SPECTROSCOPIC EVIDENCE FOR A 5.4 MINUTE ORBITAL PERIOD IN HM CANCRI

The average Keck spectrum of HM Cnc is shown in Figure 1. As previously reported (Israel et al. 2002), the spectrum is dominated by ionized helium emission lines. Their full width at half-maximum (FWHM) of ∼2500 km s⁻¹ is well resolved by the 300 km s⁻¹ resolution of our spectra. Lines of neutral helium are also present but much weaker. Our higher-quality spectrum confirms earlier findings that the even-term transitions of the He ii Pickering series are stronger than the odd-term ones (see Figure 1), which was interpreted by Norton et al. (2004) and Reinsch et al. (2007) as evidence for the presence of hydrogen. In particular, the absence of He ii 4200 in our spectrum is striking, and most easily explained if one assumes that there is indeed hydrogen in HM Cnc (for simplicity, we will refer to He ii Pickering lines in the remainder of this Letter).

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In addition to the N iii 4640 line in the Bowen blend identified by Israel et al. (2002), the strong N iii 4379 line appears to be present to the red of He ii 4338. Apparent emission lines near 4600 Å and 4800 Å could be due to N iv, while a faint feature to the red of He i 4471 may be N iii 4514. The presence of several nitrogen lines appears to be at odds with the suggestion (based on fits to X-ray spectra) that HM Cnc may have an unusual chemical abundance pattern and be underabundant in nitrogen in particular (Strohmayer 2008). We note that nitrogen-rich matter would be expected due to the CNO-cycle if HM Cnc is an ultracompact binary, unless the material experienced helium burning (Yungelson 2008), and that nitrogen indeed appears to be abundant in most AM CVn stars (see Roelofs et al. 2009).

When we phase-fold our spectra using the ephemeris of Barros et al. (2007), we observe continuum flux variations that lag slightly (∼30° in phase) behind Barros et al. (2007)'s photometry. This is not unexpected given, in particular, the uncertainty in the period derivative measured by Barros et al. (2007), which translates into a phase uncertainty of about this magnitude.

By matching the phases of the variable g-band flux in our time-resolved spectra with the g-band photometry of Barros et al. (2007), we can obtain a refined ephemeris for HM Cnc. This will not be a very precise refinement since our spectra only marginally sample the orbital period, and since atmospheric transparency and seeing were variable. However, because we have hundreds of spectra and the folding period is much shorter than the typical timescales for sky quality variations, we can still resolve this shift. The easiest way to correct the phase shift is to reduce the frequency ν and its time derivative from Barros et al. (2007) by approximately 1.3σ:

$$\begin{eqnarray} t_0\,\mathrm{(barycentric\ TDB)} &\equiv & 53 009.889 943 753,\nonumber \\ \nu \,\mathrm{(Hz)} &=& 0.003 110 138 11(10),\nonumber \\ d\nu /dt\,\mathrm{(Hz\ s^{-1})} &=& 3.57(2)\times 10^{-16}, \end{eqnarray} \tag{ 1 }$$

where numbers in parentheses are the uncertainties in the corresponding number of last decimals. The new error on the frequency derivative is estimated as the error that gives the same phase uncertainty (at our epoch) as the error on the frequency. The error on the frequency has been kept the same. This ephemeris is used throughout this Letter.

The left panels of Figure 2 show the time-resolved (trailed) spectrum of HM Cnc around the He i 4471 line, folded on the above ephemeris (Equation (1)). The continuum flux is seen to follow broadly the same pattern as in Barros et al. (2007) after aligning the phases: during a cycle the continuum flux increases slowly and decreases more rapidly. The scatter in our continuum fluxes, shown in the left-most panel of Figure 2, is due to differences in the average sky quality between phase bins rather than due to noise in individual spectra.

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Clear spectral line variations are seen in the He i 4471 line, in the form of an "S-wave" feature that is visible for about half the period while it moves from redshifted to blueshifted wavelengths. The intensity of the S-wave in the He i 4471 line follows the intensity of the variable continuum flux, suggesting that they originate from the same region. In the remaining panels of Figure 2, the intrinsically variable continuum flux has been fitted and subtracted to highlight the line variability. Here, the He ii 4686 line is plotted as well as the sum of the three strongest He ii Pickering series lines (4100, 4338, and 4859 Å; see Figure 1). The He ii lines behave quite differently from the He i 4471 line. The He ii 4686 line is double-peaked and appears to wobble in anti-phase with respect to the He i 4471 line. The He ii Pickering lines have a strongly modulated, narrow component which may be moving from blueshifted to redshifted in phase with (and at the same radial velocity of) the He ii 4686 line. In addition, the He ii Pickering lines have a fairly constant, broad component with a FWHM of ∼2500 km s⁻¹, which matches the FWHM of the (double-peaked) He ii 4686 profile.

We measure, from a linear back-projection Doppler tomogram (Marsh & Horne 1988), a radial velocity semi-amplitude of 390 ± 40 km s⁻¹ for the S-wave feature in the He i 4471 line (corrected for a ∼5% bias due to finite exposure times, which we modeled using synthetic data). The error was estimated from a large ensemble of Doppler tomograms calculated from bootstrap samples of our 400 spectra. For the double-peaked He ii 4686 line moving in anti-phase, we similarly measure a radial velocity semi-amplitude of 260 ± 40 km s⁻¹. The two peaks in the profile are redshifted and blueshifted by 650 ± 50 km s⁻¹.

Finally, the spectra were phase-folded on trial periods of up to 6 hr, but no radial velocity or intensity variations in the spectral lines were seen on periods other than 5.4 minutes.

The observed radial velocity variations in the spectral lines strongly suggest that HM Cnc has a 5.4 minute orbital period. We favor the semi-detached binary white dwarf (or AM CVn) model, shown in Figure 3. It is the only model that can account for the spectroscopy presented in this Letter. The IP model predicts an orbital period of hours rather than minutes, and the observed line kinematics do not match predictions by Norton et al. (2004) for spectral line variability on the accretor's spin period (if any). We see the broad, fairly constant (and in the case of He ii 4686, double-peaked) He ii emission as a clear signature of accretion, which is not predicted by the UI model.

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We conclude that HM Cnc is by far the shortest period binary star known. Its direct progenitor was probably a detached binary white dwarf, since evolutionary models with initially non-degenerate donor stars cannot reach such short orbital periods (e.g., Yungelson 2008; van der Sluys et al. 2005).

The observed decrease of HM Cnc's 5.4 minute period, considered the main obstacle for the AM CVn model, may be expected naturally if it is a semi-detached binary white dwarf with this orbital period, due to the relatively long mass-transfer turn-on timescales predicted for fairly low-mass helium donors (Deloye & Taam 2006; G. H. A. Roelofs & C. J. Deloye 2010, in preparation).

Based on the radial velocities of the He i 4471 and He ii 4686 emission lines moving in anti-phase, we can estimate the ratio of donor to accretor mass q ≡ M₂/M₁ in HM Cnc. The He i 4471 line seems to originate from the irradiated face rather than the exact mass center of the donor star, and so we must estimate a "K-correction" (e.g., Muñoz-Darias et al. 2005) to obtain the donor's actual projected orbital velocity from the observed emission-line velocity. Without a K-correction, we get an upper limit of q ⩽ 0.67 ± 0.12. Assuming that the He i line originates from the inner Lagrange point (the maximum conceivable K-correction) gives a lower limit q ⩾ 0.33 ± 0.06. The true value will be in between; assuming a flat probability density distribution between our upper and lower limits gives q = 0.50 ± 0.13.

Figure 4 shows a combined Doppler tomogram of the He i 4471 and He ii 4686 lines, showing the (putative) irradiated donor star and ring-like emission centered on the (proposed) accreting star. The mass ratio derived above can easily be estimated from this figure. A further interesting feature is the "bright spot" in the He ii 4686 emission, occurring in the lower left quadrant of the Doppler tomogram. The velocity vectors from the center of mass to the donor star and to this He ii 4686 bright spot make a 110° angle approximately. Kinematically, this bright spot matches with a spot approximately on the side of the accreting star as seen from the donor, if the material in the bright spot is moving at roughly half the Keplerian (breakup) velocity near the surface of the accretor (but depending on its mass). An accretion stream impact spot at this location was proposed by Barros et al. (2007) to explain the relative phases of the optical and X-ray light curves. Assuming M₂ = 0.13 M_☉, which is the minimum donor star mass corresponding to a Roche lobe-filling degenerate helium object, an accretor mass M₁ ≈ 0.55 M_☉ is required to get an accretion stream impact spot sufficiently on the side of the accretor, as shown by Barros et al. (2007). Our mass ratio q = 0.50 ± 0.13 implies that both the donor and the accretor have to be more massive than this to still produce an impact spot sufficiently on the side (but see Wood 2009 for a possible relaxation of this constraint).

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The measured rate of change of the orbital period in HM Cnc (Equation (1)) also suggests that the donor star has to be significantly more massive than its fully degenerate value, if one assumes that the rate of change matches the secular rate set by gravitational-wave radiation (neglecting mass transfer). Given q ≈ 0.50, masses M₂ = 0.27, M₁ = 0.55 would be required to yield the observed period derivative. We note that these are quite typical masses for binary white dwarfs according to theoretical models (Nelemans et al. 2001). Assuming instead M₂ = 0.13 M_☉, which is again the minimum mass of a Roche lobe-filling helium donor, an accretor mass M₁ = 1.3 M_☉ would be required. This is incompatible with our mass ratio measurement. Note that mass transfer reduces the orbital period change rate relative to the case of a detached binary, so that higher masses would be required; but in practice, it is likely that the mass transfer is not yet sufficiently developed to have a big influence (Deloye et al. 2007). We can conclude that the donor star is probably at least twice as massive as a fully degenerate helium donor would be. For masses M₂ = 0.27 M_☉, M₁ = 0.55 M_☉ and assuming that the radial velocity semi-amplitude of the wobbling He ii 4686 line represents the projected orbital velocity of the accretor, we obtain an inclination i ≈ 38° of the orbital plane.

With the system parameters derived in the previous section, we can calculate the gravitational-wave strain amplitude at Earth. The distance to HM Cnc is the largest remaining uncertainty; it is probably ∼5 kpc based on the expected sizes and measured temperatures of the white dwarf components (Barros et al. 2007). This gives a dimensionless gravitational-wave strain amplitude h ≃ 1.0 × 10⁻²². This, together with the design sensitivities of space-borne gravitational-wave detectors like LISA, makes HM Cnc one of the easiest detectable sources of gravitational waves known (see Nelemans 2009).