You are to put a spacecraft into a synchronous circular orbit around the Martian equator, so that its orbital period is equal to the planet's rotation period. Such a spacecraft would always be over the same part of the Martian surface. (a) Find the radius of the orbit and the altitude of the spacecraft above the Martian surface. (b) Suppose Mars had a third moon that was in a synchronous orbit. Would tidal forces make this moon tend to move toward Mars, away from Mars, or neither? Explain.

Orbital mechanics, also known as celestial mechanics, is a branch of astrophysics that deals with the motions of celestial objects, such as stars, planets, moons, and artificial satellites. It involves the application of physical laws to predict and understand the orbital behavior of these bodies.

When placing a spacecraft into a synchronous orbit around a planet, such as Mars, the goal is to have the spacecraft's orbital period match the rotation period of the planet. This intricate dance between a spacecraft and the celestial body it orbits relies on calculating the precise velocity and altitude for the spacecraft to remain in a fixed position relative to the planet's surface.

Using the principles of orbital mechanics, one can determine the radius and speed of an orbit that fulfills specific conditions, such as a geostationary or in this case, a synchronous orbit around Mars. These calculations take into account the mass of the planet, its radius, and the gravitational constant, which influences the gravitational force exerted on the spacecraft.

Kepler's Third Law of Planetary Motion states that the square of the orbital period (\( T \)) of a planet is directly proportional to the cube of the semi-major axis (\( r \)) of its orbit. The law can be mathematically expressed as:
\( T^2 = \frac{4\text{\textpi}^2}{G*M_m} * r^3 \),
where \( G \) is the gravitational constant, and \( M_m \) is the mass of the central body around which the satellite orbits.

This law is used to calculate the radius of an orbit given the period, which is the time it takes for one complete orbit. In the provided exercise, Kepler's Third Law helps us find the necessary orbital radius for a spacecraft to maintain a synchronous orbit around Mars. By reorganizing the law and inserting the known values, the orbit's radius that allows it to synchronize with Mars' rotation can be computed, thereby determining where the spacecraft will orbit in relation to the Martian surface.

Tidal forces are the result of the varying gravitational pull exerted by a celestial body on different parts of another body. These forces are responsible for phenomena such as the ocean tides on Earth due to the Moon's gravitational influence.

The specificity of tidal forces is that they affect bodies differently depending on their location within the gravitational field. In the case of a synchronous orbit, the tidal forces work to distort the shape of the orbiting body, stretching it along the line toward the planet it orbits.

However, in a perfect synchronous orbit, tidal forces are in a sort of equilibrium with the gravitational force that maintains the body's orbit. This equilibrium means that the object would neither fall towards the planet nor drift away, so long as no other perturbations or forces act on it. Thus, with Mars' hypothetical third moon in a similar state, it would not experience a net movement towards or away from Mars due to tidal forces alone, suggesting a stable orbit, barring external influences.