The MATLAB programs we created whose names begin with "orbit 2" simulate using a centered scheme the motion of a body which feets only one force, that of the gravity of the Earth assuming Newton's Law of Gravity which is assumed spherically symmetrical Kepler's Three Laws predict: I. The orbit is an ellipse with the Earth at one focus II. The line from the Earth to the satellite sweeps out equal areas in equal times III. The square of the period of the orbit is proportional to the cube of the semi major axis semi major axis. The semi-major axis, a, is one half of the long dimension of the ellipse. If the initial position is along the x axis and the initial velocity is along the y axis, the major axis, that is the long dimension of the ellipse, will be on the x axis, if the speed in this case is greater than the circular speed, v then the initial position will be the closest point in the orbit (perigee for orbit about the Earth). The equation for an ellipse in the xy plane whose major axis lies along the x axis is: ((x - x_c)^2/a) + y/b = 1 where the center of the ellipse is at [x, 0]. a > b and b is the semi-minor axis (half the width of the ellipse measured perpendicular to the major axis), and one focus of the ellipse is at (k_c + c, 0) and the other is at (x_c c, 0), where c = (a^2 middot b^2). The rate at which area is swept out is (1/2), where v = ||v - v || and v = (v - r/r)(r/r), r = (x, y) and r = ||r||. Write a simulate using a centered scheme and a 15 second timestep the motion of a satellite which feels only the force of a spherically symmetric Earth with initial conditions of a distance of 6700 km from the Earth's center and a velocity Earth with speed OPTION 1 (1. 1 v_c) OR OPTION2(1. 2 V_c) OR OPTION3 (1.3 v_c) OR OPTION4 (1.4 v_c). Run for at least one orbit. Plot x'(t), y'(t) and the trace of the path in the xy plane. Plot the orbit of an ellipse with major axis on the x axis extending from the time = 0 one of the satellite to the position at the next time step when it reaches or crosses the x axis and with one focus at the region. Calculate and plot the rate at which area is swept out. Do a simulation the same as in part 1 except the initial distance 26, 800 km and the initial speeds is 1/2 of what you used in part 1. determine the semimajor axis and the period for this orbit and the one you are your simulation in part 1. Comment on whether Kepler's Three Laws hold. Repeat steps 1-6 for 150 second timestep. The MATLAB programs we created whose names begin with "orbit 2" simulate using a centered scheme the motion of a body which feets only one force, that of the gravity of the Earth assuming Newton's Law of Gravity which is assumed spherically symmetrical Kepler's Three Laws predict: I. The orbit is an ellipse with the Earth at one focus II. The line from the Earth to the satellite sweeps out equal areas in equal times III. The square of the period of the orbit is proportional to the cube of the semi major axis semi major axis. The semi-major axis, a, is one half of the long dimension of the ellipse. If the initial position is along the x axis and the initial velocity is along the y axis, the major axis, that is the long dimension of the ellipse, will be on the x axis, if the speed in this case is greater than the circular speed, v then the initial position will be the closest point in the orbit (perigee for orbit about the Earth). The equation for an ellipse in the xy plane whose major axis lies along the x axis is: ((x - x_c)^2/a) + y/b = 1 where the center of the ellipse is at [x, 0]. a > b and b is the semi-minor axis (half the width of the ellipse measured perpendicular to the major axis), and one focus of the ellipse is at (k_c + c, 0) and the other is at (x_c c, 0), where c = (a^2 middot b^2). The rate at which area is swept out is (1/2), where v = ||v - v || and v = (v - r/r)(r/r), r = (x, y) and r = ||r||. Write a simulate using a centered scheme and a 15 second timestep the motion of a satellite which feels only the force of a spherically symmetric Earth with initial conditions of a distance of 6700 km from the Earth's center and a velocity Earth with speed OPTION 1 (1. 1 v_c) OR OPTION2(1. 2 V_c) OR OPTION3 (1.3 v_c) OR OPTION4 (1.4 v_c). Run for at least one orbit. Plot x'(t), y'(t) and the trace of the path in the xy plane. Plot the orbit of an ellipse with major axis on the x axis extending from the time = 0 one of the satellite to the position at the next time step when it reaches or crosses the x axis and with one focus at the region. Calculate and plot the rate at which area is swept out. Do a simulation the same as in part 1 except the initial distance 26, 800 km and the initial speeds is 1/2 of what you used in part 1. determine the semimajor axis and the period for this orbit and the one you are your simulation in part 1. Comment on whether Kepler's Three Laws hold. Repeat steps 1-6 for 150 second timestep