Why is the common envelope ejected in some accretor-donor systems?

Question

As an example, let us consider a binary system of a neutron star and an evolved star (e.g. red giant) that has expanded, filled its roche lobe, and started the mass transfer onto the neutron star. Under certain conditions such mass transfer can become a runaway process, the accretion onto the neutron star is engulfed, and the matter that leaves the outer layers of the donor quickly surround the binary system and the two objects find themselves orbiting around each other in a dense gaseous environment. Such phase is generally referred to as Common Envelope phase. In many papers I have read the sentence:

The common envelope phase ends either with the ejection of the common envelope or by the merger of the two systems still inside the envelope.

I understand the latter possibility, that is that the dynamical interaction of the two objects in a dense environment would lead the system to the lost of orbital energy, the shrinking of their orbital separation, and due to further loss through gravitational wave emission, to merging.

What puzzles me is: why, and how is it possible that the common envelope is instead being ejected?

Answer 1

The envelope of the larger star may be rather weakly bound. Roughly speaking, the gravitational binding energy of the envelope is $-GMm/r$, where $M$ is the mass interior to the envelope, $m$ is the mass of the envelope and $r$ is its characteristic radius.

When material is accreted onto the neutron star, then a fraction of the kinetic energy on impact with the surface will be released as radiation - roughly $Gm_{\rm NS} m_{\rm acc}/2r_{\rm NS}$, where $m_{\rm NS}$ and $r_{\rm NS}$ are the mass and radius of the neutron star and $m_{\rm acc}$ is the accreted mass - and absorbed in the envelope.

The envelope could be ejected if the radiated energy exceeds the (modulus of) the envelope binding energy. This could happen because even though $m_{\rm acc}$ could be small compared with the mass of the engulfing star, $r_{\rm NS}\ll r$.