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Up: Spectropolarimetry of magnetic

4. Results

4.1. Generalities

Magnetic field moments have been derived as described in the previous section from 95 observations of 44 stars. These measurements are presented in Table 3 (click here). The columns give, in order: the star HD (or HDE) number, the heliocentric Julian date of mid-observation, the rotation phase (for stars whose rotation period is reliably known), the longitudinal field tex2html_wrap_inline2667 and its estimated uncertainty tex2html_wrap_inline2873, the crossover tex2html_wrap_inline2875 and its estimated uncertainty tex2html_wrap_inline2877, the quadratic field tex2html_wrap_inline2781 and its estimated uncertainty tex2html_wrap_inline2881, and the number of spectral lines used for the determination of the field moments. The absence of entries in the last two columns corresponds to cases when the fit of the measurements of tex2html_wrap_inline2847 by a function of the form given in Eq. (3) yields a negative value of tex2html_wrap_inline2885. As discussed in Paper V, this may happen when the square field is small compared to the accuracy which can be achieved in its diagnosis.

    Table 4: Variation of the longitudinal field: least-squares fit parameters

    Table 5: Variation of the crossover: least-squares fit parameters

    Table 6: Variation of the quadratic field: least-squares fit parameters

For the stars for which observations have been obtained repeatedly throughout the rotation cycle (including observations already reported in Papers III to V), the variations of the field moments can be well represented either by a cosine wave:
equation1547
or by the superposition of a cosine and of its first harmonic:
eqnarray1549
In these expressions, tex2html_wrap_inline3245 stands for any of the measured quantities tex2html_wrap_inline2667, tex2html_wrap_inline2875, or tex2html_wrap_inline2781. tex2html_wrap_inline3253 is the rotation phase, determined using the values of the period P and of the Julian date HJD0 of the phase origin which appear in Table 2 (click here). The mean value A0, the amplitude(s) A1 (and when significant A2), and the phase(s) tex2html_wrap_inline3265 (and possibly tex2html_wrap_inline3267) of the variations are derived through a least-squares fit of the field moment measurements by function (4) or (5). This fit is weighted according to the uncertainty of the individual measurements.

The results of those fits are presented in Table 4 (click here) (for tex2html_wrap_inline2667; the amplitudes are denoted Hi, tex2html_wrap_inline3273, and the phases tex2html_wrap_inline3275, tex2html_wrap_inline3277), Table 5 (click here) (for tex2html_wrap_inline2875, amplitudes Xi and phases tex2html_wrap_inline3283), and Table 6 (click here) (for tex2html_wrap_inline2781, amplitudes Qi and phases tex2html_wrap_inline3289). Columns 7 to 9 of these tables give the number of degrees of freedom about the fit tex2html_wrap_inline2897, the reduced tex2html_wrap_inline3293 of the fit tex2html_wrap_inline2899, and the multiple correlation coefficient R. Graphical representations of the fits also appear in several of the phase diagrams plotted below. In those diagrams, the meaning of the symbols used to represent the data points is as follows. Filled squares correspond to our measurements of Papers III to V. Our new data appear either as open squares (for those obtained with the long camera of CASPEC) or as filled triangles (corresponding to observations performed with the short camera of CASPEC). When data of other authors are included in the figures for comparison, crosses are used to distinguish them. As in the previous papers of this series, the error bars shown for our measurements correspond to tex2html_wrap_inline3299.

The results obtained will now be discussed star by star.

4.2. Stars already studied in Papers II to V

4.2.1. HD 24712

Only 2 observations of the rapidly oscillating Ap (roAp) star HD 24712 were discussed in the previous papers of this series. Five new observations have now been obtained. The resulting 7 measurements of the longitudinal field are plotted together with Preston's (1972) data in Fig. 1 (click here), against the phases computed from the value of the rotation period (124572) favoured by Kurtz & Marang (1987). The phase diagram shown in Fig. 2 (click here) rests on the value of the period (124610) proposed in Paper II. As suspected in that paper, comparison of Figs. 1 (click here) and 2 (click here) strongly supports the view that the actual value of the rotation period is (close to) 124610. It has been shown in Paper II that this value is also consistent with 7 unpublished tex2html_wrap_inline2667 measurements

  figure512
Figure 1: Plot of our measurements of the longitudinal field of HD 24712 and of those of Preston (1972) against the rotation phase computed assuming that the rotation period is 124572 (see the text for the meaning of the symbols)

  figure517
Figure 2: Same as Fig. 1 (click here), but assuming a value of 124610 for the rotation period

of J.D. Landstreet (private communication). Reasons for the possible inaccuracy of the value derived by Kurtz & Marang (1987) have been discussed in Paper II.

Crossover and quadratic field remain below the detection limit in HD 24712, not surprisingly given its low tex2html_wrap_inline2777 and the small upper limit of its field modulus (Mathys & Lanz 1992).

  figure524
Figure 3: Phase diagram of our measurements of the mean longitudinal magnetic field (lower panel), of the crossover (centre), and of the mean quadratic magnetic field (upper panel) of HD 83368 (see the text for the meaning of the symbols). The curves are least-squares fits of the data by sinusoids

4.2.2. HD 83368

HD 83368 is also a roAp star. Its fairly rapid rotation (tex2html_wrap_inline3317 tex2html_wrap3323) makes magnetic field diagnosis difficult. This accounts for our present inability to derive a quadratic field from our only new observation of this star, and for the rather poor accuracy of the crossover determination reported here. The latter, however, is not inconsistent with our previous data, as can be seen in Fig. 3 (click here). The same figure also shows that our new tex2html_wrap_inline2667 measurement agrees quite well with our 12 earlier determinations. We confirm that, to the achieved accuracy, the curves of variation of the three field moments discussed in this paper do not significantly depart from sinusoids. Nevertheless, the variations of the quadratic field are at the limit of significance, and their shape cannot be regarded as well established.

4.2.3. HD 94660

Resolved magnetically split lines have been discovered in HD 94660 by Mathys (1990). On the basis of photometric observations, Hensberge (1993) suggested that the rotation period of this star may be close to 2700 d. This is well supported by the mean magnetic field modulus measurements of Mathys et al. (1996; hereafter MHLLM). The latter authors also argued that no longitudinal field variations are detected either in our own measurements (2 in Paper III and 2 here) or in those of other authors (Borra & Landstreet 1975; Bohlender et al. 1993). Given the good accuracy (of the order of 100 G) of our 4 tex2html_wrap_inline2667 determinations and their reasonable distribution in phase (assuming that the rotation period is indeed close to 2700 d), it appears unlikely that tex2html_wrap_inline2667 may vary with a peak-to-peak amplitude exceeding 300 or 400 G. This is amazing, considering that the peak-to-peak amplitude of variation of the mean field modulus tex2html_wrap_inline2691 of HD 94660 must be about 300 G or larger, and that large variations of tex2html_wrap_inline2691 are seldom observed, even in stars where tex2html_wrap_inline2667 is strongly variable and/or undergoes polarity reversal. In fact, HD 94660 appears as a unique case of a star where one observes definite variations of tex2html_wrap_inline2691 but not of tex2html_wrap_inline2667. This, consistently with the very anharmonic shape of the field modulus variations (see Fig. 26 of MHLLM) hints at a rather unusual geometrical structure of the magnetic field.

We neither detect any significant variation of the quadratic field. This is not unexpected: the relative amplitude of the latter should be at most of the same order as the relative amplitude of variation of the field modulus, that is, about 5%. Such variations are below the detection limit at the accuracy achieved in the quadratic field diagnosis. Note also that the ratio between tex2html_wrap_inline2781 and tex2html_wrap_inline2691 is approximately 1.3.

Finally, crossover can, of course, not be measured in a star rotating so slowly.

4.2.4. HD 98457

Our attention had been called to HD 98457 by its unusually large photometric variations (Waelkens 1985). Our first two observations of this star, obtained at phases 0.152 and 0.983, had not allowed us to detect definitely a magnetic field from the consideration of any of the three field moments that we are studying (Papers II to V), although values of the quadratic field were measured at the 2.6 and tex2html_wrap_inline3343 levels. We have obtained one new observation, at phase 0.446 (the uncertainty of the relative phasing with respect to our previous measurements should be less than 0.03). Again, we do not detect any significant longitudinal field or crossover, while we are unable to diagnose the quadratic field. Given the improvement of the phase sampling resulting from this new observation, it seems increasingly unlikely that HD 98457 may have a very strong magnetic field.

  figure541
Figure 4: Phase diagram of our measurements of the mean longitudinal magnetic field (lower panel) and of the mean quadratic magnetic field (upper panel) of HD 116458 (see the text for the meaning of the symbols). The curves are least-squares fits of the data by sinusoids

4.2.5. HD 116458

Like HD 94660, HD 116458 is a star in which magnetically resolved lines have been discovered by Mathys (1990). The rotation period, tex2html_wrap_inline3347, has been determined from photometry by Hensberge (1993), who has shown that it is consistent with our 5 measurements of tex2html_wrap_inline2667 presented in Paper II. These are complemented here by 5 new determinations. All our longitudinal field measurements are plotted against rotation phase in Fig. 4 (click here). The variations have a fairly small amplitude but are undisputable. The reduced tex2html_wrap_inline3293 obtained when fitting them by a cosine curve is somewhat high, but there is no definite indication of anharmonicity.

HD 116458 rotates too slowly to allow detection of any crossover.

  figure551
Figure 5: Phase diagram of our measurements of the mean longitudinal magnetic field (lower panel) and of the mean quadratic magnetic field (upper panel) of HD 119419 (see the text for the meaning of the symbols). The curves are least-squares fits of the data by a cosine wave (thin curves) and by the superposition of a cosine wave and of its first harmonic (thick curves)

From the consideration of the upper panel of Fig. 4 (click here), the quadratic field of HD 116458 shows some hint of variability. However the amplitude of the corresponding sinusoidal fit is at most marginally significant (see Table 6 (click here)). As a matter of fact, the field modulus of this star is not detectably variable (MHLLM), so that we do not expect tex2html_wrap_inline2831 to be variable either. Any variation of tex2html_wrap_inline2781 should thus arise from the contribution of tex2html_wrap_inline2833. It can also be noted that the ratio between the mean value Q0 of the quadratic field and the mean magnetic field modulus (see Table 3 (click here) of MHLLM) is 1.33.

4.2.6. HD 119419

Two new observations of HD 119419 have been obtained. When plotted together with our 20 previous measurements against the phase computed with the period 26006 (Lanz & Mathys 1991), the two new data show a definite phase shift, indicating that the period is no longer accurate enough to represent data that span a timebase of almost 2500 d (compared to less than 750 d for the measurements reported in the previous papers of this series). Accordingly, we revised the period determination, deriving an improved value
displaymath3363
from our whole set of longitudinal field measurements. This value lies within the uncertainty range of the period obtained by Lanz & Mathys (1991). Note also that our magnetic data are definitely inconsistent with the value of the period (260562) proposed by Catalano & Leone (1993) on the basis of photometric data.

The contribution to the curve of variation of tex2html_wrap_inline2667 of a small but definite term with twice the rotation frequency of the star is confirmed (compare the fit of the data by a single sine curve and by the superposition of a sine and of its first harmonic in the lower panel of Fig. 5 (click here)). The variations of the quadratic field occur with an amplitude too small compared to the accuracy with which tex2html_wrap_inline2781 can be measured to decide definitely whether they show any departure from harmonicity.

As mentioned in Paper IV, crossover is not clearly detected in HD 119419 (somewhat surprisingly). The upper limit given in Paper IV for this quantity remains unchanged.

4.2.7. HD 125248

In Paper II, we had shown that with the value of 92954 proposed by Babcock (1960) for the rotation period of HD 125248, our 19 longitudinal field measurements could be adequately phased together with those obtained by Babcock (1951) 35 years before or more. By contrast, the use of the revised period value of 929571 suggested by Catalano et al. (1992) introduces a relative phase shift of 0.07 between Babcock's (1951) data and ours, hence quite significantly degrades their agreement. Therefore, we stick to the value of the period proposed by Babcock (1960).

The 2 new measurements of the field moments reported here agree well with our 19 previous determinations. In particular, tex2html_wrap_inline2667, tex2html_wrap_inline2779, and tex2html_wrap_inline2781 all vary nearly sinusoidally (see Fig. 6 (click here)).

4.2.8. HD 126515

A refined value of the rotation period of HD 126515, P=12995, has recently been derived from the consideration of measurements of its mean magnetic field modulus spanning 38 years (MHLLM). The same authors also pointed out the remarkable anharmonicity of the variations of the longitudinal field, from the simultaneous consideration of measurements of the latter obtained by Preston (1970) and van den Heuvel (1971), and of our data published in Paper III (6 determinations) and reported here (3 additional points). The anharmonicity is fully confirmed by unpublished Htex2html_wrap_inline2723 photopolarimetric determinations of tex2html_wrap_inline2667 (kindly provided by D. Bohlender & G. Hill), which are shown together with our data in Fig. 7 (click here). The agreement between the two sets of measurements is quite good. This is not trivial: significant, as yet not fully understood differences are observed for some stars between CASPEC ("photographic'') and Balmer-line photopolarimetric tex2html_wrap_inline2667 determinations.

  figure579
Figure 6: Phase diagram of our measurements of the mean longitudinal magnetic field (lower panel), of the crossover (centre), and of the mean quadratic magnetic field (upper panel) of HD 125248 (see the text for the meaning of the symbols). The curves are least-squares fits of the data by sinusoids

Our measurements alone are not sampling the stellar rotation cycle quite well enough to constrain the shape of the variation curve. Therefore, the fitted amplitude of the first harmonic is hardly significant (see Table 4 (click here)). The contribution of this harmonic is nonetheless quite obvious in the lower panel of Fig. 8 (click here).

  figure588
Figure 7: Phase diagram of our measurements of the longitudinal field of HD 126515 and of unpublished determinations of Bohlender & Hill (see the text for the meaning of the symbols)

  figure593
Figure 8: Phase diagram of our measurements of the mean longitudinal magnetic field (lower panel) and of the mean quadratic magnetic field (upper panel) of HD 126515 (see the text for the meaning of the symbols). The curves are least-squares fits of the data by a cosine wave and its first harmonic (for the longitudinal field) and by a cosine alone (for the quadratic field)

Similarly, although a simple sinusoid gives an excellent fit to the variations of tex2html_wrap_inline2781, our measurements are somewhat too scarce and not quite accurate enough to rule out definitely a possible anharmonicity. The amplitude of the variations of the quadratic field is unusually large, consistently with the large amplitude of variation of the field modulus, which has long been known (Preston 1970). However, the ratio between the maximum and the minimum of tex2html_wrap_inline2781 appears smaller than the ratio between the extrema of tex2html_wrap_inline2691 (MHLLM). Accordingly, the ratio tex2html_wrap_inline2781/tex2html_wrap_inline2691 significantly varies along the rotation cycle, ranging approximately from 1.25 at field modulus maximum to close to 1.50 at field modulus minimum. This indicates that, at least at some phases, the contribution of tex2html_wrap_inline2833 to the quadratic field must be rather considerable.

Crossover is not detected in HD 126515 as a result of its slow rotation.

4.2.9. HD 128898

After many unsuccessful attempts by various authors, the rotation period of HD 128898, the brightest of all roAp stars known, has been determined by Kurtz et al. (1994): P=44790. Using this value of the period, our 2 new observations can be phased together with our 5 older ones within an accuracy of 0.01 cycle. However, when our magnetic measurements are plotted against phase, no systematic trend is seen for any of the studied field moments, even though our observations sample rather well the rotation period. This is not too surprising, given the low level of rotational variability shown by the star in photometry or in spectroscopy. As a consequence, no progress is made in the knowledge of the magnetic field of HD 128898, with respect to the results already reported in Papers II to V: tex2html_wrap_inline2667 and tex2html_wrap_inline2779 are below the detection threshold, while our observations are consistent with a fairly constant quadratic field of the order of 7.5 kG.

4.2.10. HD 137509

HD 137509 has one of the strongest magnetic fields known in an Ap star (Paper V). The distribution in phase of our 9 observations of this star obtained until 1988 was far from ideal (half of the rotation cycle remained unsampled), so that the variation curves of the various field moments were poorly constrained. The 5 new observations reported here improve this situation, and reveal the contribution of a very significant term with twice the rotation frequency in the variations of the longitudinal field and of the crossover (see Fig. 9 (click here)). The first harmonic is possibly present in the quadratic field curve too, but for this field moment, our data are not quite conclusive yet. A marked double-wave character of the tex2html_wrap_inline2667 variations such as we find in HD 137509 had been observed so far in the stars HD 37776 (Thompson & Landstreet 1985) and HD 133880 (Landstreet 1990). This indicates that the magnetic field of HD 137509 has an unusually important quadrupolar component.

  figure606
Figure 9: Phase diagram of our measurements of the mean longitudinal magnetic field (lower panel), of the crossover (centre), and of the mean quadratic magnetic field (upper panel) of HD 137509 (see the text for the meaning of the symbols). The curves are least-squares fits of the data by a cosine wave (thin curves) and by the superposition of a cosine wave and of its first harmonic (thick curves)

4.2.11. HD 137909

The 4 new observations of the famous cool Ap star HD 137909 that we present here fill only partly the gap in phase coverage of our 11 older data. But, thanks in particular to their better accuracy, they bring a significant improvement in the definition of the variation curves. For all three field moments considered here, these curves show no departure from sinusoids, in contrast with the very anharmonic behaviour of the field modulus (MHLLM). The quadratic field is approximately 1.25 times larger than the latter close to maximum, while the ratio between both goes down to very nearly 1.0 at their minimum. This remarkably small value of tex2html_wrap_inline3411, which had already been pointed out in Paper V, is unique among the stars that we have studied. It points to a rather unusual structure of the magnetic field. In particular, it indicates that at quadratic field minimum, the magnetic field of HD 137909 must be seen almost purely transversally. This view is supported by the relative phasing of the curves of variation of tex2html_wrap_inline2667 and of tex2html_wrap_inline2781 (which are in quadrature with respect to each other - see Fig. 10 (click here) and Tables 4 (click here) and 6 (click here)). But this apparent simplicity is misleading: the actual geometrical structure of the field of this star appears very complex and its detailed modelling is quite challenging (Leroy 1995; Wade 1995).

  figure621
Figure 10: Phase diagram of our measurements of the mean longitudinal magnetic field (lower panel), of the crossover (centre), and of the mean quadratic magnetic field (upper panel) of HD 137909 (see the text for the meaning of the symbols). The curves are least-squares fits of the data by sinusoids

  figure628
Figure 11: Phase diagram of our measurements of the mean longitudinal magnetic field (lower panel), of the crossover (centre), and of the mean quadratic magnetic field (upper panel) of HD 147010 (see the text for the meaning of the symbols). The curves are least-squares fits of the data by a cosine wave (thin curves) and by the superposition of a cosine wave and of its first harmonic (thick curves)

4.2.12. HD 147010

Our 2 new observations of HD 147010 and the derivation of a refined value of its rotation period by Catalano & Leone (1993) do not significantly change the picture of its magnetic field that we had obtained from our previous 17 observations. Accordingly, the reader is referred to the previous papers of this series for a detailed discussion. The updated variation curves are shown in Fig. 11 (click here).

  figure640
Figure 12: Phase diagram of our measurements of the mean longitudinal magnetic field (lower panel), of the crossover (centre), and of the mean quadratic magnetic field (upper panel) of HD 153882 (see the text for the meaning of the symbols). The curves are least-squares fits of the data by cosine waves with the rotation frquency of the star (longitudinal field and crossover) or with twice that frequeny (quadratic field)

4.2.13. HD 153882

Our 14 observations of HD 153882 published so far were not sampling the rotation period ideally, with a gap in the coverage left between phases 0.1 and 0.5. The 3 new observations reported here all fall in this gap, which allows the variation curves of the magnetic field moments to be better defined. The resulting fits are shown in Fig. 12 (click here). While the longitudinal field and the crossover are adequately represented by a sinusoid with the rotation frequency of the star, we fully confirm the suspicion expressed in Paper V that the best fit to the quadratic field variations is given by a sinusoid with twice the rotation frequency, alone.

4.2.14. HD 165474

Preston (1971) has discovered resolved magnetically split lines in the spectrum of HD 165474 observed in unpolarized light. The mean magnetic field modulus of this star has been studied by MHLLM. According to these authors, tex2html_wrap_inline2691 appears to undergo low-amplitude (200 to 300 G peak-to-peak) variations about a mean value of 6.5 kG, over a (rotation) period which could not be uniquely determined, but which may plausibly be quite short (the most probable value being 254).

Given the fairly large value of the field modulus, it is surprising that our 3 determinations of the longitudinal field all yield null values. In particular, the 2 new measurements reported here are significantly more accurate than the one published in Paper III and allow us to set a rather conservative tex2html_wrap_inline3425 upper limit of the order of tex2html_wrap_inline3427 (in absolute value) for the longitudinal field of HD 165474. However, preliminary visual examination of an additional CASPEC observation of HD 165474 obtained in July 1996 shows the unmistakable signature of a definite longitudinal field. Thus our 3 null measurements so far just appear to be very unfortunately phased.

We have not detected any crossover in HD 165474 (with a quite stringent tex2html_wrap_inline3425 upper limit of 1.5 tex2html_wrap3433 in our last observation). This is not quite unexpected for a star having resolved magnetically split lines, but the result is nonetheless non-trivial if the rotation period is as short as 254 (note that HD 137909, another star with magnetically resolved lines with a rotation period of 185, does show significant crossover).

Our 2 new determinations of the quadratic field yield values of this field moment of the order of 7.0 kG, while a value of 10.1 kG had been obtained in Paper V. However, the uncertainty of the latter was larger, so that there is no major inconsistency between it and the more recent measurements, within the limits of their respective accuracies. Tentatively adopting 7.0 kG as the representative value of the quadratic field, the ratio between this field moment and the mean field modulus (MHLLM) is of the order of 1.08.

  figure651
Figure 13: Phase diagram of our measurements of the mean longitudinal magnetic field (lower panel), of the crossover (centre), and of the mean quadratic magnetic field (upper panel) of HD 175362 (see the text for the meaning of the symbols). The curves are least-squares fits of the data by a cosine wave (thin curves) and by the superposition of a cosine wave and of its first harmonic (thick curves)

4.2.15. HD 168733

From photometric observations, Manfroid & Renson (1994) have derived an unambiguous value of the rotation period of HD 168733 (P=63540), which however seemed inconsistent with the longitudinal magnetic field measurements of Jones & Wolff (1974). The 4 observations of this star reported here were obtained as part an effort to solve this discrepancy. Together with 3 former observations discussed in Papers II to V, they sample well the rotation cycle. Still, the standard deviation of the resulting 7 tex2html_wrap_inline2667 measurements is only 130 G, that is, quite comparable to the typical uncertainty of the individual determinations. Thus, we are not detecting any significant tex2html_wrap_inline2667 variation. We are led to speculate that, as is known to have occurred frequently for tex2html_wrap_inline2667 diagnosis from photographic plates, Jones & Wolff (1974) somewhat underestimated the uncertainty of their measurements (they quote a typical probable error of 150 G), and that the standard deviation of their data (310 G) only reflects their random errors. Thus we argue that the longitudinal field of HD 168733 is not detectably variable, and that its mostly constant value is close to the average of our 7 determinations, -636 G (which is consistent with the average of all the measurements of Jones & Wolff: -688 G).

The absence of a measurable crossover in HD 168733 is not surprising, given that fairly sharp spectral lines had been observed at high dispersion by Mathys &\ Lanz (1992). We had been unable to determine the quadratic field in Paper V, and we achieve only one marginally significant detection here (3.7 kG at the tex2html_wrap_inline3449 level).

  figure665
Figure 14: Phase diagram of our measurements of the mean longitudinal magnetic field (lower panel) and of the mean quadratic magnetic field (upper panel) of HD 187474 (see the text for the meaning of the symbols). The curve is a least-squares fit of the longitudinal field data by a sinusoid

4.2.16. HD 175362

The anharmonicity of the variations of the longitudinal field of HD 175362, first suspected by Borra et al. (1983), has been definitely established in Paper II. A similar anharmonicity was found in Paper IV for the crossover. By contrast, the quadratic field variations could be satisfactorily represented by a sine wave alone (Paper V), although a small contribution of the first harmonic could not be excluded. A relative weakness of the 24 observations of HD 175362 described in Papers II to V is their uneven distribution in phase: 3 groups of points concentrated resp. between phases 0.355 and 0.417, between phases 0.545 and 0.682, and between phases 0.821 and 0.939, plus 3 points between phases 0.087 and 0.132. In other words, less than half of the cycle was sampled densely, and the shape of the variation curves in the remaining large gaps was poorly constrained. The 5 new observations presented here partly fill those gaps and provide some of the missing constraints, but the conclusions about the general shapes of the curves are not altered (see Fig. 13 (click here) and Tables 4 (click here) to 6 (click here)).

4.2.17. HD 187474

Resolved magnetically split lines have been discovered in HD 187474 by Didelon (1987). Our 4 new observations of this star are rather unfortunately phased, since they sample essentially the same half of the 2345 d rotation period of this star as the 7 observations discussed in Papers II to V. This is not critical for the longitudinal field, for which excellent measurements had been secured by Babcock (unpublished, see Paper II) throughout the whole cycle. But the gap in the quadratic field curve between phases 0.22 and 0.70 appears particularly regrettable in view of the quite unusual behaviour of the field modulus in this phase range - note especially the sharp raise of tex2html_wrap_inline2691 starting at phase 0.2 (MHLLM). In these conditions, it does not appear suitable to try to find a mathematical function representing the tex2html_wrap_inline2781 data obtained so far. The fit of a cosine to our tex2html_wrap_inline2667 measurements, shown in the lower panel of Fig. 14 (click here) (coefficients in Table 4 (click here)), must also be regarded as provisional, although consideration of Babcock's data (see Fig. 33 of Paper II) suggests that it is probably not too far from reality.

The ratio between the quadratic field and the mean field modulus between phases 0.8 and 0.2 (where tex2html_wrap_inline2691 is fairly constant) is approximately 1.50.

Of course, the very slow rotation of HD 187474 rules out any possibility of detection of crossover.

4.2.18. HD 188041

HD 188041 is another long period (2239, see Hensberge 1993) star with resolved lines (Preston 1971). Our 7 observations (4 of which had already been discussed in the previous papers of this series) are insufficient to characterize the behaviour of its longitudinal field by themselves, but they appear roughly consistent with Babcock's (1954, 1958) data.

No significant crossover nor quadratic field was detected in this star.

4.2.19. HD 201601

The slow drift of the longitudinal field of HD 201601 from 1946 to 1988 has been illustrated in Fig. 37 of Paper II. Leroy et al. (1994) have presented compelling arguments (based on broad-band linear polarization measurements) definitely establishing that the observed tex2html_wrap_inline2667 variation does result from stellar rotation with an extraordinarily long period exceeding 70 years. Some ambiguity is left on the exact value of this period. According to Leroy et al. (1994), its smallest plausible value appears close to 77 years, but it might possibly be as long as 110 years. For a period close to the lower limit of this range, the negative extremum of tex2html_wrap_inline2667 was expected in 1994, while longer periods allow the longitudinal field to keep becoming more negative for a while still. The 4 new measurements of tex2html_wrap_inline2667 reported here show some hint of a flattening of the variation curve, which may be indicating that the star is indeed approaching its negative extremum and would accordingly suggest that the period is close to the low end of the acceptable range. That the mean field modulus (diagnosed from the observation of magnetically resolved lines) also seems to be reaching its maximum (MHLLM) may be another indication of this - but one has to remember that the tex2html_wrap_inline2667 and tex2html_wrap_inline2691 extrema do not necessarily coincide in phase.

Given the long rotation period, one cannot expect to observe crossover in HD 201601.

For the reasons exposed in Sect. 3, quadratic field diagnosis is more difficult and less accurate here than in Paper V. This explains why, for HD 201601, quadratic field could be diagnosed only from the spectra taken with the long camera of CASPEC, and why in spite of the higher dispersion of the latter, the uncertainty affecting those 2 determinations is larger than that of the measurements of Paper V. Accordingly, no real progress is achieved here in the knowledge of this field moment. Based on the (better) data of Paper V alone and on the only field modulus measurement of MHLLM contemporaneous with them, the ratio tex2html_wrap_inline3411 is found to be of the order of 1.85.

4.3. Additional stars

4.3.1. HD 2453

MHLLM have derived a refined value of 521 d for the rotation period of HD 2453, an Ap star with resolved magnetically split lines (Mathys & Lanz 1992). They have shown that with this value of the period, the measurement of tex2html_wrap_inline2667 reported here is consistent with those of Babcock (1958) and of Wolff (1975). The slow rotation of the star does not allow crossover to be observed and the quadratic field cannot be determined.

4.3.2. HD 5737

Our single observation of HD 5737 yields null values of tex2html_wrap_inline2667 and tex2html_wrap_inline2779, while we could not diagnose the quadratic field. The longitudinal field had already been studied in detail by Shore et al. (1990). Their 16 Htex2html_wrap_inline2723 photopolarimetric determinations of this field moment, together with 7 older measurements of Borra et al. (1983), revealed that it varies mostly sinusoidally between -0.3 and +0.5 kG over the stellar rotation period. The most accurate value of the latter, 21652, has been derived by Manfroid & Renson (1994) from photometric observations in the Strömgren system. Using it, our observation can be phased with respect to those of Shore et al. (1990) with an accuracy of 0.03 rotation cycle. Our tex2html_wrap_inline2667 value of tex2html_wrap_inline3487 G, derived at phase 0.312, is quite consistent with the value of -314 G predicted for that phase from the best fit cosine to the data of Borra et al. (1983) and of Shore et al. (1990).

4.3.3. HD 6532

HD 6532 is a roAp star with a short rotation period: 194 (see Kurtz et al. 1996 and references therein). As a result of this fast rotation, its spectral lines show significant Doppler broadening, which complicates the magnetic field diagnosis. Our observation is, to our knowledge, the first attempt to detect the magnetic field of this star. While it yielded null values of the longitudinal field and of the crossover, a quadratic field of about 22 kG was measured. That tex2html_wrap_inline2667 and tex2html_wrap_inline2779, which generally vary in phase quadrature, are both small at a given phase suggests that the longitudinal field never gets very large. Although it remains to be confirmed by observations at other phases, we note that this behaviour is similar to that of the only two other fast rotating roAp stars whose magnetic field has been studied so far. Indeed, these stars, HD 83368 and HD 128898, also have relatively modest longitudinal fields and quite strong quadratic fields (see Sects. 4.2.2 and 4.2.8).

4.3.4. HD 19918

HD 19918 is another roAp star (Martinez & Kurtz 1994) whose magnetic field has never been studied. Here we report the detection of a longitudinal field at the tex2html_wrap_inline3495 level. Null values are derived for both the crossover and the quadratic field. But the constraint on the latter is very weak, with a tex2html_wrap_inline3425 upper limit of 21 kG. A high-resolution (tex2html_wrap_inline3499) observation in unpolarized light, performed with the ESO Coudé Echelle Spectrograph (CES) fed by the 3.6 m telescope is significantly more stringent. Indeed, from this spectrum, the lines of HD 19918 appear quite sharp, indicating that the field modulus of this star is unlikely to exceed much 1 kG.

4.3.5. HD 22920

Borra et al. (1983) concluded that HD 22920 almost certainly has a weak longitudinal field, from the consideration of the average of four measurements performed by Htex2html_wrap_inline2723 photopolarimetry, none of which taken alone yields a firm detection. Our single attempt to determine tex2html_wrap_inline2667 is not more successful. But it is noteworthy that it gives a small positive tex2html_wrap_inline2667 value quite consistent with the data of Borra et al. (1983), thus that it supports these authors' argument about the reality of the field. We detect no crossover in HD 22920 and we cannot diagnose its quadratic field.

4.3.6. HD 36485

6 determinations of the longitudinal field of HD 36485 have been performed through Htex2html_wrap_inline2723 photopolarimetry by Bohlender et al. (1987), and these authors have concluded that tex2html_wrap_inline2667 in this star is constant, at a value of -3.4 kG. However, our single tex2html_wrap_inline2667 determination yields a value of -1.9 kG. We cannot decide at present whether the discrepancy with the conclusion of Bohlender et al. (1987) means that these authors have misinterpreted their data or if it results from the use of different measurement techniques. Indeed, inconsistencies between longitudinal fields diagnosed with the Zeeman analyzer of CASPEC and with the Htex2html_wrap_inline2723 photopolarimeter have been found for several stars and are not fully understood (see Paper II; see also Mathys 1989). But they seldom are so extreme as the difference obtained here for HD 36485.

No significant crossover is detected in HD 36485, and its quadratic field cannot be diagnosed.

4.3.7. HD 37058

Sargent et al. (1967) performed the first determination of the longitudinal field of HD 37058 and derived a value of tex2html_wrap_inline3519 G. Conti (1970) obtained 5 measurements ranging from +30 to +1300 G, with uncertainties of 400-600 G. None of Borra et al.'s (1983) 3 individual values of tex2html_wrap_inline2667 (all negative) quite reach the tex2html_wrap_inline3425 level (the tex2html_wrap_inline2855 of each of the 3 measurements is just slightly larger than 300 G), but their average indicates that a field is detected at the 99.9% confidence level. Therefore Borra et al. (1983) argue that all observations are consistent with a reversing longitudinal field varying with a peak-to-peak amplitude of 2 to 3 kG. If this interpretation is correct, our determination of tex2html_wrap_inline2667, which is much more accurate than the previous ones, must have been very unfortunately phased, since it yields a null result too. The value of the rotation period derived by Pedersen (1979) is regrettably too coarse to test the relative phasing of our observation with respect to older ones. But the reality of the field of HD 37058 is supported by our finding of a fairly significant crossover (at the tex2html_wrap_inline3535 level). We also measure a marginally significant quadratic field, at the tex2html_wrap_inline3537 level.

4.3.8. HD 50169

The measurements of the mean magnetic field modulus of HD 50169 by MHLLM indicate that the period of rotation of this star with magnetically resolved lines, discovered by Mathys & Lanz (1992), must be much longer than 4 years. Consistently with this, Babcock (1958) observed a slow increase of tex2html_wrap_inline2667 from +670 G in 1953 to +2120 G three years later. Our own measurement falls between these two values.

Comparison of the value of tex2html_wrap_inline2781 found here with contemporaneous tex2html_wrap_inline2691 data of MHLLM indicate that the ratio between the two quantities is of the order of 1.7. Of course, no crossover is detected.

4.3.9. HD 55719

HD 55719 is one of the three double-lined spectroscopic binaries definitely known to contain a magnetic Ap star. The latter has resolved magnetically split spectral lines (Mathys 1990). Both the characteristics of the binary system and the longitudinal field of the Ap component have been studied in detail by Bonsack (1976). This author's tex2html_wrap_inline2667 measurements and his determination of the rotation period have been rediscussed in detail by MHLLM. Indeed, none of the two values of the period that he favours is consistent with the field modulus measurements. MHLLM could not definitely establish the value of the rotation period, but 847 or 775 days appear as most plausible. Nevertheless, it seems unavoidable that 2 or 3 of Bonsack's (1976) measurements are bound to be inconsistent with the bulk of his data, regardless of what the rotation period is (they were already discrepant with Bonsack's preferred period values). Except for these discrepant values, all of Bonsack's (1976) tex2html_wrap_inline2667 data are larger than our 3 measurements. If the interpretation proposed by MHLLM that the rotation period is longer than 2 years is correct, this hints at the existence of systematic differences betwen Bonsack's (1976) and our longitudinal field determinations. This is not unusual but it implies that we need to accumulate more data before we can hope to use the tex2html_wrap_inline2667 measurements to constrain the rotation period.

The crossover measured in our third observation is too marginal (at the tex2html_wrap_inline3449 level) to question the long period hypothesis. The first two quadratic field determinations (based on spectra recorded with the long camera of CASPEC) are significantly more accurate than the third one (for which the short camera was used). Their ratio to the average field modulus of HD 55719 (MHLLM) is close to 1.3.

4.3.10. HD 70331

The rotation period of HD 70331, one of the hottest Ap stars with magnetically resolved lines, could not be uniquely determined from the consideration of its mean magnetic field modulus, but it must probably be short, two of the most plausible values being 303 and 365 (MHLLM). Longitudinal field measurements may prove very useful to establish the value of the period more reliably. This field moment is determined here for the first time, yielding a large negative value of -2.8 kG. As is often the case for stars with magnetically resolved lines, no significant crossover is detected (although one cannot exclude to observe it at other phases). The large quadratic field is about 1.1 to 1.2 times larger than the field modulus.

4.3.11. HD 81009

Our single determination of the longitudinal field of HD 81009 is fully consistent with an unpublished variation curve of this moment, obtained by G. Hill and D. Bohlender (Hill, private communication; see also MHLLM). Rather remarkably, given the rather long rotation period of 3396 (Waelkens 1985), crossover is detected at a fairly significant level (tex2html_wrap_inline3495). This is fully consistent with the observation of some Doppler distortion in the split components of the magnetically resolved lines of this star (resolved magnetically split lines have first been observed in this star by Preston 1971). The ratio tex2html_wrap_inline3411 at the phase of our observation is approximately 1.75.

4.3.12. HD 93507

HD 93507 is another star with magnetically resolved lines discovered by MHLLM, whose longitudinal field has never been measured before. The two determinations presented here are separated by 0.302 rotation cycle (the period is 556 d). That they differ by 1 kG suggests that tex2html_wrap_inline2667 undergoes quite sizeable variations. The second determination of the quadratic field, based on a spectrum taken with the long camera, is considerably more accurate than the first one, for which the short camera of CASPEC was used. We only compare the former to the field modulus at the same phase: the ratio between them is 1.07. HD 93507 rotates too slowly to show observable crossover.

4.3.13. HD 116114

The observation of magnetically resolved lines in HD 116114 has first been reported by Mathys et al. (1993). From the mean field modulus measurements, it was inferred that the rotation period must be much longer than 3 years (MHLLM). The first measurement of tex2html_wrap_inline2667 which we report here yields a large negative value (-1.9 kG), while the quadratic field is 1.2 times larger than the field modulus. No crossover is detected.

4.3.14. HD 134214

The roAp star HD 134214 (Kreidl 1985) has one of the smallest field moduli measured by MHLLM in any star with resolved magnetically split lines. These authors were unable to derive a definite value of the period, but the latter may well be short. A tentative value of 41456 is suggested by MHLLM. Should this value prove correct, the two observations reported here would be separated by only 0.104 rotation cycle. The fact that no significant measurement of either tex2html_wrap_inline2667, or tex2html_wrap_inline2779, or tex2html_wrap_inline2781 is obtained from any of them would then only be a weak constraint.

4.3.15. HD 137949

MHLLM have shown that all previous measurements of tex2html_wrap_inline2667 (Babcock 1958; van den Heuvel 1971; Wolff 1975) in the roAp star HD 137949 (Kurtz 1982), together with our two measurements presented here, consistently indicate that the star has a very long rotation period (possibly more than 75 years). The crossover measured from our first observation is not significant enough (at the tex2html_wrap_inline3449 level) to challenge this conclusion. Our two spectra were taken with the short camera of CASPEC, and the tex2html_wrap_inline2781 determinations have uncertainties too large to be really meaningful.

4.3.16. HD 144897

We are reporting the first determination of the longitudinal field of HD 144897, an Ap star with resolved magnetically split lines (MHLLM). The phase of this observation is about mid-way between the maximum and the minimum of the field modulus. If the tex2html_wrap_inline2667 and tex2html_wrap_inline2691 extrema roughly coincide in phase (which is not infrequent, even though some stars show large departures from this phase relation), we expect tex2html_wrap_inline2667 to reach at its maximum a value significantly greater than the already large +2.0 kG derived here. Not surprisingly, since the rotation period is fairly long (4843), we observe no crossover. Comparing our good measurement of tex2html_wrap_inline2781 with tex2html_wrap_inline2691 at the same phase, the ratio of the two is found to be 1.13.

4.3.17. HDE 318107

Magnetically resolved split lines had been sought (and found - see MHLLM) in HDE 318107 following North's (1987) report that the rotation period of this star is 52 d. Somewhat ironically, the field modulus measurements proved inconsistent with this value of the period. They were however too noisy to allow MHLLM to establish what the actual value of the period is. Longitudinal field measurements may prove most useful to this effect, since the variations of this field moment are often (relatively) larger than those of tex2html_wrap_inline2691. The first determination of tex2html_wrap_inline2667 reported here is promising, since a large positive value (almost 2.0 kG) is obtained, with a reasonably small uncertainty (230 G). The achieved determination of the quadratic field is pretty accurate too. The ratio of tex2html_wrap_inline2781 to the average of the tex2html_wrap_inline2691 measurements of MHLLM is close to 1.45.

4.3.18. HD 166473

Our interest in HD 166473, originally motivated by the fact that it is a roAp star (Kurtz & Martinez 1987), grew when resolved magnetically split lines were discovered in its spectrum (MHLLM). The mean magnetic field modulus of this star varies slowly, with a period still unknown, but definitely much longer than 3.2 years, and a large relative amplitude. Our 3 measurements of tex2html_wrap_inline2667 (the first ones ever obtained for this star) have been performed over 2 months, close to the maximum of tex2html_wrap_inline2691. Not surprisingly, tex2html_wrap_inline2667 shows no significant variation over that time interval. No crossover is detected. From the last two, more accurate (long camera) determinations of the quadratic field, the ratio of the latter to the field modulus at the time of our observations was 1.30.

4.3.19. HD 176232

The first study of the longitudinal field of HD 176232, a roAp star (Heller & Kramer 1988), was performed by Babcock (1958). If the uncertainties he quotes for his 6 determinations of tex2html_wrap_inline2667 are not underestimated (as has unfortunately often been the case for photographic tex2html_wrap_inline2667 measurements), these determinations rank among the most accurate ones ever achieved, and all of them indicate that HD 176232 has a small (between -315 to +440 G) but definite longitudinal field. The accuracy of our measurement of tex2html_wrap_inline2667 is significantly worse than the accuracy claimed by Babock (1958), and we do not detect this field moment, nor crossover or quadratic field.

4.3.20. HD 193756

We neither detect any of the three field moments considered in this study in the roAp star HD 193756 (Martinez & Kurtz 1990). We are not aware of any other attempt to diagnose the magnetic field of this star.

4.3.21. HDE 335238

HDE 335238 has long been known to have magnetically resolved spectral lines, from which a large, variable field modulus is derived (Preston 1971). But probably because of its faintness, it has not been paid much attention until MHLLM's recent systematic study of the magnetic field modulus. These authors were unable to derive an unambiguous value of the rotation period, but they showed that it must without doubt be between 40 and 50 days. The phase of our observation is accordingly unknown, and in particular, we cannot relate the quadratic field that we determine (+10.5 kG) to the field modulus. We also detect a fairly large negative longitudinal field, and more unexpectedly, a (somewhat marginal) crossover (at the tex2html_wrap_inline3619 level).

4.3.22. HD 203932

HD 203932 is another roAp star (Kurtz 1984) whose magnetic field has never been studied before. Our two attempts yield no detection of the longitudinal field or of the crossover, and we were unable to diagnose the quadratic field.

4.3.23. HD 216018

We find no significant variation of tex2html_wrap_inline2667 nor of tex2html_wrap_inline2781 between our 3 observations (spread over slightly less than one year) of HD 216018. We do not detect any crossover either. All this is consistent with the very long rotation period (much longer than 3 years) inferred from the variations of the mean field modulus of this Ap star with magnetically resolved lines (MHLLM). The ratio of the quadratic field to the field modulus is of the order of 1.25.

4.3.24. HD 217522

The only hint of a magnetic field that we find in the roAp star HD 217522 (Kurtz 1983) is provided by a very marginal (tex2html_wrap_inline3625) quadratic field measurement. Our single observation (which is the first one aimed at detecting a field in this star) does not show any significant longitudinal field or crossover.

4.3.25. HD 218495

HD 218495 is another roAp star (Martinez & Kurtz 1990) whose magnetic field has never been studied. That both the tex2html_wrap_inline2667 and tex2html_wrap_inline2779 determinations yield nonzero values at a low level of significance (tex2html_wrap_inline3631 and tex2html_wrap_inline3633, resp.) may be purely coincidental, but it may also indicate that the star indeed has measurable longitudinal field and crossover and that our single observation was unfortunately phased approximately mid-way between the extrema of both those field moments. The quadratic field could not be diagnosed.


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