Orbital Mechanics and Design
Transcript Orbital Mechanics and Design
Mr. Ken Ramsley
(508) 881- 5361
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
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)
Geodesics: The Science and Art
of 4D Curved Space Trajectories.
All objects in
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
with the Sun
Kepler’s Second Law
A line between an orbiting
body and primary body
sweeps out equal areas in
equal intervals of time.
Kepler’s Third Law
This defines the
Orbital Period &
for any two
bodies in orbit.
For a given body,
the orbital period
distance for the
P = 1 Year
R = 1 AU
P = 1.88 Years
R = 1.52 AU
P 2 = R3
P = Orbital Period
R = Average Radius
Vernal Equinox – The Celestial Baseline
When the Sun
passes over the
south to north.
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).
Defines a fixed
vector in space
through the center
of the Earth to a
Conic Sections (shape) Eccentricity
e < 1 Orbit is ‘closed’ – recurring path (elliptical)
e > 1 Not an orbit – passing trajectory (hyperbolic)
e, a, and v (3 of 6)
(0.0 to 1.0)
(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
defines ellipse shape
defines ellipse size
defines satellite angle from perigee
Intersection of the
(4th Keplerian Element)
( defined by Earth’s equator )
Ascending Node is where a
satellite crosses the equatorial
plane moving south to north
0° -- Geostationary
52° -- ISS
98° -- Mapping
Right Ascension  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
ω = angle from
ascending node to
perigee on the
Right Ascension is the astronomical
term for celestial (star) longitude.
The Six Keplerian Elements
= Semi-major axis (usually in
kilometers or nautical miles)
= Eccentricity (of the elliptical
= True anomaly The angle
between perigee and satellite in
the orbital plane at a specific time
= 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
angle measured between the
ascending node and perigee
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
Catalog Number: 25544
Epoch time: 09061.52440963 = yrday.fracday
Element set: 900
Inclination: 51.6398 deg
RA of ascending node: 133.2909 deg
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
NonKeplerian Coordinate System
Cartesian x, y, z, and 3D velocity
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).
Alt-Az only (best fit from many observations).
Launch From Vertical Takeoff
• Raising your altitude from 0 to 300 km
– 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
– ∆V (velocity) = 7000 m/s
– ∆V (altitude) = 1715 m/s
– ∆V (total)
= 8715 m/s 
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 drift
westward as the Earth
rotates below an orbit.
Each orbit type has a
signature ground tract.
More Astronomy Facts
Drifts east in the sky ~1° per day.
Rises 0.066 hours later each day.
(because the earth is orbiting)
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
“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.
can be very useful.
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
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 Plane Changes
Burn must take place where the
initial and target planes intersect.
Even a small amount of plane
change requires lots of ΔV
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
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
1. launch to
Initial orbit has greater
apogee than standard
Plane change at much
higher altitude requires
far less ΔV.
PRO: Less overall ΔV
from higher inclination
CON: Takes longer to
establish the final orbit.
2. Plane change
1. Launch to
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.
Launch when the
orbital plane of the
target vehicle crosses
(Ideally) launch as the
target vehicle passes
Smaller transfer orbits
slowly overtake target
(because of shorter
designed to arrive in
the same orbit at the
same true anomaly.
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.
Cassini – Saturn orbit
insertion using good ‘ol
fashion rocket power.
Using Lagrange Points to ‘stay put’
Halo Orbits (stability from motion)
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)
Complex Orbital Trajectories
…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
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?
Orbit suitability for mission.
Cost to get there – and stay there.
Space environment (e.g., radiation).
Engineering 176 Orbits