Eppur si eclissa: Eccentric low-mass companions and time-in-dust selection explain long secondary periods

TL;DR

This work addresses Long Secondary Periods (LSPs) in pulsating red giants by proposing a binary–dust geometry: a close-in, eccentric, low-mass companion with a co-orbiting dusty wake that periodically obscures the giant. It combines analytic geometry with a time-in-dust gate, Monte-Carlo sampling, and 3D hydrodynamical simulations to show that the fraction of the orbit spent in the dust formation zone, together with line-of-sight projection, can reproduce the observed $\omega$-bias, the roughly equal incidence of mid-IR eclipses, and the phase lag between radial-velocity and photometric light curves, while naturally yielding the higher LSP occurrence in SRVs than in Miras. The model predicts a detectable LSP fraction of $f_{\rm LSP} \sim 0.32$ for SRVs and $\sim 0.03$ for Miras, with a secondary mid-IR eclipse fraction around $0.45$, and ties these to dust-condensation radii $R_{\rm cond}$ around 2.5–3 au (SRVs) and larger values for Miras. By linking dust physics, orbital dynamics, and radiative transfer, the study provides a cohesive explanation for several longstanding LSP constraints and offers concrete, testable predictions for future observations of wind speeds, wake widths, and companion masses.

Abstract

[abbreviated] Long Secondary Periods (LSPs) are observed in about one third of pulsating red giants yet remain unexplained. Four key observational constraints anchor the discussion: (i) a roughly 30 percent occurrence rate in semi-regular variable AGB stars (SRVs), with a much lower rate or absence in regularly pulsating Mira-type AGB stars (Miras), (ii) about 50 percent of LSP stars show a secondary mid-infrared minimum, (iii) Keplerian fits to radial-velocity curves favor argument of periastron greater than 180 degrees, and (iv) the radial-velocity to light-curve phase lag clusters around minus 90 degrees. We test whether a close-in, eccentric, low-mass companion that spends only part of its orbit within the giant's dust-formation zone can match all four empirical constraints. Guided by observed radial-velocity amplitudes and periods of about 500 to 1500 days, we adopt companion masses of 0.08 to 0.25 solar masses, orbital separations of 1.5 to 3 AU, eccentricities up to 0.6, and dust condensation radii of about 2.5 to 3 AU for SRVs (larger for Miras via scaling with luminosity). We compute the fraction of the orbit spent outside the condensation radius and apply line-of-sight criteria for detectability. We test the first three empirical constraints analytically, then model the phase offset using three-dimensional hydrodynamical simulations. Our scenario explains the observed excess of systems with argument of periastron greater than 180 degrees and yields LSP detectability of about 27 percent for SRVs and about 2.5 percent for Miras, with a conditional secondary mid-infrared eclipse fraction of about 44 percent. The hydrodynamical models place the optical-depth maximum downstream of the companion near apastron and then shift it forward by roughly 90 to 225 degrees later in the orbit, consistent with the observed radial-velocity to light-curve offsets.

Figures (4)

• Figure 1: Optical-depth maps along the $z$ axis for a slice through the orbital plane for a binary setup with $M_1\!=\!1.5\,M_\odot$, $M_2\!=\!0.125\,M_\odot$, $a\!=\!2.3$ au, $e\!=\!0.3$, $R_{\rm cond}\!=\!3$ au and mass-loss rate of $5\!\times\!10^{-7}$$M_\odot$ yr$^{-1}$. Plots are in the co-moving frame, with the center of mass at $(0,0, 0)$ and the low-mass companion to the right. Video available at Video \ref{['Video:optical_depth']}.
• Figure 1: Evolution of $k \equiv w_{\rm eff}/R_{\rm BHL}$ over $\sim3$ orbits for cut angles: $\theta=-\mu$ (red), $\theta=-\psi$ (purple), and $\theta=-90^\circ$ (blue). Bottom x-axis: time; top x-axis: orbital radius $r$. Left panel: rays in the orbital plane. Right panel: rays perpendicular to the plane ($z$-direction). In each panel, the apastron passage is indicated with a grey dashed vertical line. Curves are computed from discrete hydro snapshots (only every few time steps retained), so the small-scale jaggedness reflects the sampling cadence rather than intrinsic variability.
• Figure 2: Relation between the orbital light curve (solid) and the radial-velocity curve (dashed) for an eclipsing binary with $e = 0.3$, $\omega = 225^\circ$, and $i = 60^\circ$. The bottom axis shows the normalized orbital phase, while the top axis indicates the corresponding mean anomaly $M$. Dotted vertical lines mark the epochs of primary and secondary eclipses. For a circular, edge-on orbit, the primary's RV minimum is a quarter of an orbital cycle ($\pi/2$) later in phase than the light curve minimum (primary eclipse); for eccentric orbits, this offset depends on both $e$ and $\omega$.
• Figure 3: 3D hydrodynamical simulation of an eccentric binary with a red-giant primary ($M_1=1.5\,M_\odot$) and a low-mass companion ($M_2=0.125\,M_\odot$). The orbit has semi-major axis $a=2.3\,\mathrm{au}$ and eccentricity $e=0.3$; the dust–condensation radius is $R_{\rm cond}=3\,\mathrm{au}$, the orbital period is 1000 days. The video shows the optical-depth maps in a slice through the orbital plane. The red-giant primary is at the origin $(0,0)$; the low-mass companion lies to the right in each panel. If the embedded video does not render, a copy is available on request from leen.decin@kuleuven.be.