Ratio of the power in the sidebands to the total power in conventional AM
this is John BG jgb2012@sky.com

In AM modulation, as shown in

https://en.wikipedia.org/wiki/Amplitude_modulation#Modulation_index

the modulation index is the ratio between the amplitude of the modulated and the amplitude of the carrier.

So the sought modulation index $a=\frac{1}{Ac}$

modulating signal power: $\frac{1}{2}$

carrier power: $\frac{Ac^2}{2}$

side bands to total power ratio:   $\frac{1}{(1+\frac{Ac^2}{2})}$

solving:  $\frac{1}{(1+\frac{Ac^2}{2})} =\frac{1}{3}$ gives $Ac=2$

so the answer is $a=\frac{1}{2}$