An evolutionary model for SAX J1808.4-3658

An evolutionary scenario to explain the transient nature and short total duration of the X-ray burst of SAX J1808.4 -- 3658 is proposed. An optical companion of the neutron star (a ``turn-off'' Main - Sequence star) fills its Roche lobe at the orbital period ($P_{orb}$) $\sim$ 19 hours. During the initial high mass--transfer phase when the neutron star is a persistent X-ray source, the neutron star is spun up to a millisecond period. Due to its chemical composition gradient, the secondary does not become fully convective when its mass decreases below 0.3 $\msun$, hence a magnetic braking remains an effective mechanism to remove orbital angular momentum and the system evolves with Roche - lobe overflow towards a short orbital period. Near an orbital period of two hours the mass transfer rate becomes so small ($\sim$ $10^{-11}\msun$/yr) that the system can not continue to be observed as a persistent X-ray source. During further Roche - lobe filling evolution deep mixing allows the surface of secondary to become more and more helium rich. Since the accreted matter is helium rich, it is easy to explain observed short total duration of the burst . This evolutionary picture suggest that radio emission can be observed only at shorter wavelength's. Our model predicts a faster orbital period decay than expected if the orbital evolution is driven only by gravitational wave radiation