V407 Vul: a direct impact accretor

Abstract

V407 Vul (= RX J1914.4+2457) shows pulsations in X-ray flux on a period of 9.5 min, which have been ascribed to accretion on to a magnetic white dwarf, with the X-ray pulses seen as the accreting pole moves into and out of view. The X-ray flux drops to zero between pulses, and no other periods are seen, suggesting that V407 Vul is a type of system known as a ‘polar’ in which the white dwarf has a strong enough field to lock to the orbit of its companion. If so, then V407 Vul has the shortest orbital period known for any binary star. However, unlike other polars, V407 Vul shows neither polarization nor line emission. In this paper we propose that V407 Vul is the first example of a new type of X-ray emitting binary in which the mass transfer stream directly hits a non-magnetic white dwarf as a result of the very compact orbit. Our model naturally explains the X-ray and optical pulsations, as well as the absence of polarization and line emission. We show that direct impact will occur for plausible masses of the accreting star and its companion, e.g. M₁≈0.5, M₂≈0.1 M_⊙. In our model V407 Vul retains its status as the binary star with the shortest known orbital period, and is therefore a strong source of low-frequency gravitational waves. V407 Vul is representative of an early phase of the evolution of the AM CVn class of binary stars and will evolve into the normal disc-accretion phase on a time-scale of 10⁶ to 10⁷ yr. The existence of V407 Vul supports the double-degenerate route for the formation of AM CVn stars.

accretion, accretion discs, gravitational waves, binaries: close, stars: individual: V407 Vul, novae, cataclysmic variables, white dwarfs

1 Introduction

Accreting white dwarfs in binary stars fall into two groups according to the magnetic field of the white dwarf. The flow of material close to strongly magnetic white dwarfs is controlled by the magnetic field, and matter is channelled on to one or both magnetic poles. In this case, accretion energy is released as the material crashes into the white dwarf, emitting copious X-rays and optical cyclotron emission. The X-ray and cyclotron emission are modulated on the spin period of the white dwarf as the accreting poles rotate into and out of our view. Moreover, the cyclotron emission is circularly and linearly polarized. In the non-magnetic case, by contrast, the accreting material, having some initial angular momentum, forms a disc. The disc material accretes on to the white dwarf via an equatorial boundary layer where the kinetic energy of the gas is released. The resulting radiation is not significantly modulated, and there is no polarized cyclotron radiation. Pulsed X-ray emission from white dwarf binaries has therefore been regarded as a secure indication that the accretor is magnetic.

When X-ray pulsations on a period of 9.5 min were discovered from the star V407 Vul (=RX J1914.4+2457), they were immediately interpreted in terms of an accreting, magnetic white dwarf spinning on the same period (Motch et al. 1996). Similar spin periods are commonly seen in the ‘intermediate polar’ class of cataclysmic variable star in which a relatively weakly magnetized white dwarf spins faster than the binary orbit. However, in these stars one also sees other periods, related to the orbital period and the ‘beat’ period between the spin and orbital periods. In V407 Vul, however, both X-ray and optical data show just the one period of 9.5 min. Moreover, the X-ray flux from V407 Vul drops to zero between pulses, which is difficult to account for on an intermediate polar model. This led Cropper et al. (1998) to suggest, instead, that V407 Vul is a ‘polar’, in which the magnetic field of the white dwarf locks its spin to the orbit of its companion star. If so, then V407 Vul has an orbital period of only 9.5 min, the shortest known for any binary system. Such a period implies that the donor is a helium-rich degenerate star, making V407 Vul the first magnetic member of the AM CVn stars. The AM CVn stars are a select group of eight mass-transferring binary stars (other than V407 Vul), which have periods ranging from 17 min (AM CVn, Nelemans, Steeghs & Groot 2001b) to 65 min (CE 315, Ruiz et al. 2001). As for V407 Vul, their short periods imply that the donor stars are hydrogen-deficient, and indeed no hydrogen appears in their spectra. They are thought to form from initially detached double white dwarf systems, from systems with helium-star donors or from mass transfer initiated when a ∼1 M_⊙ donor starts to transfer mass to a white dwarf at the end of its core hydrogen burning (Nelemans et al. 2001a; Podsiadlowski, Han & Rappaport 2001). Double white dwarfs that fail to become AM CVn stars are possible Type Ia supernova progenitors.

A problem with the polar model is that V407 Vul shows no optical polarization (Ramsay et al. 2000), possible on the polar model only if the white dwarf has either a very strong or relatively weak field (while remaining synchronized). In an effort to explain this, Wu et al. (2001) proposed a model in which the spin of the magnetic white is not synchronised with the orbit leading to dissipation of electric currents in the donor which produces unpolarized optical flux. However, the dissipation also leads to synchronization on a short time-scale, which makes the chance of such a configuration low.

In this paper we present an alternative model for V407 Vul, in which the white dwarf need not be magnetic, but the X-rays will still be strongly modulated. V407 Vul may thus be the first example of a new class of X-ray emitting binary star. We start by describing the observational characteristics of V407 Vul that need explaining.

2 Observed Features of V407 Vul

While there have been relatively few observations of V407 Vul, any model of the system must satisfy the following constraints.

• (i)

  The 9.5-min X-ray pulsations. The X-ray bright phase occupies half of the pulsation period; for the other half of the cycle, the X-ray flux is undetectable (Cropper et al. 1998). No other periodic signals are seen in X-rays.

• (ii)

  The 9.5-min optical pulsations. Again, no other periodic signals are seen. The optical pulsations have a peak-to-peak amplitude of 0.07 mag. The optical flux peaks 0.4 cycles before the X-ray flux (Ramsay et al. 2000).

• (iii)

  The X-ray spectrum. This is soft and can be fitted with an absorbed black-body spectrum of temperature 40–55 eV (Motch et al. 1996; Wu et al. 2001).

• (iv)

  The lack of optical polarization. Ramsay et al. (2000) measured 0.3 per cent circular polarization, a level consistent with zero given the systematic uncertainties.

• (v)

  The optical spectrum. Unfortunately this has not yet been published, but it is reported to be devoid of emission lines (Wu et al. 2001). Without having seen it, it is hard to judge the importance of this, but we will consider it as another possible constraint to satisfy.

• (vi)

  The distance. V407 Vul is heavily absorbed from which Ramsay et al. (2000) obtain d<100 pc. They further find that d>400 pc from a de-reddened I-band magnitude of 15.5, although the ‘reddening’ is deduced from the X-ray column, and may be too high since the colours end up being too blue even for a Rayleigh–Jeans spectrum. Ramsay et al. (2000) used A_V=5.6, but also state that colours deduced from A_V=4 do fit blackbodies, from which we estimate that the de-reddened I-magnitude lies in the range 15.5>I>16.1.

It was the singly periodic signal and the on/off nature of the X-ray light-curve that led Cropper et al. (1998) to suggest their polar model. The absence of both polarization and optical line emission presented difficulties that led Wu et al. (2001) to develop their unipolar inductor model by analogy with the Jupiter–Io system. As outlined in the introduction, in their model, the magnetic white dwarf (the accretor in our model) is slightly asynchronous with the binary orbit and the resulting electric field drives currents that run between the two stars, leading to energy dissipation in both of them. Dissipation on the magnetic white dwarf powers the X-ray emission, while dissipation at the donor plus irradiation powers the optical flux. The irradiation is predicted to be comparable to the ohmic dissipation, which they suggest can explain the relatively weak optical modulation and the lack of line emission. Since the non-magnetic star dominates the optical emission, the absence of polarization is also explained.

The major problem with Wu et al.'s model is that it is short-lived: they estimate that it will last only ∼1000 yr. Even if all AM CVn systems pass through this stage, there would only be 1–10 such systems in our Galaxy at any one time, according to the formation rates of Nelemans et al. (2001a) and Podsiadlowski et al. (2001). The chance of finding one within our neighbourhood is therefore small. Magnetic systems may in fact comprise only a small fraction of the total, making the probability of finding such a system very low. It is therefore worth searching for longer-lived models.

3 A New Model for V407 Vul

Our idea is simple: in very close binary systems, the mass transfer stream can plough straight into the accretor even in the absence of the magnetic field. This happens if the minimum distance of the ballistic gas stream from the centre of mass of the accretor is smaller than the radius of the accretor, which is most famously the case in Algol binary stars, where the accretors are main-sequence stars. We propose that V407 Vul is the first instance of Algol-like direct impact in the case of a white dwarf accretor.

In Fig. 1 we show that the direct impact is possible using typical system parameters (Section 4). The figure shows the ballistic path of the mass transfer stream calculated for an accretor mass of M₁=0.5 M_⊙ and a donor mass of M₂=0.1 M_⊙. The separation of the binary is fixed through Kepler's laws by the masses and the orbital period, P=9.5 min, while we have used Nauenberg's (1972) analytic formula for the radius of the white dwarf. If the 9.5-min pulsations truly reflect the orbital period of V407 Vul, direct stream impact is possible.

Figure 1.

Path of the stream in V407 Vul in the case M₁=0.5 M_⊙, M₂=0.1 M_⊙. The dashed line is tangent at the impact point, to show that the impact is hidden from the donor in this case.

This model simmediately explains the absence of polarization and the X-ray pulsations. There is no polarization because the white dwarf is non-magnetic, while the impact point is fixed in the rotating frame of the binary and will produce one pulse per orbit. Our model nicely explains the absence of X-ray flux for half the cycle since the impact will be on the equator of the accretor; this is a key problem with intermediate polar models of V407 Vul, as Cropper et al. (1998) point out.

There are two possible explanations for the optical pulsations. First, we do not expect the white dwarf to be synchronized with the orbit, but instead to spin rapidly as the result of the accretion of material of high specific angular momentum. The surface of the white dwarf will therefore move under the point of impact, and one expects a heated area to trail downstream from the spot. If this area is responsible for the optical pulsations, they would then peak at an earlier phase than the X-rays, as observed. The amplitude of the optical pulsations in this case depends upon the extent of the heated region and the brightness of the unheated photosphere of the white dwarf. In order to dominate the optical flux, given the lower limit on the distance of 100 pc (Ramsay et al. 2000) and the unreddened I magnitude between 15.5 and 16.1 (section 2), we require M_I>10.5–11.1. For a 0.5-M_⊙ white dwarf this implies an effective temperature T_eff<15 000–21 000 K (Bergeron, Wesemael & Beauchamp 1995), which is nothing out of the ordinary. Indeed, given an accretion rate of 10⁻⁸ M_⊙ yr⁻¹ (section 5), we would expect a much higher temperature from compressional heating alone: Townsley & Bildsten (2002) give a surface temperature of 27 000 K from compressional heating for a 0.6-M_⊙ white dwarf accreting at 10⁻⁹ M_⊙ yr⁻¹.

Alternatively, the optical pulsations may come from the heated face of the donor star. This could be heated by irradiation from the impact spot (but see the next section), or by photospheric emission from the white dwarf. Although the latter is normally ignored, once again the very close orbit of V407 Vul means that it is not just possible, but likely. The geometry of the impact site means that the heated face of the donor star will also produce a peak optical flux in advance of the X-ray flux, as observed, although the impact site would have to be on the side facing the donor to explain the 0.4 cycle shift seen. The temperature of the heated face of the donor due only to photospheric emission from the accretor is T_h≈(R₁/a)^1/2T₁≈0.33T₁, where R₁ and T₁ are the radius and temperature of the accretor. Given that the donor is about three times the size of the accretor, and the temperature of the accretor estimated above, the donor could easily produce the optical pulsations.

Of the observational facts above, we are left to explain the reported lack of optical line emission and the softness of the X-ray spectrum.

3.1 Optical line emission

The direct-impact model provides a beautiful way to avoid X-ray irradiation of the donor star, and therefore any associated emission lines, altogether, because it is possible for the impact point to be hidden from the donor star. Fig. 1 illustrates just such a case; we will examine the parameter constraints needed for this to be the case in the next section. Nevertheless, it is not clear whether it is really necessary to exclude irradiation of the donor because any emission lines would execute high-amplitude sinusoidal motion on a period of only 9.5 min and would be smeared in wavelength, especially given that V407 Vul is optically faint (the orbital velocity of the donor in Fig. 1 is 800 km s⁻¹). Given the hydrogen deficiency and the proximity of the two stars, one would only expect to see ionized helium emission. Since V407 Vul is only easily observable in the I band, it is not clear whether any emission lines should have been detected, even if irradiation is taking place. There is, of course, some irradiation of the stream, but only where it nears the impact site. By this time it is moving very fast, and will be highly ionized, and since emission from this part is hard to detect even in much brighter and longer period polars, we do not regard this as a problem.

3.2 The soft X-ray spectrum

Direct impact accretion differs significantly from magnetically-confined accretion. First, the accretion stream will be much narrower in the direct impact case. The reason is that in the magnetic case, threading occurs over a range of radii leading to impact in a relatively large arc close to one or both poles of the white dwarf. On the other hand, as Lubow & Shu (1976) and Lubow (1989) showed, as the ballistic stream nears its closest approach to the centre of mass of the accretor, it becomes comparable in width to the hydrostatic scaleheight that an accretion disc would have at the same radius. At the white dwarf this is a width of order for T=20 000 K (stream from the heated face of donor) and M₁=0.5 M_⊙. Therefore, accretion will take place over a fraction ƒ∼0.01 per cent of the surface of the white dwarf.

Together with an accretion rate of M˙∼10⁻⁸ M_⊙ yr⁻¹ (section 5), some 10–100 times higher than in polars, the narrow stream width makes the accretion in V407 Vul comparable to the blob accretion model (Kuijpers & Pringle 1982; Frank, King & Lasota 1988; King 2000), in which the soft X-ray excess of polars is explained by the stream breaking into dense blobs which are able to penetrate below the photosphere and therefore become thermalized, giving soft X-ray emission. This happens for blob densities ρ<10⁻⁷ g cm⁻³ (Frank et al. 1988). For a stream width of 10⁻⁴ R_⊙, an accretion rate of M˙=10⁻⁸ M_⊙ yr⁻¹, and a free-fall velocity v_ff=4×10⁸ cm s⁻¹, we find ρ≈3×10⁻⁵ g cm⁻³, well in excess of Frank et al.'s limit.

In magnetically confined accretion, after the accretion shock, the material continues to flow along the field lines and can only come to a halt through cyclotron and bremsstrahlung cooling. This leads to the shock forming some height above the point at which the ram pressure of the stream matches the pressure in the atmosphere of the white dwarf. In the direct impact case, there is nothing to stop the stream expanding sideways following the shock which will cause adiabatic cooling and lead to the shock forming much closer to the point where the pressures match. This increases the chance that the shock will be buried. The final, and perhaps most important, difference is that the white dwarf moves rapidly beneath the impact site, which will sweep the heated material downstream. We suggest that it is this which spreads the emission over a large enough area to cause the spectrum to be very soft as observed. We can crudely estimate the cooling time as the thermal time-scale of a column of gas heated to the observed temperature of T≈50 eV. The column density Σ is fixed by the penetration depth according to

(1)

where is the surface gravity, ρ is the stream density and v_ff is the free-fall velocity , which is comparable to that of the stream. Given an energy of 3kT/2 per particle, and a loss rate of σT⁴, we obtain a cooling time t_c, in seconds, of

(2)

Since the white dwarf spin period may be as short as 10 s, this cooling time is the right order of magnitude to lead to significant spreading of the emission. All that is needed is that the area is spread so that ƒ∼0.1 per cent, for the energy liberated by accretion at the rate 10⁻⁸ M_⊙ yr⁻¹ to be radiated away with a temperature of T∼50 eV, as observed.

4 Parameter Constraints

We now calculate the limits upon the parameters required for our model to be viable, fixing the orbital period at P=9.5 min. We have first the orbital separation in terms of the total mass of the binary

(3)

For the radii of the white dwarfs we use Nauenberg's (1972) formula

(4)

This is a lower limit, since Nauenberg's relation is based upon zero temperature and no rotation, and departures from these assumptions act to increase the radius for a given mass. The radius of closest approach to the centre of the white dwarf relative to the separation of the binary is a function of mass ratio, for which Nelemans et al. (2001a) give a fit based upon the calculations of Lubow & Shu (1975), and confirmed by our own integrations.

Next, we have the well-known relation between the mean density of the Roche lobe filling donor and the orbital period, which gives ρ¯=4.3×10³ g cm⁻³ in the case of V407 Vul. Using Nauenberg's formula, this corresponds to a mass of M₂=0.077 M_⊙. More generally this is a lower limit because any hydrogen content or semi-degeneracy reduces the density for a given mass, and since the density increases monotonically with mass, the mass must increase to match a fixed density.

Two other constraints come from considerations of the stability of mass transfer which set upper limits to the mass ratio. To ensure that the Roche lobe does not shrink faster than the donor in the case of conservative mass transfer, the mass ratio must satisfy

e.g. Nelemans et al. (2001a). Nauenberg's formula gives

where x=M₁/1.433. This strictly only applies to the limiting case of M₂=0.077 M_⊙, but we will take it to apply to higher mass cases too. This is a conservative assumption, because radiative stars shrink on mass loss. A stricter constraint applies if accretion occurs directly on to the white dwarf and the angular momentum of the stream is not transferred back from the white dwarf to the orbit (i.e. tidal and magnetic coupling are ineffective). The limit then becomes

(5)

(Nelemans et al. 2001a), where r_h is the circularization radius as a fraction of a, which is a function of mass ratio for which we use the approximation of Verbunt & Rappaport (1988), accounting for their inverted definition of q.

Finally, if the reported absence of emission lines is significant (see Section 3 for a discussion of this), there is another upper limit on the mass of the donor for a given accretor mass, required in order for the impact spot to be hidden from the donor star. We carried out our own integrations in the Roche potential to derive this upper limit as a function of accretor mass.

These constraints give the allowed parameter space indicated in Fig. 2 which shows that direct stream impact in a stable binary is possible if the masses of the two stars lie in either the large triangle ABC for the standard stability limit, or in the smaller triangle A′BC′ if the more rigorous stability criterion, equation (5), applies. This smaller region is also consistent with the requirement that the donor not be irradiated (curved dashed line in Fig. 2). Moreover, this happens for M₁≈0.5 M_⊙, a typical mass for a white dwarf. Thus this model is feasible without the need for fine tuning of the parameters. If the donor is degenerate, then, for no irradiation, the parameter space is restricted to M₂=0.077 M_⊙, M₁=0.42–0.49 M_⊙.

Figure 2.

Parameter space constraints. The almost-straight solid and dashed lines are upper limits from mass transfer stability. The dashed line applies if the angular momentum of the accreting material is not fed back to the orbit. The horizontal line is the lower limit on the donor mass. The curved solid line is a lower limit required for direct stream impact. The curved dashed line is the upper limit for the impact site to be out of sight looking from the donor star.

5 Future Evolution

V407 Vul's destiny is to become a standard member of the AM CVn systems. As the mass of the donor decreases, even if the accretor fails to increase in mass, the increase in the orbital period and the decrease in mass ratio, both act to make it less likely that the stream will hit the accretor directly. Eventually, the stream will orbit the accretor and a disc will form. Depending upon the precise parameters, it will do so after the loss of 0.01–0.1 M_⊙ from the donor. The mass transfer rate for an orbital period of 9.5 min is ≈10⁻⁸ M_⊙ yr⁻¹ (Nelemans et al. 2001a), so this phase will last 10⁶–10⁷ yr. If all AM CVn systems pass through such a phase, then, according to the formation rates of Nelemans et al. (2001a) and Podsiadlowski et al. (2001), there should be 10³–10⁵ systems like V407 Vul in the Galaxy. The nearest of these should be between 100 and 400 pc away, consistent with estimates of the distance to V407 Vul. This agreement favours the detached double white dwarf route for the formation of AM CVns over either the helium star or the semi-degenerate routes, since the latter rarely reach orbital periods as short as 10 min (Nelemans et al. 2001a; Podsiadlowski et al. 2001), whereas Nelemans et al. (2001a) predict that most double white dwarfs will pass through a direct-impact phase if they avoid merging.

6 Gravitational Waves

V407 Vul is a very promising source of gravitational waves. Following Meliani, de Araujo & Aguiar (2000), who calculate strain amplitudes for many cataclysmic variable stars, and using M₁=0.5 M_⊙ and M₂=0.1 M_⊙, V407 Vul will produce a strain amplitude h=1.2×10⁻²¹(d/100 pc)⁻¹ at Earth. This is comparable to the largest values listed by Meliani et al. (2000). In addition, the very short period of V407 Vul will lead to gravitational wave emission at a frequency of ƒ_gw=2/P=3.5×10⁻³ Hz, where the background noise of an instrument such as LISA is predicted to be much lower than at the lower frequencies characteristic of other cataclysmic variable stars. V407 Vul is thus one of the brightest prospects for space-based gravitational wave detection.

7 Conclusions

We have shown that plausible masses for the two components of V407 Vul suggest that the mass transfer stream in this system strikes the accreting white dwarf directly. The resulting spot explains the X-ray pulses observed from this system, without the need for a magnetic white dwarf. This is consistent with the lack of polarization from the system, and with the complete disappearance of X-ray flux in between pulses. Our model has a lifetime of 10⁶ to 10⁷ yr compared with the 10³ yr of Wu et al.'s unipolar inductor model, making it easier to reconcile with estimated formation rates for the AM CVn systems, as long as a substantial fraction of these systems pass through a direct impact-phase, as predicted by Nelemans et al. (2001a,b) on the basis of a double white dwarf origin for these systems. V407 Vul is the first example of a new class of stream-fed, non-magnetic white dwarf accretors.

Acknowledgments

We thank Gijs Nelemans for discussions on AM CVn systems. DS was supported on a PPARC post-doctoral grant.