orbital mechanics and design_rev17

8/8/2019 Orbital Mechanics and Design_Rev17

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Engineering 176

Orbital Design

Mr. Ken [email protected]

(508) 881- 5361


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The Ancients

Aristotle (384 BC 322 BC) Claudius Ptolemaeus (AD 83 c.168)


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Copernicus and Tycho

Nicolaus Copernicus (1473 - 1543) Tycho Brahe (1546 - 1601)


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The Copernicus Solar System

Tycho Brahe's Uraniborg

Observatory and 90

Star Sighting Quadrant

Image: Courtesy of tychobrahe.com


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Kepler and Galileo

Johannes Kepler (1571 - 1630) Galileo Galilei (1564 - 1642)


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Newton and LaGrange

Isaac Newton (1643 - 1727) Joseph Louis Lagrange (1736-1813)


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Einstein


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Geodesics: The Science and Artof 4D Curved Space Trajectories.

All objects in

motion conserve

momentum

through a

balance of

Gravity Potential

and

Velocity Vector

(think rollercoaster)


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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?


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Keplers First Law

Every orbit is

an ellipsewith the Sun

(main body)

located at

one foci.


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Keplers Second Law

A line between an orbiting

body and primary body

sweeps out equal areas in

equal intervals of time.

Day 0

Day 10

Day 20

Day 30Day 40

Day 50

Day 60

Day 70

Day 80

Day 90

Day 100

Day 110Day 120


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Keplers Third Law

P2 = R3

P1 P2

R2

R1

EXAMPLE:

Earth

P = 1 Year

R = 1 AU

Mars

P = 1.88 Years

R = 1.52 AU

This defines the

relationship of

Orbital Period &

Average Radius

for any two

bodies in orbit.

For a given body,

the orbital period

and average

distance for the

secondorbiting

body is:

P = Orbital Period

R = Average Radius


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Vernal EquinoxThe Celestial Baseline

First some

astronomy

When the Sun

passes over the

equator movingsouth to north.

Vernal Equinox(March 20th)

Defines a fixed

vector in space

through the center

of the Earth to a

known celestial

coordinate point.

June 21st

December 22nd

Sun

The Vernal Equinox drifts ~0.014

/ year. Orbits are thereforecalculated for a specified date

and time, (most often Jan 1,

2000, 2050 or today).

Epoch 2000


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Conic Sections (shape) Eccentricity

e=0 -- circle

e1 -- hyperbola

e < 1 Orbit is closed recurring path (elliptical)

e > 1 Not an orbit passing trajectory (hyperbolic)


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Keplerian Elements e, a, and v(3 of 6)

Perigee

0

Apogee

180

e defines ellipse shape

a defines ellipse size

v defines satellite angle from perigee

Semi-major

axis(nm or km)

True anomaly

(angle)

Eccentricity

(0.0 to 1.0)

Apo/Peri gee Earth

Apo/Peri lune Moon

Apo/Peri helion Sun

Apo/Peri apsis non-specific

90120

a

ev

150

e=0.8 vrs e=0.0


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Inclination i (4th Keplerian Element)

Inclination(angle)

Equatorial Plane

( defined by Earths equator )

Intersection of the

equatorial and

orbital planes

(below)

(above)

Sample inclinations0 -- Geostationary

52 -- ISS

98 -- Mapping

Ascending

Node

Ascending Node is where a

satellite crosses the equatorial

plane moving south to north

i


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RightAscension [1]oftheascendingnode

andArgumentofperigee (5th and 6th Elements)

Vernal Equinox

Perigee Direction

= angle fromvernal equinox to

ascending node on

the equatorial plane

= angle fromascending node to

perigee on the

orbital plane

[1] Right Ascension is the astronomical

term for celestial (star) longitude.

Ascending

Node


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The Six Keplerian Elements

a = Semi-major axis (usually inkilometers or nautical miles)

e = Eccentricity (of the ellipticalorbit)

v = True anomaly The anglebetween perigee and satellite in

the orbital plane at a specific time

i = Inclination The angle betweenthe orbital and equatorial planes

= Right Ascension (longitude)of the ascending node The

angle from the Vernal Equinoxvector to the ascending node on

the equatorial plane

[ = Argument of perigee Theangle measured between the

ascending node and perigee

Shape, Size,

Orientation,and Satellite

Location.


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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

Satellite: ISS

Catalog Number: 25544

Epoch time: 09061.52440963 = yrday.fracday

Element set: 900

Inclination: 51.6398 deg

RA of ascending node: 133.2909 deg

Eccentricity: .0009235

Arg of perigee: 79.9705 deg

Mean anomaly: 280.2498 degMean motion: 15.71202711 rev/day (semi-major axis derivable from this)

Decay rate: 1.05960E-04 rev/day^2

Epoch rev: 2917

Checksum: 315


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State VectorsNonKeplerian Coordinate System

Cartesian x, y, z, and 3D velocity


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Orbit determination

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).

Ground Sightings:Alt-Az only (best fit from many observations).


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Launch From Vertical Takeoff

Raising your altitude from 0 to 300 km (standing jump)

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 (orbital insertion)

V (velocity) = 7000 m/s

V (altitude) = 1715 m/s

V (total) = 8715 m/s [1]

[1] plus another 1500 m/s lost to drag during early portion of flight.


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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/sV (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.


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Ground Tracks

Ground tracks drift

westward as the Earth

rotates below an orbit.

Each orbit type has a

signature ground tract.


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More Astronomy Facts

The SunDrifts east in the sky ~1 per day.Rises 0.066 hours later each day.

(because the earth is orbiting)

The EarthRotates 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 theSidereal day notthe solar day


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Orbital Perturbations

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.

OtherFactors Gravitational irregularities such asEarth-axis wobbles, Moon, Sun, Jupiter gravity (tends to

flatten inclination). Solar photon pressure. Insignificant

for LEO primary perturbations elsewhere.


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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.


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Nodal Regression

Orbital planes

rotate eastward

over time.

(below)

(above)

Ascending

Node

Nodal Regression

can be very useful.


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Sun-Synchronous OrbitsRelies 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).


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Molniya - 12hr Period

Long loitering high latitude apogee. Once usedused for early warning by both USA and USSR


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Tundra Orbit - 24hr Period

Higher apogee than Molniya. For dwelling over

a specific upper latitude (Used only by Sirius)


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GPS Constellation ~ 20200km alt.

GPS: Six orbits with six

equally-spaced satellites

occupying each orbit.


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Orbital Plane Changes

Burn must take place where theinitial and target planes intersect.

Even a small amount of plane

change requires lots of V

Less V 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 V = 2 x Vorbit x sin(/2)

Example: Orbit Velocity = 7000m/s, Target Inclination Change = 30

Plane Change V = 2 x 7000m/s x sin(30/ 2)

Plane Change V = 3623m/s


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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 stagerestart where excess fuel is

often available.


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Geostationary Transfer OrbitGTO

Requires plane change

and circularizing burns.

Less plane changing is

required when launched

from near the equator.2. Plane change

where GTO plane

intersects GEO

plane

3. Hohmann

circularizing burn

1. launch to

GTO


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Super GTO

Initial orbit has greaterapogee than standard

GTO.

Plane change at much

higher altitude requires

far less V.PRO: Less overall V

from higher inclination

launch sites.

CON: Takes longer to

establish the final orbit.

2. Plane change

plus initial

Hohmann burn

GEOTarget

Orbit

1. Launch to

Super GTO

3. Second

Hohmann burn

circularizes at

GEO


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Low Thrust Orbit Transfer

PROs: Lower mass propulsion system. Same system used for orbital maintenance.

CONs: Weeks or even months to reach final orbit. Van Allen Radiation belts.

A series of plane and altitude changes. Continuous electric engine propulsion.


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RendezvousLaunch when the

orbital plane of the

target vehicle crosseslaunch pad.

(Ideally) launch as the

target vehicle passes

straight overhead.

Smaller transfer orbitsslowly overtake target

(because of shorter

orbit periods).

Course maneuvers

designed to arrive in

the same orbit at thesame true anomaly.

Apollo LM

and CSM

Rendezvous


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Orbital Debris a.k.a., Space Junk

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.

February 2009 Iriduim / Cosmos collision created > 1,000 items > 10cm diameter


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Deep Space

Cassini Saturn orbit

insertion using good ol

fashion rocketpower.


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Using Lagrange Points to stay put


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Halo Orbits (stability from motion)


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AeroBrakingEarth, Mars, Jupiter, etc.The poor mans Hohmann maneuver


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The Solar System Super Highwaydesigning geodesic trajectories like tossing a message bottle

into the sea at exactly the right time, direction, and velocity.


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Gravity Assist(Removing Velocity)


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Gravity Assist(adding velocity)


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Solar Escape


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Multiple Mission

Trajectories


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Complex Orbital Trajectories

Galileo (Jupiter)

Cassini (Saturn)


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Designing Deep

Space Missionsyes, there are software tools for this


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Engineering 176 Orbits

Assignments for April 2

Create a trade table to

compare orbit designs.

Trade criteria should include:Orbit suitability for mission.

Cost to get there and stay there.

Space environment (e.g., radiation).

HOMEWORK:Design minimum two,

preferably three orbits

your mission could use.

For the selected orbits:Describe it (orbital elements)

How will you get there?

How will you stay there?

Estimate perturbations

Reading on Orbits:SMAD ch 6 scan 5 and 7

TLOM ch 3 and 4 scan 5 and 17

orbital mechanics and design_rev17

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