Newton essay

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The Kepler Problem
Can Kepler’s Equations be Considered Laws of Nature?
Glen B. Alleman
Niwot, Colorado
galleman@niwotridge.com
Abstract: The essential strength of the science of physics lies in its deep conceptual
schemes that unify a broad range of knowledge about the physical universe with rela-
tively few principles. Kepler derived his laws from empirical evidence. Newton used
these laws to determine the 2
1 r dependence of the force in the law of universal gravi-
tational attraction. It can be easily shown that these two laws produce planetary orbits
that are elliptic. What is surprising though is that the inverse square law and Hooke’s
potential law are the only central force laws capable of produce periodic elliptical
orbits. Of all the possible central force laws, only the Kepler problem yields strict
clockwork–like periods for all possible values of the initial conditions. 1
One foundation of this knowledge comes from Isaac Newton and others of his time.
These scientists not only solved an important problem in the field of dynamics, they
laid the groundwork for the thought processes involved in solving these problems.
The result was a set of laws by which nature can be examined. The laws of motion
were set down in Newton’s Principia, which consists of three Books, plus an Intro-
duction. The first book is the starting point of every proposition on dynamics. It treats
the motion of bodies without resistance under various force laws. The second book
explores motion of bodies in a resisting media. The third book discusses universal
gravitation.
Proof of Kepler’s’ Laws of Motion can be found in many forms in almost any modern
text on dynamics or celestial mechanics. This essay provides a summary of these
proofs.
The intention here is to show how these proofs were derived as well as discuss why
the term the laws of nature can be applied to these discoveries. A brief introduction to
why solutions constrained by the Bertrand–Königs theorem – Kepler solutions –are
the only ones allowed by the extra global conserved quantity independent of the angu-
lar momentum and the energy equations of a planet following a Keplerian orbit.
1 Isotropic harmonic oscillators are included in this solution as well. The periodic orbit is produced independent
on energy, angular momentum, and other force constants.

2
1 Introduction
This essay is about a single although not insignificant fact. When an astronomical
object travels through space under the influence of gravity, it follows a path described
by a mathematical family of curves – a circle, an ellipse, a parabola, or a hyperbola.
Why does nature choose to have astronomical objects follow such a curve? What is
the force that creates such an orbital path? How does this force behave as a function
of distance from the force center? The answer to these simple questions has profound
scientific and philosophical significance.
1.1 Proof and Physics
Philosophers have argued for centuries about the answer to this question, and how
something can be proven – and they no continue to do so. Mathematicians, on the
other hand, have been using “working definitions” of proof to advance mathematical
knowledge for the same amount of time.
There is a long tradition of refining Newton’s mechanics into formulations of every
greater sophistication and elegance. 2
This essay does not contribute to that literature,
but it does make heavy use of authors that have, both in the past and the present. 3
The proofs of Kepler's first two laws are mathematical, but they are dealing with facts
that belong to physics. This does not rule out Kepler's preferred method of proof,
which is the geometric method of Euclid. Euclid gave definitions and axioms and then
proceeded by deduction. All mathematicians and most philosophers of the time agreed
this method was the most reliable.
The rigorous methods of deductive proof found in geometry and mathematics cannot
be applied if the basis of the proof is the messy numbers derived from observation or
experiment. One of the most important things that are learned from doing experiments
or making observations is to know the acceptable form of the “answer” in order to
deduce it from the experimental data.
Kepler was determined to make the best possible use of the huge number of observa-
tions. He used a method that involved repeated trial and error. Historically, this is
interesting because we know he preferred to model himself on Euclid and use strict
logical proofs. Kepler's methods or measurement were not known in astronomy that
used geometry. This geometric approach was generally regarded at the time as intel-
lectually superior to geometric and algebraic proofs.
Although at the time astronomers were known as “mathematicians”, their work did
not actually involve proof in the same way as Euclid. Astronomers of the time were
expected to predict the positions of the Moon. These predictions were often found to
agree, and many times not agree, with observation. Kepler's work was different since
it set out to define a geometrical description of the orbit of Mars, and of the structure
of the Solar system.
For a deductive proof of Kepler's laws the world had to wait for Newton. Newton
made use geometry and ignored the numerical variations that had to be addressed
Kepler's proofs.
2 Several famous physicists have made advances in explaining Newton’s mechanics. James Clerk Maxwell proved
Kepler’s 3rd Law in the 1877 edition of Matter and Motion. Maxwell attributes the method of proof to Sir William
Hamilton.
3 Rather than burden the reader with a detailed reference notation, a bibliography is provided. It is assumed the
material here is traceable to this bibliography.

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1.2 Forces of Nature
The term gravitational force is taken lightly these days. The term force is actually not
well understood by the layman. We see the effects of force all around us. The force of
gravity, the electric and magnetic forces of natural and manmade objects, and the me-
chanical force of machines all have well known effects. In pre–twentieth century sci-
ence, natural philosophers asked many of the same questions that are asked here —
why does nature behave in the way it does? Although these questions have the tone of
theological or philosophical inquiries, the study of these forces and their interaction
with matter is generally the domain of physics.
The development of the concept of a force marks the boundary between science and
pre–science. In early history, objects were believed to have internal powers, which
could account for their movements. The motion of the planets through the night sky
was associated with gods, and super-natural powers. It was realized during the time of
Galileo that the function of a force was not to produce the motion, but to produce a
change in the motion. This description of force was not significantly different from
the previous occult force, since the origin of the force was still not known. However,
these forces could be measured which allowed quantitative order to be brought to na-
ture.
1.3 The Role of Scientific Law
In 1687, Isaac Newton published his Principia. This volume contains a remarkable
passage on the rules of reasoning. There are four such rules, which collectively reflect
Newton’s profound faith in the unity of nature. These rules were intended to guide
scientists in the scientific process.
The first rule is called the principle of parsimony. It says that scientists should make
no more assumptions or assume no more causes than are absolutely necessary to ex-
plain their observations. The principle of parsimony is also known as Occam’s Razor,
after William of Occam, who stated his principle of economy of thought in the phrase,
“a plurality must not be asserted without necessity.”
The second rule is the principle of cause and effect, or the belief that what occurs in
nature is the result of cause–and–effect relationships. Where similar effects are seen
the same cause must be operating.
The third rule is the principle of universal qualities or the belief that those qualities,
such as mass or length, that describe bodies exposed to our immediate experience also
describe bodies removed from our immediate experience, such a starts or galaxies.
The fourth rule is the principle of induction. Induction is the process of deriving con-
clusions about a class of objects by examining a few of them, then reasoning from the
particular to the more general. 4
This rule states that concepts, hypothesis, laws, and
theories arrived at by induction should be assumed as universal both in time and place
until new evidence proves the contrary to be true. This is the means by which Kepler
developed his laws of planetary motion.
These rules for reasoning are fundamental to the process of discovery of natural or
scientific laws. The following definition will be used here for a scientific law:
As formulated by humans, natural or scientific laws are rules, preferably mathematical rules, by which we be-
lieve nature operates, and such laws can be classified as being either empirical, definitional, or derived laws.
4 Deduction is the process of reasoning from the general to the more specific.

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Empirical laws are general statements that identify regularities in many observations
without offering a theoretical explanation for these observations.
Definitional laws are a second level of physics law. These laws usually involve the
definition of fundamentally important concepts. Newton’s second law of motion and
the law of conservation of energy are examples of definitional laws.
Newton’s law of universal gravitation is derived from Kepler’s third law.
The scientific laws of nature are usually thought of as inexorable and inescapable; in
part because of the word law suggests an erroneous analogy with divine law. Scien-
tific laws, built on concepts, hypothesis, and experiments, are only as trustworthy as
those concepts and as those experiments are accurate. Since humans formulate scien-
tific laws, they are neither eternally true nor unchangeable.
2 Newton’s Formulation of Universal Gravity
In August of 1684, Edmund Halley traveled to Cambridge to speak with Isaac Newton
about celestial mechanics. Floating around Europe and England was the idea that the
motions of the planets in the solar system could be accounted for by a force that ema-
nated from the sun. This force diminished as the inverse square of the distance, but no
one had yet been able to produce a satisfactory demonstration of this principle.
Newton had hinted that he could provide such a demonstration. A demonstration that
the forces involved would lead to elliptical orbits. Johannes Kepler had deduced these
elliptical orbits 70 years earlier.
Halley asked Newton to see the demonstration, but Newton claimed to have mis-
placed the calculations. Halley left disappointed, but a few months later received a 9–
page treatise showing that the inverse square law along with some basic principles of
dynamics could account for the elliptical orbits as well as Kepler’s other laws of plan-
etary motion.
Halley knew he was holding the key to understanding the universe as it was then con-
ceived and asked Newton if he could publish the results. Newton was not yet ready
and delayed the final publication for three years. The resulting work was published in
1687 under the title Philosophiae Naturalis Principia Mathematica. This was New-
ton’s masterpiece and the foundation of modern science. 5
In the Principia Newton used a method of polygonal approximations to demonstrate
that Kepler’s law of equal areas holds for any force directed toward a fixed center.
Using these results Newton extended his dynamics to a general method of determin-
ing the nature of the force required to maintain a specific type of orbital motion about
a given center of force. These solutions included: circular, spiral and elliptical orbits.
While Kepler's laws applied only to the Sun and planets, Newton's universal theory
provided the means to calculate the gravitational force and motion of any astronomi-
cal body.
5 The source of this anecdote can be found in the notes accumulated by John Conduitt, husband of Newton’s niece
and Newton’s successor at the Royal Mint, for a proposed biography of Newton.

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2.1 The Kepler Problem
My goal is to show that the heavenly machine is not a kind of divine living being, but similar to clockwork in
so far as all the manifold motions are taken care of by one single absolutely simple magnetic bodily force, as
in clockwork all motion is take care of by a single weight. – Johannes Kepler
Kepler studied Tyco Brahe’s observations of planetary motion and discovered that the
planets do not move with uniform speed in their orbits. They travel more slowly the
greater their distance from the sun. 6
Kepler searched for a dynamical explanation of planetary motion in his Astronomia
Nova, which is considered the modern book in astronomy.
The mathematical treatment of planetary motion does not appear until Kepler’s Epit-
ome astronomiæ Copernicanæ, where he used a mechanical description of force –
“vis seu energia,” for the first time. The determination of the dependence of planetary
velocity on distance lead Kepler to the conclusion that the cause of this observation is
physical rather than spiritual.
Kepler reported in 1609 that Mars moved in an elliptical orbit with the sun at one fo-
cus of the ellipse, with the radius vector from the sun to Mars sweeping out equal are-
as in equal times. Huygens determined the force function required for uniform motion
in 1659 and independently by Newton in 1669. No one prior to Newton had demon-
strated the specific mathematical formulation of the force function required to pro-
duce elliptical orbits. [7]
Kepler’s first two laws were published in Astronomia Nova
(The New Astronomy: Based on Causes or Celestial Physics) (1609) and the third in
Harmonice Mundi (Harmony of the World) (1619). In simple form, Kepler's three
laws are:
§ Lex I: Each planet moves in an elliptical orbit, with the Sun at one focus of the
ellipse (1605);
§ Lex II: The focal radius from the Sun to a planet sweeps equal areas of space in
equal intervals of time (1604);
§ Lex III: The square of the sidereal periods of the planets are proportional to the
cube of their mean distance to the Sun. This third law can be stated as where T is
the period of the planet and A is the semimajor axis of its elliptical orbit and k
can be given in terms of Newton's gravitational constant (1618).
Kepler’s discoveries about the behavior of planets in their orbits played an essential
role in Isaac Newton's formulation of the law of universal gravitation in 1687. New-
ton's theory showed that celestial bodies were governed by the same laws as objects
on Earth. The philosophical implications of this played as key a part in the Enlight-
enment as did the theory itself in the subsequent development of physics and astron-
omy.
6 Kepler’s own formulation of the law, which was done before the 1st Law, was correct only at the apsides where
the radius vector perpendicular to the tangent.

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2.2 Newton’s Laws
Newton set about to prove Kepler’s Third Law using the mathematical tools of the
time. Although Newton invented differential and integral calculus, he had no yet pub-
lished the details due to a nasty dispute with the German philosopher and mathemati-
cian Gottfried Leibniz, who had made the same mathematical discoveries.
Newton's three laws of motion are formally given in Philosophiae Naturalis Principia
Mathematica (Mathematical Principals of Natural Philosophy) as: 7
§ Lex I (in editions of 1687 and 1713) – Corpus omne perseverare in statu suo
movendi uniformiter in directum, nisi quatenus illud a viribus impressis cogitur
statum suum mutare.
§ Lex I (in edition of 1726) – Corpus omne perseverare in statu suo quiescendi vel
movendi uniformiter in directum, nisi quantenus illud a viribus impressis cogitur
statum suum mutare. (Every body continues in its state of rest, or of uniform mo-
tion in a right line, unless it is compelled to change that state by forces impressed
upon it.)
A Body at rest remains at rest and a body in a state of uniform linear motion continues
its uniform motion in a straight line unless acted on by an unbalanced force. This law
is often called the law of inertia. This means that the state of motion in a straight line
remains at rest of continues its uniform motion unless acted on by an unbalanced
force. The presence of the unbalanced force is indicated by changes in the state of
motion of a body.
§ Lex II – Mutationem motis proportionalem esse vi motrici impressae, et fieri
secundum lineam qua vis illa imprimatur. (The change of motion is proportional
to the motive force impressed; and is made in the direction of the right line in
which that force is impressed).
An unbalanced force, F, applied to a body gives it an acceleration, a, in the direction
of the force such that the magnitude of the force divided by the magnitude of the ac-
celeration is a constant, m, independent of the applied force. This constant, m, is iden-
tified with the inertial mass of the body. The inertial mass is a derived rather than
basic quantity. Newton's equations of motion establish a procedure for measuring this
mass. This is done by applying a known force to a body and measuring its accelera-
tion. The result of this measure is the mass of the body. There is an additional inter-
pretation of the second law of motion. If a body is observed to be accelerating than a
force must be acting on it, but if no force is known to be physically applied to the
body, Newton concluded that this force must act–at–a–distance. 8
§ Lex III – Actioni contrariam semper et aequalem esse reactionem: sive corporum
duorum actiones is se mutuo semper esse aequales et in partes contrarias dirigi.
(To every action there is always opposed an equal reaction; or, the mutual actions
of two bodies upon each other are al-ways equal, and directed to contrary parts.)
If a body exerts a force of any kind on another body, the latter exerts an exactly equal
and opposite force on the former. This law introduces a symmetry that does not ap-
pear in the first two laws. It states that forces appear in equal and opposite pairs.
7 These are the well–known laws of motion, which form the starting point of every argument in classical dynamics.
The first two laws, which relate to inertia, were generalizations from Galileo’s observations. The first law, known
as the law of inertia, is a special case of the second law.
8 By alteration of motion, Newton had in mind the rate of change of momentum. For the case of constant mass, this
becomes F=ma. The second law provides a definition of force in terms of the acceleration given to a mass.

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These three laws, along with the other postulates in Principia were extensions of pre-
vious work including the laws derived by Galileo Galilei. Galileo discovered the em-
pirical basis for the law of inertia through systematic experiments. These experiments
led Galileo to assert that all bodies should accelerate at the same constant rate near the
earth’s surface. 9
3 A Simple Proof of Kepler’s Laws
The first step in discovering a universal, mathematically precise description of forces
is to start with the inverse question.
Given that a body’s orbit is elliptical, circular, parabolic, or hyperbolic with a motionless force–center, what
is the force law that produces such an orbit?
This was the question Newton answered in the Principia. In modern notation, if the
angular momentum is defines as = ´L r p and the torque defined as ´N = r F ,
where r is the position of a body, then,
dL
dt
= N, (1)
follows directly from Newton’s second law, since =md dtv the cause of deviation
from uniform rectilinear motion, where m is the coefficient of resistance to any
change. Eq. (1) means that the momentum vector p is conserved in the absence of a
net torque on the body. 10
A central force law yields angular momentum conservation. If ( )= f rF r , where r is
the unit vector in the r–direction, then the force on the mass m is always directed
along the position vector of the body relative to the force center. This implies the
presence of a second body at the force center generating the force, whose motion can
be ignored for the moment.
In this case the angular momentum given by, = ´mL r v is constant at all times and
therefore defines a fixed direction ˆL in space. Because both the velocity and linear
momentum must remain perpendicular to the angular momentum, the conservation
law confines the motion of the mass m to a fixed plane perpendicular to the angular
momentum vector L.
Planar orbits agree with the observations of planetary motions, so Newton was able to
restrict his consideration to central forces.
9 The third law was original with Newton, and is the only physical law of the three. Taken from the second, it de-
scribes the concept of mass in terms of its inertial properties. Mass cannot be defined explicitly but rather it must
be described in terms of its inertial and gravitational properties, which means it cannot be described inde-
pendently of the concept of force. Newton spoke of mass as the quantity of matter in a body, which lacks precision
because there is no definition of matter. The first two laws are best considered as definition of force. The first de-
scribes the motion of a body in equilibrium while the second describes its lotion when the forces acting upon it do
not balance one another.
10 The momentum of an isolated system is always constant. The vector sum of all the momenta mv of all the ob-
jects of a system cannot be changed by interactions within the system. This places a strong constraint on the
types of motions that can occur in an isolated system. If one part of the system is given a momentum in a given
direction, then some other part or parts of the system must simultaneously be given exactly the same momentum
in the opposite direction. It is assumed that the conservation of momentum is an absolute symmetry of nature.
That is, we do not know of anything in nature that violates it.

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If Newton’s law is written in polar coordinates ( ),qr , in the plane of motion, which
is perpendicular to the z–axis, then Newton’s second law of motion =d dtp F has
the form,
( )
2
2
,
0,
- q =
= =z
d r
m mr f r
dt
dL
dt
(2)
where 2
= qzL mr d dt is the magnitude of the angular momentum and is constant.
z
y
rx q
Figure 1
Eq. (2) can be rewritten in the form,
( )22
2 2 3
2
,
constant.
- =
= q =
z
z
f rLd r
mdt m r
L mr
(3)
By assuming the orbit is a conic section and differentiating Eq. (3),
( )
1
1 cos ,= + e qC
r
(4)
where C and e are constants and e is the eccentricity of the conic section. Eq. (4)
requires that ( )f r varies as the inverse square of the distance r of the body from the
force center, which lies at one of the two focal points of the conic section as shown in
Figure 2.
With the origin of the coordinates lying at one focus the central force function gives,
( )
2
2
.= - zCL
f r
mr
(5)
If 1e < then the conic section is an ellipse, which agrees with Kepler’s 1st Law. If
1e > , then the orbit is hyperbolic. For the special case of 0e = and 1e = circular and
parabolic orbits occur.
Newton derived the solutions to Eq. (4) geometrically by searching for the shapes of
curves that define intersections of a plane with a cone.

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3.1 The Ellipse
The ellipse is central to Kepler’s laws, as well as general orbital mechanics. One way
of defining an ellipse is through the directrix, which is a line, together with the focus
that defines a conic section as the locus of points whose distance from the focus is
proportional to the horizontal distance from the directrix. 11
The constant of propor-
tionality is called the eccentricity e .
Some useful relationships of the ellipse shown in Figure 2 and used by Kepler in-
clude:
cos ,
cos ,
,
.
= q
= - q
e =
= = e
FB r
PQ k r
c
a
FP r PQ
(6)
Using these relationships, the vector r describing the path a particle takes while fol-
lowing an ellipse is given by,
( ) ( )cos ,
.
1 cos
q = e - q
e
=
+ e q
k r
k
r
(7)
The eccentricity e is a measure of the flatness of the ellipse. It is a number between 0
and 1 which is the focus divided by the semimajor axis. If the focus is zero then the
eccentricity is zero, both foci occur at the center and the figure is a circle. If the focus
is the same length as the semimajor axis, then the eccentricity is one and the figure is
a straight–line segment equal to the semi–major axis and the focus, and the semi–
minor axis is zero.
The semi–minor, semi–major axis, focus distance from the center of the ellipse can
now be given as,
2
2
,
1
,
1
.
e
=
- e
e
=
- e
= e
k
a
k
b
c a
(8)
Using this notation, and noting that 1 1= = ep c a , the aphelion radius (the furthest
distance) is 1-ep and the perihelion radius (the closet distance) is 1+ep .
11 The ellipse was first studied by Menaechmus. Euclid wrote about the ellipse and it was given its present name by
Apollonius. The terms focus and directrix were described by Pappus. Kepler introduced the term focus.

10
r
q
FO
P
Q
x
y
a c
b
1
p
e- 1
p
e+
x k=
B
Figure 2
3.2 Are Newton’s Inverse Solutions Unique?
In order to proceed with a proof of Kepler’s Laws as they were developed by Newton,
the results of Eq. (4) must be shown to be unique. There are two central force laws
that yield close, periodic orbits for arbitrary initial conditions. Both of these close
solutions result in elliptical orbits.
Using = -kF r , if 0>k then the orbit is elliptic with the force–center at the center of
the ellipse. This is the case for an isotropic simple harmonic oscillator. In this case the
force constant k is independent of direction and is the same in all directions, since any
central force is spherically symmetric. 12
If 0>k , then the orbit is an ellipse with the
force–center at one focus, which is an idealized description of a single planet moving
around a motionless sun.
The determination of the force laws that yields an elliptical orbit for all possible initial
conditions, or closed orbits of any kind for arbitrary initial conditions is summarized
by the Bertrand–Königs Theorem. According to scholars Newton knew of this theo-
rem and its application to the solution of the central force law equation.
The conclusion so far is that Newton’s solutions to Kepler’s equations are unique. The
only missing element is the motion of the Sun itself, which will be taken into account
later.
12 An anisotropic harmonic oscillator would be represented by a force law ( ), ,F k x k y k z= - - -1 2 3 , where at least
one of the three force constants, ik differs from the other two, representing the absence of spherical symmetry.

11
4 A More Elaborate Proof of Kepler’s Laws
The first step in discovering universal, mathematically precise description of forces is
to start with the inverse question. Given that a body’s orbit is elliptical, circular, para-
bolic, or hyperbolic with a motionless force–center – the sun in this case – at one fo-
cus, what is the force law that produces the orbit?
4.1 Torque
One key to proving Kepler's 2nd law (and further laws as well) is the concept of
torque. Torque is a tendency to change an object state of rotation. Torque is the rota-
tional analog of force. If torque is applied to a wheel, the wheel has a tendency to ro-
tate. Torque is in rotational mechanics what force is in linear mechanics.
Torque, t can be defined as,
,t = ´r F (9)
where F is the impressed force and r is the lever arm over which the torque acting.
The vector r begins at the axis of rotation and ends at the point where the impressed
force is acting. 13
The vector r indicates in which direction the body tends to rotate.
While torque is usually applied to rigid bodies, it can also be applied to celestial bod-
ies. The concept of torque can be applied to any body with respect to a fixed point in
space. The vector between this fixed point and the celestial body becomes the lever
arm.
This notion of torque can be applied to a planet orbiting the Sun. The impressed force
will be gravity. The fixed reference point will be the Sun. Using = ×rr r and
( )2
= - ×GMm rF r gives,
( )
( )
2
,
,
,
0.
t = ´
æ ö
= × ´ - ×ç ÷
è ø
æ ö
= - ´ç ÷
è ø
=
GMm
r
r
GMm
r
r F
r r
r r
(10)
Any vector whose cross products is with itself is the zero vector, 0, which says the
Sun never impresses a torque on a planet.
4.2 Conservation of angular momentum
Torque t is defined as the instantaneous time rate of change of angular momentum L:
t º
d
dt
L
(11)
Angular momentum is a quantity that plays the same part in rotational mechanics as
linear momentum does in linear mechanics.
13 Vectors like the position r and the momentum p change sign under inversion. They are called polar vectors, or
ordinary vectors. But a vector product of two polar vectors such as = ´L r p will not change sign under inver-
sion. Such vectors are called axial vectors or pseudovectors. The scalar product of a polar vector and a pseudo-
vector is a pseudovector; it changes sing under inversion, where a scalar vector does not.

12
From the previous section 0t = , which says that the Sun never applies no torque to a
planet. Therefore d dtL must also be the zero vector:
0.=
d
dt
L
(12)
If the time derivative of something is zero, that means that thing does not change as
time passes; in other words, it remains constant. This is usually only applied to sca-
lars, however. In vectors, if the time derivative of a vector is the zero vector, then that
vector does not change magnitude or direction. In other words, the angular momen-
tum vector of a planet is a constant vector:
constant.=L (13)
Because the Sun does not apply a torque to a planet from its gravitational influence,
the angular momentum of the planet remains constant; it is conserved. This is the core
concept of Kepler's 2nd law.
What is the mathematical expression for angular momentum, though? We can find an
expression for angular momentum from our expression for torque, substituting in
d dtL for t :
.= ´
dL
dt
r F (14)
We can use Newton's law of motion, = mF a , and substitute this into Eq. (14) to give:
( ).= ´
d
m
dt
L
r a (15)
The acceleration of a body is equal to its instantaneous rate of change of velocity,
gives
.=
d
dt
v
a (16)
Making this substitution and exploiting the fact that the cross product is associative
with respect to scalar factors, results in,
( )
,
.
æ ö
= ´ç ÷
è ø
= ´
d d
m
dt dt
m
L v
r
r v
(17)
Solving this differential equation, gives,
( ).= ´mL r v (18)
The magnitude of the angular momentum is then give as,
( )
,
,
.
=
= ´
= ´
L
m
m
L
r v
r v
(19)
Eq. (19) relates the angular momentum of a planet to its mass, position, and velocity.

13
4.3 Kepler's Second Law
Kepler's 2nd law states that a vector directed from the Sun to a planet sweeps out
equal areas in equal times.
Starting with the planet’s position at time t = 0, and allowing it to move to a new posi-
tion a short time later = Dt t . In that time, the vector has moved by a short displace-
ment,
0.=D =D = -t t tr r r (20)
The three vectors 0=tr , Dr , and =Dt tr as shown in Figure 3, form a triangle. The area
of this triangle closely approximates the area swept out by the vector r during the time
Dt .
0t=r
t t=Dr
Dr
Figure 3
This small area represented by this triangle, DA , can be represented as one–half of
the parallelogram defined by the vectors r and Dr , or,
1
2
.D = ´DA r r (21)
Dividing both sides of Eq. (21) by Dt , and using the associative properties of the
cross product, gives,
( )
1
2
1
2
1
2
1
,
,
.
D æ ö
= ´Dç ÷
D Dè ø
´D
=
D
D
= ´
D
A
r
t t
t
r
r r
r
r
t
(22)
Letting 0D ®t , and taking the limit of both sides of Eq. (22), the approximation ap-
proaches the real value of the area, giving,
1
2
1
2
, or
= .
= ´
´
dA d
dt dt
r
r
r v
(23)
Knowing,
,= ´L m r v (24)
and dividing both sides by m, gives,
.= ´
L
m
r v (25)
Substituting Eq. (25) into the expression for dA dt gives,

14
.
2
=
dA L
dt m
(26)
That is the instantaneous time rate of change of area. Knowing that the mass of the
planet is constant, and that the angular momentum vector is constant, the time deriva-
tive of area swept out by this vector is also constant. This says that no matter where
on the orbit the planet is, the vector from the sun to the planet sweeps out the same
amount of area. This is Kepler's 2nd law.
4.4 Polar Basis Vectors
Kepler's 1st law is concerned with the shape of the orbit that a planet makes around
the Sun. This law can be developed easily using polar basis vectors.
The polar coordinate system is an effective way of representing the positions of bod-
ies with the angle they make with the origin, and the distance they are away from it.
Polar coordinates are useful for dealing with motion around a central point – as is the
case for planets moving around the Sun.
The polar coordinates r and q are related to rectangular coordinates by,
cos ,
sin .
= q
= q
x r
y r
(27)
For any plane curve, the position vector = +x yr i j shown in Figure 4 is given by,
( ) ( )
( ) ( )( )
cos sin ,
cos sin .
= q + q
= q + q
r r
r
r i j
i j
(28)
where =r r .
The vector r is a function of q ; in other words, the unit vector representing the direc-
tion in which the body is located from the Sun is dependent on the angle.
This vector is formally defined as,
.= ×rr r (29)
Our definition of r as a polar basis vector is identical to the definition found in the
plane of the orbit in Cartesian coordinates.
Since there are two Cartesian basis vectors, i and j, there are also two polar basis vec-
tors. The second basis vector is the unit transverse vector and represent with ˆθ , and is
defined as the rate of change of r with respect to q :
ˆθ ,
sin cos .
º
q
= - q + q
d
d
r
i j
(30)
This definition means that ˆθ always points orthogonal to the unit radial vector. Using
polar basis vectors, the motion of a planet can be using a radial vector along r, and a
transverse vector, ˆθ .
Taking the derivative of ˆθ with respect to q gives,

15
( )
ˆθ
cos sin ,
cos sin ,
.
= - q - q
q
= - q + q
= -
d
d
i j
i j
r
(31)
qD
( )t t+ Dr
( )tr
qDr
ˆq ˆr
ˆq
ˆr
cosx r q=
siny r q=
q
Figure 4
The velocity vector v and the angular momentum vector L can now be expressed in
polar coordinates. Velocity is the instantaneous rate of change of the position of the
planet, given as,
( )
,
,
.
=
æ ö
= ×ç ÷
è ø
= +
d
v
dt
d
r
dt
dr d
r
dt dt
r
r
r
r
(32)
Using the chain rule to expand d dtr into a form that includes qd dr , gives,
.
q
= +
q
dr d d
r
dt dt
r
v r (33)
Since qd dr is w, the unit transverse vector, and the angular speed, w, is defined as,
,
q
w =
d
dt
(34)
we can obtain our final expression for velocity in polar coordinates as
.= + w
q
dr d
r
dt d
r
v r (35)

16
A similar expression for the angular momentum vector L in polar coordinates, we go
back to the expression we found for angular momentum:
( ).= ´mL r v (36)
Substituting ×r r for r and the expression in Eq. (35) for v, gives:
( ) .
é ùæ ö
= × ´ + wç ÷ê úqè øë û
dr d
m r r
dt d
r
L r (37)
We expand this expression to obtain,
( )
( ) ( ) ( )
( ) ( )2
,
ˆθ ,
ˆθ
é ù
= × ´ + wê úqë û
æ ö
= × ´ + × ´ wç ÷
è ø
= ´ + w ´
dr d
m r r
dt d
dr
m r m r r
dt
dr
mr mr
dt
r
L r r
r r r
r r r
(38)
Since, a vector crossed with itself is the zero vector, the first term evaluates to zero
gives,
( )2 ˆθ .= w ´L mr r (39)
Since ˆ´q =r k , the final expression for the angular momentum vector is,
2
= wmrL k (40)
Taking magnitude of this vector gives,
2
2
2
,
,
,
.
=
= w
= w
= w
L
mr
mr
mr
L
k
k
(41)
4.5 Kepler's 1st Law
Each planet moves in an elliptical orbit with the sun at the focus of the ellipse. – Johannes Kepler (1605)
Starting with Newton's law of motion and Newton's law of universal gravitation gives,
2
.
æ ö
= -ç ÷
è ø
GMm
m
r
a r (42)
Dividing both sides of Eq. (42) by m gives,
2
.
æ ö
= -ç ÷
è ø
GM
r
a r (43)
Using polar basis vectors, ˆθ q = -d d r . Solving for r and applying the chain rule
gives,

17
ˆθ
,
ˆ1 θ
.
= -
q
= -
w
dt d
d dt
d
dt
r
(44)
Substituting Eq. (44) into Eq. (43) gives,
2
2
ˆ1 θ
,
ˆθ
.
æ öæ ö
= - -ç ÷ç ÷
wè øè ø
=
w
GM d
dtr
GM d
dtr
a
(45)
Multiplying the RHS of Eq. (45) by m/m (which is unity), gives,
2
ˆθ
.=
w
GMm d
dtmr
a (46)
Since 2
= wL mr Eq. (46) can be rewritten as,
ˆθ
.=
GMm d
L dt
a (47)
Multiplying both sides of Eq. (47) by L GMmgives,
ˆθ
.=
L d
GMm dt
a (48)
Knowing that = d dta v , and substituting accordingly gives,
ˆθ
.=
L d d
GMm dt dt
v
(49)
This is a differential equation that can be solved in closed form to give,
ˆθ ,= +
L
GMm
v C (50)
where C is a constant vector. Solving for v gives,
ˆθ .= +
GMm
C
L
v (51)
Eq. (51) is a general solution for the velocity, but doesn’t describe the shape of a
planet's orbit. Knowing the shape is the next step, and any restrictions on the possible
orientations of the orbit.
Starting with the perihelion – closest approach to the Sun — at time t = 0, the orienta-
tion can be restricted so that, when perihelion occurs, the planet lies along the zero
radian line from the Sun. At this point the position vector, r, of the planet, will have a
component only in the positive x-axis. The planet will be orbiting the Sun counter-
clockwise, through increasing measures of angles. The velocity, v, at the perihelion
will be orthogonal to the position vector, r, and will have only a component in the
positive y–axis.

18
0q =
0t =
r
ˆθ
Figure 5
According to the expression for v, there is a scalar times the vector quantity ˆθ + C,
0
ˆθ ,q= = j (52)
that is, the unit transverse vector points up when the unit radial vector points right, as
shown in Figure 5. Since, at t = 0, ˆθ points entirely in the y–direction, then the con-
stant vector C must only have a component in the y–axis — this is the only way to get
a resultant vector (v) that points entirely in the y–direction. So, C can be rewritten as a
scalar times the unit basis vector in the y–direction:
,= eC j (53)
where e is a scalar constant. Substituting Eq. (53) into the equation for v gives,
ˆθ .= + e
GMm
L
v j (54)
This is the specific case when the orbit is oriented so that perihelion occurs at
0.= q =t
Performing the dot–product both sides of Eq. (54) with ˆθ gives,
( )
( )( )
ˆ ˆ ˆθ θ θ,
ˆ ˆ ˆθ θ θ .
× = + e ×
= × + e ×
GMm
L
GMm
L
v j
j
(55)
A dot–product of a vector with itself produces the square of that vector's magnitude,
so ˆ ˆθ θ =1× . Simplifying ˆθ×v , gives,
( )
( ) ( )
ˆ ˆ ˆθ θ θ ,
ˆ ˆ ˆθ θ θ .
× = + w ×
= + w ×
dr dt r
dr dt r
v r
r
(56)
The dot–product of two orthogonal vectors is zero, so ˆθ× =r 0. Knowing that ˆ ˆθ θ =1×
gives,
ˆθ .× = wrv (57)

19
The final part of the development is finding an expression for ˆθ×j . Knowing that
ˆθ sin cos= - q + qi j, gives,
( )ˆθ sin cos ,
cos .
× = × - q + q
= q
j j i j
(58)
r
v^
ˆv
rv
1 1
1
pv
a
e
e
+æ ö
= ç ÷
-è ø
1 1
1
av
a
e
e
-æ ö
= ç ÷
+è ø
Figure 6
4.6 Putting Kepler’s 1st Law All Together
The solution to Kepler’s 1st
Law now available. Starting with,
( )1 cos .w = +e q
GMm
r
L
(59)
Since wr is on the LHS of Eq. (59) and knowing that
2
1w =mr , both sides of the equation can be multiplied by mr to give,
( )
2
2
1 cos .w = + e q
GMm
mr r
L
(60)
Replacing the LHS of Eq. (60) by l and moving the constants to the left side of the
equation, gives,
( )
2
1 1 cos .= + e q
GMm
r
L
(61)
There is now an explicit function in terms of r and q which is the polar equation for a
planet's orbit.
Solving for r gives,
2
2
.
1 cos
=
+ e q
L
GMmr (62)
The equation of a conic section with focus–directrix distance k and eccentricity e is
represented by the polar equation,

20
.
1 cos
e
=
+ e q
k
r (63)
This is the same as Eq. (62), given that,
2
2
.e =
L
k
GMm
(64)
The focus–directrix distance is constant, which k is: L, G, m, and M are all individual-
ly constant; therefore the expression 2 2
L GMm must also be constant. Therefore,
Newton's laws of motion and universal gravitation dictate that the orbits of planets
follow conic sections. This is Kepler's 1st law.
Kepler's 1st law actually states that planets follow the paths of ellipses. An ellipse is
only one type of conic section. One question might be – why is an ellipse allowed
while the other conic sections are not?
Others are found in the solar system – but the objects that follow them are not planets.
When Kepler said planet, he meant a body that repeatedly returns to our skies. The
curve representing the orbit is closed — it must repeatedly retrace itself.
The only two conic sections that are closed are the circle and the ellipse with the cir-
cle being a special case of the ellipse. The other two conic sections – the parabola and
hyperbola – are open curves and correspond to a position where the body has suffi-
cient velocity to escape from the Sun's gravity well. The body would approach the
Sun from an infinite distance, round the Sun rapidly, and then recede away into the
infinite abyss, never to be seen again.
So we have proved an extension of Kepler's 1st law: A body influenced by the Sun's
gravity follows a path defined by a conic section with the Sun at one focus.
4.7 Kepler's 3rd law
After generating Kepler's 1st and 2nd laws, Kepler's 3rd is now straightforward.
The 3rd law relates the period of a planet's orbit, T, to the length of its semimajor axis,
A. It states that the square of the orbit 2
T is proportional to the cube of the semimajor
axis 3
a . The constant of proportionality is independent of the individual planets.
Beginning with Kepler's 2nd law – Eq. (26) which states that equal areas are swept in
equal times,
.
2
=
dA L
dt m
(65)
Multiplying both sides by dt gives,
.
2
=
L
dA dt
m
(66)
Integrating once around the orbit (from 0 to A and from 0 to T) gives an expression
relating the total area of the orbit to the period of the orbit,
.
2
=
L
A T
m
(67)
Squaring both sides and solving for 2
T , gives,

21
2
2 2
2
4 .=
m
T A
L
(68)
The area A of an ellipse is pab , where a is the length of the semi–major axis and b
the length of the semi–minor axis. The expression of 2
T becomes,
2
2 2 2 2
2
4 .= p
m
T a b
L
(69)
Knowing from Figure 2 that b is related to a and c, the focus–center distance, by
2 2 2
= +a b c , so 2 2 2
= -b a c ,
( )
2
2 2 2 2 2
2
4 .= p -
m
T a a c
L
(70)
Since = ec a ,
( )
( )( )
( )
2
2 2 2 2 2 2
2
2
2 2 2 2
2
2
2 4 2
2
4 ,
4 1 ,
4 1 .
= p - e
= p - e
= p - e
m
T a a a
L
m
a a
L
m
a
L
(71)
Since we're dealing here with ellipses (and circles), we can use a property ellipses
geometry that indicates that,
( )2
1 .e = - ek a (72)
This relates the semi–major axis a and the eccentricity e of an ellipse to its focus–
directrix distance p. Factoring out ( )2
1- e from the expression for 2
T and replacing it
with ek gives,
( )( )
2
2 2 3
2
2
2 3
2
4 1 2 ,
4 .
= p - e
= p e
m
T a a
L
m
a k
L
(73)
Now there is an expression showing that 2
T is proportional to 3
a . But the constant of
proportionality appears to be a function of m and L, whose are different for each plan-
et.
From Kepler's 1st law 2 2
e =k L GMm . Using this to substitute into the expression for
2
T gives,
2 2
2 2 3
2 2
4 .= p
m L
T a
L GMm
(74)
The 2
m and 2
L cancel, leaving,
2
2 34
.
p
=T a
GM
(75)

22
The constant of proportionality, 2
4p GM , is the same for all planets since it depends
only on G, the constant of universal gravitation, and M, the mass of the Sun. The
square of the period of a planet is proportional to the cube of the length of the semi-
major axis, and this proportionality is the same for all planets. This is Kepler's 3rd
Law.
Quod erat demonstrandum.
5 Applying Kepler’s Law to our Solar System
The energy and angular momentum equations for a central force law system are:
2 2
2
1
,
2 2
æ ö
= + -ç ÷
è ø
dr L k
E m
dt rmr
(76)
and
2
.
q
=
d
L mr
dt
(77)
Integrating Eq. (76) and Eq. (77) results in r and q as a function of time.
Before doing these somewhat tedious integrations, it is useful to obtain r as a func-
tion of q , which describes the shape of the orbit.
Starting with,
2
,
.
q
=
q
=
q
dr dr d
dt d dt
dr L
d mr
(78)
in the energy equation. Rearranging and integrating this gives,
0 0
0
2
2 2 2
.
2 1 2
q
q
= q = q - q
- +
ò ò
r
r
dr
d
mE mk
r
L r L r
(79)
The r–integration is performed by setting 1=u r , 2
= -du dr r to give,
0
0
2
2 2
22 2 2
4 2 2
2 2
.
2
1
- =
+ -
= -
æ ö æ ö
+ - -ç ÷ ç ÷
è øè ø
ò
ò
u
u
u
u
du
mE mk
u u
L L
du
m k L E mk
u
L mk L
(80)
Use the substitutions,
2
2 2 2
2
1 cos ,- = + a
mk mk L E
u
L L mk
(81)
and,

23
2
2 2
2
1 sin .= - + a a
mk L E
du d
L mk
(82)
The integration now gives 0a = q-q , which leads to the orbit equation,
( )01 cos ,= + e q- q
p
r
(83)
with the semi–latus–rectum 2
=p L mk and eccentricity ( )2 2
1 2e = + L E mk . The
constant of integration is chosen so that 0q = q from pericenter is called the true
anomaly.
Energy Eccentricity Orbital Shape
2
0 2
2
= =
mk
E E
L
0e = Circle
0 0< <E E 0 1< e < Ellipse
0=E 1e = Parabola
0 < E 1< e Hyperbola
Figure 7
Using this concept, the energy and eccentricity can be used to construct orbital
shapes.
r
a
2a r-
2 ae
pericenter
Figure 8
The ellipse in Figure 8 has a major axis of 2a and a minor axis of a. The ratio of the
distance between the foci to the major axis is the eccentricity e of the ellipse.
Using simple trigonometry’s cosine law to the triangle in Figure 8, gives,
( )
2 2 2 2
2 4 4 cos ,- = + + e aa r r e a ar (84)
which gives the polar form of the equation of an ellipse, with the semi–latus–rectum
of ( )2
1= - ep a .
Applying this to a solar system is Kepler’s 1st Law of planetary motion.
The semi–major axis of the ellipse can be expressed in terms of the energy and angu-
lar momentum.

24
22
2
2
= =
L EL
p a
mk mk
(85)
so,
2
=
k
a
E
(86)
and,
.
2
= -
k
E
a
(87)
The semi–major axis depends only on the energy, not the angular momentum. The
energy depends on the semi–major axis, not on the eccentricity.
The time dependence of these variables can now be constructed using Kepler’s 2nd
Law. The rate at which the radius vector sweeps out an area is,
( )21
1 .
2 2
= = -e
dA L ka
dt m m
(88)
The time required to complete one orbit, the period t , is the time to sweep out a
complete area, 14
2 2
1 ,= p = p -eA ab a (89)
enclosed by the ellipse 15
and is given by,
( )
2 2
21 1
1 ,
2
p - e
= - e
t
a ka
m
(90)
which can be simplified to,
3
2
2 .t = p
m
a
k
(91)
14 There is confusion created here, since t is now being used for two purposes, the torque and the orbital time
period. This common in any work that merges two distinct fields of physics together.
15 Here b a= - e2
1 is the semi–major axis of the ellipse.

25
For the family of planets orbiting the sun, with =k m GM , where M is the mass of
the sun. The period of a planet is proportional to the 3
2 power of the semi–major axis
of the planet’s orbit. It does not depend of the mass of the planet, or the eccentricity of
the orbit. This is again Kepler’s 3rd Law. Figure 9 shows the slope of the log–log plot
as 3
2 .
( )log Ea a
( )log Et t
3
2
0
1
-1
-1 0 1 2
Pluto
Neptune
Uranus
Saturn
Jupiter
Mars
Earth
Venus
Mercury
Haley's Comet
Figure 9
Kepler’s Law with some geometry can produce an expression for how a planet moves
in orbit as a function of time. Starting with the equation for radial motion,
Planet
Semi–Major
Axis 10
10 m
Period in
Years
2 3
T a
24 2 3
10-
yr m
Mercury 5.79 0.241 2.99
Venus 10.8 0.615 3.00
Earth 15.0 1.0 2.96
Mars 22.8 1.88 2.98
Jupiter 77.8 11.9 3.01
Saturn 143 29.5 2.98
Uranus 287 84 2.98
Neptune 450 165 2.99
Pluto 590 248 2.99

26
2 2
2
1
2 2
æ ö
+ - =ç ÷
è ø
dr L k
m E
dt rmr
(92)
This can be rearranged in the form,
( )
0 0
0
2
2
2 2
2
= = -
+ -
ò ò
r t
r t
dr
dt t t
m mk L
E
r mr
(93)
If the energy, E, and the angular momentum L are expressed in terms of the semi–
major axis a and eccentricity e , the LHS of Eq. (93) becomes,
( )( )
22 2
1
2
.
-e e - -
ò
r
a
a rdr
k a r a
(94)
In Eq. (94) 0r has been chosen as the pericenter radius ( )1-ea . The time of the pas-
sage of the pericenter is then 0t . The integration of Eq. (94) can be performed by set-
ting,
cos- = -e yr a a (95)
and,
sin= e + y ydr a r d (96)
where y is a new variable call the eccentric anomaly, whose geometric meaning is
shown in Figure 10.
a
r
a
ellipse
circle
y
( )
cos cosa a r
a r
y e a
e
= +
= -
Figure 10
The integration then becomes,
( ) ( )
0
2 2
1 cos sin
y
- e y y = y - e yò
a a
a d a
k k
(97)
which results in,
1 cos ,= -e y
r
a
(98)
with,
( )0
2
sin
p
y -e y = -
t
t t (99)

27
where t is the period of the motion. Eq. (99) is known as Kepler’s Equation. It is the
relation between the eccentric anomaly y and the time t, or the so–called mean
anomaly ( )( )02p t -t t .
Finally, the relation between the eccentric anomaly y and the true anomaly ( )0q-q
can be found by eliminating r a between the Eq. (98) and the orbit equation to ob-
tain,
( )
2
0
1
1 cos .
1 cos
- e
- e y =
+ e q - q
(100)
6 Appling Kepler to Earth’s Orbit
The earth’s orbit around the Sun lies in the plane of the ecliptic, which is marked by
the apparent path of the Sun through the constellations of the zodiac over the course
of the year. The plane of the earth’s equator makes an angle of approximately 23°
with the plane of the ecliptic. And the intersection of these two planes gives a direc-
tion space.
In September of each year the sun passes through the plane of the earth’s equator go-
ing from north to south. This is known as the autumnal equinox (AE). The direction
from the sun to the earth at this time provides a convenient reference from which to
measure the angle q , so the autumnal equinox 0q = .
As the year progresses the midday sun moves lower (in the northern hemisphere) in
the sky until at 2q = p , the winter solstice (WS) in December, it reaches its lowest
point. The sun then moves higher in the sky and at q = p , the vernal equinox (VE) in
March, the sun passes again through the plane of the earth’s equator, this time moving
from south to north.
The Sun moves higher in the sky and at 3 2q = p , the summer solstice (SS) in June,
the midday sun reaches it highest point.
The equinoxes, solstices, and seasons in Figure 11 are shown in detail in Figure 12.
WS
VE
SS
AE
2q p=
q p=
3 2q p=
0q =
Sun
Earth
Figure 11 16
16 Drawing not to scale!

28
Date (1) Day (2) Days (3) (4) Season
AE 23 Sept 01:19 266.0549
89.8361 0.245961 Autumn
WS 21 Dec 21:23 355.8910
88.9938 0.243654 Winter
VE 20 Mar 21:14 444.8847
92.76639 0.253977 Spring
SS 21 Jun 15:34 537.6486
93.6521 0.256408 Summer
AE 23 Sept 07:13 631.3007
Figure 12
(1) 1 Jan 94 0:00h = 1.0000
(2) Fractions of a year
(3) Season in days
(4) Season in fractions of a year
The observed times of these seasonal events using The Astronomical Almanac can be
used to find the parameters of the earth’s orbit. The eccentricity e , which determines
the shape of the ellipse, the angle 0q of the perihelion that determines the orientation
of the ellipse in the plane of the ecliptic, and the time 0t of the passage of the perihe-
lion.
Eq. (100) gives the relationship between the eccentric anomaly y and the true anom-
aly 0q-q . Expanding the RHS of Eq. (100) in a power series in the eccentricity
gives,
( )
( )
( ) ( )
0
2
0
2 2
0 0
cos cos
sin
cos sin ,
y = q-q +
+ e q-q
-e q-q q-q +
(101)
which gives,
( ) ( ) ( )2
0 0 0
1
sin sin2 ,
4
y = q-q -e q-q + q-q +e (102)
When Eq. (102) is substituted into Kepler’s Eq. (99)) it gives the relationship between
the observed quantities, the true anomaly, and the time in years of the orbit.
( ) ( )
( ) ( )
0 0
2
0 0
2 sin
3
sin 2 2 .
4
q-q - e q-q +
e q-q + = p -t t
(103)
Setting 0q = at the Autumnal Equinox, 2q = p at the Winter Solstice, q = p at the
Vernal Equinox, and 3 2q = p at the Summer Solstice, four equations are given,

29
( ) ( )
( )
( )
( )
2
0 0 0 AE 0
2
0 0 0 WS 0
2
0 0 0 VE 0
2
0 0 0 SS 0
3
2 sin sin 2 2 ,
4
3
2 cos sin 2 2 ,
2 4
3
2 sin sin 2 2 ,
4
3 3
2 cos sin 2 2 .
2 4
-q + e q - q - e q + = p -
p
- q - e q + e q + = p -
p - q - e q - e q + = p -
p
- q + e q + e q + = p -
t t
t t
t t
t t
(104)
These four equations can be combined to give,
( ) ( )
( ) ( )
0 VE AE
0 SS WS
0
0 SS VE AE WS
2
0 WS AE SS VE
2 1 1
sin Autumn+Winter ,
2 2
2 1 1
cos Winter+Spring ,
2 2
2 3
4 ,
2
1 3
sin 2 Autumn+Spring.
2 2
e
q = - - = -
p
e
- q = - - = -
p
q
= + + + + -
p
e
+ q = - + - =
p
t t
t t
t t t t t
t t t t
(105)
The first two expressions in Eq. (105) give the eccentricity and angle of perihelion;
the third expression gives the time of perihelion in 1995, and the fourth expression is
a check on the consistency of the data with the assumption of a Keplerian orbit.
Using the data in Figure 12,
0
0
2
sin 0.010385,
2
cos 0.002369,
e
q =
p
e
- q =
p
(106)
which give 0.016732e = and 0 102.85q = . The third equation gives,
( )04 1604.4792 2 102.85 180 1.5 365.2458,= + ´ - ´t (107)
which gives,
0 368.50 3 January 1995, 12h.= =t (108)
The fourth equation gives,
0.499942 0.499937.» (109)
The agreement with the almanac values 0.01673 0.00002e = ± and
0 102.87 0.08q = ± where the ± is the variation of the course of the year due to var-
ious perturbations, is excellent for such a simple set of calculations and data sources.
However there is a problem. The predicted 0t is 18 hours early. The reason is the cal-
culations of the parameters use the earth–moon barycenter perihelion. The almanacs
use the earth barycenter only. These differ by 1.3sin» f days, where f is the angular
phase of the moon near the perihelion. In 1995 perihelion occurred about a third the
way through the first quarter of the moon, so 30f » . This corrects the results by ap-
proximately 16 hours.

30
7 Winding Numbers and Stable Orbits
The class of central potential problems and the periodic orbits that produce the bound
motions of Kepler’s Laws are actually a rarity. The central potential problem can be
generalized as the sum of two terms that depend on the separation of the bodies,
( ) 2
.= - +
k
U r ar
r
(110)
The Bertrand–Königs theorem states that for bound motion, the only central potential
that yields periodic orbits for all values of the force constants and initial conditions
are the isotropic oscillator and the Kepler problem.
The Bertrand–Königs theorem proves a unique kind of periodicity for the Kepler
problem but does not explain why the solution is singular. McCauley develops the
reason in §4 of Classical Mechanics: Transformations, Flows, Integrable, and Chaot-
ic Dynamics.
The reason is there is an extra globally conserved quantity functionally independent of
the angular momentum and system energy (represented by the Hamiltonian).
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