2. Overview of the Polaris system
Polaris is a multiple stellar system, which consists of a close pair, UMi A and UMi P (= UMi a), and a distant companion, UMi B, and two distant components, UMi C and UMi D. We use here the designation `P' for a close companion of A, which was used in the IDS and was adopted by the CCDM and by the HIPPARCOS Input Catalogue, rather than the traditional version `a', which is used e.g. by the WDS and by CHARA).
2.1. The Cepheid UMi A
The main component of Polaris is a low-amplitude Cepheid with a pulsational period of about 3.97 days. This period is increasing with time (e.g. Kamper & Fernie 1998). According to Feast & Catchpole (1997), UMi A is a first-overtone pulsator (rather than a fundamental one), since UMi A is too luminous for a fundamental pulsator, if they apply their period-luminosity relation (for fundamental pulsators) to Polaris. The fundamental period of UMi A would follow as days, if the observed period is the first-overtone period (using the relation between and derived by Alcock et al. (1995) for Galactic Cepheids). An extraordinary property of UMi A among the Cepheids is that the amplitude of its pulsation has been dramatically declined during the past 100 years, as seen both in the light curve and in the radial-velocity curve (Arellano Ferro 1983; Kamper & Fernie 1998, and other references given therein). The full amplitude was about 0 .12 in and about 6 km/s in radial velocity before 1900, and seems now to be rather constant at a level of only 0 .03 in and at 1.6 km/s in radial velocity. An earlier prediction (Fernie et al. 1993) that the pulsation should cease totally in the 1990s was invalid. A discussion of the HIPPARCOS parallax and of the absolute magnitude of UMi A is given in the next Sect. 2.2.
2.2. The spectroscopic-astrometric binary UMi AP
The Cepheid UMi A is a member of the close binary system UMi AP. This duplicity was first found from the corresponding variations in the radial velocity of UMi A. However, the interpretation of the radial velocities of UMi A in terms of a spectroscopic binary is obviously complicated by the fact that UMi A itself is pulsating and that this pulsation varies with time. We use in this paper the spectroscopic orbit derived by Kamper (1996), which is based on radial velocity observations from 1896 to 1995. Kamper (1996) took into account changes in the amplitude of the pulsation and in the period of pulsation, but used otherwise a fixed sinusoid for fitting the pulsation curve. In an earlier paper, Roemer (1965) considered even `annual' changes in the form of the pulsation curve. In Table 4, we list the elements of the spectroscopic orbit of A in the pair AP given by Kamper (1996, his Table III, DDO + Lick Data). The orbital period of UMi AP is 29.59 0.02 years, and the semi-amplitude is km/s. The value of AU corresponds to about 22 milliarcsec (mas), using the HIPPARCOS parallax.
Attempts to observe the secondary component UMi P directly or in the integrated spectrum of UMi AP have failed up to now. Burnham (1894) examined Polaris in 1889 with the 36-inch Lick refractor and found no close companion to UMi A (nor to UMi B). Wilson (1937) claimed to have observed a close companion by means of an interferometer attached to the 18-inch refractor of the Flower Observatory. Jeffers (according to Roemer (1965) and to the WDS Catalogue) was unable to confirm such a companion with an interferometer at the 36-inch refractor of the Lick Observatory. HIPPARCOS (ESA 1997) has not given any indication for the duplicity of Polaris. Speckle observations were also unsuccessful (McAlister 1978). All these failures to detect UMi P directly are not astonishing in view of the probable magnitude difference of A and P of more than 6m and a separation of A and P of less than 0."2 (see Sect. 3.2.5). Roemer and Herbig (Roemer 1965) and Evans (1988) searched without success for light from UMi P in the combined spectrum of UMi AP. From IUE spectra, Evans (1988) concluded that a main-sequence companion must be later than A8V. This is in agreement with our results for UMi P, given in Table 5. A white-dwarf companion is ruled out by the upper limit on its effective temperature derived from IUE spectra and by considerations on its cooling age, which would be much higher than the age of the Cepheid UMi A (Landsman et al. 1996).
After Polaris had become known as a long-period spectroscopic binary (Moore 1929), various attempts have been made to obtain an astrometric orbit for the pair UMi AP. Meridian-circle observations were discussed by Gerasimovic (1936) and van Herk (1939). While van Herk did not find a regular variation with a period of 30 years, Gerasimovic claimed to have found such a modulation. However, the astrometric orbit of the visual photo-center of UMi AP determined by Gerasimovich (1936) is most probably spurious, since he found for the semi-major axis of the orbit mas, which is much too high in view of our present knowledge ( mas). More recent meridian-circle observations gave no indications of any significant perturbation. This is not astonishing in view of the small orbital displacements of the photo-center of AP of always less than 0."04. Long-focus photographic observations have been carried out at the Allegheny Observatory (during 1922-1964), the Greenwich Observatory, and the Sproul Observatory (during 1926-1956), mainly with the aim to determine the parallax of Polaris. The discussion of this material by Wyller (1957, Sproul data) and by Roemer (1965, Allegheny data) did not produce any significant results. The Allegheny plates were later remeasured and rediscussed by Kamper (1996), using his new spectroscopic orbital elements. Kamper also rediscussed the Sproul plates. While the Sproul data gave no relevant results for UMi AP, the Allegheny data gave just barely significant results, such as mas. For our purpose (see Sect. 3.2.3), the most important implication derived by Kamper (1996) from the Allegheny data is that the astrometric orbit of AP is most probably retrograde, not prograde.
In Sect. 3 we shall present a more reliable astrometric orbit of UMi AP by combining ground-based FK5 data with HIPPARCOS results, using Kamper's (1996) spectroscopic orbit as a basis.
The HIPPARCOS astrometric satellite has obtained for UMi AP a trigonometric parallax of mas, which corresponds to a distance from the Sun of pc. In the data reduction for HIPPARCOS, it was implicitely assumed that the photo-center of the pair AP moves linearly in space and time, i.e. a `standard solution' was adopted. This is a fairly valid assumption, since the deviations from a linear fit over the period of observations by HIPPARCOS, about 3 years, are less than 1 mas (see Sect. 4.2). Hence the HIPPARCOS parallax obtained is most probably not significantly affected by the curvature of the orbit of AP. Nevertheless, it may be reassuring to repeat the data reduction of HIPPARCOS for UMi, adopting the astrometric orbit derived here for implementing the curvature of the orbit of the photo-center of UMi AP.
The mean apparent visual magnitude of the combined components A and P is (Feast & Catchpole 1997). This agrees fairly well with the HIPPARCOS result (ESA 1997) . In accordance with most authors we assume that the reddening and the extinction of the Polaris system are essentially zero (e.g., Turner 1977, Gauthier & Fernie 1978), within a margin of in and in . Using the HIPPARCOS parallax, we find for the mean absolute magnitude of AP . If we use our results of Table 5 for component P, i.e. , and subtract the light of P from , then the absolute magnitude of the Cepheid component A is . Unfortunately, the pecularities in the pulsation of UMi A are certainly not very favourable for using this nearest Cepheid as the main calibrator of the zero-point of the period-luminosity relation of classical Cepheids.
2.3. The visual binary UMi (AP) - B
Already in 1779, W. Herschel (1782) discovered the visual-binary nature of Polaris. The present separation between AP and B is about 18."2. This separation corresponds to 2400 AU or 0.012 pc, if B has the same parallax as AP. Kamper (1996) has determined the tangential and radial velocity of B relative to AP. Both velocities of B agree with those of AP within about 1 km/s. Hence Kamper (1996) concludes that B is most probably a physical companion of AP, and not an optical component. The physical association between AP and B is also supported by the fair agreement between the HIPPARCOS parallax of AP ( pc) and the spectroscopic parallax of B (114 pc, as mentioned below).
The spectral type of B is F3V. The magnitude difference between B and the combined light of AP is (Kamper 1996). Using , this implies for B an apparent magnitude of . Adopting the HIPPARCOS parallax (and no extinction), we obtain for B an absolute magnitude of . The standard value of for an F3V star on the zero-age main sequence is . If we use this standard value for , we obtain for B a spectroscopic distance of pc. Similar values of the spectroscopic distance were derived (or implied) by Fernie (1966), Turner (1977), and Gauthier & Fernie (1978). These authors were interested in the absolute magnitude (and hence in the distance) of B in order to calibrate the absolute magnitude of the Cepheid A. Now the use of the HIPPARCOS trigonometric parallax is, of course, better suited for this purpose.
The typical mass of an F3V star is . If we use for the masses of A and P the values adopted in Table 5 (), we obtain for the triple system a total mass of ). We derive from "2 and the statistical relation an estimate for the semi-major axis of the orbit of B relative to AP of " or 2700 AU. From Kepler's Third Law, we get then an estimate of the orbital period of B, namely years.
From the data given above, we can estimate the acceleration of the center-of-mass of the pair UMi AP due to the gravitational attraction of UMi B. If we project this estimate of on one arbitrarly chosen direction, we get for AP a typical `one-dimensional' acceleration of about 0.003 (km/s)/century or 0.4 mas/century2. Therefore, we should expect neither in the radial velocity nor in the tangential motion of AP a significant deviation from linear motion due to the gravitational force of B during the relevant periods of the observations used. For all present purposes, it is fully adequate to assume that the center-of-mass of the pair UMi AP moves linearly in space and time. The same is true for the motion of B.
A modulation of the relative position of B with respect to the photo-center of AP with a period of about 30 years is not seen in the available observations of B. This is in accordance with our determination of the motion of the photo-center of AP with respect to the cms of AP, given in Table 7. The expected amplitude of the modulation is less than 0."04 and is obviously not large enough with respect to the typical measuring errors in the relative position of B.
The contribution of the orbital motion of the center-of-mass (cms) of AP, due to B, to the total space velocity of AP is of the order of a few tenth of a km/s. The expected value of the velocity of B relative to the cms of AP is of the order of 1 km/s.
2.4. UMi C and UMi D
In 1884 and 1890, Burnham (1894) measured two faint stars in the neighbourhood of UMi AB. In 1890.79, the component C had a separation of 44."68 from A, and the component D 82."83. According to the WDS Catalogue, the apparent magnitudes of C and D are 13 .1 and 12 .1.
The nature of the components C and D is unclear. The probability to find by chance a field star of the corresponding magnitude with the observed separation around UMi A (galactic latitude (Wielen 1974)) is of the order of 10 percent for each component. This favours on statistical grounds a physical relationship of the components C and D with A. If C and D are physical members of the Polaris system (instead of being optical components), their absolute magnitudes in V would be + 7 .5 and + 6 .5. Due to the low age of the Polaris system of about 70 million years (deduced from the Cepheid UMi A), they would either just have reached the zero-age main sequence, or they may still be slightly above this sequence (i.e. pre-main-sequence objects, Fernie 1966).
© European Southern Observatory (ESO) 2000
Online publication: July 27, 2000