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Orbits: Select, Achieve, Determine, Change

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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

- 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

- 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

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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

Actuator

ControlAlgorithm

Plant(satellite)

Error

Setpoint

Disturbances

Sensor

- Design Activity
- Team designations
- Mission selections
- Homework - ACS for mission

Engin 176 Meeting #5

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

v

b

r

X

F

F’

c

rp

- Reading
- SMAD 18
- SMAD 17 (if you haven’t already)
- TLOM: Launch sites

ra

a

Engin 176 Meeting #5

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

Engin 176 Meeting #5

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

Defines ellipse, circle, parabola, hyperbola

rF’

rF

F

F’

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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

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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)

Hyperbolic Asymptote

r

b

v

rp

a

Note to Orbital Racers: Lower means: - Higher velocity and - Shorter Orbit Period

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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)]

Hyperbolic Asymptote

r

b

v

rp

a

c

Orbit Elements:

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

Ω (longitude of ascending node)

w (argument of periapsis, ccw from Ω)

tp, q0… (epoch)

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Parabola: (mostly synthetic objects)

Escape (to V∞= 0)

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

Hyperbolic Asymptote

r

b

v

rp

a

c

Hyperbola: (mostly synthetic objects

Interplanetary & beyond

Escape with V∞> 0

Planetary Assist (accelerate & turn)

-> motion of M matters <-

e = 1 + V2∞ rp/µ

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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:

circlee = 0a = radius

ellipse0< e < 1a > 0

parabolae = 1a = ∞ (eq. above is useless)

hyperbolae > 1a < 0

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

Engin 176 Meeting #5

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.

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*NB: Earth axis rotation is not considered

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 flightAltitude 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 2g - 1 ]

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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

- 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

- #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

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- 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/sVelocity 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

- 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 spacecraftJn is “nth zonal harmonic coefficient”Pn is the “nth Legendre Polynomial”J1 = 1 (if there were a J1)J2 = 1.082 x10-3J3 = -2.54 x10-6J4 = -1.61 x10-6

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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

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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

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Polar Stationary

L4 (Stable)

L3 (Unstable)

L2 (Unstable)

L1 (Unstable)

Polar Stationary

L5 (Stable)

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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

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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 Termspinning 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