We calculate transonic disk accretion flows around a weakly magnetized neutron star, assuming that the neutron star is within the general relativistic marginally stable orbit and that the accretion disk extends very close to the surface of the star. We consider only a weak magnetic dipole field, the strength of which is of order of 107-108 G at the surface. We introduce a parameter χ ≡ ζ(B0/107)2 for the magnetic field, where ζ is the ratio of the dissipation timescale of the magnetic field to the Keplerian orbital period and B0 is the strength of the magnetic field at the surface. We employ a scaling law for the electronic conductivity σ so that ζ will be constant. We found that near the surface of the neutron star, Joule heating becomes the dominant source of heating when χ ≳ 10. This Joule heating balances the radiative cooling if the mass accretion rate is small. When the mass accretion rate is as large as Ṁc ≡ 32πcrg/Kes, where rg is the Schwarzschild radius and Kes is the electron scattering opacity, the advection cooling and the radiative cooling are comparable near the surface and the sum of the two coolings balances the Joule heating. Because of this heating, the radiative flux perpendicular to the disk surface is largely enhanced near the neutron star's surface. For a small mass accretion rate Ṁ = 0.1Ṁc, the sonic point shifts outward as χ increases, while for a high mass accretion rate Ṁ = Ṁc, the sonic point shifts inward as χ increases, and no transonic solution through a saddle-type critical point can be found for χ larger than some critical χc ∼ 60, the value of which may depend on parameters such as α. We also found that the Joule heating makes possible transonic solutions through a saddle-type critical point for a low mass accretion rate even for a large value of α.
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