On the stability of non-isothermal Bonnor-Ebert spheres. II. The effect
  of gas temperature on the stability
  Aims. We investigate the stability of non-isothermal Bonnor-Ebert spheres
with a model that includes a self-consistent calculation of the gas
temperature. This way we can discard the assumption of equality between the
dust and gas temperatures, and study the stability as the gas temperature
changes with chemical evolution of the gas.
  Methods. We use a gas-grain chemical model including a time-dependent
treatment of depletion onto grain surfaces, which strongly influences the gas
temperature as the main coolant, CO, depletes from the gas. Dust and gas
temperatures are solved with radiative transfer. For comparison with previous
work, we assume that the cores are deeply embedded in a larger external
structure, corresponding to visual extinction $A_{\rm V}^{\rm ext}=10$ mag.
  Results. We find that the critical non-dimensional radius $\xi_1$ derived
here is similar to our previous work where we assumed $T_{\rm dust}=T_{\rm
gas}$; the $\xi_1$ values lie below the isothermal critical value
$\xi_0\sim6.45$, but the difference is less than 10%. Chemical evolution does
not affect notably the stability condition of low-mass cores (<0.75 $M_\odot$).
For higher masses the decrease of cooling owing to CO depletion causes
substantial temporal changes in the temperature and density profiles of the
cores. In the mass range 1-2 $M_\odot$ , $\xi_1$ decreases with chemical
evolution, whereas above 3 $M_\odot$ , $\xi_1$ instead increases. We also find
that decreasing $A_{\rm V}^{\rm ext}$ increases the gas temperature especially
when the gas is chemically old, causing $\xi_1$ to increase with respect to
models with higher $A_{\rm V}^{\rm ext}$. The derived $\xi_1$ values are close
to $\xi_0$. The density contrast between the core center and edge varies
between 8 to 16 depending on core mass and the chemical age of the gas,
compared to the constant value $\sim$ 14.1 for the isothermal BES.
