How delicate are the $f(R)$ gravity models with disappearing
  cosmological constant?
  We consider stability of spherically symmetric solutions in $f(R)$ gravity
model proposed by Starobinsky. We find that the model suffers from a severe
fine tuning problem when applied to compact objects like neutron stars. The
problem can be remedied by introducing a cut off on the mass of the scalar
degree of freedom present in the model. A new mass scale associated with
neutron stars density is then required for the stabilities of $f(R)$ gravity
solutions inside relativistic stars.
