Answer Happy

Planetary motion in polar coordinates The Earth (E) moves in an elliptical orbit around the Sun (S), as shown in the figure below: b semi minor center GM a₁ = -- a semi-major In polar coordinate, the equation of motions for the radial and angular coordinate (r, 0) of E with respect to the Sun, are given by +r a*e X mr
where L = mr r² dº . L = m√/GMa(1 – e²) is a constant of the system, denoting the angular moment of the Earth about the Sun; a = 1.496 × 10¹¹ m (=1 A.U) (semi-major); m = 5.97 x 10²4 kg, mass of Earth; M = 1.987 x 1030 kg, mass of the Sun; G = 6.673 x 10-¹¹ Nm²/kg²; e = 0.017 (eccentricity). The period is T = 27₁ At the 'perihelion' position where the Earth is closest to the Sun, the radial velocity v, is 0. GM The instantaneos position of the Earth, x(t), y(t) at time t is related to r, 0 as per x(t) = r(t) cos (1) y(t) = r(t) sin 0(t) where r(t), 0(t) denote that both r and are time-dependent. Use Velocity Verlet algorithm to simulate the motion of the Earth around the Sun for a length of simulation time of 27. It is up to you whether to present your simulation in the form of a video or just display the simulation on the screen via display.display(). Hint : \ You have to design your own code that implements the Verlet algorithm based on the centripetal acceleration a, to obtain r(t), and from there translate it into the x(t), y(t) coordinates of the Earth. To this end, you also need to know how to obtain the value of the instantanous value of 0(t). It can be obtained from the angular momentum L. This is a challenging question to test your ability to translate your understanding of classical physics concept into a computer code, as well as your ability to solve an intelectually advanced problem.