Mantle Convection in Tidally-Locked Planets

Observations over the years have shown that short-period terrestrial planets appear to be rather common around stars. Most of these planets are expected to be tidally-locked and are likely to undergo processes in planetary evolution that do not occur for the terrestrial planets in our solar system. For a tidally-locked planet, one day is equal to one year. As a result, one hemisphere will always be facing its host star while the other hemisphere will never be. This creates a fixed hot substellar region and a fixed cold antistellar region.

Figure 1: Kepler-20e and Kepler-20f compared to Venus and Earth. Both Kepler-20e and Kepler-20f orbit a Sun-like star, Kepler-20, closer than Mercury orbits the Sun. The two planets are likely to be tidally-locked with scorching dayside temperatures. Credit: NASA/Ames/JPL-Caltech.

A study done by S. E. Gelman et al. (2011) identifies two main stages of mantle convection evolution for tidally-locked terrestrial planets. The first stage is a transient stage which involves a lithospheric temperature and thickness dichotomy emerging between the planet’s substellar and antistellar hemispheres. Basically, the substellar lithosphere warms and becomes thinner, while the antistellar lithosphere cools and thickens. During this stage, mantle upwellings and downwellings are pervasive throughout the planet’s interior. The second stage of mantle convection occurs with the onset of degree-1 mantle convection. Here, mantle convection occurs as a single upwelling under the substellar pole and as a single downwellings under the antistellar pole. Degree-1 mantle convection is stable and will continue until the planet cools enough to halt mantle convection entirely.

The timescale for onset of degree-1 mantle convection depends primarily on the planet’s thermal Rayleigh number (Ra) - a dimensionless number indicating the vigor of convection. For a planet with a high enough Ra, the onset of degree-1 mantle convection can happen within a few billion years (Figure 2). However, if Ra is small, the timescale for onset of degree-1 mantle convection can far exceeded the age of the universe (Figure 3). For this reason, it is expected that a large fraction of tidally-locked terrestrial planets have yet to develop degree-1 mantle convection. Red dwarf stars have lifetimes spanning hundreds of billions to trillions of years. This is why tidally-locked terrestrial planets orbiting such stars have ample time to develop degree-1 mantle convection provided these planets do not cool too much until mantle convection comes to a halt.

Figure 2: Modelled time evolution of a tidally-locked terrestrial exoplanet whose stellar flux corresponds with that of Gliese 581 c (F = 5400 W/m^2), and whose mantle properties and geometry are similar to Earth’s (Ra = 3 × 10^6). The top right corner of each panel shows the simulated time in millions of years since model initiation. Stellar flux arrives from the top of each panel. Initial mantle temperature, as well as the factor of temperature dimensionalization, is set to 1300 K. Notice that the timescale for onset of degree-1 mantle convection far exceeds the age of the universe. (S. E. Gelman et al., 2011)

Figure 3: Modelled time evolution of a tidally-locked terrestrial exoplanet whose stellar flux corresponds with that of Gliese 581 c (F = 5400 W/m^2), but whose Ra is 3 orders of magnitude greater than that of Earth’s (Ra = 1 × 10^9). The top right corner of each panel shows the simulated time in millions of years since model initiation. Stellar flux arrives from the top of each panel. Initial mantle temperature, as well as the factor of temperature dimensionalization, is set to 1300 K. Notice that the timescale for onset of degree-1 mantle convection is only a few billion years. (S. E. Gelman et al., 2011)

With the onset of degree-1 mantle convection, increased volcanism is expected around the planet’s substellar point, while the antistellar side will experience suppressed volcanism. The results by S. E. Gelman et al. (2011) show that surface temperature can force patterns of mantle convection in tidally-locked terrestrial planets. Although the models presented have surface temperatures that do not exceed ~850 K, many known exoplanets receive higher stellar fluxes and are expected to have significantly higher substellar temperatures. If the substellar temperatures are high enough, the hottest regions on the planet’s surface can melt to produce “magma ponds” or localized magma oceans.

Reference:

S. E. Gelman et al. (2011), “Effects of Stellar Flux on Tidally Locked Terrestrial Planets: Degree-1 Mantle Convection and Local Magma Ponds”, ApJ 735: 72