A Craps Tutorial

Solving The Kepler Problem

Deriving Kepler's Laws of Planetary Motion From Newton's Laws

Book and TV Program : The Mechanical Universe
Many of the ideas in this article were taken from the TV series called "Mechanical Universe"
( hosted by Professor David Goodstein ). There are several useful textbooks that accompany the series. For example, "The Mechanical Universe Mechanics and Heat Advanced Edition"
© 1986 Cambridge University Press See especially chapters 16 and 17.

Section Links For This Page

	Some Definitions
	Torque and Angular Momentum
	Central Force Fields
	The r and θ Components of Any Vector
	Derivatives of r-hat and theta-hat
	The Velocity Vector
	The Acceleration Vector
	Newton's Laws
	Kepler's Laws
	General Ellipse Properties
		The Ellipse In An X-Y Coordinate System
		The Major and Minor Axes
		Two Ellipse Properties
		Ellipse Equation in an X-Y Coordinate System
		The Ellipse As A Conic Section
		Right-side Focus As Origin of Polar Coordinates
		Left-side Focus As Origin of Polar Coordinates
		Ellipse Eccentricity
		The Circle As An Ellipse
		The Latera Recta
		The Area of an Ellipse
	Deriving Kepler's 1st Law
	Deriving Kepler's 2nd Law
	Deriving Kepler's 3rd Law
	Deriving The Vis-Viva Equation
	The Three Anomalies
	Uniform Circular Motion
	Gravitational Potential Energy
	Escape Velocity
	Apogee-Perigee Example
	Oblique Launch
	Links To Other Web Sites

In this first draft, the notation is sometimes inconsistant.
For example, the symbols L and are used interchangeably.
Depending on context, the symbol "L" could represent either
a vector named L or the length of the vector L.
Likewise, the symbol "r" could represent either the
radius vector or the length of the radius vector.

To improve legibility, we don't always end a sentence with a period.

Definitions

	Symbol	Meaning
	t	time ( a scalar quantity )
	D_t	derivative with respect to t [ where t stands for time ] ( e.g. D_tL = dL/dt = instantaneous rate of change of L )
	m	mass ( a scalar quantity )
	v	velocity ( a vector quantity )
	\| a \|	the length of vector a
	a X b	the cross product of vectors a and b
	F	Force ( a vector quantity )
	r	radius vector ( its tail is at the origin of our coordinate system )
	τ	torque ( vector cross product defined by τ = r X F )
	θ	angle from positive X-axis to radius vector ( see Figure 1.0 )
	ω	angular speed ( derivative of θ with respect to time )
	≈	is approximately equal to ( e.g. 4.99999999 ≈ 5 )
	<<	is much less than ( e.g. 2 << 100,000 )
	L	angular momentum ( L = radius vector cross linear momentum ) = r X ( mv )

		implies ( e.g. x = 6 x+1 = 7 )

	Sometimes we put an arrow over a letter to indicate a vector
		For example, indicates that r is a vector quantity

	A "'hat" ( ^ ) over a letter indicates a unit vector
		For example, means that the vector x has length one

	A "'dot" ( . ) over a letter indicates derivative with respect to time
		For example, means derivative of θ with respect to time [ By definition, ω also equals the derivative of θ with respect to time. ]

Torque and Angular Momentum

We will use an X-Y Cartesian coordinate system.
A vector r from the origin to a point in space is called the radius vector.
A Force F is a vector whose twisting effect is called torque.
Torque is symbolized by the Greek letter tau ( τ ) and is defined to be the
vector cross product r X F.
If we cross the radius vector into the linear momentum we get angular momentum.
The angular momentum L is defined to be r X ( mv ).

If you recall that

1 ) v = dr/dt by definition of velocity

2 ) F = d/dt ( mv ) Newton's 2nd law of motion

3 ) For any vector A, the cross product A X A is zero

4 ) D_t ( A*B ) = [ ( dA/dt ) * B ] + [ A * ( dB/dt ) ]

then you can easily see that torque is the rate of change of angular momentum:

dL/dt	=	D_t [ r X (mv) ]
	=	[ dr/dt X mv ] + r X [ d/dt (mv) ]
	=	[ v X mv ] + [ r X F ]
	=	m [ v X v ] + τ
	=	0 + τ

In any force field τ = dL/dt

Central Force Fields

In a central force field the force vector has the same line of action as
the radius vector, and the force magnitude is a scalar function
( call it k ) of the length of the radius vector.

F	=	k( r )

τ	=	X F
	=	X ( k( r ) )
	=	( r ) X k( r )
	=	r k( r ) ( X )
	=	0

But τ always equals dL/dt.
So in a central field dL/dt must be zero, and thus
L must be a constant.

L = length of = | r X ( mv ) | = m | r X v | = a constant vector

In a central force field
the angular momentum L
is a constant vector

r X v = L / m = a constant vector

We will use this result later ( when deriving Kepler's 2nd Law ).

Since each of and is perpendicular to the ( fixed ) direction of
, motion in a central force field will be confined to a single plane
which we take to be the X-Y plane.

The r and θ Components of Any Vector

Whenever motion is confined to a single plane we can use
polar coordinates in addition to our X-Y coordinates.
Figure 1.0 shows the r-θ components of a radius vector.
The unit vector is defined by theta-hat is the derivative of r-hat with respect to theta See Figure 2.0.

r and theta components of vector in XY plane

If the horizontal component of r has length x and the vertical component
has length y then we have the relationships shown in Figures 2.0 and 3.0.

r-hat and theta-hat equations in the XY plane

Since r-hat dot theta-hat is zero, those two vectors must be perpendicular to each other, as shown in Figure 1.0. The sense of theta-hat ( i.e. which direction it points ) can be determined by the fact that the cross product of r-hat and theta-hat is k-hat, as shown in Figure 3.0.

We will keep a list of some of the key facts we derive that will be needed later.

r-hat and theta-hat equations in the XY plane

Derivatives of and

Referring to Figure 2.0 above , we will now find ( see figure 5.0 below )
some derivatives of the r-hat and θ-hat unit vectors:

derivatives of r-hat and theta-hat

List of 3 formulas we will use later

The Velocity Vector

Next we will find an expression for the velocity vector.

r-hat and theta-hat equations in the XY plane

The Acceleration Vector

For our purposes we won't need an expression for the acceleration vector.
But just for curiosity you could differentiate the velocity vector to verify that
the acceleration is given by :
formula for acceleration vector

To find the above result you can first extend the product rule
for differentiation (ab)' = ab' + a'b
to the less familiar rule for differentiating a triple product:
(abc)' = a(bc)' + a'(bc) = a( bc' + b'c ) + a'(bc) = abc' + ab'c + a'bc

Thus, you could use this rule: (abc)' = a'bc + ab'c + abc'
together with previously computed derivatives.

Recall the distributive law for vectors a, b, and c:
a X ( b + c ) = ( a X b ) + ( a X c )
Using this, we now compute ( see Figure 8.0 below )
the value of the cross product r X v.
( Here is where we use the top formula in Figure 6.0 : r-hat cross theta-hat equals k-hat )

value of r cross v

But earlier we showed that in a central force field the magnitude
of rXv is L / m. Therefore ....
( see Figure 9.0. )

formula for L over m in a central force field

Revised list of 3 formulas we will use later

Newton's Laws

Newton laws: F = GMm/r**2 and F= derivative wrt time of mv

Kepler's Laws

Kepler's 3 laws of planetary motion

The Ellipse In An X-Y Coordinate System

We can regard an ellipse as the path traced out by a point that moves
in the X-Y plane in such a way that the sum of its distances from two
fixed points ( the foci ) remains constant. See Figure 12.0 where the
foci F₁ and F₂ lie on the X-axis, and point A is one vertex of the ellipse.
The distance from the center of the ellipse to a focus is denoted by c ,
and a is the distance from the center to point A.

the foci and distances whose sum is constant

The Major and Minor Axes

In Figure 13.0, the major axis of the ellipse is the line segment
from point A' to point A, and the minor axis is the line segment
between points B' and B.

	The length of the major axis is 2a.	( a = length of the semi-major axis )
	The length of the minor axis is 2b.	( b = length of the semi-minor axis )

ellipse in XY system with major and minor axes

Two Ellipse Properties :
Distances To Foci Sum is 2a and a² = b² + c²

In Figure 12.0, the distance from A to F₁ is a+c
and the distance from A to F₂ is a-c

Since point A is on the ellipse, the constant d₁+d₂
must equal (a+c) + (a-c) = 2a.

For any point on an ellipse, the sum of the
distances from that point to each focus is 2a.

In Figure 14.0 the point ( 0,b ) lies on the ellipse;
and so, w + w must equal 2a.
Thus w = a and ( see Figure 15.0 ) we see that a, b, and c
satisfy a pythagorean relationship.

	a² = b² + c²

ellipse with points (a,0) and (0,b) ; b = semi-minor axis

Ellipse Equation In An X-Y Coordinate System

Suppose that an ellipse with semi-major axis a and
semi-minor axis b has its center at the origin of an X-Y
coordinate system. If the foci are at ( -c,0 ) and ( c,0 ) then

	the standard ellipse equation is	x² / a² + y² / b² = 1
	the right-side directrix equation is	x = a² / c
	the left-side directrix equation is	x = - ( a² / c )

If you want to see a derivation of these results, then click here

The Ellipse As A Conic Section

To develop useful formulas for the ellipse we use two coordinate
systems -- the usual X-Y rectangular system and also a polar ( r-θ )
coordinate system. We could place the origin of the polar system at
either focus. Let's first derive formulas for the case where the r-θ
origin is placed at the right-side focus as shown in Figure 16.0.

Right-side Focus As Origin of Polar Coordinates

ellipse with directrix

The point P will move along the ellipse provided that the
ratio r / PA remains constant. The constant ratio is
denoted by ε and is called the eccentricity of the ellipse.

Let ε = r / PA = the eccentricity of the ellipse.
F is a focus and d is the distance from F to the closer directrix.

d = r cos θ + PA

Since PA = r / ε,

d = distance from focus to directrix

= PA + r cos θ

= r / ε + r cos θ

= r ( 1 / ε + cos θ )

εd = εr ( 1 / ε + cos θ )

= r ( 1 + ε cos θ )

Therefore

r = ε d / ( 1 + ε cos θ )

We can rewrite this in a more useful form by replacing εd
with the value shown in the 3rd formula below Figure 17.0 :

r =

a ( 1 - ε² )

1 + ε cos θ

It's convenient to use special symbols for the radius vector
when θ is either zero or π radians.

	The value of r at θ = 0 is denoted by r₀ and
	the value of r at θ = π is denoted by r₁ [ See Figure 17.0 ]

ellipse showing r-sub-zero and r-sub-1

Assuming that

a	=	the semi-major axis
b	=	the semi-minor axis
c	=	distance from the center of the ellipse to a focus

we have the following "right-side focus" formulas :

	r₀ = ε d / ( 1 + ε )

	r₁ = ε d / ( 1 - ε )

	ε d = a ( 1 - ε² )

	r₀ = a ( 1 - ε )

	r₁ = a ( 1 + ε )

	r₁ = a + c

	ε = c / a

	r₀ = a - c

	b = a ( 1 - ε² )^1/2

	b = ( r₀r₁ )^1/2

If you want to see how the above ten formulas were derived,
then click here

Left-side Focus As Origin of Polar Coordinates

An ellipse has two foci and two directrices.
If we had chosen the other focus ( see Figure 18.0 ) to be the origin of
our r-θ coordinate system then the equation for our ellipse would be
given by r = ε d / ( 1 - ε cos θ )

ellipse with other directrix

If you would like to see the formulas that apply to an ellipse where
the origin of the polar coordinates is placed at the left-side focus
instead of the right-side focus then click here

Ellipse Eccentricity


	In our discussion of ellipses we define
		a = length of the semi-major axis
		b = length of the semi-minor axis = ( a² - c² )^1/2
		c = distance from the Y-axis to either focus
		ε = eccentricity = c / a
	and we always require that
		a > 0
		0 <= c <= a

	Usually we restrict ε to the range 0 <= ε < 1

	If ε = 1 then our "ellipse" degenerates to a straight line.

	When we derived the standard ellipse equation
		x² / a² + y² / b² = 1
	we could not allow ε = 1 because that would cause
		b = a ( 1 - ε² )^1/2 = 0

Let's summarize what happens at the boundary endpoints.

If	ε = 1	then
	b = 0 ( the "ellipse" is totally flat )
	c = a ( foci are as far from the Y-axis as possible )
	The "ellipse" is a line segment of length 2a.
	The directrices are as close as possible to the Y-axis.
	The equation of the right-side directrix is x = a
	The equation of the left-side directrix is x = -a
	See Figure 19.0.

If	ε = 0	then
	b = a ( the ellipse is a circle of radius a )
	c = 0 ( both foci are at the center of the circle )
	The ellipse is a circle of radius a.
	The directrices are as far away as possible from the Y-axis.
	The equation of the right-side directrix is
	The equation of the left-side directrix is
	In Figure 20.0 the left-side directrix is not shown.

The Circle As An Ellipse

	The bigger the eccentricity,	the flatter the ellipse will be.
	The smaller the eccentricity,	the more circular the ellipse will be.

We can regard any circle as an ellipse of eccentricity zero.

Consider a sequence of ellipses, all having the same fixed major axis,
but with foci that start far apart and then move closer together. As the
values of c become smaller and the ellipses become more circular, the
directrices move further away ( "toward infinity" ).

Figure 21.0 shows two of the ellipses in the sequence.
The outer ellipse has smaller eccentricity and is therefore more circular.
The inner ellipse is flatter; and so, b₂ < b₁
Although the picture does not show any foci, it is also a fact that the
ellipse of smaller eccentricity would have to have its foci closer to
the Y-axis. ( To see a simple analytic proof that
ε₁ < ε₂ b₂ < b₁ and c₁ < c₂ click here )

two ellipses - one flat, one more circular

Looking at limits gives us another way to see what happens to an
ellipse whose eccentricity is at or near zero. We can easily show
what happens to r as ε approaches zero. To do this we first need
to recall that

d = distance from right-side focus to right-side directrix

= distance from left-side focus to left-side directrix

We also need to point out that in previous sections we developed
equations with the origin of the r-θ coordinate system placed at
either the left or right focus, and in both cases we found that

ε d = a ( 1 - ε² ) and stmt 10

r = ε d / ( 1 ± ε cos θ ) stmt 20

To find the limit of r as ε approaches zero we cannot simply replace
ε with zero in stmt 20 because ( see further below ) d approaches
infinity as ε approaches zero. But combining stmts 10 and 20 gives

r = a ( 1 - ε² ) / ( 1 ± ε cos θ ) stmt 30

From stmt 30 it is obvious that limit of r as epsilon approaches zero

Recall that the equation of the right-side directrix is x = a² / c = a / ε.

Considering just the right half of an ellipse, the limit as c approaches
zero from the right of a² / c is plus infinity. In symbols we write this
as shown in Figure 23.0.

To prove this statement we have to show that for any number
N > 0 ( no matter how large ), there must exist a number δ > 0
such that 0 < c < δ a² / c > N

The proof is very easy:

	Let N > 0 be given.
	Choose δ = a² / N
Then

	0 < c < δ		c < a² / N

			c N < a²

			a² / c > N

From this it is also clear that limit of a / epsilon as epsilon goes to zero from the right

So the right-side directrix is at infinity when the eccentricity is zero.
Similarly, the left-side directrix is at minus infinity when the
eccentricity is zero.

The Area of An Ellipse

Let A = area of the ellipse shown in Figures 13.0 and 14.0

Since the ellipse has equation x² / a² + y² / b² = 1
we can remove the fractions and solve for y to get

b²x² + a²y²	=	a²b²
a²y²	=	a²b² - b²x²
\| y \|	=	( b/a ) ( a² - x² )^1/2

The upper half of the ellipse has equation y = ( b/a ) ( a² - x² )^1/2

In Figure 25.0 we use this equation and the fact that the ellipse is
symmetric about both axes to find A.

deriving formula for area of ellipse

If you want to verify that the formula from the table of integrals is correct
you will need this formula for the derivative of the arcsine function:
D_xsin^-1U = D_xU / ( 1 - U² )^1/2
We will skip the verification.

The Latera Recta

A chord is a straight line that connects two distinct points on an ellipse.
A chord that passes thru a focus and is perpendicular to the major axis
is called a latus rectum.
Figure 26.0 shows an ellipse with its two latera recta
The Latin term "latus" refers to "side" or "chord", and
"rectum" can mean either "straight" or "right" ( as in "right angle" ).
So a latus rectum is a special right-angle chord.

ellipse with its two latera recta

In Figure 26.0 assume that :

a	=	length of the semi-major axis
b	=	length of the semi-minor axis
c	=	distance from the center of the ellipse to a focus
L	=	length of each latus rectum

We will find the length of L.

Since our ellipse has equation x² / a² + y² / b² = 1,

\| y \|	=	( b / a ) ( a² - x² )^1/2

At a latus rectum, x = ± c and \| y \| = L / 2 So

L / 2	=	( b / a ) ( a² - c² )^1/2
	=	( b / a ) ( a² - a ²ε² )^1/2
	=	b ( 1 - ε² )^1/2

But recall this formula for b that we previously derived :
	b = a ( 1 - ε² )^1/2
which implies that
	b / a = ( 1 - ε² )^1/2
Therefore
L / 2	=	b ( b / a ) = b² / a
L	=	2b² / a

Deriving Kepler's 1st Law

Kepler's 1st Law of Planetary Motion says that each planet
moves about the sun in an elliptical orbit with the sun at one focus.

Of Kepler's three laws, this one is probably the most difficult to prove.
If you want to see a proof then click here

Deriving Kepler's 2nd Law

The main ideas for this derivation come from pages 346-347 of the
Goodstein, et al book referenced at the top of this topic .

ellipse sector determined by r and r_plus_delta_r

In Figure 27.0 the radius vector is shown at time t and again at time t+Δt.
At time t the radius vector has length r, and at time t+Δt its length is r+Δr.

In Figures 27.0 and 27.1, φ is the acute angle between the lines of action
of the r and Δr vectors.

Let ΔA = the area of the ellipse sector swept out during time Δt.
We want to compute the limit as Δt approaches zero of ΔA / Δt.

If Δt is small, then ΔA is approximately equal to the area of triangle HGF.

To find the area of triangle HGF we will regard r as the base and
y as the height. So the area of triangle HGF is
( 1/2 ) * base * height = ( 1/2 ) * r * y.

Since the sine of φ is y / Δr, y = (Δr) sin φ and
the area of the ellipse sector = ΔA ≈ ( 1/2 ) ( rΔr sin φ ).

But rΔr sin φ is the magnitude of the cross product of the r and Δr vectors.

But recall that | r X v | = L / m = a constant value

Hence dA / dt = L / 2m = constant

From this it follows that, in time Δt,
the sector area swept out will be ( L / 2m ) Δt.

Deriving Kepler's 3rd Law

The 3rd Law says that the square of the period is
proportional to the cube of the semi-major axis of the orbit.

i.e. T² = k a³ for some constant k

To open a new window showing a 3rd Law derivation click here

The Vis-Viva Equation

The speed of an orbiting body can be found from the
Vis-Viva Equation which says that V² = GM ( 2/r - 1/a )

To open a new window showing a Vis-Viva Equation derivation click here

The Three Anomalies

This topic is still not finished, but to open a new window
showing what we have so far click here

Uniform Circular Motion

To open a new window showing our first version of this topic click here

Gravitational Potential Energy

For an object of mass m at a distance r from the center of
the Earth we define the gravitational potential energy U by
U( r ) = -GMm / r
To open a new window showing why we define U this way click here

Escape Velocity

Parabolic velocity is often referred to as the escape velocity.
At the surface of the Earth ( whose radius is R ≈ 6,378 km )
escape velocity is given by the formula
v_escape = ( 2GM / R )^0.5 = ( 2gR )^0.5
( which is about 11.180 km / sec or 25,000 mph )

To open a new window showing two derivations, a discussion of
the terms apogee & perigee, and a look at circular velocity click here

Apogee - Perigee Example

	Suppose a satellite orbits the Earth in an elliptical path. Given its speed and altitude at perigee, find the following:
		altitude at apogee
		speed at apogee
		length of the semi-major axis
		eccentricity
		period

To open a new window showing a solution for this type
of problem click here

Oblique Launch

To open a new window showing how you can modify your
calculations to handle an oblique launch click here

Links To Other Web Sites

	Motion In Two Dimensions [ vectors, projectiles, circular motion, .... ]
		Pierce College [ near Los Angeles, California ]
		Michelle Meyer 42 page pdf file

	Orbital Mechanics: Principles of Space Systems Design
		Univ of Maryland -- pdf file consisting mostly of diagrams & equations
		David Akin 49 page pdf file University of Maryland

	orbital mechanics and rockets [ 11 page pdf file ]
		McGill University on orbital mechanics and rockets
		pdf by Andrew Higgins of McGill University

	satellites, orbits, etc.
		University of Oregon -- a course by James Schombert home page for Astronomy 121 ( a course by James Schombert )

	Orbitals, Kepler Laws, Vis-Viva, etc,
		Frosty Drew Observatory
		article by Leslie S. Coleman of Frosty Drew Observatory

	download vis-viva Powerpoint slides
		Naval PostGraduate School -- Monteray, California
		Vis-viva Applications wk4 lect 1 ppt slides from nps.navy.mil

	Vis-viva and Energy
		a one page article that asks if certain forms of energy are "real" David M. Harrison -- Physics Department -- University of Toronto

	Deriving The Vis-viva Equation By Energy Considerations
		Vis-viva derivation with equations in large font a one-page derivation -- Case Western Reserve University

	Derivation of Vis-viva By Energy Conservation
		a one-page pdf file single page derivation -- California State University Northridge JPL
		More links at California State University Northridge JPL

	David P. Stern site with many pages for students and teachers
		site map for Stern's site Kepler's 3 Laws -- Overview For Science Teachers History: Tycho Brahe, Johannes Kepler, conics, and 3rd Law 2nd Law; Energy; Mean, True, and Eccentric Anomalies How Orbital Motion Is Calculated

	Tutorial on Earth/Sun Relations and Seasons
		Palomar Community College [ San Marcos, California ]
		Dr. Jane Thorngren explains how the Earth's tilt causes 4 seasons

	Mechanical Universe video "The Kepler Problem"
		CalTech Physics Lecture 22 : The Kepler Problem

	other Mechanical Universe videos
		Go to video.google.com and search for "mechanical universe"

	Rocket & Space Technology by Robert A. Braeunig
		a "must-see" web site with lots of info on many aspects of space technology
		home page for Robert Braeunig web site
		Orbital Mechanics

	Air University Space Primer
		by AU at Maxwell-Gunter Air Force Base ( Montgomery, Alabama )
		chapter 8 Orbital Mechanics ( 24 page pdf )
		Links to other Air University Space Primer documents

	Celestial Coordinates
		just one of many interesting articles by Peter Alway
		Unit 1C: Celestial Coordinates, Motions, and Time

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Part 1

Part 2

	1 )	v = dr/dt by definition of velocity
	2 )	F = d/dt ( mv ) Newton's 2nd law of motion
	3 )	For any vector A, the cross product A X A is zero
	4 )	D_t ( AB ) = [ ( dA/dt ) B ] + [ A * ( dB/dt ) ]

d	=	distance from focus to directrix
	=	PA + r cos θ
	=	r / ε + r cos θ
	=	r ( 1 / ε + cos θ )

	d	=	distance from right-side focus to right-side directrix
		=	distance from left-side focus to left-side directrix

	ε d	=	a ( 1 - ε² ) and	stmt 10
	r	=	ε d / ( 1 ± ε cos θ )	stmt 20