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.

Derive the relative two-body equations of motion in a coordinate system anchored at mass one, yielding r'' = -mu r / r^3, and explore angular momentum and the orbital plane.

Explore polar coordinates to derive the energy equation from relative motion, expressing velocity as r-dot and theta-dot, linking angular momentum to flight-path angle, and obtaining E = v^2/2 - mu/r.

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.

Explore elliptical orbits defined by semi-major axis a, eccentricity e, and true anomaly nu; learn the trajectory equation r = p/(1+ e cos nu) and perigee and apogee.

Derive perigee and apogee radii from a and e for elliptical orbits, and relate energy and angular momentum to the orbit's semi-major axis, velocity, and eccentricity, including circular orbits.

Explore Kepler's laws: ellipses with the sun at a focus, the sun–planet line sweeps equal areas in equal times, and Kepler's third law links period to orbit.

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.

Compute eccentricity from two orbit points using the trajectory equation and angular momentum. Then determine altitude of perigee, semi-major axis, and orbital period by applying h^2/μ and P relationships.

Treat circular orbits as a case of elliptical orbits with eccentricity zero, where radius equals the circle and velocity remains constant as altitude z changes in low earth orbit.

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 find roots of differentiable functions, using a starting guess, tolerance, and max iterations. Solve Kepler's transcendental equation numerically with a concrete example.

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.

Apply Kepler's equation and Newton's method to compute the satellite's radius and speed at 10 hours after perigee, using period, eccentricity, and the trajectory equation.

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.

Learn the six orbital elements and geocentric equatorial and Perry focal coordinate systems that define an orbit, including inclination, longitude of the ascending node, and the argument of Perry axis.

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.