The Solar System problem was solved in steps. First Copernicus advanced a detailed heliocentric model (this enabled calculation of the relative distances of the planets from the Sun). Tycho Brahe built an observatory which enabled the most accurate pre-telescopic observations of the positions of stars and planets - in particular very detailed observations of the apparent motion of Mars. He turned these over to Johannes Kepler, who finally realized that the heliocentric model would work very well if there was non-uniform elliptical motion of the planets around the Sun. He formulated his Three Laws:

1) the planets move in elliptical orbits, with the Sun at one focus

2) a line joining a planet and the Sun sweeps out equal areas in the ellipse in equal time intervals

3) the square of the orbital period of a planet is proportional to the cube of its semi-major axis

Explanations: an ellipse is an egg-shaped figure found by either cutting a cone at an angle, or putting a loop of string around 2 nails and moving a pencil around them keeping the string taught. The nails will be at the foci of the ellipse, and its eccentricity (flatness) will increase the further we put them from each other (if they are in the same place, we get a circle). The semi-major axis of the ellipse is half its longest dimension. The Second Law requires that a given planet move more quickly when it is near the Sun than when it is further out. The Third Law requires planets with smaller semi-major axes to move generally faster. Kepler realized these both probably implied that the Sun exerted a force on the planets which increased as they got close (so they have to move faster to avoid falling in).

It was Newton who finally formulated a physical theory which explained all this. He expressed his Three Laws of Motion:

1) a body will keep a constant velocity unless acted upon by a force

2) if acted upon by a force, a body will accelerate proportionally to that force an inversely proportional to its mass

3) if a body exerts a force on another body, it will experience an equal and opposite force

The first law expresses the property of inertia, the other two tell how forces will act on bodies. He also formulated a law which tells what kind of force gravitation is: it is proportional to the 2 interacting masses and inversely proportional to the distance between them. He developed calculus, and used it to show that Kepler's Laws are a direct consequence of the Laws of Motion and the force of Gravity. This allows us to determine the mass of any astronomical body if we can observe the orbital period and distance of another body orbiting the central one (Earth-Moon, Sun-Earth, Jupiter-Io, Star A - Star B, etc.) The general form of Kepler's Third Law is :

(orbital period)² = (semi-major axis)³ / (system mass).

The system mass is often dominated by the central body, and so essentially equivalent to it. One must be careful of the units when applying this equation; for the Sun the distance should be in AU and the orbital period in Earth years. For other systems, it often pays to take the ratio of that system to the Sun-Earth system (you can divide the equation by itself for the 2 systems and it will still express an equality). Be sure you know what is meant by this.

Newton's laws also give us a way to calculate the surface gravity and escape velocity for any body. The circular orbital velocity for a given distance is the escape velocity at that distance divided by the square root of 2. If one has more velocity than that it will produce an elliptical orbit with increasing semi-major axis, up to the escape velocity (at which point the "orbit" is parabolic, and never returns).