Generic Conical Orbits, Keplers Laws, Satellite Orbits and Orbital Mechanics

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Title: Generic Conical Orbits, Keplers Laws, Satellite Orbits and Orbital Mechanics

1
Generic Conical Orbits Keplers LawsSatellite
Orbits and Orbital Mechanics

Guido Cervone

2
Summary

Class Website
Continuation of previous lecture
Generic Conical Orbits Keplers LawsSatellite
Orbits and Orbital Mechanics
Video - Satellite Launch

3
Topics

The Remote sensing data we are treating come from
spaceborne satellites
In this lecture we will discuss the concepts of
orbiting satellites
We will discuss
Generic Conical Orbits
Keplerian Laws
Celestial Mechanics
Satellite Orbits
Mathematical Formalization

4
Approach

Mathematics is necessary to fully understand the
problem but can be complicated because of 3D
We will approach the problem in two ways
Conceptual. Without any math or formulas just
exploring the concepts
Mathematical. Formulas and equations

5
Earths Gravitational Pull

The Earths gravity pulls everything toward the
Earth. In order to orbit the Earth the velocity
of a body must be great enough to overcome the
downward force of gravity
One important fact to remember is that orbits
within the Earths atmosphere do not really
exist. Atmospheric friction caused by the
molecules of air (causing a frictional heating
effect) will slow any object that could try to
attain orbital velocity within the atmosphere.
In space with virtually no atmosphere to cause
friction satellites can travel at velocities
strong enough to counteract the downward pull of
Earths gravity
The satellite is said to orbit around the Earth

6
How do Satellites Orbit
7
Orbits

Orbit refers to the path of a smaller object
(secondary) around a bigger object (primary) as a
result of the combined effects of inertia and
gravity.

8
Conic Orbits

The orbit can be in the shape of one of four
conic sections
Circle Ellipse Parabola Hyperbola
A conic section is the shape formed on a plane
passing through a right circular cone.

9
Conic Orbits II
10
Conic Orbits III

Most satellite and planetary orbits are elliptical

11
Review of Ellipses

For an ellipse there are two points called foci
(singular focus) such that the sum of the
distances to the foci from any point on the
ellipse is a constant.
a b constant
The long axis of the ellipse is called the major
axis while the short axis is called the minor
axis.
Half of the major axis is termed a semi-major
axis.
The length of a semi-major axis is often termed
the size of the ellipse.
It can be shown that the average separation of a
secondary from the primary as it goes around its
elliptical orbit is equal to the length of the
semi-major axis.
Thus by the radius of an orbit one usually
means the length of the semi-major axis.

12
Eccentricity

The amount of flattening of the ellipse is
termed the eccentricity.
A circle may be viewed as a special case of an
ellipse with zero eccentricity while as the
ellipse becomes more flattened the eccentricity
approaches one.
Thus all ellipses have eccentricities lying
between zero and one.
The range for the eccentricities of the different
types of orbits follows circular e 0
elliptical 0 gt e lt 1 parabolic e 1 hyperbolic
e gt 1.
The eccentricity for ellipses is the ratio of
distance between the two foci and the length of
the major axis.

13
Keplers laws

In the early 1600s Johannes Kepler proposed
three laws of planetary motion
Kepler was able to summarize the carefully
collected data of his mentor - Tycho Brahe - with
three statements which described the motion of
planets in a sun-centered solar system
The laws are still considered an accurate
description of the motion of any planet and any
satellite

14
Keplers First Law

Keplers First Law
The orbits of the planets are ellipses with the
Sun at one focus of the ellipse. (generally there
is nothing at the other focus of the ellipse).
The planet then follows the ellipse in its orbit
which means that the Earth-Sun distance is
constantly changing as the planet goes around its
orbit.

15
Keplers Second Law

The line joining the planet to the Sun sweeps out
equal areas in equal times as the planet travels
around the ellipse.
Thus a planet executes elliptical motion with
constantly changing angular speed as it moves
about its orbit.
The point of nearest approach of the planet to
the Sun is termed perihelion. The point of
greatest separation is termed aphelion.
Hence by Keplers second law the planet moves
fastest when it is near perihelion and slowest
when it is near aphelion.

16
Keplers Third Law

The ratio of the squares of the revolutionary
periods for two planets is equal to the ratio of
the cubes of their semi-major axes.
In this equation P represents the period of
revolution for a planet and R represents the
length of its semi-major axis. The subscripts 1
and 2 distinguish quantities for planet 1 and 2
respectively.
Keplers Third Law implies that the period for a
planet to orbit the Sun increases rapidly with
the radius of its orbit. Thus we find that
Mercury the innermost planet takes only 88 days
to orbit the Sun but the outermost planet (Pluto)
requires 248 years to do the same.

17
Calculations of Keplers Third Law

A convenient unit of measurement for periods is
in Earth years and a convenient unit of
measurement for distances is the average
separation of the Earth from the Sun which is
termed an astronomical unit and is abbreviated as
AU. If these units are used in Keplers 3rd Law
the denominators in the preceding equation are
numerically equal to unity and it may be written
in the simple form
P (years)2 R (AUs)3
This equation may then be solved for the period P
of the planet given the length of the semi-major
axis
P (years) R (AU)3/2
or for the length of the semi-major axis given
the period of the planet
R (AU) P (Years) 2/3

18
Calculations of Keplers Third Law

Lets calculate the radius of the orbit of Mars
(that is the length of the semi-major axis of
the orbit) from the orbital period.
The time for Mars to orbit the Sun is observed to
be 1.88 Earth years.
R P 2/3 (1.88) 2/3 1.52 AU
As a second example let us calculate the orbital
period for Pluto given that its observed average
separation from the Sun is 39.44 astronomical
units. From Keplers 3rd Law
P R3/2 (39.44)3/2 248 Years

19
How does it Relate to Satellites

We will now address the following questions
Does all this apply to satellites orbiting the
Earth
How can we use the Keplerian equations to find
the position of a satellite
How do satellites orbit around the Earth
How do we send a satellite in orbit

20
Orbital Mechanics

Orbital mechanics is the study of the motions of
artificial satellites and space vehicles moving
under the influence of forces such as gravity
atmospheric drag thrust etc.
Orbital mechanics is a modern offshoot of
celestial mechanics which is the study of the
motions of natural celestial bodies such as the
moon and planets.
The root of orbital mechanics can be traced back
to the 17th century when mathematician Isaac
Newton (1642-1727) put forward his laws of motion
and formulated his law of universal gravitation.
The engineering applications of orbital mechanics
include ascent trajectories reentry and landing
rendezvous computations and lunar and
interplanetary trajectories.

21
Orbital Mechanics II

Orbital mechanics remain a mystery to most people
Difficulty in thinking in 3D
Cryptic names given by astronomers
To make matters worse sometimes several
different names are used to specify the same
number.
Vocabulary is one of the hardest part of
celestial mechanics!

22
Perigee and Apogee

The point where the secondary is closest to the
primary is called perigee although its
sometimes called periapsis or perifocus.
The point where the seconday is farthest from
primary is called apogee (aka apoapsis or
apifocus).

23
Vernal Equinox

For some of our calculations we will use the term
vernal equinox
Teachers have told children for years that the
vernal equinox is the place in the sky where the
sun rises on the first day of Spring.
This is a horrible definition. Most teachers and
students have no idea what the first day of
spring is (except a date on a calendar) and no
idea why the sun should be in the same place in
the sky on that date every year.

24
Vernal Equinox II

Consider the orbit of the Sun around the Earth.
Although the Earth does not orbit around the sun
the math is equally valid either way and it
suits our needs at this instant to think of the
Sun orbiting the Earth.
The orbit of the sun has an inclination of about
23.5 degrees. (Astronomers use an infinitely more
obscure name The Obliquity of The Ecliptic.)
The orbit of the Sun is divided (by humans) into
four equally sized portions called seasons.
In other words the first day of Spring is the
day that the sun crosses through the equatorial
plane going from South to North.

25
Keplerian Elements

Severn numbers are required to define a satellite
orbit. This set of seven numbers is called the
satellite orbital elements or sometimes
Keplerian elements
These numbers define an ellipse orient it about
the Earth and place the satellite on the ellipse
at a particular time. In the Keplerian model
satellites orbit in an ellipse of constant shape
and orientation
The real world is slightly more complex than the
Keplerian model and tracking programs compensate
for this by introducing minor corrections to the
Keplerian model
These corrections are known as perturbations and
are due to the unevenness of the earths
gravitational field (which luckily you dont have
to specify) and the drag on the satellite due
to atmosphere.
Drag becomes an optional eighth orbital element

26
Keplerian Elements II

Epoch
Orbital Inclination
Right Ascension of Ascending Node (R.A.A.N.)
Argument of Perigee
Eccentricity
Mean Motion
Mean Anomaly
Drag (optional)

27
Epoch

A set of orbital elements is a snapshot at a
particular time of the orbit of a satellite.
Epoch is simply a number which specifies the time
at which the snapshot was taken.

28
Orbital Inclination

The orbit ellipse lies in a plane known as the
orbital plane. The orbital plane always goes
through the center of the earth but may be
tilted any angle relative to the equator.
Inclination is the angle between the orbital
plane and the equatorial plane.
By convention inclination is a number between 0
and 180 degrees.
Orbits with inclination near 0 degrees are called
equatorial orbits or Geostationary. Orbits with
inclination near 90 degrees are called polar.
The intersection of the equatorial plane and the
orbital plane is a line which is called the line
of nodes.

29
Right Ascension of Ascending Node

Two numbers orient the orbital plane in space.
The first number was Inclination RAAN is the
second.
After weve specified inclination there are
still an infinite number of orbital planes
possible. The line of nodes can intersect
anywhere along the equator.
If we specify where along the equator the line of
nodes intersects we will have the orbital plane
fully specified.
The line of nodes intersects two places of
course. We only need to specify one of them. One
is called the ascending node (where the satellite
crosses the equator going from south to north).
The other is called the descending node (where
the satellite crosses the equator going from
north to south).
By convention we specify the location of the
ascending node.

30
Right Ascension of Ascending Node II

The Earth is spinning. This means that we cant
use the common latitude/longitude coordinate
system to specify where the line of nodes points.
Instead we use an astronomical coordinate
system known as the right ascension /
declination coordinate system which does not
spin with the Earth.
Right ascension is another fancy word for an
angle in this case an angle measured in the
equatorial plane from a reference point in the
sky where right ascension is defined to be zero.
Astronomers call this point the vernal equinox.
Finally right ascension of ascending node is
an angle measured at the center of the earth
from the vernal equinox to the ascending node.

31
Right Ascension of Ascending Node III

Draw a line from the center of the earth to the
point where our satellite crosses the equator
(going from south to north). If this line points
directly at the vernal equinox then RAAN 0
degrees.
By convention RAAN is a number in the range 0 to
360 degrees.

32
Argument of Perigee

Argument is yet another fancy word for angle. Now
that weve oriented the orbital plane in space
we need to orient the orbit ellipse in the
orbital plane. We do this by specifying a single
angle known as argument of perigee.
If we draw a line from perigee to apogee this
line is called the line-of-apsides or major-axis
of the ellipse (Green dotted line).

33
Argument of Perigee II

The line-of-apsides passes through the center of
the Earth.
Weve already identified another line passing
through the center of the earth the line of
nodes.
The angle between these two (green dotted) lines
is called the argument of perigee. The argument
of perigee is the angle (measured at the center
of the earth) from the ascending node to perigee.
Example When ARGP 0 the perigee occurs at the
same place as the ascending node. That means that
the satellite would be closest to earth just as
it rises up over the equator. When ARGP 180
degrees apogee would occur at the same place as
the ascending node. That means that the satellite
would be farthest from earth just as it rises up
over the equator.
By convention ARGP is an angle between 0 and 360
degrees.

34
Eccentricity

In the Keplerian orbit model the satellite orbit
is an ellipse. Eccentricity tells us the shape
of the ellipse.
For our purposes eccentricity must be in the
range 0 lt e lt 1.

35
Mean Motion

So far weve found the orientation of the orbital
plane the orientation of the orbit ellipse in
the orbital plane and the shape of the orbit
ellipse.
Now we need to know the size of the orbit
ellipse. In other words how far away is the
satellite
Keplers third law of orbital motion gives us a
precise relationship between the speed of the
satellite and its distance from the earth.
Satellites that are close to the earth orbit very
quickly. Satellites far away orbit slowly. This
means that we could accomplish the same thing by
specifying either the speed at which the
satellite is moving or its distance from the
Earth!
Satellites in circular orbits travel at a
constant speed. We just specify that speed and
were done. Satellites in non-circular (i.e.
eccentricity gt 0) orbits move faster when they
are closer to the Earth and slower when they are
farther away.
The common practice is to average the speed. You
could call this number average speed but
astronomers call it the Mean Motion.
Mean Motion is usually given in units of
revolutions per day.

36
Mean Motion II

In this context a revolution or period is
defined as the time from one perigee to the next.
Sometimes orbit period is specified as an
orbital element instead of Mean Motion. Period is
simply the reciprocal of Mean Motion. A satellite
with a Mean Motion of 2 revs per day for
example has a period of 12 hours.
Sometimes semi-major-axis (SMA) is specified
instead of Mean Motion. SMA is one-half the
length (measured the long way) of the orbit
ellipse and is directly related to mean motion
by a simple equation.
Typically satellites have Mean Motions in the
range of 1 rev/day to about 16 rev/day.

37
Mean Anomaly

Now that we have the size shape and orientation
of the orbit firmly established the only thing
left to do is specify where exactly the satellite
is on this orbit ellipse at some particular time.
Our very first orbital element (Epoch) specified
a particular time so all we need to do now is
specify where on the ellipse our satellite was
exactly at the Epoch time.
Anomaly is yet another astronomer-word for angle!
Mean anomaly is simply an angle that marches
uniformly in time from 0 to 360 degrees during
one revolution.
It is defined to be 0 degrees at perigee and
therefore is 180 degrees at apogee.

38
Mean Anomaly II

If you had a satellite in a circular orbit
(therefore moving at constant speed) and you
stood in the center of the Earth and measured
this angle from perigee you would point directly
at the satellite.
Satellites in non-circular orbits move at a
non-constant speed so this simple relation
doesnt hold. This relation does hold for two
important points on the orbit however no matter
what the eccentricity. Perigee always occurs at
MA 0 and apogee always occurs at MA 180
degrees.

39
Drag

Drag caused by the Earths atmosphere causes
satellites to spiral downward. As they spiral
downward they speed up.
The Drag orbital element simply tells us the rate
at which Mean Motion is changing due to drag or
other related effects.
Drag is one half the first time derivative of
Mean Motion.
Its units are revolutions per day per day. It is
typically a very small number.
Common values for low-Earth-orbiting satellites
are on the order of 10-4.
Common values for high-orbiting satellites are on
the order of 10-7 or smaller.
Can you tell me why

40
Drag II

Occasionally published orbital elements for a
high-orbiting satellite will show a negative
Drag!
There are several potential reasons for negative
drag.
First the measurement which produced the orbital
elements may have been in error.
A satellite is subject to many forces besides
Earths gravity and atmospheric drag
Some of these forces (for example gravity of the
Sun and Moon) may act together to cause a
satellite to be pulled upward by a very slight
amount.
This can happen if the Sun and Moon are aligned
with the satellites orbit in a particular way.
If the orbit is measured when this is happening
a small negative Drag term may actually provide
the best possible fit to the actual satellite
motion over a short period of time.

41
Drag III

You typically want a set of orbital elements to
estimate the position of a satellite reasonably
well for as long as possible often several
months. Negative Drag never accurately reflects
whats happening over a long period of time. Some
programs will accept negative values for Drag
but I dont approve of them. Feel free to
substitute zero in place of any published
negative Drag value.

42
Other Satellite Parameters

The following parameters are optional. They allow
tracking programs to provide more information
that may be useful or fun

43
Epoch Rev

This tells the tracking program how many times
the satellite has orbited from the time it was
launched until the time specified by Epoch.
Epoch Rev is used to calculate the revolution
number displayed by the tracking program. Dont
be surprised if you find that orbital element
sets which come from NASA have incorrect values
for Epoch Rev.

44
Attitude

The spacecraft attitude is a measure of how the
satellite is oriented in space. Hopefully it is
oriented so that its antennas point toward you!
There are several orientation schemes used in
satellites. The Bahn coordinates apply only to
spacecraft which are spin-stablized.
Spin-stabilized satellites maintain a constant
inertial orientation i.e. its antennas point a
fixed direction in space.
The Bahn coordinates consist of two angles often
called Bahn Latitude and Bahn Longitude. Ideally
these numbers remain constant except when the
spacecraft controllers are re-orienting the
spacecraft. In practice they drift slowly.

45
How do Satellite Elements look like

Satellite elements can be downloaded from NORAD
http//www.celestrak.com/NORAD/elements/
They are in TLE format (Two Line Elements)
NOAA 18
1 28654U 05018A 06216.35688869 -.00000204 00000-0
-89227-4 0 5825
2 28654 98.7889 158.6074 0015435 83.8593 276.4346
14.10977823 62171

46
TLE Format

Data for each satellite consists of three lines
in the following format
AAAAAAAAAAAAAAAAAAAAAAAA1 NNNNNU NNNNNAAA
NNNNN.NNNNNNNN .NNNNNNNN NNNNN-N NNNNN-N N
NNNNN2 NNNNN NNN.NNNN NNN.NNNN NNNNNNN NNN.NNNN
NNN.NNNN NN.NNNNNNNNNNNNNN
Line 0 is a twenty-four character name

47
Satellite Orbit

One of the most important aspect of a satellite
orbit is its inclination
The inclination limits the types of coverage and
data that a satellite can acquire
The velocity of the satellites determines the
height above the geoid

48
Satellites Orbit
49
Geosyncronous Satellites

GEO are circular orbits around the Earth having a
period of 24 hours.
A geosynchronous orbit with an inclination of
zero degrees is called a geostationary orbit.
A spacecraft in a geostationary orbit appears to
hang motionless above one position on the Earths
equator. For this reason they are ideal for some
types of communication and meteorological
satellites.
A spacecraft in an inclined geosynchronous orbit
will appear to follow a regular figure-8 pattern
in the sky once every orbit.
To attain geosynchronous orbit a spacecraft is
first launched into an elliptical orbit with an
apogee of 35786 km (22236 miles) called a
geosynchronous transfer orbit (GTO). The orbit is
then circularized by firing the spacecrafts
engine at apogee.

50
Typical Geostationary Coverage
51
Metereological Satellites
52
World Clouds
53
Polar Orbits

PO are orbits with an inclination of 90 degrees.
Polar orbits are useful for satellites that carry
out mapping and/or surveillance operations
because as the planet rotates the spacecraft has
access to virtually every point on the planets
surface
Most PO are circular to slightly elliptical at
distances ranging from 700 to 1700 km (435 - 1056
mi) from the geoid.
At different altitudes they travel at different
speeds.

54
(Near) Polar Orbiting Satellites
55
Ascending Vs. Descending
56
Daily Coverage
57
Polar Regions

The satellite doesnt pass directly over the pole
due to the slight inclination of the orbital
plane.
The transparent overlay identifies the 3000 km
wide swath that is viewed by the AVHRR imaging
instrument on the satellite.
The yellow curves delineate the limits of the 60
degree viewing arcs from the six standard
geostationary satellites included in these
discussions.

58
Multiple Passes
59
Sun Synchronous Orbits

SSO are near polar orbits where a satellite
crosses periapsis at about the same local time
every orbit.
This is useful if a satellite is carrying
instruments which depend on a certain angle of
solar illumination on the planets surface.
In order to maintain an exact synchronous timing
it may be necessary to conduct occasional
propulsive maneuvers to adjust the orbit.
Most research satellites are in Sun Syncronous
Orbits
There is a special kind of sun-synchronous orbit
called a dawn-to-dusk orbit. In a dawn-to-dusk
orbit the satellite trails the Earths shadow
(Why do you think this could be convinient)

60
Molniya Orbits

They are highly eccentric Earth orbits with
periods of approximately 12 hours (2 revolutions
per day).
The orbital inclination is chosen so the rate of
change of perigee is zero thus both apogee and
perigee can be maintained over fixed latitudes.
This condition occurs at inclinations of 63.4
degrees and 116.6 degrees. For these orbits the
argument of perigee is typically placed in the
southern hemisphere so the satellite remains
above the northern hemisphere near apogee for
approximately 11 hours per orbit. This
orientation can provide good ground coverage at
high northern latitudes.

61
Molniya Orbits
62
Tundra Orbits

Tundra orbit is a class of a highly elliptic
orbit with inclination of 63.4 and orbital
period of one sidereal day (almost 24 hours).
A satellite placed in this orbit spends most of
its time over a designated area of the earth a
phenomenon known as apogee dwell.

63
Different Orbital Distances
64
Orbital Distances

A low Earth orbit (LEO) is an orbit around Earth
between the atmosphere and the Van Allen
radiation belt. The boundaries are not firmly
defined but are typically around 200 - 1200 km
(124 - 726 miles) above the Earths surface
Intermediate circular orbit (ICO) also called
Medium Earth Orbit (MEO) is used by satellites
between the altitudes of Low Earth Orbit (up to
1400 km) and geostationary orbit (35790 km)
A rather vaguely defined orbit which usually
means anything from geosynchronous orbit up

65
Satellite Constellation

A group of electronic satellites working in
concert is known as a satellite constellation.
Such a constellation can be considered to be a
number of satellites with coordinated ground
coverage operating together under shared
control synchronised so that they overlap well
in coverage and complement rather than interfere
with other satellites coverage.

66
Satellite Formation
67
Demonstration
68
Formalization

In the followings we will formalize all the
concepts we have discussed
When you measure what you are speaking about and
express it in numbers you know something about
it but when you cannot express it in numbers
your knowledge is of a meager and unsatisfactory
kind. (Lord Kelvin British Scientist) William
Thompson Lord Kelvin Popular Lectures and
Addresses 1891-1894 in Bartletts Familiar
Quotations Fourteenth Edition 1968 p. 723a.

69
Newtons Laws of Motion and Universal Gravitation

The first law states that if no forces are
acting a body at rest will remain at rest and a
body in motion will remain in motion in a
straight line. Thus if no forces are acting the
velocity (both magnitude and direction) will
remain constant.
The second law tells us that if a force is
applied there will be a change in velocity i.e.
an acceleration proportional to the magnitude of
the force and in the direction in which the force
is applied. This law may be summarized by the
equation
F ma
where F is the force m is the mass of the
particle and a is the acceleration.
Remember that both F and a are vector quantities

70
Newtons Laws of Motion and Universal Gravitation
II

The third law states that if body 1 exerts a
force on body 2 then body 2 will exert a force
of equal strength but opposite in direction on
body 1. This law is commonly stated for every
action there is an equal and opposite reaction.
In his law of universal gravitation Newton
states that two particles having masses m1 and m2
and separated by a distance r are attracted to
each other with equal and opposite forces
directed along the line joining the particles.
The common magnitude F of the two forces is
where G is an universal constant called the
constant of gravitation We will use this formula
often

71
Newtons Laws of Motion and Universal Gravitation
III

Lets now look at the force that the Earth exerts
on an object. If the object has a mass m and the
Earth has mass M and the objects distance from
the center of the Earth is r then the force that
the Earth exerts on the object is GmM /r2 . If we
drop the object the Earths gravity will cause
it to accelerate toward the center of the Earth.
By Newtons second law (F ma) this
acceleration g must equal (GmM / r2 )
/ m or
At the surface of the Earth this acceleration has
the valve 9.80665 m/s2
Many of the upcoming computations will be
somewhat simplified if we express the product GM
as a constant which for Earth has the value
3.986005x1014 m3/s2

72
Problem

Your professor is asking to send CEOSR-1
satellite into orbit using a rocket
The weight of the rocket satellite is 500000
kg loaded with fuel and 320000 kg with no fuel
The rocket creates a thrust of 10000000N
Approximating g at 10m/s2 what is the
acceleration at (1) launch and at (2) burn out

73
Solution

Lets assume upward direction to be positive and
downward to be negative so we can work with
numbers rather than vectors
At launch two forces act on the rocket
T Positive thrust 10000000N
W Negative mg 500000kg 10m/s2
-5000000N
The total Force is TW 5000000N
By Newtons second law
aF/m 5000000N / 500000kg 10m/s2 10g

74
Solution II

At burnout
W Negative mg 320000kg 10m/s2
-3200000N
The total Force is TW 6800000N
By Newtons second law
aF/m 6800000N / 320000kg 10m/s2 21.25g

75
Uniform Circular Motion

In the simple case of free fall a particle
accelerates toward the center of the Earth while
moving in a straight line. The velocity of the
particle changes in magnitude but not in
direction.
In the case of uniform circular motion a particle
moves in a circle with constant speed. The
velocity of the particle changes continuously in
direction but not in magnitude.
From Newtons laws we see that since the
direction of the velocity is changing there is
an acceleration.
This acceleration called centripetal
acceleration is directed inward toward the center
of the circle and is given by
where v is the speed of the particle and r is the
radius of the circle.
Every accelerating particle must have a force
acting on it defined by Newtons second law (F
ma). Thus a particle undergoing uniform circular
motion is under the influence of a force called
centripetal force whose magnitude is given by
The direction of F at any instant must be in the
direction of a at the same instant that is
radially inward.

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Uniform Circular Motion II

A satellite in orbit is acted on only by the
forces of gravity.
The inward acceleration which causes the
satellite to move in a circular orbit is the
gravitational acceleration caused by the body
around which the satellite orbits.
Hence the satellites centripetal acceleration
is g that is g v2/r.
From Newtons law of universal gravitation we
know that g GM /r2. Therefore by setting these
equations equal to one another we find that for
a circular orbit

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Problem

Calculate the velocity of a satellite orbiting
the Earth in a circular orbit at an altitude of
200 km above the Earths surface.
Radius of Earth 6378.140 km
GM of Earth 3.986005x1014 m3/s2
Given r (6378.14 200) x 1000 6578140 m
v SQRT GM / r
v SQRT 3.986005x1014 / 6578140
v 7784 m/s

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Motions of Planets and Satellites

Now we are going to formalize Newtons three laws
1. All planets move in elliptical orbits with
the sun at one focus. 2. A line joining any
planet to the sun sweeps out equal areas in equal
times. 3. The square of the period of any
planet about the sun is proportional to the cube
of the planets mean distance from the sun.

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Motions of Planets and Satellites II

Although all planets move in elliptical orbits
their eccentricity is very small. We can learn
much about planetary motion by considering the
special case of circular orbits.
We shall neglect the forces between planets
considering only a planets interaction with the
sun.
These considerations apply equally well to the
motion of a satellite about a planet.

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Motions of Planets and Satellites III

Lets examine the case of two bodies of masses M
and m moving in circular orbits under the
influence of each others gravitational
attraction
The center of mass lies along the line joining
them at a point C such that mr MR
M and m move in an orbit of radius R and r
respectively with the same angular velocity
For this to happen the gravitational force
acting on each body must provide the necessary
centripetal acceleration.
Both forces are equal but opposite in direction.

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Motions of Planets and Satellites IV

That is mw2r must equal Mw2R. The specific
requirement then is that the gravitational
force
acting on either body must equal the centripetal
force needed to keep it moving in its circular
orbit
that is
If one body has a much greater mass than the
other as is the case of the Sun and a planet or
the Earth and a satellite its distance from the
center of mass is much smaller than that of the
other body.
If we assume that m is negligible compared to M
then R is negligible compared to r then we can
simply the above formula into

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Motions of Planets and Satellites V

If we express the angular velocity in terms of
the period of revolution w 2 /P we obtain
where P is the period of revolution.
This is a basic equation of planetary and
satellite motion.
It also holds for elliptical orbits if we define
r to be the semi-major axis of the orbit.
A significant consequence of this equation is
that it predicts Keplers third law of planetary
motion that is P2r3

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Problem

Geostationary are a special class of satellites
that orbit the Earth with a period of one day.
Anwer the following
How will the satellites motion appear when
viewed from the surface of the Earth
What type of satellites use this orbit and why is
it important for them to be located in this
orbit (Keep in mind that this is a relatively
high orbit. Satellites not occupying this band
are normally kept in much lower orbits.)
Determine the orbital radius at which the period
of a satellites orbit will equal one day. State
your answer in
kilometers
multiples of the Earths radius
fractions of the moons orbital radius

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(3) Solution

The period of the Earths rotation is
approximately equal to the mean solar day
(24 x 3600 s 86400 s) but for best results
use the sidereal day (86164 s).
We know that P2 4 x p 2 x r3 / GM
r P2 x GM / (4 x p 2 ) 1/3
r 86164.12 x 3.986005x1014 / (4 x p 2 ) 1/3
r 42164170 m

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(3) Solution II

Convert to Earth radii r 4.216 107 m /
6378140 m 6.610 7 Earth radii
Convert to Earth-Moon distances
r 4.216 107 m / 384400000 m9
0.1097 1 distance from earth to moon
Homework
Answer 1 and 2
Can you find the same solution using Keplers
third law

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What About for Elliptical Orbits

The figure shows a particle revolving around C
along some arbitrary path
The area swept out by the radius vector in a
short time interval t is shown shaded
This area (neglecting the small triangular region
at the end) is one-half the base times the height
or approximately r(rwDt)/2
This expression becomes more exact as t
approaches zero i.e. the small triangle goes to
zero more rapidly than the large one.
For any given body moving under the influence of
a central force the value wr2 is constant

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it gets a bit more complicated

Lets now consider two points P1 and P2 in an
orbit with radii r1 and r2 and velocities v1 and
v2.
Since the velocity is always tangent to the path
it can be seen that if f is the angle between r
and v then
And multiplying by R
or for two points P1 and P2 on the orbital path
What happens at periapsis and apoapsis (Hint f
90 degrees)

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Consarvation of Energy

Lets now look at the energy of the above
particle at points P1 and P2.
Conservation of energy states that the sum of the
kinetic energy and the potential energy of a
particle remains constant.
The kinetic energy T of a particle is given by
mv2/2
The potential energy of gravity V is calculated
by the equation -GMm/r. Applying conservation of
energy we have

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and finally
The eccentricity will be given by
90
Problem

An artificial Earth satellite is in an elliptical
orbit which brings it to an altitude of 250 km at
perigee and out to an altitude of 500 km at
apogee.
Calculate the velocity of the satellite at both
perigee and apogee.

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Solution

Rp (6378.14 250) x 1000 6628140 m
Ra (6378.14 500) x 1000 6878140 m
Vp SQRT 2 x GM x Ra / (Rp x (Ra Rp))
Vp SQRT 2 x 3.986005x1014 x 6878140 /
(6628140 x (6878140 6628140))
Vp 7826 m/s
Va SQRT 2 x GM x Rp / (Ra x (Ra Rp))
Va SQRT 2 x 3.986005x1014 x 6628140 /
(6878140 x (6878140 6628140))
Va 7542 m/s

92
Problem

A satellite in Earth orbit passes through its
perigee point at an altitude of 200 km above the
Earths surface and at a velocity of 7850 m/s.
Calculate the apogee altitude of the satellite.

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Solution