Tidal decay and circularization of the orbits of short-period planets

It is well-known that tidal friction produces variations in the orbital elements of a close-in companion. In the case of an interacting pair with two extended (and tidally deformed) bodies, the tidal evolution depends on the rotational state of the bodies. We can identify two important examples: the satellite and the exoplanet cases. In the first case, the rotation of the central body (a parent planet) is generally much faster than the mean orbital motion of the satellite orbiting the primary. In the exoplanet case, the rotation of the central body (a parent star) is generally much slower than the mean orbital motion of the planet. Changes in orbital elements are also accompanied by variations in the rotation of each deformed body. The reader is referred to Ferraz-Mello et al. (2008) for further details.

Consider a two-body system formed by a single star and a short-period companion planet. We suppose that both bodies are able to be tidally deformed due to the mutual interaction. Tides on each body provokes variations in the elements of the relative orbit and their rotations. The equations that govern the mean orbital changes in the astrocentric orbital elements of an exoplanet are, following Ferraz-Mello et al. (2008) (correcting a misprint in equation ( 93) and neglecting the inclinations and the terms proportional to Ω/n):

where n, a and e are the mean orbital motion, semi-major axis and eccentricity while m * ,R * , k d * and m p ,R p , k dp are the masses, radii and dynamical Love numbers of star and planet, respectively. We stress that the above equations are valid only in the case of star-exoplanet interacting pair. The main equations for the planetsatellite case are different (Ferraz-Mello et al. 2008).

The parameter D is defined as

where ǫ ′ 0 * and ǫ ′ 2p are lag angles associated to the tidal waves whose frequencies are 2Ω * -2n (on the star) and 2Ω p -n (on the exoplanet), respectively. Here, Ω is the angular velocity of rotation of the tidally deformed body. The lag angles come from the delay in their response to the tide raising potentials (see Ferraz-Mello et al., 2008 for details). Several hypotheses were done to obtain the above equations. The planetary rotation was assumed in a quasi-synchronous state (Ω p ∼ n) and the star rotation verifies Ω * ≪ n. For the star, a linear model is assumed in the relationship between lag angles and frequencies (Darwin 1880;Mignard 1979). In this case, each lag angle is proportional to the frequency of the corresponding tidal wave, and the coefficient of proportionality ∆t * (time delay), is the same for all frequencies. No model needs to be assumed for the planetary tide lags, except that equal frequencies are assumed to span equal lags. The motion is supposed planar, i.e the reference and orbital planes coincide (zero obliquities). The equations are valid only up to second order in the eccentricity.

In order to simplify the notation, we define the following parameters:

where Q are quality factors defined by

(in general k d and Q are poorly known quantities for both stars and planets). Hence,

(2.5) Introducing (2.4) into equations (2.1) and (2.2) and using the third Kepler law to relate ȧ to ṅ, ȧ = -2a 3n ṅ , we have

(2.7)

Note that, for each equation, the terms proportional to ŝ and p are the contribution to the total variation of the tides raised on the star and planet, respectively.

It is clear from equations (2.6) and (2.7) that the effect on the orbital elements due to the tidal interaction is to reduce the size and eccentricity of the relative orbit. In this section, we discuss the decreasing timescales of semi-major axis and eccentricity.

The decreasing timescale of the semi-major axis can be defined as τ a ≡ a/| ȧ|. Using equation (2.6),

If we need to know the timescale only due to planetary tides, it is enough to put ŝ = 0 (or Q -1 * = 0) in the above equation to obtain

Note that lim τ p a = ∞ as e → 0, indicating that, when the only tide is the planetary, the semi-major axis stops decreasing after circularization. The contribution of the stellar tide follows a similar analysis putting p = 0 (or

< ∞ as e → 0. This shows that after circularization the semi-major axis continues to decrease due to the stellar tide.

The timescale of orbital circularization can be defined as τ e ≡ e/| ė|. Using (2.7)

As in the case of the semi-major axis, we can compute the individual timescale due to each tide. For the planetary and stellar tides we have, respectively In order to have an idea of the numerical values of the above quantities, we give a simple example. Consider a planet-star system with masses and radii equal to those of the Sun and Jupiter. We have p ŝ ≈ 50

In the above equation, typical values are 2.5 × 10 4 < Q p /k dp < 2.5 × 10 5 (Laskar & Correia 2004) and 2 × 10 7 < Q * /k d * < 1.5 × 10 9 (Carone & Pätzold 2007). Hence

Qp ≥ 1 and p ŝ ≫ 1. Then from (3.6), τ p e ≪ τ * e , meaning that the tidal evolution of the eccentricity is m

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