Sunday, November 24, 2013


Kepler's Laws gracefully
determine elapsed time of flight
from perihelion (q) to any orbital position.
Uses following concepts.
Kepler's Laws.
Orbital positions can be expressed as anomalies (or differences) from a reference point (usually the perihelion, orbit's nearest point to the Sun). These anomalies are usually expressed as angles (degrees or radians).
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 in-orbit velocities vary between a range of values. Kepler's Laws present an elegant, general solution for predicting orbital positions in elliptical orbits.
File:Johannes Kepler 1610.jpgA 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 pre-telescope 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 egg-shaped 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 focusKepler's first law of planetary motion. 

Kepler's First Law

The orbit of each Solar satellite is an ellipse with the Sun at one focus.
"a, b, c" of an Ellipse
a: semimajor axisb: semiminor axisc: focus
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 = (a2 - c2
Thus, b = (1.52 - .52)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:
Ellipse Area

Circle Area
= a b π

a a π

= b

 Semilatus Rectum
Semi-latus rectum ()  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 YOrb (ellipse point, ye).
 Thus, segment, S-YOrb, is orbit's semilatus rectum.
Extend this segment to intercept AC at point, YAC (circle point, yc).
Above values all relate via the a:b ratio:




Rewrite semilatus rectum in terms of e, eccentricity:
= b2/a = a (1 - e2)  = 1.5(1-.3332) AU = 1.333 AU
Define True Anomaly (ν)
 Polar Coordinates: (ν, r)
Express object's position  as angle and distance: (ν, r).

Let eccentricity, e  = .333; let semilatus rectum, ℓ =1.333 AU.
Angle, ν, is the "True Anomaly".
For objects orbiting the Sun, the true anomaly, ν (Greek letter, "nu"), is the asteroid's actual angular position on its orbit. Since the reference ray (Sun to the perihelion, q) is 0°, ν is the angle from reference ray to the positional vector, R, from Sun to object's position. 

Also, Cartesian Coordinates: (X, Y)
COMPUTE: X = r × Cos(ν); Y = r × Sin(ν)
 Polar Coord.Cartesian Coord.
TA (ν)Radius (r)x(ν) y(ν)
0.998 AU0.9975 AU0.0000 AU
30°1.032 AU0.8938 AU0.5160 AU
45°1.076 AU0.7611 AU0.7611 AU
60°1.140 AU0.5700 AU0.9873 AU
90°1.330 AU0.0000 AU1.3300 AU
120°1.596 AU-0.7980 AU1.3822 AU
135°1.740 AU-1.2305 AU1.2305 AU
150°1.870 AU -1.6193 AU0.9349 AU
180°1.995 AU -1.9950 AU0.0000 AU

r × Cos(ν) r × Sin(ν)
For every angle, ν, compute distinct set of X,Y coordinates.

SUMMARY: Easily generate values for r, x and y
for any angle (ν) in any orbit.
Define Eccentric Anomaly (E)
To measure an object's Eccentric Anomaly (E),
determine YAC, object's superposition on the orbit's Auxiliary Circle (AC).
 Every orbiting object's position
has a corresponding YAC directly above YOrb.
 AC has orbit's semi-major axis (a) as its radius.

Every orbital position has following values.


for following:
Thus,  we
Orbit:Orbital Y:Get ratio:


(a2 - x2)b


 = yOrb

Circle:Circular Y:Get relation:


(a2 - x2)
yAC =yOrb × a ÷ b
YAC  = (1.05×1.5 / 1.414)AU = 1.118 AU
Ref: Fundamentals of Astronautics, Bate, Mueller, White. pg 183-184).

For any YOrb, readily determine YAC (=YOrb×a÷b). 
Compute E
AC radius = a = length of semimajor axisHYPotenuse
of Right Triangle
YAC is perpendicular distance from C-q ray
to orbiting object's superposition on Auxiliary Circle (AC).
OPPosite leg
of Right Triangle
NOTE: Points  X-C-YAC form a right angle; thus, use trig function.










a = 1.5 AU
b= 1.41 AU
  c = 0.5 AUe = 1/3 =1.325 AU
Givenc = (a²-b²)e = c/aℓ = b²/a
True AnomalyEccentric Anomaly
νRadius (r)YOrbE
0.000 AU0.0000 AU0.0°
15°1.009 AU0.2610 AU10.6°
30°1.032 AU0.7611 AU21.4°
45°1.076 AU0.8068 AU 32.6°
60°1.140 AU0.9873 AU 41.0°
75°1.227 AU1.1856 AU 44.4°
90°1.330 AU1.3300 AU 70.6°
105°1.459 AU1.4095 AU85.3°

r × Sin(ν)

SUMMARY: For every True Anomaly, ν,
there is a distinct YOrb and corresponding Eccentric Anomaly, 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 contemporaries--see 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 arc-minutes.

Diagram of the geocentric trajectory of Mars through several periods of apparent retrograde motion. Astronomia nova, Chapter 1, (1609).
Then, Kepler attempted to fit the data to an ovoid orbit (eventually an ellipse). Kepler assumed the motive power radiated by the Sun weakens with distance; thus, planets move slower as they move farther from Sol.  After considerable work, Kepler created a formula to show that a planet's rate of motion is inversely proportional to its distance from the Sun. Verifying this relationship throughout the orbital cycle, however, required very extensive calculation; to simplify this task, by late 1602, Kepler reformulated the proportion in terms of geometry: planets sweep out equal areas in equal timesKepler's second law of planetary motion. 

Kepler's Second Law

The 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, t1,
asteroid orbits from first position, p1, to second position, p2,
and vector, R, sweeps an area, A1.

R vector varies in both angle and length.
By inspection, one can observe the length S-p1 as less then S-p2.

 Equal Areas Show Equal Times

For any two sectors of equal areas (A1 = A2);
corresponding orbit times must also be equal (t1 = t2).

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 YAC.

Thus, E sector area is a simple fraction of a circular area:
AE = (E ÷ 360°) × πa2

For every angle E, there is a corresponding ν.
ν sector is bounded by an ecliptic arc and off-center 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.
Call this sector's angle, M, the Mean Anomaly (MA).
Recall the goal is to compute area  of True Anomaly (TA);
QUESTION: If so, w
hy do we need the Mean Anomaly???
ANSWER: MA has a regular shape, TA has an irregular shape:
1) TA's outer edge (orbit) is ecliptic.
2) TA apex is off center.
However, the MA is a circular sector with area equal to TA area.
If we knew the MA sector angle, we could easily compute the area.
Long ago, Johannes Kepler, discovered a formula to determine this angle.

M = E - e ×  Sin(E)
For a great derivation, see Fundamentals of Astronautics, Bate, Mueller, White. pg 185
M = E - e ×  Sin(E)
Define Terms:
Mean Anomaly
(angle in radians)
Eccentric Anomaly
(angle in radians).
(ratio of c : a).
Sin(E):ratio of YAC : a
(equals ratio of YOrb:b).
Anomaly (E)
νRadius (r) EM
10°1.004 AU 7.1° 4.7°
20°1.015 AU 14.2° 9.5°
30°1.035 AU21.5°14.5°
70°1.197 AU52.7°37.5°
80°1.260 AU61.4° 44.6°
90°1.333 AU 70.5° 52.5°
100°1.415 AU 80.2°61.4°
109.47°1.500 AU 90.0° 70.9°


E - e ×  Sin(E)
Example: Given following values:
CONSTANT: e = c/a = 0.5 AU/1.5 AU =  0.333
CONSTANT: = b²/a= (1.414 AU)²/1.5 AU = 1.33333 AU
VARIABLE: ν= 60°
Table shows: E = 52.7° = .9198 rad; thus, Sin(E)= 0.7955
Compute: M=.9198 rad-.333 × .7955 rad=.6546 rad=37.5° 
Determine area value held in common
between True Anomaly (ATA) and Mean Anomaly(AMA).
AMA = ∠MA/360° ×  π × a × a = ATA

AMA =37.5°/360°×3.14×1.5AU×1.5AU= .736 AU2 = ATA
Geometrical harmonies in the
perfect solids from Harmonices Mundi (1619)
 Harmonices Mundi ("Harmony of the World").
After he published his Second Law of Orbits, Kepler went through many personal trials. Following the death of his first wife, Barbara, Kepler married the 24-year-old Susanna Reuttinger after considering 10 other matches over two years.  He eventually chose Reuttinger who "won me over with love, humble loyalty, economy of household, diligence, and the love she gave the stepchildren".
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 Law

Square of orbit's period (T2) is proportional to the cube of its semimajor axis (a3).
FOR EVERY VALUE "a", there is one and only one orbital period.
T = (a³)
Regardless of shape,
orbit's period depends only on "a".

If semimajor axis, a, is 1.5 AU, orbital period is:
T = (1.5³) =1.84 yr
Orbit's shape is determined by the orbit's semiminor axis, "b"; which can range from 0 AU to a max of "a".
 GIVEN "a", any possible semiminor axis, b, has a corresponding
perihelion, q, and aphelion, Q. 
Regardless of shape, time from q to Q is the same duration,
tSemi = T/2 = (a³)/2
Same period orbits can have different shapes;
each with a corresponding q and Q on the orbit's major axis. Since any orbit is symmetric about the major axis, the top semiorbit has same area as bottom. Thus, Kepler's 2nd Law compels us to conclude that orbital travel time between q and Q will always be T/2.
Determine E for Different Quadrants
180° ≥ E > 90°
E =180°-

90° ≥ E > 0°
E =0+

180° ≤ E < 270°

270° ≤ E < 360°

Anomaly (ν)
Anomaly (E)

ν EMt
 0.0° 0.0° 0 dy
60.7° 45°31.5°58.7 dy
109.5°90°70.9°132.2 dy
 147.3°135° 121.5°236.5 dy
180°180°180° 335.5 dy 
212.6°225°238.5°444.5 dy 
250.5°270° 289.2° 539.0 dy 
 299.3°315° 329.3° 612.4 dy 
360°360°360.0°671.0 dy 

E - e ×  Sin(E)

Example: Given following values:
CONSTANT: b = 1.414 AU
CONSTANT: TOrb = = √(1.5³) yr =  671 days
VARIABLE: ν= 60.7°
Table shows: E = 45° = .785 rad; thus, Sin(E)= 0.707
Table shows: M = (.785  - .333 × .707) rad = .553 rad = 31.5°
Determine travel time (t) from ν= 0° to ν= 45°.

× 671 days = 58.7 days = t

(ν) True
Time (tν) 
 (ν) True
Time (tν) 
(ν)  True
Time (tν) 
15°13.26 dy135°191.28 dy255°547.46 dy
30°26.97 dy150°234.56 dy270°573.12 dy
45°41.65 dy165°283.38 dy285°594.68 dy
60°57.88 dy180°335.51 dy300°613.15 dy
75°76.34 dy195°387.64 dy315°629.37 dy
90°97.90 dy210°436.47 dy330°644.05 dy
105°123.56 dy225°479.74 dy345°657.77 dy
120°154.38 dy240°516.64 dy360°671.02 dy
tν =M×TOrb

M = E-e×Sin(E)
E =Sin-1(rν×Sin(ν)

rν =



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, rν, distance from Sol to the orbit.
3. Each ν and rν, 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, TOrb, 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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