Planets and the Celestial Sphere:

The planets (Greek for `wanderers') were important to the new science of astrology, the belief that the position of the planets in the sky foretold important events. There were only seven objects visible to the ancients, the Sun and the Moon, plus the five planets, Mercury, Venus, Mars, Jupiter and Saturn.

It was obvious that the planets were not on the celestial sphere since the Moon clearly passes in front of the Sun and planets, plus Mercury and Venus can be seen to transit the Sun. Plato first proposed that the planets followed perfect circular orbits around the Earth. Later, Heraclides (330 B.C.) developed the first Solar System model, placing the planets in order from the Earth it was is now called the geocentric solar system model.

Note that orbits are perfect circles (for philosophical reasons = all things in the Heavens are "perfect"). Heraclides model became our first cosmology of things outside the Earth's atmosphere.


Ptolemy was an ancient astronomer, geographer, and mathematician who took the geocentric theory of the solar system and gave it a mathematical foundation (called the "Ptolemaic system").

Ptolemy accepted the following order for celestial objects in the solar system: Earth (center), Moon, Mercury, Venus, Sun, Mars, Jupiter, and Saturn. The Sun appears to describe a yearly circular path called the ecliptic. However, when the detailed observations of the planets in the skies is examined, the planets undergo motion which is impossible to explain in the geocentric model, a backward track for the outer planets. This behavior is called retrograde motion.

He realized, as had Hipparchus, that the inequalities in the motions of these heavenly bodies necessitated a system of deferents and epicycles in order to account for their movements in terms of uniform circular motion. In the Ptolemaic system, deferents were large circles centered on the Earth, and epicycles were small circles whose centers moved around the circumferences of the deferents. The Sun, Moon, and planets moved around the circumference of their own epicycles.

Ptolemy's solar system model looked like the following, although the planets had as many as 28 epicycles to account for all the details of their motion.

This model, while complicated, was a complete description of the Solar System that explained, and predicted, the apparent motions of all the planets. The Ptolemic system began the 1st mathematical paradigm or framework for our understanding of Nature.


After the destruction of the Library of Alexandria, the Roman Catholic Church absorbed Ptolemy's Solar System model into its own doctrine. Unfortunately, the geocentric model was accepted as doctrine and, therefore, was not subjected to the scientific method for hundreds of years.

Copernicus (1500's) reinvented the heliocentric theory and challenged Church doctrine. The heliocentric model had a greater impact than simply an improvement to solve retrograde motion. By placing the Sun at the center of the Solar System, Copernicus forced a change in our worldview that started a paradigm shift or science revolution.

However, Copernicus, like Ptolemy, also used circular orbits and had to resort to epicycles and deferents to explain retrograde motions. In fact, Copernicus was forced to use more epicycles than Ptolemy, i.e. a more complicated system of circles on circles. Thus, Copernicus' model would have failed our modern criteria that a scientific model be as simple as possible.

Tycho Brahe:

Tycho Brahe (1580's) was astronomy's 1st true observer. He built the Danish Observatory (using sextants since telescopes had not been invented yet) from which he measured positions of planets and stars to the highest degree of accuracy for that time period (1st modern database). He showed that the Sun was much farther than the Moon from the Earth, using simple trigonometry of the angle between the Moon and the Sun at 1st Quarter.

Tycho Brahe argued that, if an object is near the Earth, its position relative to the background stars should change over night (see definition of parallax). Since he failed to measure the parallax for a supernova in 1572 and a comet in 1577, he concluded that these objects are far from the Earth. Tycho's measurements were used to show that there was ``change'' in the celestial sphere, in contradiction with the assumption of a perfect, immutable Heaven which was one of the pillars of the Greek cosmological model.


Kepler (1600's) a student of Tycho who used Brahe's database to formulate the Laws of Planetary Motion which corrects the problems of epicycles in the heliocentric theory by using ellipses instead of circles for orbits of the planets. This is a key mathematical formulation because the reason Copernicus' heliocentric model has to use epicycles is due to the fact that he assumed perfectly circular orbits. With the use of ellipses, the heliocentric model eliminates the need for epicycles and deferents.

Kepler developed, using Tycho Brahe's observations, the first kinematic description of orbits formulated in his three laws of planetary motion. Newton will later develop a dynamic description that involves the underlying influence (gravity)

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


    Kepler's laws are a mathematical formulation of the solar system. But, is the solar system `really' composed of elliptical orbits, or is this just a computational trick and the `real' solar system is geocentric. Of course, the answer to questions of this nature is observation.

    The pioneer of astronomical observation in a modern context is Galileo. Galileo (1620's) developed laws of motion (natural versus forced motion, rest versus uniform motion). Then, with a small refracting telescope (3-inches), destroyed the the idea of a "perfect", geocentric Universe with the following 5 discoveries:

    spots on the Sun

    mountains and "seas" (maria) on the Moon

    Milky Way is made of lots of stars

    These first three are more of an aesthetic nature. Plato requires a `perfect' Universe. Spots, craters and a broken Milky Way are all features of imperfection and at odds with Plato's ideas on purely philosophical grounds. However, the laws of motion are as pure as Plato's celestial sphere, but clearly are not easy to apply in the world of friction and air currents etc. So these observations, by themselves, are not fatal to the geocentric theory. The next two are fatal and can only be explained by a heliocentric model.

    Venus has phases

    Jupiter has moons (Galilean moons: Io, Europa, Callisto, Ganymede)

    Notice that planets with phases are possible in a geocentric model. But for a planet to change in apparent size with its phases, like Venus is impossible if the planet orbits the same distance from the Earth. And, lastly, if all bodies orbit around the Earth, then the moons of Jupiter, which clearly orbit around that planet, are definitive proof that the geocentric model is wrong.


    Newton (1700's) expanded on the work of Galileo to better define the relationship between energy and motion. He expanded the Galileo's work on the motion of objects introducing the concept of accleration:

    and developed the following laws of motion:

    From Newton's 1st law we know that an object travels in a straight line unless acted upon by an external force. A circular orbit is clearly not a straight line, what is the force? Newton showed that the planets are acted on by the force of gravity arising from the Sun.

    Each orbit is a constantly changing velocity where gravity adds a small ``delta-vee'' at each moment. This ``delta-vee'' is what produces the elliptical curvature that is the orbit. This is best shown using vectors, a mathematical formulation Newton first introduced.

    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

    All masses, regardless of size, attract other masses with gravity. You don't notice the force from nearby objects because their mass is so small compared to the mass of the Earth. Consider the following example:

    To play a java applet and see the effects of an attractive force varying as 1/r2 click here.

    Escape Velocity:

    A perfect circular orbit occurs if the centrifugal force exactly balances the gravitational force. At the Earth's surface, this value is 7.86 km/s.

    A circular orbit for object A occurs when its burnout velocity is vc. When the velocity is v=1.44vc, then a parabolic orbit is achieved and we say the object has reached escape velocity. For a greater velocity, the orbit is hyperbolic.

    With vector calculus, Newton was able to develop a cosmology which included the underlying cause of planetary motion, gravity, explained Kepler's laws, and 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.

    Scientific Method:

    Science is any system of knowledge that is concerned with the physical world and its phenomena and entails unbiased observations and/or systematic experimentation. In general, a science involves a pursuit of knowledge about the operations of fundamental laws of nature.

    Science can be separated from pseudo-science by the principle of falsifiability, that is, ideas must be capable of being proven false in order to be scientifically valid. In this regard, science is also separated from religion. Example of pseudo-science are some theories behind crop circles and astrology.

    The keystone to science is proof or evidence/data, which is not to be confused with certainty. Except in pure mathematics, nothing is known for certain (although much is certainly false). Central to the scientific method is a system of logic. The scientific method has four steps:

    Note that there is an emphasis on falsification, not verification. If a theory passes any test then our confidence in the theory is reinforced, but it is never proven correct in a mathematically sense. Thus, a powerful hypothesis is one that is highly vulnerable to falsification and that can be tested in many ways.

    The goal of the scientific method is the construction of models and theories, all with the final goal of understanding.