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Astron. Astrophys. 360, 399-410 (2000)

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3. Astrometric orbit of [FORMULA] UMi AP

In this section we determine the astrometric orbit of the photo-center of the pair [FORMULA] UMi AP (i.e. essentially of A) with respect to the center-of-mass of AP. We adopt all the elements of the spectroscopic orbit of A in the system AP, derived by Kamper (1996). The remaining elements, i.e. the orbital inclination i and the nodal length [FORMULA], are basically obtained from the following considerations:

The observed difference [FORMULA] between the instantaneous proper motion of [FORMULA] UMi A, provided by HIPPARCOS for an epoch [FORMULA], and the mean proper motion of [FORMULA] UMi A, provided by long-term, ground-based observations, summarized in the FK5, is equal to the tangential component of the orbital velocity of A with respect to the center-of-mass of the pair AP. Using the spectroscopic orbit of A and the HIPPARCOS parallax, we can predict [FORMULA] for various adopted values of i and [FORMULA]. Comparing the predicted values of [FORMULA] with the observed difference [FORMULA], we find i and [FORMULA]. The length of the two-dimensional vector of [FORMULA] gives us the inclination i; the direction of [FORMULA] fixes then the ascending node [FORMULA]. Unfortunately, two values of i, namely i and [FORMULA], predict the same value for [FORMULA] (see Fig. 1). This ambiguity corresponds to the fact that [FORMULA] itself does not allow us to differentiate between a prograde orbit and a retrograde one. In the case of [FORMULA] UMi AP, it is fortunate that the ground-based observations of the Allegheny Observatory strongly favour the retrograde orbit over the prograde one.

[FIGURE] Fig. 1. Astrometric orbit (prograde or retrograde) of the photo-center of [FORMULA] UMi AP. The retrograde orbit is our preferred solution. For detailed explanations see Sect. 3.2.4

3.1. The determination of [FORMULA]

3.1.1. The proper motion of the center-of-mass of AP

We determine first the proper motion [FORMULA] of the center-of-mass of the pair AP. (µ is used here for [FORMULA] or for [FORMULA]). The proper motion [FORMULA] of [FORMULA] UMi given in the FK5 should be very close to [FORMULA], since the ground-based data are averaged in the FK5 over about two centuries, which is much larger than the orbital period of AP of about 30 years. In Table 1, we list [FORMULA] in the FK5 system and, by applying appropriate systematic corrections, in the HIPPARCOS/ICRS system. The mean errors of [FORMULA] in the HIPPARCOS system include both the random error of [FORMULA] and the uncertainty of the systematic corrections.


[TABLE]

Table 1. Mean proper motion of the photo-center of [FORMULA] UMi AP


Another determination of [FORMULA] is based on the positions [FORMULA] and [FORMULA] at the central epochs [FORMULA] and [FORMULA] of the HIPPARCOS Catalogue and of the FK5. The designation x stands for [FORMULA] or [FORMULA], where [FORMULA] is the right ascension and [FORMULA] the declination of [FORMULA] UMi. The position [FORMULA] represents a time-averaged, `mean' position in the sense of Wielen (1997). Before being used, [FORMULA] must be reduced to the HIPPARCOS/ICRS system.

The HIPPARCOS position is (approximately) an `instantaneously' measured position of the photo-center of AP. Before combining the HIPPARCOS position with [FORMULA] to a mean proper motion [FORMULA], we have to reduce [FORMULA] to the mean position [FORMULA] of the photo-center of AP at time [FORMULA]. This is done by going first from [FORMULA] to the center-of-mass [FORMULA] by subtracting from [FORMULA] the orbital displacement [FORMULA] predicted by the astrometric orbit of the photo-center of AP. Then we have to add to [FORMULA] the (constant) off-set between the mean position of the photo-center [FORMULA] and the center-of-mass (see Fig. 1). Using now [FORMULA] and [FORMULA], we obtain

[EQUATION]

where [FORMULA] is the semi-major axis of the orbit of the photo-center of AP around the center-of-mass of AP. The other elements of this orbit are: eccentricity e, inclination i, longitude of periastron [FORMULA], position angle of the ascending node [FORMULA], orbital period P, epoch of periastron passage [FORMULA]. The quantities in the Eqs. (1) and (2) which follow after [FORMULA] are just the Thiele-Innes elements B and A. The equations use the fact that, in the orbital plane, the time-averaged position is located on the major axis, towards the apastron, at a distance of [FORMULA] from the center-of-mass.

If [FORMULA] and [FORMULA] would change linearly with time, we could determine the mean proper motion [FORMULA] from

[EQUATION]

However, for Polaris we should use more accurate formulae because it is so close to the celestial pole. We determine [FORMULA] strictly by requiring that [FORMULA] is that proper motion which brings the object from [FORMULA] to [FORMULA]. For calculating the (small) foreshortening effect, we have adopted the radial velocity of the center-of-mass of AP, [FORMULA] km/s (Kamper 1996).

The agreement between the two mean proper motions [FORMULA] and [FORMULA] is rather good (Table 1). For determining the best value [FORMULA] of the mean motion of the photo-center, which is equal to the proper motion of the center-of-mass, we take the weighted average of [FORMULA] and [FORMULA]. Since the orbital corrections to [FORMULA] are different for the prograde and retrograde orbits, we have two values for [FORMULA] and hence for [FORMULA]. In both cases, we had to iterate the determinations of the orbital elements (i and [FORMULA]) and of [FORMULA] (and hence [FORMULA]), since [FORMULA] depends on the orbital corrections. The values for [FORMULA] finally adopted are listed in Table 1.

3.1.2. The HIPPARCOS proper motion [FORMULA]

The HIPPARCOS proper motion [FORMULA] of Polaris (ESA 1997) refers to the photo-center of AP. Basically, [FORMULA] is the sum of the proper motion [FORMULA] of the center-of-mass (cms) of AP and of the orbital motion [FORMULA] (abbreviated as [FORMULA]) of the photo-center of AP with respect to the cms of AP at time [FORMULA]:

[EQUATION]

During the reduction of the HIPPARCOS data, a linear `standard' solution was applied to Polaris. The variation of [FORMULA] during the period of observations of about three years was neglected. This slightly complicates the comparison of the observed [FORMULA] with the orbital ephemerides. In Sect. 3.1.5 we assume that [FORMULA] is obtained from a linear fit to quasi-continuously measured true positions over a time interval [FORMULA], centered at time [FORMULA]. From the correlation coefficients given in the HIPPARCOS Catalogue, we derive for the central epochs [FORMULA] and [FORMULA]. We neglect the slight difference between [FORMULA] and [FORMULA] and use the average of both, namely [FORMULA]. From the epochs of the individual observations of Polaris by HIPPARCOS, we estimate [FORMULA] years.

3.1.3. The observed value of [FORMULA]

The observed value of [FORMULA] is derived from

[EQUATION]

The values of [FORMULA] in [FORMULA] and [FORMULA], derived from Eq. (5), are listed in Table 1 and Table 2. Table 2 gives also the total length [FORMULA] of the [FORMULA] vector,

[EQUATION]

and the position angle [FORMULA] of the [FORMULA] vector,

[EQUATION]


[TABLE]

Table 2. Proper-motion difference [FORMULA] between [FORMULA] and [FORMULA] of the photo-center of [FORMULA] UMi AP at the epoch [FORMULA]


All the values are valid for the equinox J2000 in the HIPPARCOS/ICRS system at the epoch [FORMULA].

Since [FORMULA] depends on the direction of motion in the orbit (prograde or retrograde), this is also true for [FORMULA], and we obtain therefore two values for [FORMULA]. Table 2 shows that [FORMULA] mas/year and [FORMULA] are statistically quite significant and rather well determined. A value of [FORMULA] mas/year corresponds to a tangential velocity of 3.05 km/s. Hence the `instantaneous' HIPPARCOS proper motion of Polaris has a significant `cosmic error' (Wielen 1995a, b 1997; Wielen et al. 1997, 1998, 1999a, b) with respect to the motion of the center-of-mass. If Polaris were not already known as a close binary, our [FORMULA] method (Wielen et al. 1999a) would have detected Polaris to be a [FORMULA] binary because of its large test parameter [FORMULA] 6.18 for [FORMULA].

3.1.4. The photo-center of [FORMULA] UMi AP

The HIPPARCOS observations refer to the photo-center of [FORMULA] UMi AP, since the pair is not resolved by HIPPARCOS. The `phase' used in constructing the HIPPARCOS Catalogue is practically identical to the phase of the photo-center, because the magnitude difference [FORMULA] of more than 6 mag between A and P is quite large and because the separation between A and P at [FORMULA] was rather moderate (about 93 mas). It can also be shown that the component B does not significantly affect the HIPPARCOS measurements of AP, because of [FORMULA] 6.61, in spite of its separation [FORMULA]. The HIPPARCOS observations have been carried out in a broad photometric band called Hp . The photo-center refers therefore to this photometric system.

The spectroscopic orbit, however, refers to component A. We have therefore to transform the value [FORMULA] of the spectroscopic orbit into [FORMULA] for obtaining an astrometric orbit of the photo-center of AP. The relation between [FORMULA] and [FORMULA] is given by

[EQUATION]

where B and [FORMULA] are the fractions of the mass [FORMULA] and the luminosity L of the secondary component P:

[EQUATION]

[FORMULA] is the magnitude difference between A and P:

[EQUATION]

Using the results given in Table 5, we find for [FORMULA] UMi AP

[EQUATION]

with an estimated error of about [FORMULA]. For calculating [FORMULA], we have assumed that [FORMULA] is the same in Hp as in V. This approximation is fully justified for our purpose. We derive (see the end of Sect. 3.1.6) from Kamper (1996) for component A:

[EQUATION]

The HIPPARCOS parallax of Polaris (ESA 1997) is

[EQUATION]

This leads to

[EQUATION]

Using Eqs. (8), (12), and (15), we obtain for the photo-center

[EQUATION]

3.1.5. The predicted value of [FORMULA]

For predicting [FORMULA], we use four elements [FORMULA] of the spectroscopic orbit derived by Kamper (1996), and [FORMULA] according to Eq. (16), all listed in Table 4. In addition we adopt various values of i and [FORMULA] in order to produce predicted values of [FORMULA] at time [FORMULA] as a function of i and [FORMULA].

Since the observed value of [FORMULA] is not an instantaneously measured tangential velocity, we mimic the HIPPARCOS procedure of determining [FORMULA]. We calculate the positions [FORMULA] of the photo-center of AP with respect to the cms of AP as a function of time, using standard programs for the ephemerides of double stars. We then carry out a linear least-square fit to these positions over a time interval of length [FORMULA] years, centered at [FORMULA]:

[EQUATION]

Tests have shown that especially [FORMULA] is not very sensitive against small changes in the slightly uncertain quantity [FORMULA]. Actual numbers for the predicted values [FORMULA] are given in the Tables 2 and 3.

3.1.6. The problem of [FORMULA] and of [FORMULA]

In his paper, Kamper (1996, his Table III) gives for his best orbit (DDO+Lick Data) a value for [FORMULA]. This is exactly the value derived by Roemer (1965) from the Lick Data and also quoted in Kamper's Table III under `Lick Data'. There are three possibilities for this coincidence: (1) Kamper has adopted this value of [FORMULA] as a fixed input value from Roemer. Nothing is said about this in his paper. (2) Kamper found from a full least-square solution by chance the same values for [FORMULA] and its mean error as quoted for Roemer. Such a mere accident is highly improbable. (3) The identical values of [FORMULA] and its mean error in the two columns of Kamper's Table III occured due to a mistake or misprint. However, Kamper has not published any erratum in this direction.

Dr. Karl W. Kamper died in 1998 (Bolton 1998). We tried to get clarification on the problem of [FORMULA] from colleagues of Dr. Kamper, but they were unfortunately unable to help us in this respect. Hence we are inclined to accept the possibility (1). However, even then there is an additional problem with the mean error of [FORMULA]. Kamper has obviously overlooked that Roemer (1965) gave probable errors instead of mean errors. Hence the mean error of [FORMULA] according to Roemer should read [FORMULA] 0.12 in Kamper's Table III.

For our purpose, a value of [FORMULA] closer to [FORMULA] should be chosen. Using [FORMULA] and [FORMULA] years, we obtain an alternative value (two periods later) of

[EQUATION]

We have tested this value by carrying out an unweighted least-square fit to the mean radial velocities of [FORMULA] UMi A listed in Table II of Kamper (1996). In this solution we solved for [FORMULA] only, while we adopted all the other spectroscopic elements as given by Kamper (1996). We obtained [FORMULA], in good agreement with Eq. (19). However, the formally most accurate radial velocity listed in the last line of Kamper's Table II does not fit perfectly (O-C = - 0.15 km/s) his final orbit with [FORMULA] according to Eq. (19), but rather indicates the value of [FORMULA]. The independent radial-velocity data published by Dinshaw et al. (1989) lead us to [FORMULA] with a very small formal error. This is in good agreement with Eq. (19). Hence we have finally adopted [FORMULA] as given by Eq. (19). An error of [FORMULA] 0.13 years introduces errors of [FORMULA] in i and of [FORMULA] in [FORMULA], which are small compared to the errors in i and [FORMULA] due to the uncertainties in [FORMULA] and in the observed value of [FORMULA].

The values of [FORMULA], and e given by Kamper (1996) in his Table III under DDO+Lick Data are unfortunately not consistent. If we accept [FORMULA], and e, we find [FORMULA] AU, while Kamper gives in his Table III 2.90 AU. In the text of his paper, Kamper gives 2.9 AU for [FORMULA]. Has he rounded 2.934 to 2.9 and later inserted this rounded value as 2.90 into his Table III? We prefer to trust [FORMULA] and e, and hence we use for [FORMULA] the value of 2.934 AU (see Sect. 3.1.4) in our investigation.

3.2. The astrometric orbit

3.2.1. Determination of the inclinationi

In Table 3 we compare the observed values of [FORMULA] (from Sect. 3.1.3 and Table 2) with the predicted values of [FORMULA] (using the procedures described in Sect. 3.1.5 and the elements [FORMULA] and [FORMULA] given in Table 4) for different trial values of the inclination i. The length [FORMULA] of the vector [FORMULA] is obviously not a function of the nodal direction [FORMULA]. The mean error of the predicted value of [FORMULA] includes the uncertainties in all the orbital elements except in i and [FORMULA].


[TABLE]

Table 3. Determination of the inclination i from [FORMULA]



[TABLE]

Table 4. Orbital elements of [FORMULA] UMi AP


The best agreement between the observed and predicted values of [FORMULA] occurs for [FORMULA] (prograde orbit) and for [FORMULA] (retrograde orbit). The uncertainties in the observed value of [FORMULA] and in the orbital elements (mainly in [FORMULA]) lead to an uncertainty in i of [FORMULA].

Our values for the inclination i of the two orbits do not fulfill strictly the expected relation [FORMULA]. The reason is the following: i is determined (Table 3) from two slightly different values [FORMULA] for the prograde and retrograde orbits (Table 2). The difference in the [FORMULA] values stems from a slight difference in [FORMULA] and hence in [FORMULA] (Table 1), and this difference in [FORMULA] is caused by a difference in [FORMULA] (Eq. (3)). The difference in the position [FORMULA] of the mean photo-center is due to the small, but totally different corrections which have to be added to the observed HIPPARCOS position [FORMULA] in order to obtain the mean photo-center (see Table 6 and Fig. 1). As mentioned already at the end of Sect. 3.1.1, we had to iterate our procedure of determining i and [FORMULA], since the corrections depend on these orbital elements.

The fit between the observed and predicted values of [FORMULA] is rather pleasing. It is not granted that such a fit is always possible. In the case of Polaris, for example, the spectroscopic orbit and the HIPPARCOS parallax together require a minimum value of [FORMULA] of 2.00 mas/year, which occurs for [FORMULA]. There is no formal upper limit for [FORMULA] for [FORMULA]. However, the requirement that the component P is not visible in the combined spectrum of AP gave for a main-sequence companion P a spectral type later than A8V (Sect. 2.2), or [FORMULA]. Combined with the mass function of the spectroscopic orbit, [FORMULA], and with a reasonable estimate of [FORMULA], this gives a lower limit for i of about [FORMULA], which corresponds to [FORMULA] mas/year. Our observed value of [FORMULA] of about 5 mas/year fulfills nicely the range condition of 2 mas/year [FORMULA] mas/year.

3.2.2. Determination of the nodal length [FORMULA]

Having fixed the inclination i in Sect. 3.2.1, we now determine [FORMULA] from a comparison of the observed and predicted values of the direction [FORMULA] of the vector [FORMULA]. The difference (modulo 360o) between the observed value of [FORMULA] and the predicted value of [FORMULA] for [FORMULA] gives just that desired value of [FORMULA] for which the observed and predicted values of [FORMULA] agree. We find [FORMULA] for the prograde orbit and [FORMULA] for the retrograde orbit. The uncertainties in the observed value of [FORMULA] and in the orbital elements (now mainly in i and [FORMULA]) lead to an uncertainty in [FORMULA] of [FORMULA] or [FORMULA].

The quality of the fit in the components of [FORMULA] in [FORMULA] and [FORMULA] can be judged from the data given in Table 2. The overall agreement is quite good.

3.2.3. The ambiguity problem ofi

If we know only the vector [FORMULA] at one epoch and the spectroscopic orbit of a binary, then there is an ambiguity (i or [FORMULA]) in the inclination i, i.e. in the direction of motion in the astrometric orbit. In the prograde (or `direct') orbit ([FORMULA]), the position angle of P relative to A increases with time, in the retrograde orbit ([FORMULA]) it decreases. The reason for the ambiguity is the fact that [FORMULA] itself does not indicate whether the orbit will turn to the left-hand side or to the right-hand side (see Fig. 1).

In principle, the knowledge of the mean position of the photo-center predicted by the FK5 for [FORMULA] would resolve the ambiguity. However, the mean errors of this predicted position of [FORMULA] 72 mas in [FORMULA] and [FORMULA] 66 mas in [FORMULA] are so large with respect to the differences between [FORMULA] and [FORMULA], which are less than 26 mas (Table 6), that this method is not useful in our case.

At present, the best solution of the ambiguity problem is provided by the results of the photographic observations carried out at the Allegheny Observatory, which we discussed already in Sect. 2.2. While the full astrometric orbit based on the Allegheny data (Kamper 1996) is not very trustworthy, the Allegheny data give strong preference for a retrograde orbit (in contrast to a direct one). This can be seen best in Fig. 3 of Kamper (1996): The minimum of the residuals (dashed line) occurs for [FORMULA], and for this range of i the semi-major axis derived from the Allegheny data is quite reasonable. For our preferred value of [FORMULA], we read off from Kamper's Fig. 3 a value of [FORMULA] mas with an estimated uncertainty of [FORMULA] 9 mas. This is in very good agreement with our result, 28.7 [FORMULA] 2.8 mas. Even the nodal length [FORMULA] derived by Kamper [FORMULA] is compatible with our result [FORMULA]. Kamper's determination of [FORMULA] is very uncertain and therefore not in contradiction to our value (130o). He himself says in the text of the paper that `all inclinations between 135o and 180o are equally satisfactory' in fitting the Allegheny data. (There is a small mistake in Kamper's discussion of this point: He claims in the text `that the minimum scatter is for an inclination of almost 90o, which results in a face-on orbit'. The relative clause after 90o, his own Fig. 3 and his Table III all indicate that `90o' should be replaced by `180o'.)

A new astrometric space mission will immediately resolve the ambiguity, since it shall then be clear to which side of our [FORMULA] vector (Fig. 1) the orbit will have turned over. Probably the much higher accuracy of a new space mission will allow to determine the direction (and amount) of the instantaneous acceleration (i.e. to obtain a `G solution' in the HIPPARCOS terminology, if not even a full orbital `O solution').

3.2.4. Resulting orbits of [FORMULA] UMi AP

The resulting orbits of the photo-center of [FORMULA] UMi AP are listed in Table 4. As explained in Sect. 3.2.3, the retrograde orbit should be preferred. The semi-major axes of the orbits of [FORMULA] UMi A and P itself, relative to the center-of-mass of AP, and that of P relative to A, are given in Table 5.


[TABLE]

Table 5. Physical properties of [FORMULA] UMi A and P


In Fig. 1, the two orbits (prograde and retrograde) of the photo-center of AP are illustrated. The zero-point of the coordinates [FORMULA] and [FORMULA] is the HIPPARCOS position [FORMULA] at epoch [FORMULA]. The zero-point is then comoving with the center-of-mass (cms) of either the prograde orbit or the retrograde one. Therefore the orbits stay fixed in these coordinates. Since the proper motion [FORMULA] of the cms of the two orbits differs slightly (Table 6), the linear motion of the position [FORMULA] predicted from the HIPPARCOS Catalogue, differs slightly for the two cases. The indicated motion of [FORMULA] corresponds to [FORMULA] (Table 2). Hence by construction, the motion of [FORMULA] is a tangent to the corresponding orbit, except for the slight difference between the averaged position and the instantaneous position of the photo-center at [FORMULA]. The dots on the orbits mark the positions in intervals of one year, the years 1990, 1995, 2000, etc. being accentuated by a larger dot. We indicate also the true major axis, on which periastron, center-of-mass, mean photo-center, and apastron are located. In addition we plot the line of nodes. The position of the ascending node is indicated by [FORMULA]. Fig. 1 demonstrates clearly that the position predicted by HIPPARCOS is drifting away from the actual position of the photo-center of AP.


[TABLE]

Table 6. Proper motion [FORMULA] and position [FORMULA] of the center-of-mass of [FORMULA] UMi AP.
Notes:
Explanation for (+): To the quantities marked with (+) in the second part of Table 6, one has to add the HIPPARCOS positions [FORMULA] at the corresponding epochs which are given in the first two lines of the third part of Table 6.


3.2.5. Derived physical properties of [FORMULA] UMi P

In Table 5, we summarize some physical properties of the components A and P of [FORMULA] UMi.

The mass of [FORMULA] UMi A is derived from the mass-luminosity relation for Cepheids given by Becker et al. (1977). Since we use the luminosity based on the HIPPARCOS distance (132 pc), our value of 6 [FORMULA] is higher than that of other authors who have used a smaller distance.

The age of [FORMULA] UMi A, and therefore of the whole system of Polaris, can be estimated from the period-age relation for Cepheids (Becker et al. 1977, Tammann 1969). Using [FORMULA] days (see Sect. 2.1), we derive an age [FORMULA] of about [FORMULA] years.

The spectroscopic orbit provides the mass function [FORMULA]. Adopting the inclination [FORMULA] of the retrograde orbit and [FORMULA] for the Cepheid, we obtain [FORMULA] for the component P. Using this value for [FORMULA], we estimate for a star on the zero-age main sequence an absolute magnitude of [FORMULA] and a spectral type of F0V. The magnitude difference [FORMULA] between A and P is then about 6 .[FORMULA]5. As mentioned in Sect. 2.2, a White Dwarf is ruled out by the IUE spectra and the low age of Polaris. Our estimate for [FORMULA] itself would not violate the Chandrasekhar limit for White Dwarfs, if we consider the uncertainty in [FORMULA] of [FORMULA]. A neutron-star nature of P is possible, but not very likely. In any case, the adopted main-sequence nature of P is a rather probable solution which is in agreement with all observational constraints. Our derived astrometric orbit does not depend sensitively on the nature of P, since all the possible solutions indicate a very small value of [FORMULA], so that the difference between the positions of A and of the photo-center of AP (see Sect. 3.1.4) is small in any case.

Using [FORMULA] and [FORMULA] as derived above, the predicted semi-major axis of the orbit of P relative to A is 142 [FORMULA] 21 mas. Our Table 7 provides a prediction of the position of P relative to A, if the ephemerides for [FORMULA] are multiplied by about -4.95. The separation between A and P should be presently about 160 mas, is slightly increasing to 186 mas until 2006, and is then decreasing to about 38 mas in 2017. Hence the next decade is especially favourable for resolving the pair [FORMULA] UMi AP. Of course, the large magnitude difference of more than 6m makes a direct observation of [FORMULA] UMi P rather difficult. Since A and P seem to have nearly the same colour (as judged from the spectral types given in Table 5), the magnitude difference should be (unfortunately) rather the same in all the photometric bands. Nevertheless we hope that modern interferometric techniques or the use of other devices may be able to resolve the pair [FORMULA] UMi AP during the next decade. Our paper provides hopefully a fresh impetus for such investigations.


[TABLE]

Table 7. Orbital corrections [FORMULA] for the photo-center of [FORMULA] UMi AP


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Online publication: July 27, 2000
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