How do I make an Earth-like habitable planet tidally locked to a red supergiant (class L or M)?

Question

Assuming tidally locked planets have the potential to be inhabitable: Is there any way for a planet of Earth-like size to become tidally locked to a red supergiant of class L or M?

Answer 1

It's probably not going to happen; tidal locking simply takes too much time.

The time it takes a planet to become tidally locked is roughly $$\tau\approx6\frac{a^6R\mu}{mM^2}\;10^{10}\;\mathrm{years}$$ with $a$ the semimajor axis and $R$ the planet's radius in meters, $m$ and $M$ the mass of the planet and star in kilograms, and $\mu$ the rigidity, in Newtons $\cdot$ meters squared. The takeaway is that the timescale depends really strongly on the distance to the star and somewhat less strongly on the star's mass.

Let's say we have a red supergiant with a mass of $20M_{\odot}$ and a radius of $100R_{\odot}$ -- not atypical. That radius is actually about 1.2 times the distance between the Sun and Mercury! To lower the timescale as much as possible, let's say we place this planet at 1.5 times Mercury's semimajor axis, or 0.58 AU -- still close to the star, but not inside it. Plugging in the numbers (and assuming the planet has the same mass and radius as Earth), we get a tidal locking timescale of about 10 million years.

This puts it roughly around the midpoint of the red supergiant's lifespan (depending on the mass of the star, its metallicity, and processes like mass loss through a stellar wind). Since we plopped our Earth-like planet pretty close to supergiant, can we move it further out? Well, if we even increase it to just twice Mercury's orbital radius -- or 0.77 AU -- we get a tidal locking timescale of 58 million years, at which point the star will have long exploded in a supernova.

Since the habitable zone's size scales with the square root of the star's luminosity, the habitable zone should be much, much further out than the Sun's, and therefore far from the region where a planet can become tidally locked around this red supergiant before the star dies. To put numbers on this, I would expect the star to have -- at minimum -- a luminosity of $\sim1000L_{\odot}$, placing the inner edge of its habitable zone around 30 AU. The requirement of habitability then increases the tidal locking timescale by a factor of something like $\sim10^{10}$ over a planet orbiting at 0.58 AU.

Granted, all of this does assume that the initial rotation period of the planet is similar to Earth's and that there are no other influences on its spin. Venus has a notably long day, which may be due to tidal forces from the Sun but might also be due in part to atmospheric effects. A giant impact could also play a role. Unfortunately, the luminosity of the star and the tidal locking timescale's extremely strong dependence on semimajor axis mean these factors likely would be of little help.