SCATTER Common Envelope Formalism for Triples

TL;DR

This work extends the SCATTER common-envelope formalism from binaries to triple-star systems, enabling predictions of post-CE orbital configurations and merger outcomes in hierarchical and non-hierarchical triples. It treats CE evolution as angular-momentum exchange between each star and the envelope, introducing the functions $\mathcal{F}(q)$ and $\mathcal{Q}(q)$ and a mass-ratio–dependent efficiency $\eta$, calibrated from WD post-CE data, to map pre-CE to post-CE states. The authors derive analytic expressions for the orbital changes of inner and outer binaries, present hierarchical and non-hierarchical implementations, and illustrate the results with detailed examples showing mergers, SN Ia channels, and potential ejections. The framework broadens population-synthesis capabilities, improving predictions of WD mergers, gravitational-wave sources, and other energetic transients in systems with higher-order multiplicity, while noting the need for further calibration and simulations.

Abstract

Many stars are components of triple-star systems, or of higher-order multiples. In such systems mass transfer is common, and when the transfer is dynamically unstable, a common envelope forms. As such, it is important to be able to compute the post-common-envelope orbital separations among the various stars comprising the system, and to determine whether the common envelope induces mergers and/or makes later mergers inevitable. In this paper we compute the results of common-envelope evolution for triples. We employ the SCATTER formalism, a new approach to the computation of post-common-envelope separations. This work has applications to gravitational mergers, Type Ia supernovae, and a broad range of highly energetic phenomena.

Figures (10)

• Figure 1: Top panel: Initial and final orbital separations versus $M_3^c$. The initial (post-CE) orbit of Star 3 has radius $a_3(0)$ ($a_3(f)$). Values of $a_3(f)$ range from tens of solar radii, for the smallest core masses, to just under $10\, R_\odot$ for the largest possible core mass. The star occupying the post-CE orbit is a WD, with radius too small to merge with other stars, unless the other stars are large. In this case, the other stars are Star 1 and Star 2, whose orbital radius (i.e., the radius of the inner orbit) shrinks by a factor as small as, or even smaller than, $0.0003$. Thus, the inner binary will merge. Bottom panel:$\log_{10}\!\left[a_{\mathrm{in}}(f)/a_{\mathrm{in}}(0)\right]$ versus $\log_{10}\!\left[a_{\mathrm{out}}(f)/a_{\mathrm{out}}(0)\right]$ for varying tertiary core mass.
• Figure 2: Triple-CE results versus core mass of Star 3, the outer-orbit star that serves as donor. Each point on each curve represents a system in which the inner binary would not merge within a Hubble time, but which will merge sometime after a CE phase triggered by the RL filling of Star 3. Top panel: the logarithm of the time (in units of the Hubble time) required for the inner binary to merge without the intervention of a CE is plotted versus the minimum core mass needed to satisfy all of the conditions described in the text. Middle panel: The pre-CE and post-CE orbital separations of the inner binary are plotted versus the core mass. Bottom panel: The pre-CE and post-CE orbital separations of the outer binary are plotted versus the core mass.
• Figure 3: Values of ${\cal {P}}_{\max}$ are plotted versus $q_{1,2}$ for each of six values of $q_{3,{\mathrm bin}}$. The values of $q_{3,{\mathrm bin}}$ start at $1/3$ for the bottom curve and increase by $1/3$ for each curve above, reaching the value of $2$ for the topmost (dark blue) curve.
• Figure 4: The initial (pre-CE) orbit of Star 3 has radius $a_{\mathrm{out}}(0)$. Values of $a_{\mathrm{out}}(f)$ range from tens of solar radii, for the smallest core masses, to just under $10\, R_\odot$ for the largest possible core mass. The star occupying the post-CE orbit is a WD, with radius too small to merge with other stars, unless the other stars are large. In this case, the other stars are Star 1 and Star 2, whose orbital radius (i.e., the radius of the inner orbit) changes by a factor of $0.0003$, or even less. Thus, the inner binary will merge.
• Figure 5: The initial (post-CE) orbit of Star 3 has radius $a_3(0)$ ($a_3(f)$). Values of $a_3(f)$ range from tens of solar radii, for the smallest core masses, to just under $10\, R_\odot$ for the largest possible core mass. The star occupying the post-CE orbit is a WD, with radius too small to merge with other stars, unless the other stars are large. In this case, the other stars are Star 1 and Star 2, whose orbital radius (i.e., the radius of the inner orbit) shrinks by a factor of $0.0003$, or even less. Thus, the inner binary will merge.