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We investigate the thermal stability of nuclear burning on the accretingX-ray pulsar GRO J1744-28. The neutron star's dipolar magnetic field is<3\times 10^{11} G if persistent spin-up implies that the magnetospheric radiusis less than the co-rotation radius. After inferring the properties of theneutron star, we study the thermal stability of hydrogen/helium burning andshow that thermonuclear instabilities are unlikely causes of the hourly burstsseen at very high accretion rates. We then discuss how the stability of thethermonuclear burning depends on both the global accretion rate and the neutronstar's magnetic field strength. We emphasize that the appearance of theinstability (i.e., whether it looks like a Type I X-ray burst or a flarelasting a few minutes) will yield crucial information on the neutron star'ssurface magnetic field and the role of magnetic fields in convection. Wesuggest that a thermal instability in the accretion disk is the origin of thelong (~300 days) outburst and that the recurrence time of these outbursts is>50 years. We also discuss the nature of the binary and point out that avelocity measurement of the stellar companion (most likely a Roche-lobe fillinggiant with m_K>17) will constrain the neutron star mass.Comment: 19 pages, 3 PostScript figures, uses aaspp4.sty and epsfig.sty, to  appear in the Astrophysical Journa

Thermonuclear Burning on the Accreting X‐Ray Pulsar GRO J1744−28