# If two stars collide, what is the probability that they merge to form a single star?

After looking at What are the odds that the Sun hits another star? and answering it (crudely), now I'd like to ask the following:

What is the probability that if two stars collide, their cores merge to form one larger, more massive star?

• I'm not sure how to do the calculations correctly, so I won't write an answer, but I suspect that if 2 normal (Sun-like) stars fall into each other, even from a relatively small distance, the total KE in the rest frame would exceed their gravitational binding energies. So I seriously doubt that a peaceful merger could occur. – PM 2Ring Jan 22 at 22:37
• @PM2Ring I don't care how violent the merger is and how messy the debris is after the first strike, just what is the chance that the stars don't destroy each other and actually merge their cores to form a larger star without a common envelope being shed? – fasterthanlight Jan 23 at 0:06
• Sure. And I'm saying the KE of the collision is probably enough to completely unbind both stars. And that's totally ignoring their current thermal energy content, or any energy released via nuclear reactions caused by the increased pressure. Hopefully, one of the professional astrophysicists will be able to give us a more definitive answer. – PM 2Ring Jan 23 at 0:26
• That might depend on the other star's mass and size – Jonas Jan 23 at 12:47

## 1 Answer

Two stars of mass $$M$$ falling from infinity straight towards each other until they merge at distance $$2R$$ will get kinetic energy $$GM^2/R$$. This is a lot, for two suns it is $$1.8978\times 10^{41}$$ J. However, compared to the binding energy of even a single star, $$\approx 3GM^2/5R$$ this is less(the sun has binding energy $$2.2774\times 10^{41}$$ J, and a double mass $$2^{1/3}R$$ radius same density merged star $$7.2302\times 10^{41}$$ J, 3.17 times more). So there is not enough energy released to blow up a star, but is is about a quarter of it: a lot of matter is going to get ejected or end up in orbits through a heated envelope that will take a while to simmer down.

The key issue is whether the cores get slowed down enough by the encounter to remain bound, becoming a common envelope binary. A direct hit clearly would work, but glancing collisions may allow the cores to miss each other: now the question is whether the envelope can absorb enough kinetic energy. A rough estimate may be that there is significant slowing if the mass scooped up/pushed aside $$\pi r_{core}^2 \rho_{envelope} R$$ becomes comparable to $$m_{core}$$. For two sun-like stars with $$r_{core}=0.2R_\odot$$ this seem to happen, but much hydrodynamics may occur complicating things.

Glebbek's dissertation on stellar mergers estimates a rough condition for the orbital angular momentum to exceed the maximum spin angular momentum of the merged star as $$\frac{r_p}{R_1+R_2} > k^4\frac{(1+q)^{\xi+4}}{2q^2}$$ where $$k^2\approx 0.05$$, $$\xi\approx 0.6$$, $$r_p$$ the periastron distance, and $$q=M_2/M_1$$. This typically is exceeded: there is a lot of angular momentum that needs to be shed (for example by blowing off a lot of heated gas). For example, two sun-like stars having $$r_p=R_\odot/2$$ has a LHS of 1/4 and a RHS of 0.0303.

That dissertation also contains numerical simulations of various merger cases.