  Numerical simulations of planets embedded in protoplanetary gaseous discs are
a precious tool for studying the planetary migration ; however, some
approximations have to be made. Most often, the selfgravity of the gas is
neglected. In that case, it is not clear in the literature how the material
inside the Roche lobe of the planet should be taken into account. Here, we want
to address this issue by studying the influence of various methods so far used
by different authors on the migration rate.
  We performed high-resolution numerical simulations of giant planets embedded
in discs. We compared the migration rates with and without gas selfgravity,
testing various ways of taking the circum-planetary disc (CPD) into account.
Different methods lead to significantly different migration rates. Adding the
mass of the CPD to the perturbing mass of the planet accelerates the migration.
Excluding a part of the Hill sphere is a very touchy parameter that may lead to
an artificial suppression of the type III, runaway migration. In fact, the CPD
is smaller than the Hill sphere. We recommend excluding no more than a 0.6 Hill
radius and using a smooth filter. Alternatively, the CPD can be given the
acceleration felt by the planet from the rest of the protoplanetary disc.
  The gas inside the Roche lobe of the planet should be very carefully taken
into account in numerical simulations without any selfgravity of the gas. The
entire Hill sphere should not be excluded. The method used should be explicitly
given. However, no method is equivalent to computing the full selfgravity of
the gas.
