  Recent observations of a large number of DA and DB white dwarfs show evidence
of debris disks, which are the remnants of old planetary systems. The infrared
excess detected with \emph{Spitzer} and the lines of heavy elements observed in
their atmospheres with high-resolution spectroscopy converge on the idea that
planetary material accretes onto these stars. Accretion rates have been derived
by several authors with the assumption of a steady state between accretion and
gravitational settling. The results are unrealistically different for DA and DB
white dwarfs. When heavy matter is accreted onto stars, it induces an inverse
$\mu$-gradient that leads to fingering (thermohaline) convection. The aim of
this letter is to study the impact of this specific process on the derived
accretion rates in white dwarfs and on the difference between DA and DB. We
solve the diffusion equation for the accreted heavy elements with a
time-dependent method. The models we use have been obtained both with the IRAP
code, which computes static models, and the La Plata code, which computes
evolutionary sequences. Computations with pure gravitational settling are
compared with computations that include fingering convection. The most
important result is that fingering convection has very important effects on DAs
but is inefficient in DBs. When only gravitational settling is taken into
account, the time-dependent computations lead to a steady state, as postulated
by previous authors. When fingering convection is added, this steady state
occurs much later. The surprising difference found in the past for the
accretion rates derived for DA and DB white dwarfs disappears. The derived
accretion rates for DAs are increased when fingering convection is taken into
account, whereas those for DBs are not modified. More precise and developed
results will be given in a forthcoming paper.
