Kepler developed, using Tycho Brahe's observations, the first kinematic description of orbits, Newton will develop a dynamic description that involves the underlying influence (gravity)

• 1st law (law of elliptic orbits): Each planet moves in an elliptical orbit with the Sun at one focus.

Ellipses that are highly flattened have high eccentricity. Ellipses that are close to a circle have low eccentricity.
• 2nd law (law of equal areas): a line connection the Sun and a planet (called the radius vector) sweeps out equal areas in equal times

Objects travel fastest at the low point of their orbit, and travel slowest at the high point of their orbit.
• 3rd law (law of harmonics): The square of a planet's orbital period is proportional to its mean distance from the Sun cubed.

The 3rd law is used to develop a ``yardstick'' for the Solar System, expressing the distance to all the planets relative to Earth's orbit by just knowing their period (timing how long it takes for them to go around the Sun).

Newton:

Newton expanded on the work of Galileo to better define the relationship between energy and motion. In particular, he developed the following concepts:

• change in velocity = acceleration -> caused by force
• inertia = resistance to change in velocity and is proportional to the mass of the object
• momentum = quantity of motion energy and is equal to mass times velocity
• law of conservation of momentum = total momentum (mass x velocity) of an interaction is conserved -> is the same before and after

Newton's laws of motion:

• 1st law: a body remains at rest or moves in a straight line of constant velocity as long as no external forces acts on it

• 2nd law: a body acted on by a force will accelerate such that force equals mass times acceleration (F=ma)
  

• 3rd law: for every action there is an equal and opposite reaction

Newton's Law of Universal Gravitation:

Galileo was the first to notice that objects are ``pulled'' towards the center of the Earth, but Newton showed that this same force (gravity) was responsible for the orbits of the planets in the Solar System.

Objects in the Universe attract each other with a force that varies directly as the product of their masses and inversely as the square of their distances

Vectors:

Newton went beyond his simple laws of motion and gravitation to develop a whole set of mathemathics to describe and calculate orbits. Today we can this mathematics calculus. The key to calculus is the use of vectors. A vector is a quantity that has both magnitude and direction. It is typically represented symbolically by an arrow in the proper direction, whose length is proportional to the magnitude of the vector. Although a vector has magnitude and direction, it does not have position. A vector is not altered if it is displaced parallel to itself as long as its length is not changed.

Because vectors are different from ordinary (i.e., scalar) quantities, all mathematical operations involving vectors must be carefully defined. If vector A is added to vector B, the result is another vector, C, written A + B = C. The operation is performed by displacing B so that it begins where A ends. C is then the vector that starts where A begins and ends where B ends.

Vector subtraction is defined by A - B = A + (-B), where the vector -B has the same magnitude as B but the opposition direction. A vector may be multiplied by a scalar. Thus, for example, the vector 2A has the same direction as A but is twice as long.

Newton applied vectors in terms of force. A body is added on by a vector force as shown above. Forces can be added just like vectors, so that force 1 and force 2 add together to produce the total force, F. Total force F can also be broken into components x and y that correspond to the forces in the x and y directions (for example, along a road and with gravity).

A particle moving with constant velocity v suffers a displacement s in time t given by s = vt. The vector v has been multiplied by the scalar t to give a new vector, s, which has the same direction as v but cannot be compared to v in magnitude (a displacement of one metre is neither bigger nor smaller than a velocity of one metre per second). This is a typical example of a phenomenon that might be represented by different equations in differently oriented Cartesian coordinate systems but that has a single vector equation (for all observers not moving with respect to one another).

For a particle of mass m, a force is applied with results in an accerlation a. The accerlation changes the velocity vector by a small amount, delta v, every interval of time, delta t. This results in trajectories, a vector map of the changes in position from an origion, the vector x and the velocities, vector v.

With vector calculus, Newton was able to develop a cosmology which included the underlying cause of planetary motion, gravity, completed the solar system model begun by the Babylonians and early Greeks. The mathematical formulation of Newton's dynamic model of the solar system became the science of celestial mechanics, the greatest of the deterministic sciences.