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

Context. Long secondary periods (LSPs) are observed in about one-third of pulsating red giants, yet this phenomenon remains unexplained. Four key observational constraints anchor the discussion: (i) a ∼30% occurrence rate in semi-regular variable AGB stars (SRVs) with a much lower rate (or complete absence thereof) in regularly pulsating Mira-type AGB stars (Miras); (ii) ∼50% of LSP stars show a secondary mid-infrared (MIR) minimum; (iii) Keplerian fits to radial-velocity (RV) curves favour the argument of periastron ω > 180°; and (iv) the RV-light curve phase lag clusters around −π/2. Aims. We test whether a close-in, eccentric low-mass companion that only spends part of its orbit within the giant’s dust-formation (wind-launching) zone can match all four empirical facts. Methods. Guided by observed RV amplitudes and periods of ∼500–1500 days, we adopted a companion mass of M2 ∈ 0.08, 0.25 M⊙, orbital separation of a ∈ 1.5, 3 au, and eccentricty of e ≤ 0.6. Next, we took the dust condensation radius of Rcond ∼ 2.5 − 3 au for SRVs (larger for Miras when scaling with luminosity). We computed the time-in-dust fraction fdust (time with r ≥ Rcond) and applied line-of-sight criteria: an LSP requires an orbital inclination of i ≥ iLSP and fdust ≥ fmin, while a secondary MIR minimum interpreted as secondary eclipse further requires i ≥ iecl > iLSP and a superior conjunction. We tested the first three empirical facts analytically, then modelled the RV-light phase offset with 3D hydrodynamical simulations. Results. Our proposed scenario explains the observed excess of ω > 180°. For SRV-like parameters, we obtained an LSP detectability of ∼31.6 ± 0.1%, while Mira-type conditions yield ∼3.0 ± 0.1%; for both scenarios, the conditional secondary MIR eclipse fraction is ∼44%. Our hydrodynamical models place the optical-depth peak just downstream of the companion near apastron, then shift it to ∼90 − 225° phase offsets later in the orbit. This result is consistent with the RV-light offsets. Conclusions. A time-in-dust geometric selection for low-mass companions in close eccentric orbits is sufficient to explain the four key empirical facts constraining the LSP mechanism.