Elective 3 | midTERMS

Tycho Brahe

• Produced the most accurate pre-telescope measurements of planetary positions.

• His data revealed inconsistencies with circular orbits, especially for Mars.

• He proposed the Tychonic system, a hybrid model that preserved Earth’s central position while allowing planets to orbit the Sun.

Johaness Kepler

• Formulated the three laws of planetary motion that mathematically describe how bodies orbit the Sun.

• He proved that planetary orbits are elliptical rather than circular using Tycho Brahe’s high-precision Mars data.

• His third law established a quantitative relationship between orbital period and orbital size, forming the basis of modern orbital mechanics.

Isaac Newton

• Derived Kepler’s laws from his law of universal gravitation and laws of motion.

• He showed that elliptical orbits naturally result from an inverse-square gravitational force.

• His work transformed Kepler’s empirical laws into a predictive physical theory applicable to planets, comets, and satellites.

The Law of Equal Areas

A satellite sweeps out equal areas in equal times around their center of attraction

Kepler’s First Law of Orbital Motion

An elliptical orbit can be described in terms of its eccentricity and the major and minor lengths.

Kepler’s Second Law of Orbital Motion

It states that a line drawn between the central mass and a satellite sweeps out an equal area in an equal amount of time.

The Law of Harmonies

The square of the orbital period is directly proportional to the cube of the average distance between the satellite and the center of attraction.

Kepler’s Third Law of Motion

Quantifies that the orbital period is proportional to the cube of the semi-major length of the elliptical orbit.

Mirror Theorem

If the system passes through a mirror configuration at time t0, then the motion after that instant is the exact mirror image in time of the motion before.

Newtonian N-body Gravitational System

• Motion is described relative to the center of mass

• Each body’s velocity is perpendicular to its position vector

• Motion is purely tangential

• The system has no radial expansion or contraction

Periodic

At mirror configuration t2 the orbit is then again symmetric with the same position, orbital velocity, and no radial expansion or contraction.