## GmM

is radiated quasi-instantaneously from the disk or from near its surface. From (12.5) one can derive the accretion rate M if the mass and radius of the white dwarf and the bolometric accretion luminosity are known. Deriving Lacc requires knowledge of the distance, of the overall spectral energy distribution, and (at least an estimate) of the wavelength-dependent angular distribution of the emitted radiation. The obvious observational difficulties explain why accretion rates are notoriously uncertain. Attempts, based on accurate HSTFGS parallaxes and X-ray/FUV/optical fluxes are described in [5,6].

An entirely different approach considers compressional heating of the accreting white dwarf and derives M from the effective temperature of the white dwarf, which has the invaluable property of being independent of the distance. For bright X-ray sources such as polars, one obtains a check on M derived from the X-ray fluxes and the notoriously uncertain distances. Accretion of AM compresses the nondegener-ate envelope of the white dwarf and drives a corresponding mass at its bottom into degeneracy, increasing the core mass by AM. While the mass of the accreted envelope, Macc = 10-6 - 10-3Mq, is negligible compared with the core mass Mc, the geometrical thickness AR = R - Rc of the envelope, is not and the white dwarf radius R exceeds the core radius Rc noticeably,2 R ~ 1.06Rc (1.17Rc) for an 0.6MQ white dwarf with a hydrogen envelope of 10-4MQ and Teff = 104 (3 x 104) K. Accretion at a rate M compresses the envelope and produces an equilibrium luminosity

which is rooted in the envelope and appears in addition to the external accretion luminosity of (12.5). The term GMM(1/Rc - 1/R) describes the energy release by the addition of the accreted mass to the core. The fraction (1 - Venv/Vad) ~ 0.48 is available to be radiated, while Venv/Vad serves to heat the accreted matter to the core temperature as it sinks. Here, V = d log T/d logP is the logarithmic gradient of the temperature variation with pressure in the envelope and the indices "env" and "ad" refer to the actual gradient in the envelope and the adiabatic gradient, respectively.3 Equation (12.6) has been used by [18] to interprete the low temperature of the white dwarf in the long-period polar RXJ1313-32 in terms of a low accretion rate. A similar form was also discussed in [80] to explain the temperatures of accreting white dwarfs in CVs. Since the cooling time of the envelope is of the order of a million years, compressional heating measures a long-term mean accretion rate.

A comprehensive study of compressional heating, including an internally consistent treatment of the path to ignition of hydrogen burning, has been presented by Townsley & Bildsten [85,86]. Their numerical results on the dependence of the effective temperature of the white dwarf on M, Macc and (negligibly) on M [85, their Fig. 1] can be approximated as

where M10 is the accretion rate in units of 10-10 MQ yr-1 and f the mass of the envelope accumulated since the last nova outburst in units of the ignition mass Mign leading to the next nova event. Figure 12.8 displays a collection of HST-derived effective temperatures of white dwarfs in CVs [1] selected to avoid as far as possible the effects of recent heating in dwarf nova outbursts. The right-hand ordinate provides the equivalent long-term mean accretion rates from (12.7) using f = 0.5 corresponding to a time midways between two nova outbursts. The fact that the white dwarf experiences short-term heating episodes in nova and, if applicable, in dwarf

2 The core has approximately the Chandrasekhar radius for fe = 2. AR = R — Rc ~ kTc/(VenvJumuGMc) ~ 5HP, with Venv ~ 0.21 and HP the barometric pressure scale height for the temperature Tc at the bottom of the envelope.

3 Note that Lenv vanishes if rapid accretion leads to adiabatic heating and a convective envelope.

For a monatomic gas, the adiabatic gradient is Vad = 2/5, the radiative gradient is Vad ~ 0.21 and depends on the opacity law k(T, P).

2 30

10-9

10-io

rorb

Fig. 12.8 Observed effective temperatures of white dwarfs in CVs vs. orbital period (mostly from [1]). Different subtypes of CVs are indicated by the shading of the symbols. Statistical error bars are typically smaller than the symbol size. The right-hand scale gives a long-term average {M) from (12.7). Also shown is the 20-yr average M of AM Her [30] as a cross and the luminosity-derived M of EX Hya and V1223 Sgr [5,6] as open triangles nova outbursts, in addition to the long-term compressional heating suggests that one has to interprete the results with caution. Nevertheless, the result looks convincing in the sense that the derived accretion rates of short-period CVs - in particular, of the polars - agree with those expected for gravitational radiation and the accretion rates for the long-period dwarf novae increase with orbital period as expected from magnetic braking. The low long-term M of the long-period polars may be due to a lower efficiency of the angular momentum loss by the wind the secondary in the presence of the strong field of the primary. We recall that the so derived M values represent averages over the Kelvin-Helmholtz cooling time scale of the envelope. For comparison, we have added the 20-yr average accretion rate for AM Her, M ~ 1.2 x 10-10 Mq yr-1 [30], which represents a mean between high and low states (cross). The reff-derived accretion rate (lightly filled circle) is higher by about a factor of two and suggests that AM Her has spent the last million years on average in a state rather corresponding to the present high than the low state. Also added are the present accretion rates of the intermediate polars EX Hya [5] and V1223 Sgr [6] derived from accurate parallaxes and the bolometric - that is largely X-ray - fluxes (triangles). They fall in the same range as the temperature-derived rates and support the basic validity of the compressional heating model. The high M-values of the three novalike variables near an orbital period of 3 h demonstrate that there is no simple M-P relationship. These systems as well as long-period dwarf novae reach up to beyond 10-8 MQ y-1 into the region where surface hydrogen burning can occur.

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