4. Proper motion and position of Polaris
4.1. Center-of-mass of UMi AP
The proper motion of the center-of-mass (cms) of the closest components A and P of UMi has already been derived in Sect. 3.1.1 for the epoch . This proper motion is then transformed to the other epochs by using strict formulae, assuming a linear motion of the cms of AP in space and time. The values of for the epochs 1991.25 and 2000.0 are given in Table 6.
In order to derive the position of the center-of-mass of UMi AP (Table 6), we first transform the HIPPARCOS position of the photo-center of AP from epoch 1991.25 to using , since corresponds best to the effective mean epoch of the HIPPARCOS observations. Then we subtract from the orbital displacements , where is calculated from the derived astrometric orbits (prograde and retrograde), using the averaging method described by Eq. (17). This gives us the position at the epoch . Using the proper motion , we transform from the epoch to the standard epoch 2000.0. For the convenience of those users who like to use the HIPPARCOS standard epoch, , we give also the position for this epoch . The values which should be used for predicting the position and its mean error are given in Table 6 in bold face. The right ascension is given alternatively in the classical notation (h, m, s) and, as done in the HIPPARCOS Catalogue, in degrees and decimals of degrees. As discussed in Sect. 3.2.3, we propose to use preferentially the retrograde orbit.
The position at an arbitrary epoch t can be derived by using the strict formulae for epoch transformation, using the epochs 2000.0 or 1991.25 as a starting epoch. The mean error of should be derived from
This equation assumes that and are not correlated. This assumption is not strictly true. However, for most applications it is not neccessary to allow for correlations, because for epoch differences larger than a few years, the second term in Eq. (20) is fully dominating. The correlation between and is negligably small (only caused by the tiny correlation between and ).
All the quantities given in Table 6 refer to the HIPPARCOS/ICRS system and to the equinox J2000 (but to various epochs).
4.2. Orbital corrections for the photo-center of UMi AP
In order to obtain a prediction for the instantaneous position of the photo-center of UMi AP at an epoch t, one has to add the orbital correction to the position of the center-of-mass :
The ephemerides for the orbit of the photo-center of AP are given in Table 7. The orbital elements used in calculating the ephemerides are those listed in Table 4. Usually it is allowed to neglect the effect that the system is slightly rotating (/year), due to the motion of Polaris on a great circle. Table 7 lists also the position of the intantaneous photo-center at periastron, apastron, and at . The small difference between the instantaneous position and the averaged position (Sect. 3.1.5) of the photo-center at shows that the deviations of the fitting straight line from the actual orbits remain mostly below 1 mas within the interval of years of the HIPPARCOS observations, since these deviations reach their maximum at the borders of , namely about twice the deviation at . The very small deviations from a straight line explain also why we were, during the HIPPARCOS data reduction, unable to obtain an orbital (O) solution or an acceleration (G) solution for Polaris, although we tried to do so.
At the end of Table 6, we give the (constant) off-set between the mean photo-center and the center-of-mass. All values are valid for the equinox J2000.0, and for the orientation of the system at epoch 1991.31 (which differs from that at epoch 2000.0 by only).
The typical mean error of , due to the uncertainties in the orbital elements (mainly in ), is about 5 mas. It varies, of course, with the orbital phase, approximately between 2 mas and 7 mas. However, a detailed calculation of this mean error is often unnecessary for deriving the mean error of the prediction for , since the mean error of is governed by the mean error of for epoch differences larger than about 20 years.
4.3. Comparison of positions
In Table 8 we compare positions predicted by our results with those predicted by HIPPARCOS and by the FK5.
Table 8. Comparison between predicted positions for the photo-center of UMi AP
At epoch , the positions of the photo-center of AP predicted by our results (for both types of orbits) agree with the HIPPARCOS position by construction (except for the slight difference between the instantaneous and averaged position).
From Fig. 1 we see that the HIPPARCOS predictions for small epoch differences , say for years, are also in good agreement with our predictions, since the HIPPARCOS data are essentially a tangent to our astrometric orbits. In other words, the HIPPARCOS data are a good short-term prediction (relative to ) in the terminology of Wielen (1997). For larger epoch differences (Table 8), the HIPPARCOS prediction for starts to deviate significantly from our predictions. Going to the past, e.g. to , the differences reach large values of about 300 mas = 0."3 in each coordinate. Such differences are already larger than the measuring errors of some meridian circles at that time, especially for Polaris. (The formal mean errors in and of the position predicted by the linear HIPPARCOS solution at the epoch 1900 are 43 mas and 50 mas only.) The reason for the failure of a linear prediction based directly on the HIPPARCOS Catalogue is the fact that the quasi-instantaneously measured HIPPARCOS proper motion of Polaris contains an orbital motion of about 5 mas/year as a `cosmic error'.
Our data reproduce rather well the FK5 positions at the central FK5 epochs. This is to be expected, since we have made use of these positions in determining (and hence ). For , the FK5 prediction deviates rather strongly from our values. This is in accordance with the mean error of and the large epoch differences . (The mean errors in and of the FK5 position (reduced to the HIPPARCOS system) are at the central epochs (given in Table 8) 37 mas and 34 mas, and at epoch 1991.31 72 mas and 66 mas.)
How large are the differences between the positions which we predict if we use either the retrograde orbit or the prograde one ? At , the differences are nearly zero by construction. At other epochs, the orbital differences can be seen in Fig. 1. To these differences in the orbital corrections, we have to add the slight positional differences which are due to the differences in of both orbits. The total differences between the prograde and retrograde orbit are shown at the end of Table 8 for some epochs. An extremum in these differences occurs in (+ 68 mas) and in (- 59 mas) at about the year 2012.
4.4. Space velocity of Polaris
From the derived proper motions of the center-of-mass of UMi AP (Table 6, retrograde orbit), from the radial velocity (Table 4), and from the HIPPARCOS parallax (Eq. 14), we derive the three components U, V, W of the space velocity v of Polaris (Table 9). We neglect a possible intrinsic K term in the pulsating atmosphere of the Cepheid UMi A (Wielen 1974). This is probably justified, especially in view of the very small amplitude of the radial velocity due to pulsation.
Table 9. Space velocity of the center-of-mass of UMi AP. For detailed explanations see Sect. 4.4.
The velocity component U points towards the galactic center, V in the direction of galactic rotation, and W towards the galactic north pole. The velocity is measured relative to the Sun. The velocity refers to the local standard of rest. For the solar motion we use = (+ 9, + 12, + 7) km/s, proposed by Delhaye (1965). The velocity is the peculiar velocity of Polaris with respect to the circular velocity at the position of Polaris (see Wielen 1974). For the required Oort constants of galactic rotation, we adopt (km/s)/kpc and (km/s)/kpc. As mentioned in Sect. 2.3, the velocity of the center-of-mass of UMi AP may differ from that of UMi AP+B by a few tenth of a km/s.
The peculiar velocity of Polaris is reasonable for a classical Cepheid. According to Wielen (1974), the velocity dispersions (, , ) for nearby classical Cepheids are (8, 7, 5) km/s. Hence only the V component of of Polaris is slightly larger than expected on average.
© European Southern Observatory (ESO) 2000
Online publication: July 27, 2000