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Trajectories & Orbits

Describing & Using Orbits

COEs, Ground Tracks and Real World Issues

Orbit Ground Tracks Classical Orbital Elements Freefall

FOU - A

Describing & Using Orbits - 1

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Section Overview Objectives

Outline

¾ Define the classic orbital elements ((COEs)) used to describe the size,, shape, and orientation of an orbit and the location of a spacecraft in that orbit ¾ Explain p and use orbital g ground tracks ¾ Understand issues associated with satellite tracking, including prediction and p perturbations ¾ Appreciate the complexity of orbit design for real-world missions

¾ Classical Orbital Elements Semi-major major axis, a • Semi • Eccentricity, e • Inclination, I • RAAN, Ω • Argument Arg ment of Perigee, Perigee ω • True Anomaly, υ ¾ Satellite Ground tracks ¾ Satellite Tracking • Prediction • Perturbations ¾ Orbit Design

References & further reading: Understanding g Space: p An Introduction to Astronautics ((US)) by y Sellers, Chapter p 5, 8

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Describing & Using Orbits - 2


Classical Orbital Elements(COE) Preview • •

How many pieces of information will we need? We’ll use the COEs to tell us important things about an orbit – – – –

Size Shape Orientation » Orbital plane in space » Orbit within the plane p Location of Spacecraft in the orbit

The COEs tell us – – – – –

Orbital O bit l size, i i represented is t db by th the semimajor i j axis, i a Orbital shape, is represented by eccentricity, e Orientation of the orbital plane in space, defined by » inclination,, i » right ascension of the ascending node, Ω Orientation of the orbit within the plane defined by » argument of perigee, ω Spacecraft’s location in the orbit is represented by true anomaly anomaly, ν Describing & Using Orbits - 3

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Orbit Size: Semimajor axis, a • The size si e of an orbit is described b by the semimajor a axis, is a – Half the distance across the orbit’s major (long) axis,

• The size of the orbit tells us

– The specific mechanical energy energy, ε » the bigger the orbit the more energy it has – The period, P, the time it takes to go once around the orbit » the bigger the orbit, the longer the period (see Kepler’s third law)

ε =−

μ

US: Eq. 5-1

2a

P = 2π

a3

μ

US: Eq. 4-27

For the Space Shuttle, the semimajor axis is about 6678 km and the period is about 90 minutes minutes. It orbits the Earth US: Fig. 5-2

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about 16 times each day! Describing & Using Orbits - 4


Orbital Shape: Eccentricity, e • We described the “out of roundness” of a conic section in terms of its eccentricity, e.

.7 .5(circular) ee == 0.8 Circle Ellipse Parabola Hyperbola

e = 0.0 0.0 < e <1.0 e = 1.0 e > 1.0 © DO NOT REPRODUCE WITHOUT PERMISSION

Describing & Using Orbits - 5

Different Eccentricities

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Describing & Using Orbits - 6


Orbital Tilt: inclination, i • Inclination Inclination, i, i specifies the tilt of the orbital plane with respect to the fundamental plane—the equatorial plane for Earth Inclination h

K

i

Angular momentum vector

Equatorial Plane

Ascending Node

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Describing & Using Orbits - 7

Inclination and Orbit Type Inclination, i

Orbit Type

0o or 1800

Equatorial

90o

Polar

0o ≤ i < 900

Direct or prograde (satellite moves in direction of Earth’s rotation)

90o < i ≤ 1800

Indirect or retrograde (satellite moves in opposite direction of Earth’s rotation) © DO NOT REPRODUCE WITHOUT PERMISSION

Describing & Using Orbits - 8


Orbital Swivel: RAAN, Ω The right ascension of the ascending node (RAAN), (RAAN) Ω, describes the orbital plane’s orientation with respect to the principal direction—the vernal equinox direction. Measured Meas red east eastward ard from the principal direction to the ascending node.

Ω

Ascending Node

Vernal Equinox Direction © DO NOT REPRODUCE WITHOUT PERMISSION

Describing & Using Orbits - 9

Different RAANs

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Describing & Using Orbits - 10


Orbital Rotation: Argument of Perigee Argumentt off P A Perigee, i ω, specifies ifi the th orientation i t ti off the th orbit within the plane—where is perigee? Measured in the direction of satellite motion in the orbital plane from the ascending node to perigee

ω

Perigee (Point Closest to the Earth)

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Describing & Using Orbits - 11

Different Arguments of Perigee

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Describing & Using Orbits - 12


Satellite Location: True Anomaly True anomaly, T l υ, specifies ifi a spacecraft’s ft’ location l ti along l its orbital path—where is the satellite? Measured in the direction of satellite motion in the orbital plane from perigee to the satellite’s current position

ν Perigee (Point Closest to the Earth)

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Describing & Using Orbits - 13

Classical Orbital Elements Summary • Size: Si h how bi big ((energy): ) a • Shape: eccentricity (how squashed the orbit is) e

• • • •

Tilt: inclination (tilt to equatorial plane) i Swivel: ascending node location Ω Perigee location: perigee relative to the ascending node ω Satellite location: satellite location relative to perigee ν © DO NOT REPRODUCE WITHOUT PERMISSION

Describing & Using Orbits - 14


NORAD Two-Line Orbital ELement SET (ELSET) Intl Designator

2nd Derivative Epoch Time: Mean motion Line No Y Year, Julian J li Day.dec D d Ephemeris Type Sat No. 1st Derivative Elset No Class Drag Mean motion Check S Sum 1 24753U 97012A 99248.47511223 .00000343 00000-0 20608-3 0 1270 2 24753 98.8218 296.0202 0007755 274.6467 85.2621 14.13296418124820 Argument of Perigee g

Inclination Right Sat No. Ascension

Eccentricity

Mean Motion Mean Anomaly

Check Sum

Rev No

This example: p DMSP F-14 © DO NOT REPRODUCE WITHOUT PERMISSION

Describing & Using Orbits - 15

Example - Mission Identification Try to guess a possible mission for each of the four satellites listed below: Orbit

a

e

i

ω

1

6678 km

0

28°

N/A

2

6678 km

0

96.67°

N/A

3

26600 km

0.75

63.4°

270°

4

42160 km

0

N/A

Mission

Radius of the Earth = 6378.135 km FOU - A

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Describing & Using Orbits - 16


Satellite Ground Tracks • Most satellites are for missions focused on the Earth – Taking pictures, communications, navigation – Need to know what path the satellite traces over the Earth’s Earth s surface

• Let’s look at a typical Shuttle mission – Period is about 90 minutes (~1.5 hours to go once around the Earth) – Inclination is normally between 28o and 51o – As a satellite revolves around the Earth the Earth rotates under the Earth, satellite – Now, try to imagine what the trace of the satellite looks like on the surface of the Earth

US: Fig. 5-24 © DO NOT REPRODUCE WITHOUT PERMISSION

Describing & Using Orbits - 17

Satellite Ground Tracks • If the Earth didn’t rotate, the ground track would look like this:

US: Fig. 5-30

• Why does a circular orbit look like a sine wave on a global l b l map? ? © DO NOT REPRODUCE WITHOUT PERMISSION

Describing & Using Orbits - 18


Satellite Ground Tracks • Si Since the th E Earth th d does rotate t t underneath d th th the satellite orbit, each successive ground track shifts t the to th left l ft and d is i “scrunched” “ h d” b by th the orbital bit l period i d times the Earth’s rotation rate (15 deg per hour)

US: Fig. 5-31 © DO NOT REPRODUCE WITHOUT PERMISSION

Describing & Using Orbits - 19

Satellite Ground Tracks

Orbit O bit A A: P P=2.67 2 67 h hr Orbit C: P=18 hr O bi E Orbit E: P P=24 24 h hr

O Orbit bit B B: P P=8 8h hr Orbit D: P=24 hr

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US: Fig. g 5-33

Describing & Using Orbits - 20


Satellite Ground Tracks • These orbits have the same period (90 minutes) but each minutes), has a different inclination

• O Orbit bit A and d orbit bit B both have the same period (11.3 (11 3 hours) and inclination (50o) – What makes them different? US: Fig. 5-33 © DO NOT REPRODUCE WITHOUT PERMISSION

Describing & Using Orbits - 21

Low Earth Orbit (LEO)

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Describing & Using Orbits - 22


Geosynchronous Orbit (GEO)

Describing & Using Orbits - 23

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Space Mission Geometry •

Depending on the orbital elements, a satellite (and the payloads on it) can “see” or “access” different parts of the Earth depending on

Satellite FOR

– Geometry – Physical capabilities of the sensor or antenna

Field of Regard (FOR)

Sensor FOR

Sensor FOV

– The angle that describes the potential cone of visibility for a satellite or sensor limited by geometric constraints (e.g. edge of radio horizon)

Field of View (FOV)

True horizon

True horizon

– The angle that describes the actual cone (or other shape) of visibility for a sensor limited by physical constraints (e.g. the lens, image plane or antenna beam pattern)

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Describing & Using Orbits - 24


Space Mission Geometry • •

Swath is the projection of the satellite FOR, the sensor FOR or the sensor FOV onto the Earth These depend on – Satellite altitude – Minimum elevation angle, ε, angle measured from local horizon at the edge of the FOR to the satellite or sensor True Horizon, ε = 0°

Sensor swath for FOR to ε = 60 deg

ε Local h i horizon

λ

SL Altitude

REarth

ε

Field of Regard=2η

Swath for sensor FOR of 1 deg

SL = Slant Range to edge of FOR η= nadir angle ρ = earth central angle © DO NOT REPRODUCE WITHOUT PERMISSION

Describing & Using Orbits - 25

Satellite Tracking Requirements •

D il ttracking Daily ki enables: bl – Planning access and pointing from a given tracking site to different satellites – Analysis of the coverage a satellite will have over a given area on the ground – Assessment of the threat of collisions with debris

US Fig. 8-2

To do this effectively, we need ways to… – Estimate the satellite’s current COEs—from ground tracking or on-board GPS – Accurately predict (propagate) the satellite’s COEs into the future » Requires two changes of variables, plus a solution to a transcendental equation » This is referred to as Kepler’s Problem, see Chapter 8 in US

– Account for environmental effects that will “perturb” the orbit © DO NOT REPRODUCE WITHOUT PERMISSION

Describing & Using Orbits - 26


Orbit Perturbations • Two-body equation assumptions – Gravity is the only force – Earth’s mass is much greater than the spacecraft’s mass – Earth is spherically symmetric with uniform density, so it could be treated as a point mass – The spacecraft’s mass is constant, so ∆m = 0

• Changes to the satellite satellite’s s COEs due to other forces we call perturbations. • The most important perturbations for low-Earth orbit are: – Atmospheric Drag – Earth’s Oblateness

• Other perturbation sources (more important for GEO and interplanetary trajectories are: – Solar radiation pressure – Third-body gravitational effects (Moon, Sun, planets, etc.) – Unexpected thrusting

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Describing & Using Orbits - 27

Perturbations Due to Atmospheric Drag •

Drag is a non-conservative non conservative force—it force it takes energy away from the orbit in the form of friction on the spacecraft. – Orbital energy is a function of semimajor axis » the semimajor j axis,, a,, shrinks over time

– The eccentricity also decreases, since the orbit becomes more circular.

Drag is very difficult to model because of the many factors affecting Earth’s upper atmosphere – – – – – – –

Earth’s day-night cycle Seasonal tilt Variable solar distance Fluctuating magnetic field The Sun’s 27-day rotation Solar Cycle Spacecraft’s coefficient of drag and frontal area

Drag is a significant factor below 600 km

US Fig. 8-7

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Describing & Using Orbits - 28


Perturbations Due to Earth’s Oblateness • The earth may appear spherical, but it is actually kind of squashed. – We call this squashed shape oblateness

• What does an oblate Earth look like? – Imagine spinning a ball of gelatin and you can visualize how the middle (or equator) of the spinning gelatin would bulge out Earth is about 22 km fatter at the equator than at the poles

Oblateness (a (a.k.a. k a “J2” J2 or the “J2 J2 effect”) effect ) causes a slight shift in the direction that gravity pulls a spacecraft – the spacecraft’s orbit precess’ , changing » right ascension of the ascending node node, Ω » argument of perigee, ω © DO NOT REPRODUCE WITHOUT PERMISSION

Describing & Using Orbits - 29

Sun-synchronous Orbits • Ascending Node and Perigee precession give rise to a unique orbits that have very practical applications. applications • Sun-synchronous Orbits take advantage of the rate of change in right ascension of the ascending node. – Select the proper inclination and altitude,

» match the rotation of Ω with the movement of Earth around the Sun. – The same angle between the orbital plane and the Sun will be maintained maintained.

US Fig. 8-12.

» Useful for remote sensing applications because shadows cast by targets on Earth stay the same. © DO NOT REPRODUCE WITHOUT PERMISSION

Describing & Using Orbits - 30


Sun Synchronous Orbit (SSO)

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Describing & Using Orbits - 31

Molniya Orbits •

Molniya (HEO) orbits—named after the Russian word for lightning (as in “quick-as-lightning”) also use Earth’s oblateness to advantage. – Typically y y ~ 12-hour orbits with very y high g eccentricity y ((e = ~0.7)) and a perigee g location in the Southern Hemisphere. The inclination is 63.4°—why? – At this inclination the perigee doesn’t precess so the spacecraft “hangs” over the Northern Hemisphere for nearly 11 hours of its 12-hour period – The Russians (and others) use these orbits to provide coverage at high latitudes that can’t “see” satellites at GEO

US Fig. 8-13. © DO NOT REPRODUCE WITHOUT PERMISSION

Describing & Using Orbits - 32


Molniya Orbit (HEO)

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Describing & Using Orbits - 33

GEO, MEO, Molniya, LEO, SSO Orbits

Orbits Demo

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Describing & Using Orbits - 34


Orbit Design Designing the right orbit for a given mission is a complex juggling act between competing requirements—some pulling low, some pulling you high—some pulling you into low inclination, others into high inclination. D i i seek Designing k a compromise i ““middle iddl ground” d” b between t thi this ttug-a-war

Characteristic

All Allowed d Range R (km) (k )

Comments

Launch Capability

Launch Vehicle Limit

Radiation

Inner Radiation Belt

Coverage

Higher is Better

Payload Resolution

Lower is Better

Communications

Higher is Better (for coverage)

Lifetime

Trade with Launch Limit 0

1000

2000

3000 Adapted from SMAD, Fig. 3-1 Describing & Using Orbits - 35

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Review of Satellite Missions & Orbits Orbit

Semi-major Semi major axis (altitude)

Period

Inclination

Other

1

42,158 km (35,780 km)

~24 hr

~0o

e = ~0

2

6628 - 7378 km (250 – 1000 km)

~95 to 105 min

95-105o

e = ~0

26,610 km (20,232)

~12 hr

~55o

~6700 km (~300)

~90 min

28o to 51o

e = ~0

26,571 km

~12 hr

63.4o

e = 0.7 ω = 270o

3

4

5

(perigee = 1593 km, apogee = 38,792 km)

6

6628 - 7378 km (250 – 1000 km)

~95 to 105 min

~45° to 90o

e = ~0

e = ~0

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Orbit Type

Mission(s)

Geostationary Constellations (GEO)

•Communications •Early warning •Nuclear detection •Weather W th

Sun-Synchronous

•Remote sensing

Medium Earth Orbit Constellations (MEO) Low Earth Orbit (LEO)

•Navigation

•Space Shuttle •ISS •Parking orbit

Molniya

•Communications •Intelligence g

Low Earth Orbit Constellations (LEO)

•Communications

Describing & Using Orbits - 36


Firesat Orbital Requirements: Trade-offs Step

FireSat Example

1. Select trade parameter (typically a system driver) 2. Identify factors which affect the parameter or are affected by it

3. Assess impact of each factor

Altitude Coverage (deployment strategy – coverage evolution; orbit period; time in view; eclipse fraction; response time; number of spacecraft needed) Launch capability (resolution; payload weight) Radiation environment (survivability; jamming susceptibility; communications) Lifetime Best coverage above 400 km Resolution – lower is better micro FireSAT Can launch up to 1800 km Survivability not an issue

4 Document and summarize results 4. Characteristic Launch Capability Radiation Coverage C Coverage E Evolution l ti Payload Resolution Communications Lifetime

Allowed Range (km)

0

1000

5. Select parameter value and possible range

2000

Comments Launch Vehicle Limit Inner Radiation Belt Higher is Better M j Pl Major Plateau t att 375 kkm Lower is Better Lower is Better Trade with Launch Limit 3000

Altitude = 700 km; 600 to 800 is okay SMAD, Table 3.4 and Fig 3.1

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Describing & Using Orbits - 37

3-Big Ideas: Describing & Using Orbits The classical orbital elements (a, e, i, Ω, ω, ν) are a convenient, intuitive way to describe orbits. They tell us six things about orbits: size, shape, orientation (three angles) and location A ground track is the path a spacecraft traces on Earth’s surface as it orbits.

1.

2. • • •

3.

The orbital plane slices through the center of the Earth. Earth Orbital planes are fixed in inertial space and Earth rotates beneath them Ground tracks appear to shift westward during successive orbits

Perturbations resulting from small disturbing forces cause our two-body orbit to vary. • •

These include: drag drag, oblateness (J2) (J2), solar pressure, and 3rd body effects. We take advantage of these effects for sun-synchronous and Molniya orbits

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Describing & Using Orbits - 38


Intro to Aerospace at NASA