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Ever since the origin of DN outbursts was known to arise in the disc surrounding the white dwarf there has been an effort to understand the processes involved in causing the observed outbursts. The Disc Instability Model (DIM) contains all the understanding gleaned from observational and theoretical studies. This model is an attempt to characterise and predict the outburst cycle. For detailed reviews on the disc instability model see Smak (1984b); Cannizzo (1993); Osaki (1996); Lasota (2001) and references therein.

An accretion disc is not a rigid body; each annulus is in a Keplerian orbit. There exists differential rotation between the disc annuli and each annulus orbits with a different angular velocity. Bulk flow of fluid is accompanied with turbulent and chaotic motion that generates

stresses in the disc. The shear of viscosity transports angular momentum outwards enforcing co- rotation between annuli (H¯oshi, 1979). The slower outer annuli are sped up and the faster inner annuli are slowed down causing angular momentum to be transported outwards. During this process the inner annuli lose angular momentum causing them to drop further into the potential well releasing gravitational potential energy.

In 1973 Shakura & Syunyaev hypothesised that the viscosity was due to vertical eddies inside the disc. The eddies cannot exceed the vertical scale height of the disc, H, in size. The tur- bulent velocity of these eddies must also be below the sound speed, cs, otherwise the turbulence will be thermalised. Known as theα-prescription, it allows for ignorance about the viscosity,ν, to be encapsulated inside of the parameterαby:

ν=αcsH (1.11)

In 1974 Osaki proposed a model that suggested dwarf nova outbursts were caused by intermittent accretion of matter onto the white dwarf by some, unknown at that time, instabilities within the disc. He proposed that the mass-transfer rate from the secondary is constant, but mass is not accreted onto the central white dwarf during quiescence and is stored in the outer parts of the disc. If the mass stored reaches a critical amount, an instability within the disc sets in and a significant amount of mass is then suddenly accreted onto the white dwarf.

Thermal Instability

The physical mechanism responsible for the disc outburst was proposed by H¯oshi (1979) and Meyer & Meyer-Hofmeister (1981). They showed that the accretion disc becomes thermally unstable due to partial hydrogen ionisation and outbursts occur when the disc transitions from a convective to radiative energy transport mechanism exhibiting bi-stable states. This is due to the increasing ionisation at higher temperatures. The accretion disc moves between these two states discontinuously showing a limit-cycle characterised by a well known S-shaped thermal equilibrium curve, see Figure 1.7. The behaviour seen in the disc is that of a limit cycle with alternating long-lived low viscosity states with low accretion rates and short-lived high viscosity states with high accretion rates. The solid line shows the two thermally stable equilibrium states,

M

sec

.

Temperature / Mass flow rate / viscosity

Hydrogen un−ionized Hydrogen partially ionized

Σ

Σ

B

D

C

A

Σ

Figure 1.7: The limit cycle, showing the local behaviour of an annulus in the accretion disc. The branch AB is the low equilibrium state, branch CD is the high equilibrium state and lines BC and DA represent the path taken by the disc when it moves offa stable branch.

the dashed line shows the thermally unstable equilibrium state. This discussion is purely local to each annuli in the disc.

The low branch, AB on Figure 1.7, represents the cool, convective state. The annulus is formed from neutral hydrogen giving it a low opacity, thus it is optically thin. This branch is thermally stable, a small perturbation in viscosity in a region of the disc would drain that region lowering the density and reducing the viscous heating. The temperature then drops back to equilibrium. The annulus has low viscosity causing the mass transferred into it to accumulate. As material builds up the surface density,Σ, increases until it reaches a critical surface density, Σ_max, where the annulus becomes partially ionised. At this point the thermal instability develops. The annulus moves to the middle branch where it is thermally unstable and heating dominates cooling; a small increase in temperature will cause the annulus to quickly transition into its hot state on the upper branch.

With the disc partially ionised, the opacity,κ, becomes very important and is very sen- sitive to temperature (κ _∝ T10_{) (H¯oshi, 1979). A small increase in the temperature of the disc} will make the opacity rapidly increase, trapping heat and feeding back, making the temperature rise even further until the disc is fully ionised. At this stage the opacity loses its sensitivity to temperature.

Once in the high state, CD on Figure 1.7, the annulus is hot and fully ionised causing it to have a high opacity. It is therefore optically thick and as a result there are a greater number of interactions between photons and ionised particles in the disc. This impedes the flow of radiation and gives rise to high viscosity. This branch is radiative and cooling dominates heating, the annulus is thermally stable, as on the low branch. In the hot, highly viscous state the inward flow of material through the disc exceeds that being added from the secondary. The state cannot be sustained and eventually matter drains from the annulus until the surface density reaches the critical surface density,Σmin. At this point, D, the temperature drops and recombination of the ionised gas occurs, the disc returns to its cold state.

Between outbursts, the material supplied by the secondary component is stored in the relatively cool disc. This eventually leads to a local thermal instability which results in the increase of the viscosity and of the local accretion rate. As the instability propagates across the disc, in the form of a heating wave, the global increase in the accretion rate is observed as an outburst. Similarly, a cooling wave spreads when an annulus has been drained sufficiently to have its surface density drop below the minimum critical surface density. This always occurs at the outer disc since this is where the surface density is the highest.

During the period when all annuli in the disc are on their hot branch, with high accretion rates, only _∼ 10 per cent of the material in the disc has time to be accreted onto the central white dwarf. Much of the remaining material simply flows inward due to the heating wave and expands as the cooling wave propagates through the disc.

Balbus-Hawley Instability

A weak field instability was first discussed by Chandrasekhar (1960, 1961) and Shakura & Syun- yaev (1973), but it was not until the work of Balbus & Hawley (1991) that it received fresh im-

petus as a mechanism of transporting angular momentum in accretion discs. They showed that a magnetic field is unstable if the angular momentum of the fluid increases outward. A magnetic field line permeating the disc will be coupled with ionised gas. A perturbation will result in the field line becoming radially distorted; this distortion increases in size as the field line experi- ences opposing forces from neighbouring annuli stretching it. The increased magnetic tension acts to enhance the instability increasing the efficiency of angular momentum transport between adjacent annuli. This instability is an effective way of transporting mass and angular momentum in the disc.

The Balbus-Hawley instability is strongly dependent on the coupling between the mag- netic field in the disc and the disc material. This occurs when the disc is hot and ionised. When the disc is cold the instability effectively shuts down. In the cold state the secondary transfers material into the accretion disc, as it does the surface density grows causing the temperature to rise. At near 104 K the hydrogen in the disc begins to ionise, allowing the Balbus-Hawley instability to increase the magnetic viscosity in the disc. This significantly increases the mass flow through the disc, or accretion rate, and effectively transports angular momentum outwards.

Tidal Instability

The outburst types that are seen in many dwarf nova cannot be fully explained by the standard DIM using the thermal instability explained above. Short period dwarf nova display standard disc instability behaviour with normal outbursts lasting_∼3 days but also show less frequent su- peroutbursts lasting_∼10₋18 days. However, the rise to superoutburst cannot be distinguished from the rise of a normal outburst. Superoutbursts are longer in duration and_∼1₋2 magnitude brighter than the normal outburst. A hump shaped modulation also appears near the superout- burst maximum with an amplitude of 20₋30 per cent of the total brightness. The superhump period is_∼ 2₋3 per cent Patterson et al. (2005) longer than the orbital period of the system. Therefore, by observing the superhumps, an orbital period of the system can obtained.

To reproduce these observed characteristics modifications need to be made to the stan- dard instability model. Vogt (1982) first proposed that superhumps were caused when the disc becomes elliptical during superoutburst and that the elliptical disc would precess on a time-scale

much longer than the orbit. Simulations by Whitehurst (1988) showed that for systems with extreme mass ratios, q. 0.33, the disc becomes elliptical and begins to precess. In almost all cases, systems of this type have been found to have orbital periods of less than two hours.

With each successive outburst the size of the accretion disc increases until the proximity of the edge of the disc gets closer to the orbiting secondary. The secondary tidally interacts with the disc producing a significant torque that acts as a sink severely depleting the angular momen- tum in the disc. This limits the radial extent of the outer disc to the tidal truncation radius. Most of the disc mass is then accreted onto the white dwarf producing a brighter and longer super- outburst. In this scenario resonances of the separate orbits of the disc and secondary become important, however, due to restrictions of mass ratios attainable in binary systems it is the 3:1 resonance ratio of the disc to secondary orbit that is important. For every 3 orbits made by the gas in the disc the secondary makes one complete orbit providing a driving force, known as the tidal instability. The tidal interaction causes material in Keplerian orbits in the disc to intersect leading to shearing in the disc. The shearing interaction of the material produces dissipation which is observable as a photometric modulation. This additional instability is thought to be the origin of elliptical discs in outburst.