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The gravitational force is a fundamental concept in physics, especially in orbital mechanics. It is the force of attraction between two masses. This force is crucial for celestial bodies, like planets and stars, as it keeps them in orbit around each other. In the context of the planet orbiting Rho Cancri, the star's gravitational pull keeps the planet in motion around it.

The mathematical expression for gravitational force is given by the equation:

• Gravitational Force: \( F = \frac{G M m}{r^2} \)

Here, \( G \) represents the gravitational constant \( (6.674 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2) \), \( M \) is the mass of the star, \( m \) is the mass of the planet, and \( r \) is the distance between the centers of the two bodies.

This gravitational force acts as the centripetal force, which is necessary for maintaining the planet’s circular orbit. By equating gravitational force to centripetal force (

• Centripetal Force: \( F = \frac{m v^2}{r} \)

), we can calculate the orbital speed of the planet.

Orbital speed is the velocity that a planet must have to remain in a stable orbit around a star. It is derived from the balance between the gravitational pull from the star and the centrifugal force due to its motion. For the planet orbiting Rho Cancri, the orbital speed is determined using the formula:

• \( v = \sqrt{\frac{G M}{r}} \)

Here, \( G \) is the gravitational constant, \( M \) is the mass of the star, and \( r \) is the radius of the planet's orbit. Substituting the known values:

• \( M = 0.85 \times M_{\text{sun}} \)
• \( r = 0.11 \times r_{\text{earth}} \)

This equation shows that the speed depends on the mass of the star and the distance of the planet from the star.

The derived orbital speed for this planet is approximately \( 4.56 \times 10^{4} \, \text{m/s} \). This speed ensures that it stays in a stable orbit, preventing it from drifting away into space or spiraling into the star.

The orbital period is the time it takes for a celestial body, like a planet, to complete one full orbit around a star. This period is influenced by the orbital speed and the radius of the orbit. For the planet orbiting Rho Cancri, you can calculate this period using the formula:

• \( T = \frac{2 \pi r}{v} \)

Where \( r \) is the orbital radius, and \( v \) is the orbital speed. By substituting the previously known values:

• \( r = 0.11 \times 1.496 \times 10^{11} \, \text{m} \)
• \( v \approx 4.56 \times 10^{4} \, \text{m/s} \)

This results in an orbital period \( T \approx 16 \times 10^{6} \, \text{seconds} \), which converts to about 189 days.

Thus, it takes approximately 189 Earth days for the planet to travel once around its star. Understanding the orbital period is crucial for astronomers, as it helps in predicting the planet's position at different times, which is essential for further exploration of planetary systems.