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Magnetic stars, 2004, 221-224

Averaged magnetic phase curves. A 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 collected published o f s t a r s o n t h e m a i n s eq u en ce a n d Be variations. We present magnetic p a r a m et er s . M o s t o f t h e ca t a l o g ed o st a r s ( 1 3 4 st a r s) .

measurements of the effective magnetic field above it and compiled a catalog of periodic phase curves for 139 stars and tables of their b jects are chemically peculiar A- and B-type

1

Intro duction

Variability of the effective magnetic fields Be in Ap stars was discovered over 50 years ago (Babcock & Bard 1952), and a large number of observational data have been collected ever since. We have extensively searched literature and selected all the available measurements of the effective (longitudinal) magnetic field Be for main sequence stars and a few other stars. Averaged magnetic phase curves Be () have been constructed for those stars for which we know the magnetic (i.e. rotational) periods, or can determine them. The picture of a star with a large-scale magnetic field can be described by the oblique rotator model in which the axis of the magnetic dipole is inclined to the rotation axis. The dipole field itself is not time variable. The periodic variability of the effective magnetic field Be is caused by changes in aspect during rotation of the star. Therefore the period of magnetic Be variations, Pmag , can be identified with the rotational period Pr o t . The above model was proposed to explain the behavior of magnetic CP stars which exhibit periodic variations of Be (Stibbs 1950; Preston 1967). In this model the dipole magnetic field is frozen into the stellar atmosphere and intrisically constant at each point. Our catalog presents averaged magnetic phase curves in a homogeneous form. We have also determined other parameters of magnetic variability of all stars in the catalog. The list of these parameters is briefly described below.

2

Parameters of magnetic variability

In this Section we present a list of all parameters of the magnetic phase curves, and also the parameter r, which was defined by Stibbs (1950). These are: B0 , B1 , B2 , T0 , P , and r.

3

Sine wave

For all stars with an adequate number of Be determinations, and for which the period of magnetic variability Pmag was known, we have determined, by the least squares method, the best fit for the relation of Be vs. phase Bei () = B0 + B1 cos , (1) where = 2 Ti - T P
0

.

(2)

c Special Astrophysical Observatory of the Russian AS, 2010


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BYCHKOV, BYCHKOVA, MADEJ

. Here B0 is the average field, B1 stands for the half-amplitude, Ti is the time of measurement, P is the period, and T0 is the zero epoch, i.e. the time corresponding to the zero phase . We have chosen the zero epoch T0 in such a way that the phase = 0 corresponds to the minimum of the best fit magnetic curve for all listed stars.

4

Parameter r

The parameter r relates both the angle between the magnetic dipole axis and the rotation axis, and the angle i between the rotation axis and the line of sight r= Alternatively, one can write r= Be (min) . Be (max) (4) cos cos i - sin sin i . cos cos i + sin sin i (3)

5

Double wave

In the case where the shape of the magnetic phase curve is more complex than a simple cosine, we include the second harmonic wave Bei () = B0 + B1 cos( + z1 ) + B2 cos(2 + z2 ) . (5)

6

Error analysis

For each star in the sample we performed a 2 test to evaluate goodness of the assumed fit given either by Eq. 1 or Eq. 4 and estimated the scatter of the available Be measurements. The statistical test 2 can indicate a large discrepancy between the observed points and the assumed fitting curve if either the fitting curve is intrinsically inconsistent with observations, or errors of observations (i.e. values of Be ) are overestimated. Error estimates of all parameters, T0 , B0 , B1 , B2 , and r, were performed in the following way. For each sec Bei measurement with known standard error i we generated a series of secondary Bei values with a random sec number generator. The values of Bei had a normal distribution around the observed Bei with the width i . (In the case where the authors did not provide the i estimate, we used the error typical of the given method of observation.) This method generated a set of artificial values of Be for which the secondary parameters T0 , B0 , B1 , B2 , and r were determined. The above computations were repeated many times (usually 1000 times or even more). In such a way we obtained numerous sets of fitting parameters and were able to estimate errors of T0 , B0 , B1 , B2 , and r separately.

7

Tabular data

The full catalog of investigated stars and the corresponding parameters of the best fit sine magnetic curves (Eq. 1) is presented in Table 1, which will be available on the Internet. The columns of Table 1 list: HD number, B0 , B0 , B1 , B1 , period P , T0 , T0 , N ­ the number of individual points, ­ the average scatter of Bei observation around the fitting curve, r, r , value of 2 (for one degree of freedom), the reference number, and (in some cases) the running number of brief comments. Table 2 of our catalog presents for each star: the HD number, the parameter r and its standard error r , NV = N - 3 or N - 5 (for sine wave or double wave fits, respectively), 2 for one degree of freedom, the method and the reference numbers. There exist 18 magnetic stars which display more complex phase curves Be (). For these stars the phase curves were fitted by the expansion in harmonic series with the second cosine term, see Eq. 5 (double wave). Table 3 presents additional parameters necessary for defining the double wave. These are the coefficients B2 , the phase shifts z1 and z2 , and their errors B2 , z1 , z2 , NV , the values of 2 , the number of references and comments. Here NV = N - 5.


AVERAGED MAGNETIC PHASE CURVES. A CATALOG

223

8

Sample phase curve: single wave

HD37017
1000

0

Be (Gs)

-1000

-2000

-3000

0,0

0,5

1,0

Phase

Figure 1: HD 37017. Open circles ­ Bohlender et al. 1987 (H lines), filled circles ­ Borra, Landstreet 1979 (H lines), open squares ­ Bohlender et al. 1987 (He 5867 line).

9

Sample phase curves: double wave

HD112413

500

Be (G)

0

-500

0,0

0,5

1,0

Phase

Figure 2: HD 112413. The figure presents the phase curve Be () derived from Be measurements of Wade et al. 2000 (metal lines, LSD method). Averaged phase curve Be () has form of double wave.


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BYCHKOV, BYCHKOVA, MADEJ

HD137509
3000

2000

1000

Be (G)

0

-1000

-2000 0,0 0,5 1,0

Phase

Figure 3: HD 137509. Magnetic phase curve is described best by a double wave. Fil led squares ­ Mathys, Hubrig 1997, open circles ­ Bohlender et al. 1993, fil led circles ­ Mathys 1991.

10

Summary

We have compiled a catalog of magnetic phase curves Be () for 139 stars which exhibit periodic variations of the effective magnetic field Be . Most of the cataloged ob jects, 134 stars, are chemically peculiar A- and B-type stars. The catalog consists of figures which display individual Be measurements and error bars, and phase curves approximated either by a sine wave, or by a double wave. The catalog also presents a list of the following parameters of magnetic phase curves: the coefficients B0 , B1 , and B2 of the harmonic expansion of Be (), the period P (in days) and the Julian Date of the zero phase T0 , and the coefficient r defined by Stibbs (1950).
Acknowledgements. Our research is based on data compiled 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
Babcock H.W., Burd S., 1952, Astrophys. J., 116, 8 Bohlender D.A., Brown D.N., Landstreet J.D., 1987, Astrophys. J., 333, 325 Bohlender D.A., Landstreet J.D., Thompson I.B., 1993, Astron. Astrophys., 269, 355 Borra E.F., Landstreet J.D., 1979, Astrophys. J., 228, 809 Bychkov V.D., Bychkova L.V., Madej J., in preparation Mathys G., 1991, Astron. Astrophys. Suppl. Ser., 89, 121 Mathys G., Hubrig S., 1997, Astron. Astrophys. Suppl. Ser., 124, 475 Preston G.W., 1967, Astrophys. J., 150, 547 Stibbs D.W., 1950, Mon. Not. R. Astron. Soc., 110, 395 Wade G.A., Donati J.F., Landsreet J.D., Schorlin S.L.S., 2000, Mon. Not. R. Astron. Soc., 313, 851