Modelling of the polarized cyclotron emission from magnetic cataclysmic variables (MCVs) has been a powerful technique for determining the structure of the accretion zones on the white dwarf. Until now, this has been achieved by constructing emission regions (for example arcs and spots) put in by hand, in order to recover the polarized emission. These models were all inferred indirectly from arguments based on polarization and X-ray light curves. Potter, Hakala & Cropper (1998) presented a technique (Stokes imaging) which objectively and analytically models the polarized emission to recover the structure of the cyclotron emission region(s) in MCVs. We demonstrate this technique with the aid of a test case, then we apply the technique to polarimetric observations of the AM Her system V347 Pav. As the system parameters of V347 Pav (for example its inclination) have not been well determined, we describe an extension to the Stokes imaging technique which also searches the system parameter space (GOSSIP)

Cropper, M

Hakala, P

Potter, SB

English

UCL Discovery

arXiv:astro-ph/9809317v1  24 Sep 1998The GOSSIP on the MCV V347 Pav.S.B. Potter,1 Mark Cropper1 and P. J. Hakala21Mullard Space Science Laboratory, University College London,Holmbury St. Mary, Dorking, Surrey, U.K.2Observatory and Astrophysics Laboratory, FIN-00014, University ofHelsinki, Finland.Abstract. Modelling of the polarized cyclotron emission from magneticcataclysmic variables (MCVs) has been a powerful technique for deter-mining the structure of the accretion zones on the white dwarf. Untilnow, this has been achieved by constructing emission regions (for ex-ample arcs and spots) put in by hand, in order to recover the polarizedemission. These models were all inferred indirectly from arguments basedon polarization and X–ray light curves.Potter, Hakala & Cropper (1998) presented a technique (Stokes imag-ing) which objectively and analytically models the polarized emission torecover the structure of the cyclotron emission region(s) in MCVs. Wedemonstrate this technique with the aid of a test case, then we apply thetechnique to polarimetric observations of the AM Her system V347 Pav.As the system parameters of V347 Pav (for example its inclination) havenot been well determined, we describe an extension to the Stokes imagingtechnique which also searches the system parameter space (GOSSIP).1. IntroductionMany authors (e.g. Potter et al. 1997; Wickramasinghe & Ferrario 1988; Beuer-mann, Stella & Patterson 1987; Cropper 1986) have found that the emissionregion in MCVs must be extended into an arc or comma shape in order toexplain the shape of the polarized cyclotron light curves. Until now, the pa-rameters that define the shape and location of the emission regions have beenadjusted by a trial-and-error approach until the model gave a good fit. Clearlythe number of free parameters describing the location, shape and structure ofthe emission regions becomes very large.The technique of Potter, Hakala & Cropper (1998, PHC) objectively mod-elled the polarimetric data and obtained maps of the emission regions for thefirst time. Their model had the anisotropies of polarized emission incorporatedinto it. This allowed them to model the intensity, circular polarization, linearpolarization and position angle data. Instead of using the maximum-entropymethod and the conjugate optimization (used on intensity data of Cropper &Horne 1994) they used Tikhonov regularization and a genetic algorithm for op-timisation.11.1. The cyclotron modelThe first step in modelling cyclotron emission from MCVs is to calculate theviewing angle, i.e., the angle between the line of sight and the magnetic field linefrom where the emission emanates. This was done by using a dipole magneticfield formalism for the white dwarf, together with a system inclination andoffset dipole parameters (see Cropper 1989). For a given magnetic longitudeand latitude of the emission point on the white dwarf, the viewing angle wasobtained for different phases of the spin cycle. The local magnetic field wascalculated and subsequently the local optical depth parameter. The intensityspectrum and percentage of circular and linear polarization was then calculatedfrom interpolating on the cyclotron opacity calculations of Wickramasinghe andMeggitt (1985). Extended sources were modelled by summing the componentsof many such emission points as a function of spin phase.The optimisation of the model to the data proceeds by adjusting the numberand distribution of emission points across the surface of the white dwarf. Duringthe optimisation the inclination, magnetic dipole offset (in latitude and azimuth)and the magnetic field strength at the poles are kept fixed.1.2. Stokes ImagingPHC used a genetic algorithm (GA) in order to optimise the fit to the data.The GA works by first generating a set of random solutions. The fitness of eachsolution is then calculated usingF (p) = χ2 + λ∑i‖∇pi‖2 (1)where ‖∇pi‖ is the mean gradient of the number of emission points at pointi and χ2 is the chi-square fit. The fitness is a measure of how good the fit is tothe data plus the smoothness of the image solution. The solutions are ranked inorder of their fitness. The next generation of solutions are then produced by atype of natural selection procedure – the solutions that were ranked best are moreprobable to breed the next generation and the next population is generated byapplying genetic crossover and mutation operators to the selected parent pairs.Eventually, after many generations, the improvement in fitness of the GAsolutions will start to level out and a more analytical approach is required toimprove the fit further. For the final approach to minimum the Powell’s methodline minimisation routine (see Press et al. 1992) was used.1.3. The test caseTo demonstrate the technique we have generated synthetic Stokes parametercurves with 50 points over the orbital phase (typical datasets have roughly thisnumber). This gives us 200 data points to fit. In this paper we use a 6 degreeresolution map on the white dwarf surface, which in turn yields 1740 free param-eters (60× 29 surface grid elements on the white dwarf surface). The simulatedpolarized light curves presented below were produced by calculating the emis-sion that would arise from two extended sources (Fig. 1a) in the B–band. Theresulting light curves (Fig. 1b) has infinite signal–to–noise ratio. The inclination2     0500100015002000Intensity Fluxa.b.c.Magnetic LatitudeMagnetic Longitude10     050100150200Linear Flux     -1000-50005001000Circular Flux0.0 0.5 1.0 1.5 2.0Phase-100-50050100P.A. DegreesModel Prediction          Test Image          Figure 1. The test case a. The emission on the surface of the whitedwarf. b. The test emission light curves (histogram) and the model fit(solid smooth curve). c. The optimised image.was 80o, the magnetic dipole offset was 10o, the magnetic polar field strengthwas 60 MG and the polar optical depth parameter Λ was 1.0× 105.Figure 1b shows the model fit (thick smooth curves) to the input test data(histogram curves). The optimisation technique has reproduced the complexfeatures arising from the extended regions. These include the linear and intensityfeatures such as dips and peaks. It has also correctly modelled both the positiveand negative polarized light curves.Figure 1c shows the model prediction for the shape and location of theemission regions. The grey scale is a measure of the optical depth across theemission region. As can be seen from the figure, the optimisation routine hasaccurately located and mapped the emission regions across the surface of thewhite dwarf.2. Genetically Optimised ‘Super’ Stokes Imaging Procedure (GOS-SIP)For the test case described above we created polarimetric light curves using thecyclotron model. Therefore, the system parameters (inclination, magnetic field3orientation and strength) were known. In the case of real data these parametersare generally unknown.PHC explained that the system parameters were not included in the opti-misation procedure. This is because these parameters are qualitatively differentfrom those describing each point on the derived map. Therefore, the systemparameter space has to be searched separately from that searched during the‘Stokes imaging’ technique, either by trial and error, or by the use of a ge-netic algorithm operating as an outer loop to the ‘Stokes imaging’ procedure.The outer optimisation loop optimises the final fitness values produced by sev-eral optimised solutions. In effect, the optimised solutions produced by severalruns of the ‘Stokes imaging’ technique become solutions for genetic optimisationthemselves.2.1. The MCV V347 PavFigure 2a,b shows the blue and red polarimetric observations of the MCV V347Pav (RE J1844-741) from Ramsay et al (1996). Bailey et al (1995) suggested thatthe positive and negative excursions of the circular polarization over the orbitalperiod of ∼ 90 mins was due to emission arising from two regions located at ornear opposite ends of the magnetic field. The high signal-to-noise observationsof Ramsay et al. (1996) make them ideal for the optimisation technique.The technique predicts a best fit for an inclination of 62o, dipole offset of32o and magnetic field strengths of 28 and 24MG for the upper and lower polesrespectively. From Figure 2 it can be seen that the optimised model solutionhas reproduced the red polarization data remarkably well. The morphology ofall the red polarized light curves has been accurately predicted, in particular,the variation in position angle, the relative amounts of linear and circular flux,the peaks and dips in the circular polarization and the overall variation in theintensity curve. It has also predicted the gross features of the blue polarizedvariations and the correct amount of relative flux between the two wave bands.The technique finds a solution that fits both wave bands simultaneously.However, it has not modelled the finer details of the blue polarimetric varia-tions. This has arisen because of the less accurate determination for the magneticfield strengths, thus affecting the quality of the fit to one wave-band more thanthe other (the relative fluxes between the two wave-bands depends on the mag-netic field strength). Also, the red polarimetric observations have significantlymore counts than the blue band. Therefore, the optimisation routine will bemore biased towards fitting the red data. The fit to the blue data can be im-proved by increasing the magnetic field strength on the upper pole. Maintaininga dipole field constraint, however, makes the fit to the red data significantlyworse. Any temperature structure within the shock will also affect the finer de-tails of the polarized light curves in different bands. We have assumed a constanttemperature of 10keV for the emitting gas.Figure 3 shows the predicted emission regions mapped onto a sphere rep-resenting the surface of the white dwarf as viewed from Earth for a completeorbital rotation. The observed features of the polarimetric observations (Figure2a,b) can be explained with reference to Figure 3. The most prominent featureof the observations is the bright and faint phase. The main emission region inthe lower hemisphere is responsible for the bright phase emission during φ ∼ 0.0–4OG570 (red) filter          Intensityb.BG39 (blue) filter     05.0•1041.0•1051.5•1052.0•105ctsa.           Linear Pol.     0200040006000800010000cts          Circular Pol.     -3•104-2•104-1•10401•104cts0.0 0.5 1.0 1.5 2.0Phase     P.A.0.0 0.5 1.0 1.5 2.0Phase050100150200DegreesFigure 2. The ‘Stokes imaging’ solution to the polarimetric observa-tions of Ramsay et al. (1996) for an inclination = 62o and a dipoleoffset = 36o. a. The blue data. b. The red data.0.5 and the secondary smaller emission region is responsible for the faint phaseemission during φ ∼ 0.5–1.0. The south west–north east orientation of the mainemission region means that it gradually appears over the limb of the white dwarfat φ ∼ 0.0 and then rapidly disappears at φ ∼ 0.5 resulting in a large linearlypolarized pulse. The secondary emission region is visible for a larger fraction ofthe orbital period. It first appears just before φ ∼ 0.4 and therefore contributesto the large linearly polarized pulse. At φ ∼ 0.8 the magnetic field lines feed-ing the secondary emission region are most pointing towards the line of sightresulting in depolarization of the radiation by cyclotron self-absorption. This isevident from the dip in the positive circular polarization. The secondary regionthen moves away from the line of sight, resulting in an increase in emission (seenas the contribution to the rapid increase in the intensity) and the brief increasein positive circular polarization at φ ∼ 1.0 . The secondary emission region con-5φ=0.0 φ=0.1 φ=0.2 φ=0.3 φ=0.4φ=0.5 φ=0.6 φ=0.7 φ=0.9φ=0.8Figure 3. The position of the cyclotron emission regions as viewedfrom Earth for a complete orbital rotation. The dot and the crossrepresent the spin and magnetic poles respectively.tinues to stay on the upper limb of the white dwarf and gives rise to the linearpulse at phase ∼ 0.2 . The more gradual decline in the intensity from the brightphase at φ ∼ 0.5 (seen especially in IR observations in Bailey et al. 1995) canbe attributed to the presence of the secondary region.2.2. The accretion regionIn Figure 4 we compare our prediction for the shape and location of the cyclotronemission region with that of previous work. The figures are mapped onto a flatsurface. As a result, the large differences in longitude near the polar regions aremuch smaller on the surface of a sphere than they appear on the maps. Theemission region is mapped onto the same coordinate system used in each case.Figure 4b shows the prediction for the shape and location of the emission regionfrom Ramsay et al. (1996). The main difference is that Ramsay et al. (1996)located the main emission region responsible for the negative polarization onthe upper hemisphere of the white dwarf. They also used a much higher valuefor the inclination (80o) as an explanation for the equal orbital coverage of thebright and faint phases and the variation in the position angle. Furthermore,they used two long thin arcs. The only similarity is that both models predictenhanced brightness on the trailing edge of the main emission region.Figure 4c shows the model prediction of Bailey et al. (1995). They alsopredicted two long thin arcs located in approximately the same locations (largechanges in magnetic longitude near the poles, translate to small changes inactual position) as that of Ramsay et al. (1996). However, they locate the mainemission region in the lower hemisphere in agreement with the ‘Stokes imaging’technique. Furthermore, Bailey et al. (1995) estimated an orbital inclination ofi = 60o and a magnetic dipole offset of β = 50o, also in closer agreement withthe ‘Stokes imaging’ technique.6Magnetic Latitude (degrees)0901800Magnetic Longitude (degrees)180360 270 90Magnetic Latitude (degrees)180270900360 270 180 90 0Magnetic Longitude (degrees)ab018090Magnetic Latitude (degrees)Magnetic Longitude (degrees)c360 180 90 0Figure 4. A comparison between: a. The optimised image solution.b. The model prediction in Ramsay et al. (1996) c. The modelprediction of Bailey et al. (1995).3. SummaryIn this paper we have applied an extension of the Stokes imaging technique,developed and discussed in PHC, simultaneously to the blue (3500–6500Å ) andred (5500–9500Å ) polarimetric observations of the AM Her star V347 Pav fromRamsay et al. (1996). We have also described an extension to the Stokes imagingtechnique, which also searches the system parameter space.Our technique has predicted an extended main cyclotron emission region,located in the lower hemisphere of the white dwarf, and a secondary, smallerless dense region diametrically opposite. The main emission region is foundto consist of a higher density region whose center is located ∼ 25o from themagnetic equator. It has a broken extended region extending ahead in phasetowards but trailing the magnetic pole. The technique also predicts a secondaryemission region in the upper hemisphere which is relatively less dense than themain emission region. It is smaller in extent and almost diametrically opposedto the center of the main emission region. These maps are different from thosefrom previous less objective techniques in which two long thin arcs were used toreproduce the polarimetric variations.7Acknowledgments. SBP acknowledges the PPARC for financial supportin the form of a research studentship. PJH is supported by an Academy ofFinland research fellowship.ReferencesBailey, J. A. et al., 1995, MNRAS, 272, 579Beuermann, K., Stella, L. & Patterson, J., 1987 ApJ, 316, 360Cropper, M. S., 1989, MNRAS, 236, 935Cropper, M, & Horne, K. 1994, MNRAS, 267, 481Potter, S. B., Cropper, M, Mason, K. O., Hough, J. H. & Bailey, J. A., 1997,MNRAS, 285, 82Potter, S. B., Hakala, P. J. & Cropper, M, (PHC) 1998, MNRAS297, 1261Press, W. H., Teukolsky, S. A., Vetterling W.T. & Flannery, B. P., 1992, Nu-merical Recipes in FORTRAN, Cambridge University Press, p406Ramsay, G., Cropper, M., Wu, K. & Potter, S., 1996, MNRAS, 282, 726Wickramasinghe, D. T., & Ferrario, L., 1988 ApJ, 334, 412Wickramasinghe, D. T., & Meggitt, S. M. A., 1985, MNRAS, 214, 6058

The GOSSIP on the MCV V347 Pav

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The GOSSIP on the MCV V347 Pav

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