Binary mass transfer in 3D: Mass Transfer Rate and Morphology

TL;DR

This study uses high-resolution 3D hydrodynamic simulations in the corotating frame to investigate mass transfer through both inner and outer Lagrangian points in binaries, explicitly including the Coriolis force. By comparing to analytic prescriptions (Ritter 1988; Kolb & Ritter 1990) across a broad mass-ratio range ($q rom 10^{-6}$ to 10) and for adiabatic and isothermal envelopes, it reveals a non-axisymmetric, trailing-side stream morphology and a concave, sometimes non-intersecting sonic surface, while showing that mass-transfer rates deviate from analytic predictions by only factors of a few. The authors derive fitting formulas and an extended dynamical term that can be readily implemented in stellar evolution codes (e.g., MESA), improving mass-transfer rate calculations for both $L_{ m in}$ and $L_{ m out}$ and enabling exploration of stability and angular-momentum consequences. They highlight the practical applicability of their findings while noting limitations due to simplified envelope physics and the approximate Roche potential, and outline directions for incorporating more realistic stellar structures and physics in future work.

Abstract

Mass transfer is crucial in binary evolution, yet its theoretical treatment has long relied on analytic models whose key assumptions remain debated. We present a direct and systematic evaluation of these assumptions using high-resolution 3D hydrodynamical simulations including the Coriolis force. We simulate streams overflowing from both the inner and outer Lagrangian points, quantify mass transfer rates, and compare them with analytic solutions. We introduce scaling factors, including the overfilling factor, to render the problem dimensionless. The donor-star models are simplified, with either an isentropic initial stratification and adiabatic evolution or an isothermal structure and evolution, but the scalability of this formulation allows us to extend the results for a mass-transferring system to arbitrarily small overfilling factors for the adiabatic case. We find that the Coriolis force -- often neglected in analytic models -- strongly impacts the stream morphology: breaking axial symmetry, reducing the stream cross section, and shifting its origin toward the donor's trailing side. Contrary to common assumptions, the sonic surface is not flat and does not always intersect the Lagrangian point: instead, it is concave and shifted, particularly toward the accretor's trailing side. Despite these structural asymmetries, mass transfer rates are only mildly suppressed relative to analytic predictions and the deviation is remarkably small -- within a factor of two (ten) for the inner (outer) Lagrangian point over seven orders of magnitude in mass ratio. We use our results to extend the widely-used mass-transfer rate prescriptions by Ritter(1988) and Kolb&Ritter(1990), for both the inner and outer Lagrangian points. These extensions can be readily adopted in stellar evolution codes like MESA, with minimal changes where the original models are already in use.

Figures (12)

• Figure 1: Curvature of the gravitational (Roche) potential near the inner and outer Lagrangian points around the donor star. The middle panel depicts an overall shape of the Roche potential in a binary in the corotating frame. The inner and outer Lagrangian points of the donor, indicated by a violet star and magenta cross, respectively, are where our computational domain is located. The $\tilde{x}$ axis of the domain is aligned with the binary axis and the $z$ axis is parallel to the binary orbit axis. The left panels illustrate the significantly steeper curvature and shallower potential depth -- both perpendicular (top) and parallel (bottom) to the binary axis -- near the inner Lagrangian point than the outer Lagrangian point for a system where the donor and accretor have comparable masses (for instance, stellar binaries). The right panels show the same quantities for a case where the donor mass is significantly smaller than the accretor mass (for example, stellar extreme mass ratio inspiral). In this case, similar to the comparable mass ratio case, the potential near the outer Lagrangian point exhibits shallower curvature. However, the depths become comparable. In fact, the depths become equal as the ratio of the donor mass to the accretor mass approaches zero. The curvature of the potential near the Lagrangian points is primarily determined by the coefficients of the quadratic terms in the Taylor expansion of the Roche potential (Eq. \ref{['eq:phi']}).
• Figure 2: Trajectories of particles for a steady state solution of an equal-mass adiabatic case (left: without Coriolis force and middle: with Coriolis force) and an equal-mass isothermal case (right) in the midplane around the inner Lagrangian point, plotted over the density distribution. The particles are initially distributed following the same distribution of the initial density profile over a region with $10^{-3}\lesssim\tilde{\rho}\lesssim 20$ and their trajectories are integrated for $\tilde{t}\simeq 20$ since the flow reaches a steady state, assuming the particles are advected with the gas. Crosses at one end of the lines indicate the initial locations of the particles. We distinguish particles that reach beyond the Lagrangian point (yellow) from those that remain inside the donor (orange) over the integration duration.
• Figure 3: Shape and location of the sonic surface near the inner Lagrangian point. The 3D images of the binary near the Lagrangian point in the center show the orientation of the 2D plane in which the 2D distribution of Mach number is depicted -- mid- (top) and vertical (bottom) slices. The panels on the left show the distributions for the equal-mass adiabatic case, while those on the right correspond to the equal-mass isothermal case. In the panels showing the Mach number distribution, the dashed diagonal orange lines depict the shape of the Roche lobe.
• Figure 4: Vertical and lateral motion of overflowing stream, relative to the motion toward the accretor, in the plane normal to the binary axis and intersecting the Lagrangian point (magenta cross), with the Coriolis force. Gas with $\tilde{\rho}<10^{-3}$ (white background), which contributes negligibly to the overflowing stream, is masked out for better visualization. $\tilde{v}^{i}$ ($i=x,y,z$) is the $i^{\rm th}$ velocity component relative to the Lagrangian point. In both cases -- adiabatic (left) and isothermal (right), we assume an equal-mass binary. The hatched areas indicate the velocity ratios less than 0.05, suggesting that the gas may be assumed to be in hydrostatic equilibrium. The arrows show the overall motion of the stream.
• Figure 5: Angle of the transferring stream relative to the binary axis for adiabatic (left) and isothermal cases (right) in the presence of the Coriolis force. The angle is defined as $\tan\theta=\langle\tilde{v}^{y}\rangle/\langle\tilde{v}^{x}\rangle$, where $\langle\tilde{v}^{x}\rangle$ and $\langle\tilde{v}^{y}\rangle$ are mass-weighted averages of $\tilde{v}^{x}$ and $\tilde{v}^{y}$ measured at the right $x-$boundary. The solid curves following the dots indicate the fitting formulae given in Eq. \ref{['eq:fitting_theta']}. The vertical solid gray lines indicate equal-mass binary. In the right panel, the dotted curves indicates the predictions of the tilt angle for the isothermal case by LubowShu1975.