The essential strength of the science of physics lies in the depth of its conceptual schemes, in the relatively few principles that to unify a broad range of knowledge about the physical universe. One foundation of this knowledge comes from Isaac Newton and those on whom he based his work. 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.

1. Proof of the Kepler Problem
Can Kepler’s Equations be Proved and then Considered a Law of Nature?
Glen B. Alleman
Niwot, Colorado
galleman@niwotridge.com
0 Abstract
The essential strength of the science of physics lies in the
depth of its conceptual schemes, in the relatively few princi-
ples that to unify a broad range of knowledge about the
physical universe.
One foundation of this knowledge comes Isaac Newton and
those on whom he based his work. 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 exam-
ined. The laws of motion were set down in Newton’s Prin-
cipia. Laid out in three books, plus an Introduction. The first
book is the starting point of every proposition on dynamics,
and treats the motion of bodies without resistance under
various force laws. The second book explores motion in a
resisting media. The third book discusses universal gravita-
tion.
Proof of Kepler’s’ Laws of Motion can be found in many
forms in these and other texts. This essay provides a sum-
mary of these proofs in modern notation.
The intention is to show how these proofs were derived as
well as confirm the laws of nature can be discovered using
this approach.
1 Introduction
This essay is about a single fact, although not a small one.
When a planet or other astronomical object travels through
space under the influence of gravity, the path it takes follows
a mathematical curve – a circle, an ellipse, a parabola, or a
hyperbola. These paths belong to a family of curves called
conic sections.
Why does nature choose to have astronomical objects fol-
low such a curve? The answer to this simple question turns
out to have profound scientific and philosophical signifi-
cance.
The motion of material bodies was the subject of the earliest
researches of science. From these efforts there evolved a
filed known as analytical mechanics or dynamics.
1.1 Proof of Newton’s and Kepler’s Laws
There is a centuries long tradition of refining Newton’s me-
chanics into formulations of every grater sophistication and
elegance. This essay does not contribute to that literature,
but does make heavy use of authors in the past and present.
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.
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 mechanical force of machines all have well
known effects. In pre–twentieth century science, 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 bound-
ary between science and pre–science. In early history, ob-
jects 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 nature.
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 rules, which collectively reflect Newton’s
profound faith in the unity of nature. These rules were in-
tended 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 explain 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 ne 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.
1

2. The third rule is the principle of universal qualities or the
belief that those qualities, such as mass or length, that de-
scribe bodies exposed to our immediate experience also de-
scribe 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 conclusions about a class of objects by
examining a few of them, then reasoning from the particular
to the more general. [1]
This rule states that concepts, hy-
pothesis, laws, and theories arrived at by induction should be
assumed as universal both in time and place until new evi-
dence 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 defini-
tion will be used here for a scientific law:
As formulated by humans, natural or scientific laws are rules,
preferably mathematical rules, by which we believe nature oper-
ates, and such laws can be classified as being either empirical,
definitional, or derived laws.
Empirical laws are general statement that identifies a regu-
larity in many observations with offering a theoretical ex-
planation for these observations.
Definitional laws are a second level of physics law. These
laws usually involve the definition of fundamentally impor-
tant 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 Ke-
pler’s third law.
The scientific laws of nature are usually thought of as inexo-
rable and inescapable, in part because of the word law sug-
gests an erroneous analogy with divine law. Scientific laws,
built on concepts, hypothesis, and experiments, are only as
trustworthy as those concepts and as those experiments are
accurate. Since humans formulate scientific 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 the emanated 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 princi-
ple.
Newton had hinted that he could provide such a demonstra-
tion. A demonstration that the forces involved would lead to
elliptical orbits. Johannes Kepler had deduced these ellipti-
cal orbits 70 years earlier.
Halley asked Newton to see the demonstration, but Newton
begged off claiming to have misplaced the calculations. Hal-
ley left disappointed, but a few months later received a 9–
page treatise which showed 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 plane-
tary motion.
Halley knew he was holding the key to understanding of the
universe as it was then conceived. He asked Newton if he
could publish the results. Newton was not yet ready and de-
layed the final publication for three years. The resulting
work was published in 1687 under the title Philosophiae
Naturalis Principia Mathematica. This was Newton’s mas-
terpiece and the foundation of modern science.
In the Principia Newton used a method of polygonal ap-
proximations 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 determining the nature of the force required to
maintain a specific type of orbital motion about a given cen-
ter 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 astronomical body.
2.1 Quick Overview of Kepler’s Law of Planetary
Motion
My goal is to show that the heavenly machine is not a kind of
divine living being but similar to a clockwork in so far as all
the manifold motions are taken care of by one single absolutely
simple magnetic bodily force, as in a clockwork all motion is
taken care of by a simple weight. And indeed I also show how
this physical representation can be represented by calculation
and geometrically. Johannes Kepler
Kepler reported in 1609 that Mars moved in an elliptical
orbit with the sun at one focus of the ellipse, with the radius
vector from the sun to Mars sweeping out equal areas 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 demonstrated the specific
mathematical formulation of the force function required to
produce elliptical orbits. [7] Kepler’s first two laws were pub-
lished in Astronomia Nova (The New Astronomy: Based on
Causes or Celestial Physics) (1609) and the third in Har-
monice 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 pe-
riod of the planet and A is the semimajor axis of its el-
liptical 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 formulation1
Deduction is the process of reasoning from the general to the more
specific.
2

3. of the law of universal gravitation in 1687. Newton's theory
showed the celestial bodies were governed by the same laws
as objects on Earth. The philosophical implications of this
played as key a part in the Enlightenment as did the theory
itself in the subsequent development of physics and astron-
omy.
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 published the
details due to a nasty dispute with the German philosopher
and mathematician Gottfried Leibniz, who had made the
same mathematical discoveries.
Newton's three laws of motion are formally given in Phi-
losophiae Naturalis Principia Mathematica (Mathematical
Principals of Natural Philosophy) as: [2]
Lex I (in editions of 1687 and 1713) – Corpus omne
perseverare in statu suo movendi uniformiter in direc-
tum, 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 direc-
tum, nisi quantenus illud a viribus impressis cogitur
statum suum mutare. (Every body continues in its state
of rest, or of uniform motion 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 uni-
form linear motion continues its uniform motion in a straight
line unless acted on by an unbalanced force. This law is of-
ten called the law of inertia. This means that the state of mo-
tion in a straight line remains at rest of continues its uniform
motion unless acted on by an unbalanced force. The pres-
ence of the unbalanced force is indicated by changes in the
state of motion of a body.
Lex II – Mutationem motis proportionalem esse vi mo-
trici 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 accel-
eration, a, in the direction of the force such that the magni-
tude of the force divided by the magnitude of the accelera-
tion is a constant, m, independent of the applied force. This
constant, m, is identified 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 acceleration. The result of this
measure is the mass of the body. There is an additional in-
terpretation of the second law of motion. If a body is ob-
served 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. [3]
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 for-
mer. This law introduces a symmetry that does not appear in
the first two laws. It states that forces appear in equal and
opposite pairs.
These three laws, along with the other postulates in Principia
were extensions of previous work including the laws derived
by Galileo Galilei. Galileo discovered the empirical basis for
the law of inertia through systematic experiments. These
experiments led Galileo to assert that all bodies should ac-
celerate at the same constant rate near the earth’s surface. [4]
3 A Simple Proof of Kepler’s Laws
The first step is discovering a universal, mathematically pre-
cise description of forces is to start with the inverse question.
Given that a body’s orbit is elliptical, circular, parabolic, or hy-
perbolic with a motionless force–center, what is the force law
that produces the orbit?
This was the question Newton answered in the Principia. In
modern notation, if the angular momentum is defines as
and the torque is defined as N = , where r is
the position of a body, then,
= ×L r p ×r F
,
d
dt
=
L
N (1)
follows directly from Newton’s second law, since
md dt =v 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 con-
served in the absence of a net torque on the body.
3
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 accel-
eration given to a mass.
4
The third law was original with Newton, and is the only physical
law of the three. Taken from the second, it describes 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 describes the motion of a body in
equilibrium while the second describes its lotion when the forces
acting upon it do not balance one another.
2
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 observa-
tions. The first law, known as the law of inertia, is a special case of
the second law.
3

4. A central force law yields angular momentum conservation.
If , 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.
( )f r=F a
p
b
p
+ ε1
p
− ε1
r
v
ˆ
)
In this case is constant at all times and therefore
defines a fixed direction 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.
m= ×L r
L
Figure 1
3.1 Are Newton’s Inverse Solutions Unique?
In order to proceed with a proof of Kepler’s Law as they
were developed by Newton, the results of Eq. (4) must 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.
Planar orbits agree with the observations of planetary mo-
tions, so Newton was able to restrict his consideration to
central forces.
If Newton’s law is written in polar coordinates , in the
plane of motion, which is perpendicular to the z–axis, then
Newton’s second law of motion
( ,θr
d dt =p F has the form,
( )
2
2
,
0,z
d r
m mr f r
dt
dL
dt
θ− =
= =
(2)
Using , if 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 where
the force constant k is independent of direction and is the
same in all possible directions, since any central force is
spherically symmetric.
k= −F r 0k >
[5]
If , 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.
0k >
where 2
zL mr d dθ= t is the magnitude of the angular mo-
mentum and is constant.
Eq. (2) can be rewritten in the form,
( )22
2 2 3
2
,
constant.
z
z
f rLd r
mdt m r
L mr θ
− =
= =
(3)
The determination of the force laws that yield elliptical or-
bits 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 theorem and its application to the solution of
the central force law equation.
By assuming the orbit is a conic section and differentiating
Eq. (3) ,
(
1
1 cosC
r
ε θ= + ), (4)
The conclusion so far is that Newton’s solutions to Kepler’s
equations are unique. The only missing element is the mo-
tion of the Sun itself, which will be taken into account later.
4 A More Elaborate Proof of Kepler’s Laws
The first step in discovering universal, mathematically pre-
cise 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 – the
sun in this case – at one focus, what is the force law that
produces the orbit?
where C and ε are constants andε is the eccentricity of the
conic section. Eq. (4) requires that must vary 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 1.
( )f r
4.1 TorqueThe origin of the coordinates lies at one focus giving,
( )
2
2
.zCL
f r
mr
= − (5)
Essential to proving Kepler's second law (and further laws)
is the concept of torque. A torque is a tendency to change
something's state of rotation; it is the rotational analogue of
force. For instance, if I apply torque to a wheel, I'm provid-
ing a tendency to rotate that wheel. Torque is in rotational
mechanics what force is in linear mechanics.
If then the conic section is an ellipse, which agrees
with Kepler’s First Law. If , then the orbit is hyper-
bolic. For the special case of and ε = circular and
parabolic orbits orrcur.
1ε <
1ε >
ε = 0 1
)
Newton actually derived the solutions to Eq. (4) geometri-
cally by asking for the shapes of the curves that define in
intersections of a plane with a cone.
5
An anisotropic harmonic oscillator would be represented by a force
law , where at least one of the three force
constants, differs from the other two, representing the absence
of spherical symmetry.
( , ,F k x k y k z= − − −1 2 3
ik
4

5. Torque, τ can be employed as,
(6),
r
τ = ×r F
If the time derivative of something is zero, that means that
thing does not change as time passes; in other words, it re-
mains constant. This is usually only applied to scalars, how-
ever. In vectors, if the time derivative of a vector is the zero
vector, then that vector does not change magnitude or direc-
tion. In other words, the angular momentum vector of a
planet is a constant vector:
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. [6]
Note that torque is a vector quantity with direction
and quantity. The torque vector r indicates in which direc-
tion the body tends to rotate.
(10)constant.=L
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.
But while torque is usually applied to rigid bodies, such as
wheels and levers, it can also be applied in celestial mechan-
ics. 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 body then becomes the lever arm, al-
though it is by no means a solid one.
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 τ :We shall apply this notion of torque to a planet orbiting the
Sun. Here, however, the impressed force will be gravity. Our
fixed reference point will be the Sun itself. We know that
andr= ⋅r ( 2
GMm r= − ⋅F ) r so we can get,
.
dL
dt
= ×r F (11)
( )
( )
2
,
,
,
0.
GMm
r
r
GMm
r
τ = ×
 
= ⋅ × − ⋅ 

 
= − × 
 
=
r F
r r
r r

(7)
We can use Newton's law of motion, F , and substitute
this into Eq. (11) to give:
m= a
( ).
d
m
dt
= ×
L
r a (12)
The acceleration of a body is equal to its instantaneous rate
of change of velocity; that is,
.
d
dt
=
v
a (13)
We know that any vector crossed with itself is the zero vec-
tor, 0, so the Sun never impresses a torque on a planet. This
makes perfect sense: if you can only pull radially on bucket
(as the Sun can only pull radially on a planet), you won't be
giving the bucket rotating about an axis a tendency to speed
up or slow down in its rotation.
Making this substitution (and also exploiting the fact that the
cross product is associative with respect to scalar factors),
we find that,
4.2 Conservation of angular momentum
Torque τ is defined as the instantaneous time rate of change
of angular momentum L:
d
dt
τ ≡
L
(8)
( )
,
.
d d
m
dt dt
m
 
= ×

= ×
L v
r
r v


)
(14)
If we solve this differential equation, we find that,
(15)( .m= ×L r v
The magnitude of the angular momentum is,
Angular momentum is a quantity that plays the same part in
rotational mechanics as linear momentum does in linear me-
chanics. ( )
,
,
.
L
m
m
=
= ×
= ×
L
r v
r v
(16)
From the previous section τ = , which says that the Sun
never applies no torque to a planet. Therefore
0
d dtL must
also be the zero vector:
0.
d
dt
=
L
(9)
This relates the angular momentum of a planet to its mass,
position, and velocity.
4.3 Kepler's Second Law
We now proceed to directly address Kepler's second law, the
one which states that a ray from the Sun to a planet sweeps
out equal areas in equal times. This ray is simply the vector r
that we've been using. (And we shall continue to use it; r,
remember, is defined as the vector from the Sun to the
planet.)
6
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 , will not
change sign under inversion. Such vectors are called axial vectors or
pseudovectors. The scalar product of a polar vector and a pseu-
dovector is a pseudovector; it changes sing under inversion, where a
scalar vector does not.
= ×L r p
What we're looking for is the area that this vector sweeps
out. Imagine the planet at some time t = 0, and then imagine
5

6. 4.4 Polar Basis Vectorsat a short time afterward t . In that time, the vector has
moved by a short displacement,
t= ∆
(17)0.t t t=∆ =∆ = −r r r
Kepler's first law concerns itself with the shape of the orbit
that a planet makes around the Sun. This law can be devel-
oped easily using polar basis vectors.
The three vectors r , , and form a triangle. The
area of this triangle closely approximates the area swept out
by the vector r during that short time .
0t= ∆r t t=∆r
t∆
The polar coordinate system is an effective way of represent-
ing the positions of bodies with the angle they make with the
origin, and the distance they are away from it. Polar coordi-
nates are useful for dealing with motion around a central
point – just the case we have with planets moving around the
Sun.
We can write this small area represented by this triangle,
, as one–half of the parallelogram defined by the vectors
r and , or,
A∆
∆r
1
2
.A∆ = ×∆r r (18) However, to continue with our use of vectors, we must de-
fine a few polar basis vectors. The polar coordinates r and θ
are related to rectangular coordinates by the equations,
We'll divide both sides of this equation by , the short time
involved. Because of this, and the associative properties of
the cross product, we find:
t∆
( )
1
2
1
2
1
2
1
,
,
.
A
r
t t
t
∆  
= × 
∆ ∆ 
×∆
=
∆
∆
= ×
∆
r
r r
r
r
t
∆
(19)
(24)
cos ,
sin .
x r
y r
θ
θ
=
=
For any plane curve, the position vector r i shown in
Figure 2 is given by,
x y= + j
)
( ) ( )
( ) ( )(
cos sin ,
cos sin .
r r
r
θ θ
θ θ
= +
= +
r i j
i j
(25)
where r = r .
As we choose smaller and smaller values of ∆ , we get bet-
ter and better approximations of the area swept out by the
ray. If we let taking the limit of both sides of Eq.
(19), the approximation approaches the real value and we
find that,
t
0t∆ →
Note that this vector is a function of θ ; in other words, the
unit vector representing the direction in which the body is
located from the Sun is naturally dependent on the angle.
This is a fortunate definition; according to it,
1
2
1
2
, or
= .
dA d
dt dt
= ×
×
r
r
r v
(20)
(26)r= ⋅r r
is a relation that we were already using! Therefore we need
make no change of notation. Our definition of r as a polar
basis vector merely quantifies our work in the plane of the
orbit.
Knowing
,L m= ×r v (21)
and dividing both sides by m, we find
.
L
m
= ×r v (22)
Since we have two Cartesian basis vectors, i and j, we
should also have two polar basis vectors. The second basis
vector, which we shall call the unit transverse vector and
represent with θ , is defined as the rate of change of r with
respect to θ :
ˆ
We can substitute this into our expression for dA dt and find
that,
.
2
dA L
dt m
= (23)
ˆθ ,
sin cos .
d
dθ
θ θ
≡
= − +
r
i j
(27)
This definition means that always points orthogonal to the
unit radial vector. This makes it easy to talk about the com-
ponent of a vector along r, the radial direction, and the com-
ponent along , the transverse direction.
ˆθ
ˆθ
That is, the instantaneous time rate of change of area is the
magnitude of the angular momentum divided by twice the
mass of the planet. But we know that the mass of the planet
is constant, and we also know from our work earlier that the
angular momentum vector is constant (and thus its magni-
tude certainly is). Therefore, the time derivative of area
swept out by this ray is constant. In other words, no matter
where on the orbit the planet is, its ray still sweeps out the
same amount of area. This is Kepler's second law.
Note that if we again take the derivative of with respect to
we find that,
ˆθ
θ
(
ˆθ
cos sin ,
cos sin ,
.
d
d
θ θ
θ
θ θ
= − −
= − +
= −
i
)
j
i j
r
(28)
6

7. To find a similar expression for the angular momentum vec-
tor L in polar coordinates, we go back to the expression we
found for angular momentum:
θ∆
( )t t+ ∆r
( )tr
θ∆r
ˆθ ˆr
ˆθ
ˆr
cosx r θ=
siny r θ=
θ
(33)( .m= ×L r )v
We can substitute r for r and the expression we just found
for v, and get:
⋅r
( ) .
dr d
m r r
dt d
ω
θ
 
= ⋅ × +
  
r
L r

 (34)
We expand this expression to obtain,
( )
( ) ( ) ( )
( ) ( )2
,
ˆ ,
ˆ
dr d
m r r
dt d
dr
m r m r r
dt
dr
mr mr
dt
ω
θ
ω
ω
 
= ⋅ × + 
 
 
= ⋅ × + ⋅ × 
 
= × + ×
r
L r r
r r r
r r r
θ
θ
(35)
Figure 2 Since, again, a vector crossed with itself is the zero vector,
the first term evaluates to zero and we find that,This express will be used later in the solution of the Kepler
problem.
(36)(2 ˆθ .L mr ω= ×r )
k
k
Let us, for the sake of an example, see what our velocity
vector v and our angular momentum vector L would look
like in terms of this new polar system. (We shall require
them later in the proof anyway.)
Since , our final expression for the angular momen-
tum vector is,
ˆ×θ =r
(37)2
mr ω=L
Velocity is the instantaneous rate of change of the position
of the planet:
( )
,
,
.
dr
v
dt
d
r
dt
dr d
r
dt dt
=
 
= ⋅ 
 
= +
r
r
r
(29)
As we took the magnitude of this vector before, we shall do
it again:
2
2
2
,
,
,
,
L
mr
mr
mr
ω
ω
ω
=
=
=
=
L
k
k
(38)
as the magnitude of a unit vector is, by definition, unity.
But this looks like something we've already dealt with! You
may be tempted immediately to substitute θ in forˆ d , but
remember the definition:
dtr
ˆθ d dθ= r , something considera-
bly different. We use the chain rule to expand d dtr into a
form which includes d dθr :
4.5 Kepler's First Law
Now that we have polar basis vectors (and the polar repre-
sentations of velocity and angular momentum), we are ready
to proceed with the proof of Kepler's first law -- that the or-
bits of planets are ellipses with the Sun at one focus.
.
dr d d
r
dt dt
θ
θ
= +
r
v r (30)
To begin with, we will start off by applying Newton's law of
motion and Newton's law of universal gravitation together to
find that,
Since d dθr is , the unit transverse vector, and the angu-
lar speed, ω , is defined as,
ω
,
d
dt
θ
ω = (31)
2
,
GMm
m
r

= −
 
a

r (39)
and, dividing both sides of the equation by m,
2
.
GM
r
 
= − 
 
a r (40)we can obtain our final expression for velocity in polar co-
ordinates as
.
dr d
r
dt d
ω
θ
= +
r
v r (32) Recalling our work with polar basis vectors, we know
ˆθd dθ = −r . Solving for r and applying the chain rule, we
find that,
7

8. ˆθ
,
ˆ1 θ
.
dt d
d dt
d
dt
θ
ω
= −
= −
r
(41)
Substituting this into our equation for a we find,
2
2
ˆ1 θ
,
ˆθ
.
GM d
dtr
GM d
dtr
ω
ω
  
= − −  
  
=
a
(42)
If we multiply the right side of the equation by m/m (which
is unity), we obtain,
2
ˆθ
.
GMm d
dtmr ω
=a (43)
But (I told you this would come in handy as well),
so we can rewrite this as,
2
L mr σ=
ˆθ
.
GMm d
L dt
=a (44)
We can multiply both sides by L GMm and find
ˆθ
.
L d
GMm dt
=a (45)
But we know that d dt=a v , and can substitute accordingly:
ˆθ
.
L d d
GMm dt dt
=
v
(46)
This is a differential equation that we can now solve. Upon
solving it, we find that,
ˆθ ,
L
GMm
= +v C (47)
where C is some constant vector. We'll solve this for v and
find that,
ˆθ .
GMm
C
L
=v +
j
(48)
This is a general solution to the differential equation.
But we're not finished. This doesn't tell us much about the
shape of a planet's orbit, although all the pieces are there.
This is the general solution, and it could be an orbit of any of
the possible shapes (though we can't be sure what they are
yet) or any of the possible orientations. We're interested in
knowing the shape, of course, so we want to restrict the pos-
sible orientations.
To do that, we'll take a special case. It makes sense to have
perihelion -- that is, closest approach to the Sun — at time t
= 0. We'll restrict the orientation so that, when perihelion
occurs, the planet lies along the zero radian line from the
Sun (or, in Cartesian terminology, along the positive x-axis)
— that is, θ = . At this point, r, the position vector of the
planet, will have only a component in the positive x-axis.
We'll also assume that the planet orbits the Sun counter-
clockwise, through increasing measures of angles. If this is
the case, then the velocity v at the instant of perihelion
should be orthogonal to the position vector r, and it should
have only a component in the positive y–axis.
0
According to our expression for v, we have a scalar times
the vector quantity + C.ˆθ
(49)0
ˆθ ,θ = = j
that is, the unit transverse vector points “up” when the unit
radial vector points “right.” Since, at t = 0, θ points entirely
in the y–direction, then our constant 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,
we can rewrite C as a scalar times the unit basis vector in the
y–direction:
ˆ
(50),ε=C
where ε is some scalar constant. (it will be clear later reason
for choosing the letter e in this case.) Substituting this into
our equation for v, we get,
ˆθ .
GMm
L
ε= +v j (51)
This is the specific case when we want the orbit oriented so
that perihelion occurs at t 0.θ= =
Now we are ready to finish up the problem. We can dot–
product both sides of the equation with and get:ˆθ
( )
( )( )
ˆ ˆ ˆθ θ θ,
ˆ ˆ ˆθ θ θ .
GMm
L
GMm
L
ε
ε
⋅ = + ⋅
= ⋅ + ⋅
v j
j
(52)
A vector dotted with itself yields the square of that vector's
magnitude, so . Simplifying , we findˆ ˆθ θ = 1⋅ ˆθ⋅v
( )
( ) ( )
ˆ ˆ ˆθ θ θ ,
ˆ ˆθ θ .
dr dt r
dr dt r
σ
ω
⋅ = + ⋅
= +
v r
ri ˆθ⋅
0
θ j
)
(53)
But the dot product of two orthogonal vectors is zero, so
. We also already know that θ θ . Therefore,ˆθ⋅ =r ˆ ˆ = 1⋅
(54)ˆθ .rω⋅ =v
The last part of our problem is finding an expression for
. We know that (by definition), so,Φji ˆθ sin cosθ= − +i
(ˆθ sin cos ,
cos .
θ θ
θ
⋅ = ⋅ − +
=
j j i j
(55)
4.6 Putting It All Together
Three pieces of the solution are now available. They can be
assembled starting with,,
(1 cos
GMm
r
L
ω ε= + ).θ (56)
Since is on the LHS of Eq. (56) and knowing thatrω
8

9. 2
1mr ω = , both sides of the equation can be multiplied by
mr to give,
(
2
2
1 cos
GMm
mr r
L
ω ε= + ).θ (57) A
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 is proportional to the cube of the semimajor axis
. The constant of proportionality is independent of the
individual planets; in other words, each planet has the same
constant of proportionality.
2
T
3
Replacing the LHS of Eq. (57) by l and moving the con-
stants to the left side of the equation, gives,
(
2
1 1 co
GMm
r
L
ε θ= + )s . (58)
Starting with the expression derived for the rate of change of
the area that the Sun–planet ray is sweeping out (Kepler's
2nd law),
1
2
dA
dt m
= . (62)There is now an explicit function in terms of r and θ which
is the polar equation for a planet's orbit.
Solving for r gives, Multiplying both sides by dt gives,
2
2
.
1 cos
L
GMmr
ε θ
=
+
(59)
1
.
2
dA dt
m
= (63)
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,
The equation of a conic section with focus–directrix distance
p and eccentricity ε is represented by the polar equation,
.
1 cos
p
r
ε
ε θ
=
+
(60)
1
.
2
A
m
= T (64)
This is the same as Eq. (59), given that, Squaring both sides and solving for T , gives,2
2
2
.
L
ep
GMm
= (61)
2
2
2
4
m
T
L
= 2
.A (65)
The focus–directrix distance should be a constant, which p
is: L, G, m, and M are all individually constant; therefore the
expression 2
L GMm ust also be constant. Therefore, New-
ton's laws of motion and universal gravitation dictate that the
orbits of planets follow conic sections. This is Kepler's first
law.
The area A of an ellipse is π , where a is the length of the
semi–major axis and b the length of the semi–minor axis.
Thus our expression of T becomes,
ab
22
m
2
2 2 2
2
4
m
T
L
π= 2
.a b
2 2
(66)
We know that b is related to a and c, the focus–center dis-
tance, by , so b a ,2 2
a b c= + 2 2
c= −
Kepler's first 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? (
2
2 2 2 2 2
2
4
m
T a a
L
π= )c− . (67)
Other are found in the solar system – but the object that fol-
low them are not planets. When Kepler said planet, he meant
a body that repeatedly returns to our skies. The curve repre-
senting the orbit is closed — it must repeatedly retrace itself.
Since ,c aε=
( )
( )( )
( )
2
2 2 2 2 2 2
2
2
2 2 2 2
2
2
2 4 2
2
4 ,
4 1
4 1 .
m
T a a a
L
m
a a
L
m
a
L
π ε
π
π ε
= −
=
= −
,ε−
).ε
)
(68)
The only two conic sections that are closed are the circle and
the ellipse with the circle 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 sufficient 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.
Since we're dealing here with ellipses (and circles), we can
use a property ellipses geometry which indicates that,
(69)( 2
1p aε = −
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.
This relates the semimajor axis a and the eccentricity ε of
an ellipse to its focus–directrix distance p. If we factor out a
from our expression of T and replace it with ε
we find that,
( 2
1 ε− 2
p4.7 Kepler's third law
After generating Kepler's 1st and 2nd laws Kepler's 3rd is
straightforward.
9

10. ( )( )
2
2 2 3
2
2
2 3
2
4 1
4 .
m
T a a
L
m
a p
L
π ε
π ε
= −
=
2 ,
(70)
Here we have an expression that indicates that T is propor-
tional to ; but the constant of proportionality doesn't look
like it is the same quantity for all planets – after all, it seems
to be a function of m and L, which are certainly different for
each planet.
2
3
a
Thus, we'll continue on with our proof. We know from Ke-
pler's first law that 2
p L GMmε = 2
. We can use this infor-
mation to substitute into our expression for T to find that,2
2 2
2 2
2 2
4
m L
T
L GMm
π= 3
.a (71)
Date
Day
1 Jan 94
0:00h =
1.0000 In Days
Fractions
of a year 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
The m and cancel, and we are left with,2 2
L
Figure 32
2 4
.T
GM
π
= 3
a (72) 6 Bibliography
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Glen Alleman is a member of Niwot Ridge Consulting specializ-
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