The sample is too small to estimate reliably from the RR Lyrae
the distance to
the Sagittarius dwarf galaxy. Furthermore, we are looking at the edge
of the dwarf galaxy and the RR Lyrae sample might be
a mixture of stars from the galactic halo and the dwarf galaxy.
An indication that we are possibly
dealing with a mixed sample can be obtained as follows.
Select from the sample those RRab stars for
which extinction corrections are possible from their colours
at minimum light (i.e. Q=0:
both the period and the classification are correct).
There are only four of those stars in the sample.
If the extinction correction is done as described by
Wesselink (1987) and Alard (1996) we have to discard two
stars from our consideration, because one
is too blue (#1371) and the other one is probably too
faint (#1524). For the two remaining stars we apply the
mean reddening correction mentioned in Sect. 2.
This gives a mean distance modulus for these stars of
or a distance of 
kpc.
The distance modulus is in good agreement with
obtained by Mateo et al. (1995). The distance
is within the uncertainties of 
kpc,
obtained by Alard (1996) from a much larger sample of stars.
The mean distance of the RRab stars in Table 1 is about 32 kpc.
This might be due to a contribution of stars from the galactic halo
or it might be due to an extension along the line of sight of the dwarf galaxy.
Radial velocities of the stars might help to distinguish the two
scenarios from each other.
Figure 2: Period-
relation for Mira (open square) and
semiregular (triangle) variables in PG3.
The open circle indicates a carbon star among these variables.
The thick solid line is the relation
obtained by Glass et al. (1995) for the
Miras in Sgr I.
The long dashed area shows the contribution in the galactic bulge
and the dashed line is the relation from Glass et al., shifted 
(i.e. 26 kpc)
The Mira and semiregular variables in our sample of possible members
of the dwarf galaxy are best looked at in a period-luminosity diagram
(Fig. 2 (click here)). Whitelock et al. (1991) demonstrated that the
period-luminosity relation is
independent of the metallicity of the Miras. This might also
apply to the semiregulars. In Paper II it is
argued that they are the short period extension to this relation.
In Fig. 2 (click here) we plotted the variables in Table 1, together with the
galactic contribution in PG3. We refer to Paper II
for details about the galactic contribution.
We also show the 
-relation of Glass et al. (1995) shifted
, i.e. to a distance of 26 kpc.
At this distance the stars are about 4 kpc out of the galactic
plane, where one expects to find mainly old metal-poor stars,
from the metal-poor thick disc and the halo (Ng et al. 1997).
In these populations one does not expect
to find long period variable stars, because they
have not been found in the old metal-poor globular clusters.
Therefore, the semiregular variables and Miras cannot be of galactic origin
and should belong to a considerable younger population.
But a dwarf galaxy, which has interaction with our Galaxy,
can contain younger populations.
This demonstrates that the long period variables
could indeed be located in the dwarf galaxy, except for
variables #192 and #1128 which might have a galactic origin.
Additional photometry is required to determine this.
One of the variables (#283)
is a carbon star, see Fig. 3 (click here).
Details about the spectroscopic observations and the spectral classification
of the Miras and
the semiregular variables will be given in a forthcoming paper
(in preparation).
This star appears to be comparable with the carbon
stars found in other dwarf galaxies (Aaronson et al. 1983;
Azzopardi et al. 1985 & 1986) and
some of the carbon stars found by Azzopardi et al. (1991).
Unfortunately, the spectrum of variable #283 did not include the
sodium D doublet and it is not clear if this star is comparable
with those found in the "bulge" or the low-metallicity objects
in the SMC and the dwarf galaxies.
The "bulge" carbon stars have been a mystery (Lequeux 1990;
Tyson & Rich 1991; Westerlund et al. 1991), because they are about
in bolometric luminosity too faint to be regarded
as genuine AGB stars, if located inside the metal-rich bulge.
But if some or all of them
are located in the dwarf galaxy, just like variable #283,
there is no need for a metal-rich origin.
They are in that case
just ordinary metal-poor to intermediate metallicity
carbon stars.
Figure 3: Medium resolution spectrum smoothed to
2.6Å/pixel for variable star #283 (C 3,2), which has a period
P = 137.39 days (Wesselink 1987)
Membership of the dwarf galaxy has important consequences.
It implies that the carbon stars are at least younger than
approximately 4 Gyr (Marigo et al. 1996 and references cited therein).
It indicates that
at least 2 major epochs of star formation occurred in the
dwarf galaxy, just like the recurrent star formation epochs
in the Carina dwarf spheroidal (Schmecker-Hane et al. 1996).
It possibly traces the tidal star formation tail due to its passage through
our Galaxy. This tail extends at least from 
to 
, which is far more larger than previously
thought. This tail most likely indicates that the Sagittarius
dwarf galaxy has passed the galactic plane not so long ago.
It would explain the rather blue colours and
might also explain the large velocity dispersion of the carbon stars
obtained by Tyson & Rich (1991).
Alksnis (1990) showed that the majority of the carbon stars are
SRa-type long period variables. A detailed monitoring of the
Azzopardi et al. (1991) carbon stars is required to determine
if they are variable and to determine their periods. Together
with the period-luminosity relation their membership to the
dwarf galaxy can be secured.
This can be compared with the properties of the carbon stars studied
by Whitelock et al. (1996). Their study indicates the presence of two
different groups. One group with
is comparable with the
carbon stars from Azzopardi et al. (1991), while another group
has significantly redder colours 
.
The two groups could be an indication for two different star formation
epochs from recent passages through the galactic plane.
AGB stars, like the carbon stars, are the progenitors of planetary nebulae (PN). Along the trail of the Azzopardi et al. (1991) carbon stars one would expect to find long period variables and PNs. The question arises if the PNs found at low galactic latitude with velocities near to that of the Sagittarius dwarf galaxy (Zijlstra & Walsh 1996) should be considered as true bulge members? Membership of the dwarf galaxy would support the proposition that some or all of the Azzopardi et al. carbon stars do indeed trace the tidal tail.
Acknowledgements
The research of MS is supported by a grant from the Austrian Science Fund under project number P9638-AST and S7308. YKN is supported by HCM grant CHRX-CT94-0627 from the European Community.