von Zeipel-Kozai-Lidov oscillations in nearby bright stars. I. Lambda Ophiuchi

TL;DR

This study solves the orbital architecture of the nearby hierarchical triple Lambda Ophiuchi by combining historical astrometry of the outer A+B orbit (P_out ≈ 129 yr) with high-precision VLTI/GRAVITY interferometry of the inner Aa+Ab binary (P_in ≈ 42 d). Using a quadrupole, double-averaged framework that accounts for General Relativity, tidal, and rotational precession, the authors show the inner and outer orbits are retrograde and mutually inclined, with i_mut ≈ 88.5° or 113.5°, placing the system in a regime of von Zeipel-Kozai-Lidov oscillations whose amplitude is modulated by fast slaved precession of the stellar spins. The analysis yields an inner-eccentricity range Δe ≈ 0.15–0.70 and demonstrates that a semi-analytical solution is feasible, pending measurements of the spin-orbit angles, which could break key degeneracies. The work highlights the value of combining historical outer-orbit data with precise inner-orbit interferometry to constrain triple dynamics in intermediate-mass stars and outlines observational paths to achieve a fully unique dynamical solution.

Abstract

The challenge of constraining both the inner and the outer orbits in multiple stars has resulted in a growing abyss between the rich theoretical and the sparse observational studies of von Zeipel-Kozai-Lidov (ZKL) oscillations in stellar systems. Here we solve for the full orbital architecture of the bright intermediate-mass nearby system Lambda Ophiuchi based on astrometric measurements of the outer orbit (period of 129 years) compiled in the Sixth Catalog of Orbits of Visual Binary Stars and new VLTI/GRAVITY interferometric measurements that are used to determine the inner orbit (period of 42 days). The orbits are retrograde and misaligned by either $88.5\pm1.9^o$ or $113.5\pm1.9^o$, which in either case results in the inner binary currently undergoing ZKL oscillations. While pure Newtonian point source evolution would have predicted the stars in the inner binary to have merged long ago, in reality the eccentricity oscillations are significantly modulated by general relativistic, tidal and rotational bulge precession. We show that due to the effect of ``slaved'' precession the dynamics can still be solved semi-analytically. We find that the (currently unknown) inclination angles between the stellar spins axes and the inner orbital axis play a very important role in the amplitude of the ZKL oscillations, which is at a minimum $Δe = e_{\mathrm{max}} - e_{\mathrm{min}} \simeq 0.15$ and could be as high as $Δe \simeq 0.70$. We argue that currently feasible spectroscopic and interferometric observations could allow for a complete and unique dynamical solution for this system.

Figures (10)

• Figure 1: Astrometric data (colored) and best fit orbital solution (solid black) for Lambda Ophiuchi A + B (top) and Aa + Ab (bottom). The arrows show the direction of revolution of the secondaries in the sky plane. The line of apsides ($\omega$) and the line of nodes ($\Omega$) are shown in magenta and orange respectively. The numbers next to each point in the lower plot show the observation date (MJD).
• Figure 2: VLTI/GRAVITY data (colored) for $\lambda$ Oph Aa+Ab and best fit binary model (solid black) for epoch 2024-05-19. The upper panel shows the squared visibilities for the six baselines and the lower panel shows the closure phases for the four triangles. The dashed lines show the expected values for a single unresolved star.
• Figure 3: von Zeipel-Kozai-Lidov oscillations for the inner binary in Lambda Ophiuchi for the case in which the current mutual inclination is $88.5^{\hbox{$^\circ$}}$ (top) and $113.5^{\hbox{$^\circ$}}$ (bottom). The labels for the red lines refer to the (currently unknown) inclination between the stellar spin axes and the inner orbital axis.
• Figure 4: Maximum eccentricity ($e_{\mathrm{max}}$), minimum eccentricity ($e_{\mathrm{min}}$) and amplitude ($\Delta e = e_{\mathrm{max}} - e_{\mathrm{min}}$) in the von Zeipel-Kozai-Lidov oscillations of Lambda Ophiuchi Aa+Ab as a function of the spin axes misalignment relative to the inner orbital axis ($i_A$). The two plots correspond to the two possible current mutual inclinations of the inner and outer orbits.
• Figure 5: Predicted radial velocity curves for components Aa and Ab (two possible solutions for $\omega$ in blue and $\omega+180^{\hbox{$^\circ$}}$ in red) with a systemic velocity of zero. The reported measurements in Abt80 for the net spectrum are shown in black.