Q.Both earth and moon are subject to the gravitational force of the sun. As observed from the sun, the orbit of the moon

The key idea here relates to the Equivalence Principle and the superposition of gravitational forces.

1. Both the Earth and the Moon are subject to the Sun's gravitational force. Due to the Equivalence Principle, they experience nearly the same gravitational acceleration from the Sun, as their distances from the Sun are very similar.
2. This causes the entire Earth-Moon system to orbit the Sun.
3. However, the Moon is also under the significant gravitational influence of the Earth.
4. From the Sun's perspective, the total gravitational force on the Moon is the vector sum of the Sun's pull (which is central to the Sun) and the Earth's pull (which is not central to the Sun). …

To understand the Moon's orbit as observed from the Sun, we need to consider all significant gravitational forces acting on the Moon. An orbit is strictly elliptical only under specific conditions, which are not met in this three-body system.

Concept: Conditions for an Elliptical Orbit

An object follows a strictly elliptical orbit around a central body if and only if:

1. It is subject to a single gravitational force.
2. This force is always directed towards the central body (a central force).
3. The magnitude of this force is inversely proportional to the square of the distance from the central body ($F \propto 1/r^2$).

This is the essence of Kepler's laws of planetary motion, which describe the motion of a single planet around a single star. When additional bodies are present, the situation becomes more complex, leading to what is known as the three-body problem.

1. Identify the forces acting on the Moon:

   The Moon is primarily subject to two significant gravitational forces:

   • The gravitational force from the Sun ($\vec{F}_{SM}$).
   • The gravitational force from the Earth ($\vec{F}_{EM}$). The total gravitational force on the Moon is the vector sum of these two forces: $\vec{F}_{total} = \vec{F}_{SM} + \vec{F}_{EM}$.

2. Analyze the nature of these forces:

   • The force $\vec{F}_{SM}$ is always directed towards the Sun. This is a central force with respect to the Sun.
   • The force $\vec{F}_{EM}$ is always directed towards the Earth. This is a central force with respect to the Earth.

3. Determine if the total force is central with respect to the Sun:

   For the Moon's orbit around the Sun to be strictly elliptical, the total force $\vec{F}_{total}$ must always be directed towards the Sun. However, since the Earth is not located at the Sun's position, the force $\vec{F}_{EM}$ (which points towards the Earth) is generally not directed towards the Sun.

   Therefore, the vector sum $\vec{F}_{total} = \vec{F}_{SM} + \vec{F}_{EM}$ is generally not directed towards the Sun. This means the total gravitational force on the Moon is not a central force with respect to the Sun.

   Watch out

   A common misconception is that if the Sun's force is stronger, the Earth's influence can be ignored. While the Sun's gravitational pull on the Moon is indeed stronger than the Earth's pull on the Moon (by a factor of about 2), the Earth's force is still significant enough to perturb the Moon's orbit around the Sun, making it deviate from a perfect ellipse.

4. Evaluate the options based on this analysis:

   • (A) will be elliptical. This is incorrect because the total force on the Moon is not central with respect to the Sun. …

Step 1: From the Sun's frame, the Moon feels two significant pulls: the Sun's own (central — always directed toward the Sun) and Earth's (NOT central with respect to the Sun, since it's directed toward Earth's instantaneous position, which itself orbits the Sun).

Step 2: A gravitationally bound orbit is a perfect closed ellipse (Kepler's laws) ONLY when the net force is purely central (inverse-square, directed at a fixed focus). Here the net force on the Moon (Sun's pull + Earth's pull) is the vector sum of a central term and a non-central perturbation. …