A recent reanalysis of Einstein data, and new ROSAT observations, have
revealed the presence of at least two components in the X-ray spectra
of X-ray faint early-type galaxies: a relatively hard component (kT >
1.5 keV), and a very soft component (kT approximately 0.2-0.3 keV). I
n this paper we address the problem of the nature of the very soft com
ponent, and whether it can be due to a hot interstellar medium (ISM),
or is most likely originated by the collective emission of very soft s
tellar sources. To this purpose, hydrodynamical evolutionary sequences
for the secular behavior of gas flows in ellipticals have been perfor
med, varying the Type Ia supernovae rate of explosion, and the dark ma
tter amount and distribution. The results are compared with the observ
ational X-ray data: the average Einstein spectrum for six X-ray faint
early-type galaxies (among which are NG-C 4365 and NGC 4697), and the
spectrum obtained by the ROSAT pointed observation of NGC 4365. The ve
ry soft component could be entirely explained with a hot ISM only in g
alaxies such as NGC 4697, i.e., when the depth of the potential well-o
n which the average ISM temperature strongly depends-is quite shallow;
in NGC 4365 a diffuse hot ISM would have a temperature larger than th
at of the very soft component, because of the deeper potential well. S
o, in NGC 4365 the softest contribution to the X-ray emission comes ce
rtainly from stellar sources. As stellar soft X-ray emitters, we consi
der late-type stellar coronae, supersoft sources such as those discove
red by ROSAT in the Magellanic Clouds and M31, and RS CVn systems. All
these candidates can be substantial contributors to the very soft emi
ssion, though none of them, taken separately, plausibly accounts entir
ely for its properties. We finally present a model for the X-ray emiss
ion of NGC 4365, to reproduce in detail the results of the ROSAT point
ed observation, including PSPC spectrum and radial surface brightness
distribution. The present data may suggest that the X-ray surface brig
htness is more extended than the optical profile. In this case, a stra
ightforward explanation in terms of stellar sources could be not satis
factory. The available data can be better explained with three differe
nt contributions: a very soft component of stellar origin, a hard comp
onent from X-ray binaries, and a approximately 0.6 keV hot ISM. The la
tter can explain the extended X-ray surface brightness profile, if the
galaxy has a dark-to-luminous mass ratio of 9, with the dark matter v
ery broadly distributed, and a SN Ia explosion rate of approximately 0
.6 the Tammann rate.