A B S T R A C TThe evolution of helium stars with masses of 1:5 -6:7 M ( in binary systems with a 1.4-M ( neutron star companion is presented. Such systems are assumed to be the remnants of Be/X-ray binaries with B-star masses in the range 8 -20 M ( which underwent a case B or case C mass transfer and survived the common-envelope and spiral-in process. The orbital period is chosen such that the helium star fills its Roche lobe before the ignition of carbon in the centre. We distinguish case BA (in which mass transfer is initiated during helium core burning) from case BB (onset of Roche-lobe overflow occurs after helium core burning is terminated, but before the ignition of carbon). We found that the remnants of case BA mass transfer from 1:5 -2:9 M ( helium stars are heavy CO white dwarfs. This implies that a star initially as massive as 12 M ( is able to become a white dwarf. CO white dwarfs are also produced from case BB mass transfer from 1:5-1:8 M ( helium stars, while ONe white dwarfs are formed from 2:1-2:5 M ( helium stars. Case BB mass transfer from more massive helium stars with a neutron star companion will produce a double neutron star binary. We are able to distinguish the progenitors of type Ib supernovae (as the high-mass helium stars or systems in wide orbits) from those of type Ic supernovae (as the lower-mass helium stars or systems in close orbits). Finally, we derive a 'zone of avoidance' in the helium star mass versus initial orbital period diagram for producing neutron stars from helium stars.The total change in orbital angular momentum is assumed to be affected by gravitational wave radiation and by the loss of mass with angular momentum from the system, i.e.where the rate of change of orbital angular momentum arising from gravitational wave radiation is given by (Landau & Lifshitz 1958)

The double pulsar J0737–3039 has become an important astrophysical laboratory for testing fundamental physics. Here we demonstrate that the low measured mass of Pulsar B can be used to constrain the equation of state of neutron star matter under the assumption that it formed in an electron‐capture supernova. We show that the observed orbital parameters as well as the likely evolutionary history of the system support such a hypothesis, and discuss future refinements that will improve the constraints this test may provide.