1

Satellite orbits.

• Where is the satellite ?

• May we see it ?

Satellite geophysics, 2013-11-10

2

CTS Referencesystem. Fixed with respect to the Earth.

/

Satellite geophysics, 2013-11-10

3

Inertial system. Newtons laws valid.

Center in

Gravity center

Fixt in relation to

The Fix-stars.

Connection to CT

Through siderial

Time.

Satellite geophysics, 2013-11-10

4

Satellite movent around ideal Earth.

Spherical, homogeous, no athmosphere

Newtons law of attraction: Force= F =G(Mm)/r2

M=Earth mass, m = satellitte mass

G= gravitational constant, r distance from C./

Satellite geophysics, 2013-11-10

5

Orbit is curve in 3D-space.

Orbital curve:

Acceleration Force

2. order differental equation

If in ONE point we know:

Velocity-vector (3 numbers)

Position (3 numbers)

Determines orbit ! (6 numbers)

)(

)(

)(

)(

3

2

1

tc

tc

tc

tc

32

2

/)(

rrGMmdt

tcd

State-vector

Satellite geophysics, 2013-11-10

6

The Kepler laws as consequences of the law of attraction

1. Law: Orbit is elliptic, with 1 focus in the gravity center of the Earth. Orbital plane fix in inertial coordinate-system – tree constants fixed.

With a, e 5 constants fixed !

/ b

aC

f

2

222

a

bae

Satellite geophysics, 2013-11-10

7

Kelper’s 2. law.

Areas covered by the position-vector is proportional with time, t.

Velocity of Satellite is NOT constant.

Minumum: Apogee

Maximum: Perigee

Satellite geophysics, 2013-11-10

8

Kepler’s 3. law.

ant),(

)T time,Revolution(3

2

constaaxismajorsemi

anomalymeanTtnM

velocityangularmeanaGMn

GMaT

)(

/

,4/

3

232

Satellite geophysics, 2013-11-10

9

3. law:

Consequence: 2 satellites with same semi-major axis will have same revolution time, T, independent of the excentricity.

/

Satellite geophysics, 2013-11-10

10

6 Kepler-elements

Position given by

statevector or

6 Kepler- elements

= Ascending nodes

rectancention,

i: orbit inclination,

= perigee

argument

a= semi major axis,

e: excentricity, f=latitude,

Satellite geophysics, 2013-11-10

11

Computation of state-vector from Kepler-elementer

Coordinat system in Orbital plane, center in C. Polar coordinates f, r.

E: excentric

anomaly

Satellite geophysics, 2013-11-10

12

Velocity and angular velocity

Linar in time !

Orbit is straight line expressed in Kepler-elementes in the 6-dimensional space

EeEM

TtnMMedeE

Eef

Eear

sin

)(cos

sin1)tan()tan(

)cos1(

2

Satellite geophysics, 2013-11-10

13

To Inertial system by Rotations:

Position = Rxqq, Velocity = Rxqq’

Composed of 3 rotations

/)()()( 313 RiRRRxq

Satellite geophysics, 2013-11-10

14

Satellitorbits GPS, i= 55 - Torge 5.2.

Satellite geophysics, 2013-11-10

15

Forces acting on the satellite.

• Fc= Ideal spherical Earth,

• Fnc= deviation from ideal

• Fn,Fs from Sun and Moon

• Fr , solar pressure

• Fa=atmosphere,

• Tides,• Magnetic Field

/

Satellite geophysics, 2013-11-10

16

Satellite orbits – influence of non-central force.

•

/

Satellite geophysics, 2013-11-10

17

Satellit orbits, solar pressure, atmosphere

Forces depend on shade/non shade of sun.

Relationship masse/surface area. Variations of 2 m.

Depends on density of atmosphere, satellite diameter, mass and velocity.

v=7500 m/s, force 0.000001 m/s2

Neglicible for GPS.

/

Satellite geophysics, 2013-11-10

18

Satellit-orbits – other bodies and mass changes.

• Moon most important, Planets small effect• Earth deformation, tides/loading• Seasonal masse-changes.

Satellite geophysics, 2013-11-10

19

Satellit orbits – description of changes.

16 parametres,

Update

Every hour.

Satellite geophysics, 2013-11-10

20

Satellite orbital parameters for GPS

• Mean anomaly• Mean movement difference• Excentricity

• Square-roor of a• Right acension• Inclination at t0e

• Perigee argument• Time derivative of rectac.• Time derivative of i• Correction to f • Correction to r• Corrections to i• Reference-time

e

rcrs

isic

usuc

t

CC

CC

CC

i

i

a

e

n

M

0

0

0

,

,

,

Satellite geophysics, 2013-11-10

21

Computation of position, Torge p. 132.

GM=3.98608x1014 m3/s2, =7.292115147x10-5 rad/s2

True anomaly fk from time-difference tk=t-t0e

Mean-anomali:

Solution iterativly wrt Ek,

e

kk tnaGMMM )/( 30

)sin( kkk EeEM

Tkkkkkeekek

kiskickk

krskrckk

kuskuckk

k

kk

ruRiRRXthentt

Longitude

fCfCtiii

fCfCEear

fCfCfu

eE

Eeaf

)0,0,)(()()()(

:node ascending of

)(2sin)(2cos

)(2sin)(2cos)cos1(

)(2sin)(2cos

cos

sin1tan

1300

0

2

Satellite geophysics, 2013-11-10

22

Satellit orbits .

• LEO: Low Earth Orbit h < 2000 km• MEO: Medium Earth Orbit 5000-20000 km• GEO: Geostationary, h=36000 km

• IGSO: Inclined Geo-syncronous Orbit• HEO: Highly Elliptic Orbit

Satellite geophysics, 2013-11-10