$\beta$ Pictoris' inner disk in polarized light and new orbital parameters for $\beta$ Pictoris b

We present $H$-band observations of $\beta$ Pic with the Gemini Planet Imager's (GPI's) polarimetry mode that reveal the debris disk between ~0.3" (~6 AU) and ~1.7" (~33 AU), while simultaneously detecting $\beta$ Pic $b$. The polarized disk image was fit with a dust density model combined with a Henyey-Greenstein scattering phase function. The best fit model indicates a disk inclined to the line of sight ($\phi=85.27{\deg}^{+0.26}_{-0.19}$) with a position angle $\theta_{PA}=30.35{\deg}^{+0.29}_{-0.28}$ (slightly offset from the main outer disk, $\theta_{PA}\approx29{\deg}$), that extends from an inner disk radius of $23.6^{+0.9}_{-0.6}$ AU to well outside GPI's field of view. In addition, we present an updated orbit for $\beta$ Pic $b$ based on new astrometric measurements taken in GPI's spectroscopic mode spanning 14 months. The planet has a semi-major axis of $a=9.2^{+1.5}_{-0.4}$AU, with an eccentricity $e\leq 0.26$. The position angle of the ascending node is $\Omega=31.75{\deg}\pm0.15$, offset from both the outer main disk and the inner disk seen in the GPI image. The orbital fit constrains the stellar mass of $\beta$ Pic to $1.60\pm0.05 M_{\odot}$. Dynamical sculpting by $\beta$ Pic $b$ cannot easily account for the following three aspects of the inferred disk properties: 1) the modeled inner radius of the disk is farther out than expected if caused by $\beta$ Pic b; 2) the mutual inclination of the inner disk and $\beta$ Pic $b$ is $4{\deg}$, when it is expected to be closer to zero; and 3) the aspect ratio of the disk ($h_0 = 0.137^{+0.005}_{-0.006}$) is larger than expected from interactions with $\beta$ Pic $b$ or self-stirring by the disk's parent bodies.