Direct N-body simulations of globular clusters -- III. Palomar\,4 on an
  eccentric orbit
  Palomar 4 is a low-density globular cluster with a current mass $\approx30000
M_{\odot}$ in the outer halo of the Milky Way with a two-body relaxation time
of the order of a Hubble time. Yet, it is strongly mass segregated and contains
a stellar mass function depleted of low-mass stars. Pal 4 was either born this
way or it is a result of extraordinary dynamical evolution. Since two-body
relaxation cannot explain these signatures alone, enhanced mass loss through
tidal shocking may have had a strong influence on Pal 4. Here, we compute a
grid of direct N-body simulations to model Pal 4 on various eccentric orbits
within the Milky Way potential to find likely initial conditions that reproduce
its observed mass, half-light radius, stellar MF-slope and line-of-sight
velocity dispersion. We find that Pal 4 is most likely orbiting on an eccentric
orbit with an eccentricity of $e\approx 0.9$ and pericentric distance of
$R_p\approx5$ kpc. In this scenario, the required 3D half-mass radius at birth
is similar to the average sizes of typical GCs ($R_h\approx4-5$ pc), while its
birth mass is about $M_0\approx10^5 M_{\odot}$. We also find a high degree of
primordial mass segregation among the cluster stars, which seems to be
necessary in every scenario we considered. Thus, using the tidal effect to
constrain the perigalactic distance of the orbit of Pal 4, we predict that the
proper motion of Pal 4 should be in the range $-0.52\leq\mu_\delta\leq-0.38$
mas\,yr$^{-1}$ and $-0.30\leq\mu_{\alpha\cos\delta}\leq-0.15$ mas\,yr$^{-1}$.
