AM CVn star
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An AM CVn star, or AM Canum Venaticorum star, is a type of cataclysmic variable star. These binary stars are named after their type star, AM Canum Venaticorum. In these variables, a white dwarf accretes hydrogen-poor matter from a compact companion star. These binaries have extremely short orbital periods (shorter than about one hour), and are predicted to be strong sources of gravitational radiation, strong enough to be detected with the Laser Interferometer Space Antenna.
AM CVn stars differ from most other cataclysmic variables (CVs) in the lack of hydrogen lines from their spectra. Their ultra-short orbital periods of less than about 70 minutes indicate that both the donor star and accretor star in the binary are degenerate or semi-degenerate objects. Where the accretor is always a white dwarf, the donor star can be either a (helium or hybrid) white dwarf, a low-mass helium star or an evolved main-sequence star. Although all three types of donor star have been predicted to exist, observations suggest that only helium-star donors may have been found so far.
Formation scenarios 
The three possible types of donor stars each have a different formation scenario.
AM CVn stars with a white-dwarf donor can be formed when a binary consisting of a white dwarf and a low-mass giant evolve through a common-envelope (CE) phase. The outcome of the CE will be a double white-dwarf binary. Through the emission of gravitational radiation, the binary loses angular momentum, which causes the binary orbit to shrink. When the orbital period has shrunk to about 5 minutes, the least-massive (and the largest) of the two white dwarfs will fill its Roche lobe and start mass transfer to its companion. Soon after the onset of mass transfer, the orbital evolution will reverse and the binary orbit will expand. It is in this phase, after the period minimum, that the binary is most likely to be observed.
AM CVn stars with a helium-star donor are formed in a similar way, but in this case the giant that causes the common envelope is more massive and produces a helium star rather than a second white dwarf. A helium star is more expanded than a white dwarf, and when gravitational radiation brings the two stars into contact, it is the helium star which will fill its Roche lobe and start mass transfer, at an orbital period of roughly 10 minutes. As in the case of a white-dwarf donor, the binary orbit is expected to 'bounce' and start expanding soon after mass transfer is started, and we should typically observe the binary after the period minimum.
The third type of potential donor in an AM CVn system is the evolved main-sequence star. In this case, the secondary star does not cause a common envelope, but fills its Roche lobe near the end of the main sequence (terminal-age main sequence or TAMS). An important ingredient for this scenario is magnetic braking, which allows efficient angular-momentum loss from the orbit and hence a strong shrinkage of the orbit to ultra-short periods. The scenario is rather sensitive to the initial orbital period; if the donor star fills its Roche lobe too long before the TAMS the orbit will converge, but bounce at periods of 70-80 minutes, like ordinary CVs. If the donor starts mass transfer too long after the TAMS, the mass-transfer rate will be high and the orbit will diverge. Only a narrow range of initial periods, around this bifurcation period will lead to the ultra-short periods that are observed in AM CVn stars. The process of bringing the two stars into a close orbit under the influence of magnetic braking is called magnetic capture. AM CVn stars formed this way may be observed either before or after the period minimum (which can lie anywhere between 5 and 70 minutes, depending on exactly when the donor star filled its Roche lobe) and are assumed to have some hydrogen on their surface.