# How do I determine the Luminosity with a half-life decay?

I know there is a proportionality between the luminosity and $$\frac{dN}{dt}$$, but I am not sure why they are proportional and why there is a need for an initial mass to compute the luminosity if the half-life equation is:

$$N(t) = N_{0} \cdot (\frac{1}{2})^{t/t_{half-life}^{1/2}}$$

and how does this apply to supernovae?

I tried to use Cobalt decay to help me understand, but I do not see how to progress from:

$$\frac{dN}{dt} = \frac{N_{0}}{N(t)} \cdot (\frac{1}{2})^{t/t_{half-life}^{1/2}}$$

• The luminosity is proportional to the number of atoms decaying per unit time, and obviously depends on how many atoms there are at the start (the $N_{0}$ in your equation). Dec 9, 2022 at 12:26

You need an indication of how much mass is decaying, in order to have an idea of how much energy will be released. That $$N_0$$ indicates quantity of substance, which you can express in grams, moles, number of atoms or whatever units you prefer.