The circumstances mentioned in the previous paragraph stress the importance of studying AM CVn-stars in all possible wavebands. LISA will measure a combination of all the parameters that determine the GWR signal (frequency, chirp mass, distance, position in the sky, and inclination angle; see, e.g., ), so if some of these parameters (period, position) can be obtained from optical or X-ray observations, the other parameters can be determined with higher accuracy. This is particularly interesting for the distances, inclinations, and masses of the systems, which are very difficult to measure with other methods.
In the optical, the total sample of AM CVn-type stars is expected to be dominated by long-period members of the class due to emission of their disks. But the shortest periods AM CVn-type stars that are expected to be observed with LISA may be observed both in optical and X-rays thanks to high mass-transfer rates (see Figure 9). A model for electromagnetic-emission properties of the ensemble of the shortest orbital period was constructed by Nelemans et al. . In , only systems with He-WD or “semidegenerate” He-star donors were considered (see Figure 5). Systems with donors descending from strongly evolved MS-stars were excluded from consideration, since their fraction in the orbital period range interesting for LISA is negligible. The “optimistic” model of  was considered, which assumes efficient spin-orbital coupling in the initial phase of mass-transfer and avoids edge-lit detonations of helium accreted at low . Average temperature and blackbody emission models in V -band and in the ROSAT 0.1 – 2.4 keV X-ray band were considered, taking into account interstellar absorption. The ROSAT band was chosen because of the discovery of AM CVn itself  and two candidate AM CVn systems as ROSAT sources (RXJ0806.3+1527  and V407 Vul ) and because of the possibility for a comparison to the ROSAT all-sky survey.
One may identify four main emission sites: the accretion disc and boundary layer between the disc and the accreting white dwarf, the impact spot in the case of direct impact accretion, the accreting star, and the donor star.
Optical emission. The luminosity of the disk may be estimated as.
The emission from the donor was treated as the emission of a cooling white dwarf, using approximations to the cooling models of Hansen .
The emission from the accretor was treated as the unperturbed cooling luminosity of the white dwarf16.
A magnitude-limited sample was considered, with , typical for observed short-period AM CVn-type stars. Interstellar absorption was estimated using Sandage’s model  and Equation (67).
X-ray emission. Most AM CVn systems experience a short (106 – 107 yr) “direct impact” stage in the beginning of mass-transfer [144, 286, 251]. Hence, in modeling the X-ray emission of AM CVn systems one has to distinguish two cases: direct impact and disk accretion.
In the case of a direct impact a small area of the accretor’s surface is heated. One may assume that the total accretion luminosity is radiated as a blackbody with a temperature given by and the observed X-ray emission of V407 Vul, a known direct-impact system .
In the presence of a disk, X-ray emission was assumed to be coming from a boundary layer with temperature –13 erg s–1 cm–2 were selected. Then, the intrinsic flux in this band, the distance and an estimate of the Galactic hydrogen absorption  provide an estimate of detectable flux.
Figure 16 presents the resulting model. In the top panel there are 220 systems only detectable in X-rays and 330 systems also detectable in the V -band. One may distinguish two subpopulations in the top panel: In the shortest period range there are systems with white-dwarf donors with such high that even sources close to the Galactic centre are detectable. Spatially, these objects are concentrated in a small area on the sky. At longer periods the X-rays get weaker (and softer) so only the systems close to the Earth can be detected. They are more evenly distributed over the sky. Several of these systems are also detectable in the optical (filled symbols). There are 30 systems that are close enough to the Earth that the donor stars can be seen as well as the discs (filled squares). Above the systems with helium-star donors show up and have a high enough mass transfer rate to be X-ray sources, the closer ones of which are also visible in the optical, as these systems always have a disc. The bottom panel shows the 1,230 “conventional” AM CVn systems, detectable only by optical emission, which for most systems emanates only from their accretion disc. Of this population 170 objects closest to the Earth also have a visible donor. The majority of the optically detectable systems with orbital periods between 1,000 and 1,500 s are expected to show outbursts due to the viscous-thermal disc instability  which could enhance the chance of their discovery.
© Max Planck Society and the author(s)