We present the first VLTI/GRAVITY observations at R$_\lambda \sim 4000$ of $\beta$ Pic b. These four high S/N ($\sim$20) K-band spectra conserve both the pseudo-continuum and molecular absorption patterns. We analyze them with four self-consistent forward model grids (Exo-REM, ATMO, BT-Settl, Sonora) exploring $T_{ \mathrm{ eff}}$, log(g), metallicity, C/O, and $^{ 12}$CO/$^{ 13}$CO ratio. We also upgrade our forward modeling code ForMoSA to account for the data multi-modality and combine the GRAVITY epochs with published 1-5 $\mu$m photometry, low- to medium-resolution spectra (0.9-7 $\mu$m), and high-resolution echelle spectra (2.1-5.2 $\mu$m). Sonora and Exo-REM are statistically preferred. Exo-REM yields $T_{ \mathrm{ eff}}$ $=1607.45^{ +4.85}_{ -6.20}$ K and log(g) $=4.46^{ +0.02}_{ -0.04}$ dex from GRAVITY alone, and $T_{ \mathrm{ eff}}$ $=1502.74^{ +2.32}_{ -2.14}$ K and log(g) $=4.00\pm0.01$ dex when including all datasets. Archival data significantly affect the retrieved parameters. C/O remains solar ($0.552^{ +0.003}_{ -0.002}$) while [M/H] reaches super-solar values (0.50$\pm$0.01). We report the first tentative constraint on log($^{ 12}$CO/$^{ 13}$CO) $\sim$1.12, though this remains inconclusive due to telluric residuals. Additionally, we estimate the luminosity to be log(L/L$_\odot$) $=-4.01^{ +0.04}_{ -0.05}$, implying a heavy-element content of up to $\sim$5% (20-80 M$_\oplus$) given the system age and dynamical mass measurements. Access to both continuum and molecular lines at K-band significantly impacts the metallicity, possibly owing to collision-induced absorption shaping the continuum. Echelle spectra do not dominate the final fit with respect to lower resolution data. Future multi-modal frameworks should include weighting schemes reflecting bandwidth and central wavelength coverage.
Categories: astro-ph.EP