Introduction to Orbital Mechanics for Engineering Students

Explore the inertial coordinate system, derive the equations of motion for orbiting bodies, and apply Newton's law of gravitation to two-body systems using vector form and the gravity constant.

Examine the ellipse geometry, its equation x^2/a^2 + y^2/b^2 = 1, and define the semi-major axis, semi-minor axis, and focus. Derive the trajectory equation r = a(1−e^2)/(1+e cos theta) for orbital motion.

This example demonstrates calculating an elliptical Earth orbit with perigee altitude 400 kilometers and eccentricity 0.6, deriving rp, ra, a, v at perigee and apogee, true anomaly, and orbital period.

Relate mean, eccentric, and true anomalies with time using Kepler's equation and Kepler's second law; prepare for Newton-Raphson solution in the next lecture.

Apply Newton's method to Kepler's equation to solve for the eccentric anomaly E from M and e. Use f(E)=M-E+e sin E and its derivative, iterating to convergence.

Compute eccentricity and semi-major axis from perigee and apogee radii, then use Kepler's equation to determine time of flight from perigee to a true anomaly of 120 degrees.

Derive orbital elements from a satellite’s position and velocity by formulating the eccentricity and node vectors and using angular momentum to compute a, e, i, omega, and nu.

Compute the six orbital elements from the given position and velocity vectors using the eccentricity vector and angular momentum in a geocentric equatorial frame.

Convert six orbital elements into the position and velocity vectors in the geocentric equatorial coordinate system using the Perry focal coordinate system and a transformation matrix.

Convert orbital elements to position and velocity vectors by computing perifocal coordinates from p = a(1−e^2) and r = p/(1+e cos nu), then transform to the geocentric equatorial frame.