Download - Orbital Mechanics and Design_Rev17
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Engineering 176
Orbital Design
Mr. Ken [email protected]
(508) 881- 5361
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The Ancients
Aristotle (384 BC 322 BC) Claudius Ptolemaeus (AD 83 c.168)
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Copernicus and Tycho
Nicolaus Copernicus (1473 - 1543) Tycho Brahe (1546 - 1601)
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The Copernicus Solar System
Tycho Brahe's Uraniborg
Observatory and 90
Star Sighting Quadrant
Image: Courtesy of tychobrahe.com
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Kepler and Galileo
Johannes Kepler (1571 - 1630) Galileo Galilei (1564 - 1642)
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Newton and LaGrange
Isaac Newton (1643 - 1727) Joseph Louis Lagrange (1736-1813)
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Einstein
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Geodesics: The Science and Artof 4D Curved Space Trajectories.
All objects in
motion conserve
momentum
through a
balance of
Gravity Potential
and
Velocity Vector
(think rollercoaster)
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Defining Simple 2-Body Orbits
This is all we need to know
Shape More like a circle, or stretched out?
Size Mostly nearby, or farther into space? Orbital Plane Orientation Pitch, Yaw, and Roll
Satellite Location Where are we in this orbit?
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Keplers First Law
Every orbit is
an ellipsewith the Sun
(main body)
located at
one foci.
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Keplers Second Law
A line between an orbiting
body and primary body
sweeps out equal areas in
equal intervals of time.
Day 0
Day 10
Day 20
Day 30Day 40
Day 50
Day 60
Day 70
Day 80
Day 90
Day 100
Day 110Day 120
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Keplers Third Law
P2 = R3
P1 P2
R2
R1
EXAMPLE:
Earth
P = 1 Year
R = 1 AU
Mars
P = 1.88 Years
R = 1.52 AU
This defines the
relationship of
Orbital Period &
Average Radius
for any two
bodies in orbit.
For a given body,
the orbital period
and average
distance for the
secondorbiting
body is:
P = Orbital Period
R = Average Radius
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Vernal EquinoxThe Celestial Baseline
First some
astronomy
When the Sun
passes over the
equator movingsouth to north.
Vernal Equinox(March 20th)
Defines a fixed
vector in space
through the center
of the Earth to a
known celestial
coordinate point.
June 21st
December 22nd
Sun
The Vernal Equinox drifts ~0.014
/ year. Orbits are thereforecalculated for a specified date
and time, (most often Jan 1,
2000, 2050 or today).
Epoch 2000
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Conic Sections (shape) Eccentricity
e=0 -- circle
e1 -- hyperbola
e < 1 Orbit is closed recurring path (elliptical)
e > 1 Not an orbit passing trajectory (hyperbolic)
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Keplerian Elements e, a, and v(3 of 6)
Perigee
0
Apogee
180
e defines ellipse shape
a defines ellipse size
v defines satellite angle from perigee
Semi-major
axis(nm or km)
True anomaly
(angle)
Eccentricity
(0.0 to 1.0)
Apo/Peri gee Earth
Apo/Peri lune Moon
Apo/Peri helion Sun
Apo/Peri apsis non-specific
90120
a
ev
150
e=0.8 vrs e=0.0
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Inclination i (4th Keplerian Element)
Inclination(angle)
Equatorial Plane
( defined by Earths equator )
Intersection of the
equatorial and
orbital planes
(below)
(above)
Sample inclinations0 -- Geostationary
52 -- ISS
98 -- Mapping
Ascending
Node
Ascending Node is where a
satellite crosses the equatorial
plane moving south to north
i
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RightAscension [1]oftheascendingnode
andArgumentofperigee (5th and 6th Elements)
Vernal Equinox
Perigee Direction
= angle fromvernal equinox to
ascending node on
the equatorial plane
= angle fromascending node to
perigee on the
orbital plane
[1] Right Ascension is the astronomical
term for celestial (star) longitude.
Ascending
Node
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The Six Keplerian Elements
a = Semi-major axis (usually inkilometers or nautical miles)
e = Eccentricity (of the ellipticalorbit)
v = True anomaly The anglebetween perigee and satellite in
the orbital plane at a specific time
i = Inclination The angle betweenthe orbital and equatorial planes
= Right Ascension (longitude)of the ascending node The
angle from the Vernal Equinoxvector to the ascending node on
the equatorial plane
[ = Argument of perigee Theangle measured between the
ascending node and perigee
Shape, Size,
Orientation,and Satellite
Location.
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Sample Keplerian Elements (ISS)
TWO LINE MEAN ELEMENT SET - ISS
1 25544U 98067A 09061.52440963 .00010596 00000-0 82463-4 0 9009
2 25544 51.6398 133.2909 0009235 79.9705 280.2498 15.71202711 29176
Satellite: ISS
Catalog Number: 25544
Epoch time: 09061.52440963 = yrday.fracday
Element set: 900
Inclination: 51.6398 deg
RA of ascending node: 133.2909 deg
Eccentricity: .0009235
Arg of perigee: 79.9705 deg
Mean anomaly: 280.2498 degMean motion: 15.71202711 rev/day (semi-major axis derivable from this)
Decay rate: 1.05960E-04 rev/day^2
Epoch rev: 2917
Checksum: 315
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State VectorsNonKeplerian Coordinate System
Cartesian x, y, z, and 3D velocity
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Orbit determination
On Board GPS
Ground Based Radar:Distance or Range (kilometers).
Elevation or Altitude (Horizon = 0, Zenith = 90).
Azimuth (Clockwise in degrees with due north = 0).
On board Radio Transponder Ranging:
Alt-Az plus radio signal turnaround delay (like radar).
Ground Sightings:Alt-Az only (best fit from many observations).
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Launch From Vertical Takeoff
Raising your altitude from 0 to 300 km (standing jump)
Energy = mgh = 1 kg x 9.8 m/s2 x 300,000 m
V = 1715 m/s
7 km/s lateral velocity at 300 km altitude (orbital insertion)
V (velocity) = 7000 m/s
V (altitude) = 1715 m/s
V (total) = 8715 m/s [1]
[1] plus another 1500 m/s lost to drag during early portion of flight.
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Launch From Airplane at 200 m/s
and 10 km altitude
Raise altitude from 10 to 300 km (flying jump)Energy = mgh = 1 kg x 9.8 m/s2 x 290,000 m
V = 1686 m/s (98% of ground based launch V)(96% of ground based launch energy)
Accelerate to 7000 m/s from 200 m/sV (velocity) = 6800 m/s (97% of ground V, 94% of energy)
V (Height)= 1686 m/s (98% of ground V, 96% of energy)
V (total, with airplane) = 8486 m/s + 1.3 km/s drag loss = 9800 m/s
V (total, from ground) = 8715 m/s + 1.5 km/s drag loss = 10200 m/s
Total Velocity savings: 4%, Total Energy savings: 8%
Downsides: Human rating required for entire system, limited launch vehicle
dimension and mass, fewer propellant choices, airplane expenses.
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Ground Tracks
Ground tracks drift
westward as the Earth
rotates below an orbit.
Each orbit type has a
signature ground tract.
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More Astronomy Facts
The SunDrifts east in the sky ~1 per day.Rises 0.066 hours later each day.
(because the earth is orbiting)
The EarthRotates 360 in 23.934 hours
(Celestial or Sidereal Day)
Rotates ~361 in 24.000 hours(Noon to Noon or Solar Day)
Satellites orbits are aligned to theSidereal day notthe solar day
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Orbital Perturbations
All orbits evolve
Atmospheric Drag (at LEO altitudes, only) Worse during increased solar activity.
Insignificant above ~800km.
Nodal Regression The Earth is an oblate spheroid.This adds extra pull when a satellite passes over the
equator rotating the plane of the orbit to the east.
OtherFactors Gravitational irregularities such asEarth-axis wobbles, Moon, Sun, Jupiter gravity (tends to
flatten inclination). Solar photon pressure. Insignificant
for LEO primary perturbations elsewhere.
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LEO < ~1,000km (Satellite Telephones, ISS)
MEO = ~1,000km to 36,000km (GPS)
GEO = 36,000km (CommSats, HDTV)
Deep Space > ~GEO
LEO is most common, shortest life. MEO difficult due to radiation belts.
Most GEO orbit perturbation is latitude drift due to Sun and Moon.
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Nodal Regression
Orbital planes
rotate eastward
over time.
(below)
(above)
Ascending
Node
Nodal Regression
can be very useful.
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Sun-Synchronous OrbitsRelies on nodal regression to shift the ascending node ~1 per day.
Scans the same path under the same lighting conditions each day.
The number of orbits per 24 hours must be an even integer (usually 15).
Requires a slightly retrograde orbit (I = 97.56 for a 550km / 15-orbit SSO).
Each subsequent pass is 24 farther west (if 15 orbits per day).
Repeats the pattern on the 16th orbit (or fewer for higher altitude SSOs).
Used for reconnaissance (or terrain mapping with a bit of drift).
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Molniya - 12hr Period
Long loitering high latitude apogee. Once usedused for early warning by both USA and USSR
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Tundra Orbit - 24hr Period
Higher apogee than Molniya. For dwelling over
a specific upper latitude (Used only by Sirius)
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GPS Constellation ~ 20200km alt.
GPS: Six orbits with six
equally-spaced satellites
occupying each orbit.
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Orbital Plane Changes
Burn must take place where theinitial and target planes intersect.
Even a small amount of plane
change requires lots of V
Less V required at higher altitudes
(e.g., slower orbital velocities).Often combined with Hohmann
transfer or rendezvous maneuver.
Simple Plane Change Formula (No Hohmann component):
Plane Change V = 2 x Vorbit x sin(/2)
Example: Orbit Velocity = 7000m/s, Target Inclination Change = 30
Plane Change V = 2 x 7000m/s x sin(30/ 2)
Plane Change V = 3623m/s
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Fast Transfer Orbit
Requires less time due to
higher energy transfer orbit.
Also faster since transfer is
complete in less 180.Can be used to reduce or
increase orbit altitudes.
Less common than Hohmann
Typically an upper stagerestart where excess fuel is
often available.
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Geostationary Transfer OrbitGTO
Requires plane change
and circularizing burns.
Less plane changing is
required when launched
from near the equator.2. Plane change
where GTO plane
intersects GEO
plane
3. Hohmann
circularizing burn
1. launch to
GTO
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Super GTO
Initial orbit has greaterapogee than standard
GTO.
Plane change at much
higher altitude requires
far less V.PRO: Less overall V
from higher inclination
launch sites.
CON: Takes longer to
establish the final orbit.
2. Plane change
plus initial
Hohmann burn
GEOTarget
Orbit
1. Launch to
Super GTO
3. Second
Hohmann burn
circularizes at
GEO
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Low Thrust Orbit Transfer
PROs: Lower mass propulsion system. Same system used for orbital maintenance.
CONs: Weeks or even months to reach final orbit. Van Allen Radiation belts.
A series of plane and altitude changes. Continuous electric engine propulsion.
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RendezvousLaunch when the
orbital plane of the
target vehicle crosseslaunch pad.
(Ideally) launch as the
target vehicle passes
straight overhead.
Smaller transfer orbitsslowly overtake target
(because of shorter
orbit periods).
Course maneuvers
designed to arrive in
the same orbit at thesame true anomaly.
Apollo LM
and CSM
Rendezvous
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Orbital Debris a.k.a., Space Junk
Currently > 19,000 items 10cm or larger. ~ 700 (4%) functioning S/C.
In as few as 50 years, upper LEO and lower MEO may be unusable.
February 2009 Iriduim / Cosmos collision created > 1,000 items > 10cm diameter
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Deep Space
Cassini Saturn orbit
insertion using good ol
fashion rocketpower.
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Using Lagrange Points to stay put
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Halo Orbits (stability from motion)
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AeroBrakingEarth, Mars, Jupiter, etc.The poor mans Hohmann maneuver
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The Solar System Super Highwaydesigning geodesic trajectories like tossing a message bottle
into the sea at exactly the right time, direction, and velocity.
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Gravity Assist(Removing Velocity)
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Gravity Assist(adding velocity)
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Solar Escape
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Multiple Mission
Trajectories
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Complex Orbital Trajectories
Galileo (Jupiter)
Cassini (Saturn)
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Designing Deep
Space Missionsyes, there are software tools for this
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Engineering 176 Orbits
Assignments for April 2
Create a trade table to
compare orbit designs.
Trade criteria should include:Orbit suitability for mission.
Cost to get there and stay there.
Space environment (e.g., radiation).
HOMEWORK:Design minimum two,
preferably three orbits
your mission could use.
For the selected orbits:Describe it (orbital elements)
How will you get there?
How will you stay there?
Estimate perturbations
Reading on Orbits:SMAD ch 6 scan 5 and 7
TLOM ch 3 and 4 scan 5 and 17