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.

💡 Research Summary

This paper reports the discovery of a second massive giant planet, HD 164604 c, orbiting the K3.5 dwarf HD 164604, and provides a detailed dynamical analysis of the two‑planet system. The authors combine 21 years of high‑precision radial‑velocity (RV) measurements from the MIKE and PFS spectrographs with absolute astrometry from the re‑processed Hipparcos catalog and Gaia Data Releases 2 and 3. By jointly fitting the RVs, Hipparcos–Gaia proper‑motion anomalies (PMA), and Gaia’s two‑body orbital solutions, they derive a full three‑dimensional orbit for the outer companion: semi‑major axis a_c = 5.556 ± 0.10 au, eccentricity e_c = 0.196 ± 0.078, and a true mass of 9.5 ± 1.2 M_Jup (or 7.6 ± 1.0 M_Jup depending on the inclination solution).

The inner planet, previously known from a 650‑day RV signal, is refined to a_b = 1.362 ± 0.012 au, e_b = 0.479 ± 0.024, and a true mass of 13.2 ± 1.7 M_Jup (or 8.8 ± 1.7 M_Jup). The joint astrometric analysis reveals a degeneracy in the orbital orientation of HD 164604 c, leading to two possible mutual inclinations between the planets: ψ_bc = 5.0° + 3.7°/‑2.2° for a prograde configuration, or ψ_bc = 162.1° + 7.1°/‑4.7° for a retrograde configuration.

To assess the long‑term stability of both configurations, the authors perform N‑body integrations with MERCURY6 over 10 Myr. The retrograde configuration remains stable in the vast majority of simulated realizations, whereas the prograde configuration often becomes chaotic or results in the ejection of the outer planet within a few Myr. This stark contrast suggests fundamentally different dynamical histories. A retrograde architecture points to a violent past, possibly involving von Zeipel–Lidov–Kozai cycles induced by a distant perturber or three‑body scattering events that pumped the mutual inclination to high values. In contrast, a prograde, near‑coplanar system could be the outcome of smooth disk‑driven migration and modest dynamical interactions.

The paper also presents a thorough stellar characterization: spectral energy distribution fitting and isochrone analysis give M_* = 0.77 ± 0.02 M_⊙, R_* = 0.78 ± 0.02 R_⊙, and an age of ~10 Gyr (with gyrochronology suggesting a younger component). A rotation period of 19 ± 3 days is derived from TESS photometry. Periodogram analyses of the RVs, activity indices (S‑index, Hα), and ASAS‑SN photometry show no significant stellar activity signals at the planetary periods, justifying a white‑noise RV model.

The authors emphasize that upcoming Gaia Data Release 4, with its improved astrometric precision, will directly measure the inclination of HD 164604 c and resolve the prograde/retrograde degeneracy. This will enable a decisive test of the proposed formation pathways and provide a benchmark case for studying mutual inclinations in non‑transiting multi‑planet systems.

In summary, the study introduces HD 164604 c as a long‑period (~15 yr) massive giant planet, refines the inner planet’s orbit, quantifies the possible mutual inclination, demonstrates the dynamical feasibility of a retrograde configuration, and outlines how future Gaia astrometry will conclusively determine the system’s three‑dimensional architecture.