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Gravitational attraction is the force that draws two masses toward each other. It is governed by Newton's law of universal gravitation, which states that every point mass in the universe attracts every other point mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.

The formula can be expressed as: \[\begin{equation} F = G \frac{m_1 m_2}{r^2} \end{equation}\]where:

• F is the gravitational force between the masses,
• G is the gravitational constant ( 6.674 \times 10^{-11} \ m^3kg^{-1}s^{-2}),
• m_1 and m_2 are the masses of the two objects, and
• r is the distance between the centers of the two masses.

In the orbital mechanics problem described, the two masses in question are Earth and the Sun. The gravitational force between them keeps Earth in a stable orbit around the Sun. If Earth were to stop in its orbit, the gravitational attraction would cause the Earth to begin falling directly toward the Sun.

Kepler's laws of planetary motion describe how planets orbit around the Sun. These laws are critical to understanding orbital mechanics and solving related problems.

• First Law (The Law of Ellipses)

  Planets move in elliptical orbits with the Sun at one focus. The path of the Earth around the Sun is not a perfect circle but an ellipse, though it is fairly close to circular.

• Second Law (The Law of Equal Areas)

  A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. This implies that a planet moves faster when it is closer to the Sun and slower when it is farther away.

• Third Law (The Law of Harmonies)

  The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit. This law provides a relation between the time a planet takes to orbit the Sun (its period) and its average distance from the Sun.

In our exercise, the third law is particularly relevant. It allows us to calculate the average distance between the Earth and the Sun, referred to as the semi-major axis ( a), from the period of Earth's orbit, thus setting up the basis for determining the time it would take for the Earth to fall into the Sun if it stopped in its orbit.

The free-fall time calculation refers to determining how long it would take an object to fall a certain distance under the influence of gravity alone. This is an important concept in physics and is particularly relevant to understanding how objects behave under the influence of gravitational forces in space.The time ( t) it takes for an object starting from rest to fall a distance ( d) under constant acceleration ( g) is given by: \[\begin{equation} t = \sqrt{\frac{2d}{g}} \end{equation}\]For the Earth falling towards the Sun, the distance is the average distance from the Earth to the Sun, and the acceleration due to gravity ( g) varies as we move away from the Earth's surface. The acceleration due to gravity at any distance from the Sun can be calculated using the modified gravitational formula: \[\begin{equation} g = \frac{GM}{r^2} \end{equation}\]where ( M) is the mass of the Sun, and ( r) is the distance between the center of the Sun and the Earth. After calculating ( g) and using the average distance as our ( d), we can determine the free-fall time. This exercise illustrates the application of these principles in a hypothetical scenario where the Earth stops its orbit and experiences free fall towards the Sun.