Rotational Spectroscopy as a Tool to Study Vibration-Rotation Interaction: Investigations of $^{13}$CH$_3$CN and CH$_3$$^{13}$CN up to $v_8 = 2$ and a Search for $v_8 = 2$ Transitions toward Sagittarius B2(N)
Methyl cyanide, CH$_3$CN, is present in diverse regions in space, in particular in the warm parts of star-forming regions where it is a common molecule. Rotational transitions of $^{13}$CH$_3$CN and CH$_3$$^{13}$CN in their $v_8 = 1$ lowest excited vibrational states ($E_{\rm vib} \approx 520$ K) are quite prominent in Sagittarius B2(N). In order to be able to search for transitions of the next higher vibrational state $v_8 = 2$, we recorded spectra of samples enriched in $^{13}$CH$_3$CN and CH$_3$$^{13}$CN up to $v_8 = 2$ in the 35 to 1091~GHz region and reinvestigated existing spectra of CH$_3$CN in its natural isotopic composition between 1085 and 1200 GHz. Perturbations caused by near-degeneracies in $K = 4$ of $v_8 = 2^0$ and $K = 2$ of $v_8 = 2^{-2}$ yielded accurate information on the energy spacing of 22.93 and 21.79 cm$^{-1}$ between the $l$-components of $^{13}$CH$_3$CN and CH$_3$$^{13}$CN, respectively. Fermi-type interaction between $K = 13$ and 14 of $v_8 = 1^{-1}$ and $v_8 = 2^{+2}$ probe the energy differences between the two states of both isotopomers. In addition, a $\Delta K \pm2$, $\Delta l \mp1$ interaction between the ground vibrational state of $^{13}$CH$_3$CN and $v_8 = 1^{+1}$ provides information on their energy spacing. Furthermore, we obtained improved or extended ground state rotational transition frequencies of $^{13}$CH$_3$$^{13}$CN and extensive data for $^{13}$CH$_3$C$^{15}$N and CH$_3$$^{13}$C$^{15}$N. Finally, we report the results of our search for transitions of $^{13}$CH$_3$CN and CH$_3$$^{13}$CN in their $v_8 = 2$ states toward Sagittarius B2(N).