The thermal emissivity of crystalline silicon photovoltaic (PV) solar cells plays a role in determining the operating temperature of a solar cell. To elucidate the physical origin of thermal emissivity, we have made an experimental measurement of the full radiative spectrum of the crystalline silicon (c-Si) solar cell, which includes both absorption in the ultraviolet to near-infrared range and emission in the mid-infrared. Using optical modelling, we have identified the origin of radiative emissivity in both encapsulated and unencapsulated solar cells. We find that both encapsulated and unencapsulated c-Si solar cells are good radiative emitters but achieve this through different effects. The emissivity of an unencapsulated c-Si solar cell is determined to be 75% in the MIR range, and is dominated by free-carrier emission in the highly doped emitter and back surface field layers; both effects are greatly augmented through the enhanced optical outcoupling arising from the front surface texture. An encapsulated glass-covered cell has an average emissivity around 90% on the MIR, and dips to 70% at 10 pm and is dominated by the emissivity of the cover glass. These findings serve to illustrate the opportunity for optimising the emissivity of c-Si based collectors, either in conventional c-Si PV modules where high emissivity and low-temperature operation is desirable, or in hybrid PV-thermal collectors where low emissivity enables a higher thermal output to be achieved.