Transcript PPT with Animation

Introduction to Orbital Mechanics
What is an Orbit?
A closed “path” around which a planet or satellite travels.
Graphic obtained from
Astronautics Primer by Jerry
Sellers.
Johannes Kepler discovered in the 1600’s that planets did not orbit
in circles but rather in ellipses. Satellites, natural or man-made, also
orbit the earth in ellipses.These elliptical orbits remain FIXED in
space (the earth spins under a fixed orbit of a satellite).
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
A
A circle is a set of points that are a
fixed, constant distance from a center
point (or focus). A = constant
B
Instead of a center, an ellipse has
two focii. The sum of distances
from each focii is a constant.
A+B = constant
A circle is simply an ellipse with both focii located at the same spot.
For satellites orbiting the earth, one of the focii is at the center of the earth. The
other is simply an “empty” point. It may be within the boundaries of the earth or it
may be beyond it.
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 colocated at the center of the
ellipse
Orbits can be very diverse
Click to display Diverse.avi
How are orbits described?
Orbits are described by a set of parameters called
orbital elements or the Keplerian elements.
The Keplerian element set consists of 6 parameters
(plus a time stamp):
2 of these describe the size and shape of an orbit
3 of these describe the orientation of the orbit in space
1 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 fat to wide it is.
Semi-minor axis, b
Semi-major axis, a
e=
2
b
1– 2
a
Eccentricity (cont.)
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.
Eccentricity, continued
Values between 0 and 1 mean the orbit is “elliptical”
e = .74
e = .60
e = .4
e=0
Click to display Eccentricity.avi
Beyond eccentricity
Orbits may have the
same eccentricity
(e) but be different
sizes. There must
be a Keplerian
element which
describes the SIZE
of an orbit.
Semi-major axis
Major axis, 2a
Semi-major axis, a
Center
of
ellipse
Semi-major axis,
a, describes the
size of the
ellipse. It is half
of the largest
diameter (the
major axis) of the
orbit.
The semi-major axis originates from the center of the orbit, but we are located on earth.
This makes semi-major axis difficult for us to visualize from our reference point.
Important points on the orbit
**The “gee” part of
the words refer to
earth. Generic terms
are apoapsis and
periapsis.
Apogee
Apogee altitude
Perigee
altitude
Perigee
Apogee defines the point in an orbit that is farthest from earth.
Perigee describes the point in an orbit that is closest to earth.
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
“Apogee”
Apogee Altitude = Perigee Altitude
Apogee
altitude
Perigee
altitude
“Perigee”
• In a circular orbit, apogee altitude and perigee altitude are the same! (So,
technically, a perfectly circular orbit has neither an apogee nor perigee. They
are undefined. However, in reality perfectly circular orbits cannot be achieved.)
• In general, circular orbits can simply be described by their altitude. Semimajor axis is rarely used to describe circular orbits.
Semi-major axis
(Altitude for circular orbits)
Semi-major axis is the
only (see below
equation) orbital
parameter that
determines the
orbital period.
Period = 2p * a3/2
m
Click to display Altitude.avi
m is a constant called the gravitational parameter.
Or 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.
Semi-major axis (cont.)
• These orbits all
have the same
semi-major axis
(a), but their
eccentricities (e)
and their
orientations
around the earth
are different.
• Observe their
orbital periods.
Click to display semi_maj_Ax.avi
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.
3 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
???
Click to display Inclination.avi
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.
(More on this
later!)
Click to display GroundTrace.wmv
Ground Traces (cont.)
After 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 the
ground trace!
Inclination
(cont.)
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
(cont.)
A satellite in an equatorial
orbit will only pass directly
over the equator.
A satellite in polar obit
will pass over the entire
earth.
What Do Ground Traces Reveal?
2nd pass, 25 degrees
west longitude
1st pass, 0 degrees
longitude
Based on what we have already learned about orbital parameters, we can
determine both inclination and orbital period from a ground trace.
• 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.
Determining a satellite’s orbital
period from its ground trace
1. 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 still
be oriented differently in space—they can be “rotated” or “twisted” about the earth in various
amounts.
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.
Click to display raan.avi
RAAN (cont.)
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).
Click to display raan2.avi
W = 0o
W = 30o
W = 60o
W = 90o
Argument of Perigee (w)
Orbits may have the same e, a, i and W yet still have different orientations around the earth.
The location of their perigee point can vary within the orbital plane.
Argument of perigee describes
the orientation of the orbit within
the orbital plane (where is apogee
and where is perigee?).
It is measured as the angle from
the ascending node to the perigee
point in the direction of the
satellite’s motion.
w= 0o
w = 90o
w = 180o
w = 270o
Click to display ArgPer.avi
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.
Click to display TrueAnomaly.avi
Keplerian Elements in Review
The Keplerian element set consists of 6 parameters:
2 of these describe the size and shape of an orbit:
•Eccentricity (e)
•Semi-major axis (a)
3 of these describe the orientation of the orbit in space:
•Inclination (i)
•Right ascension of the ascending node (W)
•Argument of perigee (w)
1 of these describes the location of the satellite within the orbit:
•True anomaly (u)
A time stamp, referred to as “epoch” must also be included when providing a Keplerian
Element Set so that it is known WHEN this set of values was correct for the satellite or
when the “snapshot” of the orbit was taken.
Kepler’s Laws
Kepler’s 1st Law: Satellites will travel around Earth in elliptical paths with the center of
Earth at one of the foci.
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
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 its
slowest.
Kepler’s 3rd Law: The period of an orbit (T) is related its semi-major axis (a) by:
T2 = 4p2 * a3
m
“Special” Types of Orbits
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 cut-off altitude, but LEO orbits are
relatively close to the earth (several hundred km).
• LEO orbits are characterized by
short orbital periods (roughly 90
minutes), many revolutions per day
and limited swath areas (what the
satellite can see) on the Earth’s
surface.
• All manned space missions
except for the lunar missions have
been LEO.
• Many earth-observing satellites
(weather, imagery, etc.) are
in LEO orbits. Why is this?
The International Space Station’s orbit
is at an altitude of about 350 km. This
picture shows the height of its orbit to
scale.
GEO (Geostationary)
•What’s in a name?
A geostationary satellite stays in one spot with respect to the earth.
• This is achieved by placing it at an altitude where its orbital period is exactly equal to
one day (roughly 36,000 km or 22,200 miles above the earth!) AND its inclination is
exactly zero degrees.
• Therefore, a geo satellite can ONLY exist
at a location directly above the equator.
Such an orbit is often described simply by
the sub-satellite longitude.
• A geostationary satellite can “see”
about 70 degrees north and south of
the equator (more on this later).
• Geostationary satellites are used
mainly for communications.
They provide a “permanent relay
station” in space.
GEO (cont.)
• Since there is just one altitude above the earth for which the period of an orbit is
24 hours, all geostationary orbits are in a “ring” around the earth. This “ring” is
called the geostationary belt.
• Therefore, the geostationary belt is a limited resource!!
• When a “geobird” dies, it must be removed from its slot in the geobelt to
make room for another satellite. It is usually boosted to a slightly higher orbit.
GEO (cont.)
• In actuality, it is very difficult to achieve an orbit with an exact 24-hour period and zero inclination.
Usually an orbit has a slight inclination (which causes the satellite to drift slightly north and south of the
equator) and a slight east or west drift (due to the period).
• Regular small orbit-adjustment burns (station-keeps) must be performed to maintain the satellite’s
location.
Satellites with a 24 hour period and a non-zero inclination are called geosynchronous.
The terms “geostationary” and “geosynchronous” are often used interchangeably.
The “Real” Geobelt
This slide shows the ground traces of all operational geostationary satellites projected out to
geostationary altitude (these are NOT the actual orbits). Those with large inclinations (figure 8
type orbits) have run out of station-keeping fuel. Those with sine wave type orbits are being
drifted to a new location. (Orbit color corresponds to participation in specific data sharing
program. It does not represent any orbit-related data.)
GEO, continued
A short lesson in “urban navigation”
Q.
A.
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? (And it is too cloudy to see the
sun, and no moss is growing anywhere! Think about what
you have learned about orbits.)
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!
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 longterm communications over their northern land mass.
Molniya (cont.)
A Molniya orbit is a highly inclined, highly elliptical orbit. Its high inclination allows it
to cover the northern parts of Russia. Due to its high eccentricity, it has a large apogee
altitude which results in a very slow velocity at apogee. If apogee occurs over Russia,
the satellite “hangs” over Russia for a long period (remember Kepler’s 2nd Law?).
Molniya (cont.)
The Molniya ground trace looks quite different from most conventional ground traces. It clearly
illustrates the “hang time” of the satellite over Russia.
Click to begin animation
Polar
Because 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 that earth at roughly the same local time every day. Why
might this be useful?
Constellations
Often times a single satellite is 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
Use your new understanding of orbital mechanics to
answer the following questions.
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?
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/academicresources/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