The Distance to the Galactic Center
ABSTRACT
The Optical Gravitational Lensing Experiment (OGLE) data on high‐amplitude δ Scuti stars (HADS) and RR Lyrae stars have been analyzed to determine the distance to the Galactic bulge. Individual color excesses and V absorption for each individual variable star have been computed, which lead to accurate 
magnitudes. The period‐luminosity relation of the HADS is utilized to calculate distance moduli. We find that the metal‐strong HADS are concentrated strongly to the bulge. The best distance modulus to the bulge from the HADS data is 
mag. The dependence of 
on period for the bulge RR Lyrae stars can be found from the reddening‐free magnitude 
. The zero point of the 
–
relation is set with the horizontal branches of Oosterhoff type I globular clusters. 
mag (long‐distance scale) in the 
interval −0.29 to −0.22 as inferred by main‐sequence fitting of the clusters to local subdwarfs and utilizing short‐period HADS in the clusters. The best distance modulus to the bulge from the RR Lyrae (ab) stars is 14.45 mag, which compares favorably with the distance modulus of 14.49 mag found from the HADS. We adopt the mean value 
mag for the bulge distance modulus. A small additive correction leads to 
kpc as the best distance to the Galactic center.
Received 1999 August 24; accepted 1999 October 28
1. INTRODUCTION
Many attempts have been made to find the distance to the Galactic center since Shapley (1918) mapped out the three‐dimensional distribution of the globular clusters known at the time of his investigation. His study showed, of course, that the clusters formed a spherical system with the highest concentration in the direction of Sagittarius, some 13 kpc distant. He made the assumption, since proven correct, that this center corresponds to the center of the galaxy. Many of the later attempts to improve Shapley’s distance determination have involved improving the calibration of the luminosities of standard candles in clusters to improve the distance to the Galactic center, 
. At the same time, the identification of RR Lyrae and Mira variables in “windows” in the direction of the Galactic bulge has led to a more direct determination of 
.
Reid (1993) has written a recent review on the Galactic distance scale in which he discusses not only the optical determination of 
but also 
determinations from other regions of the electromagnetic spectrum including X‐ray, radio, and infrared. Reid’s review discusses 
determinations primarily in the time frame of 1974–1993. The discussion is valuable because he gives a best estimate of 
considering both statistical and systematic uncertainties.
Our discussion of the distance to the Galactic center is based on recent observations of variable stars in the Galactic bulge obtained primarily in the direction of Baade’s window by the Optical Gravitational Lensing Experiment (OGLE). We have employed both RR Lyrae stars and high‐amplitude δ Scuti variables, hereafter referred to as HADS, to obtain the distance. The latter includes both the metal‐poor variables referred to as SX Phe variables as well as the metal‐strong variables. These variables are frequently referred to as dwarf Cepheids.
2. OBSERVATIONAL DATA
OGLE’s main goal is the detection of microlensing events, but a by‐product of the investigation is the detection and collection of V, I photometry (with a CCD 
chip, 15 μm pixels) of variable stars. All the data were secured with the 1 m telescope at the Las Campanas Observatory. A description of the catalog, reductions, period‐search technique employed, and other details are given by Udalski et al. (1994). Subsequent data that extend the catalog can be found in additional papers by Udalski et al. (1995a, 1995b, 1996, 1997). The catalog contains I magnitude light curves, V−I color indices at light maximum, and periods of variable stars with 
in the range 14–18, including variable stars with periods of 0.1–100 days. The error of the zero point of the photometry is estimated to be ≤0.04 mag. Udalski et al. (1993) have compared their photometry with similar photometry of Terndrup (1988) and Walker & Mack (1986). About 30 (and 25) common stars indicated mean difference of 0.007 (−0.084) in V and 0.029 (0.026) in V−I respectively. The differences are in the sense of OGLE minus others. Thus, as suggested by Udalski et al., the systematic errors of the photometry are probably ≤0.04 mag.
Most of the observational data were secured in the I band. The errors of the individual observations of the stars depend on the magnitude of the stars and the quality of the seeing. For the most part the errors are typically ±0.02 mag. This may increase to ±0.03 and possibly ±0.04 mag at 
mag. Similar or slightly larger errors are appropriate for the V data.
3. INTRINSIC COLOR INDICES, COLOR EXCESSES, AND ABSOLUTE MAGNITUDES OF THE VARIABLE STARS
3.1. High‐Amplitude δ Scuti Variables
The intrinsic color indices of the HADS on the b−y color index scale are well known from uvbyβ photometry of 26 stars (McNamara 1997b). The variables exhibit a period‐color relation in the sense that the longer period variables have more positive color indices. The intrinsic b−y values at mean light (mag mean) were transferred to the V−I color index system by the equation 
given by Cousins (1987). The I magnitude is 
, that is, the I magnitude on the Cousins system similar to the I magnitude in the OGLE data. It was found that the (V−I)0 color indices of the field stars could by adequately represented by the equation 
The mean color index 
, was formed for all the HADS and the color excess calculated. We are assuming that the difference between 
and 
is negligible. Following the suggestion of Woźniak & Stanek (1996), we calculated the visual absorption from the equation 
The period‐luminosity (P‐L) relation of the HADS given by McNamara (1997b) 
was used to calculate the absolute magnitudes of the variables. We here assume that all of the variables are pulsating in the fundamental mode. Table 1 lists the HADS according to OGLE designation (Baade’s window field and star number). The 
value and mean 
and 
magnitudes, 〈V〉−〈I〉 and (V−I)0 color indices, color excess E(V−I), 
magnitude, 
magnitude, and distance modulus 
are given in columns (2)–(10) of Table 1.
3.2. RR Lyrae Stars
The photometric parameters of the RR Lyrae variables measured in the OGLE experiment have been published by Olech (1997). The data of interest in our discussion are found in his Table 7. He gives, in addition to the star designation and period of the variables, the mean 
and 〈V〉−〈I〉 magnitudes of the variables along with the designation (i.e., RRab or RRc). He also lists the reddening‐free magnitude 
, defined as 
To estimate the reddening of the variables we have adopted the equation 
to calculate the intrinsic 〈(V−I)0〉 values of the RRab variables. This equation is based on the reddening‐free 〈(V−I)0〉 values of RRab variables listed by McNamara (1997a). We have adopted 〈(V−
for all the RRc variables. The color excess is given by the difference between 〈V〉−〈I〉 and (V−I)0. The absorption in V is given by 
(V−I). The typical residual of an individual data point from the (V−I)0 values predicted by either equation (2) or equation (6) is ∼0.015 mag. The mean color excess values found from the three types of variables over Baade’s window are HADS (31 stars), 〈E(V−
; RRab (112 stars), 〈E(V−
; and RRc (49 stars), 〈E(V−
. Thus, while there is a large range in color excess from star to star, on average we find similar color excess values from the three types of variables.
4. ANALYSIS AND DISCUSSION
4.1. HADS
The distance moduli of the HADS have been grouped into 0.1 mag intervals. Histograms of the variables are shown in Figure 1. Note that we have divided the sample into the short‐period, 
, and long‐period, 
, variables. This is equivalent to dividing the sample according to abundance since the short‐period variables are metal poor, 
, and the longer period variables tend to be more metal strong, 
. See McNamara (1997b) for a discussion of how the [Fe/H] values depend on period. Note that the short‐period (metal poor) variables exhibit a much larger spread in distance modulus than the long‐period (more metal strong) variables. This is probably primarily a consequence of the concentration of metal‐strong variables in the bulge. At 7000 pc the line of sight to Baade’s window is ∼300 pc above the Galactic plane. This is well within the halo, where we would anticipate few if any metal‐strong (long‐period) HADS but some metal‐poor variables. This concept is given additional support by the lack of long‐period HADS with distance moduli greater than 15.1, which is on the other side of the Galactic bulge. Although the long‐period HADS have luminosities similar to RR Lyrae stars, they are absent while, as we shall see, RR Lyrae stars are found beyond the bulge. No short‐period variables (fainter variables) are found beyond the bulge because their magnitudes are too faint for them to show up in the data set.
Fig. 1.— Number of HADS in distance moduli, 
, bins in the direction of the Galactic bulge. Note that the shorter period (
) metal‐poor stars (
) are found over a larger range of distance moduli (some foreground variables) than the longer period (
) metal‐strong variables (
). The latter seem to be restricted to the bulge only.
In Table 2 we have computed the mean distance modulus, 
, under different assumptions based on 
, [Fe/H] values. N is the number of stars in the various bins. If we ignore the second row, which includes some metal‐poor stars, we note that the 
values differ by only 0.24 mag, with the value increasing as the [Fe/H] values increase. The second‐row entry in the table is the average value for all stars with 
values in the range 13.89–15.06 mag regardless of their [Fe/H] values. It is the metal‐poor stars that bring the average down. Since there is a real possibility that the metal‐poor variables give a 
value biased toward a small value because of their extreme faintness, near the limit of the magnitude of the survey, we discount this value. Weights of the other four values have been assigned based on the [Fe/H] values, number of stars, and the standard deviations of the mean values. With this weighting system we find that the best value of 
is 
mag where the internal error is estimated. This corresponds to a distance of 7834 pc to the bulge in the direction of Baade’s window.
We emphasize again that we have assumed all the HADS are pulsating in the fundamental mode. Of 26 field variables discussed by McNamara (1997b), only one may be pulsating in the first overtone. Thus, this assumption is probably close to being fulfilled. If first‐overtone pulsators are present in the data set, their periods would be smaller than the fundamental period (
), and we would have underestimated their 
values by ∼0.2 mag. Further study beyond the scope of this investigation is required to assign pulsation modes. If a considerable number of first overtone pulsators are present in the data set (for example 
the total), the distance modulus of the bulge would have to be increased by ∼0.05 mag.
4.2. RR Lyrae Stars
In order to understand the dependence of the absolute magnitudes of the RR Lyrae stars on period and on metal abundance, we have binned the variable stars in the bulge into period intervals and computed the mean reddening‐free magnitude, 
, from the data given by Olech (1997). We have selected stars in the interval 
mag, where the data exhibit a Gaussian distribution with a well‐defined maximum for this purpose. The data are presented in Table 3 according to the period intervals designated in column (1). Columns (2)–(5) list the number of stars in the period interval used to calculate the average, the type of variable, average period, and average reddening‐free magnitude along with the standard deviation of the mean value, respectively. As expected, the longer period variables are the most luminous. The 
values of Table 3 are plotted against 
in Figure 2. The RRab data are plotted as circles and the RRc variables as crosses. The run of absolute magnitude with 
must be similar to 
.
Fig. 2.— Average 
reddening‐free magnitudes vs. 
. It is evident that the longer period variables are the more luminous. The circles are RRab variables and the crosses are RRc variables. The absolute magnitude scale on the right has been forced to fit the data points such that 
in the 
interval −0.29 to −0.22. The vertical lines through the data points are the errors (standard deviation) of the mean values. The stepped solid line, dashed line, and straight solid line refer to the three solutions described in the text to obtain the distance to the Galactic bulge.
We consider next the choice of absolute magnitudes of the variables. Since the metal abundance of the field variables is constant or nearly so in the 
interval of the Oosterhoff type I variable, 
to −0.22 (McNamara 1999), we have used RR Lyrae stars in this interval to fix the zero point of the 
‐P relation. We assume that 
, that is, the 
values of the bulge variable are identical to the 
values of RR Lyrae stars of similar period on the horizontal branches (HBs) of globular clusters.
Three investigations of the 
(HB) have yielded about the same 
values of the HBs of Oosterhoff type I globular clusters. Two of the studies are based on fitting of the main sequence of globular clusters to local subdwarfs (Reid 1997, 1998; Gratton et al. 1997). The other (McNamara 1997b) is based on utilizing the SX Phe (HADS) variables in the globular clusters to fix the luminosity of the horizontal branch 
(HB), or RR Lyrae stars. The three studies are in excellent agreement. For typical Oosterhoff clusters we find the following values: Reid (seven clusters), 
mag, 
; Gratton et al. (four clusters), 
mag, 
; McNamara (three clusters), 
mag, 
. The latter would be 
mag if the Petersen (1999) P‐L relation were utilized instead of the McNamara (1997) P ‐ L relation. In the light of these data, we adopt 
mag at 
. This should be the appropriate 
value that spans the 
range of RR Lyrae stars from −0.29 to −0.22 where the [Fe/H] values are nearly constant (McNamara 1999). Our first solution for the distance moduli of the RRab variables is based on the following equations: 
These equations are based on the assumption that 
in the 
interval −0.29 to −0.22; that is, the bulge variables are identical in absolute magnitude to globular cluster variables of like period. Outside this period interval, we assume that the absolute magnitude is given by 
where we have translated this dependence to the 
–
plane. The stepped solid line in Figure 2 shows the dependency of 
on 
in the three period intervals. The Olech Table 7, referred to previously, has been modified to our Table 4 to include 
in column (2), 
, 
, 
, 
, 
in columns (6)–(10), respectively, and three solutions for 
from the absolute magnitudes given under columns (11), (13), and (15). The entries in columns (6)–(10) in the table have been calculated with the aid of equations and assumptions described earlier (i.e., eq. [6] and following). Note that the entries in columns (11), (12), (13), and (14) for the RRc stars of Table 4 are based on different assumptions than those made for the RRab stars.
The distance modulus of each variable calculated from the 
values given by equations (7)–(9) is noted in column (12) of Table 4. The stars have been divided into 0.1 mag bins of 
and the numbers plotted in Figure 3. A Gaussian fit, shown as the solid curve, yields a maximum at 
mag with a dispersion of 0.24 mag.
Fig. 3.— Filled circles are the number of RR Lyrae (ab) stars as a function of the distance modulus, 
. The bins are 0.1 mag in width. The solid curve is a Gaussian fit (solution [1]) with 
and 
.
A second solution is based on the three equations: 
The stepped dashed line in Figure 2 shows the dependence of 
on 
in the three period division intervals. These equations give the best fit to the 
data. The absolute magnitude and distance modulus of each variable is given in columns (13) and (14) of Table 4. As in solution (1), the stars have been divided into 0.1 bins of 
. The number of stars in each bin is plotted in Figure 4. A Gaussian fit to the data yields a maximum at 
mag and a dispersion of 0.23 mag.
Fig. 4.— Filled circles are the number of RR Lyrae (ab) stars as a function of the distance modulus, 
. The bins are 0.1 mag in width. The solid curve is a Gaussian fit (solution [2]) with 
and 
.
A third solution is based on the short‐distance scale. Fernley (1998) suggests that the absolute magnitudes of the RR Lyrae stars can be represented by the equation 
If we translate this to the 
–
plane we find 
The lower solid line in Figure 2 is based on this equation. Note the 
scale on the right of Figure 2 is adjusted to fit solutions (1) and (2) and not solution (3). If the 
data were superimposed on the lower solid line, the fit would be poor: the slope of the line is too small. The present data support the suggestion made earlier by McNamara (1999) that a slope of ∼0.31 rather than 0.18 is more appropriate for 
[Fe/H], at least for stars with 
.
The distance modulus of each variable has been calculated with the aid of 
values given by equation (14). The 
and distance modulus values are listed in columns (15) and (16) of Table 4. As previously, the stars have been divided into 0.1 mag bins of 
and the numbers plotted in Figure 5. The Gaussian fit shown in the figure yields a maximum at 
mag (
kpc) and a dispersion of 0.27 mag. The fit is not as good as for solutions (1) and (2) and yields a much smaller distance modulus.
Fig. 5.— Filled circles are the number of RR Lyrae (ab) stars as a function of the distance modulus, 
. The bins are 0.1 mag in width. The solid curve is a Gaussian fit (solution [3]) with 
and 
.
A fourth solution has also been obtained utilizing the RRc variables. They appear to exhibit little if any dependence of 
on period (see Fig. 2). We have adapted 
mag and carried out a solution to find the best distance modulus of the bulge. The relevant data employed are given in Table 4 for the RRc variables (cols. [11] and [12]). We restricted our solutions to only RRc variables in the period interval 
days. The number of stars in each 0.1 mag bin are plotted in Figure 6. The solid curve in the figure is the Gaussian fit to the data. We find a maximum at 
mag and a dispersion of 0.36 mag. The fit and the dispersion are not nearly as good as the RRab solutions. Distance moduli of RRc variables are also given in Table 4 assuming 
mag (cols. [13] and [14]).
Fig. 6.— Filled circles are the number of RRc variables as a function of the distance modulus, 
. The bins are 0.1 mag in width. The solid curve is a Gaussian fit [RRc solution with 
] that yields 
and 
.
We now reexamine our basic assumption that the bulge variables of a given period (at least in the 
interval −0.29 to −0.22) have luminosities similar to the RR Lyrae stars (or HB) found in globular clusters. In Figure 7 we plot the Walker & Terndrup (1991) [Fe/H] values of RR Lyrae bulge variables against 
. The stepped solid line is the behavior of the mean [Fe/H] values of the field stars as a function of 
(see McNamara 1999, Fig. 1). It is clear that the bulge variables, unlike the field RRab variables, show little variation of [Fe/H] with period. They are, as pointed out by Walker & Terndrup (1991), very homogeneous with a sharp peak at 
. Note that the [Fe/H] values are ∼−1.10 in the 
interval −0.29 to −0.22 compared to 
for the Oosterhoff type I clusters. Is the difference real or can we attribute this difference to a systematic error? Since Blanco (1984) also concludes that bulge variables are more metal rich than variables in M3 and M5, we conclude that the difference is real. This implies the use of 
mag for the stars in the 
interval −0.29 to −0.22 is too luminous. Since 
, we should have used 
mag. This would systematically reduce the 
values of solutions (1) and (2) to 14.47 and 14.44 mag. These numbers should yield the best distance to the bulge. We adopt 14.45 mag (7762 pc). As indicated previously, the HADS gave 14.47 mag (7834 pc). If we had used the Petersen (1999) P‐L relation we would have obtained 14.51 mag. We adopt the mean, 14.49 mag (7907 pc), as the best value. Finally, we adopt the mean of the RR Lyrae and HADS distance moduli, 
mag (
pc), as our best value.
Fig. 7.— [Fe/H] values of Walker & Terndrup (1991) of Galactic bulge variables (dots) plotted against 
. The stepped solid line is the behavior of the mean [Fe/H] values of the field RR Lyrae variables. The bulge variables appear to be more homogeneous than the field variable in [Fe/H]. In the 
interval −0.29 to −0.22 the bulge variables have smaller [Fe/H] values than the field RR Lyrae stars as well as [Fe/H] values of variables in Oosterhoff type I globular clusters.
Thus, the HADS and RR Lyrae data give bulge distances that are in good agreement. We place little confidence in the distance derived from the RRc variables. The number of stars is small, the dispersion of the solution is large, and the assignment of 
mag to these stars is uncertain. We also regard the short distance (
kpc) as simply too small, inconsistent with other data that indicate 
kpc. A contrary view regarding the luminosities of RR Lyrae stars and the viability of the short‐distance scale can be found in Popowski & Gould (1998) and Fernley (1998).
In order to try to estimate the uncertainty of the mean‐distance modulus, we have tried to estimate the possible systematic errors inherent in our analysis which certainly must be larger than random errors. These are listed in Table 5. Our estimate of the uncertainty of the 
value is only the uncertainty of the 
value based on the long‐distance scale. If the short‐distance scale 
value is included, the uncertainty of 
would be much higher. The largest source of error is the uncertainty in the absolute magnitudes (RR Lyrae variables or HADS), which we have estimated to be ±0.07 mag. If we consider that all of the errors are additive, we find that the uncertainty in the distance modulus (RR Lyrae variables or HADS) is ±0.11 mag. Since we have obtained the distance modulus by two independent methods, the average distance modulus and its uncertainty is 
mag, which translates into 
pc.
Baade’s window, in which all the HADS and the majority of RR Lyrae stars in our discussion are found, is located at 
, 
9. The two bar fields MM5 and MM7 that were also used in the RR Lyrae analysis are located at 
2 and 
4. To correct our bulge distance to the Galactic center requires increasing the bulge distance by 30 pc. Our Galactic center distance thus becomes 
kpc.
In the process of completing our manuscript, we became aware of two papers that appeared that have a direct bearing on our results. They are studies that employ the same data with the same aim—to determine the distance to the Galactic center.
The first investigation by Morgan Sinet, & Bargenguast (1998) utilizes the δ Scuti variables and an early P‐L relation of Petersen & Høg (1998) to derive a distance of 7.6 kpc. Their analysis differs from ours in several respects. They used V−I at light maximum instead of mean values, the Stanek (1996) absorption maps were used to estimate the absorption of the variables, and finally they used a larger sample of stars including many of the short‐period variables that our study indicates are foreground objects. The inclusion of these variables would of course lead to a shorter distance to the Galactic center.
The second investigation by Udalski (1998) utilizes the RR Lyrae stars in the OGLE data set to derive a distance modulus of 14.53 mag similar to our value of 14.47 mag. As in the previous study, the interstellar extinction maps of Stanek (1996) were used to obtain 
and (V−I)0. On the other hand, Udalski used 
mag for the mean period, 
days, of the RR Lyrae stars. This is clearly in the short‐distance scale domain, similar to equation (14), which gives 
mag at 
days. How do we account for similar distance moduli when such dissimilar absolute magnitudes have been used in the analyses? In our view the answer lies in the extinction values used. Udalski’s Galactic bulge extinction‐free (V−I)0 mean color index of the bulge RR Lyrae sample is (V−
. For the mean period, 
days, equation (6) yields 0.478 mag. Udalski’s mean (V−I)0 values of the Large Magellanic Cloud and Small Magellanic Cloud RR Lyrae stars are 0.45 and 0.48 mag, similar to our Galactic bulge mean value. Clearly the mean color index utilized by Udalski for the bulge variables is too large, implying that the Stanek absorption maps of the Galactic bulge give on average too small of absorption. The difference, Δ(V−I), is 0.16 mag. Since the absorption in V is 2.5 E(V−I), this implies his 
magnitudes should be 0.40 mag brighter. This would reduce 
mag Udalski (1998) to 15.01 mag. Since Udalski adopted 
mag at the mean period of the RR Lyrae stars, 
mag (
pc). If we adopt 
mag rather than 
mag, 
mag, which compares well with 14.24 mag found in our short‐distance analysis.
Since our best estimate of 
at the mean period is 
mag for the long‐distance scale, Udalski’s corrected mean 
magnitude would yield 
mag. This compares favorably with our adopted value of 14.45 mag. Thus, the two sets of data and distance moduli are consistent if similar extinction values and absolute magnitudes are used.
We note that we find larger color excesses and hence also absorptions than many previous observers (see Gould, Popowski, & Terndrup 1998 for a discussion of this topic). We emphasize that there are large variations in extinction on the small scale as well as from one Baade’s window subregion to another. For example, we find that the RRab variables in the BWC field yield 〈E(V−
mag (22 stars) while the same variables in the BW 9, 10 fields yield 〈E(V−
mag (33 stars). These values compare with the overall average of 〈E(V−
mag. Most of the previous studies have been in the BWC field, where the extinction is less than the overall average. A very large part of the discrepancies in color excess and absorption from observer to observer can be accounted for by the particular region observed and star sample studied. Stutz, Popowski, & Gould (1998) claim that the (V−I)0 values of Baade’s window RR Lyrae stars are redder in (V−I)0 than the local RR Lyrae stars. In our view this is extremely unlikely since we obtain the same extinction with the HADS as with the RR Lyrae variables by assuming normal color indices. It is much more likely that they have not adapted the correct extinction in their study.
As pointed out previously, Reid (1993) has discussed the various methods and investigations involved in determining the distance, 
, to the Galactic center. We will not attempt to repeat in detail his discussion here. He concludes that the investigations cited in his review lead to a best value of 
kpc. In a more recent study Carney et al. (1995) concluded that the distance is 
kpc. These values may be compared with our value of 
kpc.
We note that both long‐ and short‐period HADS are found in the bulge. Their abundance, [Fe/H], must cover the range ∼0.00 to −1.5. This range may be compared with that found for other stars with abundances of −1.6 to 1.2 by Sadler, Rich, & Terndrup (1996). We note also that the HADS are blue stragglers and the evolutionary ages of the longer period variables are ∼109 years. They may be much younger than most of the stars in the bulge.
5. SUMMARY
RR Lyrae and high‐amplitude δ Scuti stars in the direction of the Galactic bulge in the Optical Gravitational Lensing Experiment have been utilized to determine the distance to the Galactic bulge. Individual color excesses and visual absorptions are calculated for each individual variable star at mean light. Our absorptions are somewhat larger (0.16 mag) than those given by the Stanek (1996) absorption maps in the bulge direction.
The 
‐magnitude data of the HADS are utilized in conjunction with absolute magnitudes computed from the period‐luminosity relation to infer their distances. We find that the metal‐strong variables are strongly concentrated to the bulge and yield a mean distance modulus of 14.49 mag. Petersen (1999) has pointed out that HADS should yield distance moduli with an accuracy of ±0.1 mag.
The bulge RR Lyrae stars show that the longer period variables are brighter than the shorter period variables by 
0.35 mag. The dependence of 
on 
is taken into account for three of our solutions to obtain the bulge distance. We adopt the long‐distance scale in setting the zero point of the 
–
relation. For this purpose we adopt for the mean absolute magnitude of RR Lyrae stars, the 
(HB) of Oosterhoff type I globular clusters, 
mag at 
. This mean value is based on fitting the main sequence of the globular clusters to local subdwarfs (Reid 1997, 1998; Gratton et al. 1997) and inferring the 
(HB) from short‐period HADS in the clusters. The two methods give the same result to less than 0.1 mag. Direct application of our equations leads to a distance modulus of 14.56 mag (mean of solutions [1] and [2]). We find the globular cluster stars are more metal poor than bulge variables in the 
interval −0.29 to −0.22. We assume the 
[Fe/H] and consequently reduce the distance modulus to 14.45 mag. This value is combined with the HADS value of 14.49 mag to yield 
mag. A small additive correction yields 
kpc as the best distance to the Galactic center. This agrees favorably with Reid’s (1993) estimated of 8.0 kpc based on an analysis of many earlier studies.
We also utilized RRc variables and a short‐distance scale to estimate 
. We discount the RRc analysis because of a limited number of stars and a poor Gaussian fit to the observational data and uncertainty in the proper choice of 
. The short‐distance scale (fainter 
values for the RR Lyrae stars) yields a bulge distance of 7.0 kpc. This is very small, certainly inconsistent with our estimate based on the HADS data and with the Reid estimate. In our view the long‐distance scale is to be preferred.
Good counterarguments supporting the short‐distance scale can be found in papers by Popowski & Gould (1998) and Fernley (1998).
We are greatly indebted to the OGLE team for making their observations readily available.
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