Laws two and three - more amazing discoveries

The discovery of the elliptical orbit concept propelled me towards another simple but important conclusion.  Again, using Brahe's observational data coupled with my own study, I determined that the line from the sun to a given planet sweeps out equal areas of the ellipse of its orbit during equal time intervals.  As shown below, at the aphelion the planet covers less distance along its orbital path, while more distance is covered along its orbital path when rounding the perihelion.  This relationship holds because when planets are further from the sun they move at slower orbital rates than when they are closer to the sun.  In turn, the two shaded areas are in fact equal!

This concept has gone down in the history books as my "Second Law of Planetary Motion."  More importantly, in my opinion, this analysis led me to ponder the relationship between the size of a given planet's orbit and the amount of time it takes said planet to make one complete revolution about this orbit, and therefore, one complete revolution around the sun (call this time period, P).

To better understand this, first note that the major axis of an ellipse is the longest diameter one can take of the shape, running through both foci and the center.  A semi-major axis, a, is simply 1/2 the length of the major axis, as shown to the right. Amazingly, if we cube this distance, it is always equal to the square of the amount of time it takes for the given planet to orbit the sun (P)!  The following equation can be used to summarize these data points: 
                         P² = a³

Here is an example that should inspire the reader's belief, if in fact it is lacking.  For simplicity, let's assign a value of 1 astronomical unit (AU) to the length of earth's semi-major axis, a.  Also, let  P = 1 earth year.  Given these values of course the equality holds: (1)² = (1)³ .  Keep in mind that one earth year is equal to 365 24-hour periods we conventionally call "days."  Trust me when I tell you that Brahe's observational data provides us with the length of Mars' semi-major axis with relation to the earth's, a value of 1.524 AU's (a = 1.524).  Cubing this value yields 3.54.  By our equation, recall that this equals, P² .  In order to solve for P we must take the square root of 3.54; it is 1.88.  In other words, the time it should take Mars to orbit the sun is equivalent to 1.88 earth years.  You now know by much more rigorous observational data and measurements than I was ever privy to that Mars makes one complete orbit about the sun every 687 earth days, which is precisely equal to 1.88 earth years!  See the steps below of the aforementioned mathematical operations:

1) P² = a³
2) a = 1.524 AU, so a³ = (1.524)³ = 3.54
3) Therefore P² = 3.54. 
4) Take the square root of both sides (or raise both sides to the 1/2 power) to solve for P.
5) (3.54)^(1/2) = 1.88 earth years ~ 687 earth days!

So therein lies the third and final planetary motion law that I derived in my lifetime.  Note that if we wish to use another type of measure such as kilometers rather than the simple units of earth years and astronomical units, I did in fact derive the appropriate constant that would allow us to do so (go here for more information), which still yields the same proportional relationship.

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Thank you for indulging this humble, hind-sighted look at my planetary motion laws. I am proud and so thankful to God that he enabled me to do all of this work and that it seems to have paid off for the good of humanity and His creation.  As I said when I started, it would not have been possible without Him, or without those who came before me.  What's even more rewarding than having my name attached to these laws in the history books is the view I've gotten from up here as others have continued to build on my work.  As a matter of fact, as I write this, one Sir Isaac Newton is close to proving that these laws hold not just in our solar system, but anytime one object moves under the influence of another object's pulling force - something he calls gravity.  That is to say, these laws may very well hold as general universal principles - who would've thought!  Sir Isaac, I suppose...

For kicks, I'll leave you with a picture that I found of myself.  Someone photo-shopped me over a pic of our solar system, as if I created it or something.  A little presumptuous but it certainly gave me a good laugh!  Anyway, take care for now and God bless.

Sources:

1. http://oneminuteastronomer.com/8626/keplers-laws/

2. https://en.wikipedia.org/wiki/Semi-major_and_semi-minor_axes
3. http://www.windows2universe.org/the_universe/uts/kepler3.html
4. https://www.youtube.com/watch?v=KbXVpdlmYZo
5. http://whoisbiography.com/johannes-kepler-biography/

The first law - treading lightly

My first law states that planets move around the sun in an elliptical orbit with the sun at one focus of the ellipse.  I realize this seems simple enough, especially when stated so succinctly.  But believe me when I say it took a lot of time and energy to arrive at such a conclusion.

As stated last time, I began by piecing together the clues discovered by my predecessors.  The Greeks, famous for their geometry and other Pythagorean legacies, believed a particular idea known as "harmony of the spheres."  Their worldview was such that the world had perfect organization and order; indeed this extended to the heavens.  Having detected at least some extraterrestrial objects, the Pythagorean's in particular proposed that the earth was spherical (perfectly shaped object!) and these discovered objects revolved with the earth about a central fire - the sun.  Not only so, but the different heavenly objects were separated from one another by spherical intervals of harmonic lengths.  The movement of the spheres, therefore, was thought to produce a beautiful musical arrangement known as the "harmony of the spheres."  I marvel at the wonder and intellect that generated such a notion!

I interpreted the "harmony of the spheres" through the lenses of Copernicus and Brahe, and added to it some other geometrical shapes that I hypothesized would be involved, which resulted in the representation shown below (also in the updated edition of Mysterium Cosmographicum).  However, in the early 1600s I abandoned the picture perfect idea, to the dismay of many, because the numbers just weren't adding up.  Once I derived accurate calculations of the earth's orbit (from Brahe's observational data), I turned my attention towards Mars, the planet nearest earth.  My calculations showed that the center of Mars' presumably circular orbit was not equidistant from the planet itself.  At the aphelion and perihelion (the points where the planet is farthest from the sun and closest to the sun, respectively) the measurements from the planet to the estimated center were distinctly larger than the measurements during the rest of the planet's orbit.  Thus, at least one thing was for certain - the shape of the orbit was not circular or spherical in nature.  Now, it is important to note here that treading lightly is a critical skill to practice when your newly found information rails against essentially everything (i.e. the perfect order of the universe and the earth within it) the most influential people and institutions have held dear for millennia.  So instead of getting my name out there I focused on solving the answer to the question that this presented - if planetary orbits weren't circular, then what were they?

Enter William Gilbert's 1601 publication On Magnets, and the plot thickens.  Gilbert's musings highlighted the fact that not only did I care about the shape of planetary orbits, but I also wondered how in the world (pun intended 😉) these celestial spheres remained suspended in space, in their respective patterns, accomplishing the remarkable feat of circumnavigating die Sonne, as we Germans say.  My spiritual leanings made me happy to hypothesize that there was a perfectly good reason, or better yet, Logos, perhaps radiating from the sun that acted on satellites in such a way that everything just held together (see Colossians 3:17).  Regardless of the reason, it was clear, after about two years of laborious calculations (which can be examined elsewhere), that the orbital curves precisely resembled ellipses.  Hence, I arrived at the now succinct first law of planetary motion with which I started, as pictured below.

Sources:

1. https://www.oakweb.ca/harmony/pythagorean/pythagorean.html

2. http://www.keplersdiscovery.com/Harmonies.html

3. A History of Mathematics: An Introduction (Second Edition), by Victor Katz
4. http://radio.astro.gla.ac.uk/a1dynamics/ellproof.pdf
5. http://oneminuteastronomer.com/8626/keplers-laws/

Credit where it is due

You know, I never expected them to be such a big deal.  All I did was to simply follow my inquisitive instincts, and I was only able to do so because of the mathematical, theological, philosophical, and scientific geniuses who came before me.  But as often times happens when studying in these fields, one observation led to another, then to another, and so on.  I just never stopped to think that the conclusions I drew would one day be known by my own last name - "Kepler's Laws of Planetary Motion" - what an honor.

I was born in 1571 in Weil der Stadt, a small town in Germany that is about 500 kilometers (a mere 5 1/2 hour drive by today's standards!) southwest of the 1517 site of Martin Luther's Reformation (Wittenberg, Germany).  Little did I know how the religious livelihood--or what some would call volatility--of the day would shape the trajectory of my life and studies.  As a young boy and early teen I set out to be a Protestant minister, but in my young adult years I took a recommendation from the University of Tubingen to be a stand-in math professor in Austria.  Suffice it to say there was no looking back.

It didn't take long for me, compelled by my faith, to start asking questions about God's design of not only our great, green earth, but also about the universe in which it resides.  It shouldn't be hard for any reader of my now famous 1596 publication, Mysterium Cosmographicum (The Secrets of the Universe, for you English speakers), to believe that I side with the Pythagorean idea that, "...the universe is made up of number."  What this meant to me was that through deliberate theological analysis and mathematical computations, it was within my reach "...to demonstrate the numerical relationships with which God created the universe."  We'll get to those calculations over the course of the next few posts, but it is first worth acknowledging how I was able to arrive at such a place.

I had the privilege of working with Tycho Brahe for the last two years of his life (God rest his soul). Rather than continuously problem solve, Tycho found solitude in simple observation.  It just so happened that his astronomical data collected by rigorous observation over a 25-year period (roughly 1575-1600) proved to be the tipping point for me and my work.  Before Tycho, my most notable inspiration (who actually inspired both of us) was none other than Nicolaus Copernicus.  Recall that it was his 'heretical' heliocentric theory that dared to challenge conventional wisdom--and the Catholic Church (as implied by the choice adjective)--only to be proved correct in due time.  Thankfully Nic stuck to his guns, and more than once I might add.  Not only did he claim that the earth was not actually the center of the universe, and was obviously right, but he also proposed the idea of the rotation of the earth about its axis.  To him it was simply more plausible to believe that the relatively small earth moved rather than the larger spheres of stars in the heavens.  It goes without saying he was correct in this hypothesis as well.  He was clearly on to a thing or two, which is why I take much of what he said to heart.

Collectively, it is these ideas and more that served as my point of departure for my planetary laws--thanks gentlemen!

Sources:
1.  A History of Mathematics: An Introduction (Second Edition), by Victor Katz
2.  http://www.google.com/maps