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When radiation from a colder source arrives at a warmer surface there is some debate about what happens next. To make the question more concrete lets say that the colder source is at temperature 288K. The warmer surface is at 888K and has emissivity of 1. 3 possibilities

  1. We ignore such radiation because it cannot happen.
  2. The radiation is subtracted from the much larger radiation of every wavelength leaving the hotter surface.
  3. The radiation is fully absorbed and its effect is to be re radiated at characteristic temperature of 888K (plus infinitesimally small T increase due to radiation absorption).

I would have thought that 2. and 3 are more plausible than 1.

Both 2 and 3 satisfy the Stephan Boltzmann equation. 3 however seems to imply that the radiation from colder object is transformed into much higher quality radiation and a possible second law of thermodynamics infringement.

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Obviously, the option 3 is the right option. There is no violation of the 2nd law, since the emission rate is higher than the absorption rate for the hotter body.

Consider a closed system of two objects. One is hotter and the other is colder. What will happen? Both will radiate energy and both will absorb energy from the incident radiation. The hotter object will radiate energy at a higher rate than it will absorb energy. Opposite is for the colder object. As a result after sufficient time both will come to same temperature and thermal equilibrium will be established. In this condition both object will radiate and absorb energy at the same rate.

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The option three is valid -- the outgoing radiation is only dependent on the radiating object. Second law of thermodynamics is not violated since it must be applied to the whole system (and so to the net radiative exchange).

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The hotter object absorbs external radiation as well as it absorbs its own radiation inside the body before it reaches its surface.

The heat loss of a hot object (radiated power) is determined with the object temperature but the rate of cooling down (if there is cooling down $dT/dt < 0$) is smaller in case of additional external source of energy in form of radiation coming from exterior.

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