Solar wind electrons are often observed to consist of two distinguishable c
omponents, a thermal, more dense core and a suprathermal, less dense halo.
In this core/halo model linear Vlasov theory for the whistler heat flux ins
tability predicts dimensionless heat flux thresholds which decrease as the
electron core beta, <(beta)over tilde>parallel to(c), increases. It has bee
n proposed that this theoretical threshold corresponds to an observable upp
er bound on the electron heat flux. Linear theory also predicts that there
is a critical value of <(beta)over tilde>parallel to c below which the whis
tler heat flux instability does not have appreciable growth in the solar wi
nd; there is another suggestion that this corresponds to an observable lowe
r bound on <(beta)over tilde>parallel to(c). These two proposals are examin
ed by comparison of linear theory and data from the initial in-ecliptic pha
se of the Ulysses mission. The instability threshold does provide a statist
ical constraint on observed solar wind heat fluxes, and the critical <(beta
)over tilde>parallel to(c) Of theory is not inconsistent with a statistical
lower bound on the observations of that parameter.