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when ancient man looked to the heavens
for guidance from the gods
he noticed star patterns and began to
document their movement across the
heavens the ancients believed that the
earth was flat but around 350 BC
Aristotle proved that the earth was
round later about 150 AD Ptolemy
presented the geocentric theory the
belief that the earth is stationary at
the center of the universe with the Sun
Moon stars and planets revolving around
it in complex orbits in the 1500s
Nicholas Copernicus of Poland presented
the heliocentric theory the belief that
the Earth revolves around the Sun as it
rotates on its axis this aspect of
astronomy evolved into an intricate
study of planetary motion known as
orbital mechanics
today orbital mechanics is applied to
spaceflight and satellites that orbit
the Earth or travel beyond our solar
system
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in the early 1600s Johann Kepler a
German mathematician using the data on
planetary observations collected by the
Danish scientist Tycho Brahe he
developed three laws of planetary motion
Kepler's first law states all planets
move in ellipses there's a Sun at one
focus and the other focus empty
applied to earth satellites the center
of the earth becomes one focus with the
other focus empty
or circular orbits the two foci coincide
Kepler's second law the law of areas
states the line joining the planet to
the Sun sweeps over equal areas in equal
time intervals when a satellite orbits
the line joining it to the earth sweeps
over equal areas in equal periods of
time if areas one two and three are
equal times one two and three are also
equal therefore the speed of the
satellite changes depending on its
distance from the center of the earth
speed is greatest at the point in the
orbit closest to the earth called
perigee and is slowest at the point
farthest from the earth called Apogee
it is important to note that the orbit
followed by a satellite is not dependent
on its mass a large heavy satellite
could be in the same orbit with a small
light one each sweeping out equal areas
in equal periods of time Kepler's third
law the law of periods relates the time
required for a planet to make one
complete trip around the Sun to its mean
distance from the Sun for any planet the
square of its period of revolution is
directly proportional to the cube of its
mean distance from the Sun
to earth satellites Kepler's third law
explains that the farther a satellite is
from the earth the longer it will take
to complete an orbit the greater the
distance it will travel to complete an
orbit and the slower its average speed
will be
Isaac Newton the father of classical
mechanics laid the groundwork for
orbital mechanics he combined the work
of Kepler and others to formulate the
law of universal gravitation and the
three Newtonian laws of motion
while Kepler's laws provided a
conceptual model of orbital motion
Newton's laws provided the foundation
for the mathematical description of
orbits they explain why a satellite
stays in orbit
Newton's law of universal gravitation
any two objects in the universe such as
the earth and the moon attract each
other with a force directly proportional
to the product of their masses and
inversely proportional to the square of
the distance between them
stated more simply the more massive the
objects are or the closer they are the
greater the gravitational pull between
them
Newton's first law of motion a body in
motion will keep moving in the same
speed and in the same direction and less
exid upon by an external force
a satellite moves in a curved path
around the earth because the Earth's
gravitational pull
acts as an external force on it
Newton's second law of motion if the sum
of the forces acting on an object is not
zero the object will have an
acceleration proportional to the
magnitude and in the direction of the
net force newton's second law states
that force equals mass times
acceleration it is this mathematical
equation and the equation for universal
gravitation that forms the basis for
calculating orbits Newton's third law of
motion explains how a satellite gets
into orbit for every action there is an
equal and opposite reaction
if you blow up a balloon and let it go
the balloon is pushed forward by the
action of the air rushing out of it
a rockets exhaust gases are like the air
rushing out of the balloon
the following illustrates how a
satellite stays in orbit
if a man stands on a mountain and fires
a projectile horizontally gravity will
cause the path of the projectile to
curve downward and it will strike the
earth
however if the man fires the projectile
fast enough at a specific speed the
curvature of its path due to gravity
will match the curvature of the earth
under it the projectile will then fall
around the earth becoming an earth
orbiting satellite a projectile fired
even faster will have a flight path away
from the earth
but gravity will act to slow the
projectile down change its flight path
and pull it back toward Earth
if the projectiles velocity increased
enough a velocity sufficient to escape
the Earth's gravitational pull will be
reached this velocity is known as the
escape velocity it is equal to about
seven miles per second at the Earth's
surface the preceding description did
not consider atmospheric drag and the
Earth's rotation both of which will
affect the trajectory of the projectile
it Illustrated the principles governing
a satellites orbit
there are six numbers called the orbital
elements which specify the size shape
and orientation of an orbit in space as
well as the location of the spacecraft
in the orbit based on an orbit which is
an ellipse the six orbital elements are
length of the semi-major axis
eccentricity inclination right ascension
of the ascending node argument of
perigee time of perigee passage the
major axis of an elliptical orbit is the
line joining the perigee and Apogee this
line is also referred to as the line of
APSA DS the first orbital element is the
semi-major axis it is simply one-half
the major axis circular orbits have no
Apogee or perigee therefore the
semi-major axis is simply 1/2 the
diameter of the orbit the semi-major
axis is used to define the size of the
orbit from this the orbital period or
time that it takes for the satellite to
complete one orbit can be calculated the
shape of an orbit is defined by the
second orbital element called
eccentricity for all ellipses the value
of eccentricity lies between 0 and 1 the
larger the value the more elliptical the
orbit a spacecraft in Earth orbit with
an eccentricity equal to or greater than
1 will escape the Earth's gravitational
field when orienting an orbit in space a
three dimensional coordinate system must
be defined the coordinate system
commonly used is the geocentric
equatorial coordinate system which has
its origin at the earth
Center
this coordinate system is a non-rotating
reference system in which a satellites
orbital plane tends to remain fixed
relative to the stars while the Earth
turns beneath it
the xy-plane is the Earth's equatorial
plane the positive x-axis points to the
vernal equinox
this is the point where the Sun appears
to cross the earth's equator on its way
north on the first day of spring each
year the z axis is along the Earth's
spin axis toward the North Pole
nodes are points in a satellites orbit
which intersect the Earth's equatorial
plane the ascending node is the point at
which the spacecraft crosses the equator
going from south to north
the descending node is where the
spacecraft crosses the equator going
from north to south the line joining the
two nodes is called the line of nodes
the orientation of an orbit is
determined by three orbital element
angles the right ascension of the
ascending node is the angle between the
x-axis and the ascending node it is
always measured eastward from the
direction away from the vernal equinox
in the Earth's equatorial plane
the argument of perigee is the angle
between the ascending node and the point
of perigee it is measured in the orbital
plane in the direction of spacecraft
motion inclination is the angle between
the equatorial plane and the orbital
plane a satellite which has an eastward
velocity component at the ascending node
has an orbital inclination lying between
0 and 90 degrees such an orbit is called
a pro-grade orbit a satellite which
moves due north at the ascending node is
in a polar orbit polar orbits have an
orbital inclination of exactly 90
degrees a satellite with a westward
velocity component at the ascending node
is in a retrograde orbit and has an
orbital inclination between 90 and 180
degrees
the five orbital elements explained thus
far described the size shape and
orientation of the orbit in space the
final element is a time value used to
locate the satellite in its orbit a
satellite moves in a very predictable
manner
it stays on schedule thus if the time at
which a satellite passes a particular
point is known the time when it will
pass any other point can be determined
the particular point chosen is perigee
and the time of perigee passage is the
last of the six orbital elements the six
orbital elements depict a spacecrafts
orbit in non rotating coordinates to
visualize an orbit relative to the
rotating earth a projection traces the
spacecraft's position on the Earth's
surface the projected path is called the
ground track as a satellite orbits the
earth the ground track shifts westward
there are two causes for this first the
primary contributor is the Earth's
rotation toward the east under the
orbital plane second because the earth
is not a uniform sphere and bulges at
the equator its gravity is greatest at
the equator
this causes the orbital plane to rotate
slowly around the Earth's polar axis in
a motion called precession precession is
toward the west for pro-grade orbits and
toward the east for retrograde orbits
for low Earth orbits such as those of
the space shuttle at 150 miles altitude
the westward shift of the ground track
due to the Earth's rotation is about 22
and a half degrees while the shift due
to precession is only about a half
degree the inclination of a satellite
orbit determines the north and south
latitude limits of its ground track the
minimum orbital inclination is equal to
the latitude of the launch site and is
achieved by launching due east for
example if a satellite is launched due
east out of the Kennedy Space Center
which is located at 28 and a half
degrees north latitude its orbital
inclination will be 28 and 1/2 degrees
and the limits of its ground track will
vary between 28 and 1/2 degrees north
latitude and 28 and 1/2 degrees south
latitude if launch azimuth or direction
of flight at launch measured eastward
from due north is increased from due
east
the orbital inclination angle increases
as well as the maximum latitude of the
north-south ground track therefore the
latitude limits of the ground track
equal the new launch inclination
similarly if launched azimuth is
decreased from due east orbital
inclination once again increases as well
as the latitude limits of the ground
track
the maximum practical inclination from a
Kennedy Space Center launch is 57
degrees this limit is imposed for safety
considerations in order to keep the
spacecraft and its booster system from
flying over land masses during the
ascent phase to obtain an orbit with an
inclination greater than 57 degrees the
spacecraft is launched from Vandenberg
Air Force Base in California
Vandenberg offers the opportunity for
southerly launches with orbit
inclinations between approximately 70
degrees pro-grade through 138 degrees
retrograde a significant advantage of
launching from Vandenberg is the
capability to economically achieve polar
orbits with ground tracks covering all
latitudes from the North Pole to the
South Pole the earth is constantly
turning and all points on its surface
have an eastward velocity with the
greatest velocity occurring at the
equator the farther the launch site is
from the equator or as launch azimuth is
increased or decreased from due east
less of the Earth's rotational velocity
will be imparted to the launch vehicle
this requires more fuel to get into
orbit or payload weight will have to be
decreased
launches due east from a position on or
near the equator such as the kuru launch
site in French Guiana used by the
European Space Agency acquire the
advantage of a free velocity gain of
about 1500 feet per second this compares
to the approximate 1300 feet per second
gain available at the further north
latitude of the Kennedy Space Center
launching from an equatorial site offers
a significant advantage in payload
weight capability and minimizes the
amount of fuel needed to achieve an
equatorial orbit since many satellites
operate in equatorial orbits these are
important considerations
spacecraft are launched
specified time interval called the
launch window some of the factors
affecting the launch window are launched
in orbit lighting conditions Sun angles
payload orbit requirements rendezvous
phasing if a rendezvous is planned
tracking and communication requirements
and collision avoidance with other
orbiting objects to name a few one of
the factors defining the launch window
for the Space Shuttle is launch lighting
conditions which can be illustrated by
plotting time versus day of year on this
plot we see daylight and darkness at the
launch site
the longer daylight hours occur in the
middle of the year summer time
if daylight conditions are required for
a convenient emergency landing site for
the space shuttle the launch window
would now look like this during the
winter months
the available launch window for lighting
conditions alone can be as little as
three hours per day when combined with
the many other launch factors the launch
window becomes even more constrained the
choice of a particular launch vehicle
for a mission depends upon the weight
and size of the payload and the desired
orbit expendable rockets used to place
spacecraft in orbit usually consist of
several stages that may incorporate both
solid and liquid propellants for
propulsion when the fuel in each stage
is depleted the spent stage is
jettisoned staging offers the advantage
of discarding weight when it is no
longer needed the Space Shuttle is a
two-stage system at liftoff the two
solid rocket boosters and three Space
Shuttle main engines are all producing
thrust after approximately two minutes
of flight at an altitude of 25 miles the
fuel and the solid rocket boosters is
depleted and they are jettisoned the
three main engines fueled by liquid
oxygen and liquid hydrogen carried in
the external tank continue to burn for
several minutes until the shuttle
reaches its cutoff velocity at this time
the main engines are shut down and the
external tank is jettisoned to
additional burns using the orbiters
maneuvering system referred to as ohm's
are required to place the orbiter in its
final orbit the ohm's one burn occurs
about two minutes after main engine
shutdown and establishes the orbital
Apogee point the ohms to burn takes
place approximately 30 minutes later and
circular eise's the orbit once
satellites are launched and put into
orbit
it is often necessary to change the
orbit with an on-orbit burn the common
term used in describing on-orbit burns
or engine firings is Delta V Delta V is
the incremental change in spacecraft
velocity measured in feet per second
resulting from the burn the amount of
fuel used during a burn depends on the
desired Delta V change and the mass of
the spacecraft because the amount of
fuel carried is limited fuel consumption
is one of the primary considerations in
spacecraft mission planning and is
critical to orbit lifetime on orbit a
spacecraft can thrust in any direction
burns along the flight path forward and
backward are the most common a unique
feature of any orbital burn is that if
no other burns occur the spacecraft will
later always pass again through the
point of burn forward burns increase the
spacecraft's velocity and are known as
posit burns with positive rate burns the
flight path of the vehicle will be
raised at all points except the burn
point burns opposite the direction of
flight which slow the spacecraft down
are called retrograde burns for
retrograde burns the orbit will be
lowered at all points except the burn
point the greater the Delta V the
greater the difference between the pre
burn and post burned orbits
burns can be combined into maneuver
sequences to change orbit size shape or
orientation one of the most common
manoeuvre sequences is made up of two
burns and is used to accomplish an orbit
transfer between two circular orbits in
the same orbital plane the most energy
efficient transfer between two orbits of
this type is the Hohmann transfer the
Hohmann transfer is actually one half of
an elliptical orbit with its perigee in
one of the orbits and its Apogee in the
other the burns occur at the perigee and
Apogee of the transfer orbit the use of
the Hohmann transfer minimizes the Delta
V required thus having the advantage of
using minimum fuel the disadvantage of
the Hohmann transfer is that it takes
longer than most other transfers the
type of the transfer sequence depends on
the mission and the amount of fuel
available for example a space rescue
where time is critical might use a fast
transfer while a routine satellite
deployment where fuel saved for later
use is important would most likely use a
Hohmann transfer the burns discussed so
far have all been maneuvering the
original orbital plane and do not affect
orbit inclination or node position
there are situations which require an
orbital plane change such as setting up
a rendezvous or placing a satellite in
an equatorial orbit to change the
inclination the thrust vector must be
directed at an angle to the orbital
plane a thrust with a component that is
perpendicular to the orbital plane at
either the ascending or descending node
will rotate the orbital plane about the
line of nodes northerly out of plane
thrust at the ascending node will
increase the inclination of a pro-grade
orbit while a southerly thrust will
decrease it out of plane thrusts require
considerable amounts of fuel and are
performed only when absolutely required
the Space Shuttle for example using all
of its onboard propellant is capable of
an on-orbit plane change of less than 3
degrees
satellite orbital planes and altitudes
are determined by their design mission
which very often includes a field of
view requirement for optical or
communications purposes the field of
view of a satellite is defined as the
area of the Earth's surface that is in
view from the satellite at any given
time satellites in high orbits have
greater fields of view than those in
lower orbits for example a satellite at
an altitude of 800 nautical miles has a
circular field of view with a diameter
of about 40 100 nautical miles a
satellite at 200 nautical miles has a
circular field of view with a diameter
of about 2,000 nautical miles low orbit
satellites are often used for
photography and other types of Earth
observation a satellite placed in a low
inclination circular orbit at an
altitude of about 19,000 300 nautical
miles will have an angular velocity
exactly equal to that of the earth the
satellite would seem to remain
stationary in longitude as viewed from
the ground such orbits are called
geosynchronous and are used to provide a
continuous communications capability
among any system of ground stations
within their field of view the
geosynchronous orbit field of view is
constant and is limited to a latitude
zone of about 70 degrees north and south
of the equator effective satellite
communications from geosynchronous orbit
is not possible at either Pole however
because of their altitude their field of
view covers nearly half the globe a
special type of geosynchronous orbit
with an inclination of 0 degrees is
called a geostationary orbit it appears
to hover over a fixed point on the
Earth's surface at the equator
most us communication satellites are in
geosynchronous orbits providing near
worldwide communications coverage for
effective communications at high
latitudes the molniya orbit is used
mahlia is the Russian word for lightning
and is an orbit used extensively by the
Soviet Union for its communication
satellites pneumoniae orbit is highly
eccentric with an Apogee that is near
the geosynchronous altitude and an
inclination of about 63 degrees the
satellite slows down at Apogee in the
northern hemisphere and which through
perigee in the southern hemisphere this
provides communications in the northern
hemisphere for up to 75% of its orbital
period several satellites properly
spaced in Monia orbits can provide
constant communications at the northern
latitudes
navigation satellites such as the US
Navy's transit system and the joint
service Navstar GPS Global Positioning
System use lower orbits so that a user
can receive signals from more than one
satellite at any time another frequently
used orbit is known as a sun-synchronous
orbit these take advantage of the
precession of the orbital plane caused
by the earth not being a perfect sphere
all Sun synchronous orbits are highly
inclined retrograde orbits which precess
eastward around the Earth's polar axis
at the rate of one revolution per year
since the Earth's Sun line also revolves
eastward at the rate of one revolution
per year the orbital plane will maintain
a constant orientation relative to the
Earth's Sun line if the satellites
period is then synchronized with the
rotation of the earth it will pass over
the same point on the Earth's surface at
the same local time at a regular
interval a Sun synchronous satellite
ensures that a constant Sun angle and
uniform lighting exist for the same
field of view from pass to pass
satellites such as those in the defense
meteorological satellite program and
Landsat our Sun synchronous imaging the
entire Earth on a regular schedule
the gravitational attraction of the
earth on a spacecraft causes it to move
in its orbit around the Earth there are
other much smaller forces which will
cause a spacecraft to deviate from its
desired orbit these forces cause what
are known as orbital perturbations
orbital precession which is used to
obtain Sun synchronous orbits results
from the perturbing effects of the
Earth's non spherical shape other
perturbing forces are the gravitational
pull of the Sun the moon and planets and
solar winds which are charged streams of
protons and electrons that heat the
Earth's atmosphere and increase
atmospheric drag in most cases
perturbing forces can be compensated for
in the spacecraft and orbit design and
present no major problems if the forces
disturb the orbit too much thrusters can
be fired to re-establish its desired
orbital orientation or altitude this is
particularly true for spacecraft
orbiting at very low altitudes where the
effects of atmospheric drag are greater
and if not compensated for will
eventually cause the spacecraft to
deorbit a spacecraft's operational
lifetime is frequently limited only by
the amount of fuel available to maintain
its desired orbit when its useful life
is complete a satellite is left in orbit
or is deorbited burning up when
re-entering the Earth's atmosphere when
the Space Shuttle completes its orbital
mission it executes a precise retrograde
burn to initiate its controlled return
to earth this burn occurs nearly halfway
around the Earth from the landing site
the new orbit established by the
retrograde burn causes the orbiter to
enter the Earth's atmosphere about four
thousand miles from the landing site
during the period the orbiter descends
from its orbital altitude to
miss Furyk reentry its attitude is
maintained by the use of reaction
control jets located in the nose and
tail of the orbiter once the orbiter
enters the Earth's atmosphere its wing
and tail arrow surfaces begin to become
effective and gradually replace the Jets
for attitude control as the orbiter
nears the landing field it maneuvers to
a long straight in approach at an angle
of 17 to 19 degrees nearing the runway
it executes a flare maneuver to reduce
its sink rate and glides to a touchdown
at approximately 230 miles per hour as
the orbiter rolls to a stop our journey
into the world of orbital mechanics
comes to an end for now this is only the
basics of orbital mechanics and
intricate study of planetary and
satellite motion the next time you see a
launch you will see it from a different
somewhat knowledgeable perspective you
will understand the fundamentals of
spaceflight
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