Engin 176 Meeting #5 Meeting #5 Page 1 Orbits: Select, Achieve, Determine, Change Physical Background –Newton, Kepler et al. –Coordinate systems –Orbit

Engin 176 Meeting #5 Meeting #5 Page 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– 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


• 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

(Re) Orientation• 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


Actual

• 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

Plant(satellite)

Setpoint Error

ControlAlgorithm

Sensor

Disturbances

Actuator

• Design Activity– Team designations

– Mission selections

– Homework - ACS for mission

Review of Last time


FF’

ra

r

X

b

v

crp

a

v

• 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

Assignments for February 21

• Reading– SMAD 18– SMAD 17

(if you haven’t already)

– TLOM: Launch sites

Meeting #5 Page 5Engin 176 Meeting #5

16x magnificationGEO:

36,100 km(22,400 mi)

LEO vs. GEO Orbit•LEO: 1000 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


Describing Orbits

FF’

rF’ rF

Kepler’s first law: All orbits are described by a Conic Section. rF + rF’ = constant

Defines ellipse, circle, parabola, hyperbola


Elliptical Orbit Parameters

FF’

ra

r

X

b

v

crp

a

r’’ = G(M +m)/r2 (r/r)

= -µ/ r2 (r/r)

Two-Body equation V (true anomaly)


Circles, Ellipses and Beyond

r

a rp

vb

Hyp

erbo

lic A

sym

ptot

e

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, 0… (epoch)

Note to Orbital Racers:

Lower means: - Higher velocity and - Shorter Orbit Period



r

a rp

vb

Hyp

erbo

lic A

sym

ptot

e

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)]

cOrbit Elements:

a (or p), e (geometry) plus ip= a(1-e2)

Ω (longitude of ascending node)

(argument of periapsis, ccw from Ω)

tp, 0… (epoch)



r

a rp

vb

Hyp

erbo

lic A

sym

ptot

eParabola: (mostly synthetic objects)

Escape (to V∞= 0)

V(parabola) = Vesc = [2p/r]1/2

cHyperbola: (mostly synthetic objects

Interplanetary & beyond

Escape with V∞> 0

Planetary Assist (accelerate & turn)

-> motion of M matters <-

e = 1 + V2∞ rp/µ


r = a (1-e2) / (1 + e cos )

Position, r, depends on: a (semi-major axis)e(eccentricity = c/a (= distance between foci /major axis) (polar angle or true anomaly)

4 major type of orbits:circle e = 0 a = radiusellipse 0< e < 1 a > 0parabola e = 1 a = ∞ (eq. above is

useless)hyperbola e > 1 a < 0

2-Body 2-D SolutionNB: 3 terms, a, e, v, completely define position in planar orbit - all that’s left is to define the orientation of that plane


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.*NB: Earth axis rotation is not

considered


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 tableX & 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 sin = R . V/RV ( is flight path angle to local horizon):

tan = (RV 2/µ)sincos / [ (RV 2 /µ)cos 2 - 1 ]


Orbit facts You Already Know• Must be geosynch at the equator (=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

¿So how do they do this?

Escaping the solar system


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.


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= 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

• #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:

∆V = gIsp ln(Mo/Mbo)

=> Mbo / Mo = 1/

exp[∆V/gIsp])

For Isp 420, Mbo = 10% Mo


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) = 1686 m/s (98% of ground ∆V, 96% of energy)

– ∆V (total, with airplane) = 8486 m/s + 1.3 km/s loss = 8800 m/s– ∆V (total, from ground) = 8715 m/s + 1.5 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)


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, ) = -µ/r + B(r, ) => U = -(µ/r)[1 - ∑2

∞(Re/r)nJnPn (cos)]Re = earth radius; r = radius vector to spacecraftJn 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


Earth Oblateness Perturbation

-15

-10

-5

0

5

10

15

0 50 100 150 200

Inclination (°)

100 x 100 km Orbit2000 x 2000 km Orbit

Orbital Trick #3: Sun Synch (continued)

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 Equato

r

Extra Pull

Extra Pull


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


Lagrange Points

L1 (Unstable)

L2 (Unstable)

L3 (Unstable)

L4 (Stable)

L5 (Stable)

Polar Stationary

Polar Stationary


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


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 caused by applied torque

Documents

Engin 176 Meeting #5 Meeting #5 Page 1 Orbits: Select, Achieve, Determine, Change Physical Background –Newton, Kepler et al. –Coordinate systems –Orbit