinations of the
M$_{v}$ of type {\it ab}~RR Lyrae stars (Popowski \& Gould, 1999).

Kov\'{a}cs \& Walker (2001) have given an empirical expression for M$_{v}$ in
terms of the period (P) and Fourier coefficients (A1 and A3) of the variables:
\begin{equation} 
  M_{v} = -1.876\log P - 1.158A1 +  0.821A3 + K         
\end{equation} 
Here K is a constant which must be determined. 
Benedict {\it et al.} (2002) derived an M$_{v}$ of +0.61$\pm$0.10 for RR Lyrae
itself from a parallax derived from HST data; this corresponds to a modulus 
(corrected for extinction) of 7.06. RR Lyrae shows a 41-day modulation of its
amplitude (Blazhko effect) and the amplitude of this modulation also varies 
over a 4-year period (Detre \& Szeidl, 1973). Jurcsik {\it et al.} (2002) have
examined the Fourier coefficients of RR Lyrae and shown that they only 
correspond to those of a normal type {\it ab} star when both RR Lyrae has the
maximum amplitude of its 41-day cycle and when the amplitude of the 4-year
cycle is at a minimum. Observations that fulfil this condition were made by
Hardie (1955) in the interval JD 2\,434\,553 to JD 2\,434\,560. The corresponding
cycles in Hardie's data are 34749 to 34761 (using Walraven's period of
0.56683735 days). The Fourier coefficients A1 and A3 derived from this portion 
of Hardie's data are 0.31539 and 0.09768, respectively. Using these in eqn. (2), 
one finds K = 0.43. 

One can  check this result by comparing the absolute magnitudes derived by eqn. 
(2) 
against those obtained from eqn. (1) using a group of nearby RR Lyrae stars that
have both well-determined Fourier coefficients and also known metallicities.
There are 73 type {\it ab} stars whose Fourier coefficients are given by Jurcsik
\& Kov\'{a}cs (1996) that also have [Fe/H] given by Layden {\it et al.} (1996). 
For these one can calculate both M$_{v}$(Four) from eqn. 2 with K = 0.43,  
and also M$_{v}$(Fe) from eqn. (1) using Cacciari's coefficients.
 The differences ($\Delta$
M$_{v}$ = M$_{v}$(Four) $-$ M$_{v}$(Fe)), are  plotted in Fig. 1 against (a)
$\log$ P  and (b) metallicity [Fe/H]. Eight of these variables (shown by
crosses in Fig. 1) have peculiar relations between their Fourier components
according to Jurcsik \& Kov\'{a}cs (1996); two of these AN Ser ( P = 0\fday52
days, [Fe/H] = $-$0.04) and TV Lib ( P = 0\fday27 days, [Fe/H] = $-$0.27) 
clearly show abnormally large $\Delta$M$_{v}$ in Fig. 1. 
The scatter in $\Delta$M$_{v}$ is 
larger for stars of longer period and lower metallicity. This is understandable
since the longer period stars have lower amplitudes and so their Fourier 
coefficients will be less well determined for a given photometric accuracy.
Similarly, the metallicities will be more poorly determined in the
lower metallicity stars whose lines are weaker. Additionally, there may be
physical differences between the Oosterhoff I and II populations that
contribute to this increased dispersion.   Omitting the eight stars that have
peculiar relations between their Fourier coefficients, the remaining
65 stars have a 
mean value of  $\Delta$M$_{v}$ of $+$0.010$\pm$0.008 with an {\it rms}
deviation for a single star of 0.063 mag. If one includes all these stars
except AN Ser and TV Lib, the mean value of $\Delta$M$_{v}$ is $+$0.007$\pm$
0.008 with an {\it rms} deviation for a single star of 0.066 mag. 
Thus the two ways of estimating
M$_{v}$ for type {\it ab} RR Lyrae stars give very similar results in this
sample of 71 stars.

There will be cases where neither [Fe/H] is known nor is it possible to get
the Fourier components from the light curves with sufficient accuracy. One
cannot simply replace the expression containing the Fourier coefficients (in 
eqn. 2) with 
a constant because this expression shows some correlation with $\log$ P. 
 An adjustment must be made to the    
coefficient of $\log$ P to allow for this; one then gets:
\begin{equation}
 M_{v} = -1.619\log P +0.20  
\end{equation}  
Calling this absolute magnitude M$_{v}$(Per), the differences 
$\delta$M$_{v}$ = M$_{v}$(Per) $-$ M$_{v}$(Fe) were calculated and are 
plotted in Fig. 2 against (a) $\log$ P and (b) [Fe/H] for {\it all} 73 stars in
the Jurcsik \& Kov\'{a}cs sample. None of the eight stars that have
peculiar Fourier coefficients (shown in Fig. 2 by crosses) are anomalous 
in this plot. For all 73 stars the mean value of $\delta$M$_{v}$ is $-$0.003$\pm$0.006
mag. with an {\it rms} deviation for a single star of 0.048 mag. For this
sample of stars therefore, eqn. (3) is at least as good an estimator of 
M$_{v}$ as eqn (2), although there appears to be a slight trend of 
$\delta$M$_{v}$ with [Fe/H]. 

 A semi-empirical relation for M$_{v}$ in terms of $\log$ P and the blue 
amplitude of the variable ($A_{B}$) has been given by Castellani \& De Santis 
(1994) and was 
further discussed by De Santis \& Cassisi (2002). Now $A_{B}$ correlates with 
$\log$ P, and so eqn. (3) can also be regarded as a simplified version of 
such relations\IBVSfoot{For early discussions of such relations see Sandage
(1958) and Kinman (1959).}. 

The three empirical relations given by equations (1), (2) and (3) allow 
alternative ways of finding M$_{v}$ for a sample of RR Lyrae stars that may be
used (and compared) depending on the observational data that is available. 
They are generally consistent with a widely used modulus (18.50) for the LMC. 

{\it Acknowledgements:}~~I would like to thank Dr Carla Cacciari for helpful
comments on this work and also Dr Christine Clement for the use of her program
for the Fourier analysis of the light-curves. 


\IBVSfig{9cm}{5354-f1.eps}{The magnitude difference $\Delta$M$_{v}$ as a function (a) of $\log P$  and (b) [Fe/H]}

\IBVSfig{9cm}{5354-f2.eps}{The magnitude difference $\delta$M$_{v}$ as a function (a) of $\log P$  and (b) [Fe/H] }

\clearpage

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\endreferences

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