TESS observations of the asynchronous polar CD Ind: mapping the changing accretion geometry

TL;DR

This study uses TESS data to monitor an asynchronous polar star, revealing dynamic changes in its magnetic accretion geometry over a beat cycle, supported by modeling and simulations.

Contribution

It provides the first detailed optical mapping of accretion geometry changes in an asynchronous polar over its beat cycle using TESS data.

Findings

Accretion geometry shifts from single to two poles and back.

Cyclotron emission maps show changing accretion regions.

Simulations support the observed accretion scenario.

Abstract

We present the results of near continuous TESS optical observations of the asynchronous polar CD Ind (RX J2115-5840). The 27.9 d long light curve, with 2 min resolution, reveals remarkable changes in the magnetic accretion geometry of the system over the 7.3 d beat period. We have modelled the changes in the optical spin period pulse shape using a cyclotron emission mapping technique. The resulting cyclotron emission maps of the magnetic white dwarf reveal how the accretion geometry changes from single to two pole accretion and back over the beat cycle. Finally, we present the results from particle based numerical magnetic accretion simulations, that agree with our interpretation of the changing accretion scenario.

Equations

$$F={\chi }^2+\lambda \sum _i{S}_i$$

$$S_{i}=p_{i}-q_{i}-p_{i}\cdot ln\bigg{(}\frac{p_{i}}{q_{i}}\bigg{)}$$

$$a_{mag}=-k(\mathbf{r}){\mathbf{v}}_{\perp }$$

Full text

††pagerange: TESS observations of the asynchronous polar CD Ind: mapping the changing accretion geometry–References

1 Introduction

Magnetic cataclysmic variables (mCVs) are stellar binaries in which a magnetic (B$$\buildrel{\lower 3.0pt\hbox{>}}\over{\lower 2.0pt\hbox{\sim}}1 MG) white dwarf accretes material from a late-type main sequence star through Roche Lobe overflow. The magnetic field of the white dwarf is sufficient to disrupt the formation of an accretion disc. Instead, either a truncated disc is formed or the accretion flow is channeled directly onto the magnetic pole regions of the white dwarf. Those mCVs which have $B>$10 MG are called AM Herculis systems or ’Polars’, on account of their high levels of optically polarised light (see Cropper (1990) and Warner (1995) for reviews).

The magnetic field strength is also high enough to lock the spin of the white dwarf with the binary orbital period. Around 100 Polars are currently known and they have orbital periods between $\sim$80 min to 8 hrs. Of these, four (V1500 Cyg, V1432 Aql, BY Cam and CD Ind) are ‘asynchronous’ in that the spin period is not synchronised with the orbital period. The ‘beat’ period, $P_{b}$, is defined as $\frac{1}{P_{b}}=\frac{1}{P_{s}}-\frac{1}{P_{o}}$, where $P_{o}$ is the binary orbital period and $P_{s}$ is the spin period of the white dwarf. Over one beat cycle the accretion flow from the secondary star rotates around the magnetic field of the white dwarf, i.e. the orientation of the white dwarf magnetic field respect to the secondary star repeats at the beat period. Depending on the orientation of the magnetic field (and its geometry) the accretion flow will follow different magnetic field lines and potentially switch from one pole to the other.

In principle, if observations can be made over a beat cycle then insight can be gained on the magnetic field geometry of the white dwarf and how the accretion stream interacts with it. In practice this is difficult for various reasons: the length of the beat cycle can be long (50 days for V1432 Aql) which will normally result in data gaps and secondly, most information is gained from polarimetry, which is often not so easily undertaken.

CD Ind (also known as EUVE J2115–586 and RX J2115.7–5840) was discovered as a likely mCV by Vennes et al (1996) using EUVE data and then as a X-ray source by Schwope et al (1997). The latter also first identified it as a possible short period asynchronous polar with the white dwarf having a magnetic field of 11$\pm$2 MG. Optical polarimetry and X-ray observations Ramsay et al (1999, 2000) confirmed CD Ind as an asynchronous polar with a beat period of approximately a week. Using ten years of data, Myers et al (2017) determined an orbital period of 110.812 min and a white dwarf spin period of 109.652 min, giving a beat period of 7.299 d, making it the asynchronous polar with the shortest beat period. It is therefore the most useful system to more easily study the interaction of an accretion stream with a rotating magnetic field.

Optical photometric observations of CD Ind have recently been obtained using the Transiting Exoplanet Survey Satellite (TESS) satellite which cover 27 days, equivalent to four beat cycles. Littlefield et al. (2019) have recently submitted a paper which uses the TESS data of CD Ind to make an in-depth study of the power spectra over the beat phase and infer the location of the accretion regions using the time of the spin and orbital maxima in the light curve. In this paper, we use the TESS data to map the accretion regions on the surface of the white dwarf using a cyclotron emission model and maximum entropy approach over the beat cycle. We also compare these maps with simulations of the accretion flow in this system.

2 Observations

TESS was launched on 18 April 2018 into a 13.7 d orbit. It has four small telescopes which cover an instantaneous field-of-view of 2300 square degrees (see (Ricker et al, 2015) for details). Compared to Kepler it goes less deep but has a instantaneous sky coverage 20 times greater. In contrast to Kepler, it stays on a single field for 28 days, and in contrast to K2, it will survey nearly the entire sky over the initial 2 year mission.

The brightness of CD Ind varies considerably over the beat cycle. Myers et al (2017) report observations of CD Ind made over ten years which show a white light magnitude in the range 15.5–18.5 mag and will therefore show a lower signal-to-noise than most of the targets which are being searched for exo-planets, the faintest being $I\sim 13$ (Ricker et al, 2015). Each full-frame image is downloaded every 30 min. However, a number of targets were selected for higher cadence photometry, giving one photometric point every 2 min, including some which were included as a result of an open call for proposals. CD Ind was observed using TESS between 25 July and 22 August 2018 (Sector 1).

We downloaded the calibrated light curve of CD Ind from the MAST data archive111https://archive.stsci.edu/tess/. We used the data values for PDCSAP_FLUX which the Simple Aperture Photometry values, SAP_FLUX, after it has been corrected for systematic trends common to all stars. Each photometric point is assigned a QUALITY flag which indicates if the data may have been compromised to some degree. We removed those points which did not have QUALITY=0 and normalised it by dividing the flux of each point by the mean flux. The resulting light curve is shown in Figure 1 (the gap in the middle is due to the data being downloaded to Earth at this point).

The full light curve shows variability on many different timescales, the most obvious being the repeating features every $\sim$7.3 days. The variation in the flux corresponds to $\sim$2.5 mag, which is comparable with the variation found by Myers et al (2017). In Figure 2 we show the light curve covering the first beat cycle. On short time scales the effect of the pronounced orbital and spin period of $\sim$110 min can be seen. Our subsequent analysis uses the light curve shown in Figure 1.

3 Modelling of the light curves

In our following analysis we have assumed $P_{orb}$ = 110.8124 min and $P_{spin}$ = 109.6525 min (Myers et al, 2017), which imply a beat period of 7.3 d. It should be noted, that the orbital period value (110.82005 min) quoted in (Myers et al, 2017) is not compatible with the value implied by their quoted orbital frequency (110.81237 min). Moreover, their quoted spin frequency (13.1324 cyc/d, section 3.3) is not in agreement with the spin period quoted in section 3.1 (109.6564 min). For our time baseline of 27.9 d, the small difference in choice of correct spin/orbital period is not significant.

We folded the TESS light curve of CD Ind on consecutive spin cycles and stacked them over the course of the observations (Figure  3). The start of the observations show a broad pulse shape which advances in spin phase as a function of time. At $\sim$4 days the pulse appears to momentarily split into two local maxima after which the pulse jumps to an earlier spin phase. The broad modulation over the beat cycle is indicated by the zig-zag line in Fig. 3. This is broadly consistent with the optical intensity spin folded light curves shown in Ramsay et al (2000).

3.1 The Cyclotron mapping

Given the clear change in the spin modulation over the beat cycle and the high quality of the data, we aim to map the cyclotron emission from the surface of the white dwarf by fitting the beat phase resolved spin light curves with a cyclotron emission model. We seek to find the most likely cyclotron emission distribution on the white dwarf surface that can fit the observed light curve at each beat phase. We have previously employed very similar mapping techniques (i.e. Stokes imaging; Potter, Hakala & Cropper 1998) based on the cyclotron emission from the magnetic white dwarf surface. Of course, in the case of TESS data, we do not have the benefit of polarisation information, which we employed in Potter, Hakala & Cropper (1998). This complicates the separation of cyclotron emission from opposite magnetic poles.

Firstly, we need to identify an appropriate local cyclotron emission model to be used for the mapping. CD Ind has an mean magnetic field strength of 11$\pm$ 2 MG (Schwope et al, 1997), although there is also evidence for a non-dipole geometry, with one pole being stronger than the other (Ramsay et al, 2000). Since the wavelength of the cyclotron emission (and its harmonics) depend on the magnetic field strength, we chose to use the 10 keV, optical depth parameter $\Lambda=10^{5}$ - $10^{7}$ and Nharm = 8 models from Wickramasinghe & Meggit (1985) as a basis for the local cyclotron emission. The modelling was carried out with two different $\Lambda$ parameter values, as these enable us to investigate the sensitivity of our results to the chosen model. Cyclotron models with different $\Lambda$ values exhibit different strength of cyclotron beaming. The chosen models are also consistent with the X-ray emission properties of CD Ind (Ramsay et al, 2000). These emission models produce a cyclotron hump centered in the TESS band (which covers $\sim$6000-10000Å; Ricker et al 2015).

We are still required to make several further assumptions for the actual modelling (or cyclotron mapping). Firstly, we assume radial accretion on the white dwarf surface — i.e. the magnetic field line orientation at the cyclotron emission site is parallel to the surface normal. This is strictly true only at the magnetic poles. However, our simulations (see section §4) suggest that this assumption is reasonable for our purposes. It is well known, though, that in some cases non-radial accretion onto the WD seems to provide better fits to the position angle behaviour of linear polarisation from the cyclotron region(s) (Cropper, 1989). In our case, we do not have polarisation data to verify this and as assuming some ad hoc magnetic field configuration would only introduce extra unverifiable bias in the modelling, we prefer to work with radial accretion. It should be noted that whilst radial accretion is strictly true only at the magnetic poles, a predefined field orientation would also most certainly differ from reality in most locations over the WD surface, as indicated by an offset dipole suggested for this system by Ramsay et al (2000). Secondly, especially in the blue end of the optical spectrum, the accretion stream can have a large contribution to the optical spectrum. However, this should not be an important factor in the TESS band, so we can assume the cyclotron emission dominates the emission. Furthermore, the contribution by the accretion stream would not be modulated at the spin period and would appear as a diluting, constant emission component. Also, there is no evidence for emission from the secondary star in the optical spectrum (Schwope et al, 1997). Finally, in order to compute a light curve from a given cyclotron emission map, we also require the system inclination. CD Ind is not eclipsing, so the inclination must be $<75^{\circ}$. On the other hand, earlier studies (e.g. Ramsay et al, 1999, 2000) have suggested the inclination is likely to be in excess of 60∘. With these constraints in mind, we adopt a value of 70∘ for our modelling.

3.2 The fitting procedure

Mapping cyclotron emission on the white dwarf surface based on light curve data is intrinsically an ill-posed problem, since the total number of data-points is far smaller than the number of free parameters (i.e. the brightness grid on the white dwarf surface). Hence, we will need to regularise our solution. The most commonly used regularisation for similar problems is to search for maximum entropy maps that can still fit the data. We therefore seek to minimise:

[TABLE]

where $\chi^{2}$ is the least squares fit to the data, $\lambda$ the Lagrangian multiplier and

[TABLE]

is the entropy term per pixel (Cropper & Horne, 1995). $p_{i}$ denotes the individual pixel values and $q_{i}$ the default pixel value for each $p_{i}$, taken as the geometric mean of the four nearest pixels to $p_{i}$, again following Cropper & Horne (1995).

In our earlier work (Potter, Hakala & Cropper, 1998) we used genetic algorithms (along the lines introduced in Hakala (1995) for eclipse mapping) for the Stokes imaging of the cyclotron regions. That required a separate ‘conventional’ optimization phase to polish the final solution. Here, instead, we have adopted a global optimisation approach based on the differential evolution (DE) algorithm (Storn & Price, 1997). DE is a population based optimisation algorithm for real-valued functions, that has been successfully used for global optimisation problems. It evolves a population of solutions based on differences within that population (with optional mutation) and tournament selection of parent solutions that biases the next generation of solutions towards the fitter ones. The algorithm also self regulates its search space, as the differences within the population diminish during the fitting. In our implementation we have used a population of 150 cyclotron maps, that are typically evolved for 10000-30000 generations until the termination criterion is reached. This occurs when the merit function values have a standard deviation below a pre-selected limit (10*-5* of the best fit value). We refer the reader to the original work of Storn & Price (1997) for the detailed description of the optimisation algorithm.

3.3 Results

We split the light curve into twenty beat phase intervals and then phased the data in each beat phase bin on the spin period and then folded and binned the data into 100 spin phase bins. The small error on the orbital and spin periods imply that we can fold and bin the data on these periods with a very small uncertainty in the phasing ($<$0.005 in phase for the spin and even less for the orbital). The results of our cyclotron mapping are shown in Fig 4 for system inclination of 70∘. They show that for the first half of the beat cycle the cyclotron emission region on the surface of the white dwarf is extended, but the second region, to which the accretion is directed during the second half of the beat cycle, is much more confined. Both emission regions are located mostly in the ’southern’ hemisphere of the white dwarf. The emission maps become more complex during the pole switching phases at beat phases 0.45-0.5 and 0.95-1.0.

Ramsay et al (2000) used a similar approach to map the emission regions on CD Ind, but using circularly polarised optical light. These data have a more direct signature of whether the emission region is predominately in the upper or lower hemisphere. Interestingly, despite the TESS data having no polarisation information, we find that this data produces very similar results to that of Ramsay et al (2000).

4 Numerical simulations

It is fairly intuitive to think that given the small degree of asynchronism between the orbital and spin periods, the footprints of the magnetic accretion flow should not be stationary on the white dwarf surface over the beat period. Direct observational evidence for this is shown in Fig 3, where the phasing of the spin pulse maxima changes over the beat period (see also Littlefield et al. (2019)). However, as the ionised accretion flow interacts with the white dwarf magnetic field, it is worth examining if such behaviour can be reproduced through simulations using our best estimate for the system parameters. With this in mind, we have carried out particle based diamagnetic 3D simulations of the mass accretion flow in the system using the HYDISC simulation code (King, 1993; Wynn & King, 1995).

HYDISC has been previously used for simulating magnetic accretion flows in several systems including polars, intermediate polars and the magnetic propeller system AE Aqr (see e.g. Wynn, King & Horne (1997), Norton et al. (2008)). At the heart of HYDISC lies a Lagrangian, particle, or blob, based hydrodynamic code with added treatment for the magnetic interaction. This is achieved by applying a magnetic field dependent acceleration (magnetic drag) term in the equation of motion for the particles. This term, $a_{mag}$, depends on the relative shear velocity of the particles with respect to the local (rotating) magnetic field:

[TABLE]

where ${\bf v_{\perp}}$ is the relative shear velocity and, in addition to that, $k({\bf r})\propto B^{2}C_{A}^{-1}l_{b}^{-1}\rho_{b}^{-1}$, where $C_{A}$ is the Alfvén speed in the surrounding plasma, $l_{b}$ is the blob length scale and $\rho_{b}$ is the blob density. We refer the reader to (King, 1993; Wynn & King, 1995) and references therein for more detailed discussion of the simulation method.

For our purposes, we have adopted the orbital and spin periods of CD Ind as determined by Myers et al (2017) as a basis of our simulations. Furthermore, we have assumed M2=0.21$M_{\odot}$ based on the orbital period and M1=0.7$M_{\odot}$ leading to the mass ratio q=0.3. The simulations were then set up using typical HYDISC parameters for the viscosity, length scale and k-value distribution to match the magnetic field properties in Polars. A 70∘ tilted offset dipole field was fixed on the primary white dwarf and the system was evolved for several beat periods. We employed an offset of 0.5 WD radius for the dipole axis to force it “below" the orbital plane, as suggested by the cyclotron maps.

Sample results from our simulations are shown in Figure 5. The beat phases of the chosen samples are shown. These correspond approximately to the phasing shown in Fig 4. The six different columns show the results in X-Y (orbital) and X-Z planes at three different length scales, as indicated by the size of the WD. It is clearly evident that the accretion stream flips between the two magnetic poles, and during pole switching, a short duration ($\sim$ 0.05 in beat phase) double pole accretion phase also takes place. The radially offset dipole directs the accretion flow to two different, predominantly ‘southern’, regions.

5 Discussion

Ramsay et al (1999) analysed circular polarimetry data taken over two weeks. They found that over the beat cycle the sign of the circular polarisation reversed, indicating that the accretion stream changed from accreting onto one and then the other magnetic pole. They considered two scenario’s, one in which the accretion flow from the secondary star is directed onto one or other footprint of the same set of field lines over the whole beat cycle. In the second scenario the flow is directed on to roughly diametrically opposite field lines at different beat phases. Based on observations of the spin phase of the positive and negative polarisation peaks, Ramsay et al (1999) concluded that the first scenario was much more likely (see Fig 5 of Ramsay et al 1999). Additional polarimetry and X-ray data allowed Ramsay et al (2000) to conclude that white dwarf had a non-dipole field with the upper pole having a significantly higher magnetic field strength than the lower pole.

Our modelling of cyclotron emission over the WD surface does not pick any clearly significant emission located in the ‘northern’ hemisphere. Instead, during the first half of the beat cycle, we detect an extended emission arc/curtain in the ‘southern’ hemisphere, which is replaced by emission from a more compact (and more ‘southern’) region during the second half of the beat period. There is some indication for a latitudinally extended emission during the beat phases 0.95-1.05 and 0.40-0.50, when the pole switching takes place (Fig 4). Our trials with inversion of synthetic maps that included both ‘northern’ and ‘southern’ emission regions clearly showed that should a ‘northern’ region exist, it would be picked up by the modelling. This is, however, not observed in our real data maps (Fig 4).

Since both accreting regions appear to be in the ‘southern’ hemisphere, but at somewhat different latitudes, we consider a model, where the (presumably) dipole field of the WD is tilted 70∘ from the spin axis and also offset by 0.5 RWD from the center of the WD (Fig. 6). This would produce two ‘southern’ emission regions that could account for the observed cyclotron maps. Indeed, our numerical simulations with such a field geometry manage to produce the expected locations for the accretion regions. However, we are not able to reproduce the pronounced longitudinal extent difference in emission from the two poles, which is suggested by the cyclotron maps (Fig. 4.). The difference of spin pulse profiles from the two poles is real though, as it can be clearly seen in Fig. 3 and Fig. 4 that the spin period folded light curves in the beat phase ranges 0.1-0.4 and 0.6-0.9 are very different. Explaining this might require a dipole field that is also offset along its magnetic axis, or a more complicated field geometry.

It is also apparent that the accretion maps based on the numerical simulations (Fig. 5) produce much more confined accretion footprints on the WD surface than our cyclotron modelling (Fig. 4) implies. There are at least two possible reasons for this. Firstly, the maximum entropy regularisation applied to the inversion of the light curves, together with the chosen pixel grid resolution, smooths out the resulting maps to some extent. Secondly, the accretion footprints based on the numerical simulations depend to some extent on the chosen strength and distribution of the particle-by-particle magnetic drag force coefficient, $k$. Wider distribution of $k$-coefficients would lead to more extended accretion arcs on the WD surface. Both the cyclotron maps and the accretion footprints as a function of the beat phase are shown in Fig. 4.

6 Conclusions

We have employed a near continuous (over 27.9 d) TESS high cadence light curve of CD Ind in order to study the changes in the magnetic accretion process over several beat cycles of this asynchronous polar. Our cyclotron emission maps, based on evolutionary maximum entropy inversion show that accretion onto the magnetic white dwarf primary flips between two magnetic poles. Furthermore, the accretion footprints seem to closely follow the locations of magnetic poles over the beat cycle, before the stream trajectory switches to the other pole. Double pole accretion occurs for a maximum of 0.05 in beat phase during each of the pole transitions. This is confirmed by both the numerical simulations and the cyclotron mapping. We conclude that either the system contains a tilted offset dipole, the axis of which lies entirely below the orbital plane (Fig 6.) or alternatively a higher order magnetic field configuration is required to account for the accretion geometry and its changes over the beat period.

Acknowledgements

This paper includes data collected by the TESS mission, which are publicly available from the Mikulski Archive for Space Telescopes (MAST). Funding for the TESS mission is provided by NASA’s Science Mission directorate. Armagh Observatory and Planetarium is core funded by the Northern Ireland Executive. SBP and DAHB acknowledge support of the National Research Foundation of South Africa. APB acknowledges support from the UK Space Agency. Since the submission of this paper a preprint by Littlefield et al. (2019) was submitted to the http://arxiv.org, where they analyse the same dataset and suggest a different identification for the spin and orbital periods. We thank Colin Littlefield for communicating their results in advance.