He, of less astrophysical importance (\[He\]/\[H\]$\sim 0.1$), only quantum calculations were performed and were limited to the low temperature regime ($T$=5$-$20 K and $J_1<10$).

Rotational inelastic cross sections with <span style="font-variant:small-caps;">Molscat</span> {#par:molscat}
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All calculations were made using the rigid rotor approximation, with rotational constants $B_{\rm HC_3N}=0.151739$ cm$^{-1}$ and $B_{\rm
H_2}=60.853$ cm$^{-1}$, using the <span style="font-variant:small-caps;">Molscat</span> code [@molscat]. All quantum calculations for [$\mathrm{HC_3N}$]{}– ortho-[$\rm H_2$]{}were performed with $J_{\rm H_2}\equiv J_2=1$. Calculations for [$\mathrm{HC_3N}$]{}– para-[$\rm H_2$]{}were performed with $J_2=0$. We checked at $E_{\rm
tot}=E_{\rm coll}+E_{\rm rot} = 30 \,\rm cm^{-1}$ that the inclusion of the closed $J_2=2$ channel led to negligible effects.

The energy grid was adjusted to reproduce all the details of the resonances, as they are essential to calculate the rates with high confidence [@dubernet02; @dubernet03; @wernli06]. The energy grid and the quantum methods used are detailed in table \[tab:param\]. Using this grid, we calculated the whole resonance structure of all the transitions up to $J_1=15$ for the [$\mathrm{HC_3N}$]{}– para-[$\rm H_2$]{}collisions. At least 10 closed channels were included at each energy to fully converge the [$\mathrm{HC_3N}$]{}rotational basis. We used the hybrid log-derivative/Airy propagator [@alexander87]. We increased the parameter <span style="font-variant:small-caps;">STEPS</span> at the lowest energies to constrain the step length 