Another look at AM Herculis – radio-astrometric campaign with the e-EVN at 6 cm

Abstract

1 INTRODUCTION

Cataclysmic variable stars are a broad class of binary systems that consist of a white dwarf (primary object) and a mass-transferring secondary star. The secondary component fills the Roche sphere and loses matter via the L1 point. Among these objects, we may distinguish a subclass of magnetic cataclysmic binary systems called polars. In polars, the very strong magnetic field of the primary star (10–230 MG) prevents the creation of an accretion disc. The matter transferred from the secondary component must follow the magnetic field lines to fall finally into the magnetic pole/poles of the primary star. Such mass transfer leads to the formation of strong shocks near the surface of the white dwarf. As a result, strong radiation is produced mainly by the bremsstrahlung and the electron cyclotron maser processes (Melrose & Dulk 1982; Dulk, Bastian & Chanmugam 1983). An irregular variability in the polars’ luminosity on time-scales from days to months is one of their main characteristics. As polars have no accretion disc, the changes in luminosity reflect variations in the mass-transfer rate. The origin of the mass-transfer rate instability is probably connected with the local magnetic activity of the secondary component. When the active region on the secondary star episodically drifts into the front of the inner Lagrangian point, mass transfer may cease due to the magnetic pressure of a large star-spot (Livio & Pringle 1994; Kafka & Honeycutt 2005). Alternatively, changes in the mass-transfer rate could reflect variations in the size of the active chromosphere during which the secondary star does not fill the Roche lobe (Howell et al. 2000).

AM Herculis (AM Her) is one of the most intensively studied and intriguing of magnetic cataclysmic binary systems and the prototype of polars, which are named AM Herculis-type stars. The irregular changes in the brightness of AM Her of Δm_v ≃ 2–3 mag (high and low states) are observed on time-scales from weeks to months. Tapia (1977) suggested that AM Her contains a compact star with a magnetic field of about 200 MG. After this, AM Her became a target of many observational campaigns conducted at different wavelengths (e.g. Szkody 1978; Bunner 1978; Fabbiano et al. 1981), including also radio bands (Melrose & Dulk 1982; Dulk et al. 1983; Bastian, Dulk & Chanmugam 1985). Bailey, Ferrario & Wickramasinghe (1991), using infrared cyclotron features, estimated the magnetic field strength of the white dwarf to be B ≃ 14.5 MG, which is in good agreement with previous results based on the Zeeman shifts of the photospheric absorption lines (e.g. Wickramasinghe & Martin 1985). The inclination of the system orbit i ≃ 50° (Wickramasinghe et al. 1991; Davey & Smith 1996) and the period P ≃ 3.094 h (e.g. Davey & Smith 1996; Kafka et al. 2005) were also derived. Estimates of the white dwarf mass are in the range 0.35–1.0 M_⊙ (e.g. Mouchet 1993; Gänsicke et al. 1998) with a preferred value of M_wd = 0.6–0.7 M_⊙ (Wu, Chanmugam & Shaviv 1995; Gänsicke, Beuermann & de Martino 1995). The secondary component is believed to be a M4⁺–M5⁺ type star (e.g. Gänsicke et al. 1995; Southwell et al. 1995) with a mass M_s = 0.20–0.26 M_⊙ (Southwell et al. 1995).

The first detection of the AM Her radio emission was achieved using the Very Large Array (VLA) at 4.9 GHz (Chanmugam & Dulk 1982). The measured flux density of AM Her was 0.67 ± 0.05 mJy, with no evidence of circular polarization. Dulk et al. (1983) confirmed this detection, obtaining a flux density of 0.55 ± 0.05 mJy and also specifying the upper limits of the flux density at 1.4 and 15 GHz (0.24 and 1.14 mJy, respectively). In addition, these authors also discovered a radio outburst at 4.9 GHz with a maximum flux density of 9.7 ± 2.3 mJy, which was 100 per cent right-handed circularly polarized. Dulk et al. (1983) also attributed the quiescent emission to a gyrosynchrotron process caused by mildly relativistic electrons with energies of ∼500 keV trapped in the magnetosphere of the white dwarf. An electron-cyclotron maser located near the red dwarf was proposed as a likely source of the radio outbursts. The same origin for the radio flares was suggested by Melrose & Dulk (1982).

Young & Schneider (1979) derived the first distance estimation to AM Her of d ≃ 75 pc. Their finding was based on an analysis of various M-dwarf features in the optical spectrum. These authors additionally suggested that the secondary component in the system must be an M-dwarf with a spectral type between M4 and M5. In a later paper, Young & Schneider (1981) presented another AM Her distance estimation (d = 71 ± 18 pc) using near-infrared CCD spectra and a TiO band analysis, which also revealed the presence of the M4⁺ companion. However, there is an indication that the red dwarf is illuminated by the white dwarf and the spectral type of the secondary component may be modulated with the orbital phase (Davey & Smith 1992). Dahn et al. (1982) determined the trigonometric parallax to 97 stellar systems, including AM Her (⁠|$d=108_{-28}^{+41}$| pc). Gänsicke et al. (1995) used the so-called K-band surface-brightness method (Bailey 1981; Ramseyer 1994) and calculated the distance to AM Her as |$91 _{-15}^{+18}$| pc. The most recent distance estimate to the system of AM Her was made by Thorstensen (2003) using an optical trigonometric parallax measurement made with the 2.4-m Hiltner Telescope (⁠|$d =79_{-6}^{+8}$| pc).

In this paper, we present a new astrometric campaign with the European Very Large Baseline Interferometry (VLBI) Network (EVN) at a wavelength of 6 cm, which was dedicated to the precise estimation of the AM Her annual parallax. This new value may be crucial for further modelling of the physical processes in this system. The paper is structured as follows. In Section 2, we describe our observations and the data reduction. In Section 3, we present a new astrometric model of AM Her. In Section 4, we discuss the observed AM Her radio properties and the orbital phase dependence of the radio emission. Finally, in Section 5, we summarize our conclusions.

2 OBSERVATIONS AND DATA REDUCTION

The interferometric observations of AM Her in the 5-GHz band were carried out in five epochs, spread over 12 months from 2012 December 5 to 2013 December 3, with the EVN in the e-VLBI mode of observations using the phase-referencing technique. The stations at Effelsberg, Jodrell Bank (Mk II), Medicina, Noto, Onsala, Toruń, Yebes and Westerbork (phased array) participated in our observations (proposal code EG069). The data were recorded at the rate of 1 Gb s⁻¹ providing a total bandwidth of 128 MHz, divided into eight base-band channels with a bandwidth of 16 MHz each. The fourth epoch was separated into two parts due to time allocation and both segments were treated during the reduction of data as separate epochs. The observation details of all the epochs are summarized in Table 1.

Each observational epoch spanned ∼3 h covering scans of AM Her, a bright bandpass calibrator, the phase-reference source (J1818+5017) and the secondary calibrator (J1809+5007). During the astrometric calculations, we used the position of J1818+5017 in the Radio Fundamental Catalog,¹ version rfc 2016d (RA = |$18^{\rm {h}}18^{\rm {m}}30{^{\rm s}_{.}}519224$|⁠, Dec. = 50°17΄19|${^{\prime\prime}_{.}}$|74353, J2000.0). This means that we corrected the original position measurements based on radio maps by −0.13 mas in RA and −0.14 mas in Dec., respectively. These shifts compensate for the difference between the positions of J1818+5017 in catalogues rfc 2016d and rfc 2012b (rfc 2012b was used during our observations and the correlation process).

J1809+5007 is a compact radio source selected from the Cosmic Lens All-Sky Survey² (e.g. Myers et al. 2003) in the proximity of AM Her (GB6J180913+500748, F_8.4 GHz = 25.2 mJy). J1809+5007 was observed to examine the phase-referencing success. The three observed sources are separated on the sky plane as follows: AM Her and J1818+5017 by 0|${^{\circ}_{.}}$|6, AM Her and J1809+5007 by 1|${^{\circ}_{.}}$|2, and J1818+5017 and J1809+5007 by 1|${^{\circ}_{.}}$|5. The observations were made in 5-min-long cycles comprising 3.5 min for AM Her or J1809+5007 and 1.5 min for the phase calibrator. The main loop of the observations contains five such cycles. The first cycle dedicated to J1809+5007 was followed by four cycles, which included integrations of AM Her. Note that the first two epochs of the observations overlap with a decrease of the optical luminosity (blocks A and B). The rest of the measurements were recorded at the optical low state of AM Her (blocks D, E and F). We show the AM Her optical light curve during the campaign and the moments of our observations in Fig. 1. The visual optical observations are taken from the American Association of Variable Star Observers (AAVSO) data base.³

Figure 1.

Optical light curve of AM Her during our campaign (data from AAVSO). The epochs of the e-EVN observations presented in this paper are indicated by arrows.

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The whole data reduction process was carried out using procedures in the standard National Radio Astronomy Observatory (NRAO) package aips (e.g. Greisen 2003).⁴ The maps of the phase calibrator J1818+5017 were created by self-calibration in phase and amplitude, and they were used as a model for final fringe-fitting. We applied the imagr task to produce the final total intensity images of all observed sources. During the mapping process, the natural weighting was used. AM Her appears point-like in the radio maps, but we detected a weak jet for J1818+5017 pointing to the east at all epochs. J1809+5007 also seems to be resolved in our maps with a hint of a jet suggestively directed to the north-east.

A sample of the radio maps obtained during our observations is shown in Fig. 2. The radio fluxes and astrometric positions of all observed targets were then measured by fitting Gaussian models, using the aips task jmfit. For J1809+5007 and J1818+5017, we estimated the fluxes of the core only, since a detailed modelling of these two sources is beyond the scope of this work. Moreover, these core fluxes completely dominate the resolved structures. The flux variability was tracked using the task dftpl with an averaging interval equal to the length of the scans. Before application of the dftpl task, we searched the area within a radius of 3 arcsec around the target’s position for background sources, but none were found. If a background object were detected, its model should be removed from the uv data before radio flux estimation with dftpl.

Figure 2.

An example of the radio maps obtained from our campaign. Top left: AM Her. Top right: J1809+5007. Bottom left: J1818+5017. The data were collected during the third epoch of the observations (EG069D). The successive contours show an increase of the flux density by a factor of 2, where the first contour corresponds to the detection limit of ∼3σ. The insets show the size of the restoring beam. The bottom right plot shows an example of the uv-plane coverage of a typical AM Her observation, here also for the EG069D part of the campaign.

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To expand the time span of the observations and to improve the proper motion estimation, we added two archival VLA observations at 8.4 GHz, made in BnA configuration (1988 January 10 and 2003 October 17, observational codes AC206 and AM783). The VLA observations were reduced with the aips package. Sources J1808+4542 and J1800+7828 were used as phase calibrators during AM783 and AC206, respectively. All positions collected in this paper are presented in Table 2. Short scans and distant phase calibrators were used for the VLA observations. Under the typical conditions of VLA observations, an astrometry accuracy of ∼10 per cent for the restoring beam could be achieved (∼0.1 arcsec for AC206 and AM783). In the archival observations, the conditions are worse than during a standard observation; hence, we made a crude estimate for the systematic error of 0.12 arcsec for VLA astrometry. Note that the positions of the VLA calibrators used during the observations are from different catalogues. This results in additional systematic effects in astrometry, which, in general, should be taken into account during calculations as the position of calibrators is relevant to the different global astrometric solutions. However, this effect is at the level of ∼1 mas (the typical discrepancy between a given source position in different catalogues are sub-milliarcsecond), and hence, it is negligible in comparison to the systematic errors assumed by us for VLA position measurements.

3 ASTROMETRIC MODEL AND AN ESTIMATE OF THE ABSOLUTE PARALLAX

We analyse the |$\log {\cal L}$| function in terms of the Bayesian inference. We sample the posterior probability distribution |${\cal P}(\boldsymbol {\xi } | \cal {D})$| of astrometric model parameters ξ in equation (1). Given the data set |${\cal D}$| of astrometric observations (understood as α_i and δ_i components): |${\cal P}(\boldsymbol {\xi }|{\cal D}) \propto {\cal P}(\boldsymbol {\xi }) {\cal P}({\cal D}|\boldsymbol {\xi })$|⁠, where |${\cal P}(\boldsymbol {\xi })$| is the prior, and the sampling data distribution |${\cal P}({\cal D}|\boldsymbol {\xi }) \equiv \log {\cal L}(\boldsymbol {\xi },{\cal D})$|⁠. For all parameters, we define non-informative priors by constraining the model parameters, i.e. α₀ > 0 h, δ₀ > 0°, |$\mu ^*_{\alpha }, \mu _\delta \in [-1000,1000]$| mas yr⁻¹, π > 0 mas and σ_f > 0 mas.

We used the emcee package developed by Foreman-Mackey et al. (2013) for the posterior sampling with the Markov chain Monte Carlo (MCMC) technique. In all experiments, we increased the MCMC chain lengths to up to 72 000 samples and 2240 random walkers selected initially in a small-radius hyperball around a preliminary astrometric solution found with the simplex algorithm. The MCMC acceptance ratio was near 0.5 in all cases.

We performed a few fitting experiments for three sets of measurements (see Table 2). The first set comprises all six EVN detections. The second set contains all EVN and VLA epochs. The third set comprises a minimal number of four EVN epochs that make it possible to determine the absolute parallax. We optimized the five-element model with and without the error floor. For the EVN data, we computed the astrometric parameters at the GAIA DR1 epoch JD 2457023.5, to have a direct link to the forthcoming GAIA catalogue (Lindegren et al. 2016).

The derived astrometric parameters for different data sets and epochs are displayed in Table 3. Since the MCMC posteriors look similar for all data sets, we illustrate only the astrometric model for six EVN epochs (see Table 2 and column (2) in Table 3). The results are illustrated in Fig. 3 and Fig. 4 as one- and two-dimensional projections of the posterior probability distributions for the astrometric model with and without accounting for the error floor correction, respectively. They may be compared with the posterior derived for a model with the error floor included (Fig. 4). The error floor parameter is redundant for the AM Her e-EVN data, since its posterior probability has a maximum at σ_f = 0 mas, and it is small and almost flat elsewhere with a median around 0.15 mas. Indeed, the five-parameter astrometric model yields χ² ∼ 1, and including the additional parameter does not improve the astrometric fit.

Figure 3.

One- and two-dimensional projections of the posterior probability distributions of the astrometric best-fitting parameters for all EVN detections (Table 2), expressed through the median values and marked with the crossing blue/grey lines. Δα₀ and Δδ₀ represent offsets relative to the position at the reference epoch t₀ = JD2456457.5. Contours indicate the 16th, 50th and 84th percentiles of the samples in the posterior distributions, also marked between vertical dotted lines in the one-dimensional histograms. A single quasi-Gaussian peak of the posterior appears clearly for all parameters. The model parameters do not exhibit strong pairwise correlations, though weak near-linear correlations between π and |$\mu _{\rm RA*} \equiv \mu ^*_{\alpha }$| as well as between π and Δδ are apparent.

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Figure 4.

The same as Fig. 3 but for six parameters, this time including the error floor parameter (σ_f). Note that the parameter ranges in both plots are the same.

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In the best-fitting solution for all e-EVN data in Table 2, the AM Her parallax π = 11.29 ± 0.08 mas, which is equivalent to the distance d = 88.6 ± 0.6 pc. This is formally the most accurate and absolute determination of the AM Her distance, compared to previous estimates (e.g. Gänsicke et al. 1995; Thorstensen 2003). This new estimate roughly agrees with the most recent determination based on the optical observations (⁠|$d = 79_{-6}^{+8}$| pc; Thorstensen 2003), yet our uncertainty is one order of magnitude smaller.

The top panel of Fig. 5 illustrates the synthetic parallactic motion of the target over-plotted with the original measurements as blue (grey-white) filled circles. In this scale, the error bars are smaller than the circle diameter. The best fitting model (red/dark grey curve) is over-plotted on 100 randomly sampled models from the posterior data (Fig. 3) for the time interval ±465 d w.r.t. the first and last data epochs, respectively.

Figure 5.

Sky-projected parallactic motion of the target for all e-EVN observations for the epochs in Table 2 (top panel) and for all radio data (bottom panel). Red curves are for nominal solutions in Table 3. Thin grey curves are for 100 randomly selected samples from the MCMC-derived posterior, as illustrated in Fig. 3. Note that in all cases, the synthetic curves are plotted for ±465 d, prior and beyond the first and last data epochs, respectively.

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We also compared the inferred model positions with AM Her radio maps from our campaign, and in all cases, they agree well. The residuals to the final astrometric model based on the EVN observations in Table 2 are illustrated in Fig. 6. Curiously, a systematic trend for the residuals is apparent; however, the sparse sampling makes it hardly possible to interpret the pattern of the residuals. Given the small error floor parameter, there is unlikely to be a systematic meaning.

Figure 6.

Right: Residuals to the best-fitting model in the (Δα, Δδ)-plane for all e-EVN measurements (see the model parameters in column (2) in Table 3).

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Given that the EVN observations are very expensive in terms of human power and telescope time, we also ran a simple experiment to estimate the minimum number of observations required to determine the parallax of relatively distant AM Her-like targets reliably. We assume that observations with sub-milliarcsecond-level uncertainties are scheduled to cover the whole year time-window. We found that three observations are not sufficient to determine the parallax. With six (α, δ) datums, the astrometric model in equation (1) is closed, but we could not find any reliable solution using the MCMC optimization procedure. With four observations, the results are very similar to the parameters obtained for the whole EVN data set, both in terms of the numerical values (Table 3) and the posterior distribution.

Finally, we extended our e-EVN measurements with two archival VLA observations from the NRAO data base.⁵ The VLA data, from 1988 and 2003, could improve the determination of the proper motion. For the same reason, we could use optical measurements in Thorstensen (2003); however, they are not available in source form and the reference paper reports large uncertainties ∼1 arcsec. We also found an infrared position from the 2MASS catalogue (Skrutskie et al. 2006), which has a similar substantial uncertainty of ∼80 mas.

We added the VLA measurements to improve the mean motion parameters, since the EVN data cover only 1 yr. Unfortunately, the uncertainties are much larger for this set than for the e-EVN data, and the VLA measurements stand out from the model (the bottom panel in Fig. 5). However, the spread of models randomly selected from the posterior is quite limited for an interval of almost three decades. We found that the posterior distribution looks the same as for the EVN data. We did not find any improvement in the mean motion parameters too (see Table 3).

4 PROPERTIES OF THE OBSERVED RADIO EMISSION

AM Her appears unresolved on all our maps with SNR detections in the range ∼7–20. This implies that the minimum resolvable size is equal to the upper size of the emission region, and hence the lower limit of T_b could be estimated. Our observations give r₁₁ = 21.7–36.4 (0.15–0.25 au), which translates to T_b ≳ 0.4–2.4 × 10⁶ K. A VLBI detection with a brightness temperature above 10⁶ K is usually interpreted as a signature of non-thermal radiation; however, ∼10⁶ K does not exclude the thermal emission. Therefore, this particular estimate is ambiguous.

On the other hand, the thermal radiation may come only from a relatively large emission zone ≳ 30 r_orb (T_b ≲ 10⁶ K), where r_orb ≃ 0.005 au is the radius of the AM Her orbit. This radius was calculated by assuming that the mass of the primary component is M_WD ≃ 0.7 M_⊙, the secondary star mass is M_RD ≃ 0.3 M_⊙ and the orbital period is P_orb = 3.1 h. These values agree with the data in the literature (e.g. Gänsicke et al. 2006). Such a large emission region could be considered for thermal bremsstrahlung in a stellar wind. However, using the wind analysis of Wright & Barlow (1975), we estimate that the observed flux could be achieved only for unrealistically high mass loss rates. For a slow wind v_w = 400 km s⁻¹, the required mass loss rate is |$\dot{M}\sim 3 \times {\,\,}10^{-8}{\,\,}$|M_⊙, and for a fast wind v_w = 1500 km s⁻¹, the required mass loss rate is |$\dot{M}\sim 1\times {\,\,}10^{-7}{\,\,}$|M_⊙. Therefore, it is unlikely that the observed AM Her radio emission has a thermal origin, which is in agreement with previous conclusions (e.g. Dulk et al. 1983; Mason & Gray 2007).

Curiously, the radio fluxes measured during our campaign (180–370 μJy) appear to be lower than those reported in the literature (so far). Moreover, all our observational epochs were during the decreasing and low state of AM Her optical activity. This observational result suggests that the AM Her radio luminosity may be correlated with the mass-transfer rate, which reflects the high and low states of the optical and the X-ray activity (e.g. de Martino et al. 2002). To study a possible correlation between the optical and the radio activity of AM Her, we collected all flux measurements at 5 and 8 GHz available in the literature, and compared them with the optical observations from the AAVSO archive (Fig. 7). We selected these particular radio bands because the AM Her radio spectrum appears flat at these frequencies (Chanmugam & Dulk 1982). The high value for the VLA measurement during the low optical state (15-min integration f_8.4 GHz ≃ 0.63 mJy, m_V ≃ 15.1, Mason & Gray 2007) is most likely due to a radio flare, as during the next 15-min integration, the detected flux was f_8.4 GHz ≃ 0.37 mJy. To test the significance of a possible correlation, we calculated the Pearson correlation coefficient for the available data. We removed the probable flare presented in Mason & Gray (2007) from our sample and took only detections into account. The derived correlation coefficient σ = 0.62 represents a moderate correlation, and p = 0.03 is a strong indication that the relationship is real. When the non-detections from Bastian et al. (1985) are added to the upper limits, we obtained σ = 0.45 and p = 0.11. The p value indicates that we cannot reject the hypothesis that the two AM Her physical properties discussed are unrelated. However, this is a small sample and more data are required to make any decisive statements on this issue.

Figure 7.

Comparison of the measured AM Her radio fluxes from VLA (Chanmugam & Dulk 1982; Dulk et al. 1983; Bastian et al. 1985; Mason & Gray 2007), MERLIN (Pavelin, Spencer & Davis 1994) and EVN (this paper) with optical observations from AAVSO data. For AAVSO data, we assumed an error of 0.1 arcmin for the three-dimensional average around the epoch of the e-EVN observations.

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The noted difference between the archival data and the new EVN measurements of the quiescent emission can be explained in two ways. First, we consider the extended diffuse component of the AM Her radio emission, which could be resolved on VLBI scales. However, it is difficult to identify its origin. This may be a strong stellar wind from the system, but this hypothesis needs a very high mass loss rate. It also could be due to a slowly expanding and decelerating shell after a nova outburst (e.g. RS Oph, Eyres et al. 2009), although this requires quite a recent thermonuclear runaway event in AM Her (∼1–10 yr ago). This is clearly not the case, since with the distance about of ∼88 pc and with a typical absolute magnitude of −8 mag during the maximum (della Valle & Livio 1995), the visual optical brightness of the AM Her nova at the maximum would be m_max ∼ −3 mag. This would be hard to overlook nowadays.

An alternative scenario assumes that there is a correlation between the quiescent radio luminosity and the activity level of AM Her. Dulk et al. (1983) proposed that the quiescent emission emerges through the gyrosynchrotron process, which is caused by mildly relativistic electrons (E ∼ 500 keV) in the magnetosphere of the primary star. Such a correlation implies that the accretion stream provides, at least partially, the electrons responsible for the gyrosynchrotron emission, even when the accretion occurs at a very low rate. The relation may not be a strong one, as other effects (e.g. local magnetic activity of the secondary star) could impact the quiescent emission. We prefer this scheme from the two presented, because the first of them has additional problems.

Due to the quality of the e-EVN observations, we were able to track the evolution of the AM Her radio flux from short (∼5 min) to long (∼ months) time-scales. To check the reliability of the AM Her flux measurements, we obtained fluxes for the phase calibrator J1818+5017 and the secondary calibrator J1809+5007. The fluxes for AM Her and J1809+5007 are presented in Fig. 8. The scatter of the J1809+5007 flux in individual scans is within ∼10 per cent around the mean value, and for J1818+5017 within ∼5 per cent around the mean with a few rare outliers. We also compared fluxes based on the upper half and the lower half of the frequency band, and these agree within the error bars, though a few points have a noticeable deviation. This gives us confidence that the measured AM Her fluxes are valid.

Figure 8.

Variability of the radio emission obtained from our interferometric observations. Each data point represents a single scan during the phase-referencing observations. Round bullets represent J1809+5007 and squares represent AM Her. The error bars are of length ±1σ.

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We did not detect any short-term radio outbursts, which were previously reported by Dulk et al. (1983). We observed only flux variations around an average value, which may reflect short-term changes of AM Her radio luminosity. We also checked if the radio emission is modulated with the orbital period. The phase-resolved light curve is shown in Fig. 9. We used AM Her orbital ephemerids taken from Kafka et al. (2005) for the calculations. Two minima are visible in the phased radio flux, a sharp one around ϕ ∼ 0.1 and a wider one with a local minimum at ϕ ∼ 0.6. We checked the reliability of the phased radio light curve and repeated the measurements for the upper and the lower halves of the radio band used. In all cases, both minima were visible and located at the same orbital phase. This is a strong indication that the noted dependency between the orbital phase and the quiescent radio luminosity is real; however, new and more sensitive observations are needed to support this finding. We also investigated if the pattern in the phase-resolved light curve could arise from a sample of random data. We calculated reduced χ² for the binned data relative to the mean based on all individual measurements, which gives χ² ≃ 2.6. Next, we derived χ² for 10 000 iterations for binned data with a randomized time for each single measurement. We found that a random set of data gives χ² > 2.5 with a probability that is less than 1 per cent. We conclude that it is unlikely that the observed light curve could be the result of accidental measurements.

Figure 9.

AM Her quiescent radio flux at 5 GHz phased with the orbital motion of the system. Cyan points represent measurements based on individual scans and black points binned values.

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This observational result contradicts the explanation proposed by Dulk et al. (1983) for the AM Her quiescent radio emission. That model assumes that the emission region is comparable to or larger than the physical size of the binary. The new e-EVN data suggest that there is a correlation between the observed radio flux and the AM Her orbital phase. Moreover, the radio light curve is similar to that observed for V471 Tau (Nicholls & Storey 1999), a pre-CV eclipsing binary with an orbital period of 12.51 h. Nicholls & Storey (1999) proposed that an emission mechanism like that of RS CVn binary systems could explain the observed radio properties of V471 Tau, where the gyrosynchrotron emission originates from ∼400 keV electrons near the surface of the secondary component. This model assumes that electrons are accelerated to mildly relativistic energies in the region where the magnetic fields of both stars are reconnecting. The accelerated electrons trapped in the K dwarf’s magnetosphere are responsible for the radio emission. This interaction of fields is caused by a differential rotation of both components in V471 Tau. The radio emission arises in wedge-like magnetic structures, which connect the acceleration region with the photosphere of the secondary component (see for details Nicholls & Storey 1999). Note that the observed V471 Tau flux variations are much more prominent in comparison to AM Her. This could be just a pure geometrical effect, since in V471 Tau there is an eclipse of the radio-emitting region by the K1V secondary photosphere. As the AM Her orbital inclination is relatively high (i ≃ 50°, e.g. Davey & Smith 1996), the probable dependency between the radio flux and the orbital phase likely arises due only to the different orientations of the magnetic field structures in AM Her and the line of sight. If this model is valid for AM Her, one important condition should be met: the secondary red dwarf should have a strong large-scale magnetic field (∼kG).

Recently, Williams et al. (2015) showed that the well-studied M9-type dwarf TVLM 513-46546 hosts a stable dipole magnetic field of about 3 kG at the surface. Therefore, it is plausible to assume that other low-mass red dwarfs are also able to create such strong magnetic fields. The flaring and the spectroscopic activity of the red dwarf observed during the low states of AM Her (Kafka et al. 2005; Kafka, Honeycutt & Howell 2006) supports the idea of strong magnetic fields on the red dwarf’s surface, because the flaring is a signpost to magnetic and star-spot activity. Kafka et al. (2006) also concluded that the observed spectroscopic variations in H α profiles are consistent with motions in large-loop magnetic coronal structures on the secondary star.

However, the synchronous rotation of AM Her components causes problems for the V471 Tau model and the process of electron acceleration may be different in AM Her. We postulate that the acceleration may originate from the interaction between local magnetic fields of the red dwarf frozen into the transferred plasma and the white dwarf magnetosphere. The magnetic reconnection takes place near the L1 point, where the plasma accumulates during the accretion. As the local magnetic activity is very variable for active red dwarfs, this may naturally explain the observed variations in the quiescent radio flux at short and long time-scales. This is also in agreement with the observed probable correlation between the radio luminosity and the high and low states of AM Her activity, since during the increased mass-transfer rate, the electron reservoir is simply much larger (∼3 × 10⁻¹¹ M_⊙ yr⁻¹ during the high state and at least one order of magnitude lower during the low state, de Martino et al. 1998). The observed variations in the quiescent radio flux on time-scales of minutes or hours may also be interpreted as changes in the mass-transfer rate. Such rapid changes in the accretion are observed in the optical and X-ray domains (e.g. de Martino et al. 1998; Bonnet-Bidaud et al. 2000).

Mason & Gray (2007) discovered a second persistent radio polar, AR Ursae Majoris (AR UMa), which has different physical properties than AM Her. AR UMa is a binary system with an orbital period of 1.93 h (Remillard et al. 1994). The primary white dwarf in this system has a magnetic field strength of about 230 MG (Schmidt et al. 1996) and its mass is in the range 0.91–1.24 M_⊙ (Bai et al. 2016). Using infrared spectroscopy, Harrison et al. (2005) estimated the spectral type of the AR UMa secondary red dwarf (M5.5 V). Mason & Gray (2007) also noted that the AR UMa phased radio light curve at 8.4 GHz suggests a minimum near the orbital phase ϕ ∼ 0. If we assume that the source of the quiescent radio emission is the same in both polars, then we may conclude that the emission does not depend on the physical properties of the primary white dwarf. Our findings support Mason & Gray (2007), who postulated that the quiescent radio emission is a signpost to a magnetized secondary star and this distinguishes AM Her and AR UMa from other polars, for which no such emission has been detected. A more precise phased radio light curve of AR UMa and of both systems in their high and low states of activity would shed new light on this puzzle.

5 CONCLUSIONS

We report our results from the recently conducted e-EVN astrometric campaign at 6 cm for AM Her. These observations were conducted in 2012–2013. AM Her was detected on all six scheduled observational epochs with a quiescent radio flux in the range 0.18–0.37 mJy. We calculated a new AM Her astrometric model, and we determined an improved annual absolute parallax of π = 11.29 ± 0.08 mas with an uncertainty one order of magnitude less than that in the literature. This places AM Her almost 10 pc ≡ 10 per cent farther than predicted in the most recent estimate by Thorstensen (2003). The sub-milliarcsecond accuracy of the derived astrometric positions may be similar to the outcome expected from the GAIA mission, and our results could be used as an independent technique for the GAIA measurements. We demonstrated that e-EVN makes it possible to measure the AM Her parallax with only four epochs during 1 yr while still providing a very low uncertainty.

We found observational evidence that the AM Her radio flux is likely modulated with the orbital phase and its behaviour resembles the radio light curve noticed for V471 Tau. This behaviour could be explained if the origin of the AM Her radio emission is similar to that proposed for V471 Tau and generally for RS CVn. We also postulate that the quiescent radio emission distinguishes AM Her and AR UMa from other polars, which are systems with magnetized red dwarfs. We also proposed that the correlation between the quiescent radio luminosity and the mass-transfer rates (high and low states of activity) could explain the noted difference between the AM Her flux based on our new EVN observations and the archival data. This may indicate that the accretion stream provides electrons, which are further accelerated to produce photons in the gyrosynchrotron process, but new sensitive radio observations of AM Her during the high state and AR UMa in both activity states are needed to validate this hypothesis.

ACKNOWLEDGEMENTS

We are grateful to the Polish National Science Centre for financial support (grant 2011/01/D/ST9/00735). The EVN is a joint facility of European, Chinese, South African and other radio astronomy institutes, funded by their national research councils. KG gratefully acknowledges the Poznan Supercomputing and Networking Centre (Poland) for continuous support and computing resources through grant 313. This research has made use of the SIMBAD data base operated at CDS, Strasbourg, France. We acknowledge with thanks the variable star observations from the AAVSO International Database contributed by observers worldwide and used in this research.

Footnotes

REFERENCES