If Melatonin's half life is only 20 min (or up to 2 hours), why does drowsiness persist?

ok I’m no math genius here but from a chemistry formula I have to know,

Mass(after time t) = Mass(original) X (.5)^(t/(half life))

so if I took 1mg of melatonin 12 hours ago

m(in kg) = (10^-6)(.5)^(720/20)

=

10^-6 (.5)^36 = (10^-6)(1.5 X 10^-11)

or 1.5 X 10 ^-17 kg

or about 1.5 X 10^-11 mg

or basically nothing

if 2 hours
10^-6 (.5)^6
10^-6 (1.6 X 10^-2) or 1.6X 10^-8 kg or 1.6 X 10^-2 mg which is still very little.

Unless the amount of melatonin necessary to have any effect is smaller than the true amount which I expect to be somewhere between those 2 extremes?

Melatonin doesn’t cause drowsiness directly. Its job is to trigger the onset of sleep. Once the trigger has been pulled its job is done.

Any complete answer will contain phrases like this:
Involvement of the nitric oxide cascade in melatonin-induced inhibition of long-term potentiation at hippocampal CA1 synapses.
The short and sweet of it is that melatonin acts to trigger other processes that operate on timescales longer than melatonin’s half-life.

Wait. Melatonin actually does work to help people go to sleep? I thought the evidence was inconclusive.

Last I looked for it, that was true. However, melatonin does trigger some long lasting biochemical effects.

Internally generated melatonin is known to mediate the synchronisation of the circadian rhythm. How useful taking additional melatonin is in trying to modify the rhythm, or resynchronise it, is a different problem.

Neurotransmitters can act as catalysts, too.