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The dimensionless orbital compactness $GM/(R{c}^2)$ compares the gravitational length scale with the orbital size. For two equal component masses $M$, each moving at radius $R$ about the centre of mass, the quadrupole formula givesHere the total mass is $2M$ and the separation is $2R$. Newtonian gravity gives $v^2=GM/(4R)$ for each component, so increasing orbital compactness also increases the orbital speed relative to the speed of light. Compact stellar objects permit small noncontact orbits and large gravitational wave luminosities. Near merger, the slow-motion quadrupole formula requires a relativistic replacement.

A close binary star can produce two compact stellar objects if both initial stellar masses are sufficiently large. The initially more massive star evolves first, transfers material or loses its envelope, and leaves a neutron star or black hole after core collapse. The system must remain bound despite mass loss and any natal kick. Subsequent evolution of the companion can produce a common envelope: orbital energy transferred to the envelope can expel it and greatly reduce the separation. A second collapse must again leave a bound pair. A sufficiently close surviving binary star then loses orbital energy and angular momentum through gravitational waves and may merge. Stable transfer, winds and kicks make this a possible channel rather than an inevitable outcome; dynamical encounters in dense stellar systems provide another channel.

In a close binary star, the rotating effective gravitational potential defines two Roche lobes meeting at the inner Lagrange point. Stellar expansion or orbital shrinkage can bring the donor into Roche-lobe overflow. Gas then flows through the inner Lagrange point; its angular momentum can form an accretion disk around the companion. wind mass transfer in a binary star is also possible without Roche-lobe filling. For a circular, synchronized binary, the Eggleton Roche-lobe radius approximation isThis describes the lobe geometry, not the donor's dynamical response.^[ref]

conservative binary mass transfer has fixed total mass and orbital angular momentum, neglecting stellar spin. Using $J=M_d{M}_a\sqrt{Ga/(M_d+M_a)}$ givesSince ${\dot{M}}_d<0$, transfer from the more massive donor initially shrinks the orbit; after binary mass ratio reversal it expands. Stability depends on both stars and their changing lobes. Let ${\zeta }_{\ast }=d\ln R_d/d\ln M_d$ and ${\zeta }_L=d\ln R_L/d\ln M_d$. A small mass loss reduces the relative overfill if ${\zeta }_{\ast }>{\zeta }_L$. Use the adiabatic donor response for dynamical stability, and its thermal response for slower adjustment; an equilibrium mass-radius law alone cannot decide both.

Stable transfer can proceed on nuclear or thermal timescales, while dynamically unstable overflow may create a common envelope, orbital spiral-in and either envelope ejection or a merger. Escaping gas, gravitational radiation and magnetic braking make evolution nonconservative by changing orbital angular momentum. accretion onto a compact stellar object releases power of order $G{M}_a\dot{M}/R_a$, producing bright X-ray binaries; radiation-driven outflows can limit retained mass. Eclipses, radial velocities and disk emission constrain the geometry and masses, but inclination, irradiation and assumptions about mass or angular-momentum loss affect the inference. The essential feedback is the donor's radius relative to its evolving Roche lobe.