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
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 or 'but 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 for 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 1 2 & 3
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 applied 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 let's exit
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 zero and
one the larger the value the more
elliptical the orbit a spacecraft in
Earth orbit with an eccentricity equal
to or greater than one 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's 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
as 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 where 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 1,300
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
within a 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 launched
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
paws agreed burns with paws agreed 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 burn orbits
burns can be combined into maneuver
sequences to change orbits size shape or
orientation one of the most common
maneuver 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 at 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 in 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 three
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 whips 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 molniya 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 of 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 past 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 spacecrafts 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 atmospheric
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 an 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
Oh
you