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

TL;DR

This work addresses how to measure true mutual inclinations in non-transiting multi-planet systems by combining high-precision radial velocities with Hipparcos-Gaia absolute astrometry. Applying this approach to the HD 164604 system, the authors refine the inner planet b and report the discovery of a long-period outer planet c, obtaining two viable orbital configurations that yield a mutual inclination of $\psi_{bc}=5.0^{+3.7}_{-2.2}\,^{\circ}$ (prograde) or $162.1^{+7.1}_{-4.7}\,^{\circ}$ (retrograde). Stability analysis via MEGNO favors a retrograde architecture, suggesting a violent dynamical history possibly involving ZLK cycles or scattering, while the prograde route remains plausible under disk-driven migration. Gaia DR4 is expected to break the inclination degeneracy, enabling a definitive reconstruction of the system’s dynamical past and informing models of giant-planet formation and evolution.

Abstract

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.

Figures (17)

• Figure 1: Top panel: Best-fit SED to archival photometry of HD 164604. Circles mark the observed photometric points, while diamonds indicate the corresponding synthetic fluxes generated by the model. Horizontal bars denote the effective width of each filter. Bottom panel: Residuals between the observations and the model, normalized to the photometry uncertainties.
• Figure 2: Top panel: Detrended TESS light curves of HD 164604 along with a sinusoid model (red dashed line) at $P=19.11$ days. Bottom panel: the Lomb-Scargle periodogram of the light curves with a peak at 19.11 days. A Gaussian model (blue line) is fitted to the peak to obtain the uncertainty of the rotation period.
• Figure 3: Generalized Lomb-Scargle (GLS) periodograms for HD 164604. Panel (a): the GLS periodograms of all the observed RVs. Panel (b): the residual RVs after subtracting the signal of the inner planet, HD 164604 b. Panel (c): the residuals after subtracting both planet signals. Panel (d): window function. Panel (e): stellar activity indices extracted from PFS spectra. Panel (f): photometry of ASAS-SN. The signal of planet b and c are respectively denoted with yellow and red shade regions. The horizontal grey lines in each panel, top to bottom, indicate the 0.001, 0.01, 0.1 False Alarm Probability (FAP) levels. A monthly (30 days) and yearly (365 days) signals are denoted with vertical dashed lines.
• Figure 4: Gaia PMa SNR map across the $m-a$ space. The blue, white and black contour lines respectively correspond to SNR=$1$, $3$ and $6$. The two cyan stars denote the currently measured positions of two planets in $m-a$ space. It is evident that the PMa of HD 164604 between GDR2 and GDR3 is dominated by the inner planet. We find the simulated SNR of HD 164604 b is within $3\sim6$, slightly lower than the observed value of $6.7$. This discrepancy is attributed to the uniform sampling of its inclination. If the inclination is fixed to a lower value ($<20^\circ$), corresponding to a nearly face-on configuration, the contour line will shift towards a lower mass and could align well with the observation. In other words, the SNR map presented here reflects only the average behaviour and may differ from any one specific orbital configuration.
• Figure 5: Best-fit orbit to HD 164604 RVs from our joint fitting. Panel (a) shows the best-fit two-planet Keplerian orbit (thick black line) to the RV measurements from MIKE and PFS. Panel (b) displays the associated residuals. Panel (c) presents the phase-folded orbit of the inner planet HD 164604 b, with the signal of the outer planet HD 164604 c being subtracted. Similarly, Panel (d) shows the phase-folded orbit of the outer planet HD 164604 c, after correcting for the signal of HD 164604 b.