met, even neglecting hyperfine effects[^1] [@hunt99]. In order to see the density conditions giving rise to population inversion, we solved the steady-state equations for the population of the $J=0,1,\dots,15$ levels of [$\mathrm{HC_3N}$]{}, including collisions with [$\rm H_2$]{}(densities ranging from $10^2$ to $10^6 \rm\; cm^{-3}$), a black-body photon bath at 2.7 K, in the optically thin approximation, [@goldsmith72] : $$\begin{aligned}
\label{eq:ss}
    \frac{\textmd{d}n_i}{\textmd{d}t}=0&=&
    +\sum_{j\neq i}n_j\,\left[ A_{ji} +
    B_{ji} \;n_\gamma\left(\nu_{ji}\right)+k_{ji}\;
    n_{\rm H_2}\right] \nonumber \\
    & & - n_i\,\sum_{j\neq i}
    \left[A_{ij}+B_{ij}\,n_\gamma\left(\nu_{ij}\right)+k_{ij}\; n_{\rm
    H_2}\right]\end{aligned}$$ where $i,j$ are the levels, $n_\gamma$ is the photon density at temperature $T_\gamma$ and $n_{\rm H_2} $ is the hydrogen density at kinetic temperature $T_{\rm H_2}$. Figure \[fig:invers\] shows the results at $T_{\rm H_2} = 40$ K. The lines show the population per sub-levels $\left|J_1, m_{J_1}\right>$. For a consequent range of [$\rm H_2$]{}densities, $10^4\lesssim n_{\rm H_2}
\lesssim 10^6$, population inversion does occur, for $0\leq
J_1\leq 2, 3, 4$. Our new rates are expected to improve the interpretation of the lowest-lying lines of [$\mathrm{HC_3N}$]{}, especially so in the 9 - 20 GHz regions (cm-mm waves), see for example @walms86 [@takano98; @hunt99], and @kal04 for a recent study. Moreover, from the knowledge of both collision coefficients $k_{ij}$ and Einstein coefficients $A_{ij}$, it is 