# Why circularization of an orbit has longer time scale than tidal locking?

I'm trying to understand the basic physics of orbital evolution. I know that in a two-body system (a planet orbiting a sun for example), eccentric orbits become circular, and the spin of the planet becomes tidally locked (like the Earth-moon system). However, I don't understand why the timescales of these two processes are different (as stated for example in Storch & Lai (2013)), and if there's a simple way to quantify the timescale of circularization (kind of back of the envelope calculation, like the one for tidal locking found here)?

Tidal locking is a primary effect due to a direct torque on a body from its tidal bulge lagging its rotation. This is pictured nicely on wikipedia with an example from the Earth/Moon system.

Tidal Orbital Circularization is a secondary effect. To quote Daddy Kropotkin's excellent answer to

Is the moon's orbit circularizing? Why does tidal heating circularize orbits?

tidal torque drives dissipation, and this dissipation brings the binary to a minimum kinetic energy state, i.e. circular orbit, synchronized spins with the orbit, aligned spins with the orbit.

There is really no reason to expect the timescales for these two effects to be similar since there are different mechanisms at play.

Another way to think about tidal circularization is to imagine a tidally locked planet in an eccentric orbit. From Kepler's 2nd law, we know the planet moves fastest at periapsis and slowest at apoapsis. So the planet's tidal bulge won't always quite face its star. Instead, the tidal bulge will lead and lag its orientation to the star. This offset will reduce orbital energy at periapsis and increase orbital energy at apoapsis, circularizing the orbit. Tidal heating will also continue to occur, since the tidal bulge will continue to move until the orbit comes to a minimum energy state (a circular orbit).

Is there a simple way to quantify the timescale of circularization?

No. There is a way, but it is not simple. From Rodriguez and Ferraz-Mello 2009, the time scale for orbital circularization of a short period planet in a 2-body system is

$$\tau_e=\frac{3n^{-1}a^5}{18\hat{s}+7\hat{p}}$$

Here, $$n$$ is the mean orbital motion, $$a$$ is the orbital semi-major axis, and $$\hat{s} = \frac{9k_{d*}m_p}{4Q_*m_*}R_*^5$$ $$\hat{p} = \frac{9k_{dp}m_*}{2Q_pm_p}R_p^5$$

where $$Q$$ are quality factors defined by $$Q_*=|\epsilon'_{0*}|^{-1}$$ and $$Q_p=|\epsilon'_{2p}|^{-1}$$. Here, $$\epsilon'_{0*}$$ and $$\epsilon'_{2p}$$ are lag angles associated to the tidal waves whose frequencies are $$2\Omega_*-2n$$ for the star and $$2\Omega_p-n$$ for the planet. The angular velocity of the rotation of the tidally deformed body is $$\Omega$$. The masses, radii, and dynamical Love numbers for the star and planet are respectively $$m_*$$,$$R_*$$,$$k_{d*}$$ and $$m_p$$,$$R_p$$,$$k_{dp}$$.

The above equations also assume no obliquities. That is, the axis of rotation of each body is perpendicular to the orbital plane. Without this assumption, things really start to get complicated.

The locking time equation for a planet around a star is much simpler: $$\tau_{l}=\frac{2\Omega_p a^6m_pQ}{15Gm_*^2k_{dp}R_p^3}$$

Here, $$G$$ is the gravitational constant, and $$Q$$ is the dissipation function of the planet.

Notes:

1. The Earth-Moon system is not an ideal system to discuss tidal orbital circularization since the gravitational effects of the Sun are acting against such circularization.

2. My tidal locking time equation looks different from Wikipedia because I wanted to be consistent with notation internal to this answer.

3. In general, tidal lock will occur faster than orbital circularization because tidal lock is dependent on the amount of lag in the tidal bulge, whereas circularization is dependent on the change in the lag in the tidal bulge as a function of orbital speed change. However, there may be cases in which tidal circularization occurs sooner than tidal lock. Take Venus, for example, which has the least eccentric orbit of the planets, but may be prevented from tidal lock with the Sun due to torque from its atmospheric tides.

• Thank you! I see what you say about the timescales being different, but can you please expand on why the secondary effect (circularization) will always be longer timescale? Also, why circular orbit is minimum energy compared to eccentric orbit? Feb 23, 2022 at 19:34