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Astron. Astrophys. 350, 1085-1088 (1999)

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2. Data analysis

The Hipparcos catalogue (ESA 1997) includes measurements of the parallax, [FORMULA], and the components of proper motion in right ascension and declination, [FORMULA] and [FORMULA], of 117,955 stars, together with the statistical uncertainty associated with each measurement. The total proper motion is µ = [FORMULA] toward equatorial position angle [FORMULA] = Arc tan([FORMULA]/[FORMULA]). 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 ([FORMULA], [FORMULA]) to (µ, [FORMULA]) are formally identical to those relating the Stokes parameters (q,u) to the total linear polarization (p, [FORMULA]). Dolan & Tapia (1986) give the equations for the propagation of error from ([FORMULA], [FORMULA]) to (µ, [FORMULA]) in terms of their polarimetric counterparts. The probability density functions (pdf) of the Stoke parameters and ([FORMULA], [FORMULA]) are identical, and so the pdf of polarization and proper motion measurements must also be identical. By analogy with the Stokes parameters, ([FORMULA] and [FORMULA]) are normally distributed at high signal to noise ratios R = µ/[FORMULA] [FORMULA], 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 [FORMULA] is more complicated and depends on both [FORMULA] 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[FORMULA]. 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 [FORMULA]? 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 ([FORMULA] and [FORMULA]) are not well represented by a normal distribution when R [FORMULA]. The Rice distribution becomes nearly Gaussian for R [FORMULA] (Simmons & Stewart 1985), and all estimators of the best value of µ are consistent for R [FORMULA]. The distribution of measured values of [FORMULA] becomes nearly Gaussian for R [FORMULA] (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 [FORMULA], 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 [FORMULA] (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 [FORMULA] (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 [FORMULA]. 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[FORMULA] 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, [FORMULA], 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 [FORMULA] km s-1 kpc- 1 and B = -12.37 [FORMULA] km s- 1 kpc-1 derived by Feast & Whitelock (1997) from Hipparcos observations of Galactic Cepheids. For the solar motion we used ([FORMULA], [FORMULA], [FORMULA]) = (11, 15, 7) [FORMULA] 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[FORMULA] upper limit at a distance of 2 kpc.


Table 2. Peculiar Tangential Velocities.
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)

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Online publication: October 14, 1999