Engineering 176 Orbital Design

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Engineering 176 Orbital Design
Mr. Ken Ramsley kenneth_ramsley_at_brown.edu (508)
881- 5361
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Class Topics
When Orbits Were Perfect (and politically
dangerous) Einsteins Geodesics (the art and
science of motion) Keplers Three Laws (based on
Tychos meticulous data) Orbital Elements
Defined and Illustrated Useful Orbits and
Maneuvers to Get There Interplanetary Space and
Beyond
EN176 Orbital Design
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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
Image Courtesy of tychobrahe.com
Tycho Brahe's Uraniborg Observatory and 90
Star Sighting Quadrant
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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 Art of 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 ellipse with the Sun (main
body) located at one foci.
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Keplers Second Law
Day 40
Day 30
Day 50
Day 60
Day 20
Day 70
Day 80
Day 90
Day 10
Day 100
Day 110
Day 0
A line between an orbiting body and primary body
sweeps out equal areas in equal intervals of
time.
Day 120
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Keplers Third Law
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 second orbiting body is
EXAMPLE Earth P 1 Year R 1 AU Mars P
1.88 Years R 1.52 AU
R2
R1
P1
P2

• P2 R3

P Orbital Period R Average Radius
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Vernal Equinox The Celestial Baseline
First some astronomy
June 21st
When the Sun passes over the equator moving south
to north. Vernal Equinox (March 20th) Defines a
fixed vector in space through the center of the
Earth to a known celestial coordinate point.
Sun
Epoch 2000
The Vernal Equinox drifts 0.014 / year. Orbits
are therefore calculated for a specified date and
time, (most often Jan 1, 2000, 2050 or today).
December 22nd
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Conic Sections (shape) Eccentricity

• e0 -- circle
• elt1 -- ellipse
• e1 -- parabola
• egt1 -- hyperbola

e lt 1 Orbit is closed recurring path
(elliptical)  e gt 1 Not an orbit passing
trajectory (hyperbolic)
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Keplerian Elements e, a, and v (3 of 6)
e
120
150
90
Eccentricity (0.0 to 1.0)
v
True anomaly (angle)
a
Apogee 180
Perigee 0
Semi-major axis (nm or km)
e0.8 vrs e0.0
e defines ellipse shape
a defines ellipse size
v defines satellite angle from perigee
Apo/Peri gee Earth Apo/Peri lune Moon
Apo/Peri helion Sun Apo/Peri apsis
non-specific
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Inclination i (4th Keplerian Element)
Intersection of the equatorial and orbital planes
i
Inclination (angle)
(above)
(below)
Ascending Node
Equatorial Plane ( defined by
Earths equator )
Sample inclinations 0 -- Geostationary
52 -- ISS 98 -- Mapping
Ascending Node is where a satellite crosses the
equatorial plane moving south to north
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Right Ascension 1 of the ascending node O and
Argument of perigee ? (5th and 6th Elements)
O angle from vernal equinox to ascending node
on the equatorial plane
Perigee Direction
? angle from ascending node to perigee on the
orbital plane
?
O
Ascending Node
1 Right Ascension is the astronomical term for
celestial (star) longitude.
Vernal Equinox
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The Six Keplerian Elements

• a Semi-major axis (usually in kilometers or
  nautical miles)
• e Eccentricity (of the elliptical orbit)
• v True anomaly The angle between perigee and
  satellite in the orbital plane at a specific time
• i Inclination The angle between the orbital
  and equatorial planes
• ? Right Ascension (longitude) of the ascending
  node The angle from the Vernal Equinox vector to
  the ascending node on the equatorial plane
• w Argument of perigee The angle 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 deg
• Mean motion 15.71202711 rev/day (semi-major axis
  derivable from this)
• Decay rate 1.05960E-04 rev/day2
• Epoch rev 2917
• Checksum 315

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State Vectors NonKeplerian 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/s ?V
  (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 Sun Drifts east in the sky 1 per day.
Rises 0.066 hours later each day. (because the
earth is orbiting) The Earth Rotates 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
the Sidereal day not the 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.
• Other Factors Gravitational irregularities
  such as Earth-axis wobbles, Moon, Sun, Jupiter
  gravity (tends to flatten inclination). Solar
  photon pressure. Insignificant for LEO primary
  perturbations elsewhere.

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LEO lt 1,000km (Satellite Telephones, ISS)
MEO 1,000km to 36,000km (GPS) GEO
36,000km (CommSats, HDTV) Deep
Space gt 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.
(above)
Ascending Node
(below)
Nodal Regression can be very useful.
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Sun-Synchronous Orbits

• Relies 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 used
used 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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Hohmann Transfer Orbit
Hohmann transfer orbit intersects both
orbits. Requires co-planar initial and ending
orbits. After 180, second burn establishes the
new orbit. Can be used to reduce or increase
orbit altitudes. By far the most common orbital
maneuver.
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Orbital Plane Changes
Burn must take place where the initial 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 stage restart where excess
fuel is often available.
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Geostationary Transfer Orbit GTO
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
1. launch to GTO
3. Hohmann circularizing burn
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Super GTO
3. Second Hohmann burn circularizes at GEO
GEO Target Orbit
Initial orbit has greater apogee 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.
1. Launch to Super GTO
2. Plane change plus initial Hohmann burn
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Low Thrust Orbit Transfer
A series of plane and altitude changes.
Continuous electric engine propulsion.
PROs Lower mass propulsion system. Same system
used for orbital maintenance.
CONs
Weeks or even months to reach final orbit. Van
Allen Radiation belts.
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Rendezvous
Launch when the orbital plane of the target
vehicle crosses launch pad. (Ideally) launch as
the target vehicle passes straight
overhead. Smaller transfer orbits slowly overtake
target (because of shorter orbit periods). Course
maneuvers designed to arrive in the same orbit at
the same true anomaly.
Apollo LM and CSM Rendezvous
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Orbital Debris a.k.a., Space Junk
February 2009 Iriduim / Cosmos collision created
gt 1,000 items gt 10cm diameter
Currently gt 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.
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Deep Space
Cassini Saturn orbit insertion using good ol
fashion rocket power.
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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 Highway designing
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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Assignments for April 2
Reading on Orbits SMAD ch 6 scan 5 and 7 TLOM
ch 3 and 4 scan 5 and 17
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
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).
Engineering 176 Orbits