convective envelope, such as $\gamma$Dor stars. In the limit of a non-rotating star, where $s = 0$, this expression reduces to $$P_{nl} =
  \frac{\Pi_0}{\sqrt{l(l+1)}}(n + \alpha_{g}),\label{eq:5}$$ in agreement with Eq.(\[eq:1\]) and as derived by @Tassoul1980. For each of the models in our grid, we computed the $l=1$ and $l=2$ mode frequencies for radial orders ranging from 5 to 120.

  parameter                               begin    end    step size
  -------------------------------------- ------- ------- -----------
  mass $M$ \[$M_\odot$\]                   1.4     2.0      0.05
  metallicity $Z$                         0.010   0.018     0.004
  exp. core overshooting $f_{ov}$         0.001   0.03     0.0075
  step core overshooting $\alpha_{ov}$    0.01     0.3      0.075
  initial hydrogen abundance $X_i$        0.69    0.73      0.02

  : \[tab:mesa\] The parameter values of the computed grid of 1170 MESA evolutionary tracks, consisting of some 900,000 models.

![\[fig:Spacing\_dist\]The distributions of the asymptotic period spacing values $\Delta\Pi_l$ for spherical degree $l=1$ and $2$, computed for the MESA evolution tracks with the input parameters provided in Table \[tab:mesa\]. For the computation of the distributions the ages and evolution rates of the stellar models were taken into account.](Spacing_dist_grey.pdf){width="88mm"}

![\[fig:kernels\] The Brunt-Väisälä frequency $N$ (*top*) and the rotational kernel $K_{nl}$ (*bottom*) for the lowest- and highest-order mode of the stellar model discussed in Sec.\[subsec:synthdata\]. The inset shows a zoom of $K_{nl}$. Both 