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Magnetic stars, 2004,225-229

Averaged magnetic phase curves. Analysis of the catalog
Bychkov V.D.
1 2 3

1,2

, Bychkova L.V.1 , Madej J.

3

Special Astrophysical Observatory of the Russian AS, Nizhnij Arkhyz 369167, Russia, Isaac Newton Institute of Chile, SAO Branch, Nizhnij Arkhyz, 369167 Russia Warsaw University Observatory, Al. Ujazdowskie 4, 00-478 Warsaw, Poland

Abstract. We analyze parameters of the averaged phase curves of the effective magnetic field Be for 139 main sequence and other stars. Most of them, 134 ob jects, are chemically p eculiar A and B typ e stars. This contribution presents statistical prop erties of fitting co efficients B0 , B1 , and B2 , and the distribution of the parameter r = Be (max)/Be (min).

1

Intro duction

We have selected all individual measurements of the effective magnetic field Be , which have been published over the recent 50 years, and have made a catalog of magnetic phase curves B e () for the stars which exhibit periodic variability of Be with the rotational phase . A total of 139 stars are included in the catalog, 134 of them being Ap stars (Bychkov et al., in preparation). Most of the stars collected in the catalog exhibit periodic Be variations which can be satisfactorily approximated by a simple sine wave (first order term in the harmonic expansion). In some cases, this is due to scarce or inaccurate Be observations. However, 18 stars distinctly exhibit more complex Be () curves, which we have also fitted using the second order harmonic term (double wave). The latter group of stars accounts for about 13 % of our sample. The above number, 13 %, is the lower limit which may increase in the future when new averaged phase curves Be () of higher accuracy become available. The catalog presents 166 various magnetic phase curves. This is due to the fact that magnetic stars sometimes exhibit quantitatively different phase curves when Be were measured by different methods or in different spectral lines.

2

Magnetic variability parameters

Fig.1 shows the number distribution of the cataloged stars vs. coefficient B0 for those stars in which phase curve Be () was approximated by a sine wave. Fig.1 does not include HD 215441 with its exceptionally strong Be field since this single point exceeds the scale bounds of the Figure. The average value of B 0 over all single-wave stars equals 21 ± 138 G, the latter value representing the broadening of the peak, rather than the error. Fig.2 displays a similar number distribution of the cataloged stars vs. coefficient B1 (half-amplitude of Be variability). This figure also shows only the stars which exhibit a sine wave Be ().

It can readily be noted, that there exists huge deficiency of stars with small half-amplitudes B 1 of the effective magnetic field variations, see Fig.2. In the oblique rotator model the constancy or low B e variations
c Sp ecial Astrophysical Observatory of the Russian AS, 2004


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BYCHKOV, BYCHKOVA, MADEJ

40

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N stars

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-4000

-2000

0

2000

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Be(G)

Figure 1: Number distribution of cataloged stars vs. coefficient B0 for those stars for which phase curve Be () was approximated by a sine wave.

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N stars

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10

5

1000

2000

3000

4000

5000

Be(G)

Figure 2: Number distribution of cataloged stars vs. coefficient B1 (half-amplitude of Be variability). This figure also shows only those stars which exhibit a sine wave phase function.


AVERAGED MAGNETIC PHASE CURVES. ANALYSIS OF THE CATALOG

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4

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N stars

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1

-4000

-3000

-2000

-1000

0

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B0 (G)

Figure 3: Number distribution of stars vs. coeffcient B0 for those stars which exhibit a magnetic phase curve with a double wave. of a magnetic star can occur if the angle between the magnetic and rotational axes is small or even zero. I.e. this observation suggests that magnetic stars avoid configurations with a small angle . Fig.3 shows the number distribution of stars vs. coeffcient B0 for those stars which exhibit a magnetic phase curve with a double wave. The average value of B0 equals -473 ± 296 G. Again, the value of 296 G represents the broadening of the peak distribution. The total number of stars which make up the figure is small (18 ob jects), therefore statistics is poor. However, we conclude that stars which show complex B e phase curves (double waves) more frequently exhibit negative values of Be .

3

Distribution of p erio ds

Fig.4 presents the distribution of stars in our catalog vs. decimal logarithm of period P , separately for all Ap stars with known periods (upper panel), and for stars with known phase curves (lower panel). The width of a single bin equals 0.3 dex. Approximately 60 % of all Ap stars have their periods P between 1 and 5 days. The same is true for stars with known phase curves.

4

Distribution of the parameter r

The parameter r puts constraints on the angle between the magnetic dipole axis and the rotation axis and the angle i between the rotation axis and the line of sight (Stibbs 1950) r= cos cos i - sin sin i . cos cos i + sin sin i (1)

One can express this parameter by observed quantities r= Be (min) . Be (max) (2)


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BYCHKOV, BYCHKOVA, MADEJ

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N stars (all)

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0

1

2

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N stars (with curve)

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10

0

1

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log(P)

Figure 4: Number distribution of stars in our catalog vs. decimal logarithm of period P , for al l Ap stars with known periods (upper panel), and for stars with known phase curves (lower panel).

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N stars

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10

5

-0,8

-0,6

-0,4

-0,2

0,0

0,2

0,4

0,6

0,8

r

Figure 5: Measured distribution of the parameter r vs. theoretical distribution, the latter corresponding to the random stratification of and i angles.


AVERAGED MAGNETIC PHASE CURVES. ANALYSIS OF THE CATALOG

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N stars

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b

Figure 6: Number distribution of stars in our catalogue vs. the angle between magnetic and rotational axes. Fig.5 presents the number distribution of stars in our catalog vs. the observed parameter r (the histogram). The solid line shows the theoretical distribution obtained under an arbitrary assumption that the angle is distributed uniformly. One can see that the figure demonstrates substantial differences between the observed and theoretical distributions of the parameter r. Finally, we attempted to collect or recompute the number distribution of the angle itself in the group of cataloged stars, based on our collection of phase curves Be (). Such a distribution is shown in Fig.6. Note that there exists an excess number of stars with the angle 0, which contradicts the previous results obtained (Fig. 2).

5

Summary

We have collected in a single cataloge all the available magnetic phase curves B e () and expressed them in the uniform manner. The catalog lists phase curves for 139 stars, 134 of them are magnetic Ap stars. Such a catalog allowed us to list all the parameters of stellar magnetic variations and perform homogeneous analysis of these data. We note the apparent deficiency of stars with small variations of Be , see Fig.2. We also conclude that there exists deficiency of stars with the lowest and intermediate values of the parameter r, i.e. in particular, at the limit r -1, as compared with the theoretical model of uniform distribution of the angle .
Acknowledgements. Our research is based on data collected and posted in the SIMBAD, ADS, and CDS databases. We acknowledge support from the Polish Committee for Scientific Research, grant No. 2 P03D 021 22.

References
Bychkov V.D., Bychkova L.V., Madej J., in preparation Stibbs D.W., 1950, Mon. Not. R. Astron. Soc., 110, 395