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The radiation resistance is not easy to calculate, because it depends
critically on how well coupled the oscillator is to the radiation fields,
i.e. on how good an antenna the circuit is. This will depend, in turn on,
the frequency of the oscillation. If some part of the circuit has a
dimension of order c/f, it will radiate reasonably well. Otherwise, most
of the energy will indeed end up as heat.
If you are looking for some universal radiation resistance that is fixed
once R, L and C are specified, there is none! The radiation resistance (and
the fraction of starting energy that is radiated) are determined by how
good an antenna your circuit is! This depends on the geometry; bending a
wire will change it!