Let the star grow toward a special place
Picture a close pair in which one star begins to expand. At first its outer gas still surrounds that star, but the shape is no longer an isolated sphere. The side facing the companion stretches outward, rather like the pointed end of a pear. We call the star supplying gas the donor and the star receiving it the accretor. These names describe their present roles; a binary can exchange those roles during its life.
The shape becomes easier to understand if we turn with the binary. Assume a circular orbit and a donor rotating so that the same side faces the companion. A little parcel temporarily at rest in this rotating picture responds to the gravity of both stars and to the inertial effects associated with rotation. Along the line joining the stars there is a special location where the donor’s last individually closed region narrows to a point.
That point is the inner Lagrange point, usually called L1. The critical surface passing through it bounds the donor’s Roche lobe. This is no solid membrane, and gravity does not change its law at the boundary. It is a geometrical statement about a particular rotating potential. When the expanding outer layers approach the opening, pressure and the local gas conditions can allow material to pass through.
Reading: Eggleton (1983), Approximations to the radii of Roche lobes
Gas does not simply fall straight down
By the time gas reaches the opening, it is already moving with the binary. It carries sideways motion into the passage. In the rotating picture this produces a curved stream. A sufficiently large accretor can intercept it directly. Around a smaller accretor the path may miss the surface, opening the possibility of a disc. Whether a disc actually forms also involves collisions, pressure and dissipation; one test-particle orbit cannot settle the whole problem.
A useful first calculation is therefore a route map: in this frozen orbit, where would a massless parcel launched here go? That map says nothing by itself about the mass passing each second, the shocks between neighboring parcels, or the time required to remove a stellar envelope. A drawing with more tracers is not automatically a denser physical flow. The trajectory is informative precisely when we keep its question this narrow.
Reading: Soberman, Phinney & van den Heuvel (1997), astro-ph/9703016v1
The star and its available space change together
Now remove a small parcel from the donor and let the accretor keep it. For the moment, suppose no matter or orbital angular momentum escapes the pair. The masses have changed, so the orbit must adjust. The critical Roche volume adjusts with it. Asking only whether the donor gets larger or smaller misses half the experiment: we also need to know how quickly its available space changes.
In one example a lighter donor transfers to a heavier accretor. The orbit widens and the donor’s Roche lobe grows. The donor itself can even expand slightly, yet become less overfilled if the lobe grows faster. Saying that the star “retreats inside its lobe” describes this relative fit. It need not mean that its absolute radius decreases.
A heavier donor under another stated radius-response assumption can do the opposite. Its orbit tightens, its lobe contracts, and losing mass makes the donor expand. The small removal then increases overfill and supplies positive feedback for further loss. This is a local tendency worth understanding. Whether transfer really becomes rapid, how long it persists, and what halts it require the star’s structure, thermal adjustment and the gas dynamics.
Reading: Soberman, Phinney & van den Heuvel (1997), astro-ph/9703016v1
Two cores inside one envelope
One possible later configuration puts the companion and the donor’s compact core inside the same dense gaseous envelope. Their motion transfers orbital energy and angular momentum to the gas, and their separation may shrink dramatically. Some deposited energy can expand or unbind material; some can escape as light. The energy balance records the release; its spatial and temporal distribution shapes where parcels go.
A common starting point assigns a fraction of the released orbital energy to overcoming the envelope’s gravitational binding. Under those choices it selects a final separation. Even then, we must ask whether the supplied core and companion radii directly overlap and whether each radius fits its own final Roche lobe. Those are different geometrical tests. A further difficulty is that a core boundary cut from the original giant is not necessarily the radius of the remnant after thermal relaxation.
“Is there enough energy?” and “what happens to the gas?” are related questions with different demands. The course first calculates the energy balance, then follows gas and radiation in fluid calculations. There we can ask why losing some heat as light may greatly alter how much envelope escapes. The physical competition is whether deposited heat can become expansion work before it leaves.
Reading: Ivanova et al. (2013), Common envelope evolution; arXiv:1209.4302v2 · Lau et al. (2025), arXiv2503.20506v2; DOI10.1051/0004-6361/202554782