Carry it or return to it
A battery you carry gets heavier the longer you must run. A decay source does not, its mass is fixed and its output is set by half-life. Somewhere they cross, and where they cross decides what every autonomous machine on earth is built around.
Predict first
Make the robot twice as powerful. Everything it does needs double the watts. What happens to the point where a decay source becomes lighter than a battery?
The machine
What it carries
Strategy
Decay source modelled as Sr-90 at 930 W thermal per kg with
5% thermoelectric conversion, so 46.5 W electrical per kg. Source mass
only, no shield, structure or conversion hardware, all of which make it worse.
The derivation
Battery mass m = P·t / e rises with duration. Decay mass
m = P / p does not. Setting them equal gives t* = e / p, and P cancels. The crossover is a property of the two energy technologies, not of the
machine. Figures are representative; this is a design bench, not a datasheet.