New Tidal Evolution Model
Matija Ćuk, Doug Hamilton, Simon Lock, and Sarah Stewart published a new model for the tidal evolution of the Moon: Tidal Evolution of the Moon from a high-obliquity, high-angular-momentum Earth, Nature, published online October 31, 2016.
Open access ReadCube version of the paper.
The model begins with a giant impact that tilts the Earth’s spin axis between 60 to 80 degrees from the ecliptic and leaves the Earth with a 2 to 3 hour day. Thre is about a 30% probability that a giant impact would leave the Earth tilted in this range. From this initial state, the Moon has two dramatic stages of evolution, illustrated in the cartoon and movies below.
Viewing the Earth-Moon system from the Sun in the plane of the solar system (ecliptic plane).
A. After the giant impact, the moon forms in the Earth’s equatorial plane with zero inclination. The obliquity of the Earth is 70 degrees from the ecliptic plane. The Earth is spinning so quickly that its equator is twice the length of its pole (2 to 3 hour day).
B. After the Laplace plane transition (Video 1), angular momentum is transferred away from the Earth-Moon system and the Earth becomes spherical. The Earth’s obliquity is near present day, 23.5 degrees and the inclination of the Moon’s orbit is 30 degrees.
C. After the Cassini state transition (Video 2), the Moon’s orbit is lowered to the present-day 5 degrees from the ecliptic plane.
Orbital distances not to scale. Image credit: Sarah Stewart (high resolution jpg, vector PDF)
Video 1: Laplace Plane Transition
Caption: Animation of relative orientations of Earth’s spin and the Moon’s orbit during the Laplace plane transition, following the simulation plotted in black in Fig. 1 of Cuk et al. 2016. The system is seen from the direction of Earth’s vernal equinox, the blue arrow is plotted along Earth’s spin axis and points to the north, while the lunar orbit is plotted in red. Initially the Earth has a high obliquity, the Moon has low inclination and the Laplace plane is close to Earth’s equator. As the animation progresses and the lunar orbit grows due to tidal dissipation, the Laplace plane shifts to the ecliptic plane (horizontal in this view). The moon acquires a large inclination during the Laplace plane transition, while Earth’s obliquity decreases. The labels show time, Earth’s spin period, total angular momentum (scaled to the present value) and angular momentum of the Earth-Moon system where only the ecliptic component (i.e. that along the vertical axis) of the lunar orbital momentum is taken into account. Unlike total angular momentum, this ecliptic component will be conserved during the Cassini state transition.
Video 2: Cassini State Transition
Caption: Animation of relative orientations of lunar figure and orbit during the Cassini state transition, following the simulation plotted in Fig. 4 of Cuk et al. 2106. The Moon is seen from the direction of the ascending node of lunar orbit, with the ecliptic plane (i.e. the Moon’s Laplace plane at this time) parallel to the horizontal axis. The red arrow shows the orientation of the Moon’s orbit normal. At first the Moon’s orbit normal and spin axis are on the same side of the normal to the ecliptic, indicating that the Moon is in Cassini state 1. Once the Cassini state 1 is destabilized, after some wobbling, the Moon settles in a non-synchronous state somewhat similar to the Cassini state 2 (with the orbit normal and the spin axis being on opposite sides of the normal to the ecliptic). During this time both the inclination and obliquity (which is forced by inclination) are being damped by strong obliquity tides. At the semimajor axis of 35.1 Earth radii, the Moon becomes synchronous again and enters the Cassini state 2, where it stays for the rest of the simulation (this event is visible as a 5-degree jump in obliquity).
Is this still the Giant Impact model? Yes but …..
Our model for the origin of the Moon still begins with a giant impact. But we have changed everything about the event from what is found in textbooks today. The energy is larger (by about a factor of 10 or more), the angular momentum is larger (by about a factor of 2), the tilt of the Earth is larger (between 60-80 degrees, instead of near the present-day 23 degrees).
One of the elegant features of our model is that a single event leads to the unique chemical relationship between Earth and the Moon, the present length of day (angular momentum), and the present inclination of the lunar orbit.
Image from eskipaper.com
What’s changed from the previous models?
In the Giant Impact model of the moon’s formation, the young moon began its orbit within Earth’s equatorial plane. In the standard variant of this model (top panel), Earth’s tilt began near today’s value of 23.5 degrees. The moon would have moved outward smoothly along a path that slowly changed from the equatorial plane to the “ecliptic” plane, defined by Earth’s orbit around the sun. If, however, Earth had a much larger tilt after the impact (~75 degrees, lower panel) then the transition between the equatorial and ecliptic planes would have been abrupt, resulting in large oscillations about the ecliptic. The second picture is consistent with the moon’s current 5-degree orbital tilt away from the ecliptic. Image credit: Doug Hamilton
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What are obliquity tides?
Obliquity tides result from the apparent North-South motion of the Earth as viewed from the Moon. Figure from Matija Ćuk (not to scale).
For a synchronously rotating moon with a non-zero obliquity the tidal bulge moves North-South once every orbit.
Normal tides cause the Moon’s orbit to increase and Earth’s day to decrease.
New Clues from Potassium Isotopes
A new study by Kun Wang and Stein Jacobsen finds a very slight difference in potassium isotopes between the Earth and Moon. The new measurements find that lunar rocks are slightly enriched in the heavy potassium isotope. This work supports the moon formation model by Lock et al. (see LPSC presentations below), where lunar material condenses from Earth vapor after a high-energy, high-angular momentum giant impact.
Wang, Kun, and Stein B. Jacobsen. “Potassium isotopic evidence for a high-energy giant impact origin of the Moon.” Nature 538.7626 (2016): 487-490.