91Ó°ÊÓ

The orbital period refers to the time an object takes to complete one full orbit around another object, in this case, an asteroid orbiting the Sun. This period can be calculated using Kepler’s Third Law, which articulates a clear relationship between the period of orbit (T) and the average distance from the sun (r), often expressed in astronomical units (AU).

Kepler’s Third Law posits that the square of the orbital period is proportional to the cube of the semi-major axis of the orbit. Mathematically, this relationship is expressed as:

• For orbiting bodies around the sun: \( T^2 \propto r^3 \)

In our exercise, Earth's orbital period \( (T_{Earth}) \) is 1 year with a distance of 1 AU from the Sun. Meanwhile, the asteroid's orbit is twice the distance, 2 AU. Using Kepler’s Law:

• Calculating for the asteroid: \( T_{asteroid}^2 = (2)^3 = 8 \)
• Thus, \( T_{asteroid} = \sqrt{8} \approx 2.83 \) years

Therefore, the asteroid takes approximately 2.83 Earth years to complete a single orbit around the Sun.

Kinetic energy, in the context of objects in space, is the energy that the object possesses due to its motion. For objects moving along circular orbits, such as our asteroid and Earth, kinetic energy is significant as it describes the dynamism of their motion along these paths.

The kinetic energy \( (KE) \) formula is given by:

• \( KE = \frac{1}{2}mv^2 \)

where \( m \) is the mass and \( v \) is the velocity of the object. The orbital velocity \( (v) \) is a function of the gravitational pull from the Sun and the radius of the orbit, expressed as:

• \( v = \sqrt{\frac{GM}{r}} \)

For our exercise, to compute the ratio of kinetic energy between the asteroid and Earth, consider:

• The asteroid's mass being \( 2 \times 10^{-4} \) of Earth's mass.
• Utilize the orbital velocity relations to derive that this kinetic energy ratio simplifies to \( KE_{ratio} = 7.08 \times 10^{-5} \).

Understanding these concepts requires a grasp of how mass and velocity significantly influence the kinetic energy of orbiting bodies.

Orbital velocity is the speed required for an object to sustain an orbit around another body, counteracting gravitational pull with centripetal force. It’s vital in ensuring a stable path without the object spiraling away or crashing into the central body.

The formula to calculate orbital velocity \( (v) \) around the sun is:

• \( v = \sqrt{\frac{GM}{r}} \)

where \( G \) is the gravitational constant, \( M \) is the mass of the Sun, and \( r \) is the radius of the orbit. This velocity maintains the balance essential for consistent circular orbits.

In our scenario:

• Earth has a known orbital velocity based on its 1 AU distance.
• The asteroid, being at 2 AU, has a different orbital velocity calculated by its distance: resulting in slower velocity by factor of \( \frac{1}{\sqrt{2}} \).

The decrease in orbital velocity arises because, despite the increase in distance, gravitational pull weakens, requiring less speed to maintain orbit. Understanding orbital velocity helps predict and analyze the movement of celestial bodies in space.