KEPLER AND HIS LAWS
Kepler's Laws gracefully
determine elapsed time of flight
from perihelion (q) to any orbital position.
Predicting orbital positions in a circular orbit is straightforward because objects in circular orbits move at a constant rate; thus, extrapolating duration to future positions is a simple matter of proportionality.
Predicting positions for an elliptical orbit is more complicated because these inorbit velocities vary between a range of values. Kepler's Laws present an elegant, general solution for predicting orbital positions in elliptical orbits.
A Short Biography. Johannes Kepler was born on December 27, 1571, in Württemburg (in now Germany). Though sickly and from a poor, noble family, his obvious intelligence earned him a scholarship to the University of Tübingen to study for the Lutheran ministry. There, he learned and loved the ideas of Copernicus. During his life, Kepler worked as a teacher and mathematician, along with duties as court astrologer and astronomer. However, he is best known for his work with the last great astronomical observer of the pretelescope age, Tycho Brahe. To better understand planetary motions, Brahe built the best equipment of the time and spent over twenty years meticulously collecting precise data on all the visible planets. Tyco's data was the best available prior to the telescope. While his celestial measurements were superb for his time, Brahe lacked math skills; fortunately, Kepler had considerable mathematical expertise. Kepler Works on Motion of Mars. Brahe assigned Kepler the task of analyzing the Mars data, which proved challenging. After ten years of work, Kepler determined the orbit of Mars to be an ellipse. He then set about calculating the entire orbit of Mars, using the geometrical rate law and assuming an eggshaped ovoid orbit. After approximately 40 failed attempts, in early 1605 he at last hit upon the idea of an ellipse, which he had previously assumed to be too simple a solution for earlier astronomers to have overlooked. Finding that an elliptical orbit fit the Mars data, he immediately concluded that all planets move in ellipses, with the sun at one focus—Kepler's first law of planetary motion.  
Kepler's First LawThe orbit of each Solar satellite is an ellipse with the Sun at one focus.  
"a, b, c" of an Ellipse
 

Arbitrarily assume values: c=0.5 AU; x=1.0 AU; a=1.5AU. Distance from C (Center of orbit) to Sol is focus, c. Distance from C to perihelion, q, is a, semimajor axis. Recall Eccentricity: e = c/a = 0.5AU / 1.5 AU = 0.333 Compute b Given values a and c, compute b, semiminor axis. Recall: b = √(a^{2}  c^{2}) Thus, b = √(1.5^{2}  .5^{2})AU = 1.414 AU  
Auxiliary Circle  
Auxiliary Circle (AC)
circumscribes a designated ellipse. AC and ellipse share a common center, C, as well as the two endpoints of the ellipse's major axis, shown as a and +a.
Value of semimajor axis, a, is also the radius of AC.
Compare subsequent areas for following ratio:
 
Semilatus Rectum  
Semilatus rectum (ℓ) is the distance from focus to ellipse along a line perpendicular to the major axis. Since an orbit is an ellipse with Sol at a foci (S in the diagram), we notionally position such a line at Sol. Perpendicular line would intercept the orbit at point Y_{Orb} (ellipse point, y_{e}).
Thus, segment, SY_{Orb}, is orbit's semilatus rectum.
Extend this segment to intercept AC at point, Y_{AC} (circle point, y_{c}).
Above values all relate via the a:b ratio:
Rewrite semilatus rectum in terms of e, eccentricity:
ℓ = b^{2}/a = a (1  e^{2}) = 1.5(1.333^{2}) AU = 1.333 AU  
Define True Anomaly (ν)  

Angle, ν, is the "True Anomaly".
Also, Cartesian Coordinates: (X, Y)
COMPUTE: X = r × Cos(ν); Y = r × Sin(ν)
SUMMARY: Easily generate values for r, x and y for any angle (ν) in any orbit.  
Define Eccentric Anomaly (E)  
Every orbital position has following values.
Y_{AC = (1.05×1.5 / 1.414)AU = 1.118 AU}
Ref: Fundamentals of Astronautics, Bate, Mueller, White. pg 183184).
For any Y_{Orb}, readily determine Y_{AC }(=Y_{Orb}×a÷b).
 
Compute E  

 
Research for Kepler's Astronomia nova (A New Astronomy) began with the analysis of Mars' orbit. After many complex calculations with traditional methods, many Mars measurements still did not fit predictions based on circular orbits or even an orbit of epicycles (multiple circles upon circles, a model favored by Brahe and other contemporariessee diagram). While Kepler created a model to generally agree with Tycho's data, Kepler was not satisfied with the few inaccuracies; at certain points, the model differed from the data by as much as eight arcminutes.
 
Kepler's Second LawThe line joining the object to the Sun sweeps out equal areas in equal time durations.  
R Vector Sweeps Area
Let R be a positional vector from Sun to asteroid.
During the time duration, t_{1},
asteroid orbits from first position, p_{1}, to second position, p_{2},
and vector, R, sweeps an area, A_{1}.
R vector varies in both angle and length.
By inspection, one can observe the length Sp_{1} as less then Sp_{2}.
 
Equal Areas Show Equal Times
For any two sectors of equal areas (A_{1} = A_{2});
corresponding orbit times must also be equal (t_{1} = t_{2}).
Thus, for same duration, asteroid travels greater orbital distances when closer to the Sun and a lesser distance when further away. This accords with Isaac Newton's work in gravity which proved that Sun's closer satellites have greater speeds then those further away.
Thus, we conclude:
Once we determine True Anomaly (TA) areas;
then, we can compare corresponding flight times.
 
Eccentric Anomaly (E) Area
Easily calculate area of E, a simple circular sector.
Sector apex is at orbit's center, C; and sector arc is the AC border from q to Y_{AC}.
Thus, E sector area is a simple fraction of a circular area:
A_{E} = (E ÷ 360°) × πa^{2}
For every angle E, there is a corresponding ν.
ν sector is bounded by an ecliptic arc and offcenter apex;
thus, computing ν sector area is more difficult than E.
About 400 years ago, Johannes Kepler discovered
an ingenious way to relate the area of E to area of ν.
 
Mean Anomaly  
Mean Anomaly (MA)
relates True Anomaly (TA) to Eccentric Anomaly (EA) via Kepler's 2nd law. ASSUME a certain AC sector has area equal to TA area. 
 
1610, he briefly corresponded with Galileo after Galileo used a telescope to discover four moons of Jupiter. To supplement his income, Kepler published astrological calendars. These forecast planetary positions and weather as well as political events; the latter were often cannily accurate, thanks to his keen grasp of contemporary political and theological tensions. By 1624; however, these prophecies eventually caused trouble for Kepler himself; his final calendar was publicly burned in Graz. Kepler's sponsor, Emperor Rudolph, had died, and Kepler had to move on. In a 1615 financial dispute, a vindictive woman charged Kepler's mother, Katharina, with witchcraft; relatively common in those days. After fourteen months imprisonment, Katharina was finally released in October 1621; this was due largely to the extensive legal defense drawn up by Kepler. The accusers had no stronger evidence than rumors. Harmonices Mundi ("Harmony of the World") was published in 1619; it described geometrical harmonies in the perfect solids. Kepler was convinced "that the geometrical things have provided the Creator with the model for decorating the whole world." In Harmony, he attempted to explain the proportions of the natural world—particularly the astronomical and astrological aspects—in terms of music. The central set of "harmonies" was the musica universalis or "music of the spheres," which had been studied by Pythagoras, Ptolemy and many others before Kepler. Kepler also explored regular polygons and regular solids, including the figures that would come to be known as Kepler's solids. In the final portion of the work (Book V), Kepler addressed relationships between orbital velocity and orbital distance from the Sun. Similar relationships had been used by other astronomers, but Kepler—with Tycho's data and his own astronomical theories—treated them much more precisely and attached new physical significance to them. Among the book's many harmonies, Kepler articulated: "The square of the periodic times are to each other as the cubes of the mean distances." third law of planetary motion.  
Kepler's Third LawSquare of orbit's period (T^{2}) is proportional to the cube of its semimajor axis (a^{3}).  

 
 
Summary
For any True Anomaly (angle, ν) of any Solar orbit, we can readily determine elapsed time from the orbit's perihelion to any other orbit position. 1. Determine the constant semilatus rectum, ℓ, from easily observed orbital elements, a (semimajor axis) and b (semiminor axis). 2. Each ν has a corresponding value, r_{ν}, distance from Sol to the orbit. 3. Each ν and r_{ν}, has a corresponding E, Eccentric Anomaly. 4. Each E has a corresponding M, Mean Anomaly. 5. Finally, each M enables a corresponding portion of the total orbital period, T_{Orb}, computed by Kepler's Third Law. Since Kepler's time (early 1600's), many scholars have determined many other orbital relationships and a host of useful mathematical tools; they all depend on Kepler's Laws.  
Kepler inherited Tycho's post as Imperial Mathematician. He spent the rest of his life working with Brahe’s data and publishing many works. Johannes Kepler died in Regensburg in 1630, while on a journey from his home in Sagan. Two years later, action from the Thirty Years War demolished his gravesite. Though Kepler lived and died in turbulent times, his scientific works had a far greater effect on humanity than all the historical events during his time. 
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