View
55
Download
2
Category
Preview:
Citation preview
Introduction to Orbital Mechanics
What Is an Orbit?
•Johannes Kepler discovered in 1600s that planet orbits form
ellipses, not circles.
•Satellites (natural or human-made) orbit Earth as an ellipse.
•Elliptical orbits remain fixed in space, and Earth spins under a
fixed satellite orbit.
A closed path around which a planet or satellite travels.
Graphic obtained from
Astronautics Primer by Jerry
Sellers.
What Is an Ellipse?
• An ellipse is the two-dimensional shape that is produced by a
plane fully intersecting a cone.
• Note that a plane intersecting the cone at a angle perpendicular to
the cone’s center line will form a special ellipse called a circle.
What Is an Ellipse?
A B
Ellipse has two focii instead of a
center
Sum of distances from focii is
constant
A+B = constant
Circle is simply an ellipse with both focii located at the same spot.
Circle is a set of points fixed (constant
distance) from a center point (focus)
A = constant
A
•Satellites orbit Earth with one focus at Earth’s center.
•The other focus is an empty point, which may or may not be within
Earth’s boundaries.
What Is an Ellipse?
• a defines ½ the major axis length
• b defines ½ the minor axis length
• c is the distance from the center of the ellipse to either focal point
• For a circle, a and b are equal to the radius, and both focal points are co-located at the center of the ellipse
Diverse Orbits
Basic Orbits
How Are Orbits Described?
Orbits are described by a set of parameters called
orbital elements (i.e., Keplerian elements).
The Keplerian element set consists of 6 parameters
(plus a time stamp):
• Two of these describe the size and shape of an orbit
• Three of these describe the orientation of the orbit in space
• One of these describes the location of the satellite
within the orbit
Eccentricity (e)
Eccentricity describes the roundness of an orbit. It describes the
shape of the ellipse in terms of how wide it is.
Semi-major axis, a
Semi-minor axis, b
Calculate the eccentricity of a circle.
Eccentricity can vary from 10 e
𝑒 = 1 −𝑏2
𝑎2
Eccentricity
This value is between 0 and 1 (for “closed” orbits).
Eccentricity of 0 means the orbit is circular.
An eccentricity of 1 or
greater means the orbit is not
closed. Such would be used
for interplanetary missions.
Satellites in these types of
orbits do not come back to
their starting point.
EccentricityValues between 0 and 1 mean the orbit is elliptical.
e = .74
e = .60
e = .4
e = 0
Beyond Eccentricity
Orbits may have the
same eccentricity
(e) but may be
different sizes.
There must be a
Keplerian element
which describes the
size of an orbit.
Semi-Major AxisMajor axis, 2a
Semi-major axis, a
Semi-major axis
a describes the
size of the
ellipse. It is half
of the largest
diameter (the
major axis) of the
orbit.Center
of
ellipse
The semi-major axis originates from the center of the orbit, but we
are located on Earth. This makes semi-major axes difficult for us to
visualize from our reference point.
Important Points on the Orbit
Perigee
Apogee
Apogee defines the point in an orbit that is farthest from Earth.
Perigee describes the point in an orbit that is closest to Earth.
“gee” suffix means Earthe.g. apoapsisand periapsis.Apogee altitude
Perigee
altitude
Apogee altitude is the distance between the surface of the Earth and
apogee.
Perigee altitude is the distance between the surface of the Earth and
perigee.
Apogee, Perigee, and Circular Orbits
• In circular orbit, apogee altitude and perigee altitude are the
same.
• Perfectly circular orbit has neither an apogee nor perigee and is
undefined.
• Perfectly circular orbits cannot be achieved.
• Generally circular orbits are described by their altitude.
•Semi-major axis rarely used to describe circular orbits.
Perigee
ApogeeApogee
altitudePerigee
altitude
For circular orbit
Apogee Altitude = Perigee
Altitude
Semi-Major Axis
(Altitude for circular orbits)
Semi-major axis is the only orbital parameter that determines the orbital period.
Translated as Kepler’s 3rd Law: The square of the period of a planet is
proportional to the cube of its mean distance from the Sun.
𝑇 = 𝑂𝑟𝑏𝑖𝑡𝑎𝑙 𝑃𝑒𝑟𝑖𝑜𝑑
𝑎 = 𝑆𝑒𝑚𝑖𝑀𝑎𝑗𝑜𝑟 𝐴𝑥𝑖𝑠𝜇 = 𝐺𝑟𝑎𝑣𝑖𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙
𝑃𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟G = Universal Gravitation Constant(6.67x10-11 m3/kg*s2)𝑀 = 𝑀𝑎𝑠𝑠 𝑜𝑓 𝑐𝑒𝑛𝑡𝑟𝑎𝑙 𝑏𝑜𝑑𝑦
𝑇 = 2𝜋 ×𝑎32
𝜇= 2𝜋 ×
𝑎32
𝐺𝑀
Semi-Major Axis
Let’s Have a Race
𝑇 = 2𝜋 ×𝑎32
𝜇= 2𝜋 ×
𝑎32
𝐺𝑀
Semi-Major Axis
• These orbits all have
the same semi-major
axis (a), but their
eccentricities (e) and
their orientations
around Earth are
different.
• Observe the orbital
periods.
Describing the Orientation of the
Orbit in Space Orbits may have
identical sizes and
shapes (a and e), yet
they can vary in their
orientation in space.
Three additional
Keplerian elements
define this orientation:
• Inclination
• Right ascension of
the ascending node
• Argument of
perigee
Inclination (i)Inclination is the angle between the Earth’s equatorial plane and the
plane of the orbit. It describes the tilt of the orbit.
i = 5o
i = 25o
i = 45o
i = 75o
???
Which satellite will
complete one orbit first?
We interrupt our regularly scheduled presentation on inclination to
bring you important information regarding ground traces!
If a long string with a magic marker tied to the end of it were
hung from a satellite, the path which the magic marker would
trace over the ground is the ground trace. A ground trace is a
projection of the satellite’s orbit onto the Earth.The satellite
appears to move
westward on
(most)
conventional
orbits because the
Earth is rotating
eastward.
later!)
Ground TracesAfter a full day, the ground trace of a satellite with an approximate 90
minute orbital period would look like this. Because the Earth is continually
rotating below the orbit of the satellite, the ground trace eventually spans
all longitudes.
Back to Inclination Inclination determines the
northern and southern latitude
limits over which the satellite
orbits. For example, a satellite
with a 45o inclination will have a
ground trace ranging from 45o
north to 45o south.
You can determine
the inclination of
an orbit simply by
examining its
ground trace.
Inclination
An orbit with an
inclination of 0
degrees is called an
equatorial orbit.
An orbit with an
inclination of 90
degrees is called a
polar orbit.
Inclination
A satellite in an
equatorial orbit will
pass directly over the
equator.
A satellite in polar
orbit will pass
over the entire
Earth.
What Do Ground Traces Reveal?
• Inclination is determined simply by noting the northern and
southern latitude limits of the ground trace.
• Orbital period can be determined using a simple calculation.
Based on what we have already learned about orbital parameters, we
can determine both inclination and orbital period from a ground trace.
1st pass, 0 degrees
longitude
2nd pass, 25 degrees
west longitude
Determining a Satellite’s Orbital
Period from its Ground Trace1. Recall that the orbit of a satellite remains fixed in space,
and the Earth rotates underneath it.
2. The westward regression of the ground trace is due to
the rotation of the Earth.
3. Determine how many minutes it takes for the Earth to
rotate one degree:
1440 minutes/360 degrees = 4min/degree
4. Determine how many degrees per pass the satellite’s
orbit regresses on consecutive orbits (equatorial crossing
is a common reference point). We’ll use 25 degrees as an
example.
5. How long did it take the Earth to rotate this many
degrees? That’s the period of the satellite.
25degrees * 4min/degree = 100 minutes
Right Ascension of the Ascending
Node (RAAN, W )Satellites may have identical eccentricities, semi-major axes, and
inclinations (e, a, and i) yet may still be oriented differently in space –
they can be “rotated” or “twisted” about the Earth in various ways.
Each satellite here starts
out above a different
longitude on the Earth.
However, longitude
can’t be used as a
reference point because
the Earth will rotate
underneath the orbits,
changing the reference
longitude on each
satellite pass.
RAAN
Right ascension of the
ascending node is the
angle measured along the
equatorial plane between
a vector pointing to a
fixed reference point in
space (the first point of
Aries, also known as the
vernal equinox) and the
point on the orbit where
the orbital motion is
from south to north
across the equator (this
point is called the
ascending node).
W = 0o
W = 30o
W = 60o
W = 90o
Argument of Perigee (w)
It is measured as
the angle from the
ascending node to
the perigee point
in the direction of
the satellite’s
motion.
Orbits may have the same e, a, I, and W, yet may still have different
orientations around the Earth. The location of their perigee point can
vary within the orbital plane.
w= 0o
w = 90o
w = 180o
w = 270o
Argument of perigee
describes the orientation
of the orbit within the
orbital plane (where is
apogee and where is
perigee?).
True Anomaly (u)After an orbit and its orientation have been thoroughly described,
there must be a way to describe the satellite’s position within an orbit
at any instant. True anomaly is the angle
between the perigee point
and the satellite’s location
(measured in the direction
of the satellite’s motion).
This value is constantly
changing as the satellite
moves in its orbit.
True anomaly is 0 degrees
at perigee, 180 degrees at
apogee.
Keplerian Elements in ReviewThe Keplerian element set consists of 6 parameters:
Two of these describe the size and shape of an orbit:
Three of these describe the orientation of the orbit in space:
•Eccentricity (e)
•Semi-major axis (a)
•Inclination (i)
•Right ascension of the ascending node (W)
•Argument of perigee (w)
One of these describes the location of the satellite within the orbit:
•True anomaly (u)
A time stamp, referred to as an “epoch,” must also be included when
providing a Keplerian element set. This is so that it is known WHEN this set
of values was accurate for the satellite or when the “snapshot” of the orbit
was taken.
Kepler’s LawsKepler’s 1st Law: Satellites will travel around Earth in elliptical paths with the
center of Earth at one of the foci.
Translated, this means
the speed of a satellite
changes as the
distance between it
and Earth changes. At
perigee a satellite is
moving its fastest; at
apogee, it is moving
its slowest.
Kepler’s 3rd Law: The period of an orbit (T) is related to its semi-major axis
(a) by: T2 = 4p2
m* a3
Kepler’s 2nd Law: A line drawn between Earth and a satellite will sweep out
equal areas during equal time periods anywhere along the orbit.
Time1
Time1
Special Orbit Types
The Keplerian element set chosen for any
given satellite is highly dependent on its
mission. Certain orbits are better suited
for certain missions.
LEO (Low Earth Orbit)•No specified minimum altitude
•Relatively close to the Earth (several hundred km)
•Short orbital periods ~90 minutes
•Many revolutions per day
•Limited swath areas
•What can the satellite view on Earth’s surface?
•All manned space missions (except lunar missions)
were LEO
•Many Earth-observing satellites
•Weather and imagery
•Why is this?
LEO (Low Earth Orbit)
Image is to scale showing International Space
Station height of orbit ~ 350 km
GEO (Geostationary)What’s in a name?
• Geostationary satellite remains over one
location on Earth
• Achieved by placing the satellite in a special
orbit where period exactly equals one day
• Altitude: roughly 36,000 km (22,200 miles)
• Inclination is exactly zero degrees
GEO (Geostationary)
GEO (Geostationary)
• GEO satellite ONLY exists directly above equator
AKA sub-satellite longitude
• Geostationary satellite can see ~70 degrees north
and south of the equator
• Geostationary satellites mainly used for
communications or “permanent relay station” in
space
GEO• Only one altitude with a period of 24 hours
• All geostationary orbits are in a “ring” around the Earth
• The ring is called the geostationary belt
• Geostationary belt is a limited resource
• When a “Geobird” dies, it
• Must be removed from its slot in the geobelt
• Must make room for another satellite
• Is usually boosted to a slightly higher orbit
GEO• Difficult to orbit exactly 24-hour period and zero inclination
• Orbits typically have slight inclination
• Satellites drift slightly north and south of equator
• Slight east or west drift due to imperfect period
• Small orbit-adjustment burns performed (called station-keeps)
• Satellites with 24 hour period and non-zero inclination are
called geosynchronous
• Geostationary and geosynchronous often interchanged
Real Geobelt
•Ground traces projected out to geostationary altitude
•Large inclinations (figure 8) run out of station-keeping fuel
•Sine wave orbits are being drifted to new location
•Orbit color participation in data sharing program
GEO
A Short Lesson in Urban Navigation
How can you tell what direction is south if you’re lost
in the middle of an urban area in the United States
with no compass or GPS receiver? It is too cloudy to
see the sun, and there is no moss growing anywhere!
Think about what you have learned about orbits.
Q.
Just look for a building/house with a TV satellite dish.
Since geostationary satellites can only “hover” above
the equator, all dishes in the northern hemisphere that
are communicating with geostationary satellites must
be pointing toward the south.
A.
Molniya (Moly)Using geostationary satellites for communications posed severe
problems for Russia since so much of their land mass is near or
north of 70 degrees in latitude.
To overcome this problem, they created a type of orbit, a Molniya
orbit, to allow for long-term communications over their northern land
mass.
Molniya
•Highly inclined and highly elliptical orbit
•High inclination covers northern Russia
•High eccentricity
-- Large apogee altitude
-- Very slow velocity at apogee
•If apogee is over Russia, then satellite hangs over
Russia (Kepler’s 2nd Law)
Molniya
MolniyaThe Molniya ground trace looks quite different from most conventional
ground traces. It clearly illustrates the “hang time” of the satellite over
the Russian Federation.
PolarBecause the inclination of a polar orbit is 90 degrees, a satellite in
polar orbit will eventually pass over every part of the world. This
makes polar orbits well-suited for satellites gathering information
about the Earth, such as weather satellites.
A special type of polar orbit called a Sun-synchronous orbit passes
over the same part of the Earth at roughly the same local time
every day. Why might this be useful?
ConstellationsA single satellite is often insufficient to perform a particular mission.
Groups of satellites in various orbits will work together to accomplish
the mission. Such groupings of satellites are called constellations. GPS
(Global Positioning System) is one such example.
Now That You Know the Basics
1. If Norway wanted to obtain satellite imagery of
all of its major urban areas, what type of orbit
would be appropriate?
2. Could researchers at McMurdo Station in
Antarctica use geostationary satellites for
communications?
Use your new understanding of orbital mechanics to
answer the following questions.
Now That You Know the Basics
1. If Norway wanted to obtain satellite imagery of
all of its major urban areas, what type of orbit
would be appropriate?
For the Norwegian satellites, the satellite should
have a high inclination (since Norway is in the
northern latitude region) and low altitude,
circular orbit. The inclination is approximately
70-80 degrees with an altitude of several hundred
km.
Now That You Know the Basics
2. Could researchers at McMurdo Station in
Antarctica use geostationary satellites for
communications?
No, because the latitude of Antarctica is too far
south. However, options do exist.
Now That You Know the Basics
2. Could researchers at McMurdo Station in
Antarctica use geostationary satellites for
communications?
Option #1
Old geostationary satellites that have acquired
significant inclination (i.e., >10 degrees) can often
provide continuous communications for >6 hours
a day when they are in the southern portion of
their figure 8 ground trace.
Now That You Know the Basics
2. Could researchers at McMurdo Station in
Antarctica use geostationary satellites for
communications?
Option #2
Researchers in Antarctica can also communicate
using low Earth orbiting communication
constellations such as Iridium.
Optional Analysis Tool
STK software can be used to explore, create, and
analyze orbits in greater detail.
References
Analytical Graphics, Inc. (AGI). (2010). Educational
resources. Retrieved from
http://www.stk.com/resources/academic-
resources/for-students/access-resources.aspx
National Aeronautics and Space Administration
(NASA). (2009). Basics of flight. Retrieved from
http://www2.jpl.nasa.gov/basics/bsf3-1.php
Recommended