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Astron. Astrophys. 347, 203-211 (1999)

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4. Discussion

4.1. Similarities and differences

The results from these X-ray observations of YY Dra and V709 Cas are clearly very similar. Both systems show a single, relatively short pulse period, at 264.7 s and at 312.78 s respectively, and neither system shows any evidence for orbital or beat period modulations. The absence of an orbital modulation in YY Dra is not surprising, given the relatively low inclination of the system, and suggests that our view of the white dwarf is not obscured by material located elsewhere in the binary system. The similar absence of orbital modulation in V709 Cas suggests that the inclination angle of this system too is quite low. The absence of a signal in the X-ray power spectrum at the beat frequency (or any related sideband frequency) of either system implies that both YY Dra and V709 Cas must accrete predominantly via a disc rather than directly from a stream (Mason et al. 1988; Hellier 1991; Wynn & King 1992; Norton 1993). Consequently the accreting material has lost all knowledge of the orbital phase by the time it impacts the white dwarf.

The similarities in pulsation behaviour between the two systems are rather more subtle. The spin period of the white dwarf in YY Dra is believed to be twice the period observed in its power spectrum, and so the resulting spin pulse profile is double-peaked with two similar maxima (Fig. 3). Conversely there is no evidence to suggest that the spin period of the white dwarf in V709 Cas is anything other than the dominant period seen in its power spectrum. Nonetheless we would argue that the pulse profile of V709 Cas (Fig. 6) is also `double-peaked'. The only difference between it and the pulse profile of YY Dra is that the two maxima in the case of V709 Cas are separated by only about one-third of the pulse cycle, rather than half the pulse cycle in the case of YY Dra. The secondary minimum in the V709 Cas pulse profile is therefore `filled-in' somewhat by the proximity of the two peaks. So, whilst the power spectrum of YY Dra is dominated by the first harmonic of the spin frequency (ie. [FORMULA]), that of V709 Cas is dominated by the fundamental and the second harmonic (ie. [FORMULA] and [FORMULA]).

4.2. Double-peaked pulse profiles as an indicator of a weak magnetic field

When compared with many other intermediate polars, the `unusual' feature of both YY Dra and V709 Cas is that they display double-peaked pulse profiles. Amongst the rest, the only other intermediate polars that have shown similar behaviour are AE Aqr, DQ Her, V405 Aur, GK Per and XY Ari (the latter two only have double peaked pulse profiles in quiescence and change to single-peaked pulse profiles during outburst). What these systems have in common is that they all have relatively short white dwarf spin periods. As noted above, the periods of YY Dra and V709 Cas are about 529 s and 313 s, whilst those of the other five systems listed are about 33 s, 142 s, 545 s, 351 s and 206 s respectively. All other confirmed intermediate polars have white dwarf spin periods in excess of 700 s.

It is believed that most intermediate polars exist in a state of equilibrium rotation (eg. Warner 1996), that is to say the accretion disc is disrupted at the radius where the Keplerian period of the disc is equal to the rotation period of the white dwarf. Now, if this is the case, then a short spin period implies that the white dwarf has a weak magnetic field. In fact, the magnetic moment of the white dwarf is proportional [FORMULA] and it has been calculated that YY Dra, V405 Aur and XY Ari each have magnetic moments of about [FORMULA] G cm3, for example (Warner 1996). So, as pointed out by Hellier (1996) and Allan et al. (1996), the implication is that a short white dwarf spin period, and hence a weak magnetic field, is what determines the presence of a double-peaked pulse profile.

The common interpretation of a double-peaked spin pulse profile is to say that the system is undergoing two-pole accretion, and this is indeed the interpretation previously placed on YY Dra (eg. Patterson & Szkody 1993). However, this is a simplistic interpretation since a double-peaked pulse profile is not a unique indicator of two-pole accretion. If it were then it would imply that single-peaked pulse profiles are the result of one-pole accretion. Whilst this is probably true in the phase-locked polar systems which undergo stream-fed accretion, it is unlikely to occur in a disc-fed intermediate polar system. In this case material from the inner edge of the disc is as likely to be channelled to one pole as the other, and two-pole accretion is the normal way in which such a system will accrete. As pointed out by Hellier (1996) and Allan et al. (1996), two-pole disc-fed accretion does not generally produce a double-peaked pulse profile, although it can in some circumstances. Indeed, two-pole disc-fed accretion is believed to be the `normal' mode of behaviour in intermediate polars, yet both single-peaked and double-peaked pulse profiles are seen. It is important that the paradigm which states `single-peaked profile equals one-pole accretion; double-peaked profile equals two-pole accretion' is put to rest for intermediate polars. We describe below how both types of pulse profile can be produced by two-pole disc-fed accretion, depending on the strength of the magnetic field.

4.3. Two-pole disc-fed accretion producing a single-peaked pulse profile

Many intermediate polars, such as the canonical system AO Psc (Hellier et al. 1991), show a single-peaked pulse profile resulting from two-pole disc-fed accretion. With a relatively strong magnetic field, the accreting material attaches to the field lines whilst still quite distant from the white dwarf, as shown in Fig. 7. This results in relatively small emission regions, whose `vertical optical depth' (up the accretion curtain, along the magnetic field lines) is greater than their `horizontal optical depth' (across the accretion curtain, parallel to the white dwarf surface). So in this case minimum attenuation of the X-ray flux (minimum photoelectric absorption and electron scattering) occurs when the emission region is seen from the side. Hence, when the `upper pole' (ie. the one in the hemisphere of the white dwarf above the orbital plane as we observe it) points away from the observer, then pulse maximum is seen (Fig. 7 viewpoint B), and when it points towards the observer, pulse minimum is seen (Fig. 7 viewpoint A). The contribution to the modulation from the lower pole is in phase with that from the upper pole, since when the upper pole is pointing towards the observer, the lower pole will generally be occulted. Conversely, when the upper pole is pointing away from the observer, the lower pole is viewed essentially from the side too, and so its flux adds to the pulse giving a maximum.

[FIGURE] Fig. 7. Schematic diagram showing an intermediate polar with a relatively high magnetic field. Attenuation of the X-rays is greatest along the magnetic field lines (up the accretion curtains), and least parallel to the white dwarf surface (across the accretion curtains). When viewed from B, X-ray pulse maximum is seen, and when viewed from A, X-ray pulse minimum is seen.

Single-peaked, roughly sinusoidal, pulse-profiles are seen in many intermediate polars, and so two-pole disc-fed accretion can be considered as the `normal' mode of behaviour in these systems.

4.4. Two-pole disc-fed accretion producing a double-peaked pulse profile

In contrast to the above, Hellier (1996) and Allan et al. (1996) suggest that a double-peaked pulse profile can result if the vertical optical depth (up the accretion curtain) is lower than the horizontal optical depth (across it). As they point out, if the white dwarfs in systems showing a double-peaked pulse profile have a relatively weak magnetic field, then material threads onto the field lines much closer to the white dwarf, and the accretion area is relatively large, as shown in Fig. 8. With a large enough footprint to the accretion curtain, this optical depth reversal will be inevitable. This is reminiscent of the large accretion area model suggested by Norton & Watson (1989).

[FIGURE] Fig. 8. Schematic diagram showing an intermediate polar with a relatively low magnetic field. Attenuation of the X-rays is greatest parallel to the white dwarf surface (across the accretion curtains), and least along the magnetic field lines (up the accretion curtains). X-ray pulse maxima are seen when the system is viewed from both A and B.

Now when the upper pole points towards the observer, maximum flux is seen from it, so giving the first peak in the pulse profile (Fig. 8 viewpoint A). The contribution to the modulation from the lower pole is in anti-phase with that from the upper pole since, when the upper pole is pointing towards the observer, the lower pole will generally be occulted, and when the upper pole is pointing away from the observer, the lower pole is at its most visible, so giving a second peak in the pulse profile (Fig. 8 viewpoint B). A relatively large accretion area may also account for the fact that emission from the lower pole is seen even at the relatively low inclination angle of [FORMULA] in YY Dra.

An alternative way of producing a double-peaked pulse profile from two-pole accretion may be to have tall accretion regions, and this could also conceivably be the result of a weak magnetic field. The shock height is proportional to the size of the accreting area (eg. Frank et al. 1992), so the likelihood is that the accretion regions in intermediate polars with a weak magnetic field are both wide and tall. Even if the accretion region is not wide enough for the vertical optical depth to be lower than the horizontal optical depth, it may be tall enough for each accretion region to not entirely disappear over the rim of the white dwarf. With tall accretion regions whose vertical optical depths are greater than their horizontal optical depths (as in a conventional accretion curtain) therefore, when the upper pole points away from the observer, the lower pole is viewed essentially from the side, and so its flux adds in phase to the first flux maximum of the cycle. But, when the upper pole points towards the observer (giving minimum flux), the lower pole may still be visible and so give rise to a second flux maximum in the cycle.

Having said this, the height of the accretion region also depends inversely on the mass accretion rate. In most intermediate polars, the mass accretion rate is probably not low enough for the accretion region to have an appreciable height and so make this scenario feasible. The one exception is probably EX Hya which has an accretion rate about ten times lower than most other intermediate polars, and consequently an accretion shock height of about one white dwarf radius. In that case though, the upper pole is continuously visible and partial occultation of the lower pole is the main cause of the single-peaked spin pulse profile (Allan et al. 1998).

In summary, either wide or tall accretion regions could give rise to a double-peaked pulse profile, and both could be the result of a relatively weak white dwarf magnetic field. Such double-peaked pulse profiles might therefore be expected to arise in intermediate polars whose white dwarfs have relatively short spin periods.

4.5. A subset of intermediate polars with weak magnetic fields

Allan et al. (1996) discussed the cases of V405 Aur, AE Aqr and YY Dra in support of the theory outlined above, whilst Hellier (1996) also mentioned DQ Her and XY Ari in addition to those three. As further support for this theory we can now add V709 Cas to the list of sources and we also note that GK Per, with a spin period of 351 s, has displayed a double-peaked pulse profile on some of the occasions it has been observed.

The data presented earlier demonstrate that V709 Cas follows the pattern as an intermediate polar with a short spin period displaying a double-peaked pulse profile. The fact that the two maxima in its pulse profile appear more closely spaced than those of YY Dra is most probably due to an asymmetry in the locations of the upper and lower magnetic poles. This may be caused by a dipole magnetic field which is offset from the centre of the white dwarf, for instance. If the two poles are not separated by 180o, then the times at which maximum flux is seen from the two poles will not be separated by 0.5 in phase either.

GK Per only exhibits its double-peaked pulse profile in quiescence (Norton et al. 1988; Ishida et al. 1992), and shows a single-peaked pulse profile when in outburst (Watson et al. 1985). A similar situation exists in XY Ari, which also shows a double-peaked pulse profile when the system is in quiescence (Ginga observation: Kamata & Koyama 1993; RXTE observation: Hellier 1997), but a single-peaked pulse profile when in outburst (Hellier et al. 1997). We suggest that the cause of the changing pulse profile may be the same in both cases, and can be understood in the light of the model outlined earlier, following the explanation given by Hellier et al. (1997). The radius at which the accretion disc is truncated is proportional to (amongst other things) [FORMULA] (eg. Frank et al. 1992). Since the disc already extends fairly close to the white dwarf in both these systems, due to the relatively weak magnetic field, during outburst the increased mass accretion rate will cause the accretion disc to extend even closer to the white dwarf before it is truncated. Hellier et al. (1997) suggest that in XY Ari the accretion disc then hides the lower pole from view, and a single-peaked pulse profile remains, produced by modulation of the X-ray flux from the upper pole only. Now, XY Ari is an eclipsing system, with [FORMULA] (Hellier 1997), whilst the inclination of GK Per is believed to be within the range [FORMULA] (Reinsch 1994). In XY Ari the accretion disc extends to within four white dwarf radii of the white dwarf during outburst (Hellier et al. 1997). So, if the lower pole in GK Per is also hidden during outburst, the implication is that the accretion disc must extend even closer to the white dwarf in that case.

If the model outlined in Sect. 4.4 is correct, then we would expect all short period intermediate polars to display double-peaked pulse profiles. The seven systems described above comprise the only confirmed intermediate polars with a spin period below about 700 s. However, there are also three systems that have been proposed as intermediate polars which would fall within this subset if their classifications are confirmed.

The first of these systems is V533 Her, and although it has never exhibited X-ray pulsations, it has been suggested that it is an intermediate polar on the basis of optical photometry which shows a stable 63 s period that appears and disappears with a timescale of years (Patterson 1994). As a short period, weak magnetic field system, we might expect its X-ray spin pulse profile to be double-peaked, if such a pulsation is ever detected. In this case the previously identified pulse period may in fact represent half the true spin period of the white dwarf. A similar re-assessment of the spin period of DQ Her has recently been made (Zhang et al. 1995) resulting in the identification of its spin period as 142 s, twice the value previously assumed.

The other two proposed intermediate polars are both recently discovered systems by ROSAT . RX J0757.0+6306 has an optical photometric modulation with a period of 511 s which may represent the spin period of the white dwarf (Tovmassian et al. 1998). However, no X-ray pulsation at this period has been detected and the classification as an intermediate polar has yet to be confirmed. RX J1914.4+2456 displays an X-ray `pulsation' with a period of 569 s, but Cropper et al. (1998) suggest that this may be a double degenerate polar, rather than an intermediate polar, and so the period represents the orbital period of the system rather than the spin period of a white dwarf. If either of these systems do turn out to be intermediate polars, we predict that their pulse profiles may turn out to be double-peaked also.

4.6. Beat frequency signals

None of the seven systems that show double-peaked pulse profiles has shown evidence for beat frequency signals in their X-ray emission. Such a signal is generally taken as a signature of stream-fed or disc-overflow accretion, and may therefore be confined to the intermediate polars with higher magnetic field strengths. This is to be expected, as the accretion flow becomes attached to the magnetic field lines at larger distances from the white dwarf when the magnetic field is stronger. The further from the white dwarf, the more chance there is that some of the accretion flow is still constrained to travel as an accretion stream, so the greater the likelihood of a signature of stream-fed accretion in the X-ray power spectrum.

All the intermediate polars that have exhibited X-ray beat period signals have relatively long white dwarf spin periods. FO Aqr with a spin period of 1254 s showed a strong beat period during a Ginga observation in 1988 and in an ASCA observation in 1993, although it was absent during a second intervening Ginga observation in 1990 (Norton et al. 1992a; Beardmore et al. 1998). TX Col with a spin period of 1911 s showed a strong beat period in an EXOSAT observation in 1985 and also in a ROSAT HRI observation from 1995, however it was absent from an ASCA observation in 1994 (Buckley & Tuohy 1989; Norton et al. 1997). The case of BG CMi is more controversial. Here an 847 s period detected during a 1988 Ginga observation was interpreted by Norton et al. (1992b) as the true spin period, implying that the previously detected 913 s X-ray pulsation was at the beat period. Even more radically, Patterson & Thomas (1993) have suggested that the true spin period is 1693 s, with the 913 s X-ray pulse then corresponding to a frequency of [FORMULA]. In either interpretation the strong X-ray pulse is an indicator of a stream-fed accretion component, but the case for this is not yet proved one way or the other. The two systems AO Psc and V1223 Sgr, with spin periods of 805 s and 745 s respectively, both showed tentative evidence for beat periods in EXOSAT data from 1983-1985 (Hellier 1992), although later observations with Ginga , ROSAT and ASCA failed to detect any (Taylor et al. 1997; Hellier et al. 1996). Finally, the recently discovered intermediate polar RX J1712.6-2414 displays an X-ray pulsation only at the beat period of 1003 s, with no signal at the 927 s spin period of the white dwarf (Buckley et al. 1997). Furthermore, RX J1712.6-2414 and BG CMi are two of the three intermediate polars from which polarized emission has been detected (Buckley et al. 1995; Penning et al. 1986; West et al. 1987), which is another signature of a strong magnetic field. The correlation between X-ray beat periods, strong magnetic fields and long white dwarf spin periods is clearly apparent.

We therefore suggest that other intermediate polars with long white dwarf spin periods and strong magnetic fields might be expected to exhibit X-ray beat periods at some time in their lives. A prime candidate to search for such effects may be PQ Gem with a spin period of 833 s (Mason et al. 1992), since this is the third system for which polarized emission has been detected (Piirola et al. 1993). Observations with Ginga and the ROSAT PSPC failed to detect an X-ray beat period signal, but the upper limits to the amplitudes of such a modulation were 5% and 20% respectively (Duck et al. 1994), so it cannot be ruled out. Moreover, as the observations of FO Aqr and TX Col demonstrate, beat period signals can appear or disappear on a timescale of a few years, so may yet be found in PQ Gem.

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Online publication: June 18, 1999
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