Formation of the warped debris disc around $\beta$ Pictoris

In light of the recent confirmation of an eccentric orbit giant planet, $\beta$ Pic c, I revisit the formation and evolution of the warped debris disc in the system. $\beta$ Pic c is interior to $\beta$ Pic b, and the debris disc is exterior to both planets. Previous $N$-body simulations have shown that $\beta$ Pic b is responsible for exciting the inclination of the debris disc. With hydrodynamical simulations, I model a protoplanetary gas disc misaligned with the planets. I find that the gas disc does not exhibit significant long lasting inclination excitation from the planets even for the observed disc size. The warp that is excited by the planets propagates through the entire disc with a timescale much less than the gas disc lifetime. Therefore, the observed warp in the debris disc must be produced after the gas disc has dispersed. With analytical secular theory calculations, I show that two secular resonances are exterior to $\beta$ Pic b, located at $\sim 20\, \rm au$ and $\sim 25\, \rm au$. This agrees with my $N$-body simulations that show that these secular resonances shape the inner edge of the $\beta$ Pic debris disc at a radius that agrees with observations.