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For distinct sibling supernovae $s$ and $s^{\prime }$ in the same galaxy, the shared fluctuation is the only common random term. Independence and linearity of covariance therefore giveTheir correlation coefficient isThis is the intraclass correlation coefficient of the galaxy-level random intercept model.

A common envelope is invoked because many observed compact binaries have separations far smaller than the radii of their progenitor giants; ordinary conservative transfer cannot remove enough orbital energy and angular momentum. It begins when rapid or dynamically unstable Roche-lobe overflow engulfs the companion, or when a companion is swallowed by an expanding giant. Drag inside the envelope transfers orbital energy and angular momentum to gas, causing a fast inspiral. If the deposited energy ejects the envelope before the cores touch, a close exposed-core binary survives; otherwise the cores merge. Partial ejection or delayed thermal readjustment can complicate either outcome.

The common-envelope energy formalism writesHere $M_d,M_c,M_{\mathrm{env}}$ are donor, core, and envelope masses; $\lambda$ describes envelope structure; and ${\alpha }_{\mathrm{CE}}$ is the efficiency with which released orbital energy unbinds gas. Solving estimates $a_f$, while failure to supply $E_{\mathrm{bind}}$ predicts merger.

A carbon--oxygen white dwarf can make a Type Ia supernova when carbon ignites under degenerate conditions. Burning carbon and oxygen to iron-group and intermediate-mass nuclei releases of order $10^{44}$ J, comparable to and exceeding the white dwarf's gravitational binding energy, so the thermonuclear flame disrupts the star. Radioactive ${}^{56}\mathrm{Ni}$ and then ${}^{56}\mathrm{Co}$ decay power the optical light curve by depositing gamma-ray and positron energy. Possible triggers include near-Chandrasekhar mass central ignition and sub-Chandrasekhar detonations initiated by an accreted helium shell or a merger.

One route from zero-age masses $8M_{\odot }$ and $5M_{\odot }$ is as follows. The $8M_{\odot }$ primary evolves first, fills its Roche lobe, and transfers its envelope; if stable, the orbit usually widens after mass-ratio reversal, leaving a carbon--oxygen core that becomes the first white dwarf. The rejuvenated secondary later evolves and becomes the more massive giant. Transfer to the white dwarf then has an extreme mass ratio and is dynamically unstable, producing a common envelope. Inspiral ejects the giant envelope and leaves the first white dwarf close to the secondary's exposed helium or carbon--oxygen core, which becomes the second white dwarf. Depending on the earlier separation, a second unstable episode may be required. Once detached, gravitational-wave emission from a binary system removes energy and angular momentum, steadily shrinking the orbit until the white dwarfs contact. This is the Double-degenerate Type Ia supernova scenario.

If white-dwarf mass transfer is dynamically stable, the lighter donor is gradually disrupted or transferred at rates that may lead to quiescent burning, off-centre carbon ignition, and conversion to an oxygen--neon remnant followed by accretion-induced collapse, rather than prompt thermonuclear disruption. Dynamically unstable transfer can overcome this by producing a violent merger: rapid stream impact, tidal heating, shocks, and helium or carbon hotspots may detonate before the remnant settles and ignites carbon off centre. Whether this happens is the central ignition problem of the double-degenerate channel.