1. Orbits: Select, Achieve, Determine, Change
Physical Background
Newton, Kepler et al.
Coordinate systems
Orbit Transfers
Orbit Elements
Orbit survey
LEO / MEO / GTO / GEO
Sun Synch
Interplanetary, escapes
Capture, flyby & assist
Ask an Orbitalogist if you need to:
Stay solar illuminated
constant time of day
Maintain constant position
(over equator, pole, sun/earth)
With another satellite
Constellation configuration
Rendezvous
Escape / assist / capture
Determine orbit from observation
Determine location from orbit
Optimize Ground Station location
Estimate orbit lifetime + tell you nav strategy & ∆V
Engin 176 Meeting #5

2. (Re) Orientation 1 - Introduction 2 - Propulsion & ∆V
3 - Attitude Control & instruments
4 - Orbits & Orbit Determination
LEO, MEO, GTO, GEO
Special LEO orbits
Orbit Transfer
Getting to Orbit
GPS
5 - Launch Vehicles
6 - Power & Mechanisms
7 - Radio & Comms
8 - Thermal / Mechanical Design. FEA
9 - Reliability
10 - Digital & Software
11 - Project Management Cost / Schedule
12 - Getting Designs Done
13 - Design Presentations
Another night of F=MA?
Education is the process of realizing that you don’t know what you thought you knew...
Engin 176 Meeting #5

3. Review of Last time Attitude Determination & Control Design Activity
Feedback Control
Systems description
Simple simulation
Attitude Strategies
The simple life
Eight other approaches and variations
Disturbance and Control forces (note re CD>1)
Design build & test an Attitude Control System
Design Activity
Team designations
Mission selections
Homework - ACS for mission
Or in a word, F=MA
Plant (satellite)
Set point
Error
Control Algorithm
Sensor
Disturbances
Actuator
Engin 176 Meeting #5

4. Assignments for February 21
Orbits
Select minimum 2, preferable 3 orbits your mission could use
Create a trade table comparing them
Criteria could include:
Mission suitability (e.g. close or far enough)
Revisit or other attributes
Cost to get there - and stay there
Environment for spacecraft
For the selected orbit
Describe it (some set of orbit elements)
How will you get there?
How will you stay there?
Estimate radiation & drag v b v r X F F’ c rp ra
Reading
SMAD 18
SMAD 17 (if you haven’t already)
TLOM: Launch sites
Or in a word, F=MA a
Engin 176 Meeting #5

5. LEO vs. GEO Orbit LEO: 1000 km GEO: 36,000 km Low launch cost/risk
Short range
Global coverage (not real time)
Easy thermal environment
Magnetic ACS
Multiple small satellites / financial “chunks”
Minimal propulsion
GEO: 36,000 km
Fixed GS Antenna
Constant visibility from 1 satellite
Nearly constant sunlight
Zero doppler
Engin 176 Meeting #5

6. Describing Orbits
Kepler’s first law: All orbits are described by a Conic Section. rF + rF’ = constant
Defines ellipse, circle, parabola, hyperbola
rF’ rF F F’
Engin 176 Meeting #5

7. Elliptical Orbit Parameters
r’’ = G(M +m)/r2 (r/r)
= -µ/ r2 (r/r)
Two-Body equation v
V (true anomaly) b r X F F’ c rp ra a
Engin 176 Meeting #5

8. Circles, Ellipses and Beyond
Planets, Moons, LEOs, GEOs,
Vcirc = [µ/r]1/2
Vesc =[2]1/2 Vcirc
T = 2π (a3/µ)1/2 (kepler’s 3rd Law)
Orbit elements: r, i (T, i)
plus tp, q0… (epoch)
Hyperbolic Asymptote r b v rp a
Note to Orbital Racers: Lower means: - Higher velocity and - Shorter Orbit Period
Engin 176 Meeting #5

9. Circles, Ellipses and Beyond
Transfer, Molniya, Reconnaissance orbits
Comets, Asteroids
Real Planets, Moons, LEOs, GEOs
Kepler’s 2nd law
e = c / a
r = p / [1+ e cos(v)]
Hyperbolic Asymptote r b v rp a c
Orbit Elements:
a (or p), e (geometry) plus i p= a(1-e2)
Ω (longitude of ascending node)
w (argument of periapsis, ccw from Ω)
tp, q0… (epoch)
Engin 176 Meeting #5

10. Circles, Ellipses and Beyond
Parabola: (mostly synthetic objects)
Escape (to V∞= 0)
V(parabola) = Vesc = [2p/r]1/2
Hyperbolic Asymptote r b v rp
Hyperbola: (mostly synthetic objects
Interplanetary & beyond
Escape with V∞> 0
Planetary Assist (accelerate & turn)
-> motion of M matters <-
e = 1 + V2∞ rp/µ a c
Engin 176 Meeting #5

11. 2-Body 2-D Solution
NB: 3 terms, a, e, v, completely define position in planar orbit - all that’s left is to define the orientation of that plane
r = a (1-e2) / (1 + e cos n)
Position, r, depends on: a (semi-major axis) e(eccentricity = c/a
(= distance between foci /major axis)
n (polar angle or true anomaly)
4 major type of orbits:
circle e = 0 a = radius
ellipse 0< e < 1 a > 0
parabola e = 1 a = ∞ (eq. above is useless)
hyperbola e > 1 a < 0
Engin 176 Meeting #5

12. The 6 Classical Orbit Elements*
3 elements (previous page) describe the conic section & position.
a - semi-major axis - scale ( in kilometers) of the orbit.
e - eccentricity - (elliptical, circular, parabolic, hyperbolic)
v true anomaly - the angle between the perigee & the position vector to the spacecraft - determines where in the orbit the S/C is at a specific time.
3 additional elements describe the orbit plane itself
i - inclination - the angle between the orbit normal and the (earth polar) Z-direction. How the orbit plane is ‘tilted’ with respect to the Equator.
Ω - longitude or right ascension of the ascending node - the angle in degrees from the Vernal Equinox (line from the center of the Earth to the Sun on the first day of autumn in the Northern Hemisphere) to the ascending node along the Equator. This determines where the orbital plane intersects the Equator (depends on the time of year and day when launched).
w, argument of perigee - the angle in degrees, measured in the direction and plane of the spacecraft’s motion, between the ascending node and the perigee point. This determines where the perigee point is located and therefore how the orbit is rotated in the orbital plane.
Engin 176 Meeting #5
*NB: Earth axis rotation is not considered

13. Why 6 Orbit Elements?
1-D: Example: mass + spring like the dynamic model of last week position (1 number) plus velocity (1 number) necessary
2-D: Example: air hockey puck or single ball on pool table X & Y position, plus Velocity components along X & Y axes
3-D: Example: baseball in flight Altitude and position over field + 3-D velocity vector
Alternative Orbit determination systems
GPS: Latitude, Longitude, Altitude and 3-D velocity vector
Radar: Distance, distance rate, azimuth, elevation, Az rate, El rate
Ground sitings: Az El only (but done at many times / locations)
Breaking it down: Range R and Velocity V
R X V = h angular momentum vector = constant dot prod. with pole to get i
e2 = 1 + 2E(h/µ)2 where E = V 2 /2 - µ/r
For sing = R . V/RV (g is flight path angle to local horizon): tanq = (RV 2/µ)singcosg / [ (RV 2 /µ)cos 2 g - 1 ]
Engin 176 Meeting #5

14. Orbit facts You Already Know
Must be geosynch at the equator (q=0)
Orbit planes & inclination are fixed
Knowing instantaneous position velocity fully determines the orbit
Orbit plane must include injection point and earth’s CG (hence the concept of a launch window)
Dawn / Dusk orbit in June is Noon / Midnight in September
Escaping the solar system
¿So how do they do this?
Engin 176 Meeting #5

15. Orbital Trick #1: Orbit Transfer
Where new & old orbit intersect, change V to vector appropriate to new orbit
If present and desired orbit don’t intersect: Join them via an intermediate that does
Do V & i changes where V is minimum (at apogee)
Orbit determination: requires a single simultaneous measurement of position + velocity. GPS and / or ground radar can do this.
Engin 176 Meeting #5

16. Orbital Trick #2: Getting There
#1: Raise altitude from 0 to 300 km (this is the easy part)
Energy = mgh = 100 kg x 9.8 m/s2 x 300,000 m = x 108 kg m2/ s2 [=W-s = J] = 82 kw-hr = x 106 m2/ s2 per kg ∆V = (E)1/2 = m/s
#2: Accelerate to orbital velocity, 7 km/s (the harder part)
∆V (velocity) = m/s (80% of V, 94% of energy)
∆V (altitude) = 1715 m/s
∆V (total) = 8715 m/s (+ about 1.5 km/s drag + g loss)
Note to Space Tourists:
∆V = gIsp ln(Mo/Mbo) => Mbo / Mo = 1/ exp[∆V/gIsp]) For Isp 420, Mbo = 10% Mo
Engin 176 Meeting #5

17. Orbital Trick #2’: Getting help?
Launch From Airplane at 10 km altitude and 200 m/s
#1: Raise altitude from 10 to 300 km
Energy = mgh = 100kg x 9.8 m/s2 x (300,000 m - 10,000 m) ∆V = (E)1/2 = m/s (98% of ground based launch ∆V) (or 99% of ground based launch energy)
#2: Accelerate to 7 km/s, from 0.2 km/s ∆V (velocity) = m/s (97% of ground ∆V, 94% of energy)
∆V (∆H) = m/s (98% of ground ∆V, 96% of energy)
∆V (total, with airplane) = 8486 m/s km/s loss = 8800 m/s
∆V (total, from ground) = 8715 m/s km/s loss = 9200 m/s Velocity saving: 4% Energy saving: 8% Downsides: Human rating, limited dimension & mass, limited propellant choices, cost of airplane (aircraft doesn’t fully replace a stage)
Engin 176 Meeting #5

18. Orbital Trick #3: Sun Synch
Earth needs a belt: it is 0.33% bigger (12756 v km diameter) in equatorial circumference than polar circumference
Earth’s shape as sphere + variations. Potential, U is: U(R, q, f) = -µ/r + B(r, q, f) => U = -(µ/r)[1 - ∑2∞(Re/r)nJnPn (cosq)] Re = earth radius; r = radius vector to spacecraft Jn is “nth zonal harmonic coefficient” Pn is the “nth Legendre Polynomial” J1 = 1 (if there were a J1) J2 = x10-3 J3 = x10-6 J4 = x10-6
Engin 176 Meeting #5

19. Orbital Trick #3: Sun Synch (continued)
Extra Pull
Nodal Regression, how it works, and how well
Intuitive explanations: #1: Extra Pull causes earlier equator crossing
#2: Extra Pull is a torque applied to the H vector
Equator
Extra Pull
Engin 176 Meeting #5

20. Ordinary Orbits GEO Remote Sensing: LEO Comms: Equatorial:
Favors polar, LEO, 2x daily coverage (lower inclinations = more frequent coverage).
Harmonic orbit: period x n = 24 (or 24m) hours (n & m integer)
LEO Comms:
Same! - multiple satellites reduce contact latency. Best if not in same plane.
Equatorial:
Single satellite provides latency < 100 minutes; minimum radiation environment
• Sun Synch:
Dawn/Dusk offers Constant thermal environment & constant illumination (but may require ∆V to stay sun synch)
Elliptical:
Long dwell at apogee, short pass through radiation belts and perigee...
Molniya. Low E way to achieve max distance from earth.
MEO:
Typically 10,000 km. From equator to 45 or more degrees latitude
GEO
Engin 176 Meeting #5

21. Lagrange Points Polar Stationary L4 (Stable) L3 (Unstable)
Engin 176 Meeting #5

22. GPS in 1 slide 4 position vectors => 4 pseudo path lengths
Solve for 4 unknowns:
- 3 position coordinates of user
- time correction of user’s clock
Freebies
- Atomic clock accuracy to user
- Velocity via multiple fixes
Engin 176 Meeting #5

23. Non-Obvious Terms Nodes: ascending, descending, line of nodes
True Anomaly: angle from perigee
Inclination (0, 180, 90, <90, >90)
Ascension, Right Ascension
Conjunction = same RA (see vernal eq)
Argument of perigee (w from RA)
Declination (~= elevation)
Geoid - geopotential surface
Julian Calendar: days
Gregorian: Julian + skip leap day in 1900, 2100…
Ephemerides
Frozen Orbits (sun synch, Molniya)
Periapsis, Apoapsis
Vernal Equinox (equal night)
Solstice
Ecliptic (and eclipses)
Siderial
Terminator
Azimuth, Elevation
Oblate / J2 Term spinning about minor axis (earth)
Prolate: spinning about major axis (as a football)
Precession: steady variation in h caused by applied torque
Nutation: time varying variation in h caused by applied torque
Engin 176 Meeting #5