Lecture notes on planetary sciences and orbit determination
1. Lecture notes on Planetary sciences and Satellite Orbit
Determination
Ernst J.O. Schrama
Delft University of Technology,
Faculty of Aerospace, Astrodynamics and Satellite missions
e-mail: e.j.o.schrama@tudelft.nl
29-Aug-2017
8. Chapter 1
Introduction
In these lecture notes I bundled all material that I use for the introductory and advanced course
on planetary sciences and the course on satellite orbit determination which are part of the
curriculum at the faculty of aerospace engineering at the Delft University of technology. In the
MSc track of the faculty of aerospace engineering the course code for planetary sciences I is
ae4-890 and for the follow up course it is ae4-876. In the same curriculum the course on satellite
orbit determination comes with the code ae4-872.
A main topic in satellite orbit determination is the problem of parameter estimation which
is relates the dynamics of a space vehicle to observation techniques. From this follows a number
of scientific applications that are related to the observation techniques. In order to set-up the
framework for all lectures we start in chapter 2 with the two body problem, this material is also
mandatory for the planetary science I (ae4-890). It depends on your prior education whether or
not you need to study this chapter. Chapter 2 contains all required information from the BSc
of aerospace engineering. If you don’t feel familiar with the two-body problem then study it
entirely, if you want to test your knowledge then try the exercises at the end of this chapter.
The two-body problem is directly related to potential theory which is nowadays most likely
not part of your bachelor program; for this reason I’ve included chapter 3. For course ae4-
890 I do recommend to study the Laplace equation and the series expansion of the function
1/r in Legendre functions. An advanced topic is that the Laplace equation also comes with
higher order expansions in the potential functions. A summary of some well known properties
of Legendre functions, spherical harmonics and convolution integrals on the sphere should be
seen as a reference, that is, you should recognize spherical harmonics, and potential coefficients,
but you are not asked to reproduce for instance recursive relations of the Legendre functions.
At various points in the lecture notes we refer to Fourier’s method of frequency analysis which
is described in chapter 4, this is a general mathematical procedure of which the results are used
throughout the lecture notes. It finds its application in tidal theory, it relates for instance to
the chapter 3 on potential theory where we mentioned the convolution on the sphere, and the
solution of the Hill equations in chapter 12 depends on Fourier series which are a characteristic
solution of the system. During various lectures I noticed that Fourier’s method for frequency
analysis is often not part of the BSc curriculum, so I added the topic to these lecture notes. We
treat the continuous case to introduce the topic, but rapidly switch to the discrete case which
seems most applicable to what most people use. I included a number of examples in MATLAB
to demonstrate various properties related to the Fourier transforms.
The definition of time and coordinates is essential for all lectures; this topic is not part of
7
9. the curriculum of aerospace and for this reason I added chapter 5. This chapter discusses the
relation between the Earth center inertial (ECI) and the Earth center fixed (ECF) frame, the
role of the International Earth Rotation Service (IERS), and transformations between reference
systems. Other topics in this chapter are map projections and the consequence of special and
general relativity on the definition of time and coordinates.
In chapter 6 we discuss observation techniques and applications relevant for ae4-872. We
introduce satellite laser ranging (SLR), Doppler tracking (best known is the French DORIS
system) and the Global Positioning System (GPS). There are a number of corrections common
to all observation techniques, for this reason we speak about the light time effect, but also
refraction in the atmosphere and the ionosphere and including the phenomenon multipath which
is best known during radio tracking. The applications that we discuss are satellite altimetry,
very long baseline interferometry (VLBI) and satellite gravimetry.
For the course on satellite orbit determination I recommend to study chapter 7 where we
introduce the concept of combining observations, models and parameters, the material presented
here continues with what was presented in chapters 2 to 6. In section 7.1 we discuss the need to
consider dynamics when we estimate parameters. This brings us to chapter 8 where parameter
estimation techniques are considered without consideration of a dynamical model. The need for a
statistical approach is introduced for instance in 8.1 where the expectancy operator is defined in
8.3. With this knowledge we can continue to the least squares methods for parameter estimation
as discussed in 8.5. Chapter 10 discusses dynamical systems, Laplace transformations to solve
the initial value problem, shooting problems to solve systems of ordinary differential equations,
dynamical parameter estimation, batch and sequential parameter estimation techniques, the
Kalman filter and process noise and Allan variance analysis.
For ae4-890 we recommend to study the three-body problem which is introduced in chap-
ter 11. Related to the three-body problem is the consideration of co-rotating coordinate frames
in orbital dynamics, in these notes you can find this information in chapter 12, for the course on
ae4-890 we need this topic to explain long periodic resonances in the solar system, but also to
explain the problem of a Hill sphere which is found in [11]. During the lectures on solar system
dynamics in ae4-890 the Hill sphere and the Roche limit will be discussed in chapter 13 Both
topics relate to the discussion in chapters 2 and 13 of the planetary sciences book, cf. [11].
Course ae4-890 introduces the tide generating force, the tide generating potential and global
tidal energy dissipation. I recommend to study chapter 14 where we introduce the concept
of a tide generating potential whose gradient is responsible for tidal accelerations causing the
“solid Earth” and the oceans to deform. For planetary sciences II (ae4-876) I recommend the
remaining chapters that follow chapter 14. Deformation of the entire Earth due to an elastic
response, also referred as solid Earth tides and related issues, is discussed in chapter 15. A good
approximation of the solid Earth tide response is obtained by an elastic deformation theory.
The consequence of this theory is that solid Earth tides are well described by equilibrium tides
multiplied by appropriate scaling constants in the form of Love numbers that are defined by
spherical harmonic degree.
In ae4-876 we discuss ocean tides that follow a different behavior than solid earth tides.
Hydrodynamic equations that describe the relation between forcing, currents and water levels
are discussed in chapter 16. This shows that the response of deep ocean tides is linear, meaning
that tidal motions in the deep ocean take place at frequencies that are astronomically determined,
but that the amplitudes and phases of the ocean tide follow from a convolution of an admittance
function and the tide generating potential. This is not anymore the case near the coast where
8
10. non-linear tides occur at overtones of tidal frequencies. Chapter 17 deals with two well known
data analysis techniques which are the harmonic analysis method and the response method for
determining amplitude and phase at selected tidal frequencies.
Chapter 18 introduces the theory of load tides, which are indirectly caused by ocean tides.
Load tides are a significant secondary effect where the lithosphere experiences motions at tidal
frequencies with amplitudes of the order of 5 to 50 mm. Mathematical modeling of load tides
is handled by a convolution on the sphere involving Green functions that in turn depend on
material properties of the lithosphere, and the distribution of ocean tides that rest on (i.e.
load) the lithosphere. Up to 1990 most global ocean tide models depended on hydrodynamical
modeling. The outcome of these models was tuned to obtain solutions that resemble tidal
constants observed at a few hundred points. A revolution was the availability of satellites
equipped with radar altimeters that enabled estimation of many more tidal constants. This
concept is explained in chapter 19 where it is shown that radar observations drastically improved
the accuracy of ocean tide models. One of the consequences is that new ocean tide models result
in a better understanding of tidal dissipation mechanisms.
Chapter 20 serves two purposes, the section on tidal energetics from lunar laser ranging is
introduced in ae4-890, all material in section 20.2 should be studied for ae4-890. The other
sections in this chapter belong to course ae4-876, they provide background information with
regard to tidal energy dissipation. The inferred dissipation estimates do provide hints on the
nature of the energy conversion process, for instance, whether the dissipations are related to
bottom friction or conversion of barotropic tides to internal tides which in turn cause mixing of
between the upper layers of the ocean and the abyssal ocean.
Finally, while writing these notes I assumed that the reader is familiar with mechanics,
analysis, linear algebra, and differential equations. For several exercises we use MATLAB or
an algebraic manipulation tool such as MAPLE. There are excellent primers for both tools,
mathworks has made a matlab primer available, cf. [37]. MAPLE is suitable mostly for analysis
problems and a primer can be found in [35]. Some of the exercises in these notes or assigned as
student projects expect that MATLAB and MAPLE will be used.
E. Schrama, Delft September 29, 2017
9
11. Chapter 2
Two body problem
2.1 Introduction
The first astronomic observations were made more than two millennia ago, the quality of the
observations was constrained to the optical resolution and the sensitivity of the human eye. The
brightness of a star is usually indicated by its magnitude, a change of 1 in magnitude corresponds
to a change 2.5 in brightness. Under ideal conditions the human eye is limited to magnitude
six, and the optical resolution is roughly 15” (thus 15/3600 of a degree), while the angular
resolution of binoculars is 2.5”. The naked eye is already a very sensitive and high quality
optical instrument for basic astronomic observations, as long as there is no light pollution and
when your eyes are used to darkness. We are able to distinguish planets from Mercury to Saturn,
comets, meteors and satellites but our naked-eye lacks the resolution to observe the moons of
Jupiter, or the second star of Mizar in Ursa Major.
The discussion about the motion of planets along the night sky goes back to ancient history.
The Greeks and Romans associated the planets with various gods. Mars was for instance the God
of War, Jupiter held the same role as Zeus in the Greek Pantheon and Mercury was the God of
trade, profit and commerce. Planets are unique in the night sky since the wander relative to the
stars, who seem to be fixed on a celestial sphere for an observer on a non-rotating Earth. Before
the invention of the telescope in 1608 and its first application for astronomic observations in
1610 by Galileo Galilei the believe was essentially that the Earth was the center of the universe,
that it was flat and that you could fall over the horizon and that everything else in the universe
rotated around the Earth.
Galileo, Copernicus, Brahe and Kepler
Galileo Galilei was an Italian astronomer (1564 to 1642) renowned for his revolutionary new
concept the solar system causing him to get into trouble with the inquisition. He modified the
then existing telescope into an instrument suitable for astronomic observations to conclude in
1610 that there are four Moons orbiting the planet Jupiter. The telescope was earlier invented
by the German-born Dutch eyeglass maker Hans Lippershey who demonstrated the concept
of two refracting lenses to the Dutch parliament in 1608. After all it is not surprising that
the observation of moons around Jupiter was made in southern Europe, which on the average
has a higher chance of clear night skies compared to the Netherlands. One of Galileo Galilei’s
comments on the classical view on the solar system was that his instrument permitted him to
10
12. see moons orbiting another planet, and that the classical model was wrong.
Other developments took place around the same time in Europe. Nicolaus Copernicus was
a Polish astronomer who lived from 1473 to 1543 and he formulated the concept of planets
wandering in circular orbits about the Sun, which was new compared to the traditional geocentric
models of Claudius Ptolomaeus (87 to 150) and the earlier model of Hypparchus (190 to 120 BC).
It was the Danish astronomer Tycho Brahe (1546 to 1601) to conclude on basis of observations of
the planet Mars that there were deviations from the Copernican model of the solar system. The
observations of Tycho Brahe assisted the German mathematician, astronomer and astrologer
Johannes Kepler 1571 to 1630) to complete a more fundamental model that explains the motion
of planets in our solar system. The Keplerian model is still used today because it is sufficiently
accurate to provide short-term and first-order descriptions of planetary ephemerides in our solar
system and satellites orbiting the Earth.
Kepler’s laws
The mathematical and physical model of the solar system ican be summarized in three laws
postulated by Kepler. The first and the second law were published in Astronomia Nova in 1609,
the third law was published in Harmonices Mundi in 1619:
• Law I: In our solar system, the Sun is in a focal point of an ellipse, and the planets move
in an orbital plane along this ellipse, see plate 2.1.
• Law II: The ratio of an area swept by a planet relative to the time required is a constant,
see plate 2.2.
• Law III: The square of the mean orbital motion times the cube of the largest circle con-
taining the ellipse is constant. Thus:
n2
a3
= G.M = µ (2.1)
The constant n is the mean motion in radians per second and a the semi-major axis in some
unit of length. In this equation G is the universal gravitational constant and M is the mass of
the Sun. (both in units that correspond to the left hand side).
2.2 Keplerian model
In this section we demonstrate the validity of the Keplerian model, essentially by returning to
the equations of motion inside which we substitute a suitable gradient of a potential function.
This will result in an expression that describes the radius of the planet that depends on its
position in orbit. After this point we will derive a similar expression for the scalar velocity in
relation to the radius, the latter is called the vis-viva equation.
2.2.1 Equations of motion
In an inertial coordinate system the equations of motion of a satellite are:
¨x = − V +
i
f
i
(2.2)
11
13. Figure 2.1: Elliptical orbit of a planet around the sun in one of the focal points
Figure 2.2: Kepler’s equal area law: segment AB-Sun and segment CD-Sun span equal areas,
the motion of the planet between A and B takes as long as it would between C and D
12
14. where ¨x is an acceleration vector and V a so-called potential function and where the terms f
i
represent additional accelerations. An in-depth discussion on potential functions can be found
in chapter 3. At this point it is sufficient to assume that the equations of motion in (2.2) apply
for a planet orbiting the sun. Equation (2.2) is a second-order ordinary differential equation
explaining that a particle in a force field is accelerating along the local direction of gravity (which
is the gradient of V written as V = (∂V /∂x, ∂V /∂y, ∂V /∂z) in the model). The model allows
for additional accelerations which are usually much smaller than the gravitational effect.
A falling object on Earth like a bullet leaving a gun barrel will exactly obey these equations.
In this case gravity is the main force that determines the motion, while also air drag plays a
significant role. One way to obtain a satellite in orbit would be to shoot the bullet with sufficient
horizontal velocity over the horizon. If there wouldn’t be air drag then Kepler’s orbit model
predicts that this particular bullet eventually hits the gunman in his back. There are at least
two reasons why this will never happen. The first reason is of course the presence of air drag,
the second reason is that the coordinate frame we live in experiences a diurnal motion caused by
a rotation Earth. (It is up to you to verify that ”Kepler’s bullet” will hit an innocent bystander
roughly 2000 km west of your current location on the equator.) Air drag will keep the average
bullet exiting a barrel within about 2 kilometer which is easy to verify when you implement
eq. (2.2) as a system of first-order ordinary differential equations in MATLAB. The rotating
Earth causes a much smaller effect and you will not easily notice it. (In reality cross-wind has
a more significant effect).
Foucault’s pendulum is best used to demonstrate the consequences of rotating Earth. Jean
Bernard L´eon Foucault was a French physicist who lived from 1819 to 1868 and he demonstrated
the effect of Earth rotation on a pendulum mounted in the Meridian Room of the Paris obser-
vatory in 1851, today the pendulum can be found in the Panth´eon in Paris where it is a 28-kg
metal ball suspended by wire in the dome of this building. Foucault’s pendulum will oscillate in
an orbital plane, due to the Coriolis forces that act on the pendulum we observe a steady shift
of this orbital plane that depends on the latitude of the pendulum. Some facts are:
• The coordinate system used in equation (2.2) is an inertial coordinate system that does
not allow frame accelerations due to linear acceleration or rotation of the frame.
• Whenever we speak about gravity on the Earth’s surface, as we all know it, we refer to the
sum of gravitational and rotational acceleration. Just gravitation refers to the acceleration
caused by Newton’s gravity law.
• The potential V in equation (2.2) is thus best referred to as a gravitational potential,
sometimes it is also called the geo-potential.
The concept of potential functions is best explained in a separate lecture on potential theory.
Chapter 3 describes some basic properties to arrive at a suitable potential function for the Kepler
problem.
2.2.2 Keplerian equations of motion
A suitable potential V for the Kepler model is:
V (r) = −
µ
r
(2.3)
13
15. It is up to the reader to confirm that this function fulfills the Laplace equation, but also, that it
attains a value of zero at r = ∞ where r is the distance to the point mass and where µ = G.M
with G representing the universal gravitational constant and M the mass which are both positive
constants.
The gradient of V is the gravitational acceleration vector that we will substitute in the
general equations of motion (2.2), which in turn explains that a satellite or planet at (x, y, z)
will experience an acceleration (¨x, ¨y, ¨z) which agrees with the direction indicated by the negative
gradient − of the potential function V = −µ/r. The equations of motion in (2.2) may now be
rearranged as:
¨x =
∂V
∂x
+
i
fi
x
¨y =
∂V
∂y
+
i
fi
y (2.4)
¨z =
∂V
∂z
+
i
fi
z
which becomes:
∂ ˙x/∂t = −µx/r3 ∂x/∂t = ˙x
∂ ˙y/∂t = −µy/r3 ∂y/∂t = ˙y
∂ ˙z/∂t = −µz/r3 ∂z/∂t = ˙z
(2.5)
In this case we have assumed that the center of mass of the system coincides with the origin. In
the three-body problem we will drop this assumption.
Demonstration of the gun bullet problem in matlab
In matlab you can easily solve equations of motion with the ode45 routine. This routine will
solve a first-order differential equation ˙s = F(t, s) where s is a state vector. For a two body
problem we only need to solve the equations of motion in a two dimensions which are the in-plane
coordinates of the orbit. For the gun bullet problem we can assume a local coordinate system,
the x-axis runs away from the shooter and the y-axis goes vertically. The gravity acceleration
is constant, simply g = −9.81 m/ss. The state vector is therefore s = (x, y) and the gradient is
in this case − V = (0, −g) where g is a constant. In matlab you need to define a function to
compute the derivatives of the state vector, and in the command window you to call the ode45
procedure. Finally you plot your results. For this example we stored the function in a separate
file called dynamics.m containing the following code:
function [dsdt] = dynamics(t,s)
%
% in the function we will compute the derivatives of vector s
% with respect to time, the ode45 routine will call the function
% frequently when it solves the equations of motion. We store
% x in s(1) and y in s(2), and the derivatives go in s(3) and
% s(4). In the end dsdt receives the components of the
% gradient of V, here just (0,g)
%
14
16. dsdt = zeros(4,1); % we need to return a column vector to ode45
g = 9.81; % local gravity acceleration
dsdt(1) = s(3); % the velocity in the x direction is stored in s(3))
dsdt(2) = s(4); % the velocity in the y direction is stored in s(4))
dsdt(3) = 0; % there is no acceleration in the x direction
dsdt(4) = -g; % in the vertical direction we experience gravity
To invoke the integration procedure you should write another script that contains:
vel = 100; angle = 45;
s = [0 0 vel*cos(angle/180*pi) vel*cos(angle/180*pi)];
options = odeset(’AbsTol’,1e-10,’RelTol’,1e-10);
[T,Y] = ode45(@dynamics,[0 14],s,options );
plot(Y(:,1),Y(:,2))
The command s = ... assigns the initial state vector to the gun bullet, the options command is
a technicality, ie. probably you don’t need it but when we model more complicated problems
then it may be needed. The odeset routine controls the integrator behavior. The next line calls
the integrator, and he last command plots the flight path of the bullet that we modelled. It
starts with a velocity of 100 m/s and the gun was aimed at 45 degrees into the sky, after about
14 seconds the bullet hits the surface ≈ 1000 meter away from the gun. Note that we did not
model any drag or wind effects on the bullet. In essence, all orbit integration procedures can be
Figure 2.3: Path of the bullet modelled in the script dynamics.m
15
17. Figure 2.4: The angular momentum vector is obtained by the cross product of the position and
velocity vector.
treated as variations of this problem, except that the dimension of the state vector will change
and that also, that the dynamics.m file will become more complicated.
2.2.3 Orbit plane
So far we have assumed that x y and z are inertial coordinates, and that the motion of the
satellite or planet takes place in a three dimensional space. The remarkable observation of
Kepler was that the motion occurs within a plane that intersects the center of the point source
mass generating V . This plane is called the orbit plane, and the interested reader may ask why
this is the case. To understand this problem we need to consider the angular momentum vector
H which is obtained as:
r × v = x × ˙x = H (2.6)
where v is the velocity vector and r the position vector, see also figure 2.4. If we assume that
x = r = (x, y, 0) and that ˙x = v = ( ˙x, ˙y, 0) then:
x
y
0
×
˙x
˙y
0
=
0
0
x ˙y − y ˙x
which explains that the angular momentum vector is perpendicular to the plane spanned by r
and v. To demonstrate that ˙H = 0 we evaluate:
∂
∂t
˙x × x = ¨x × x + ˙x × ˙x
The last term is zero, due to the fact that:
¨x = −
µ
r3
x
16
18. we also find that:
¨x × x = 0
so that ˙H = 0. A direct consequence is that we conserve angular momentum, and as we will
show later, we also conserve energy. The fact that the angular momentum vector is constant
in size and direction also explains why Kepler found an equal area law and that the motion is
confined to an orbital plane.
Substitution 1
To simplify the search for a solution we confine ourself to an orbital place. A convenient choice
is in this case to work in polar coordinates so that:
x = r cos θ
y = r sin θ
In the sequel we will substitute this expression in the equations of motion that follow from the
point mass potential, see also equation (2.5). An intermediate step is:
˙x = ˙r cos θ − r ˙θ sin θ
˙y = ˙r sin θ + r ˙θ cos θ
so that:
¨x = ¨r cos θ − 2 ˙r ˙θ sin θ − r¨θ sin θ − r ˙θ2
cos θ
¨y = ¨r sin θ + 2 ˙r ˙θ cos θ + r¨θ cos θ − r ˙θ2
sin θ
which is equivalent to:
¨x
¨y
=
cos θ − sin θ
sin θ cos θ
¨r − r ˙θ2
2 ˙r ˙θ + r¨θ
(2.7)
For the gradient we have:
∂V /∂x
∂V /∂y
=
∂r/∂x ∂θ/∂x
∂r/∂y ∂θ/∂y
∂V /∂r
∂V /∂θ
(2.8)
so that:
∂V /∂x
∂V /∂y
=
cos θ −sin θ
r
sin θ cos θ
r
−µ/r2
0
(2.9)
Since the right hand sides of (2.8) and (2.9) are equal we get:
¨r − r ˙θ2
= −
µ
r2
(2.10)
2 ˙r ˙θ + r¨θ = 0 (2.11)
For the length of the angular momentum vector we get:
h = |H| = x ˙y − y ˙x
= +r cos θ( ˙r sin θ + r ˙θ cos θ) − r sin θ( ˙r cos θ − r ˙θ sin θ)
= r2 ˙θ
17
19. which demonstrates that equal areas are covered in equal units of time in Kepler’s second law.
Since h is constant we obtain after differentiation with respect to time:
˙h = 2r ˙r ˙θ + r2 ¨θ = 0 (2.12)
Since r = 0 is a trivial solution we keep:
2 ˙r ˙θ + r¨θ = 0 (2.13)
which is equal to (2.11). This consideration does not lead to a new insight in the problem.
And thus we turn our attention to eq. (2.10) which we can solve with a new substitution of
parameters.
Substitution 2
At this point a suitable parameter substitution is r = 1/u and some convenient partial derivatives
are:
∂u
∂r
= −
1
r2
∂u
∂θ
=
∂u
∂r
∂r
∂t
∂t
∂θ
= (
−1
r2
)( ˙r)( ˙θ−1
) = (
−1
r2
)( ˙r)(
r2
h
) = −
˙r
h
∂2u
∂θ2
=
∂
∂t
(
∂u
∂θ
)
∂t
∂θ
= −
¨r
h
˙θ−1
= −
¨r
h
r2
h
= −
¨r
u2h2
from which we obtain:
¨r = −u2
h2 ∂2u
∂θ2
Substitution of these partial derivatives in (2.10) results in:
−u2
h2 ∂2u
∂θ2
−
h2
r3
= −µu2
so that:
∂2u
∂θ2
+ u =
µ
h2
(2.14)
This equation is equivalent to that of a mathematical pendulum, its solution is:
u = A cos θ + B
∂u
∂θ
= −A sin θ
∂2u
∂θ2
= −A cos θ
We find:
u +
∂2u
∂θ2
= B =
µ
h2
so that A becomes an arbitrary integration constant. In most textbooks we find the following
expression that relates r to θ:
r(θ) =
a(1 − e2)
1 + e cos θ
(2.15)
18
20. This expression results in circular orbits for e = 0, or elliptical orbits for 0 < e < 1. To verify
eq. (2.15) we evaluate r at the apo-apsis and the peri-apsis.
u(θ = 0) =
1
a(1 − e)
= +A + B
u(θ = π) =
1
a(1 + e)
= −A + B
From which we get:
A =
e
a(1 − e2)
B =
µ
h2
2B =
1
a(1 − e)
+
1
a(1 + e)
=
2
a(1 − e2)
B =
1
a(1 − e2)
=
µ
h2
resulting in:
h = µa(1 − e2)
which provides us with the length of the angular momentum vector.
2.2.4 Parabolic and hyperbolic orbits
So far we have demonstrated that circular and elliptic orbits appear, but in textbooks you also
find that parabolic and hyperbolic orbits exist as a solution of the Kepler problem. A parabolic
orbit corresponds to e = 1, and in a hyperbolic orbit e > 1. The parabolic orbit is one where
we arrive with a total energy of zero at infinity, therefore it is also called the minimum escape
orbit. Another option to escape the planet is to fly in a hyperbolic orbit, in this case we arrive
with a positive total energy at infinity. The total energy for the circular and eccentric Kepler
orbit is negative.
2.2.5 The vis-viva equation
Equation (2.15) contains all information to confirm Kepler’s first and second law. We will
now switch to an energy consideration of the Keplerian motion. Because of the conservation
of momentum we can not allow that energy disappears over time. This agrees with what we
observe in astronomy; planets and moons do not disappear on a cosmologic time scale (which is
only true if we leave tidal dissipation out of the discussion). If we assume that the total energy
of the system is conserved then:
1
2
mv2
−
mµ
r
= d∗
where m and v represent mass and scalar velocity and where d∗ is constant. We eliminate the
mass term m by considering d = d∗/m so that:
v2
2
= d +
µ
r
19
21. The question is now to find d, since this would give us a relation to connect the scalar velocity in
an orbit to the radius r. This is what we call the vis-viva equation or the path-speed equation.
At the peri-apsis and the apo-apsis the velocity vectors are perpendicular to r. The length
of the moment vector (h) is nothing more than the product of the peri-apsis height and the
corresponding scalar velocity vp. The same property holds at the apo-apsis so that:
a(1 − e)vp = a(1 + e)va (2.16)
The energy balance at apo-apsis and peri-apsis is:
v2
a = 2d + 2
µ
ra
= 2d + 2
µ
a(1 + e)
(2.17)
v2
p = 2d + 2
µ
rp
= 2d + 2
µ
a(1 − e)
(2.18)
From equation (2.16) we get:
v2
p =
1 + e
1 − e
2
v2
a (2.19)
This equation is substituted in (2.18):
1 + e
1 − e
2
v2
a = 2d + 2
µ
a(1 − e)
(2.20)
From this last equation and (2.17) you find:
v2
a =
1 − e
1 + e
2
2d + 2
µ
a(1 − e)
= 2d + 2
µ
a(1 + e)
(2.21)
so that:
d = −
µ
2a
As a result we find that the total energy in the Kepler problem becomes:
v2
2
−
µ
r
= −
µ
2a
(2.22)
so that the total energy by mass for an object in orbit around a planet is constrained to:
Etot = −
µ
2a
(2.23)
The scalar velocity of the satellite follows from the so-called vis-viva (Latin: living force1)
relation:
v = µ
2
r
−
1
a
which is an important relation that allows you to compute v as a function of r for a semi-major
axis a and a solar mass µ.
1
wikipedia mentions that the vis-viva is a obsolete scientific theory that served as an elementary and limited
early formulation of the principle of conservation of energy
20
22. Orbital periods
For a circular orbit with e = 0 and r = a we find that:
v =
µ
a
If v = na where n is a constant in radians per second then:
na =
µ
a
⇒ µ = n2
a3
This demonstrates Kepler’s third law. Orbital periods for any parameter e ∈ [0, 1] are denoted
by τ and follow from the relation:
τ =
2π
n
⇒ τ = 2π
a3
µ
The interested reader may ask why this is the case, why do we only need to calculate the orbital
period τ of a circular orbit and why is there no need for a separate proof for elliptical orbits.
The answer to this question is already hidden in the conservation of angular momentum, and
related to this, the equal area law of Kepler. In an elliptical orbit the area dA of a segment
spent in a small time interval dt is (due to the conservation of angular momentum) equal to
dA = 1
2h. The area A within the ellipse is:
A =
2π
θ=0
1
2
r(θ)2
dθ (2.24)
To obtain the orbital period τ we fit small segments dA within A, and we get:
τ = A/dA =
2π
θ=0
r(θ)2
h
dθ =
2π
θ=0
˙θ−1
dθ =
2πa2
√
µa
(2.25)
which is valid for a > 0 and 0 ≤ e < 1. This demonstrates the validity of Kepler’s 3rd law.
Time vs True anomaly, solving Kepler’s equation
Variable θ in equation (2.15) is called the true anomaly and it doesn’t progress linearly in time.
In fact, this is already explained when we discussed Kepler’s equal area law. The problem is
now that you need to solve Kepler’s equation which relates the mean anomaly M to an eccentric
anomaly E which in turn is connected via a goniometric relation to the true anomaly θ. The
discussion is rather mathematical, but over the centuries various methods have been developed
to solve Kepler’s equation. Without any further proof we present here a two methods to convert
the true anomaly θ, into an epoch t relative to the last peri-apsis transit t0. The algorithms
assume that:
• The mean anomaly M is defined as M = n.(t − t0) where n is the mean motion in radians
per second for the Kepler problem.
• The eccentric anomaly E relates to M via a transcendental relation: M = E − e sin E.
• The goniometric relation tan θ =
√
1 − e2 sin E/(cos E − e) is used to complete the con-
version of E to θ.
21
23. Iterative approach
There is an iterative algorithm that starts with E = M as an initial guess. Next we evaluate
Ei = M − e sin Ei−1 repeatedly until the difference Ei − e sin Ei − M converges to zero. The
performance of this algorithm is usually satisfactory in the sense that we obtain convergence
within 20 steps. For a given eccentricity e one may make a table with conversion values to be
used for interpolation. Note however that the iterative method becomes slow and that it may
not easily converge for eccentricities greater than 0.6.
Bessel function series
There are alternative procedures which can be found on the Wolfram website, cf. [29]. One
example is the expansion in Bessel functions:
M = E − e sin E (2.26)
E = M +
N
1
2
n
Jn(n.e) sin(n.M) (2.27)
The convergence of this series is relatively easy to implement in MATLAB. First you define
M between 0 and 2π, and you assume a value for e and N. Next we evaluate E with the
series expansion and substitute the answer for M back in the first expression to reconstruct the
M that you started with. The difference between the input M, and the reconstructed M is
then obtained as a standard deviation for this simulation, it is an indicator for the numerical
accuracy. Figure 2.5 shows the obtained rms values when we vary e and N in the simulation.
The conclusion is that it is difficult to obtain the desired level of 10−16 with just a few terms,
a series of N = 20 Bessel functions is convergent for e up to approximately 0.4, and N = 50 is
convergent for e up to approximately 0.5. In most cases we face however low eccentricity orbits
where e < 0.05 in which case there is no need to raise N above 5 or 10 to obtain convergence.
The Jn(x) functions used in the above expression are known as Bessel functions of the first
kind which are characteristic solutions of the so-called Bessel differential equation for function
y(x):
x2 d2y
dx2
+ x
dy
dx
+ (x2
− α2
)y = 0 (2.28)
The Jn(x) functions are obtained when we apply the Frobenius method to solve equation (2.28),
the functions can be obtained from the integral:
Jn(x) =
1
π 0
π(cos(nτ − x sin(τ))d τ (2.29)
More properties of the Jn(x) function can be found on the Wolfram website, also, the Bessel
functions are usually part of a programming environment such as MATLAB, or can be found in
Fortran or C/C++ libraries. Bessel functions of the first kind are characteristic solutions of the
Laplace equation in cylindrical harmonics which finds its application for instance in describing
wave propagation in tubes.
2.2.6 Kepler’s orbit in three dimensions
To position a Kepler orbit in a three dimensional space we need three additional parameters for
the angular momentum vector H. The standard solution is to consider an inclination parameter
22
24. Figure 2.5: Convergence of the Bessel function expansion to approximate the eccentric anomaly
E from the input which is the mean anomaly M between 0 and 2π. The vertical scale is
logarithmic, the plateau is the noise floor obtained with a 8 byte floating point processor.
I which is the angle between the positive z-axis of the Earth in a quasi-inertial reference system
and H. In addition we define the angle Ω that provides the direction in the equatorial plane
of the intersection between the orbit plane and the positive inertial x-axis, Ω is also called the
right ascension of the ascending node. The last Kepler parameter is called ω, which provides
the position in the orbital plane of the peri-apsis relative to the earlier mentioned intersection
line.
The choice of these parameters is slightly ambiguous, because you can easily represent the
same Keplerian orbit with different variables, as has been done by Delauney, Gauss and others.
In any case, it should always be possible to convert an inertial position and velocity in three
dimension to 6 equivalent orbit parameters.
2.3 Exercises
Test your own knowledge:
1. What is the orbital period of Jupiter at 5 astronomical units? (One astronomical unit is
the orbit radius of the Earth)
2. Plot r(θ), v(θ) and the angle between r(θ) and v(θ) for θ ∈ [0, 2π] and for e = 0.01 and
a = 10000 km for µ = 3.986 × 1014 m3s−2.
3. For an elliptic orbit the total energy is negative, for a parabolic orbit the total energy
is zero, ie. it is the orbit that allows to escape from Earth to arrive with zero energy at
23
25. infinity. How do you parameterize parabolic orbits, how do you show that they are a
solution of the Kepler problem? How does this relate to the escape velocity on Earth?
4. Make a perspective drawing of the Kepler ellipse in 3D and explain all involved variables.
5. Design a problem to plot ground tracks for an arbitrary Kepler orbit, assume a constant
Earth rotation speed at a sidereal rate.
6. Implement the equations of motion for the Kepler orbit in matlab and verify the numerical
solution of r and v against the analytical formulas.
7. Demonstrate in matlab that the total energy is conserved for the Kepler problem. Your
starting point is an integrated trajectory.
24
26. Chapter 3
Potential theory
Potential fields appear in many forms in physics; in the case of solar system dynamics in planetary
sciences we consider usually potential functions related to the gravitational effect of a planet
or a star. But in physics you may also speak about magnetic or electric fields that are also
potential fields. A potential function describes the potential energy of an object at some point
in a gravitational field of another mass, which is usually the Sun or the Earth.1 Potential
energy of that object depends on the location of the object, but when we talk about the concept
”potential function” we refer to the normalized potential energy of the object in question without
consideration of its own mass. The gradient of the potential function is equal to the acceleration
vector predicted by Newton’s gravity law. Yet, in the case of Newton we would have to deal
with vectors, now we can use a scalar function which reduces the complexity of the problem.
We consider the problem where we are moving around in a gravitational force field.2 Potential
energy relates to the problem of being somewhere in a force field, whereby the field itself is caused
by the gravitational attraction of a mass source that is usually far larger than the object moving
around this source. The potential at the end of the path minus the potential at the beginning
of the path is equal to the number of Joules per kg that we need to put in the motion that takes
place in this gravitational force field. If you move away from the source mass you have to push
the object, so you spend energy. But instead, when you approach the source mass then all this
potential energy comes back again for free, and if you move along surfaces of equal potential
energy then no extra energy is required to move around. Force fields that possess this property
are said to be conservative force fields.
Mathematically speaking this means that the Laplacian of the potential V is zero, and thus
that 2V = 0. To explain why this is the case we go back to the Gauss integral theorem. The
theorem states that:
Ω
( , w) dσ =
Ω
(w, n) dσ (3.1)
Here Ω is the shape of an arbitrary body and Ω its surface. Furthermore n is an vector of
length 1 that is directed outwards on a surface element, while w is an arbitrary vector function.
If we take w as the gradient of the potential V , and if we stay outside all masses that generate
1
Potential is related to the Latin word potentia which was used to describe political influence, power of strength.
2
Gravitation is the effect caused by the mass of the Sun or a planet, gravity is the effect that you experience
on a rotating planet.
25
27. V then:
Ω
( , V ) dσ =
Ω
( V, n) dσ (3.2)
In a conservative force field the right hand side of this integral relation will vanish for any
arbitrary choice of Ω that does not overlap with the masses that generate V . If we take an
infinitesimal small volume Ω then the left hand side becomes:
2
V = ∆V =
∂2V
∂x2
+
∂2V
∂y2
+
∂2V
∂z2
= 0 (3.3)
This equation is known as the Laplace equation, potential functions V that fulfill the Laplace
equation are said to generate a conservative force field V . And within such a conservative
force field you can always loop around along closed curves without losing any energy. Non-
conservative force fields also exist, in this case the opposite would happen, namely that you lose
energy along a closed path.
In physics all electric, magnetic and gravitational field are conservative. Gravitation is
unique in the sense that it doesn’t interact with electric and magnetic fields. The latter two
fields do interact, the most general interaction between E and B is described by the Maxwell
equations that permit Electro-Magnetic waves. Gravitation does not permit waves, at least, not
in Newtonian physics. The theory of general relativity does allow for gravity waves, although
these waves have not yet been detected. Other effects caused by general relativity such as the
peri-helium precession of the planet Mercury or the gravitational bending of light have been
demonstrated. The concept ”gravity wave” is also used in non-relativistic physics, and for
instance in the solution of the Navier Stokes equations. In this case we call a surface wave in a
hydrodynamic model a gravity wave because gravity is the restoring force in the dynamics.
3.1 Solutions of the Laplace equation
A straightforward solution of V that fulfills the Laplace equation is the function V = −µ/r
where r is the radius of an arbitrary point in space relative to a source point mass. Later we
will show that this point mass potential function applies to the Kepler problem.
The minus sign in front of the gradient operator in equation 2.2 depends on the convention
used for the geopotential function V . If we start at the Earth’s surface the potential would
attain a value Va, and at some height above the surface it would be Vb. The difference between
Vb − Va should in this case be positive, because we had to spend a certain number of Joules per
kilogram to get from a to b, and this can only be the case is Vb is greater than Va. Once we
traveled from the Earth’s surface to infinity there is no more energy required to move around,
because we are outside the ’potentia’ of the Earth. Thus we must demand that V = 0 at infinity.
The V = −µ/r potential function is one of the many possible solutions of the Laplace
equation. We call it the point mass potential function. There are higher order moments of the
potential function. In this case we use series of spherical harmonics which are base functions
consisting of Legendre polynomials multiplied times goniometric functions. For the moment this
problem is deferred until we need to refine variations in the gravitational field that differ from
the central force field.
26
28. 3.2 Legendre Functions
Legendre functions appear when we solve the Laplace equation ( U = 0) by means of the
method of separation of variables. Normally the Laplace equation is transformed in spherical
coordinates r, λ, θ (r: radius, λ: longitude θ: co-latitude); this problem can be found in section
10.8 in [67] where the following solutions are shown:
U(r, λ, θ) = R(r)G(λ, θ) (3.4)
with:
R(r) = c1rn
+ c2
1
rn+1
(3.5)
and where c1 and c2 are integration constants. Solutions of G(λ, θ) appear when we apply
separation of variables. This results in so-called surface harmonics; in [67] one finds:
G(λ, θ) = [Anm cos(mλ) + Bnm cos(mλ)] Pnm(cos θ) (3.6)
where also Anm and Bnm are integration constants. The Pnm(cos θ) functions are called associ-
ated Legendre functions and the indices n and m are called degree and order. When m = 0 we
deal with zonal Legendre functions and for m = n we are dealing with sectorial Legendre func-
tions, all others are tesseral Legendre functions. The following table contains zonal Legendre
functions up to degree 5 whereby Pn(cos θ) = Pn0(cos θ):
P0(cos θ) = 1
P1(cos θ) = cos θ
P2(cos θ) =
3 cos 2θ + 1
4
P3(cos θ) =
5 cos 3θ + 3 cos θ
8
P4(cos θ) =
35 cos 4θ + 20 cos 2θ + 9
64
P5(cos θ) =
63 cos 5θ + 35 cos 3θ + 30 cos θ
128
Associated Legendre functions are obtained by differentiation of the zonal Legendre functions:
Pnm(t) = (1 − t2
)m/2 dmPn(t)
dtm
(3.7)
so that you obtain:
P11(cos θ) = sin θ
P21(cos θ) = 3 sin θ cos θ
P22(cos θ) = 3 sin2
θ
P31(cos θ) = sin θ
15
2
cos2
θ −
3
2
P32(cos θ) = 15 sin2
θ cos θ
P32(cos θ) = 15 sin3
θ
27
29. Legendre functions are orthogonal base functions in an L2 function space whereby the inner
product is defined as:
1
−1
Pn (x)Pn(x) dx = 0 n = n (3.8)
and
1
−1
Pn (x)Pn(x) dx =
2
2n + 1
n = n (3.9)
In fact, these integrals are definitions of an inner product of a function space whereby Pn(cos θ)
are the base functions. Due to orthogonality we can easily develop an arbitrary function f(x)
for x ∈ [−1, 1] into a so-called Legendre function series:
f(x) =
∞
n=0
fnPn(x) (3.10)
The question is to obtain the coefficients fn when f(x) is provided in the interval x ∈ [−1, 1].
To demonstrate this procedure we integrate on the right and left hand side of eq. 3.10 as follows:
1
−1
f(x)Pn (x) dx =
1
−1
∞
n=0
fnPn(x)Pn (x) dx (3.11)
Due to the orthogonality relation of Legendre functions the right hand side integral reduces to
an answer that only exists for n = n :
1
−1
f(x)Pn(x) dx =
2
2n + 1
fn (3.12)
so that:
fn =
2n + 1
2
1
−1
f(x)Pn(x) dx (3.13)
This formalism may be expanded in two dimensions where we now introduce spherical harmonic
functions:
Ynma(θ, λ) =
cos mλ
sin mλ
a=1
a=0
Pnm(cos θ) (3.14)
which relate to associated Legendre functions. In turn spherical harmonic functions possess
orthogonal relations which become visible when we integrate on the sphere, that is:
σ
Ynma(θ, λ)Yn m a (θ, λ) dσ =
4π(n + m)!
(2n + 1)(2 − δ0m)(n − m)!
(3.15)
but only when n = n and m = m and a = a . Spherical harmonic functions Ynma(θ, λ) are
the base of a function space whereby integral (3.15) defines the inner product. We remark
that spherical harmonic functions form an orthogonal set of basis functions since the answer of
integral (3.15) depends on degree n and the order m.
In a similar fashion spherical harmonic functions allow to develop an arbitrary function over
the sphere in a spherical harmonic function series. Let this arbirary function be called f(θ, λ)
and set as goal to find the coefficients Cnma in the series:
f(θ, λ) =
∞
n=0
n
m=0
1
a=0
CnmaYnma(θ, λ) (3.16)
28
30. This problem can be treated in the same way as for the zonal Legendre function problem, in
fact, it is a general approach that may be taken for the subset of functions that can be developed
in a series of orthogonal (or orthonomal) base functions. Thus:
σ
Yn m a (θ, λ)f(θ, λ) dσ =
σ
Yn m a (θ, λ)
∞
n=0
n
m=0
1
a=0
CnmaYnma(θ, λ) dσ (3.17)
which is only relevant when n = n and m = m and a = a . So that:
Cnma = N−1
nm
σ
Ynma(θ, λ)f(θ, λ) dσ (3.18)
where
Nnm =
4π(n + m)!
(2n + 1)(2 − δ0m)(n − m)!
(3.19)
3.3 Normalization
Normalization of Legendre functions is a separate issue that follows from the fact that we are
dealing with an orthogonal set of functions. There are several ways to normalize Legendre
functions, one choice is to rewrite integral (3.15) into a normalized integral:
1
4π σ
Y nma(θ, λ)Y n m a (θ, λ) dσ = 1 (3.20)
where we simply defined new normalized functions with an overbar which are now called the
normalized spherical harmonic functions. It is obvious that they rely on normalized associated
Legendre functions:
Pnm(cos θ) = (2n + 1)(2 − δ0m)
(n − m)!
(n + m)!
1/2
Pnm(cos θ) (3.21)
The use of normalized associated Legendre functions results now in an orthonormal set of spher-
ical harmonic base functions as can be seen from the new definition of the inner product in
eq. (3.20). It is customary to use the normalized functions because of various reasons, a very
important numerical reason is that stable recursive schemes for normalized associated Legendre
functions exist whereas this is not necessarily the case for the unnormalized Legendre functions.
This problem is beyond the scope of these lecture notes, the reader must assume that there is
software to compute normalized associated Legendre functions up to high degree and order.
3.4 Properties of Legendre functions
3.4.1 Property 1
A well-known property that we often use in potential theory is the development of the function
1/r in a series of zonal Legendre functions. We need to be a bit more specific on this problem.
Assume that there are two vectors p and q and that their length is rp and rq respectively. If the
length of the vector p − q is called rpq then:
rpq = r2
p + r2
q − 2rprq cos ψ
1/2
(3.22)
29
31. for which it is known that:
1
rpq
=
1
rq
∞
n=0
rp
rq
n
Pn(cos ψ) (3.23)
where ψ is the angle between p and q. This series is convergent when rp < rq. The proof for
this property is given in [52] and starts with a Taylor expansion of the test function:
rpq = rp 1 − 2su + s2 1/2
(3.24)
where s = rq/rp and u = cos ψ. The binomial theorem, valid for |z| < 1 dictates that:
(1 − z)−1/2
= α0 + α1z + α2z2
+ ... (3.25)
where α0 = 1 and αn = (1.3.5...(2n − 1))/(2.4...(2n)). Hence if |2su − s2| < 1 then:
(1 − 2su + s2
)−1/2
= α0 + α1(2su − s2
) + α2(2su − s2
)2
+ ... (3.26)
so that:
(1 − 2su + s2
)−1/2
= 1 + us +
3
2
(u2
−
1
3
)s2
+ ...
= P0(u) + sP1(u) + s2
P2(u) + ...
which completes the proof.
3.4.2 Property 2
The addition theorem for Legendre functions is:
Pn(cos ψ) =
1
2n + 1 ma
Y nma(θp, λp)Y nma(θq, λq) (3.27)
where λp and θp are the spherical coordinates of vector p and λq and θq the spherical coordinates
of vector q.
3.4.3 Property 3
The following recursive relations exist for zonal and associated Legendre functions:
Pn(t) = −
n − 1
n
Pn−2(t) +
2n − 1
n
tPn−1(t) (3.28)
Pnn(cos θ) = (2n − 1) sin θPn−1,n−1(cos θ) (3.29)
Pn,n−1(cos θ) = (2n − 1) cos θPn−1,n−1(cos θ) (3.30)
Pnm(cos θ) =
(2n − 1)
n − m
cos θPn−1,m(cos θ) −
(n + m − 1)
n − m
Pn−2,m(cos θ) (3.31)
Pn,m(cos θ) = 0 for m > n (3.32)
For differentiation the following recursive relations exist:
(t2
− 1)
dPn(t)
dt
= n (tPn(t) − Pn−1(t)) (3.33)
30
32. 3.5 Convolution integrals on the sphere
Spherical harmonic function expansions are very convenient for the evaluation of the following
type of convolution integrals on the sphere:
H(θ, λ) =
Ω
F(θ , λ )G(ψ) d Ω (3.34)
where dΩ = sin ψ dψ dα and ψ the spherical distance between θ, λ and θ , λ and α the azimuth.
Functions F and G are written as:
F(θ, λ) =
∞
n=0
n
m=0
1
a=0
FnmaY nma(θ, λ) (3.35)
where
Y nm,0(θ, λ) = cos(mλ)Pnm(cos θ)
Y nm,1(θ, λ) = sin(mλ)Pnm(cos θ)
and
G(ψ) =
∞
n=0
GnPn(cos ψ) (3.36)
which takes the shape of a so-called Green’s function3. It turns out that instead of numerically
computing the expensive surface integral in eq. (3.34) that it is easier to multiply the Gn and
Fnma coefficients:
H(θ, λ) =
∞
n=0
n
m=0
1
a=0
HnmaY nma(θ, λ) (3.37)
where
Hnma =
4πGn
2n + 1
Fnma (3.38)
For completeness we also demonstrate the validity of eq. (3.38). The addition theorem of Leg-
endre functions states that:
Pn(cos ψpq) =
1
2n + 1
n
m=0
Pnm(cos θp)Pnm(cos θq) cos(m(λp − λq)) (3.39)
which is equal to
Pn(cos ψpq) =
1
2n + 1
n
m=0
1
a=0
Y nm(θp, λp)Y nm(θq, λq) (3.40)
When this property is substituted in eq. (3.34) then:
H(θ, λ) =
Ω nma
FnmaY nma(θ , λ )
n m a
Gn
2n + 1
Y n m a (θ, λ)Y n m a (θ , λ ) dΩ (3.41)
3
George Green (1793-1841)
31
33. which is equal to:
H(θ, λ) =
n m a
Gn
2n + 1
Y n m a (θ, λ)
nma
Fnma
Ω
Y nma(θ , λ )Y n m a (θ , λ ) dΩ (3.42)
Due to orthogonality properties of normalized associated Legendre functions we get the desired
relation:
H(θ, λ) =
nma
4πGn
2n + 1
FnmaY nma(θ, λ) (3.43)
which completes our proof.
3.6 Exercises
1. Show that U = 1
r is a solution of the Laplace equation ∆U = 0
2. Show that the gravity potential of a solid sphere is the same as that of a hollow sphere
and a point mass
3. Demonstrate in matlab that eq. (3.23) rapidly converges when rq = f × rp where f > 1.1
for randomly chosen values of ψ and rp
4. Demonstrate in matlab that eqns. (3.14) are orthogonal over the sphere
5. Develop a method in matlab to express the Green’s function f(x) =
1 ∀ x ∈ [0, 1]
0
as a series of Legendre functions f(x) = n anPn(x).
32
34. Chapter 4
Fourier frequency analysis
Jean-Baptiste Joseph Fourier (1768–1830) was a French scientist who introduced a method
of frequency analysis where one could approximate an arbitrary function by a series of sine
and cosine expressions. He did not show that the series would always converge, the German
mathematician Dirichlet (1805-1859) later showed that there are certain restrictions of Fourier’s
method, in reality these restrictions are usually not hindering the application of Fourier’s method
in science and technology. Fourier’s frequency analysis method assumes that we analyze a
function on a defined interval, Fourier made the crucial assumption that the function repeats
itself when we take the function beyond the nominal interval. For this reason we say that the
function to analyze with Fourier’s method is periodic.
In the sequel we consider a signal v(t) that is defined in the time domain [0, T] where T is the
length in seconds, periodicity implies that v(t + kT) = v(t) where k is an arbitrary integer. For
k = 1 we see that the function v(t) simply repeats because v(t) = v(t + T), we see the same on
the preceding interval because v(t) = v(t − T). Naturally one would imagine a one-dimensional
wave phenomenon like what we see in rivers, in the atmosphere, in electronic circuits, in tides,
and when light or radio waves propagate. This is what Fourier’s method is often used for, the
frequency analysis reveals how processes repeat themselves in time, but also in place or maybe
along a different projection of variables. This information is crucial for understanding a physical
or man-made signal hidden in often noisy observations.
This chapter is not meant to replace a complete course on Fourier transforms and Signal
Processing, but instead we present a brief summary of the main elements relevant for our lectures.
If you have never dealt with Fourier’s method then study both sections in this chaper, and test
your own knowledge by making a number of assignments at the end of this chapter. In case you
already attended lectures on the topic then keep this chapter as a reference. In the following
two sections we will deal with two cases, namely the continuous case where v(t) is an analytical
function on the interval [0, T] and a discrete case where we have a number of samples of the
function v(t) within the interval [0, T]. Fourier’s original method should be applied to the
continuous method, for data analysis we are more inclined to apply the discrete Fourier method.
4.1 Continuous Fourier Transform
Let v(t) be defined on the interval t ∈ [0, T] where we demand that v(t) has a finite number of
oscillations and where v(t) is continuous on the interval. Fourier proposed to develop v(t) in a
33
35. series:
v(t) =
N/2
i=0
Ai cos ωit + Bi sin ωit (4.1)
where Ai and Bi denote the Euler coefficients in the series and where variable ωi is an angular
rate that follows from ωi = i∆ω where ∆ω = 2π
T . At this point one should notice that:
• The frequency associated with 1
T is 1 Hertz (Hz) when T is equal to 1 second. A record
length of T = 1000 seconds will therefore yield a frequency resolution of 1 milliHertz
because of the definition of equation (4.1).
• Fourier’s method may also be applied in for instance orbital dynamics where T is rescaled
to the orbital period, in this case we speak of frequencies in terms of orbital periods, and
hence the definition cycles per revolution or cpr. But other definitions of frequency are
also possible, for instance, cycles per day (cpd) or cycles per century (cpc).
• When v(t) is continuous there are an infinite number of frequencies in the Fourier series.
However, all Euler coefficients that you find occur at multiples of the base frequency 1/T.
• A consequence of the previous property is that the spectral resolution is only determined
by the record length during the analysis, the frequency resolution ∆f is by definition 1/T.
The frequency resolution ∆f should not be confused with sampling of the function v(t) on
t ∈ [0, T]. Sampling is a different topic that we will deal in section 4.2 where the discrete
Fourier transform is introduced.
In order to calculate Ai and Bi in eq. (4.1) we exploit the so-called orthogonality properties of
sine and cosine functions. The orthogonality properties are defined on the interval [0, 2π], later
on we will map the interval [0, T] to the new interval [0, 2π] which will be used from now on.
The transformation from [0, T] or even [t0, t0 + T] to [0, 2π] is not relevant for the method at
this point, but is will become important if we try to assign physical units to the outcome of the
result of the Fourier transform. This is a separate topic that we will discuss in section 4.4. The
problem is now to calculate Ai and Bi in eq. (4.1) for which we will make use of orthogonality
properties of sine and cosine expression. A first orthogonality property is:
2π
0
sin(mx) cos(nx) dx = 0 (4.2)
This relation is always true regardless of the value of n and m which are both integer whereas
x is real. The second orthogonality property is:
2π
0
cos(mx) cos(nx) dx =
0 : m = n
π : m = n > 0
2π : m = n = 0
(4.3)
and the third orthogonality property is:
2π
0
sin(mx) sin(nx) dx =
π : m = n > 0
0 : m = n, m = n = 0
(4.4)
34
36. The next step is to combine the three orthogonality properties with the Fourier series definition
in eq. (4.1). We do this by evaluating the integrals:
2π
0
v(x)
cos(mx)
sin(mx)
dx (4.5)
where we insert v(t) but now expanded as a Fourier series:
2π
0
N/2
n=0
An cos(nx) + Bn sin(nx)
cos(mx)
sin(mx)
dx (4.6)
You can reverse the summation and the integral, the result is that many terms within this
integral disappear because of the orthogonality relations. The terms that remain result in the
following expressions:
A0 =
1
2π
2π
0
v(x) dx, B0 = 0 (4.7)
An =
1
π
2π
0
v(x) cos(nx) dx, n > 0 (4.8)
Bn =
1
π
2π
0
v(x) sin(nx) dx, n > 0 (4.9)
The essence of Fourier’s frequency analysis method can now be summarized:
• The ’conversion’ of time domain to frequency domain goes via three integrals where we
compute An and Bn that appear in eq. (4.1). This conversion or transformation step is
called the Fourier transformation and it is only possible when v(x) exists on the interval
[0, 2π]. Fourier series exist when there are a finite number of oscillations between [0, 2π],
this means that a function like sin(1/x) could not be expanded. A second condition
imposed by Dirichlet is that there are a finite number of discontinuities. The reality in
most data analysis problems is that we hardly ever encounter the situation where the
Dirichlet conditions are not met.
• When we speak about a ’spectrum’ we speak about the existence of the Euler coefficients
An and Bn. Euler coefficients are often taken together in a complex number Zn = An+jBn
where j =
√
−1. We prefer the use of j to avoid any possible confusing with electric
currents.
• There is a subtle difference between the discrete Fourier transform and the continuous
transform discussed in this section. The discrete Fourier transform introduces a new
problem, namely that or the definition of sampling, it is discussed in section 4.2.
The famous theorem of Dirichlet reads according to [67]: ”If v(x) is a bounded and periodic
function which in any one period has at most a finite number of local maxima and minima and
a finite number of point of discontinuity, then the Fourier series of v(x) converges to v(x) at all
points where v(x) is continuous and converges to the average of the right- and left-hand limits
of v(x) at each point where v(x) is discontinuous.”
35
37. If the Dirichlet conditions are met then we are able to define integrals that relate f(t) in the
time domain and g(ω) in the frequency domain:
f(t) =
∞
−∞
g(ω)ejωt
dω (4.10)
g(ω) =
1
2π
∞
−∞
f(τ)e−jωτ
dτ (4.11)
In both cases we deal with complex functions where at each spectral line two Euler coefficients
from the in-phase term An and the quadrature term Bn. The in-phase nomenclature originates
from the fact that you obtain the coefficient by integration with a cosine function which has a
phase of zero on an interval [0, 2π] whereas a sine function has a phase of 90◦. The amplitude
of each spectral line is obtained as the length of Zn = An + jBn, thus |Zn| whereas the phase
is the argument of the complex number when it is converted to a polar notation. The phase
definition only exists because it is taken relative to the start of the data analysis window, this
also means that the phase will change if we shift that window in time. It is up to the reader to
show how the resulting Euler coefficients are affected.
4.2 Discrete Fourier Transform
The continuous case introduced the theoretical foundation for what you normally deal with as
a scientist or engineer who collected a number of samples of the function v(tk) where tk =
t0 + (k − 1)δt with k ∈ [0, N − 1] and δt > 0. The sampling interval is now called δt. The length
of the data record is thus T = k.δt, the first sample of v(t0) will start at the beginning of the
interval, and the last sample of the interval is at T − δt because v(t0 + T) = v(t0).
When the first computers became available in the 60’s equations (4.7), (4.8) and (4.9) where
coded as shown. Equation (4.7) asks to compute a bias term in the series, this is not a lot
of work, but equations (4.8) and (4.9) ask to compute products of sines and cosines times the
input function v(tk) sampled on the interval [t0, t0 + (N − 1)δt]. This is a lot of work because
the amount of effort is like 2N multiplications for both integrals times the number of integrals
that we can expect, which is the number the frequencies that can be extracted from the record
[t0, t0 + (N − 1)δt]. Due to the Nyquist theorem the number of frequencies is N/2, and for each
integral there are N multiplications: the effort is of the order of N2 operations.
4.2.1 Fast Fourier Transform
There are efficient computer programs (algorithms) that compute the Euler coefficients in less
time than the first versions of the Fourier analysis programs. Cooley and Tukey developed in
1966 a faster method to compute the Euler coefficients, they claim that the number of operations
is proportional to O(N log N). Their algorithm is called the fast Fourier transform, or the FFT,
the first implementation required an input vector that had 2k elements, later versions allowed
other lengths of the input vector where the largest prime factor should not exceed a defined
limit. The FFT routine is available in many programming languages (or environments) such as
MATLAB. The FFT function assumes that we provide it a time vector on the input, on return
you get a vector with Euler coefficients obtained after the transformation which are stored as
complex numbers. The inverse routine works the other way around, it is called iFFT which
36
38. stands for the inverse fast Fourier transform. The implementation of the discrete transforms in
MATLAB follows the same definition that you find in many textbooks, for FFT it is:
Vk =
N−1
n=0
vn e−2πjkn/N
with k ∈ N and vn ∈ C and Vk ∈ C (4.12)
and for the iFFT it is:
vn =
1
N
N−1
k=0
Vk e2πjkn/N
with n ∈ N and vn ∈ C and Vk ∈ C (4.13)
where vn is in the time domain while Vk is in the frequency domain, furthermore Euler’s formula
is used: ejx = cos x + j sin x. Because of this implementation in MATLAB a conversion is
necessary between the output of the FFT stored in Vk to the Euler coefficients that we defined
in equations (4.1) (4.7) (4.8) and (4.9), this topic is worked out in sections 4.3.1 and 4.3.2 where
we investigate test functions.
4.2.2 Nyquist theorem
The Nyquist theorem (named after Harry Nyquist, 1889-1976, not to be confused with the
Shannon-Nyquist theorem) says that the number of frequencies that we can expect in a discretely
sampled record [t0, t0 + (N − 1)δt] is never greater than N/2. Any attempt to compute integrals
(4.8) and (4.9) beyond the Nyquist frequency will result in a phenomenon that we call aliasing
or faltung (in German). In general, when the sampling rate 1/δt is too low you will get an
aliased result as is illustrated in figure 4.1. Suppose that your input signal contains power
beyond the Nyquist frequency as a result of undersampling, the result is that this contribution
in the spectrum will fold back into the part of the spectrum that is below the Nyquist frequency.
Figure 4.2 shows how a spectrum is distorted because the input signal is undersampled. Due
to the Nyquist theorem there are no more than N/2 Euler coefficient pairs (Ai, Bi) that belong
to a unique frequency ωi, see also eq. (4.1). The highest frequency is therefore N/2 times the
base frequency 1/T for a record that contains N samples. If we take a N that is too small then
the consequence may be that we undersample the signal, because the real spectrum of the
signal may contain ”power” above the cutoff frequency N
2T imposed by the way we sampled the
signal. Undersampling results in aliasing so that the computed spectrum will appear distorted.
Oversampling is never a problem, this is only helpful to avoid that aliasing will occur, however,
sometimes oversampling is simply not an option. In electronics we can usually oversample, but
in geophysics etc we can not always choose the sampling rate the way we would like it. Frequency
resolution is determined by the record length, short records have a poor frequency resolution,
longer records often don’t.
4.2.3 Convolution
To convolve is not a verb you would easily use in daily English, according to the dictionary
it means ”to roll or coil together; entwine”. When you google for convolved ropes you get to
see what you find in a harbor, stacks of rope rolled up in a fancy manner. In mathematics
convolution refers multiplication of two periodic functions where we allow one function to shift
37
39. Figure 4.1: Demonstration of the aliasing, suppose that the true signal was in blue, but that
we sample the blue signal at the red circles, any Fourier analysis procedure will now think that
the signal is recovered as the green function. Of course the green function is not the real signal,
instead we say that it is the aliased function. Remedies are, collect samples of v(tk) at a higher
rate or, as is done in the case of tides, assume that you know the frequency of the blue function
so that the amplitude and phase of the green function can be used to recover the blue function.
38
40. Figure 4.2: Demonstration of the aliasing. The true spectrum of your signal is in red, the graph
displays the power at each frequency computed as Pi = (A2
i + B2
i ). The Nyquist frequency
is defined by the sampling rate of the input signal. Since aliasing results in folding the red
spectrum is folded back across the black dashed line which coincides with the Nyquist frequency.
The part that aliases back is the blue dashed graph left of the dashed black line, it adds to the
true spectrum which was red, so that the result will be the blue spectrum which is said to be
affected by aliasing. To summarize the situation, red is the real signal, but blue is what we
recover because our sampling rate was too low.
39
41. along another during the operation:
h(t) =
∞
−∞
f(τ)g(t − τ)d τ (4.14)
we also say the h(t) is the result of the convolution of f(t) and g(t), the function f(t) would be
for instance a signal and g(t) could be a filter, so that h(t) is the filtered version of the signal.
The problem with direct convolution in the time domain is that the process is very slow, but
fortunately we can make use of one of the properties of the Fourier transform that greatly speeds
up the evaluation of the convolution integral.
F(ν) = FFT(f(t))
G(ν) = FFT(g(t))
H(ν) = F(ν) ∗ G(ν)
h(t) = iFFT(H(ν))
where ν is frequency and t time. Convolution is used to build, design and analyze filters in
digital communication, in physics convolution is often the result of a physical property between
two quantities. Since the advent of the FFT transform it has become possible to quickly carry
out convolutions with the help of a computer. In this sense FFT is the enabling technology for
digital signal processing.
4.2.4 Effect of a data selection window
During an analysis of a finite length data record we always deal somehow with the problem
convolution. Reason is that the length of the record itself acts like a box window that we impose
on a perhaps much longer data record. It was the choice of an observer to select a certain part
of the much longer record, and as such we could also affect the spectrum that we compute by
the choice of our window. So the spectrum that we get to see will be affected by convolution
of the box window being our selected data window. There are several ways one can handle this
problem:
• Pre-process the data by removing a long term trend function from the input signal using a
least squares regression technique, see section 8.5 for a discussion. Geophysical data may
for instance show a slow drift or it may be irregularly spaced and if we would analyze a
record without taking care of that drift (or bias) term then just the presence of a drift term
would add the spectrum of the sawtooth function, for a discussion see section 4.3.2 where
we compute its spectrum in MATLAB. This is not what we want to see, so we first remove
the trend function from the data to retain a difference signal that we subject to the FFT
method. Interpolation and approximation may be a part of the story, these methods help
to get the data presented to the FFT method in such a shape that it becomes regularly
spaced and detrended, for a discussion see chapter 9.
• The signal spectrum may be such that there is a fair amount of red signal. A spectrum is
said to be red if it contains, in analogy with optics, a significant amount of energy at the
lower frequencies compared to the rest of the spectrum. When you select a data record
then it may be such that the record itself is not a multiple of the length contained in the
red part of the spectrum. This leads to distortion of the red peaks in the spectrum, instead
40
42. Figure 4.3: Convolution: The signal in the top figure is convolved with the signal in the middle
figure and the result is presented in the bottom figure. Imagine that you slide the narrow block
function in the middle figure along the upper block function that you hold in place. At each
step you carry out the multiplication and the summation, and the result is stored in the lower
graph, this is done for all possible settings of the shift that we apply. When the middle block is
entirely positioned under the top block a value of one is plotted in the bottom graph, when the
middle block is not under the top block a zero is found, and when the middle block is partially
under the top block a partial result is found. Since the overlapping area is linearly increasing
with the applied shift we get to see linear flanks on the convolved function. Please notice that
we applied a scaling factor to redefine the maximum of the h(t) function as 1, in reality it isn’t.
41
43. of being a sharp peak the content of those peaks may smear to neighboring frequencies.
This is what we call spectral leakage. A possible remedy is to apply a window or tapering
function to the input data prior to computing the spectrum.
The choice of a taper function is a rather specific topic, tapering means that we multiply a
weighting function wn times the input data vn which results in vn that we subject (instead of
vn) to the FFT method:
vn = wn.vn where n ∈ [0, N − 1] and {wn, vn, vn} ∈ R and {n, N} ∈ N (4.15)
The result will be that the FFT(v ) will improve in quality compared to the FFT(v), one aspect
that would be improved is spectral leakage. There are various window functions, the best known
general purpose taper is the Hamming function where:
wn = 0.54 − 0.46 cos(2πn/N), 0 ≤ n ≤ N (4.16)
MATLAB’s signal processing toolbox offers a variety of tapering functions, the topic is too
detailed to discuss here.
4.2.5 Parseval theorem
In section 4.2.3 we demonstrated that multiplication of Euler coefficients of two functions in
the frequency domain is equal to convolution in the time domain. Apply now convolution of
a function with itself at zero shift and you arrive at Parseval’s identity, after (Marc-Antoine
Parseval 1755-1836) which says that the sum of the squares in the time domain is equal to
the sum of the squares in the frequency domain after we applied Fourier’s transformation to a
record in the time domain, see section 4.2.5. The theorem is relevant in physics, it says that
the amount of energy stored in the time domain can never be different from the energy in the
frequency domain:
ν
F2
(ν) =
i
f2
(t) (4.17)
where F is the Fourier transform of f.
4.3 Demonstration in MATLAB
4.3.1 FFT of a test function
In MATLAB we work with vectors and the set-up is such that one can easily perform matrix
vector type of operations, the FFT and the iFFT operator are implemented as such, they are
called fft() and ifft(). With FFT(f(x)) it does not precisely matter how the time in x is defined,
the easiest assumption is that there is a vector f in MATLAB and that we turn it into a vector g
via the FFT, the command would be g = fft(f) where f is evaluated at x that appear regularly
spaced in the domain [0, 2π], thus x = 0 : 2π/N : 2π − 2π/N in MATLAB. Before you blindly
rely on a FFT routine in a function library it is a good practice to subject it to a number of
tests. In this case we consider a test function of which the Euler coefficients are known:
f(x) = 7 + 2 sin(3x) + 4 cos(12x) − 5 sin(13x); with x ∈ [0, 2π] (4.18)
42
44. A Fourier transform of f should return to us the coefficients 7 at the zero frequency, 2 at the 3rd
harmonic, +4 at the 12th harmonic and -5 at the 13th harmonic. The term harmonic comes from
communications technology and its definition may differ by textbook, we say that the lowest
possible frequency at 1/T that corresponds to the record length T equals to the first harmonic,
at two times that frequency we have the second harmonic, and so on. I wrote the following
program in MATLAB to demonstrate the problem:
clear;
format short
dx = 2*pi/1000; x = 0:dx:2*pi-dx;
f = 2*sin(3*x) + 5 + 4*cos(12*x) - 5*sin(13*x);
g = fft(f);
idx = find(abs(g)>1e-10);
n = size(idx,2);
K = 1/size(x,2);
for i=1:n,
KK = K;
if (idx(i) > 1),
KK = 2*K;
end
A = KK*real(g(idx(i)));
B = KK*imag(g(idx(i)));
fprintf(’%4d %12.4f %12.4fn’,[idx(i) A B]);
end
The output that was produced by this program is:
1 5.0000 0.0000
4 0.0000 -2.0000
13 4.0000 0.0000
14 -0.0000 5.0000
988 -0.0000 -5.0000
989 4.0000 -0.0000
998 0.0000 2.0000
So what is going on? On line 3 we define the sampling time dx in radians and also the time
x is specified in radians. Notice that we stop prior to 2π at 2π − dx because of the periodic
assumption of the Fourier transform. On line 4 we define the test function, and on line 5 we
carry out the FFT. The output is in vector g and when you would inspect it you would see that
it contains complex numbers to store the Euler coefficients after the transformation. Also, the
numbering in the vector in MATLAB does matter in this discussion. At line 6 the indices in
the g vector are retrieved where the amplitude of the spectral line (defined as (A2
i + B2
i )1/2)
exceeds a threshold. The FFT function is not per se exact, the relative error of the Euler terms
is typically around 15 significant digits which is because of the finite bit length of a variable in
MATLAB. If you find an error typically greater than approximately 10 significant digits then
inspect whether x is correctly defined. Remember that we are dealing with a periodic function f
and that the first entry in f (in MATLAB this is at location f(1)) repeats at 2π. The last entry
in the f vector should therefore not be equal to the first value. This mistake is often made, and
43
45. it leads to errors that are significantly larger than the earlier mentioned 10 significant digits.
On line 7 the number of significant Euler pairs in the g vector are recovered, and on line 8 we
compute a scaling factor which is essential to understand what is stored in the g vector. The
part that decodes the g vector starts on line 9, integer i runs from 1 to n (the number of unique
pairs in g) and the scale factor is, depending on where we are in the g vector, adjusted on lines
10 to 13. The Euler terms for each spectral line are then recovered on lines 14 and 15 and the
result is printed on line 14. Line 15 terminates the for loop.
We learn from this program that vectors in MATLAB start at index 1, and not at zero as
they do in other programming languages. The value at g(1) = k × A0 where k = 1
N with N
denoting the number of samples on the input record f (and the definition of time in x). At the
4th index in g we find the complex number 0−2j = (0, −2) where j =
√
−1, the sine term at the
third harmonic is therefore stored as (0, −2kB), at location 13 in g we see that the cosine term
is properly put at the 12th harmonic, it is stored as (2kA,0), location 14 in g confirms again
that the sine term at the 13th harmonic is stored as (0, −2kB). Next the g vector is completely
empty until we reach the end where we find the Euler coefficients stored in a reversed order
where the last term g(N) contains k(A1, B1), it is preceded by g(N − 1) = k(A2, B2) and so on.
To summarize the content of g after we executed g = fft(f) in MATLAB:
• First define a scaling term k = 1
N for the zero frequency and k = 2
N for all other frequencies.
• The first location in the g vector contains the bias term: g(1) = k(A0, 0)
• g(i) for i > 1 and i < N/2 contains g(i) = k(Ai−1, −Bi−1)
• g(N − i + 1) for i ≥ 1 and i < N/2 contains g(N − i + 1) = k(Ai, Bi)
For this reason we say that the g vector is mirror symmetric about index N/2, and that the first
part of the vector contains the complex conjugate of the Euler coefficient pair A + jB = (A, B)
where j =
√
−1. Furthermore the scaling term k should be applied. It also leaves one to wonder
what is going on at index N/2. In factor the sine term at that frequency evaluates as zero by
definition, so it does not exist.
4.3.2 Harmonics of a sawtooth function
The sawtooth function in figure 4.4 has a Fourier transform, and the question is asked, how
many harmonics to you need to approximate the function to 95% of its total power. You can do
this analytically with the help of the earlier integral definitions, but it is relatively easy to do
in MATLAB which is what we discuss hereafter. The function is shown in figure 4.4. In order
to solve this problem you need to do two things, first, compute the FFT of the input function,
next, check with the help of the Parseval theorem how much power is contained in the spectrum.
From the 0 (or DC or bias) frequency upward we will continue to look for the point where the
power contained in the lower part of the spectrum exceeds the 95% threshold which was asked
in the assignment. The result that I found is in figure 4.5. The conclusion is therefore that you
need at least 12 harmonics to reach 95% of the power contained in the input function. Let’s go
over the MATLAB source to see how it is computed.
T = 1000; N=100; dt=T/N;
t = 0:dt:(T-dt);
44
46. Figure 4.4: The sawtooth function, also called the sweep generator function. Horizontal index
is time, vertical signal is output.
Figure 4.5: Remaining power contained of the sweep generator at a harmonic, it is expressed as
a percentage.
45
