@article{2024A&A...684A..40P,
 abstract = {\ Aims: We explored the softness parameter in the infrared,
\ensuremathη$_IR$$^′$, whose main purpose is the
characterisation of the hardness of the incident ionising
radiation in emission-line nebulae. This parameter is obtained
from the combination of mid-infrared wavelength range
transitions corresponding to consecutive ionisation stages in
star-forming regions. \ Methods: We compiled
observational data from a sample of star-forming galaxies
(SFGs), including luminous and ultraluminous infrared galaxies
(LIRGs and ULIRGs), to study the
\ensuremathη$_IR$$^′$ and its equivalent expression
in two dimensions, the softness diagram. We compared them with
predictions from photoionisation models to determine the shape
of the ionising continuum energy distribution in each case. We
also used the measured emission-line ratios as input for
HCMISTRY-TEFF-IR, a code that performs a Bayesian-like
comparison with photoionisation model predictions in order to
quantify the equivalent effective temperature (T$_*$) and the
ionisation parameter. \ Results: We found similar
average values within the errors of
\ensuremathη$_IR$$^′$ in (U)LIRGs
(\ensuremath-0.57) in the rest of the SFGs
(\ensuremath-0.51), which could be interpreted as indicative
of a similar incident radiation field. This result is confirmed
from the analysis using HCM-TEFF-IR, which simultaneously points
to a slightly lower, although similar within the errors, T$_*$
scale for (U)LIRGs, even when a higher dust-to-gas mass ratio is
considered in the models for these objects. These derived
T$_*$ values are compatible with the ionisation from massive
stars, without any need of harder ionising sources, both for
(U)LIRGs and the rest of the SFGs. However, the derived T$_*$
in (U)LIRGs do not show any correlation with metallicity. This
could be interpreted as a sign that their similar average
T$_*$ values are due to the attenuation of the energetic
incident flux from massive stars by the heated dust mixed with
the gas. This is supported by the known very large amounts of
small grains associated with the very high star formation rates
measured in galaxies of this type.},
 adsnote = {Provided by the SAO/NASA Astrophysics Data System},
 adsurl = {https://ui.adsabs.harvard.edu/abs/2024A&A...684A..40P},
 archiveprefix = {arXiv},
 author = {Pérez-Montero, E. and Fernández-Ontiveros, J.~A. and Pérez-Dı́az, B. and V\ĺchez, J.~M. and Kumari, N. and Amor\',́ R.},
 doi = {10.1051/0004-6361/202348089},
 eid = {A40},
 eprint = {2401.09765},
 journal = {åp},
 keywords = {galaxies: abundances, galaxies: star formation, galaxies: stellar content, Astrophysics - Astrophysics of Galaxies},
 month = {April},
 pages = {A40},
 primaryclass = {astro-ph.GA},
 title = {Exploring the hardness of the ionising radiation with the infrared softness diagram. I. Similar effective temperature scales for starbursts and (ultra)luminous infrared galaxies},
 volume = {684},
 year = {2024}
}