How Kepler Actually Discovered his Laws

this is Mars on the nights when we can see the planet it moves with the Stars across the sky from east to west however while the Stars remain in the same fixed pattern from night to night Mars does its own thing the word planet comes from the Greek for Wanderer Mars slowly wanders from west to east against the backdrop of stars early astronomers noticed that the wonderings of Mars depend on how close it is to the Sun in the sky when Mars is close to the Sun as it sets or Rises the planet moves quickly from west to east however over time Mars and the sun drift apart in the sky with Mars rising in the east as the sun sets in the west as Mars moves further from the Sun in the sky it's night toight wanderings become stranger it changes its mind on the whole west to east thing slows down stops goes backwards for a while and then continues on as if nothing happened in the late 1500s the astronomer Cho Bur ra spent over 20 years collecting the largest and most accurate astronomical data set captured to date if we plot too's observations for the path of Mars over time against the backdrop of stars every couple years we see these retrograde Loops but they don't happen in the same position of the sky each time and the shapes of these Loops vary from cycle to cycle what causes Mars to move like this what mathematical model of the cosmos could possibly capture this erratic Behavior as Tio Lay Dying ding in 1601 he begged his assistant Johannes Kepler to not let him die in vain too believed that the Motions of Mars and the other planets or a result of his own model of the solar system where the sun orbits around a stationary Earth and the planets orbit around the Sun but he had run out of time to prove it the 31-year-old Kepler had other ideas this video is sponsored by my Heritage and this is a 1930 census form from Harris County Texas rows 82 to 92 show the members of the Cox family living on State Street in Marshall Texas on row 83 you'll find my great grandmother Odessa Bell and on row 85 is my maternal grandmother Alice just 7 years old here as we can see she's in school but not quite reading and writing yet in researching these videos I spend a ton of time digging through old documents like too br's original journals and it's amazing to me how easy it was to find documents like this about my own family with my Heritage I started by filling out some information about my parents and grandparents with some help from my mom remarkably the my Heritage algorithm did most of the work from here automatically searching over 19 billion historical records to discover this census form and more this census document was so easy to find and really special to see I actually didn't know that my great-grandmother's name was Odessa Bell my 2-year-old daughter's middle name is Belle it's a name from my wife's side of the family that we really love like and I had no idea that it was on my side too I'd love that my daughter shares the name with my great grandmother another feature that really impressed me was my heritage's ability to automatically FInd matches and expand my family tree I was quickly able to add whole branches to the tree that I had never heard of going deep into the 1800s way back to my great great great grandfather Lim my Heritage found this old high school yearbook photo of my mom and even automatically colorized it partnering with my Heritage has been super interesting and really piqued my interest to dig into my own history more but honestly one of the best parts was connecting with my mom as we work through the process I don't always get to spend as much time with her as I would like these days and reviewing all the matches and making discoveries together has been really fun I can tell that these documents and information really bring back a lot of memories for her and it's really rewarding to walk through it together so big thank you to my Heritage if this is something you might enjoy you can try it for free for 14 days using the link in the description below I really think you'll like it now back to Kepler 6 years before Cho's death Kepler had a life-changing Epiphany while lecturing Seminary students in GRS Kepler had been searching for mathematical patterns in the spacing of the planets for completely unrelated reasons Kepler Drew this figure on the Blackboard as he lectured in a flash of insight Kepler realize that the spacing between the circles created by the inscribed triangle was almost a perfect match for the spacing of the orbits of Saturn and Jupiter the triangle is the first or simplest polygon in geometry and at the time Saturn and Jupiter were the first or outermost planets from here the pattern was obvious to Kepler an inscribed Square should give the spacing between Jupiter and Mars a pentagon for the gap between Mars and Earth and so on Computing day and night Kepler realized that the math didn't quite work out but would work better if he switched to 3D platonic solids for each pair of orbits Kepler went on to publish this model in his book The Cosmic mystery where he claimed to have discovered God's design for the universe through geometry however Kepler's 3D model still deviated significantly from The observed size of the orbits of Mercury and Jupiter Kepler wondered if the deviations could be a result of inaccurate observations there was only one person on the planet who possessed sufficiently accurate data for Kepler to be sure when Kepler arrived at bakki Castle in February 1600 to work as tiko's assistant he quickly learned that getting the data he needed was going to be difficult to guarded his data closely assigning Kepler to work on Mars he only divied out the minimum data points needed frustrated but undeterred Kepler got to work on Mars boasting and even placing a bet that he could resolve the orbit of the planet in just 8 Days the retrograde Loops of Mars seen in the data were of minimal concern to Kepler although he disagreed with Tio about what caused this Behavior retrograde motion had been observed since it at least the time of the ancient Egyptians and the ancient Greeks developed sophisticated mathematical models to describe it the toic model published around the year 150 ad fixed the Earth at the center of the universe and used epicycles to explain the backwards motion of Mars This is basically a smaller circular orbit fixed to a larger circular orbit most of the time Mars moves west to east but when it comes closer to the Earth on its epicycle the planet reverses direction as observed epicycles may seem bizarre today but as Kepler would later show are actually mathematically equivalent to a heliocentric model without epicycles this is because in toi's model the motion of the sun's orbit around Earth matches the motion of Mars around its epicycle if we fix the Sun instead of the Earth at the center of the toeg model and compute the motion of Mars relative to the Sun the vector from the Sun to the Earth and the vector from the center of Mars epicycle to Mars cancel out leaving a simple cular orbit for Mars under the heliocentric model retrograde motion has a much simpler explanation since the Earth orbits the sun faster than Mars when Earth approaches and overtakes Mars the planet appears to temporarily move backwards against the background of stars this is why the motion of Mars is connected to how close it is to the Sun in the sky when Mars appears close to the Sun we're seeing it across the solar system and Mars appears to be moving more quickly because it's headed in the opposite direction of Earth as Earth approaches Mars the Sun and Mars will move to opposite sides of the sky this is literally called an opposition and Mars will appear to be moving backwards as we pass it Kepler knew all of this but he also had the insight to realize that none of these distinctions would help him solve the real problem the real problem was that the heliocentric toake geocentric and Tios hybrid model were all equally bad at predicting the Motions of Mars relative to 's incredibly accurate observations the largest source of error was not actually the loops but the way the speed of Mars changes as it moves around its orbit as a Young Man Kepler was heavily influenced by the writings of cernus and believed strongly in the still then unpopular heliocentric model placing a fixed Sun at the center of his model Kepler looked for the best way to make use of Tio's data Tio's observations were of course from the reference frame of Earth so any errors in the estimated position of Earth would impact the estimated position of Mars relative to the sun Kepler saw a clever way to decouple the Motions of Earth and Mars in 20 or so years of observations the Earth had passed directly between the Sun and Mars 10 times when Mars is in opposition like this the angle seen in an overhead view between the Sun and Mars was the same as the angle between Earth and Mars by starting his analysis with Tio's observations taken when Mars was in opposition Kepler could pretend that these observations were taken from the Sun instead of from the earth giving a more direct view into the orbit of Mars Kepler began with the 1587 1591 1593 and 1595 oppositions measuring from the start of the Gemini constellation in the Zodiac these occurred at longitudes of 115.7 26.7 282.30 5° from these observations alone it's straightforward to show that the speed of Mars around its orbit is not constant Kepler knew the time for Mars to complete one full orbit is 687 Earth days so its average angular speed is 524 degrees per day if we multiply the number of days between our observation times our average speed we can predict where Mars would be if its speed was constant on June 8 1591 1,555 days after our first observation our model says that Mars should have advanced by 1555 * 0.524 equals 84.8 Dees landing at a position of 210.8 De however 's observation puts Mars at 26.7 degrees an enormous 3.9 Dee error relative to Tio's 0.03 Dee measurement error the constant velocity model does even worse relative to the 1593 and 1595 observations yielding errors of 7.6 and 15.5 de notice that our errors are not all in the same direction adding in more of Tio's observations it's clear that Mars moves slower than the constant velocity model in the lower left part of its orbit and faster than the constant velocity model in the upper right now the variable speed of Mars was not a new problem either the ancient Greeks even had an elegant but controversial solution called the equant in the toic model the Earth is not at the center of Mars orbit instead the Earth is off center and a point in space called an equant is added directly across Mars orbital center from the earth the key idea is that Mars orbits at a constant angular velocity relative to the equin instead of the center of its orbit this shift off center has the effect of making Mars move more quickly when it's further from the equin and more slowly when it's closer the eent was controversial because it went against the teachings of Aristotle and other ancients that the heavens were made of fundamentally different stuff than the earth the heavens were understood to be Eternal and unchanging and the motions of heavenly bodies were uniform and circular uniform circular motion was so sacran that a major motivation for cernus in developing his heliocentric model in the early 1500s was to rid to's model of the eent instead of handling the variable speed of Mars with an eent cernus used a version of epicycles complicating his model Kepler did not share these qualms about the eent putting the Sun at the center of his model as cernus had but incorporating toi's eent Kepler saw a new way forward to me had assumed that the distance between the equant and the center of the orbit was equal to the distance between the center of the orbit and the Observer this seemed arbitrary to Kepler if the equant is a madeup point in space in the first place why does it have to be directly opposite the sun in his model Kepler decided to let the sun's Center and equant Center distances be independent variables and that he would tune these values to best fit too's observations according to the model moves uniformly around the eent so just as in the constant velocity model Kepler was able to compute the angle of each of Tio's observations relative to the eent by simply multiplying Mar's average angular speed by the times of Tio's observations this approach yields two sets of angles the angles directly measured by too of Mars relative to the sun and the angles of Mars around the eent computed by using the times of Tio's observations for his model to work Kepler needed to align these two sets of lines making the predicted angles from the equant line up as closely as possible with the observed angles from the Sun to make this happen a Kepler had four parameters to tune first were the distance between the Sun and the center of Mars orbit and the distance between the center of Mars orbit and the equin the further Kepler moved the equin from the Sun the more uneven the speed of Mars became Kepler could also adjust the angle of the line passing through the sun in eent and the starting position where Mars begins its orbit Kepler took a painfully Brute Force approach starting with an initial guess at his four parameters he would perform the tedious computations to find the predicted positions of Mars compare these results to observations based on his errors modify his parameters and then start the entire process again after 70 iterations in over a year of work Kepler achieved something absolutely incredible not only did his model fit the four observations he tuned on well test in his model on eight other observations yielded a maximum error of 2 arc minutes and 12 Arc seconds or 0.037 de hold an index card at arms length its thickness will cover about two arc minutes in your visual field Kepler had produced a model nearly a 100 times more accurate than any astronomer before him this result is astounding but it's also wrong after summarizing these incredible results in his book astronomia Nova Kepler Begins the next chapter with the line who would have thought it possible this hypothesis so closely in agreement with the observations is nonetheless false in the spring of 16001 after working on Mars for over a year Kepler discovered that it was impossible for his model to describe the true path of Mars and left Tiko and batki castle in disgust tiko's sudden death a few months later would arguably be the best thing to ever happen to Kepler and to astronomy in the shuffle after too's death Kepler stole Tio's complete data set before his heirs could find it without the obligation to follow Tio's prescribed approaches and access to all of his data Kepler was free to rebuild astronomy from the ground up next time we'll see how he did [Music] it thanks again to my Heritage for sponsoring this video you can try it out for free for 14 days using the link in the description below 's model of the cosmos turned out to be wrong but it did make for some really beautiful illustrations like this amazing chart published by Johan Gabriel Damar in the early 1700s showing the loopy Paths of Mars Jupiter and Saturn from 1712 to 17713 according to Tio's model you can find original printings from the 1700s for sale online I really want one but they're a little pricey I was able to find and purchase a copy right free super highres image which I've made into a poster available for purchase at the link below I slightly modified the proportions to fit nicely into a standard 13x 15in frame in my research for this video I also came across this incredibly intricate illustration of the tonic model it's from this insane 1660 book called harmonia macrocosmica often described as the most beautiful Celestial Atlas ever published there's an original copy for sale on a books right now for just $375,000 but not to worry shipping is only 1450 you can buy a poster version of this illustration at the link below I also modified this one ever so slightly to fit into a standard 13x 16inch frame

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Length: 16min 40sec (1000 seconds)

Published: Wed May 08 2024