Lesson 2: Space Mission

Geometry

Dr. Andrew KetsdeverMAE 5595

Space Mission Geometry

• Must specify an appropriate coordinate system– Earth fixed– Spacecraft fixed– Other

• Coordinate system must have a specified– Origin– Fixed point

Earth Geometry Viewed from Space

Mission Viewing Geometry

angle. grazingor angleelevation satellitesatelliteby seen angleearth maximum

footprint) ofboundary (defines radiusearth angular n)combinatioany or Long, (Lat, angle centralearth

nadir from angle target torangeslant

satellite of altitudeearth of radius

0

========

ελρληDHRE

λρλρη

cossin1sinsintan

−=

ρηε

sinsincos =

2πελη =++

ηλ sinsin DRE =

( ) 22max ⊕⊕ −+= RHRD

( ) ( ) λcos222 HRRHRRD +−++= ⊕⊕⊕⊕

HRR+

==⊕

⊕0cossin λρ

Special Cases

Note,- If point of interest is SSP: 0 ,90 ,0 === λεη o

- If point of interest is true outer horizon: ρλερη −=== o90 ,0 ,

λ

λ0 ε D

H

ρ η

R⊕

Dmax

SSP

True outer horizon

Satellite Motion for an Earth Observer

Circular, LEO

Total Time in ViewFor a LEO satellite, assume brief pass (ignore Earth’s rotation). The time in view, T, for the satellite is given by:

( )180/coscoscos

180

maxmax

min

max1

λλλ

PeriodT

PeriodT o

=

= −

maxminmax 90 ηελ −−= o

minmax cossinsin ερη =

( ) ( ) ( ) ( ) ( )longlatlatlatlat GSpoleGSpole ∆+= coscoscossinsinsin minλ

pass of at time node ascendingorbit whereinertial)not c,(geographiEarth of longitude

90station ground and poleorbit between longitudein difference

station ground of latitude

90

=

−=

=∆=

−=

node

nodepole

GS

pole

L

Llonglong

lat

ilat

o

o

where:

Satellite Slew Rate

• Maximum Slew Rate for a Satellite

( )minmin

max2PeriodD

HRDV Esat +

==• πθ

=

min

minmin sin

sinηλ

ERD

Satellite Motion for an Earth Observer

Dominant factor:

(a) Inclination

(b) Eccentricity

center of ground track

(a) (b)

h

ω

GEO

GEO Ground Track

equator

28.50 N

10 km rectenna array—beamwidth

0.8×10-2 deg

ground track—maximum latitude

SPS in 28.50 inclined orbit

Sirius Radio Satellite Constellation

• Mission: – Continuous radio broadcasts over North

America– Radio reception to remote areas

• What mission geometry would you choose?

• Geosynchronous• 60 deg inclination• Highly eccentric• Apogee over N.

Hemisphere

Sirius Constellation

How Did XM Do It?• Named "Rock" and "Roll,"

XM Radio’s two Boeing HS 702 satellites were placed in parallel geostationary orbit (GEO), 35,764 km above Earth.

• The first XM satellite, "Rock," was launched on March 18, 2001, with "Roll" following on May 8th.

Sirius and XM Constellations

http://www.dogstarradios.com/sirasasecoma.html

Swath Width

Eclipse

• Knowing the mission geometry leads to a simple calculation for the eclipse time for a satellite (SMAD has a more complicated analysis)

+

= −

HRR

E

E1sinρ

xPeriodTEo360

2ρ=