Heating rates of the approximation for cores embedded in spherical filaments, given as function of the extinction A_V(0) through the filament center. The assumed external pressure is p_ext/k = 2  ×  10⁴   K/cm³. The heating rates correspond to the considered ISRFs without (black curves) and with dust emission from the ISM (grey lines). The heating rates obtained for an ISRF without dust emission from the Galactic disc are compared with the accurate ray-tracing calculation (symbols). The total heating rates (solid lines, filled diamonds) are separated into the several components of the radiation: attenuated external radiation field, which includes the heating by scattered radiation (long dashed lines, open squares); heating by dust emission from the filaments (dashed-three dotted curves, open diamonds); heating by dust emission from the cores (dashed-single dotted curves, open triangles); and dust emission from the Galactic disc (short dashed curve).