  The role of the auxiliary scalar field $\phi$ of Brans-Dicke theory played in
baryon number asymmetry is discussed in this paper. We consider a derivative
coupling of this gravitational scalar field to the baryon current ${J^{\mu}}_B$
or the current of the baryon number minus lepton number ${J^{\mu}}_{B-L}$ based
on a series of works of R. Morganstern about the Brans-Dicke cosmology. We find
that the spontaneous baryogenesis by this coupling is capable to yield a
sufficient baryon asymmetry $n_B/s\sim 10^{10} $ for the time of the grand
unification is in a little advanced. In addition, Davoudiasl et al have
recently introduced a new type of interaction between the Ricci scalar $R$ and
the bayon current $J^{\mu}$, $\partial_{\mu}R J^{\mu}$ and also proposed a
mechanism for baryogenesis, the gravitational baryogenesis. However, the
Einstein equation tell us that $\dot{R}=0$ in the radiation-dominated epoch of
the standard FRW cosmology. In this paper we reconsider the feasibility of
having gravitational baryongenesis with such a form of interaction in
radiation-filled Brans-Dicke cosmology. We will show that $\dot{R}$ does not
vanish in this case and the required baryon number asymmetry can also be
achieved.
