  Photoneutron emission cross sections were measured for $^{13}$C below $2n$
threshold using quasi-monochromatic $\gamma$-ray beams produced in laser
Compton-scattering at the NewSUBARU synchrotron radiation facility. The data
show fine structures in the low-energy tail of the giant-dipole resonance; the
integrated strength of the fine structure below 18~MeV is intermediate among
the past measurements with bremsstrahlung and the positron annihilation
$\gamma$ rays. We compare the photoneutron emission data with the {\sf TALYS}
statistical model calculation implemented with the simple modified Lorentzian
model of $E1$ and $M1$ strengths. We also compare the total photoabsorption
cross sections for $^{13}$C with the shell model and antisymmetrized molecular
dynamics calculations as well as the statistical model calculation. We further
investigate the consistency between the present photoneutron emission and the
reverse $^{12}$C(n,$\gamma$) cross sections through their corresponding
astrophysical rate.
