Binary Mass Transfer

In this paper, 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. Here is a compact summary of our methodology and results. Let’s first start with a short description of the geometry of the Roche potential, which plays a crucial role in stream evolution.

1. Geometry of the Roche potential

The potential of a binary in its corotating frame — assuming the binary members are point masses — is given by:

$\phi_{\mathrm{exact}}(X,Y,Z) = -\frac{GM}{\left( X^{2}+Y^{2} + Z^{2} \right)^{1/2}}-\frac{Gm}{\left((X-a)^{2}+Y^{2} + Z^{2} \right)^{1/2}} &-\frac{1}{2} \Omega^{2}\left[ \left(X-\frac{m}{M+m}a \right)^{2}+Y^{2}\right],$

where $M$ and $m$ are the masses of the primary and secondary stars, respectively, $a$ is the semimajor axis of their relative circular orbits,

$\Omega^{2}=G(M+m)/a^{3},$

$X$ is a coordinate along the line connecting the two masses, centered around $M$ , $Y$ the perpendicular coordinate within the orbital plane, and $Z$ the coordinate along the orbital axis.

Because we are interested in the potential geometry near the inner ( $L_{\mathrm{in}}$ ) and outer Lagrangian ( $L_{\mathrm{out}}$ ) points around a binary member (say the donor star), we introduce an approximate expression for $\phi$ by expanding it around the given $L$ point to second order. To do that, we define a new coordinate system ( $x$ , $y$ , $z$ ) around a given $L$ point and expand $\phi_{\mathrm{exact}}$ ,

$\phi(x,y,z) \simeq \frac{1}{2}\frac{\partial^{2} \phi_{\mathrm{exact}}}{\partial x^{2}}\rvert_{L}x^{2} + \frac{1}{2}\frac{\partial^{2} \phi_{\mathrm{exact}}}{\partial y^{2}}\vert_{L}y^{2} +\frac{1}{2}\frac{\partial^{2} \phi_{\mathrm{exact}}}{\partial z^{2}}\vert_{L}z^{2}= \frac{1}{2}\Omega^{2}(A x^{2} + B y^{2} + C z^{2}).$

The first derivatives $\partial\phi_{\mathrm{exact}}/\partial_{x}|_{\mathrm{L}}=\partial\phi_{\mathrm{exact}}/\partial_{y}|_{\mathrm {L}}=\partial\phi_{\mathrm{exact}}/\partial_{z}|_{\mathrm{L}}$ are zero because the Lagrange point is a saddle point of the potential. Here, $A$ , $B$ , and $C$ are functions of mass ratio $q$ and $B = -(A+3)/2$ and $C=B+1= -(A+1)/2$ .

The geometry of the potential near the $L_{\mathrm{in}}$ and $L_{\mathrm{out}}$ points around the donor plays a major role in determining the morphology of overflowing streams and the overflow rate. This can be better understood from the coefficients $A$ , $B$ , and $C$ because they determine the curvature of the Roche potential, which is demonstrated in the figure below.

$A$ influences how easily gas near the donor’s surface climbs the potential and how rapidly the gas accelerates after crossing the $L$ point (bottom panels). On the other hand, $B$ and $C$ determine the curvature of the potential in the direction perpendicular to the binary axis, intersecting the $L$ point (top panels). More specifically, for binaries with similar masses (e.g., stellar binaries), the potential at the $L_{\mathrm{in}}$ point is much deeper and has a steeper curvature both parallel (larger $|A|$ ) and perpendicular (larger $B$ and $C$ ) to the binary axis than at the $L_{\mathrm{out}}$ point. This implies that the donor star must significantly overfill RL for $L_{\mathrm{out}}$ overflow to occur. For example, for equal-mass binaries, $L_{\mathrm{out}}$ overflow can occur when the size of the donor is larger than that of RL by $\simeq 30\%$ (or relative overfilling factor $\gtrsim 0.3$ ). On the other hand, for binaries with very different mass ratios (e.g., stellar extreme mass ratio inspirals), the depths of the potential at the $L_{\mathrm{in}}$ and $L_{\mathrm{out}}$ points become comparable, although the potential at the $L_{\mathrm{in}}$ point remains lower. This means $L_{\mathrm{out}}$ outflow can begin even when the donor slightly overfills the RL (e.g., relative overfilling factor $\gtrsim 10^{-3}$ for stellar extreme mass ratio inspiral with mass ratio of $10^{-6}$ ). However, the curvature of the potential near the $L_{\mathrm{out}}$ point is shallower, allowing a larger cross-section of overflowing streams. This indicates that the $L_{\mathrm{out}}$ overflow rate can be comparable to or potentially larger than the $L_{\mathrm{in}}$ rate when the donor star overfills its RL sufficiently.

2. Methodology

To investigate overflowing streams in 3D, we used the finite-volume adaptive mesh refinement magnetohydrodynamics code ATHENA++(Stone+2008,Stone+2020).

A key advantage of the quadratic potential given above is its scalability. We introduced scaling factors, including the Roche-lobe overfilling factor, to make the hydrodynamics equations with the Roche potential and initial conditions dimensionless. This scalability allows us to generalize simulation results for a given mass-transferring system to arbitrarily small overfilling factors.

We made two key assumptions: (1) the donor’s surface before the onset of mass transfer is in hydrostatic equilibrium and follows a polytropic relation, and (2) the gas behaves as an ideal gas. While simplified, these assumptions are necessary for scalability and allow for fully analytical solutions for steady-state overflow following Ritter(1988) and Kolb&Ritter(1990).

See the movie below showing a 3D view of mass transfer near the inner Lagrangian point.

2. Morphology of overflowing streams

In all our models including both adiabatic and isothermal cases, the main streams originate from the “trailing” side of the donor star along its orbital path because of the Coriolis force. We illustrate the origin of the main stream in the figure below, using the trajectories of tracer particles around $L_{\mathrm{in}}$ for mass ratio $=1$ in the midplane. This main stream morphology resembles the stream motion near the Lagrangian point analytically predicted by Lubow&Shu(1975) (see their Fig. 3).

2. Mass transfer rate

In the figure below, we compare the mass transfer rate from our simulations to the analytic solution for the adiabatic and isothermal cases. 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!