Extensive photometry of the intermediate polar V1033 Cas (IGR J00234+6141)
aa r X i v : . [ a s t r o - ph . S R ] N ov Extensive photometry of the intermediate polarV1033 Cas (IGR J00234+6141)
V. P. KozhevnikovAbstract
To measure the spin period of the white dwarf inV1033 Cas with high precision, we performed extensivephotometry. Observations were obtained over 34 nightsin 2017. The total duration of the observations was143 h.We found that the spin period of the white dwarfis equal to 563 .
116 33 ± .
000 10 s. Using this period,we derived the oscillation ephemeris with a long valid-ity of 100 years. The spin oscillation semi-amplitudewas stable and was equal to 95 . ± . . ± . ± P/ d t is most probably less than − × − . This contradicts the assumption that thewhite dwarf in V1033 Cas is not spinning at equilib-rium. Our spin period and our oscillation ephemeriscan be used for further investigations of the stability ofthe spin period in V1033 Cas. V. P. KozhevnikovAstronomical Observatory, Ural Federal University,Lenin Av. 51, Ekaterinburg 620083, Russia e-mail:[email protected]
Keywords stars: individual: V1033 Cas; novae, cata-clysmic variables; stars: oscillations.
Cataclysmic variables (CVs) consist of a white dwarfthat accretes material from a late type secondary star.Accretion can occur through a bright accretion discwhen the white dwarf is non-magnetic, through a brightaccretion column when the white dwarf is strongly mag-netic and through a truncated accretion disc when thewhite dwarf is moderately magnetic. In the latter case,such CVs are called intermediate polars (IPs). Be-cause of the many phenomena of variability, IPs arevery interesting for detailed photometric investigations.In IPs, the magnetic white dwarf spins asynchronouslywith the orbital period of the system and therefore pro-duces a rapid oscillation with the spin period. Thisoscillation is coherent and shows a stable period. Be-cause this oscillation is produced not only in opticallight but also in X-rays, X-ray reprocessing by the sec-ondary star or the bright spot in the disc usually createsthe oscillation with the beat period, 1 /P beat = ω − Ω,where ω = 1 /P spin and Ω = 1 /P orb . Often, such or-bital sidebands are stronger than oscillations with thespin period. In addition, other orbital sidebands suchas ω −
2Ω and ω + Ω can be produced from amplitudemodulation (Warner 1986). So, multi-periodicity is atypical feature of IPs. Reviews of IPs are presented inPatterson (1994); Warner (1995); Hellier (2001).The spin period modulation is the main characteris-tic of belonging to the IP class. Therefore, spin periodmeasurements are a necessary condition for acceptinga CV as an IP. The spin period must be coherent andstable. In addition, the precise spin period and the pre-cise oscillation ephemeris make it possible to perform anobservational test of spin equilibrium either from direct measurements of the spin period or from pulse-arrivaltime variations by using a precise oscillation ephemeris.Although the IP nature of V1033 Cas was suspected12 years ago, to date no precise spin period and pre-cise oscillation ephemeris have been obtained. In-deed, for the first time, Bikmaev et al. (2006) notedthat V1033 Cas shows an optical oscillation with aperiod of 570 s. But, they did not even define thepossible error of this period. It seems unacceptable.Next, Bonnet-Bidaud et al. (2007) found that the pe-riod of the optical oscillation in V1033 Cas is equal to563 . ± .
62 s. The precision of this period is low be-cause this period was measured using data from a singleobservational night. Another observation of V1033 Caswas obtained in X-rays by Anzolin et al. (2009). Theyfound that the X-ray period is equal to 561 . ± .
56 s.Because the last two periods correspond to each other,this confirms the IP nature of V1033 Cas. In addition,the detection of the X-ray period suggests that thisperiod is most probably the spin period of the whitedwarf, but not the sideband period (e.g., Hellier 2001).However, both detected periods have low precision andare not suitable to investigate changes of the spin pe-riod of the white dwarf in the future. Indeed, if wesuppose an oscillation ephemeris with the spin periodmeasured by Bonnet-Bidaud et al., then the formal va-lidity of this ephemeris will be only 6 days. To measurethe spin period with high precision, to obtain a spinoscillation ephemeris with a long validity and to learnother properties of V1033 Cas, we performed extensivephotometric observations within 34 nights. These ob-servations have a total duration of 134 hours and cover11 months. In this paper, we present the results ob-tained from these observations.
For observations of variable stars, we apply a multi-channel pulse-counting photometer with photomulti-plier tubes. The photometer is attached to the 70-cmtelescope at Kourovka observatory, Ural Federal Uni-versity. This photometer makes it possible to measurethe brightness of two stars and the sky background si-multaneously and almost continuously. Short and rareinterruptions appear when we measure the sky back-ground in all three channels simultaneously. This isnecessary to define differences in the sky backgroundcaused by differences in the size of the diaphragmand other instrumental and atmospheric effects. Tomaintain the centring two stars in the photometer di-aphragms, we first carried out observations by manuallyguiding with an eyepiece and a guiding star. Later we
Table 1
Journal of the observations.Date BJD
TDB start Length(UT) (-245 0000) (h)2017 Jan. 30 7784.194728 8.02017 Jan. 31 7785.092285 9.62017 Feb. 16 7801.134348 4.02017 Feb. 17 7802.129536 1.32017 Mar. 16 7829.168016 2.62017 Mar. 19 7832.173661 5.82017 Mar. 20 7833.178435 6.82017 Mar. 21 7834.182622 2.92017 Mar. 22 7835.178989 7.62017 Mar. 24 7837.281417 1.52017 Mar. 27 7840.189258 2.72017 Mar. 31 7844.202522 2.72017 Apr. 1 7845.290858 4.32017 Apr. 25 7869.286424 2.92017 Apr. 26 7870.254826 2.32017 Apr. 30 7874.275348 3.12017 May 1 7875.306878 2.22017 Aug. 24 7990.306557 3.32017 Aug. 26 7992.349511 2.52017 Aug. 28 7994.313131 3.62017 Sep. 12 8009.221367 1.12017 Sep. 15 8012.191499 7.02017 Sep. 16 8013.405991 2.02017 Sep. 19 8016.404643 2.12017 Sep. 22 8019.161590 1.32017 Oct. 10 8037.124348 1.62017 Oct. 14 8041.121558 7.32017 Nov. 21 8079.382237 1.92017 Nov. 22 8080.206350 3.12017 Nov. 24 8082.343900 6.42017 Nov. 26 8084.219557 8.82017 Nov. 27 8085.399986 1.72017 Dec. 12 8100.056677 5.22017 Dec. 13 8101.063948 4.5 included the CCD guiding system in the photometer.Because of this guiding system, the photometer and thetelescope can work automatically under computer con-trol. Precise automatic guiding improves the accuracyof brightness measurements. In addition, it allows usto use lesser diaphragms, reducing the effect of the skybackground. The design of the photometer and its noiseanalysis are described in Kozhevnikov and Zakharova(2000).Our photometer provides high-quality photometricdata even under unfavourable atmospheric conditions(e.g., Kozhevnikov 2002). For a long time, however,we could not observe stars fainter than 15 mag, whichare invisible by eye (a 70-cm telescope). A few yearsago we understood how to make the relative centring oftwo stars in the photometer diaphragms, one of whichis invisible by eye, using the coordinates of the invis-ible star, the coordinates of the nearby reference star and computer-controlled step motors of the telescope.Using this method of centring, we can observe veryfaint stars up to 20 mag (e.g., Kozhevnikov 2018). Forsuch faint stars, continuous measurements of the skybackground in the third channel, which are possible inour photometer, are extremely important because thecounts of the sky background in the stellar diaphragmare much larger than the star counts. Therefore, photo-metric observations of very faint stars, which are invis-ible by eye, seem impossible without a sky backgroundchannel.The photometric observations of V1033 Cas wereperformed in 2017 January–December over 34 nightswith a total duration of 134 h. The data were obtainedin white light (approximately 3000–8000 ˚A). The timeresolution was 4 s. For V1033 Cas and the comparisonstar, we used 16-arcsec diaphragms. To reduce the pho-ton noise caused by the sky background, we measuredthe sky background through a diaphragm of 30 arcsec.The comparison star is USNO-A2.0 1500-00409838. Ithas B = 15 . B − R = 0 . B = 16 . B − R = 0 . TDB , which is the BarycentricJulian Date in the Barycentric Dynamical Time (TDB)standard. BJD
TDB is preferential because this timeis uniform. We calculated BJD
TDB using the online-calculator (http://astroutils.astronomy.ohio-state.edu/time/)(Eastman et al. 2010). In addition, using the BARYCENroutine in the ’aitlib’ IDL library of the University ofT¨ubingen (http://astro.uni-tuebingen.de/software/idl/aitlib/),we calculated BJD
UTC , the Barycentric Julian Date inthe Coordinated Universal Time (UTC) standard. Thedifference between BJD
TDB and BJD
UTC was constantduring our observations of V1033 Cas. BJD
TDB ex-ceeded BJD
UTC by 69 s.The obtained differential light curves have high pho-ton noise due to the low brightness of V1033 Cas(roughly 17.5 mag) and the high sky background.Fig. 1 presents the longest differential light curves ofV1033 Cas. To reduce the photon noise, the magni-tudes in these light curves were previously averagedover 80-s time intervals. According to the pulse countsof the two stars and the sky background, the photonnoise of these light curves (rms) is 0.04–0.08 mag. Asseen in Fig. 1, despite this large photon noise, the short-period oscillation with a period of 563 s, which is de-tected in X-rays (Anzolin et al. 2009) and in optical
Time (hours) D i ff e r en t i a l m agn i t ude Fig. 1
Longest differential light curves of V1033 Cas. Theshort-period oscillation is clearly visible in all light curves light (Bonnet-Bidaud et al. 2007) and corresponds tothe spin period of the white dwarf, is clearly visible inall light curves due to the large oscillation amplitude.
In addition to the short-period oscillation, the longestdifferential light curves of V1033 Cas presented inFig. 1 show obvious flickering, which is typical of allCVs. There are many complicated methods to in-vestigate flickering properties in CVs such as powerspectral density (Scaringi 2014), wavelet transforms(Fritz and Bruch 1998) etc. Here, to characterize theflickering power in V1033 Cas, we use the simplestmethod, namely, the flickering peak-to-peak ampli-tude. This allows us to compare the flickering powerin V1033 Cas with the flickering power in several IPs,
Frequency (mHz) S e m i a m p li t ude ( mm ag ) Fig. 2
Amplitude spectra of V1033 Cas. The prominentpeak, which corresponds to the 563-s period, shows an al-most constant height of about 90 mmag. The dotted linemarks the first harmonic of the 563-s period which we observed using the same technique. The flick-ering power in V1033 Cas seems weak and similar tothe flickering power we observed in most other IPs.From Fig. 1, we estimate that the flickering peak-to-peak amplitude is equal to 0.2–0.3 mag. V709 Cas,V515 And and V647 Cyg revealed similar flickeringpeak-to-peak amplitudes of less than 0.4 mag (seeFigs. 1 in Kozhevnikov 2001, 2012, 2014). The flickeringpower in these IPs seem weak compared with the flick-ering power in V2069 Cyg, which showed a noticeablylarger flickering peak-to-peak amplitude of 0.4–0.6 mag(see Fig. 1 in Kozhevnikov 2017).Our multichannel photometer allows us to obtainevenly spaced data. Such data are advantageouscompared with unevenly spaced data because theyallow us to use the classical Fourier analysis. In addition, the Fourier analysis seems preferable forsinusoidal or smooth quasi-sinusoidal signals (e.g.,Schwarzenberg-Czerny 1998). It is especially suitablefor IPs because short-period oscillations of sinusoidaland quasi-sinusoidal shapes are typical of IPs in opticallight. Using a fast Fourier transform (FFT) algorithm,we calculate individual amplitude spectra and powerspectra of the data included in common time series.Before applying the FFT algorithm, we eliminate low-frequency trends from individual light curves by sub-traction of a first-, second- or third order polynomialfit. This is a usual procedure in Fourier analysis anddoes not affect high frequencies.Despite the large photon noise, the spin oscilla-tion in V1033 Cas is clearly visible in the light curves(Fig. 1). The obvious reason for the good visibility ofthe spin oscillation is the large amplitude. Indeed, asseen in Fig. 2, which shows the amplitude spectra ofV1033 Cas, the spin oscillation semi-amplitude is ap-proximately 90 mmag. In addition, Fig. 2 demonstratesthat changes of the oscillation amplitude from night tonight are not significant. At frequencies below 1 mHz,the amplitude spectra show many small peaks. Thesepeaks are randomly distributed in frequency and areapparently caused by flickering.The individual amplitude spectra shown in Fig. 2provide comprehensive information about the ampli-tude of the spin oscillation and its behaviour in time.Using these amplitude spectra, however, we cannotdefine the precise oscillation period because their fre-quency resolution is very low. Therefore, we analysedthe data included into common time series. As men-tioned, we eliminated low-frequency trends from indi-vidual light curves by subtraction of a first-, second-or third order polynomial fit. After such eliminationof trends, the nightly averages of the individual lightcurves are equal to zero. Therefore, the gaps due todaylight and poor weather were filled with zeros ac-cording to the time of observations. This is optimalbecause gaps filled with zeros do not introduce addi-tional discontinuity to the data. In addition, zerosgive no additions to the Fourier transform. Due tothe large observation coverage, the power spectra ofthe common time series have a much higher frequencyresolution. To avoid the aliasing problem and find outchanges in the oscillation behaviour, we first divided theV1033 Cas light curves into two parts, namely, the dataof January–May and the data of August–December.These parts of data are separated by a large gap fromeach other (see Table 1). Fig. 3 shows the power spectraof two common time series containing data from thesetwo parts. These power spectra reveal distinct princi-pal peaks and one-day aliases corresponding to the spin oscillation. In addition, they show fine structures thatcoincide with the fine structures of the window func-tions. This proves the absence of the aliasing problem.In addition, this is a strong indication that the spinoscillation is coherent both during January–May andduring August–December.To find the precise values of the spin period, we useda Gaussian function fitted to the upper parts of theprincipal peaks. The errors are defined according to themethod of Schwarzenberg-Czerny (1991). This methodconsiders both frequency resolution and noise. It wastested in our previous works, where we made sure thaterrors according to Schwarzenberg-Czerny are true rmserrors (e.g., Kozhevnikov 2012). The values of the spinperiod found from the power spectra presented in Fig. 3are 563 .
116 83 ± .
000 60 and 563 .
116 90 ± .
000 50 s inJanuary–May and in August–December, respectively.These values correspond to each other, because theydiffer only by 0 . σ . Although in each of these two caseswe did not use all observational coverage, nevertheless,the precision of the oscillation periods was high. Thishigh precision is achieved due to the low noise level,which is caused by the large oscillation amplitude. Thespin oscillation semi-amplitudes found from the powerspectra shown in Fig. 3 are equal to 96 and 95 mmag inJanuary–May and in August–December, respectively.These semi-amplitudes are compatible with the heightsof the peaks visible in the individual amplitude spectra(Fig. 2).Making sure that the two parts of the data providedistinct power spectra without the aliasing problem, weincluded all 34 observational nights of V1033 Cas intothe common time series. Fig. 4 presents the power spec-trum segment of this common time series around thespin oscillation. Obviously, this power spectrum givesthe most precise spin period due to the highest fre-quency resolution and due to the lowest noise level. Asseen, the fine structure of this power spectrum is verysimilar to the fine structure of the window function.In addition, two peaks near the principal peak, whichare caused by the large gap between the two parts ofdata, have much lower heights than the height of theprincipal peak. Obviously, this eliminates the aliasingproblem. The close similarity of this power spectrumand the window function proves the complete coher-ence of the spin oscillation during the eleven monthsof our observations. The period and semi-amplitudeof the spin oscillation found from the power spectrumpresented in Fig. 4 are equal to 563 .
116 33 ± .
000 10 sand 95 mmag, respectively. They are compatible withthe periods and semi-amplitudes determined from thepower spectra of the two parts of data.Summarized information about the periods and theamplitudes of the spin oscillation is presented in Ta-
Frequency (mHz) F ou r i e r po w e r ( t he s qua r e o f m illi m agn i t ude ) JANUARY - MAYAUGUST - DECEMBER
Frequency (mHz) F ou r i e r po w e r Frequency (mHz) F ou r i e r po w e r Fig. 3
Power spectra calculated for two groups of thedata of V1033 Cas. They show the coherent oscillationwith periods of 563 .
116 83 ± .
000 60 (upper frame) and563 .
116 90 ± .
000 50 s (lower frame). Insets show the win-dow functions derived from artificial time series ble 2. Precise semi-amplitudes and their rms errorsare found from a sine wave fitted to the folded lightcurves. These semi-amplitudes are close to the semi-amplitudes found from the power spectra. The spinoscillation semi-amplitudes are surprisingly constant inthe two parts of data. In the fourth column, we givethe rms errors of the periods obtained by the methodof Schwarzenberg-Czerny (1991). The spin period er-ror found from all data is much lower than other er-rors. We found deviations of the other periods andexpressed them in units of their rms errors. This isshown in the fifth column. As seen, these deviationsare not excessively small and obey a rule of 3 σ . Asin our previous works (Kozhevnikov 2012, 2014, 2017), Table 2
The values and precisions of the spin periodTime Semi-amp. Period Error Dev.span (mmag) (s) σ (s)Jan.–May 95 . ± . . σ Aug.–Dec. 95 . ± . . σ Total 95 . ± . Frequency (mHz) F ou r i e r po w e r ( t he s qua r e o f m illi m agn i t ude ) Fig. 4
Power spectrum segment around the spin oscillationcalculated for all V1033 Cas data. It reveals a period of563 .
116 33 ± .
000 10 s. The upper frame shows the windowfunction. Note the close resemblance of the window functionand the power spectrum this confirms that the errors calculated according toSchwarzenberg-Czerny (1991) are true rms errors.To investigate the spin pulse profiles, we folded thelight curves of two parts of data with a period of563 .
116 33 s. Fig. 5 presents the results. The contin-uous curves are sine waves fitted to the folded lightcurves. These two sine waves have remarkably closesemi-amplitudes, namely, 95 . ± . . ± . σ . Although some consecutive points show one-sided deviations, these deviations are also statisticallyinsignificant. Indeed, if we combine the points of thefolded light curves into groups of three points and aver-age them, then the deviations from the sine waves willnot exceed 3 σ . Thus, the spin pulse profiles are sinu-soidal with high accuracy. This is compatible with theabsence of the first harmonic of the spin oscillation inthe amplitude spectra (Fig. 2).Using the spin period measured with high precision,we can derive an oscillation ephemeris with a long va-lidity. In addition to the spin period, we must deter-mine the phase of the oscillation. Obviously, becauseof photon noise and flickering, direct measurements of -0.5 0.0 0.5 1.0 1.5 Oscillation phase R e l a t i v e m agn i t ude b a Fig. 5
Pulse profiles of the spin oscillation obtainedfrom the data of January–May (a) and from the data ofAugust–December (b). The continuous curves are sinewaves fitted to the folded light curves. Note their veryclose semi-amplitudes: 95 . ± . . ± . oscillation phases can be obtained only with low preci-sion. Therefore, we obtained the oscillation phase fromthe folding of all data. Because the spin pulse profile issinusoidal, to find the time of spin pulse maximum, it isconvenient to use a sine wave fitted to the folded lightcurve. The resulting time of spin pulse maximum wasreferred to the middle of the observations. In addition,we used the data divided into four groups for verifica-tion. The times of spin pulse maximum for these groupswere also referred to the middle of the correspondingdata groups. These times are presented in Table 3. Us-ing the main time of spin pulse maximum, we obtainedthe following ephemeris: BJD
TDB (max) = 245 7942 .
727 207(21)+0 .
006 517 5501(12) E . (1) Table 3
Verification of the spin ephemeris
Time BJD
TDB (max) N. of O–C × span (-245 0000) cycles (days)Jan. 30–Mar. 20 7808.837170(32) –20543 –0.005(39)Mar. 21–May 1 7854.798952(28) –13491 +0.013(35)Aug. 24–Oct. 14 8015.873626(26) +11223 –0.046(33)Nov. 21–Dec. 13 8090.323680(35) +22646 +0.033(41) -25000 -15000 -5000 5000 15000 25000 Number of cycles O - C ( da ys ) - . + . Fig. 6 (O–C) diagram for all V1033 Cas data, which aresubdivided into four groups
To verify the oscillation ephemeris, we calculated( O − C ) values and numbers of cycles for the fourgroups of data (Table 3). The ( O − C ) diagram is pre-sented in Fig. 6. It reveals no significant slope anddisplacement along the vertical axis. Indeed, the cal-culated ( O − C ) values obey the relation: (O − C) = − .
000 008(67) − .
000 000 0004(33) E . All quantities inthis relation are less than their rms errors. Therefore,the ephemeris does not require any correction. Accord-ing to the rms error of the spin period, the formal va-lidity time of this ephemeris is 100 years (a confidencelevel of 1 σ ).To analyse the short-period oscillations, we removedthe low-frequency trends from the individual lightcurves by subtraction of a first-, second- or third or-der polynomial fit. To search for the orbital variabilityof V1033 Cas, such a procedure cannot be used becausemany of our individual light curves are much shorterthan the orbital period of V1033 Cas. According to theradial velocity measurements by Bonnet-Bidaud et al.(2007), the orbital period of V1033 Cas is 4 . ± .
005 h. Obviously, The low frequencies correspond-ing to the orbital variability of V1033 Cas will be re-moved from the short light curves due to this removalof trends. In addition, by performing numerical exper-iments with artificial time series, we made sure thatwe also could not remove the nightly averages from theindividual light curves because such removal would sig-nificantly reduce the orbital variability of V1033 Cas.Therefore, we removed only two common averages fromtwo groups of data.The resulting power spectra are presented in Fig. 7.As seen, at least in the data of January–May, the powerspectrum shows the prominent principal peak and itsleft-hand one-day alias, which correspond to the orbitalperiod found by Bonnet-Bidaud et al. (Fig. 7a). Fromthis power spectrum, the orbital period and the semi-amplitude are equal to 4 . ± . Frequency (mHz) F ou r i e r po w e r ( t he s qua r e o f m illi m agn i t ude ) ba A F A
Fig. 7
Power spectra of V1033 Cas in the frequencyrange around the expected orbital variability. The dataof January–May (frame a) reveal the prominent princi-pal peak which coincides with the orbital period found byBonnet-Bidaud et al. (labelled F). The left-hand one-dayalias is also prominent (labelled A). In addition, the smallright-hand one-day alias is perceptible (labelled A). In con-trast, the data of August–December (frame b) reveal nopeaks corresponding to the orbital period. Note that thevertical scale in frame b is two times less respectively. Here, the error of the orbital period isequal to the half-width of the principal peak at halfmaximum. In the data of August–December, the powerspectrum does not show signs of the orbital variabilityof V1033 Cas (Fig. 7b). The data of August–Decemberare not favourable to detect the orbital variability ofV1033 Cas because of shorter lengths of individual lightcurves and larger brightness changes from night to night(see Fig. 1), which create additional noise in the powerspectrum.To find the orbital pulse profile, we folded the dataof January–May, in which the orbital variability wasdetected, with a period of 4.0243 h. To avoid distortionsof the orbital pulse profile by the spin oscillation, thedata were previously averaged over 564-s time intervals.As seen in Fig. 8, the orbital pulse profile of V1033 Casis quasi-sinusoidal with slightly wider minima comparedwith maxima. The orbital pulse profile shows a semi-amplitude of 55 ± ω − Ω -0.5 0.0 0.5 1.0 Orbital phase R e l a t i v e m agn i t ude Fig. 8
Orbital pulse profile of V1033 Cas obtained from thedata of January–May. The pulse profile is quasi-sinusoidalwith slightly wider minima compared with maxima andshows a semi-amplitude of 55 ± and ω + Ω. These oscillations have the periods close tothe spin period. Therefore, when such oscillations havesmall amplitudes, they can be hidden in the complexstructure of the window function created by the spin os-cillation. To find orbital sidebands, we applied the com-monly used method of extraction of the main oscillationfrom the data. Fig. 9 presents the power spectra oftwo parts of data, from which the spin oscillation is ex-cluded. As seen, despite the large semi-amplitude of thespin oscillation, this oscillation is completely excluded.This, however, did not lead to the detection of anysideband oscillations. Obviously, the undetected side-band oscillations should have small amplitudes, whichdo not exceed the amplitudes of the noise peaks. Themaximum semi-amplitudes of the noise peaks are 9and 12 mmag in the data of January-May and in thedata of August–December, respectively. Thus, the side-band semi-amplitudes in V1033 Cas should not exceedroughly 10 mmag and should be at least ten times lessthan the semi-amplitude of the spin oscillation. We performed photometric observations of V1033 Caswith a total duration of 134 h that cover eleven monthsin 2017. The comprehensive Fourier analysis of thesedata allowed us to measure the spin period of thewhite dwarf in V1033 Cas with high precision, P spin =563 .
116 33 ± .
000 10 s. This high precision was achieveddue to the large observational coverage and due to thelow noise level. Although the spin period of the whitedwarf in V1033 Cas was measured previously both in X-rays (561 . ± .
56 s, Anzolin et al. 2009) and in opticallight (563 . ± .
62 s, Bonnet-Bidaud et al. 2007), theirprecision was much lower because of insufficient ob-servational coverage. Indeed, Anzolin et al. observed
Frequency (mHz) F ou r i e r po w e r( t he s qua r eo f m illi m agn i t ude ) ab T S - T S + Fig. 9
Power spectra of two parts of the V1033 Cas data,from which the spin oscillation is excluded. Both the powerspectrum of the data of January–May (frame a) and thepower spectrum of the data of August–December (frame b)do not show coherent oscillations. The expected frequenciesof the negative and positive orbital sidebands are labelled ω − Ω and ω + Ω, respectively V1033 Cas within 6.9 h, and Bonnet-Bidaud et al. ob-served V1033 Cas within 5.3 h. Such short observationshave a low frequency resolution and cannot provide highprecision of the oscillation period. In addition, suchshort observations have a large noise level. However,according to the rms errors, the spin period measuredby us is compatible with the spin periods measured byBonnet-Bidaud et al. and by Anzolin et al.During our observations, the spin oscillation semi-amplitude in V1033 Cas was stable and was equal to95 . ± .
90 mmag. The semi-amplitudes seen in other 6 IPswere in the range 0.005-0.041 mag. Although MU Camshows almost the same semi-amplitude as V1033 Cas,this IP has a noticeably larger spin period (1187 s). Tofind other IPs with large spin oscillation amplitudes andwith short spin periods, which were discovered after thework by Patterson, we used the IP homepage of Mukai(https://asd.gsfc.nasa.gov/Koji.Mukai/iphome/iphome.html)and references therein. The catalogue of IPs pre-sented in this homepage was updated at the endof 2014. We identified 17 ironclad and confirmedIPs with spin periods of less than 600 s and withknown optical amplitudes. We learned that V418 Gemshows the largest semi-amplitude, namely, 150 mmag(G¨ansicke et al. 2005). However, this is the semi-amplitude of the first harmonic of the spin oscillation inV418 Gem. If the true spin oscillations are considered,then V1033 Cas will show the larger semi-amplitude.The second largest semi-amplitude (65 mmag) belongsto HT Cam (de Martino et al. 2005). In addition, weconsidered 8 recently discovered IPs (Halpern et al.2018), where 6 of them were confirmed due to opticaloscillations. We learned that, among them, the largestsemi-amplitude was observed in 2PBC J1911.4+1412(0.25 mag). However, the spin period of this IP (747 s)is noticeably larger than the spin period in V1033 Cas.Among IPs, which were described in Halpern et al.,only 1SWXRT J230642.7+550817 has a lesser spin pe-riod of 464 s. Its oscillation semi-amplitude (50 mmag)is also noticeably lesser than the semi-amplitude weobserved in V1033 Cas. Thus, the oscillation semi-amplitude in V1033 Cas seems very large among IPswith similar and lesser spin periods. Due to the lowrelative noise level caused by flickering at higher fre-quencies, the large oscillation amplitude in V1033 Casis favourable to obtain the high precision of the spinperiod and to investigate changes of the spin period us-ing O − C . 11 years ago, however, the semi-amplitudeof the spin oscillation in V1033 Cas was two times less(see Bonnet-Bidaud et al. 2007).Although most of our individual light curves ofV1033 Cas are short compared with the orbital pe-riod detected by Bonnet-Bidaud et al. (2007) from ra-dial velocity measurements, we were nevertheless ableto find the corresponding photometric variations with aperiod of 4 . ± . ± ◦ (e.g., la Dous 1994). Anotherreason for orbital variability is the presence of orbital humps in the light curves, which are caused by a hotspot in the accretion disc. They are noticeable if theorbital inclination is larger than 50 ◦ (la Dous 1994).If the orbital inclination is less than 50 ◦ , the orbitalvariations of the novalike variables, to which most IPsbelong, are difficult to detect because their accretiondiscs are much brighter than their stellar components(e.g., Bonnet-Bidaud et al. 2001). Therefore, ellip-soidal modulations, reflection effects, etc. should be in-conspicuous. Thus, the orbital inclination of V1033 Casis probably 50 ◦ − ◦ .The estimated orbital inclination of V1033 Casroughly corresponds to its large spin oscillation am-plitude. As noted by Hellier et al. (1991), the spinoscillation amplitude is greatest when the system isviewed edge-on. As mentioned, MU Cam shows almostthe same spin oscillation semi-amplitude as V1033 Cas.In addition, MU Cam shows orbital variations with asimilar semi-amplitude of 42–59 mmag (see Fig. 5 inKozhevnikov 2016). The similar amplitudes of orbitalvariations in V1033 Cas and in MU Cam can mean thatthe inclinations of these two systems are also similar.This explains the similar spin oscillations amplitudes inthese two IPs. In addition, according to Hellier et al.(1991), the large spin oscillation amplitudes in thesetwo systems suggest that they are noticeably inclined.During our observations, the spin pulse profile wasconstant and sinusoidal with high accuracy (Fig. 5).This is compatible with the sinusoidal pulse profile thatwas observed by Bonnet-Bidaud et al. (2007) 11 yearsago. This very sinusoidal pulse profile is difficult to ex-plain. For most IPs that do not show eclipses and donot rotate very fast, the accretion curtain model canexplain optical spin pulses. According to this model,optical emission originates from two accretion curtainslocated between the inner disc and the white dwarf(Hellier 1995). Because these two accretion curtains canact in phase, the two accretion poles of the white dwarfcan create a single-peaked roughly sinusoidal pulse pro-file. However, it seems strange how two accretion cur-tains viewed at different angles can produce a strictlysinusoidal pulse profile. This strictly sinusoidal pulseprofile suggests that we see optical emission from oneaccretion curtain formed by one accretion pole.Two-pole disc-fed accretion is considered to be thenormal mode of behaviour in IPs because both mag-netic poles of the white dwarf are on equal conditionswith respect to the accretion disc (e.g., Warner 1995).Therefore, we cannot explain the strictly sinusoidalpulse profile by supposing one-pole accretion. In ad-dition, one of the two accretion curtains in an ordinaryIP cannot be continuously hidden by a white dwarf be-cause the length of accretion curtains is 4 – 12 times larger than the white dwarf radius (e.g., Ferrario et al.1993) and because accretion curtains are extended overangles greater than 100 ◦ (e.g., Hellier 1999). However,Hellier (1999) performed Doppler tomography of ac-cretion curtains in seven IPs and found that three ofthem show only the upper accretion curtains. Theseare high-inclination systems ( > ◦ ), which show lineemission only from the outer parts of the accretion cur-tains. Hence, the outer regions of the lower accretioncurtain in such IPs can be hidden due to the obscura-tion of the lower accretion curtain by the inner edge ofthe accretion disc and the obscuration by the upper ac-cretion curtain (Hellier 1999). Although our estimate ofthe V1033 Cas inclination is somewhat less (50 ◦ − ◦ ),it can be sufficient, and the lower accretion curtain canbe obscured by the disc and can be obscured by theupper accretion curtain. Thus, because the lower ac-cretion curtain in V1033 Cas can be mostly invisible,we can explain the strictly sinusoidal spin pulse profileas if it was created by a single accretion curtain.Despite the large amount of our observations, wedid not find oscillations in the orbital sidebands. Thisagrees with the previous observations by Bonnet-Bidaud et al.(2007) and Anzolin et al. (2009), who also did notfind sidebands. As seen in Fig.9, the semi-amplitudesof the undetected sideband oscillations do not exceed10 mmag and are at least ten times less than the spinoscillation semi-amplitude. Such weak signals seem un-detectable in slowly spinning IPs. Indeed, as seen inTable 1 in Patterson (1994), only two IPs show slightlylesser oscillation semi-amplitudes. These are rapidlyspinning AE Aqr and V533 Her with periods of 33.1and 63.6 s and with semi-amplitudes of 5 and 7 mmag,respectively. Hence, such weak signals may be belowthe detection threshold in slowly spinning IPs. Amongthe known IPs, there is a fairly distinct division be-tween those that have a strong spin oscillation and thosethat have a strong sideband oscillation. Nonetheless, inmany IPs, the alternative weak oscillation was detected(Warner 1995). In addition, the stable optical oscilla-tion and the presence of one or more orbital sidebands,even without any X-ray detection at all, provide strongevidence for inclusion in the IP class (Kuulkers et al.2006). Therefore, the absence of detectable orbitalsidebands in V1033 Cas at a semi-amplitude level of10 mmag is unusual and difficult to explain.Assuming a pure disk accretion, as indicated by theabsence of orbital sidebands in X-rays, Anzolin et al.(2009) suggested that the white dwarf in V1033 Cas isnot strongly magnetized and is not spinning at equi-librium. If so, the spin period of the white dwarf inV1033 Cas should change over large time spans. Usingour precise oscillation ephemeris 1 and times of spin pulse maximum that were obtained earlier, we can ver-ify the stability of the spin period. Unfortunately, inthe past, only the single time of spin pulse maximumwas obtained in optical light by Bonnet-Bidaud et al.(2007). Therefore, we also used the time of spin pulsemaximum obtained in X-rays by Anzolin et al. (2009).As noted by Anzolin et al., the time of spin pulse max-imum in X-rays coincided with the time of spin pulsemaximum in optical light. Unfortunately, in the past,other times of spin pulse maximum were not obtained.The times of spin pulse maximum, which were mea-sured by Bonnet-Bidaud et al. and Anzolin et al., wereexpressed in HJD UTC . We converted these times inBJD
TDB . Using our ephemeris 1 we calculated ( O − C ).Next, we calculated d P/ d t using the following formula(Breger and Pamyatnykh 1998):(O − C) = 0 . p d P d t t . (2)The results are shown in Table 4. Here, the upperline is calculated according to the time of spin pulsemaximum measured by Bonnet-Bidaud et al., and thelower line is calculated according to the time of spinpulse maximum measured by Anzolin et al. The fourthcolumn gives the calculated d P/ d t and their rms errors.These two d P/ d t , which are equal to − (1 . ± . × − and − (1 . ± . × − , match well each other be-cause their difference is 3 times less than the summaryrms error. In addition, these d P/ d t are only 2.7 timeslarger than their rms errors. Consequently, accordingto the triple rms error, the real d P/ d t must be lessthan − × − . However, if we change the numbersof cycles to ± ± ±
3, then these two d P/ d t willstill be compatible with each other because their dif-ference will be less than 3 σ . In the last case, d P/ d t isroughly ± × − . With larger changes of the num-bers of cycles, two d P/ d t become incompatible witheach other because their difference become larger than3 σ . Thus, the d P/ d t in V1033 Cas is most probablyless than − × − . This means that the spin periodin V1033 Cas is very stable. With a low probability,the d P/ d t can reach ± × − . However, even in thiscase, the spin period in V1033 Cas seems fairly stablebecause its possible change is less than the detectionthreshold of spin period changes in slowly spinning IPs Table 4
Stability of the spin period
BJD
TDB (max) N. of O – C d P/ d t (-245 0000) cycles (phases)3975.51859(7) –608696 − . ± . − (1 . ± . × − − . ± . − (1 . ± . × − (see Table 1 in Patterson 1994 and Table 1 in Warner1996).The high stability of the spin period in V1033 Casshould be considered as a new result. This resultseems very important because it contradict the sugges-tion that the white dwarf in V1033 Cas is not spin-ning at equilibrium (Anzolin et al. 2009). As a newresult, this should be confirmed by future observations.This confirmation can be made from new photometricobservations. Using new observations and our preciseephemeris 1, one can calculate ( O − C ) and analyse theirbehaviour. However, as shown above, a small numberof ( O − C ) can give ambiguous results because of ambi-guity in cycle numbers. Therefore, observations shouldbe regular for ten years or more. Of course, such obser-vations are difficult. Direct measurements of the spinperiod seem less difficult. By performing observationswithin 30–40 nights that cover a year, it is possibleto achieve the same rms error of the spin period as weachieved in our observations, namely, 0.0001 s. Then, ifthese observations are made ten years later, comparingtwo spin periods, one can reach d P/ d t of 1 × − (aconfidence level of 3 σ ). This d P/ d t is 4 times less thanthe upper limit of d P/ d t we obtained from the analysisof ( O − C ) described above. In addition, direct mea-surements of the spin period are devoid of ambiguity incycle numbers. We performed extensive photometric observations ofV1033 Cas over 34 nights. The total duration of obser-vations was 134 h. The observations covered 11 months.From the comprehensive analysis of these data, we ob-tained the following results:1. Due to the large observational coverage and the lownoise level, we measured the spin period of the whitedwarf with high precision. The spin period is equalto 563 .
116 33 ± .
000 10 s.2. During our observations, the semi-amplitude of thespin oscillation was stable and was equal to 95 . ± . . ± . ± P/ d t is most probably less than − × − . This highstability contradicts the assumption that the whitedwarf in V1033 Cas is not spinning at equilibrium.7. Our precise ephemeris and our precise spin periodcan be used for future investigations of the stabilityof the spin period in V1033 Cas. Acknowledgments
This work was supported in part by the Ministry ofEducation and Science (the basic part of the State as-signment, RK No. AAAA-A17-117030310283-7) and byProgram 211 of the Government of the Russian Feder-ation (contract No. 02.A03.21.0006). This research hasmade use of the SIMBAD database, operated at CDS,Strasbourg, France. This research also made use of theNASA Astrophysics Data System (ADS). References