Solution
= Solution
A <cataclysmic variable> is a close <semidetached binary> in which a <white dwarf> accretes from a cool, usually low-mass <donor star> filling its <Roche lobe>. In an ordinary hydrogen-rich system the <donor star> is often near the <lower main sequence>. In a <nonmagnetic cataclysmic variable> (a disk-fed <CV>), the <white dwarf>'s <magnetic field> is too weak to control the flow over the disk. This leaves a characteristic disk-fed geometry, summarized in the original schematic below. https://heasarc.gsfc.nasa.gov/docs/objects/cvs/cvstext.html[NASA's introduction to cataclysmic variables] describes the basic components.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2015/iii/paper-65-cataclysmic-variable.png]
{title=Nonmagnetic cataclysmic variable: Roche-lobe-filling donor, L1 gas stream, accretion disk, hot spot, white dwarf and boundary layer}
{height=700}
<Roche-lobe overflow> passes through the <inner Lagrange point> $L_1$. A nearly ballistic stream bends in the rotating frame and strikes the outer <accretion disk>, producing an <accretion-disk stream-impact hot spot>. Its retained <angular momentum> prevents direct radial infall. <Viscous evolution of an accretion disk> transports <angular momentum> outward while gas moves inward through a nearly <Keplerian accretion disk>. Close to the <white dwarf>, gas slows from orbital rotation toward the stellar rotation in an <accretion-disk boundary layer>. The disk, <accretion-disk stream-impact hot spot>, <accretion-disk boundary layer>, <white dwarf> and <donor star> all contribute to the spectrum and, where the <orbital inclination> permits, to the eclipses of an <eclipsing binary>. In particular, “nonmagnetic” describes the accretor's control of the flow; it does not imply that the <donor star> cannot sustain a <magnetic field>.
The gravitational power available from <accretion> is approximately $L_{\rm acc}=GM_{\rm WD}\dot M_{\rm acc}/R_{\rm WD}$. For a slowly rotating <white dwarf> and a thin steady <Keplerian accretion disk>, the specific energy changes from approximately zero far out to $-GM_{\rm WD}/(2R_{\rm WD})$ at the inner disk. Roughly half the available power is radiated by the <accretion disk>; the rest is released in the <accretion-disk boundary layer> as the orbital <kinetic energy> is dissipated. Stellar rotation and departures from a steady thin disk change this partition.
A <classical nova> has a different energy source. Transferred <hydrogen> accumulates on the <white dwarf>; compression heats the base of its envelope until temperature-sensitive <hydrogen burning> accelerates. <Electron degeneracy pressure> initially weakens the expansion response to heating, helping a <thermonuclear runaway> develop. The envelope subsequently expands and ejects material, producing a large optical outburst followed by a decline as the ejecta expand and residual burning ends. The <white dwarf> usually survives, so continued <accretion> can build another fuel layer. A rough recurrence scale is the ignition-envelope mass divided by the mean <accretion rate>; both this mass and the rate vary strongly among systems. The event is an envelope eruption, and the retained fraction is not automatically unity. https://arxiv.org/abs/1605.04294[Starrfield, Iliadis and Hix's nova calculations] explains this nuclear mechanism. \b[A classical nova is powered by unstable nuclear burning on the white dwarf.]
A <dwarf nova> undergoes recurrent, shorter brightenings powered principally by enhanced gravitational <accretion>. The <hydrogen-ionization disk instability> creates cold, mostly neutral and hot, ionized branches of the disk's <accretion-disk thermal S-curve>, separated by unstable equilibria. In quiescence the cool disk stores matter because inward transport is slow. Once a critical <surface density> is reached, a heating transition puts the disk into a hotter, more state with higher effective <viscosity>: the inward <accretion rate> and <luminosity> rise and the disk drains. A cooling transition returns it to quiescence, completing the cycle. A persistent increase in the <donor star>'s transfer rate is not required. Sufficiently high transfer rates can keep the disk on its hot stable branch, giving a <nova-like variable> rather than ordinary disk cycles. https://arxiv.org/html/astro-ph/0102072v1[Lasota's disk-instability analysis] and https://arxiv.org/abs/1910.01852[Hameury's disk-instability review] develop this picture. \b[A dwarf-nova outburst is a disk instability, not a white-dwarf thermonuclear explosion.] The names classify mechanisms and need not identify permanently distinct binaries: a nova-producing binary can also possess an unstable disk between nuclear eruptions.
The <cataclysmic-variable orbital-period distribution> is not smooth. For ordinary hydrogen-rich <cataclysmic variables>, prominent features are a <cataclysmic-variable period gap> around two to three hours, a <cataclysmic-variable period minimum> near eighty minutes, and an accumulation near that minimum. These are population features rather than absolute exclusions. Selection effects matter: luminous high-<accretion rate> systems are easier to find than faint evolved systems. Helium-transferring binaries have a different period range and are not described by the hydrogen-rich minimum. https://eprints.whiterose.ac.uk/id/eprint/123569/[Gänsicke and collaborators' period-minimum study] documents the observed accumulation.
The <Roche-lobe-filling period-density relation> makes the <orbital period> a measure of donor structure. Combining $R_d\simeq0.46a(M_d/M)^{1/3}$ with <Kepler's third law> gives
$$
P^2=\frac{4\pi^2R_d^3}{0.46^3GM_d},\qquad
\boxed{\bar\rho_d=\frac{3\pi}{0.46^3GP^2}\simeq112\left(\frac P{\mathrm{hour}}\right)^{-2}\mathrm{g\,cm^{-3}}.}
$$
As the <donor star> loses mass, its <stellar radius response exponent> $\zeta=d\log R_d/d\log M_d$ implies
$$
\frac{d\log P}{d\log M_d}=\frac{3\zeta-1}{2}.
$$
A donor with $\zeta>1/3$ evolves toward shorter <orbital periods>. When its effective response falls below $1/3$, continued mass loss instead lengthens the <orbital period>: this is the <cataclysmic-variable period bounce>. A very low-mass donor may be substellar and increasingly affected by <electron degeneracy pressure>; the ideal degenerate scaling $R_d\propto M_d^{-1/3}$ illustrates the reversal. The precise <cataclysmic-variable period minimum> depends on thermal disequilibrium and the strength of orbital <angular momentum> loss. Near a turning point $|\dot P|$ is small; for an approximately steady evolutionary flow of systems, the number per period interval scales as $N(P)\propto1/|\dot P|$, explaining the accumulation. https://arxiv.org/abs/1102.2440[Knigge, Baraffe and Patterson's donor-based evolutionary study] relates the <donor star>'s response to these period features.
Long-term transfer is driven mainly by losses of orbital <angular momentum>, rather than by disk outbursts. <Magnetic braking of a binary star> removes the cool <donor star>'s spin through a magnetized <stellar wind>. <Tidal synchronization> makes the orbit replenish that spin, so the wind extracts orbital <angular momentum>. This is usually the dominant standard driving mechanism above the <cataclysmic-variable period gap>. <Gravitational-wave emission from a binary system> supplies a baseline loss, especially important below the gap. For a weak-field, slowly moving <circular orbit>, the <circular gravitational-wave inspiral> gives
$$
\frac{\dot J_{\rm GR}}J=-\frac{32G^3M_{\rm WD}M_d(M_{\rm WD}+M_d)}{5c^5a^4}.
$$
Because ordinary transfer from the lighter <donor star> tends to expand its <Roche lobe> if orbital <angular momentum> is conserved, an external loss is needed to sustain contact. In the conservative contact approximation, the <binary mass-transfer contact equation> reads
$$
\frac{\dot M_d}{M_d}=\frac{2\dot J_{\rm ext}/J}{\zeta+5/3-2q},\qquad q=\frac{M_d}{M_{\rm WD}},
$$
where the <stellar radius response exponent> must match the evolutionary timescale and additional donor expansion has been neglected. The positive denominator on the stable branch makes $\dot J_{\rm ext}<0$ drive $\dot M_d<0$. Nova ejecta can introduce additional <nonconservative binary mass transfer>. https://arxiv.org/abs/1108.4716[Knigge's evolutionary discussion] describes the standard loss mechanisms and their limitations.
In the <disrupted magnetic braking model>, relatively rapid mass loss above the <cataclysmic-variable period gap> keeps the <donor star> inflated relative to <stellar thermal equilibrium>. When the donor approaches the <fully convective star> transition, the model postulates a substantial reduction in <magnetic braking of a binary star>. The <donor star> can contract within its <Roche lobe>, suppressing <Roche-lobe overflow> near the upper edge of the gap. <Gravitational-wave emission from a binary system> continues to shrink the <detached binary>; near the lower edge the <Roche lobe> again reaches the donor radius and transfer resumes. No mass transfer is needed during the detached crossing. The density relation predicts a radius ratio $(3/2)^{2/3}\simeq1.31$ between contact at three and two hours if the masses remain nearly fixed, illustrating the required inflation before detachment. The torque reduction is a model ingredient, not a claim that all <fully convective stars> lose their magnetic fields. https://arxiv.org/abs/1601.07785[Zorotovic and collaborators' detached-binary study] tests the predicted detached population in the gap.
The standard formation channel starts with an initially wider <binary star> containing two <main sequence> stars. The initially more massive component evolves first and becomes a giant. Unstable <Roche-lobe overflow> can engulf the companion in a <common envelope>. Drag causes inward orbital motion, releasing <two-body orbital energy> and transferring <angular momentum> to the envelope. If the envelope is expelled before merger, a close <detached binary> survives, containing the exposed core, which becomes a <white dwarf>, and the lower-mass companion. This explains how a binary becomes much tighter than the giant progenitor's radius would have allowed. The outcome depends on envelope binding and the efficiency of energy deposition, summarized approximately by the <common-envelope energy formalism>; ejection is not guaranteed. https://arxiv.org/abs/1209.4302[Ivanova and collaborators' common-envelope analysis] discusses the relevant physics and uncertainties.
Subsequent <magnetic braking of a binary star> and <gravitational-wave emission from a binary system> shrink the <detached binary> until its companion fills its <Roche lobe>. Stable <Roche-lobe overflow>, for a suitable <binary mass ratio> and <stellar radius response exponent>, then creates a <cataclysmic variable>. Its later nuclear eruptions, disk cycles and secular <orbital period> evolution occur on different timescales. \b[The formation sequence is a wide binary, envelope ejection, a close detached white-dwarf binary, and angular-momentum-driven contact.]