3 Synchronization of Pegasi planets

It has been shown that the tides raised by the star on Pegasi planets$\,\,$should rapidly drive them into synchronous rotation (Guillot et al. 1996; Marcy et al. 1997; Lubow et al. 1997). This can be seen by considering the time scale to tidally despin the planet (Goldreich & Soter 1966; Hubbard 1984):

$$\tau_{\rm syn} \approx Q \left({R^3\over GM}\right)(\omega-\o... ... s}) \left({M\over M_\star}\right)^2\left({a\over R}\right)^6,$$ (1)

where Q, R, M, a, $\omega$ and $\omega_{\rm s}$ are the planet's tidal dissipation factor, radius, mass, orbital semi-major axis, rotational angular velocity, and synchronous (or orbital) angular velocity. M_* is the star's mass, and G is the gravitational constant. Factors of order unity have been omitted. A numerical estimate for HD 209458b (with $\omega$ equal to the current Jovian rotation rate) yields a spindown time $\tau_{\rm syn}\sim 3Q$years. Any reasonable dissipation factor Q (see Marcy et al. 1997; Lubow et al. 1997) shows that HD 209458b should be led to synchronous rotation in less than a few million years, i.e., on a time scale much shorter than the evolution timescale. Like other Pegasi planets, HD 209458b is therefore expected to be in synchronous rotation with its 3.5-day orbital period.

Nevertheless, stellar heating drives the atmosphere away from synchronous rotation, raising the possibility that the interior's rotation state is not fully synchronous. Here, we discuss (1) the energies associated with the planet's initial transient spindown, and (2) the possible equilibrium states that could exist at present.

3.1 Spindown energies

Angular momentum conservation requires that as the planet spins down, the orbit expands. The energy dissipated during the spindown process is the difference between the loss in spin kinetic energy and the gain in orbital energy:

$$\dot{E}=-{{\rm d}\over {\rm d}t}\left({1\over 2}k^2 MR^2\omega^2 -{1\over 2}M a^2\omega_{\rm s}^2 \right),$$ (2)

where k is the dimensionless radius of gyration ( k²=I/MR², I being the planet's moment of inertia). The time derivative is negative, so $\dot{E}$, the energy dissipated, is positive. The orbital energy is the sum of the planet's gravitational potential energy and orbital kinetic energy and is negative by convention. The conservation of angular momentum implies that the rate of change of $\omega_{\rm s}$ is constrained by that on $\omega$:

$${{\rm d}\over {\rm d}t}\left(M a^2\omega_{\rm s}+ k^2 MR^2\omega\right)=0.$$ (3)

The fact that the planetary radius changes with time may slightly affect the quantitative results. However, since $\tau_{\rm syn}$ is so short, it can be safely neglected in this first-order estimate. Rbeing held constant, it is straightforward to show, using Kepler's third law, that:

$$\dot{E}=-k^2 MR^2 (\omega-\omega_{\rm s})\dot{\omega}.$$ (4)

(Note that $\dot{\omega}$ is negative, and so $\dot{E}$ is positive.)

The total energy dissipated is $E\approx k^2 MR^2 (\omega_{\rm s}-\omega)^2/2$, neglecting variation of the orbital distance. Using the moment of inertia and rotation rate of Jupiter (k²=0.26 and $\omega=1.74\times 10^{-4}\,\rm\, s^{-1}$), we obtain for HD 209458b $E\approx 4\times10^{41}\rm\, erg$. If this energy were dissipated evenly throughout the planet, it would imply a global temperature increase of 1400K.

By definition of the synchronization timescale, the dissipation rate can be written:

$$\dot E = {k^2 M R^2 (\omega - \omega_{\rm s})^2\over \tau_{\rm syn}}\cdot$$ (5)

With Q of 10⁵, a value commonly used for Jupiter, $\tau_{\rm syn}\sim 3\times 10^5$ years and the dissipation rate is then $\sim$ $10^{29}\rm\, erg\rm\, s^{-1}$, or 35000 times Jupiter's intrinsic luminosity. Lubow et al. (1997) have suggested that dissipation in the radiative zone could exceed this value by up to two orders of magnitude, but this would last for only $\sim$100 years.

The thermal pulse associated with the initial spindown is large enough that, if the energy is dissipated in the planet's interior, it may affect the planet's radius. It has previously been argued (Burrows et al. 2000) that Pegasi planets$\,\,$must have migrated inward during their first 10⁷years of evolution; otherwise, they would have contracted too much to explain the observed radius of HD 209458b. But the thermal pulse associated with spindown was not included in the calculation, and this extra energy source may extend the time over which migration was possible. Nevertheless, it seems difficult to invoke tidal synchronization as the missing heat source necessary to explain HD 209458b's present (large) radius. High dissipation rates are possible if $\tau_{\rm syn}$ is small, but in the absence of a mechanism to prevent synchronization, $\dot{E}$ would drop as soon as $t > \tau_{\rm syn}$. The most efficient way of slowing the planet's contraction is then to invoke $\tau_{\rm syn}\sim 10^{10}$ years. In that case, the energy dissipated becomes $\dot E\sim 10^{24}\rm\, erg\rm\, s^{-1}$, which is two orders of magnitude smaller than that necessary to significantly affect the planet's evolution (Paper I; Bodenheimer et al. 2001). For the present-day dissipation to be significant, an initial rotation rate 10 times that of modern-day Jupiter would be needed. But the centripetal acceleration due to rotation exceeds the gravitational acceleration at the planet's surface for rotation rates only twice that of modern-day Jupiter, so this possibility is ruled out. Furthermore, such long spindown times would require a tidal Q of $\sim$ 10⁹-10¹⁰, which is $\sim$10⁴ times the Q values inferred for Jupiter, Uranus, and Neptune from constraints on their satellites' orbits (Peale 1999; Banfield & Murray 1992; Tittemore & Wisdom 1989). Dissipation of the energy due to transient loss of the planet's initial spin energy therefore cannot provide the energy needed to explain the radius of HD 209458b.

Another possible source of energy is through circularization of the orbit. Bodenheimer et al. (2001) show that the resulting energy dissipation could reach $10^{26}\rm\, erg\rm\, s^{-1}$ if the planet's tidal Q is 10⁶ and if a hypothetical companion planet pumps HD 209458b's eccentricity to values near its current observational upper limit of 0.04. If such a companion is absent, however, the orbital circularization time is $\sim$10⁸ years, so this source of heating would be negligible at present. Longer circularization times of 10⁹-10¹⁰ years would allow the heating to occur until the present-day, but its magnitude is then reduced to $10^{25}\rm\, erg\rm\, s^{-1}$ or lower, which is an order of magnitude smaller than the dissipation required.

3.2 The equilibrium state

The existence of atmospheric winds implies that the atmosphere is not synchronously rotating. Because dynamics can transport angular momentum vertically and horizontally (including the possibility of downward transport into the interior), the interior may evolve to an equilibrium rotation state that is asynchronous. Here we examine the possibilities.

Let us split the planet into an "atmosphere'', a part of small mass for which thermal effects are significant, and an "interior'' encompassing most of the mass which has minimal horizontal thermal contrasts. Suppose (since $\tau_{\rm syn}$ is short) that the system has reached steady state. Consider two cases, depending on the physical mechanisms that determine the gravitational torque on the atmosphere.

The first possibility is that the gravitational torque on the atmosphere pushes the atmosphere away from synchronous rotation (i.e., it increases the magnitude of the atmosphere's angular momentum measured in the synchronously-rotating reference frame) as has been hypothesized for Venus (Ingersoll & Dobrovolskis 1978; Gold & Soter 1969). To balance the torque on the atmosphere, the interior must have a net angular momentum of the same sign as the atmosphere (so that both either super- or subrotate). In Fig. 3, superrotation then corresponds to a clockwise flux of angular momentum, whereas subrotation corresponds to the anticlockwise scenario.

Figure 3: Angular momentum flow between orbit, interior, and atmosphere for a Pegasi planet$\,\,$in steady state. Arrows indicate flow of prograde angular momentum (i.e., that with the same sign as the orbital angular momentum) for two cases: Anticlockwise: gravitational torque on atmosphere is retrograde (i.e., adds westward angular momentum to atmosphere). For torque balance, the gravitational torque on the interior must be prograde (i.e., eastward). These gravitational torques must be balanced by fluid-dynamical torques that transport retrograde angular momentum from atmosphere to interior. Clockwise: gravitational torque on atmosphere is prograde, implying a retrograde torque on the interior and downward transport of prograde angular momentum from atmosphere to interior. Atmosphere will superrotate if gravitational torques push atmosphere away from synchronous (as on Venus). It will subrotate if gravitational torques synchronize the atmosphere (e.g., gravity-wave resonance; cf. Lubow et al. 1997).

The second possibility is that the gravitational torque on the atmosphere tends to synchronize the atmosphere. This possibility may be relevant because, on a gas-giant planet, it is unclear that the high-temperature and high-pressure regions would be $90^\circ$ out of phase, as is expected on a terrestrial planet. If the interior responds sufficiently to atmospheric perturbations to keep the deep isobars independent of surface meteorology (an assumption that seems to work well in modeling Jupiter's cloud-layer dynamics), then high-pressure and high-temperature regions will be in phase, which would sweep the high-mass regions downwind and lead to a torque that synchronizes the atmosphere. Furthermore, if a resonance occurs between the tidal frequency and the atmosphere's wave-oscillation frequency, the resulting gravitational torques also act to synchronize the atmosphere (Lubow et al. 1997). In this case, the interior and atmosphere have net angular momenta of opposite signs. Depending on the sign of the atmosphere/interior momentum flux, this situation would correspond in Fig. 3 to either the clockwise or anticlockwise circulation of angular momentum cases.

In both cases above, the gravitational torque on the atmosphere arises because of spatial density variations associated with surface meteorology, which are probably confined to pressures less than few hundred bars. Therefore, this torque acts on only a small fraction of the planet's mass. In contrast, the gravitational torque on the gravitational tidal bulge affects a much larger fraction of the planet's mass. It is not necessarily true that the torque on the atmosphere is negligible in comparison with the torque on the interior, however, because the gravitational tidal bulge is only slightly out of phase with the line-of-sight to the star (e.g., the angle is 10^-5 radians for Jupiter). In contrast, the angle between the meteorologically-induced density variations and the line of sight to the star could be up to a radian. Detailed calculations of torque magnitudes would be poorly constrained, however, and will be left for the future.

Several mechanisms may act to transfer angular momentum between the atmosphere and interior. Kelvin-Helmholtz shear instabilities, if present, would smooth interior-atmosphere differential rotation. On the other hand, waves transport angular momentum and, because they act nonlocally, often induce differential rotation rather than removing it. The quasi-biennial oscillation on Earth is a classic example (see, e.g., Andrews et al. 1987, Chapter 8). Atmospheric tides (large-scale waves forced by the solar heating) are another example of such a wave; on Venus, tides play a key role in increasing the rotation rate of the cloud-level winds. Finally, vertical advection may cause angular momentum exchange between the atmosphere and the interior of Pegasi planets, because the angular momentum of air in updrafts and downdrafts need not be the same.

A simple estimate illustrates the extent of nonsynchronous rotation possible in the interior. Suppose that the globally-averaged flux of absorbed starlight is $F_{\tiny\RIGHTcircle}$, which is of order $10^8\rm\,erg\,s^{-1}\,cm^{-2}$ for Pegasi planets$\,\,$near 0.05 AU, and that the globally-averaged flux of kinetic energy transported from the atmosphere to the interior is $\eta F_{\tiny\RIGHTcircle}$, where $\eta$ is small and dimensionless. If this kinetic energy flux is balanced by dissipation in the interior with a spindown timescale of $\tau_{\rm syn}$, then the deviation of the rotation frequency from synchronous is

$$\omega - \omega_s = \left({4 \pi \eta F_{\tiny\RIGHTcircle}\tau_{\rm syn}\over k^2 M}\right)^{1/2}\cdot$$ (6)

This estimate is an upper limit for the globally averaged asynchronous rotation rate because it assumes that kinetic energy transported into the interior has angular momentum of a single sign. The global-average asynchronous rotation could be even lower if angular momenta transported downward in different regions have opposite signs. The spindown time is uncertain and depends on the planet's Q according to Eq. (1). To date, only one estimate for the Q of a Pegasi planet$\,\,$exists (Lubow et al. 1997), which suggests $Q\sim 100$, at least during the early stages of spindown. Orbital constraints on the natural satellites of the giant planets imply that, at periods of a few days, the tidal Q of Uranus, Neptune, and Jupiter are of order $\sim$10⁵ (Tittemore & Wisdom 1989; Banfield & Murray 1992; Peale 1999), which suggests a spindown time of a few $\times10^5$ years (Eq. (1)). Although the mechanisms that determine these planets' Q values remain uncertain, likely possibilities include friction in the solid inner core (Dermott 1979) or overturning of tidally-forced waves in the fluid envelope (Ioannou & Lindzen 1993, Houben et al. 2001). Both mechanisms are possible for Pegasi planets, and the wave-dissipative mechanism should be more effective for Pegasi planets$\,\,$than for Jupiter because of the thicker stable (radiative) layer in the former (Houben et al. 2001). These calculations suggest that spindown times of $\sim$ 10⁵-10⁶ years are likely, although larger values cannot be ruled out.

As discussed in Sect. 4, experience with planets in our solar system suggests that atmospheric kinetic energy is generated at a flux of $10^{-2} F_{\tiny\RIGHTcircle}$, and if all of this energy enters the interior, then $\eta \sim 10^{-2}$. Using a spindown time of $3\times10^5\,$years implies that $\omega - \omega_{\rm s} \sim 2\times 10^{-5}~\rm\, s^{-1}$, which is comparable to the synchronous rotation frequency. The implied winds in the interior are then of order $\sim$ $2000\rm\, m\rm\, s^{-1}$. Even if $\eta$ is only 10^-4, the interior's winds would be $200\rm\, m\rm\, s^{-1}$. The implication is that the interior's spin could be asynchronous by up to a factor of two, depending on the efficiency of energy and momentum transport into the interior. If spindown times of $\sim$10⁶ years turn out to be serious under- or overestimates, then greater or lesser asynchronous rotation would occur, respectively.