- The Moon's orbit must be concave toward the Sun.
The Moon's orbit with respect to the Sun is always convex. This is easily proven by comparing the minimum possible gravitational acceleration of the Moon toward the Sun (5.7 mm/s2) and the maximum possible gravitational acceleration of the Moon toward the Earth (3.1 mm/s2). The acceleration vector, and hence the curvature, is always inward.
This means the Moon's orbit about the Sun doesn't look like either of the two images below:
Phil Plait is normally very good. This is one of those places where he was bad. The Moon's orbit about the Sun instead looks like this:
A close-up view:
- The Moon speeds up as it goes toward the Sun, and it slows down as it moves away.
Yes and no. The “yes” part is that the dominant feature of the Moon's orbit about the Sun is that the Moon orbits the Sun with the Earth. Ignoring the Moon's acceleration toward the Earth, the Moon's acceleration toward the Sun is greatest now in early July when the Earth/Moon system is closest to the Sun than at any other time.
The “no” part: The Moon is moving slightly away from the Sun when the Moon is in the second and third quarters. This is when the acceleration toward the Earth coincides with the acceleration toward the Sun. In the fourth and first quarters, the acceleration toward the Earth is directed against the acceleration toward the Sun, slowing the Moon down a bit with respect to the Sun.
For an observer on the Earth, the Moon appears to orbit the Earth roughly 13 times a year.
This is correct. More on this below.
The Earth, the Moon, and the Sun remain in the same plane.
This is incorrect. As noted by Diracology in his answer, the Moon's orbit about the Earth is slightly inclined with respect to the Earth's orbit about the Sun.
Note that one of the sites I referenced (http://www.wired.com/2012/12/does-the-moon-orbit-the-sun-or-the-earth/) asked whether the Moon orbits the Earth or the Sun. That blog erroneously came up with the answer that the Moon orbits the Sun rather than the Earth. Looking at acceleration or the shape of the orbit is the wrong way to look at the question “Does object A orbit object X or object Y?”
One answer to the question of “does the Moon orbit the Earth or the Sun?” is “Yes!” "Orbit" is not a mutually exclusive term. If one insists on a singular answer, the right way to look at things is via energy rather than acceleration. Since the Moon is gravitationally bound to the Earth, the Moon does indeed orbit the Earth.