Temperature dependence of $p$-wave contacts in a harmonically trapped Fermi gas

We study the dependence of the $p$-wave contact on the Fermi temperature $T_F$ and reduced temperature $T\mathrm{/}{T}_F$ based on the number of closed-channel molecules. From the anisotropic pattern of dissociated molecules, we resolve the narrow $m=0$ and $m=\pm 1$ dipolar splitting of the $p$-wave Feshbach resonance in ${}^6$Li, enabling the independent determination of the contact for all three $m$ components. For each component, we identify a near-resonant scaling with $\sqrt{T_F}$, indicating the contribution of the normalized effective range $k_F{R}_{\mathrm{e}}$. In addition, we show that the peak contacts observed near resonance increase as $T\mathrm{/}{T}_F$ is lowered, a trend that is accurately captured by estimates based on the second-order virial expansion. Our results, together with estimates from the $p$-wave virial expansion, provide a route toward a complete understanding of the thermodynamics of resonantly enhanced $p$-wave Fermi gases.