  In order to accurately model giant planets, a whole set of observational
constraints is needed. As the conventional constraints for extrasolar planets
like mass, radius, and temperature allow for a large number of acceptable
models, a new planetary parameter is desirable in order to further constrain
planetary models. Such a parameter may be the tidal Love number k2. In this
paper we aim to study the capability of k2 to reveal further information about
the interior structure of a planet. With theoretical planetary models we
investigate how the tidal Love number k2 responds to the internal density
distribution of a planet. In particular, we demonstrate the effect of the
degeneracy of k2 due to a density discontinuity in the envelope of a
three-layer planetary model. The effect of a possible outer density
discontinuity masks the effect of the core mass on the Love number k2. Hence,
there is no unique relationship between the Love number k2 and the core mass of
a planet. We show that the degeneracy of k2 with respect to a layer boundary in
the envelope also occurs in existing planets, e.g. Saturn and the Hot Neptune
GJ436b. As a result of the degeneracy, the planetary parameter k2 cannot be
used to further constrain Saturnian models and for GJ436b only a maximum
possible core mass can be derived from a given k2. To significantly narrow the
uncertainty about the core mass of GJ436b the combined knowledge of k2 and
atmospheric metallicity and temperature profile is necessary.
