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This paper derives two of Kepler's Laws of planetary motion. These are that all orbits are conic sections and that the radius vector sweeps out equal areas in equal time increments.ard

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1
K K
EPLER EPLER
''
SS
FF
IRSTIRST
ANDAND
SS
ECONDECOND
LL
AWSAWS
OFOF
PP
LANETARYLANETARY
MM
OTIONOTION
BYBY
R R
ICHARDICHARD
J. PJ. P
ALMACCIOALMACCIO
Introduction
Johannes Kepler, the Einstein of his day,lived in Germany during his entire lifetimefrom December 27, 1571 to November 15,1630. While he scored many major mathe-matical and scientific accomplishments,Kepler is mostly remembered for his threelaws of planetary motion. What is not quiteso widely known is another major contribution involving the development of formulas for determining the volume of solids formed by revolving plane areas aboutaxes. He was thus able to foreshadow thedevelopment of the Calculus. Kepler alsoformulated amazingly accurate astronomicaltables of data and was a great force infostering the belief, new at the time, that thesun is the center of the solar system. Many of Kepler's letters are still inexistence so much is known about his life and philosophy . He was deeply religious and strongly believed that the Creator of the universe used mathematical principles. Thisis very understandable as you shall see when we develop two of Kepler's Laws. Even themost hardboiled person cannot help but see how amazing is the Second Law. The first law states that the orbital path of a body must be a conic section, while thesecond dictates that the radius vector (a line from the sun, or center of mass, to theorbiting body) must sweep out equal areas in equal time intervals. The above two
2illustrations visually convey this second law. At the left, the area bounded between anyconsecutive two radii and the ellipse is equal to the area bounded between any other consecutive pair of radii and the ellipse,
provided the same time period elapses from one position of the orbital body to the next position.
Briefly stated, the second law states thatequal areas are swept out by the radius vector in equal time intervals. Clearly the orbiting body cannot move at a constant speed. The illustration at the right shows that in equaltime intervals, the body must move faster when it is near the sun than when it is furthestfrom the sun in order that the blue shaded areas be equal in size. This is evident since itmoves further in its orbit, in a given time interval, than it moves when it is far from thesun during a time interval of the same duration.
Preliminary Material
It is our goal here to make an understanding of the proofs of Kepler's first two lawsavailable to you. However, you must be at least at the point of the course in which youhave completed the differential calculus and the integral calculus to the coverage of theinverse trigonometric functions. This article will present the needed material on the Polar Coordinate System required for proper understanding. You also need a rudimentaryunderstanding of vectors (provided here too). Therefore if you are reading this too earlyin your Calculus AB career, come back later to more fully enjoy the wonderfulexperience of understanding these two fabulous laws of planetary motion. A lot of work has gone into this article and you deserve a complete understanding of everything!
The Polar Coordinate System
One would think that the
xy
-coordinate system you learned about way back in AlgebraI would be sufficient. Such is definitely NOT the case when describing mathematicalevents taking place in the two-dimensional plane.
The XY-Coordinate System
(
x
,
y
)
x
y
The Polar Coordinate System ( r, θ)
r θ
In the
xy
coordinate system the coordinates are both distances as shown above. ThePolar Coordinate system, like the
xy
system, features two coordinates for every location.In this system, only the first coordinate is a distance. It is the distance
outward
from thesrcin along a ray. The second coordinate is the measure of the angle in standard positionhaving that ray as its terminal ray. Thus a pair of polar coordinates states both a distance,
r
and a direction,
θ
.
3
The Vector Concept
Polar coordinates lend themselves beautifully to the vector idea since a
vector
is anentity which simultaneously embodies distance and direction. A
unit vector
is a vector which is one unit long and can point in any direction. We use arrows to diagram vectors.The length of the arrow indicates its length or
magnitude
while the arrow head indicatesthe direction. The most prevalent application of the vector concept is that of force. Thetwo important things about a given force is its magnitude and direction. We also usevectors to indicate position, velocity, and acceleration. The blue vector in the polar coordinate system on the previous page is a
position vector
since it indicates a position inthe polar coordinate plane. The position is at a distance (magnitude) of
r
from the srcinin the direction of the terminal ray of
θ
.
The XY-Coordinate System
(
x
,
y
)
x
j
y
O
i
The Polar Coordinate System
r
cos θ =
x
P ( r, θ)
r r
sin θ =
y
j
θ O
i
Above we have placed the unit vectors
i
and
j
in both coordinate systems. Thesestandard vectors are one unit long and point along the axes. In the polar coordinate planewe have drawn perpendicular segments from point P to the horizontal and vertical axes.Let
O
be the srcin. Then the vector
OP
can be written in terms of the unit vectors asfollows:
OP
=
i
(
r
cos
θ
) +
j
(
r
sin
θ
)
You will notice that the magnitude of
OP
is
r
since
√
(
r
cos
θ
)
2
+ (
r
sin
θ
)
2
= r
.We refer to
i
(
r
cos
θ
)
and
j
(
r
sin
θ
)
as being the horizontal and vertical
components
, respectively, of
OP
. For this discussion it turns out that the xy-coordinate system is totally inadequate tohandle our analysis of planetary motion in the plane! The rest of this discussion will beexclusively in the Polar Coordinate Plane. We will be “watching” an object movingalong a curve under the influence of gravity. In the Polar System we specify a curve inthe form
r = f
(θ)
. Since you are likely not to be familiar with Polar Coordinates we willlook at a few curves stated in this form. You can graph such polar functions with your TI83 by pressing the MODE key and changing FUNC to POL. Then when you press Y =you will see a list with
\r
1
=
at the top. In this screen the key you normally used to type
x
now types
θ
as this is now the independent variable. As
θ
changes,
r
changes accordingto the function,
f
. The graph shown is the graph of the function
r =
1
+
cos
θ
. The
4graph is called a cardioid due to its heart-shaped appearance. The function is very simple.As
θ
makes a complete rotation from 0 to 2π, the distance of the graph from the srcin,which is the value of
r
, changes. This distancevaries between a maximum of 2 and a minimumof 0. The first point plotted is the point (2,0) sincethe cosine of zero is 1. In the direction of
2
,straight up from the srcin, the computed value of
r
is 1 + 0 = 1 giving the point
1
,
/
2
)
.Straight to the left of the srcin, when
θ
is
π
, wehave the distance from the srcin of 1 + (-1) or 0 providing the point
0
,
)
. This makes thecurve spiral inward to the srcin when
θ
makesthe first half rotation. Symmetry causes a repeatof the curve below the horizontal axis. Why notdraw this graph in the ordinary xy coordinatesystem? Here is why that prospect is, at best,disgusting! If we convert the polar equation
r =
1
+
cos
θ
to the xy system we wouldget,
after simplifying (!)
,
x
4
+
y
4
+
2
x
¿
−
2
x
3
−
2
xy
2
−
y
2
=
0
. Now howwould you like to graph that by point plotting since even the graphing calculator cannotdo this graph in the xy-coordinate system. (We got this equation by taking
r =
1
+
cos
θ
, multiplying both sides by
r
, and then letting
r=
√
x
2
+
y
2
and r
cos
θ=x
.) Compare the two graphs on page 3, superimposingone on the other, to see why these substitutions are made. In order to adequately describe planetary motion which will involve position, velocity,and acceleration, we need to define a second set of unit vectors used in the Polar Coordinate system. Study carefully the graph below. The new unit vectors are
u
r
which
r = f
(
θ)
u
θ
u
r
j
r
P
(
r
,
θ)
θ
i
Figure 1
is a unit vector in the direction of
r
,
and
u
θ
which is a unit vector in the direction of
5increasing
θ
. These two new perpendicular unit vectors move along with point
P
(
r
,
θ)
asit moves along the curve
r = f
(
θ)
. The two right triangles shown as dashed lines arecongruent triangles whose sides contain the horizontal and vertical components of
u
r
and
u
θ
. These two triangles each have
θ
as one of their acute angles. The importantthing about these two triangles is that their perpendicular sides contain the horizontal andvertical components of the new vectors
u
r
and
u
θ
.
We begin the next section using thisfact.
The Derivation of Kepler's Second Law
We are now ready to begin the process of developing Kepler's Second Law – that theradius vector (from the srcin to
P
(
r
,
θ)
) sweeps out equal areas in equal time intervals.In so doing we will actually be doing much of the setup for the derivation of Kepler'sFirst Law – planetary orbits are conic sections. Look back at Figure 1. As a planetmoves along its path
r = f
(
θ)
it is exceedingly convenient to describe the motion relativeto the unit vectors
u
r
,
and
u
θ
instead of only the unit vectors
i
and
j
.
It isimportant to realize that the unit vectors
r
,
and
u
θ
themselves are expressed in termsof
i
and
j
.
In this discussion we will be differentiating vectors. The vectors withwhich we will be working are really
vector functions
since their horizontal and verticalcomponents vary over time.
Vector Functions and Circular Motion
We want you to get a good feel for the process of differentiating vectors so we pausehere and use an example of circular motion which is easy to understand. In addition,much use of the chain rule is necessary in developing Kepler's Second Law. Thus wewill introduce the chain rule for differentiating vectors in this little aside discussion. Point
P
Y
P
(
x, y
)
r
j
θ
O
i
X
Figure 2
in Figure 2 moves counter-clockwise around a circle of radius
r
.
r
is the red vector

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