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
– Overfly @ 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
Meeting #5 Page 1
(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
•
•
•
•
•
•
•
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
• 5 - Launch Vehicles
• 6 - Power & Mechanisms
Engin 176 Meeting #5
Meeting #5 Page 2
Review of Last time
• Attitude Determination & Control
– 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
Set
point
Control
Plant
– Team designations
Actuator
Algorithm
(satellite)
Error
– Mission selections
Disturbances
– Homework - ACS for mission
Sensor
Engin 176 Meeting #5
Meeting #5 Page 3
Assignments for February 21
• Orbits
v
– Select minimum 2, preferable 3 orbits
your mission could use
b
– Create a trade table comparing
them
r
v
• Criteria could include:
–
–
–
–
Mission suitability (e.g. close or far enough)
X
Revisit or other attributes
F
F’to get there - and stay there
Cost
Environment for spacecraft
c
– For the selected orbit
r
a
• Describe it
(some set of orbit elements)
• How will you get there?
• How will you stay there?
• Estimate radiation & drag
rp
• Reading
Engin 176 Meeting #5
– SMAD 18
– SMAD 17
(if you
a haven’t already)
– TLOM: Launch
sites
Meeting #5 Page 4
LEO vs. GEO Orbit
• LEO: 1000 km
– Low launch
cost/risk
– Short range
– Global coverage
(not real time)
– Easy thermal
•
GEO:
36,000
environment
km
– Magnetic ACS
– Fixed GS Antenna
– Multiple small
– Constant visibility
satellites /
from 1 satellite
financial
– Nearly constant
“chunks”
sunlight
Engin 176 Meeting #5
– Minimal
– Zero doppler Meeting #5 Page 5
Describing Orbits
Kepler’s first law: All
orbits are described by
a Conic Section.
rF + rF’ = constant
Defines ellipse, circle,
parabola, hyperbola
rF’
F’
rF
F
Engin 176 Meeting #5
Meeting #5 Page 6
Elliptical Orbit Parameters
r’’ = G(M +m)/r2 (r/r)
= -µ/ r2 (r/r)
v
Two-Body equation
b
r
X
F’
(true anomaly)
F
c
rp
ra
Engin 176 Meeting #5
V
a
Meeting #5 Page 7
Circles, Ellipses and Beyond
Circle:
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)
b
r
v
rp
a
Note to Orbital Racers:
Lower means:
- Higher velocity and
- Shorter Orbit Period
Engin 176 Meeting #5
Meeting #5 Page 8
Circles, Ellipses and Beyond
Ellipse:
Transfer, Molniya, Reconnaissance orbits
Comets, Asteroids
Real Planets, Moons, LEOs, GEOs
Kepler’s 2nd law
e=c/a
r = p / [1+ e cos(v)]
Orbit Elements:
r
b
a
c
v
rp
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
Meeting #5 Page 9
Circles, Ellipses and Beyond
Parabola: (mostly synthetic objects)
Escape (to V∞= 0)
V(parabola) = Vesc = [2p/r]1/2
r
b
a
Hyperbola: (mostly synthetic objects
c
Interplanetary & beyond
v
rp
Escape with V∞> 0
Planetary Assist (accelerate & turn)
-> motion of M matters <e = 1 + V2∞ rp/µ
Engin 176 Meeting #5
Meeting #5 Page 10
2-Body 2-D Solution
NB: 3 terms, a, e, v,
completely define
position in planar orbit Position, r, depends on:
all that’s left is to define
a (semi-major axis)
the orientation of that
e(eccentricity = c/a
(= distance between foci /major axis)plane
n (polar angle or true anomaly)
r = a (1-e2) / (1 + e cos n)
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 Engin 176 Meeting
a < 0#5
Meeting #5 Page 11
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 *NB:
MeetingEarth
#5
axis rotation is not
considered
Meeting
#5 Page 12
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
Meeting #5 Page 13
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
Escaping the solar system
¿So how do they do this?
• 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
Engin 176 Meeting #5
Meeting #5 Page 14
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
Meeting #5 Page 15
Orbital Trick #2: Getting There
• #1: Raise altitude from 0 to 300 km
(this is the easy part)
= 100 kg x 9.8 m/s2 x 300,000 m
= 2.94 x 108 kg m2/ s2 [=W-s = J] = 82 kw-hr
= 2.94 x 106 m2/ s2 per kg
∆V = (E)1/2 = 1715 m/s
– Energy = mgh
• #2: Accelerate to orbital velocity, 7 km/s
(the harder part)
– ∆V (velocity)
= 7000 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:
Engin 176 Meeting #5
∆V = gIsp ln(Mo/Mbo)
=> Mbo / Mo = 1/ exp[∆V/gIsp])
For Isp 420, Mbo = 10% Mo
Meeting #5 Page 16
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 = 1686 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) = 6800 m/s (97% of ground ∆V, 94% of energy)
– ∆V (∆H)
– ∆V (total, with airplane)
– ∆V (total, from ground)
Velocity saving: 4%
Energy saving: 8%
= 1686 m/s (98% of ground ∆V, 96% of energy)
= 8486 m/s + 1.3 km/s loss = 8800 m/s
= 8715 m/s + 1.5 km/s loss = 9200 m/s
Downsides: Human rating, limited dimension & mass,
limited propellant choices, cost of airplane
(aircraft doesn’t fully replace a stage)
Engin 176 Meeting #5
Meeting #5 Page 17
Orbital Trick #3: Sun Synch
• Earth needs a belt:
it is 0.33% bigger (12756 v. 12714 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 = 1.082 x10-3
J3 = -2.54 x10-6
J4 = -1.61 x10-6
Engin 176 Meeting #5
Meeting #5 Page 18
Orbital Trick #3: Sun Synch (continued)
Nodal Regression, how it works, and how well
Extra Pull
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
Meeting #5 Page 19
Ordinary Orbits
• Remote Sensing:
– 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
Meeting #5 Page 20
Lagrange Points
L4
(Stable)
L3
(Unstable)
L1
(Unstable)
L5
(Stable)
Engin 176 Meeting #5
Polar
Stationary
L2
(Unstable)
Polar
Stationary
Meeting #5 Page 21
GPS
4 position vectors =>
in 1 slide
Solve for 4 unknowns:
4 pseudo path lengths
- 3 position coordinates of user
- time correction of user’s clock
Freebies
- Atomic clock accuracy to user
- Velocity
Engin
176 Meeting #5 via multiple fixes
Meeting #5 Page 22
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: 365.25 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
Engin 176 Meeting #5
Meeting #5 Page 23
caused by applied torque