Unlocking HST's Stellar Treasure Trove: Stellar Activity Minima for HAT-P-11 Offer Prime Windows for Transmission Spectroscopy

HAT-P-11 is a well-studied, active K dwarf hosting an eccentric, misaligned transiting sub-Neptune. As part of the HST Stellar Treasure Trove program (HST-AR-17551), we analyze absolutely calibrated out-of-transit \HST{} spectra from $\mathtt{\text{STIS}}$ and $\mathtt{\text{WFC3}}$ across the $\text{textsc}G430L$, $\text{textsc}G750L$, $\text{textsc}G102$, and $\text{textsc}G141$ bandpasses to constrain the surface heterogeneity of HAT-P-11 and its impact on transmission spectroscopy. Grid-based spectral retrievals using NewEra $\mathtt{\text{PHOENIX}}$ models robustly favor two-component photospheres in the $\mathtt{\text{WFC3}}$ G102 and G141 data, with a $\sim 4950$,K photospheric component and a cooler ($\sim$3400,K) component covering 26{–}33% of the stellar disk. By contrast, retrievals on the $\mathtt{\text{STIS}}$ optical spectra do not yield a satisfactory fit, reflecting current limitations of stellar atmosphere models in the optical regime compared to the \HST{} observational precision. We contextualize these results using long-term photometric monitoring and chromospheric activity indices. The inferred high spot covering fractions are broadly consistent with the elevated photometric variability observed during the $\mathit{\text{Kepler}}$ era ($f_{\mathrm{s}\mathrm{p}\mathrm{o}\mathrm{t}}$$\sim$10–20%) but are in tension with the much lower rotational amplitudes observed from TESS in the mid 2020s ($f_{\mathrm{s}\mathrm{p}\mathrm{o}\mathrm{t}}$$\sim$1–10%). This secular decline in variability is mirrored by a $\sim$20% decrease in the Ca,$\text{textsc}ii$ H&K index. These results imply that HAT-P-11 undergoes comparatively quiescent phases that offer more favorable windows for atmospheric characterization, which serendipitously coincided with some of the recent JWST observations. More generally, our study demonstrates that multi-epoch, space-based stellar spectra provides a physically grounded pathway for mitigating stellar contamination in high-precision transmission spectra in the JWST era.