Engineering 176
Orbital Design
Mr. Ken Ramsley
kenneth_ramsley@brown.edu
(508) 881- 5361
Class Topics
When Orbits Were Perfect (and politically dangerous)
Einstein’s Geodesics (the art and science of motion)
Kepler’s Three Laws (based on Tycho’s meticulous data)
Orbital Elements Defined and Illustrated
Useful Orbits and Maneuvers to Get There
Interplanetary Space and Beyond
EN176 Orbital Design
The Ancients
Aristotle (384 BC – 322 BC)
Claudius Ptolemaeus (AD 83 – c.168)
Copernicus and Tycho
Nicolaus Copernicus (1473 - 1543)
Tycho Brahe (1546 - 1601)
The Copernicus Solar System
Image: Courtesy of tychobrahe.com
Tycho Brahe's Uraniborg
Observatory and 90°
Star Sighting Quadrant
Kepler and Galileo
Johannes Kepler (1571 - 1630)
Galileo Galilei (1564 - 1642)
Newton and LaGrange
Isaac Newton (1643 - 1727)
Joseph Louis Lagrange (1736-1813)
Einstein
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)
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?
Kepler’s First Law
Every orbit is
an ellipse
with the Sun
(main body)
located at
one foci.
Kepler’s Second Law
Day 40
Day 50
Day 30
Day 60
Day 20
Day 70
Day 80
Day 90
Day 10
Day 100
A line between an orbiting
body and primary body
sweeps out equal areas in
equal intervals of time.
Day 110
Day 120
Day 0
Kepler’s Third Law
This defines the
relationship of
Orbital Period &
Average Radius
for any two
bodies in orbit.
EXAMPLE:
R2
Earth
R1
For a given body,
the orbital period
and average
distance for the
second orbiting
body is:
P1 P2
P = 1 Year
R = 1 AU
Mars
P = 1.88 Years
R = 1.52 AU
P 2 = R3
P = Orbital Period
R = Average Radius
Vernal Equinox – The Celestial Baseline
First some
astronomy…
June 21st
When the Sun
passes over the
equator moving
south to north.
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
Vernal Equinox
(March 20th)
Defines a fixed
vector in space
through the center
of the Earth to a
known celestial
coordinate point.
Conic Sections (shape) Eccentricity
•
•
•
•
e=0
e<1
e=1
e>1
-- circle
-- ellipse
-- parabola
-- hyperbola
e < 1 Orbit is ‘closed’ – recurring path (elliptical)
e > 1 Not an orbit – passing trajectory (hyperbolic)
Keplerian Elements
e
e, a, and v (3 of 6)
120°
150°
90°
Eccentricity
(0.0 to 1.0)
Apogee
180°
v
True anomaly
(angle)
a
Perigee
0°
Semi-major
axis
(nm or km)
e=0.8 vrs e=0.0
Apo/Peri gee – Earth
Apo/Peri lune – Moon
Apo/Peri helion – Sun
Apo/Peri apsis – non-specific
e
a
v
defines ellipse shape
defines ellipse size
defines satellite angle from perigee
Inclination i
Intersection of the
equatorial and
orbital planes
(4th Keplerian Element)
Inclination
(above)
(angle)
i
(below)
Ascending
Node
Equatorial Plane
( defined by Earth’s equator )
Sample inclinations
Ascending Node is where a
satellite crosses the equatorial
plane moving south to north
0° -- Geostationary
52° -- ISS
98° -- Mapping
Right Ascension [1] of the ascending node Ω
and Argument of perigee ω (5th and 6th Elements)
Ω = angle from
vernal equinox to
ascending node on
the equatorial plane
Perigee Direction
ω = angle from
ascending node to
perigee on the
orbital plane
ω
Ω
Ascending
Node
[1]
Vernal Equinox
Right Ascension is the astronomical
term for celestial (star) longitude.
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
Ω = 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.
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/day^2
Epoch rev: 2917
Checksum: 315
State Vectors
NonKeplerian Coordinate System
Cartesian x, y, z, and 3D velocity
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).
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.
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.
Ground Tracks
Ground tracks drift
westward as the Earth
rotates below an orbit.
Each orbit type has a
signature ground tract.
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
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.
‘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.
Nodal Regression
Orbital planes
rotate eastward
over time.
(above)
Ascending
Node
(below)
Nodal Regression
can be very useful.
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).
Molniya - 12hr Period
‘Long loitering’ high latitude apogee. Once used
used for early warning by both USA and USSR
‘Tundra’ Orbit - 24hr Period
Higher apogee than Molniya. For dwelling over
a specific upper latitude (Used only by Sirius)
GPS Constellation ~ 20200km alt.
GPS: Six orbits with six
equally-spaced satellites
occupying each orbit.
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.
Orbital Plane Changes
Burn must take place where the
initial and target planes intersect.
Even a small amount of plane
change requires lots of ΔV
θ
Less ΔV 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 ΔV = 2 x Vorbit x sin(θ/2)
Example: Orbit Velocity = 7000m/s, Target Inclination Change = 30°
Plane Change ΔV = 2 x 7000m/s x sin(30°/ 2)
Plane Change ΔV = 3623m/s
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.
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
3. Second
Hohmann burn
circularizes at
GEO
‘Super GTO’
GEO
Target
Orbit
Initial orbit has greater
apogee than standard
GTO.
Plane change at much
higher altitude requires
far less ΔV.
PRO: Less overall ΔV
from higher inclination
launch sites.
CON: Takes longer to
establish the final orbit.
2. Plane change
plus initial
Hohmann burn
1. Launch to
‘Super GTO’
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.
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
Orbital Debris a.k.a., ‘Space Junk’
February 2009 Iriduim / Cosmos collision created > 1,000 items > 10cm diameter
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.
Deep Space
Cassini – Saturn orbit
insertion using good ‘ol
fashion rocket power.
Using Lagrange Points to ‘stay put’
Halo Orbits (stability from motion)
AeroBraking
Earth, Mars, Jupiter, etc.
“The poor man’s Hohmann maneuver”
The Solar System ‘Super Highway’
…designing geodesic trajectories – like tossing a message bottle
into the sea at exactly the right time, direction, and velocity.
Gravity Assist (Removing Velocity)
Gravity Assist (adding velocity)
Solar Escape
Multiple Mission
Trajectories
Complex Orbital Trajectories
Galileo (Jupiter)
Cassini (Saturn)
Designing Deep
Space Missions
…yes, there are software tools for this
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.
Create a trade table to
compare orbit designs.
For the selected orbits: Trade criteria should include:
Describe it (orbital elements)
How will you get there?
How will you stay there?
Estimate perturbations
Orbit suitability for mission.
Cost to get there – and stay there.
Space environment (e.g., radiation).
Engineering 176 Orbits