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### Engineering 176Orbital Design

Mr. Ken Ramsley kenneth_ramsley@brown.edu (508) 881- 5361

When Orbits Were Perfect (and politically dangerous)

Einstein’s Geodesics (the art and science of motion)

Kepler’s Three Laws (based on Tycho’s meticulous data)

Orbital Elements Defined and Illustrated

Useful Orbits and Maneuvers to Get There

Interplanetary Space and Beyond

EN176 Orbital Design

The Copernicus Solar System

Image: Courtesy of tychobrahe.com

Tycho Brahe's UraniborgObservatory and 90° Star Sighting Quadrant

All objects in motion conserve momentum through a balance of

Gravity Potential andVelocity Vector

(think rollercoaster)

Geodesics:The Science and Art of 4D Curved Space Trajectories.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?
- OrbitalPlaneOrientation– Pitch,Yaw,andRoll
- Satellite Location – Where are we in this orbit?

Kepler’s First Law

Every orbit is an ellipse with the Sun (main body) located at one foci.

Kepler’s Second Law

Day 40

Day 30

Day 50

Day 60

Day 20

Day 70

Day 80

Day 90

Day 10

Day 100

Day 110

Day 0

A line between an orbiting body and primary body sweeps out equal areas in equal intervals of time.

Day 120

Kepler’s Third Law

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 second orbiting body is:

P2 = R3

EXAMPLE:

Earth

P = 1 Year

R = 1 AU

Mars

P = 1.88 Years

R = 1.52 AU

R2

R1

P1

P2

P = Orbital Period

R = Average Radius

Vernal Equinox – The Celestial Baseline

First some astronomy…

June 21st

When the Sun passes over the equator moving south to north.

Vernal Equinox(March 20th)

Defines a fixed vector in space through the center of the Earth to a known celestial coordinate point.

Sun

Epoch 2000

The Vernal Equinox drifts ~0.014° / year. Orbits are therefore calculated for a specified date and time, (most often Jan 1, 2000, 2050 or today).

December 22nd

Conic Sections(shape) Eccentricity

- e=0 -- circle
- e<1 -- ellipse
- e=1 -- parabola
- e>1 -- hyperbola

e < 1 Orbit is ‘closed’ – recurring path (elliptical)

e > 1 Not an orbit – passing trajectory (hyperbolic)

Keplerian Elementse,a,andv (3 of 6)

e

120°

150°

90°

Eccentricity(0.0 to 1.0)

v

True anomaly (angle)

a

Apogee 180°

Perigee 0°

Semi-major axis (nm or km)

e=0.8 vrse=0.0

e defines ellipse shape a defines ellipse size v defines satellite angle from perigee

Apo/Peri gee– Earth Apo/Peri lune – Moon Apo/Peri helion–Sun Apo/Peri apsis– non-specific

Inclinationi(4th Keplerian Element)

Intersection of the equatorial and orbital planes

i

Inclination (angle)

(above)

(below)

Ascending Node

Equatorial Plane ( defined by Earth’s equator )

Sample inclinations0° -- Geostationary 52° -- ISS 98° -- Mapping

Ascending Node is where a satellite crosses the equatorial plane moving south to north

Right Ascension[1] of the ascending nodeΩandArgument of perigeeω(5th and 6th Elements)

Ω = angle from vernal equinox to ascending node on the equatorial plane

Perigee Direction

ω = angle from ascending node to perigee on the orbital plane

ω

Ω

Ascending Node

[1]Right Ascension is the astronomical term for celestial (star) longitude.

Vernal Equinox

The Six Keplerian Elements

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

e=Eccentricity (of the elliptical orbit)

v=True anomalyThe angle between perigee and satellite in the orbital plane at a specific time

i=InclinationThe angle between the orbital and equatorial planes

Ω=Right Ascension (longitude) of the ascending nodeThe angle from the Vernal Equinox vector to the ascending node on the equatorial plane

w=Argument of perigee The angle measured between the ascending node and perigee

Shape, Size, Orientation, and Satellite Location.

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 deg

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

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

Epoch rev: 2917

Checksum: 315

State VectorsNonKeplerian Coordinate SystemCartesian x, y, z, and 3D velocity

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 TransponderRanging:

Alt-Az plus radio signal turnaround delay (like radar).

Ground Sightings:

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

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.

Launch From Airplane at 200 m/sand10 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/s ∆V (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.

Ground Tracks

Ground tracks drift westward as the Earth rotates below an orbit.

Each orbit type has a signature ground tract.

More Astronomy Facts

The Sun

Drifts east in the sky ~1° per day. Rises 0.066 hours later each day.

(because the earth is orbiting)

The Earth…

Rotates 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 the Sidereal day – not the solar day

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.

Other Factors– Gravitational irregularities – such as Earth-axis wobbles, Moon, Sun, Jupiter gravity (tends to flatten inclination). Solar photon pressure. Insignificant for LEO – primary perturbations elsewhere.

‘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.

Nodal Regression

Orbital planes rotate eastward over time.

(above)

Ascending Node

(below)

Nodal Regression can be very useful.

Sun-Synchronous Orbits

Relies 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).

Molniya - 12hr Period

‘Long loitering’ high latitude apogee. Once used used for early warning by both USA and USSR

‘Tundra’ Orbit - 24hr Period

Higher apogee than Molniya. For dwelling over a specific upper latitude (Used only by Sirius)

GPS Constellation ~ 20200kmalt.

GPS: Six orbits with six equally-spaced satellites occupying each orbit.

Hohmann Transfer Orbit

Hohmann transfer orbit intersects both orbits.

Requires co-planar initial and ending orbits.

After 180°, second burn establishes the new orbit.

Can be used to reduce or increase orbit altitudes.

By far the most common orbital maneuver.

Orbital Plane Changes

Burn must take place where the initial and target planes intersect.

Even a small amount of plane change requires lots of ΔV

Less ΔV required at higher altitudes (e.g., slower orbital velocities).

Often combined with Hohmann transfer or rendezvous maneuver.

θ

Simple Plane Change Formula (No Hohmann component):

Plane Change ΔV = 2 x Vorbit x sin(θ/2)

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

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

Plane Change ΔV = 3623m/s

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 stage restart where excess fuel is often available.

Geostationary Transfer Orbit ‘GTO’

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

1. launch to ‘GTO’

3. Hohmann circularizing burn

‘Super GTO’

3. Second Hohmann burn circularizes at GEO

GEO Target Orbit

Initial orbit has greater apogee than standard GTO.

Plane change at much higher altitude requires far less ΔV.

PRO: Less overall ΔV from higher inclination launch sites.

CON: Takes longer to establish the final orbit.

1. Launch to ‘Super GTO’

2. Plane change plus initial Hohmann burn

Low Thrust Orbit Transfer

A series of plane and altitude changes.

Continuous electric engine propulsion.

PROs: Lower mass propulsion system.Same system used for orbital maintenance. CONs: Weeks or even months to reach final orbit. Van Allen Radiation belts.

Rendezvous

Launch when the orbital plane of the target vehicle crosses launch pad.

(Ideally) launch as the target vehicle passes straight overhead.

Smaller transfer orbits slowly overtake target (because of shorter orbit periods).

Course maneuvers designed to arrive in the same orbit at the same true anomaly.

Apollo LM and CSM Rendezvous

Orbital Debrisa.k.a., ‘Space Junk’

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

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.

Deep Space

Cassini – Saturn orbit insertion using good ‘ol fashion rocket power.

AeroBrakingEarth, Mars, Jupiter, etc.“The poor man’s Hohmann maneuver”

The Solar System ‘Super Highway’…designing geodesic trajectories – like tossing a message bottle into the sea at exactly the right time, direction, and velocity.

Designing Deep Space Missions…yes, there are software tools for this

Assignments for April 2

Reading on Orbits:

SMAD ch 6 – scan 5 and 7

TLOM ch 3 and 4 – scan 5 and 17

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

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).

Engineering 176 Orbits

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