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I read recently about VFTS 352, an overcontact binary star system where both stars have roughly equal mass. All of the reports I've read (in mass-media type publications) have said that the system has one of two fates: either the two stars will merge, or they'll supernova. But when will this happen?

The wikipedia page for contact binaries says that they have a lifespan of millions to billions of years, but doesn't say if that's different for overcontact binaries. It also says that they're often confused with common envelopes, which have a lifespan of months to years, and I'm not sure where in that spectrum an overcontact lies (or really what the distinction is, since the page for contact binaries says they share an envelope, which sounds the definition of a common envelope). I'm also not sure whether the fact that both stars have roughly equal mass affects the lifespan.

The mass-media articles I've read have implied that the merger-or-supernova is happening soon, but I don't know if this is on a human scale (months) or galactic scale (millions of years).

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Short answer: $t \lesssim 10^5\ \mathrm{years}$ (maybe)

An "overcontact binary" is just another way of saying "common envelope binary". The two phrases are exactly the same and it's frustrating that the authors on the VFTS 352 paper decided to create their own convention - as if astrophysical classifications weren't confusing enough!

A contact binary exists on timescales predominantly dependent on stellar evolution, so figuring out how long a contact binary will exist is heavily dependent on the mass, metallicity, and rotation of the primary star among other things.

Deriving the timescale:

Let's keep the scope to systems like VFTS 352, where the primary is massive and the binary has an orbital period less than 4 years (2.5 AU separation). In order to have a common envelope event, the stars must have overflown their Roche lobes. The radius for the Roche lobe of two point masses is \begin{equation*} r_L = \frac{0.49 q^{\frac{2}{3}}}{0.6q^{\frac{2}{3}}+\mathrm{ln}(1+q^{\frac{1}{3}})}a \end{equation*} where $a$ is the separation. For close binaries, the general observed trend is a high mass ratio $q=M_2/M_1$. So, if we assume $q=1$, then $r_L = 0.38a$. Hence, for a binary with $a<2.5$ AU, \begin{align*} r_L &\lesssim 1\ \mathrm{AU}\\ r_L &\lesssim 215\ R_{\odot} \end{align*} since $q=1$ is an upper bound on the Roche lobe radius. Now, performing some trivial rearrangement of the blackbody luminosity equation $L=4\pi\sigma_{SB}R^2T^4$, we find that \begin{equation*} R \approx 3.31\times10^{7} \bigg(\frac{L}{L_{\odot}}\bigg)^{\frac{1}{2}}\bigg(\frac{1\ \mathrm{K}}{T}\bigg)^2 \ R_{\odot}. \end{equation*} Massive stars typically have roughly constant luminosity, so we will choose $L\approx10^5\ L_{\odot}$. Hence, \begin{equation*} R\approx 1\times10^{10}\bigg(\frac{1\ \mathrm{K}}{T}\bigg)^2 \ R_{\odot} \end{equation*}

The massive star needs to evolve until its radius is equal to that of the Roche lobe radius, so we find that the star reaches the common envelope phase for \begin{equation*} T \gtrsim 7000\ \mathrm{K} \end{equation*} Taking a peek at an HR diagram, this star varies from about $30000\ \mathrm{K}$ to $4000\ \mathrm{K}$ from ZAMS to end of main-sequence. Thus the primary spends roughly 3/4 of its time on the main-sequence not in the common envelope phase. Hence, this binary's common envelope phase lasts for, at most, 1/4 the primary's total lifetime, which is on the order of $10^6$ years. Thus, the upper bound for the timescale of a common envelope event with massive stars with negligible rotation is $\sim10^5$ years.

Please note that this derivation does not take into account the bulging effect that occurs as the separation decreases. This will certainly lower this upper bound, but by how much I'm not sure. It could lower it by 1 year, or $10^5\ \mathrm{years}$.

Lower bounds to this timescale are entirely ambiguous and not particularly helpful in any physical context. The stars could be spinning really fast, have high or low metallicity, the binary could have a different mass ratio, there could be another binary close by, and there may be magnetic interaction (?). The list goes on! I'm sure there's something I left out.

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The articles indicate one of two possible outcomes: merger, followed ultimately by gamma ray burst, or permanent separation, separate supernovae leading to binary black holes.

In the second case the supernovae will be in a few million years, the typical life span for massive stars.

In the first case the merger could happen sooner, perhaps hundreds of thousands of years, so "soon" in astronomical terms, but long in comparison to a human life.

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