📘 Gravitation: Classical Mechanics

What you’ll learn

1. Kepler's Laws of Planetary MotionState Kepler's three laws and explain their empirical basis from observations.Kepler replaced circular orbits with ellipses and introduced the equal-area law plus the period-squared to semi-major-axis-cubed relation. These patterns emerged directly from precise positional data. They provided the kinematic foundation Newton later explained dynamically.
2. Newton's Law of Universal GravitationWrite Newton's gravitational force law in vector form and identify the role of G.The force is F = -G m1 m2 / r^2 * r-hat. G was measured later by Cavendish at 6.67430 times 10 to the minus 11. The law is central, conservative, and satisfies Newton's third law.
3. Gravitational Field and Field LinesDefine the gravitational field and calculate it for a point mass and a spherical shell.g equals minus G M / r^2 r-hat outside a sphere. Inside a uniform spherical shell the field is zero by symmetry. The field concept separates the source from the test particle and simplifies superposition.
4. Gravitational Potential EnergyDerive the expression for gravitational potential energy and relate it to the force via the gradient.U equals minus G m1 m2 / r follows from the work integral of the inverse-square force. The negative sign indicates bound systems have negative total energy. Potential obeys superposition and satisfies Poisson's equation in differential form.
5. Conservation of Mechanical EnergyApply conservation of mechanical energy to bound and unbound trajectories.Total E equals K plus U stays constant. For circular orbits this yields the vis-viva relation v squared equals G M (2/r minus 1/a). Positive total energy corresponds to hyperbolic escape trajectories.
6. Circular Orbits and Orbital SpeedDerive the orbital speed and period for a circular orbit around a central mass.Setting G M m / r squared equal to m v squared / r produces v equals square root of G M over r. The period T equals 2 pi square root of r cubed over G M. Low-Earth-orbit speeds cluster near 7.8 kilometers per second.
7. Deriving Kepler's Laws from NewtonShow how Newton's law implies each of Kepler's three laws.Angular-momentum conservation immediately recovers the equal-area law. The orbit equation in polar coordinates becomes the conic section with eccentricity determined by energy and angular momentum. The period relation follows from the semi-major axis and Kepler's third law emerges after integration over one revolution.
8. Escape Velocity and Hyperbolic TrajectoriesCalculate escape velocity and characterize hyperbolic orbits using energy and asymptotes.Escape speed equals square root of 2 G M over R. For hyperbolic orbits the trajectory asymptotes form an angle whose cosine depends on eccentricity greater than one. Real interplanetary transfers use these excess speeds to reach other planets.
9. The Two-Body Problem and Reduced MassIntroduce reduced mass and rewrite the two-body equations in relative coordinates.The reduced mass mu equals m1 m2 over m1 plus m2. The relative vector r obeys the same inverse-square equation with gravitational parameter G times total mass. Barycentric motion follows once the relative orbit is known.
10. Elliptical Orbit Elements and EnergyRelate orbital energy and angular momentum to the six classical orbital elements.Specific energy epsilon equals minus G M over 2 a. Specific angular momentum h determines eccentricity through the orbit equation. The five additional angles orient the ellipse in three-dimensional space.
11. Real-World Perturbations and Numerical IntegrationDescribe how third-body perturbations and oblateness modify ideal Keplerian motion.Perturbing accelerations from other planets or non-spherical mass distributions cause secular drifts in orbital elements. Numerical integrators such as Runge-Kutta or symplectic methods propagate the state vector forward in time. Mission design routinely employs these techniques for accurate ephemerides.
12. Limits of Classical GravitationIdentify the regimes where classical gravitation must be replaced by general relativity.Post-Newtonian corrections become necessary when v over c or G M over c squared r approaches order 10 to the minus 5 or larger. Black-hole horizons and gravitational waves lie entirely outside classical scope. The 100-level course therefore ends by mapping the boundary between Newtonian and relativistic domains.

Questions this course answers

Grounded in trusted sources

• NASA
• NIST
• Harvard University

Every Wunder lesson is built from real, reputable sources — never invented.

Related courses