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*Astron. Astrophys. 350, 1085-1088 (1999)*
## 2. Data analysis
The Hipparcos catalogue (ESA 1997) includes measurements of the
parallax, , and the components of
proper motion in right ascension and declination,
and ,
of 117,955 stars, together with the statistical uncertainty associated
with each measurement. The total proper motion is *µ* =
toward equatorial position angle
= Arc
tan(/).
We give the measured parallax and proper motion of the sources we
observed in Table 1.
**Table 1.** Parallax and proper motion
The equations relating (,
) to (*µ*,
) are formally identical to those
relating the Stokes parameters (q,u) to the total linear polarization
(p, ). Dolan & Tapia (1986) give
the equations for the propagation of error from
(, )
to (*µ*, ) in terms of
their polarimetric counterparts. The probability density functions
(pdf) of the Stoke parameters and (,
) are identical, and so the pdf of
polarization and proper motion measurements must also be identical. By
analogy with the Stokes parameters, (
and ) are normally distributed at high
signal to noise ratios R =
*µ*/
, but deviate in a known way from a
normal distribution at small values of R (Clarke et al. 1983). The pdf
describing *µ* is the Rice distribution (Simmons &
Stewart 1985). The pdf describing is
more complicated and depends on both
and *µ* (Naghizadeh-Khouei & Clarke 1993).
The confidence interval on the measured value of *µ*
will just include zero at some critical significance level
C. The question of interest here is
the significance of any particular measured proper motion, i.e., for
what value of C should we accept the hypothesis that no proper motion
has been measured if R ? The
significance criterion C is not derivable from the statistics of the
pdf describing the measured variable. It is selected by the subjective
judgment of the individual observer based on the confidence level one
wishes to obtain. Clarke et al. (1983) show that
( and
) are not well represented by a normal
distribution when R . The Rice
distribution becomes nearly Gaussian for R
(Simmons & Stewart 1985), and
all estimators of the best value of *µ* are consistent for
R . The distribution of measured
values of becomes nearly Gaussian for
R (Naghizadeh-Khouei & Clarke
1993). The significance level used in other fields of astronomy
varies. To accept a measured polarization as significantly different
from zero, polarimetry uses C = 3 (the 99.7% confidence interval)
(Walborn 1968; Serkowski et al. 1975; Piirola 1977); if R
, the measured polarization is
considered to be zero. X-ray astronomy uses C = 3 (Forman et al. 1978;
Wood et al. 1984) for the detection of a source. Pulsar astronomy uses
C (Hulse & Taylor 1974) for the
detection of a pulsar. Following Sofia et al. (1969) for proper
motions, we adopted the significance criterion C = 2 and assumed that
no proper motion was detected if R
(i.e., if the 95% confidence level included zero).
The pdf for parallax is also non-Gaussian, but can be acceptably
represented by a normal distribution at larger values of R under
certain reasonable assumptions (Kovalevsky 1998). Kovalevsky suggests
R = 2.5 as the signal to noise ratio above which the pdf of parallax
measurements can be treated as a normal distribution. For consistency,
we adopted C = 2 for parallax measurements, and accepted the
hypothesis that no parallax was detected if R
. Under this criterion, only two of
the sources in Table 1, 4U1145-619 and HZ43, have measured
parallaxes. The other sources have parallaxes consistent with zero,
and 2 lower limits are given for
their distance.
The proper motion of each source in Table 1 except HD152667 is
significantly different from zero. For the six sources for which only
lower limits on the distance can be obtained from Hipparcos
observations, only a lower limit on their tangential velocity,
, is given in Table 1.
Table 2 gives the peculiar tangential velocity, t, each source
would have if it were at the distance given in Column 2. For the
six sources for which no Hipparcos parallax was measured, the
distances in Column 2 are estimated from spectroscopic parallax
or from the distance gradient of absorption in the field (cf. the
references given in Table 2). For galactic rotation we used the
Oort constants A = 14.82
km s^{-1} kpc^{- 1}
and B = -12.37 km s^{-
1} kpc^{-1} derived by Feast & Whitelock (1997) from
Hipparcos observations of Galactic Cepheids. For the solar motion we
used (,
, )
= (11, 15, 7) km s^{-1}
(Jaschek & Valbousquet 1994, 1993). The uncertainty in t includes
no contribution from the uncertainty in the distance for any sources
except 4U1145-619 and HZ43. The peculiar tangential velocity given for
HD152667 is a 2 upper limit at a
distance of 2 kpc.
**Table 2.** Peculiar Tangential Velocities.
Notes:
a) Lyubimkov et al. (1997)
b) Steele et al. (1998)
c) Sadakane et al. (1985)
d) Schild et al. (1971); Crawford et al. (1971)
e) Bolton & Herbst (1976)
f) Margon et al. (1973)
© European Southern Observatory (ESO) 1999
Online publication: October 14, 1999
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