  Globular clusters are of paramount importance for testing theories of stellar
evolution and early galaxy formation. Strong evidence for multiple populations
of stars in globular clusters derives from observed abundance anomalies. A
puzzling example is the recently detected Mg-K anticorrelation in NGC 2419. We
perform Monte Carlo nuclear reaction network calculations to constrain the
temperature-density conditions that gave rise to the elemental abundances
observed in this elusive cluster. We find a correlation between stellar
temperature and density values that provide a satisfactory match between
simulated and observed abundances in NGC 2419 for all relevant elements (Mg,
Si, K, Ca, Sc, Ti, and V). Except at the highest densities ($\rho \gtrsim
10^8$~g/cm$^3$), the acceptable conditions range from $\approx$ $100$~MK at
$\approx$ $10^8$~g/cm$^3$ to $\approx$ $200$~MK at $\approx$
$10^{-4}$~g/cm$^3$. This result accounts for uncertainties in nuclear reaction
rates and variations in the assumed initial composition. We review hydrogen
burning sites and find that low-mass stars, AGB stars, massive stars, or
supermassive stars cannot account for the observed abundance anomalies in NGC
2419. Super-AGB stars could be viable candidates for the polluter stars if
stellar model parameters can be fine-tuned to produce higher temperatures.
Novae, either involving CO or ONe white dwarfs, could be interesting polluter
candidates, but a current lack of low-metallicity nova models precludes firmer
conclusions. We also discuss if additional constraints for the first-generation
polluters can be obtained by future measurements of oxygen, or by evolving
models of second-generation low-mass stars with a non-canonical initial
composition.
