Final answer:
To keep a perfectly circular orbit around the star, the planet must have a speed of 4,429 m/s.
Step-by-step explanation:
To keep a perfectly circular orbit around the star, the planet's gravitational force must provide the necessary centripetal force. The centripetal force equation is given by F = (mv^2)/r, where F is the force, m is the mass of the planet, v is the velocity of the planet, and r is the distance from the planet to the star.
In this case, the mass of the planet is 4 x 10^14 kg and the distance to the star is 590 km (or 590,000 m). The mass of the star is 82 times the mass of the planet. We can use this information to calculate the speed needed for the circular orbit.
The gravitational force between the planet and the star is given by F = G(m_p)(m_s)/r^2, where G is the gravitational constant, m_p is the mass of the planet, m_s is the mass of the star, and r is the distance between them.
The centripetal force is provided by the gravitational force, so we can equate these two forces: F = (mv^2)/r = G(m_p)(m_s)/r^2.
By substituting the values given, we can solve for v: v = sqrt((G(m_s))/r), where G = 6.67430 x 10^-11 N(m/kg)^2.
Substituting the values into the equation, we get: v = sqrt((6.67430 x 10^-11 N(m/kg)^2)(82(4 x 10^14 kg))/(590,000 m)).
Calculating the numerical value gives us v = 4,429 m/s