The δScuti Star DK Virginis (HR 5005) RevisitedAuthor(s): Mikołaj Jerzykiewicz and A. A. PamyatnykhSource: Publications of the Astronomical Society of the Pacific, Vol. 112, No. 776 (October2000), pp. 1341-1349Published by: The University of Chicago Press on behalf of the Astronomical Society of the PacificStable URL: http://www.jstor.org/stable/10.1086/316628 .

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PUBLICATIONS OF THE ASTRONOMICAL SOCIETY OF THE PACIFIC, 112 :1341È1349, 2000 October2000. The Astronomical Society of the PaciÐc. All rights reserved. Printed in U.S.A.(

The d Scuti Star DK Virginis (HR 5005) Revisited1

MIKO¢AJ JERZYKIEWICZ

University Observatory, Poland ;Wroc¡aw Wroc¡aw, mjerz=astro.uni.wrc.pl

AND

A. A. PAMYATNYKH

N. Copernicus Astronomical Center, Polish Academy of Sciences, Warsaw, Poland ; andInstitute of Astronomy, Russian Academy of Sciences, Moscow, Russia

Received 2000 April 26 ; accepted 2000 June 6

ABSTRACT. An analysis of Hp photometry of DK Virginis is carried out. The analysis reveals twofrequencies, and cycles day~1. These values turn out to be good enough to bridge thef1\ 9.2095 f2\ 7.5764gap between the two available series of ground-based data, obtained in 1973 and 1980, so that the fre-quencies can be reÐned. The reÐned values are equal to 9.20945 and 7.57641 cycles day~1. The correspond-ing V amplitudes amount to 8.6 ^ 0.4 and 11.1^ 0.4 mmag, and the epochs of maximum light are equal to

2,441,715.0294^ 0.0008 and 2,441,715.1205^ 0.0007. The synthetic light curve, computed with theseJD_

parameters, correctly reproduces the large light range on JD 2,439,177, the night in 1966 when the star wasdiscovered to be variable, and the small range on JD 2,440,682, when the light variation was not detected. Inthis way, all photometric observations of DK Vir are accounted for, indicating that the two frequencies havenot changed appreciably since 1965, and that the V amplitudes were constant over the interval from 1965 to1980.

Using a photometric e†ective temperature and Hipparcos parallax, we plot the star in the H-R diagram.Comparison with evolutionary tracks then shows that DK Vir is at the end of the main-sequence stage of itsevolution or slightly beyond, with a mass of 2.1 or somewhat smaller. Finally, an examination of linearM

_nonadiabatic pulsation frequencies of modes with l ¹ 2 indicates that the two observed modes cannot bothbe radial, and that the mode must be nonradial if the star is still on the main sequence.f1

The properties of the Hipparcos spectral window are examined in the Appendix.

1. INTRODUCTION

DK Virginis\HR 5005 (F1 IV)\HIP 64769 was dis-covered to be a d Scuti star by Danziger & Dickens (1967).From V -Ðlter observations on a single night in 1966 Feb-ruary, they derived a period of 0.14 days. In 1970 March thestar was reobserved by Valtier (1971) with a y-Ðlter, also ona single night. Although his observing run covered 0.19days, Valtier (1971) failed to detect any light variation.However, y-Ðlter observations, carried out on seven nightsin 1973 February, April, and May at the Lowell Observa-tory by Jerzykiewicz (1975), conÐrmed the d ScutiÈtypevariability of the star. In addition, these observationsshowed that the light range changes from night to night,implying an interference of at least two short-period terms.Two frequencies, equal to 8.606 and 9.208 cycles day~1,were subsequently derived by Peniche, &Pen8 a, Gonza� lez(1983) from V -Ðlter observations, obtained on eight nights

ÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈ1 Based in part on data from the ESA Hipparcos astrometry satellite.

in 1980 March, April, and May at the San Pedro Ma� rtirObservatory.

2. HIPPARCOS EPOCH PHOTOMETRY OFDK VIR

2.1. The DataHipparcos Epoch Photometry Annex (EPA) (ESA 1997)

contains 124 Hp magnitudes of DK Vir. As for other starsobserved with Hipparcos, these magnitudes are mean valuesof nine measurements, obtained during the starÏs transitover the grid placed in front of the detector (see, e.g., Eyer etal. 1994). In addition to the magnitudes and the epochs ofobservations, EPA gives also the standard errors of themagnitudes, and quality Ñags. Two of the 124 Hp mag-p

Hp,

nitudes of DK Vir are Ñagged as unreliable and thereforewill not be used in the present analysis. Of the remaining122 magnitudes, 99 have quality Ñag 0, signifying an agree-

1341

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1342 JERZYKIEWICZ & PAMYATNYKH

FIG. 1.ÈTop: Maximum power histogram (MPH), i.e., the maximumvalues of the LS power over 0.2 cycles day~1 frequency intervals, max p( f ),computed from the Hp magnitudes of DK Vir. Middle : MPH after prewhit-ening with cycles day~1. Bottom: MPH after prewhiteningf1\ 9.2095with and cycles day~1.f1\ 9.2095 f2\ 7.5764

FIG. 2.ÈAmplitude spectrum of the data prewhitened with the twofrequencies in the frequency range with maximum MPH ordinates.

ment between the two consortia, NDAC and FAST, whichreduced the photometric data, while 23 have Ñag 1, meaningthat they are based on the results from the NDAC Consor-tium only. We shall assume that all of these 122 magnitudesare equally reliable.

The time distribution of the 122 Hp magnitudes of DKVir follows the usual pattern of Hipparcos data. The magni-tudes span an interval of 3.2 yr. They occur in sequences ofunequal length. The number of magnitudes in a sequenceranges from two to nine, but single, isolated magnitudesalso occur occasionally. Over the 3.2 yr interval, thesequences are distributed unevenly : the gaps between themvary from 2 to as much as 30 weeks, except for about 4 days,when the gaps between sequences are much shorter. In caseof DK Vir, the Ðrst observation was taken on JD 2,447,860,while the last one, on JD 2,449,026. The 4 days intervalwhen the star was observed almost continuously started onJD 2,448,037. The number of observations during this inter-val was 56.

This observing window is very di†erent from what obser-vers of variable stars are used to. Consequently, the associ-ated spectral window has properties which may defy theintuition of the ground-based observer. The proof is givenin the Appendix.

2.2. Frequency Analysis

In order to compute the least-squares (LS) power spectrafor this section, we slightly modiÐed the well-knownmethod of Lomb (1976). The modiÐcation consisted ingiving each Hp magnitude a weight inversely proportionalto and then multiplying each equation of condition by ap

Hp2

square root of the weight.Figure 1 presents the LS power spectra, computed using

the Hipparcos data described in the preceding section. Thecomputations were carried out with a step of 0.0001 cyclesday~1, that is, about one-ninth of the frequency resolution,equal to 1/3.2 cycles yr~1 (see the Appendix). However, inthe Ðgure we do not show individual points, but maximumvalues of the power, max p( f ), over frequency intervals of0.2 cycles day~1. In this way we avoid smearing, whichwould have been caused by plotting 10,000 points per unitfrequency interval, while preserving the overall shape of thepower spectrum. Such plots we shall henceforth refer to asthe ““ maximum power histograms,ÏÏ or MPHs. Of course,the frequency responsible for the highest ordinate in anMPH can be easily found from the full-resolution spectrum.

In the top panel of Figure 1, the highest ordinate iscaused by a peak at cycles day~1. This fre-f1 \ 9.2095quency agrees with the second frequency of et al.Pen8 a(1983) and the period given in the Hipparcos catalog.

After prewhitening with we repeated the calculations.f1,The results are shown in the middle panel of Figure 1. Now,

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DK VIRGINIS 1343

the highest ordinate is caused by a peak at f2\ 7.5764cycles day~1. A simultaneous two-sine LS Ðt with thesefrequencies yields the amplitudes (half-ranges) equal to

and The standardA1\ 13.5^ 0.007 A2\ 11.8^ 0.007.deviation (s.d.) of the Ðt amounts to 3.5 mmag.

After prewhitening with and no prominentf1 f2,maximum appears in the MPH (see Fig. 1, bottom). In thefrequency range where the MPH has the highest ordinates,the amplitude spectrum (see Fig. 2) shows no peaks higherthan 3.0 mmag. We conclude that the third and furtherterms in the light variation of DK Vir, if they exist, haveamplitudes below this limit.

3. THE GROUND-BASED DATA

3.1. ReÐning the FrequenciesAfter correcting for the di†erence in the mean light levels,

we combined the Lowell (L) and San Pedro (SPM)Ma� rtirdata (see ° 1) into a single set, henceforth referred to asL&SPM. The mean light levels were equal to 0.581 and0.000 mag for L and SPM, respectively. The set consists of381 observations. Although these data are not well suitedfor a frequency analysis because of the 7 yr gap between Land SPM, they can be used to reÐne the Hipparcos fre-quencies. For this is demonstrated in Figure 3, whichf2,compares the L&SPM power spectrum (solid line) with thepower spectrum, computed in ° 2.2 from the Hipparcos dataprewhitened with (dashed line). The reason for using thef1prewhitened Hipparcos spectrum is that at the time whenthe star was observed by Jerzykiewicz (1975) and et al.Pen8 a(1983), the second amplitude was larger than the Ðrst, A2[

(see below).A1

FIG. 3.ÈLS power spectrum of the ground-based data of Jerzykiewicz(1975) and et al. (1983) combined (solid line), and that of the Hip-Pen8 aparcos epoch photometry prewhitened with cycles day~1f1\ 9.2095(dashed line).

As expected, in the narrow frequency interval seen inFigure 3, the 0.00034 cycles day~1B 1/7 cycles yr~1 aliasesin the L&SPM spectrum are all of about the same height.The three that coincide with the Hipparcos maximum arecentered at 7.57607, 7.57641, and 7.57675 cycles day~1. Weshall adopt the middle frequency as the reÐned value of f2.Of the other two, the Ðrst one is an alternative, while thesecond probably is not. In a similar way, we found thereÐned value of to be equal to 9.20945 cycles day~1, withf19.20911 and 9.20979 cycles day~1 as the second and thirdchoices, respectively.

A simultaneous two-sine LS Ðt to the L&SPM data withthe reÐned values of the frequencies resulted in the V ampli-tudes, and the epochs of maximum light, listed inA

i, max

i,

Table 1. The s.d. of the Ðt amounted to 5.0 mmag. Thisrelatively large value is mainly due to the SPM data.Indeed, an LS Ðt to these data alone yields s.d. \ 5.5 mmag,while for the Lowell data s.d.\ 3.5 mmag.

3.2. The First Two Nights

Synthetic two-sine light curves, computed with theparameters from Table 1 and the mean light levels men-tioned in the Ðrst paragraph of ° 3.1, Ðt the data very well.This can be judged from Figure 4, whereÈin order to savespaceÈwe show only two nights, one from L and the otherfrom SPM, on which the agreement was less satisfactorythan on the remaining nights. Needless to say, Ðts of verynearly the same quality would be obtained if any of thealternative frequencies (see ° 3.1) were used instead of thevalues we adopted.

With the parameters from Table 1 we can also accountfor the starÏs behavior on the two nights which were notused in the above analysis, viz., the discovery night of Dan-ziger & Dickens (1967) and the 1970 night of Valtier (1971).As Figure 5 shows, Danziger & Dickens (1967) looked atthe star when it happened to be near the phase of maximumamplitude in the beat cycle between and while Valtierf1 f2,(1971) observed it at the phase of minimum amplitude. In asense, Valtier (1971) was luckier than Danziger & Dickens(1967) because in a beat cycle the amplitude minimum issharper than the maximum and therefore less likely to becaught by chance.

TABLE 1

PARAMETERS OF THE LS FIT TO THE GROUND-BASED DATA

fi

Ai

HJD Maxi

i (cycles day~1) (mmag) (2,441,000])

1 . . . . . . 9.20945 8.6 ^ 0.4 715.0294 ^ 0.00082 . . . . . . 7.57641 11.1 ^ 0.4 715.1205 ^ 0.0007

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1344 JERZYKIEWICZ & PAMYATNYKH

FIG. 4.ÈComparison of the Lowell (top) and San Pedro Ma� rtir(bottom) observations of DK Vir (circles) with the synthetic light curves(solid lines), computed from Table 1 parameters. Shown are the nights withthe greatest rms deviations.

The two nights can be accounted for only by the Ðrst-choice frequencies. Pairs of frequencies which include thesecond or third choice (see the preceding section) do not Ðtthese observations at all. Since there are eight such alterna-tive pairs, the good agreement between the observationsand synthetic light curves seen in Figure 5 is unlikely to bedue to chance. It is far more likely that the two frequencieshave not changed appreciably since 1965, and that theamplitudes were constant over the interval from 1965 to1980. However, the Hp amplitudes derived in ° 2.2 indicatethat has increased since 1980.A1

4. DISCUSSION

4.1. The H-R DiagramUsing mean indices from Hauck & Mermil-Stro� mgren

liod (1990) and the temperature calibration of Napiwotzki,& Wenske (1993), one Ðnds that DK Vir isScho� nbernere,

unreddened and has K. The starÏs B[V colorTeff \ 6980

index of 0.30 et al. 1995) and the calibration of Code(Garc•� aet al. (1976) yield a higher value of 7286 K. A value stillhigher, equal to 7577 K, would be obtained from the B[Vif the older calibration of Morton & Adams (1968) wereused instead of the Code et al. (1976) calibration.

According to the Hipparcos catalog, the parallax of DKVir amounts to 8.21 ^ 0.85 mas. The mean V magnitude,computed from the values given by & OlsenGrÔnbech(1976) and Olsen (1983), is equal to 6.688^ 0.007. Thesedata yield andM

V\ 1.14^ 0.22 log (L /L

_)\ 1.436

The Lutz & Kelker (1973) correction of [0.12 was^ 0.088.added to The bolometric correction of 0.01 was takenM

V.

from Code et al. (1976), and the bolometric magnitude ofthe Sun was assumed to be equal to 4.74.

Figure 6 shows DK Vir in the H-R diagram. The abscissaof the plotted point ( Ðlled circle) is the logarithm of themedian value of that we derived, while the other twoTeffvalues deÐne the extent of the error bar. An error barresulting from a mean error of ^0.02 in the color indexwould be twice as short. Thus, the uncertainty of the e†ec-tive temperature is dominated by systematic di†erencesbetween photometric calibrations. For the errorlog (L /L

_),

bar represents the above-mentioned mean error of ^0.088.Also shown in the Ðgure are the WarsawÈNew Jersey

evolutionary tracks for initial X \ 0.7 and Z\ 0.02, com-puted with or without overshooting from the convectivecore (dashed and solid lines, respectively), the same as thoseused by, e.g., Breger & Pamyatnykh (1998). Details of theevolution code can be found in Pamyatnykh et al. (1998).

FIG. 5.ÈComparison of the observations that were not used in theanalysis with the synthetic light curves, computed from Table 1 param-eters. Top: Discovery night of Danziger & Dickens (1967). Bottom: 1970night of Valtier (1971).

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DK VIRGINIS 1345

FIG. 6.ÈDK Vir ( Ðlled circle) and the WarsawÈNew Jersey evolution-ary tracks (lines) for X \ 0.7 and Z\ 0.02. The heavy solid lines along the2 track are explained in the text. FG Vir, the celebrated d Scuti starM

_with the largest number of identiÐed modes (Breger et al. 1998), is alsoplotted.

Assuming the solar composition parameters is justiÐed inthis context because the index of DK Vir isStro� mgren m1equal to 0.162, a value which is typical for an early F star(Crawford 1975).

As can be seen from Figure 6, DK Vir falls in the regionof the H-R diagram near the terminal-age main sequence(TAMS), where tracks for di†erent masses intersect becauseof loops. The loops are caused by hydrogen exhaustion inthe stellar center which is followed by disappearance of theconvective core and by signiÐcant reorganization of stellarinterior structure. Since the starÏs mass is not known, itsevolutionary state cannot be uniquely determined. If themass were equal to D2.1 the star would be still in theM

_,

main-sequence phase of its evolution, but decreasing themass to D2.0 would shift it beyond TAMS. An investi-M

_gation of the rate of change of the observed frequencies maysettle the issue and provide a better mass estimate. Unfor-tunately, the available data are insufficient for this purpose.

4.2. Pulsation Modes

As far as we are aware, no multicolor or spectrographictime-series observations of DK Vir have ever been carried

out. The spherical harmonic degrees, l, and the azimuthalorders, m, of the two modes are therefore unknown.However, the relatively large photometric amplitudes implythat the lÏs are unlikely to be greater than 2. Assuming l¹ 2,we shall now compare the observed values of and withf1 f2frequencies computed by means of a linear nonadiabaticpulsation code (for a description of the code see Pam-yatnykh et al. 1998). Although a Ðrm mode identiÐcationcannot be obtained in this way, as examples of even the bestobserved d Scuti stars such as FG Vir (Breger et al. 1998) orXX Pyx (Pamyatnykh et al. 1998) clearly show, we maylimit the number of possibilities. For the purpose of thisreconnaissance we neglect rotation and overshooting andassume the initial X \ 0.7 and Z\ 0.02. Moreover, themass will be taken to be equal to 2 Thus, the equi-M

_.

librium models will be the same as those that trace themiddle solid line in Figure 6.

Making use of the value of derived in the precedingTeff,section, and the observed values of the frequencies (seeTable 1), we selected models which met the following condi-tions : (1) between 3.836 and 3.888, i.e., in the inter-log Teffval 1.5 times longer than the error bar in Figure 6 ; (2) atleast two excited modes with frequencies in the range from6.6 to 10.2 cycles day~1 ; (3) at least one frequency ratio(regardless of l) between 0.803 and 0.843. As it turned out,such models occupy the two short segments of the 2 M

_evolutionary track, indicated in Figure 6 by heavy solidlines ; the Ðrst segment (fs) covers the end of the main-sequence stage (e†ective hydrogen burning in the core) andthe beginning of the secondary contraction, while thesecond segment (ss) covers the early phases of the postÈmain-sequence expansion stage of the shell hydrogenburning.

As the next step, we plotted a Petersen diagram with f2/f1as ordinates and as abscissae ; the ordinates had a rangef2determined by condition 3, while the abscissae varied from6.6 to 8.6 cycles day~1, in agreement with conditions 2 and3. The observed values of the frequencies deÐned a pointclose to the center of the diagram. Points corresponding tomodel frequencies could be enumerated according to andl1

the spherical harmonic degrees of the and modes. Asl2, f1 f2it turned out, there were no points with Thus,l1\ l2\ 0.the observed modes cannot both be radial. Moreover, forthe fs models there were no points with and orl1 \ 0 l2\ 12. Therefore, if the star were still in the main-sequencephase, the observed mode would be nonradial. Thef1remaining possibilities cannot be rejected at this stage.

This research has made use of NASAÏs Astrophysics DataSystem Abstract Service. A. P. acknowledges partialsupport from KBN grant 2 P03D 014 14. We are indebtedto Michel Breger for his comments on the Ðrst version of thepaper.

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1346 JERZYKIEWICZ & PAMYATNYKH

APPENDIX

HIPPARCOS SPECTRAL WINDOW

A1. OBSERVING WINDOW

We employ this term in its usual sense of time distribution of the data. The analysis of the Hipparcos observing window thatwe carry out in this section is necessary for understanding the associated spectral window, considered in the next section. Inwhat follows we assume that the reader is familiar with Fourier transform methods as presented by, e.g., Wehlau & Leung(1964) and Deeming (1975).

As already mentioned in ° 2.1, the time distribution of the Hp magnitudes of DK Vir follows the usual pattern of Hipparcosdata. The magnitudes span an interval of 3.2 yr. They occur in sequences of unequal length. Within a sequence, the successivemagnitudes are separated by or days ; after the shorter interval there follows the longer one, and*1 t \ 0.0143 *2 t \ 0.0746vice versa. The shorter interval corresponds to the time of rotation of the satellite through 58¡, while the longer onecorresponds to the time of rotation through 302¡. The number of magnitudes in a sequence ranges from two to nine, butsingle, isolated magnitudes also occur occasionally.

Note that a sequence can be represented as a sum of two truncated series of d-functions, each with a spacing equal todays, but shifted with respect to each other by or Thus, the sequence starting at will be*t \*1 t ] *2 t \ 0.0889 *1 t *2 t. t

mgiven by

Sm(t)\ ;

n/0

N1~1d(t [ t

m[ n*t) ] ;

n/0

N2~1d(t [ t

m[ *

kt [ n*t) , (A1)

where is the number of magnitudes in the sequence, with and either equal to each other or di†ering by 1. TheN1] N2 N1 N2subscript k in the second term is 1 if the interval between the Ðrst two magnitudes in the sequence is and 2 otherwise. If*1 t,this interval is equal to that between the last two magnitudes, then In case of isolated magnitudes, the sequenceN1\ N2.reduces to a single d-function, S

m(t)\ d(t[ t

m).

Equation (A1) can also be written as

Sm(t)\ [III(q) ] III(q[ *

kq)]%

m(t) , (A2)

where q\ t/*t, III(q) and are the Shah functions, and is a rectangular function, extending*kq\ *

kt/*t, III(q[ *

kq) %

m(t)

from the Ðrst to the last observation in the sequence.The observing window is the sum of all sequences, including the single-magnitude ones :

W (t) \ ;m/1

NSSm(t) , (A3)

where is the number of the sequences. In case of DK Vir,NS

NS\ 32.

Over the 3.2 yr interval, the sequences are distributed unevenly : the gaps between them vary from 2 to as much as 30 weeks,except for about 4 days, when the gaps between sequences are much shorter, of the order of 0.25 days.

A2. SPECTRAL WINDOW

Strictly speaking, the spectral window is the Fourier transform of the observing window:

w( f ) \ ;m/1

NSsm( f ) , (A4)

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DK VIRGINIS 1347

where is the Fourier transform of In general, w( f ) is a complex function of frequency. In practice, however, onesm( f ) S

m(t).

deals with the modulus, ow( f ) o , or the square of it, given by the equation

ow( f ) o2 \ 1N2GC

;j/1

Nsin (2nft

j)D2]

C;j/1

Ncos (2nft

j)D2H

, (A5)

where is the epoch of the jth observation, N is the number of observations, and f is the frequency, expressed in cycles day~1.tj

Figure 7 shows ow( f ) o2 for two frequency ranges, from [27 to 27 cycles day~1 (top), and from [0.5 to 0.5 cycles day~1(bottom). In both cases the calculations were done with a frequency step of 0.0001 cycles day~1, that is, about one-ninth of thefrequency resolution (see below), but the individual values of ow( f ) o2 are plotted only for the narrower range of frequency. Inthe wider range there are plotted the maximum values of ow( f ) o2 over frequency intervals of 0.2 cycles day~1. In this way, thetop Ðgure shows the overall shape of the main lobe and the distribution of the sidelobes, while the bottom Ðgure shows detailsof the central part of the main lobe. Using the term introduced in ° 2.2, we refer to Figure 7 (top) as the ““ maximum powerhistogram ÏÏ (MPH).

The frequencies seen in Figure 7 (bottom) correspond to timescales of 2 days or longer. At these frequencies, the details ofthe spectral window depend on the time distribution of the sequences. In particular, the width of the central spike, whichdetermines the frequency resolution of Hipparcos epoch photometry, is equal to the reciprocal of the interval of summation inequation (A3), i.e., 1/3.2 cycles yr~1\ 0.00086 cycles day~1.

FIG. 7.ÈSpectral window squared, ow( f ) o2, where w( f ) is the spectral window of Hipparcos epoch photometry of DK Vir. Top: MPH, i.e., the maximumvalues of ow( f ) o2 over 0.2 cycles day~1 frequency intervals. Bottom: Details of the central peak.

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1348 JERZYKIEWICZ & PAMYATNYKH

In order to account for the sidelobes seen in Figure 7 (top), we must turn to timescales shorter than a tenth of a day. Suchtimescales appear in equation (A2). From this equation and the convolution theorem it follows that the Fourier transform of

can be written asSm(t)

sm( f )\ T

m[1 ] exp (2ni*

kqr)]III(r) \ Mexp [2ni(t

m] T

m/2) f ] sinc (nT

mf )N , (A6)

where r\ 1/q and is the time interval spanned by i.e., the width of in equation (A2). The rotation by inTm

Sm(t), %(t

m) 2n*

kqr

the Ðrst term is introduced by the shift between the Shah functions in equation (A2), while the rotation by*kq 2n(t

m] T

m/2) f

in the second term is introduced by the median epoch of the sequence. The square of the modulus of normalized to unitysm( f ),

for f\ 0, is

o sm( f ) o2\ 0.25[2] cos (2n*1 qr) ] cos (2n*2 qr)]III(r) \ sinc2(nT

mf ) . (A7)

The convolution with the III(r) function in these equations explains why the sidelobes in Figure 7 (top) are equally spaced,with the spacing equal to 1/*t \ 11.25 cycles day~1, the frequency of rotation of the satellite. This is because the III(r)function is the same for all in equation (A4). The practical conclusion is the following : if the highest frequency expecteds

m( f )

FIG. 8.ÈSpectral window squared, ow( f ) o2, and the envelope, given by (see eq. [A7]). Top: The same MPH as in0.25[2] cos (2n*1 qr) ] cos (2n*2 qr)]Fig. 7 (top), but in a wider frequency range. Bottom: ArtiÐcial MPH, obtained under the assumption that the rotation rate of the satellite was strictly constant.Note that the peaks are now coinciding with the envelope. See text for further details.

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DK VIRGINIS 1349

to be present in the light variation of a star is the analysis of Hipparcos epoch photometry of the star can be limited tof*,

frequencies below cycles day~1. In this sense, the frequency of 1/*t is analogous to the Nyquist frequency forf*

] 11.25equally spaced data.

Unfortunately, explaining the amplitudes of the sidelobes in Figure 7 (top) is not that simple. On one hand, the impressionone may get from the Ðgure that the amplitudes of the sidelobes decrease steadily with increasing o f o is incorrect. On the otherhand, the amplitudes do not follow the envelope given by the factor in square brackets in equation (A7). These facts aredemonstrated in Figure 8, which shows the envelope compared with the actual ow( f ) o2 (top), and with the ideal ow( f ) o2, i.e.,one that would be obtained if the rotation rate of the satellite were strictly constant (bottom). The artiÐcial data for the idealcase were constructed by replacing the real gaps between sequences with ones equal to where n and m are wholen*1 t ] m*2 t,numbers chosen to represent the actual gaps to within (k \ 1 or 2).^0.5*

kt

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2000 PASP, 112 :1341È1349

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