Solar wind electrons, especially under conditions of relatively low speed flow, often can be represented as two bi-Maxwellian components, a cooler, more dense core (denoted by the subscript c) and a hotter, more tenuous halo. Solar wind observations from Ulysses between 1.5 and 2 AU further indicate that the &bgr; for electron core temperatures parallel to the background magnetic field, &bgr;∥c, has a distinct lower bound near 0.1. To seek the cause of this possible constraint, numerical solutions of the full Vlasov linear dispersion equation are used for four heat flux instabilities under a core/halo model with parameters representative of the solar wind near 1 AU. In this model the whistler heat flux instability is the growing mode of lowest threshold at most observed values of &bgr;∥c. As &bgr;∥c is decreased, however, the growth of this mode is reduced, so that at sufficiently small values of this parameter the Alfv¿n heat flux instability or the electron/ion acoustic instability becomes the fastest growing mode. The critical condition corresponding to this transition is calculated as a function of T∥c/Tp (where Tp is the proton temperature) and approximately corresponds to the observed constraint at &bgr;∥c≃0.1. The Alfv¿n and ion acoustic instabilities both resonate with core electrons; the hypothesis is proposed that core heating by these two modes at the critical condition establishes a lower bound on &bgr;∥c. ¿ 1998 American Geophysical Union