  Classical novae are thermonuclear explosions in stellar binary systems, and
important sources of $^{26}$Al and $^{22}$Na. While gamma rays from the decay
of the former radioisotope have been observed throughout the Galaxy, $^{22}$Na
remains untraceable. The half-life of $^{22}$Na (2.6 yr) would allow the
observation of its 1.275 MeV gamma-ray line from a cosmic source. However, the
prediction of such an observation requires good knowledge of the nuclear
reactions involved in the production and destruction of this nucleus. The
$^{22}$Na($p,\gamma$)$^{23}$Mg reaction remains the only source of large
uncertainty about the amount of $^{22}$Na ejected. Its rate is dominated by a
single resonance on the short-lived state at 7785.0(7) keV in $^{23}$Mg. In the
present work, a combined analysis of particle-particle correlations and
velocity-difference profiles is proposed to measure femtosecond nuclear
lifetimes. The application of this novel method to the study of the $^{23}$Mg
states, combining magnetic and highly-segmented tracking gamma-ray
spectrometers, places strong limits on the amount of $^{22}$Na produced in
novae, explains its non-observation to date in gamma rays (flux < 2.5x$10^{-4}$
ph/(cm$^2$s)), and constrains its detectability with future space-borne
observatories.
