HD 164604 c: a second giant planet on a 15-yr orbit and the constraint of the planet-planet mutual inclination

We report the discovery of a new massive giant planet, HD 164604 c ($a_c = 5.556_{-0.10}^{+0.093}$ au, $e_c = 0.196_{-0.078}^{+0.078}$ and $m_c = 9.5_{-1.25}^{+1.2}$ or $7.6_{-1.0}^{+1.0}\,M_{\rm Jup}$), orbiting a K3.5 dwarf, The result is based on the combined analysis of high-precision radial-velocity data, Hipparcos, and Gaia DR2 and DR3 astrometry. We refine the orbital parameters of the inner planet HD 164604 b to $a_b = 1.362_{-0.012}^{+0.012}$ au, $e_b = 0.479_{-0.021}^{+0.027}$, and $m_b = 13.2_{-1.5}^{+1.8}\,M_{\rm Jup}$ (or $8.8_{-1.5}^{+1.9}\,M_{\rm Jup}$). Depending on the two possible orbital orientations of HD 164604 c, the true mutual inclination between the two planets is $ψ_{bc}=5.0^{+3.7}_{-2.2}$$^\circ$ (prograde) or $162.1^{+7.1}_{-4.7}$$^\circ$ (retrograde). Long-term N-body integrations show that most orbits with the retrograde configuration remain dynamically stable for at least 10 Myr, while orbits with the prograde motion might rapidly evolve into chaos or lead to ejection. The retrograde architecture points to a violent dynamical history, possibly involving von Zeipel-Lidov-Kozai cycles or three-body scattering, while the prograde scenario might be consistent with coplanar and mild disk migration. Future Gaia DR4 astrometry will break the inclination degeneracy and distinguish between prograde and retrograde orbits for HD 164604 c.