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3.4 ORBIT REQUIREMENTS

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KEPLERIAN ORBITS

Johannes Kepler (1571-1630) • German mathematician, astronomer, astrologer, and natural philosopher, • Born in Weil der Stadt and moved to Leonberg as a boy where he attended

the local Latin School, • Studied theology and philosophy at the University of Tübingen, • Professor of mathematics and astronomy (1594-1600) at the Protestant

School (later to become University of Graz) in Graz, Austria, • Assistant to Tycho Brahe (1600-1601) in Prag, • after Tycho's unexpected deat in 1601, Kepler was appointed as Brahe’s

successor as imperial mathematician (1601-1612) to the Emperor’s court of Rudolf II,

• His mother, Katharina Kepler, was accused and imprisoned for witchcraft in Leonberg,

• Between 1612 and 1630 he taught mathematics in Linz • Grave Inscription: “Die Himmel hab ich gemessen, jetzt meß ich die

Schatten der Erde. Himmelwärts strebte der Geist, des Körpers Schatten ruht hier.” (The heavens I have measured, now I measure the shadows of the Earth. Towards the heavens strived the spirit, the body’s shadow rests here.

Tycho Brahe (1546-1601)

Keplerian Laws:

First Law: The orbit of each planet is an ellipse, with the Sun at one focus.

Second Law: The line joining the planet to the Sun sweeps out equal areas in equal times.

Third Law: The square of the period of a planet is proportional to the cube of its mean distance fro the Sun.

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KEPLERIAN ORBITS

 Johannes Kepler

Johannes-Kepler-Gymnasium Leonberg, Germany

Dem Keplerschen Geist des Suchens und Erforschens und seinem Streben nach Harmonie fühlt sich die Schule verpflichtet.

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KEPLERIAN ORBITS

 Equations of Motion for a Satellite  Assumptions:

 Two body system,

 Earth is spherical, symmetrical, homogeneous,

 Earth’s mass >> satellite mass,

 Gravity only force in system.

 Newton’s Second Law:

 Law of Universal Gravitation:

 Two-Body Equation of Motion:

 Polar Equation of Conic Sections:

 Energy Equation:

F = m

a = m

r

F = −GMm

r2≡ − µm

r2

r + µr2

r = 0

r =

a 1− e2( )1+ e cosν

ε = ε

kin+ ε

pot= v2

2− µ

r= − µ

2a

ε < 0

ε < 0

ε = 0

ε > 0

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CIRCULAR & ELLIPTICAL ORBITS

 Orbital Velocity of Circular Orbit  Forces acting on Satellite

 Orbit Velocity

 Orbital Velocity of Elliptic Orbit  Velocity from Energy Equation

 Apogee & Perigee Velocities

M R0

h

r

Fc

Fg

uorb m

u

ellip= µ

2R− 1

a

⎛⎝⎜

⎞⎠⎟

u

apogee= 2µ

ra+ r

p

⋅rp

ra

= µa⋅1− e1+ e

u

perigee= 2µ

ra+ r

p

⋅ra

rp

= µa⋅1+ e1− e

muorb2

r= mg = GmM

r2= µm

r2 F

cent=F

grav

u

circu= µ

r

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ESCAPE VELOCITY

 Kinetic energy equals work required to overcome gravity.

 Newton’s Law of Gravity:  Earth’s Surface:

 Any distance r > rE:

 Velocity Requirement:

 Escape Velocity:

 Escape Velocity for selected bodies:  Earth: 11.2 km/s

 Sun: 616 km/s

 Moon: 2.4 km/s

Ekin= E

pot= W

gravity

F

grav= m g

E=

G ⋅ME

m

rE2

=µ

Em

rE2

F

grav= m g

E

rE2

r2

Ekin= 1

2m v

z2

Epot= m g

E⋅ z

Wgravity

= F ⋅drrE

rE +z

∫ = m gE

rE2

r2⋅dr

rE

rE +z

∫

12

m vz2 = m g

ErE

zrE+ z

vz=

2gE

rE

z

rE+ z

z →∞ v

esc= 2 g

ErE=

2µE

rE

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Example

19,300 Nm Earth

400 Nm

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3.5 ORBIT MANEUVERING

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HOHMANN TRANSFER

 Hohmann Transfer:

Minimum energy elliptical transfer path for co-planar transfer between two circular orbits.

 ∆v Requirement:

Δvtotal

= ΔvA+ Δv

B=

= µ12

2rA

− 1a

tx

⎛

⎝⎜⎞

⎠⎟

12

− 1rA

⎛

⎝⎜⎞

⎠⎟

12

+ 2rB

− 1a

tx

⎛

⎝⎜⎞

⎠⎟

12

− 1rB

⎛

⎝⎜⎞

⎠⎟

12

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

Transfer Ellipse

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Hohmann Transfer Equations

 Given the initial and final circular orbit radii, we can calculate the velocity changes required to transfer between the circular orbits.

Orbit A: 6,567 km Orbit B: 42,160 km

Period of an orbit [min]

Obit Frequency [rad/s]

P = 2π

a3

µ

ω

0= µ

a3

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OTHER COPLANAR ORBIT CHANGES

 One-Tangent Burn between two Circular, Coplanar Orbits  Transfer requires less time to complete than Hohmann Transfer.

 Total Velocity Change:

 Spiral Transfer between two Circular, Coplanar Orbits  Constant low-thrust burn results in a spiral transfer.

Δvtotal

= ΔvA+ Δv

BΔv

A= v

txA− v

iA

ΔvB= v

fB2 + v

txB2 − 2v

fBv

txBcosν

Variable Hohmann Transfer One-Tangent Burn

rA 6,570 km 6,570 km

rB 42,200 km 42,200 km

atx 24,385 km 28,633 km

∆vtotal 3.935 km/s 4.699 km/s

Time-of-Flight 5.256 hr 3.457 hr

Δv

total= Δv

f− Δv

i

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Hohmann Transfer

 Example:  Initial Orbit: 5,000 km

 Final Orbit: 40,000 km

 Let’s examine and understand the meaning of some of the individual terms in the Hohmann-Transfer equation.

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ORBIT PLANE CHANGE

 Simple Plane Change: Orbit-Plane (Inclination) Change  Direction change of velocity vector,

 Maneuver with component of ∆v perpendicular to orbital plane.

 Law of Cosines:

 Plane and Altitude Change:  Hohmann & Simple Plane Change  Plane Change & Tangential Burn

 Circularization at Apogee

Δv = 2v sin

α2

v

i= v

f= v

Δv = v

i2 + v

f2 − 2v

iv

fcosα

f

i

f

i

Δv = vCircOrbit

− vapogee

vapogee

= 2µra+ r

p

⋅rp

ra

= µa⋅1− e1+ e

vCO

=µ

Earth

rCO

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∆v Budget for Orbit Plane Changes

 Example: Change low altitude (h=185 km) inclined orbit (i = 28°) to equatorial orbit at same altitude  ∆v=3.8 km/s

 At ro=6563 km, v=7.8 km/s, velocity increment requirement is nearly half the orbital velocity!

 For an inclination change of 60o, the plane change ∆v is approximately equal to the ideal ∆v required to get to orbit.

 Plane changes can be very expensive maneuvers:  Do plane change when velocity is lowest (at the apogee).  Sometimes it is more economical to boost vehicle to a higher orbit, perform plane

change at low v (apogee), and then return to lower orbit.

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Pegasus Launch Vehicle

 Effects of Inclination and Orbit Height on Payload

Technical Data: http://www.astronautix.com/lvs/pegasus.htm

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3.6 ORBIT PERTURBATION

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PERTURBATIONS

 Classic Orbit Perturbations  Third-Body Perturbations,

 Perturbations caused by Non-Spherical Earth,

 Perturbations caused by Atmospheric Drag,

 Perturbations caused by Solar Radiation.

 Perturbations include torques and forces causing deviations from desired flight path:  High altitude effects include Sun and Moon gravitational forces (third body

effects) and solar radiation,

 Medium and low altitude effects include aerodynamic drag, Earth’s oblateness and magnetic field, and ocean tides,

 Satellite asymmetry and internal accelerations.

 Trajectory corrections requiring ∆v supplied by reaction control thrusters (RCS) maintain intended flight regime.

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GEO and GSO

 GEO:  geosynchronous orbit has 24-hour period,  object in GEO is above the same point on Earth every 24 hrs.

 GSO: geostationary orbit  Circular orbit oriented in the equatorial plane; satellite appears to be stationary

from the Earth.  First noted by Tsiokolvsky (ca. 1900), popularized by Arthur C. Clarke (1945),

hence the name “Clarke Belt,” Syncom III used for ‘64 Olympics, Intelsat (‘69) first global GSO network, cable TV in ‘75.

 Orbital height is 35,785 km (22,240 mi).  Footprint of GSO satellite covers ≈ 1/3 of Earth:

  three GSO satellites give full Earth coverage from about 80° N to 80 ° S.  Polar orbiting satellites (LEO) cover holes in coverage,  Compare to 66 LEO Iridium satellites for full coverage.

 Gravitational pull of Moon and Sun, oblateness of Earth at equator cause GSO satellites to change inclination over time:  N-S stationkeeping for inclination control,  E-W stationkeeping to keep the satellite in its assigned position in the GSO belt.

 To attain GSO, launch into LEO, then use upper stage to achieve elliptical geostationary transfer orbit (GTO), then conduct ‘apogee burn’ to circularize at 35,785 km.  Atlas and Delta can deliver S/C directly to GTO – circularization accomplished by

apogee engine on S/C.