  The origin of the spiral spin-order in perovskite multiferroic manganites
$R$MnO$_{3}$ ($RE=$ Tb or Dy) is here investigated using a two $e_{\rm
g}$-orbitals double-exchange model. Our main result is that the experimentally
observed spiral phase can be stabilized by introducing a relatively weak
next-nearest-neighbor superexchange coupling ($\sim10%$ of the nearest-neighbor
superexchange). Moreover, the Jahn-Teller lattice distortion is also shown to
be essential to obtain a realistic spiral period. Supporting our conclusions,
the generic phase diagram of undoped perovskite manganites is obtained using
Monte Carlo simulations, showing phase transitions from the A-type
antiferromagnet, to the spiral phase, and finally to the E-type
antiferromagnet, with decreasing size of the $R$ ions. These results are
qualitatively explained by the enhanced relative intensity of the
superexchanges.
