Orbital Mechanics - stuba.sk Mechanics Space for Education, Education for Space 1. The two body problem 2. Orbits in three dimensions 3. Orbital perturbations 4. Orbital maneuvers

Orbital Mechanics Space for Education, Education for Space

Space for Education, Education for Space ESA Contract No. 4000117400/16NL/NDe

Specialized lectures

Orbital Mechanics

Vladimr Kuti, Pavol Valko


1. The two body problem

2. Orbits in three dimensions

3. Orbital perturbations

4. Orbital maneuvers

Contents

2


Motion in inertial frame

Relative motion

Angular momentum

Solution of problem

Energy law

Trajectories

Time and position


3



Two body problem can by defined by:

Newtons law of gravitation


4

rr

mmGFF

3

211221

position of masses gravitational forces

1m

2mr

21F

12F

2311 s kg/m 106742.6 Guniversal gravitational constant






5

rr

mmGFF

3

211221


1m

2mr

21F

12F


1st time measured by Cavendish, 1798






6

rr

mmGFF

3

211221


1m

2mr

21F

12F


conservative force can be expressed by potential energy

r

mmGE p

21

1st time measured by Cavendish, 1798






7

rr

mmGFF

3

211221

rr

mFF

3

21221

can be measured with considerable precision by astronomical observation

Central body [m3/s2]

Earth 3.98600441 x 1014

Moon 4.90279888 x 1012

Mars 4.2871 x 1013

Sun 1.327124 x 1020






8

rr

mmGFF

3

211221

2

02

zr

rg

r

mGg

E

EE






9






10





Newton's laws of motion


11

inertial frame of reference

1m

2m

r

2R

1R

rr

mmGFF

3

211221

2122 FRm

1211 FRm







12


1m

2m

r

2R

1R

rr

mmGFF

3

211221

2122 FRm

1211 FRm







13


1m

2m

r

2R

1R

rr

mmGFF

3

211221

2122 FRm

GR

center of mass

1211 FRm

21

2211

mm

RmRmRG







14


1m

2m

r

2R

1R

rr

mmGFF

3

211221

2122 FRm

GR

center of mass 21

2211

mm

RmRmRG

21

2211

mm

RmRmRG

2 x time derivative

1211 FRm







15


1m

2m

r

2R

1R

rr

mmGFF

3

211221

2122 FRm

GR

center of mass 21

2211

mm

RmRmRG

21

2211

mm

RmRmRG

2 x time derivative

0

GR

center of mass is: motionless or motion is in straight line with constant velocity

1211 FRm







16

rr

mmGFF

3

211221

21

2211

mm

RmRmRG

0

GR


2122 FRm

1211 FRm







17

rr

mmGFF

3

211221

21

2211

mm

RmRmRG

0

GR


2122 FRm

1211 FRm







18

rr

mmGFF

3

211221

2122 FRm

1211 FRm

rr

mmGRm

3

2111 r

r

mmGRm

3

2122


1m

2m

r

2R

1R

i

j

k







19

rr

mmGFF

3

211221

2122 FRm

1211 FRm

rr

mmGRm

3

2111 r

r

mmGRm

3

2122

modification of equations

03

rr

r

21 mmG

1Gmif: 21 mm


1m

2m

r

2R

1R

i

j

k

gravitational parameter






Relative motion

20

03

rr

r

kzzjyyixxr

)()()( 121212

vector defined in inertial frame of reference expressed in coord. system

r

kji


1m

2m

r

2R

1R

i

j

k






Relative motion

21

03

rr

r


1m

2m

r

2R

1R

i

j

k

1i

1j

1k

vector can be expressed in coord. system , that rotates about inertial coord. system with instant angular velocity and instant angular acceleration

r

111 kji

121121121 )()()( kzjyixr






Relative motion

22

03

rr

r

121121121 )()()( kzjyixr

2 x time derivative in inertial frame of reference

relrel vrrrr

2


1m

2m

r

2R

1R

i

j

k

1i

1j

1k






Relative motion

23

03

rr

r

121121121 )()()( kzjyixr

2 x time derivative in inertial frame of reference

relrel vrrrr

2

if is not rotating coord. system

111 kji

relrr


1m

2m

r

2R

1R

i

j

k

1i

1j

1k






Relative motion

24

03

rr

r

relrr

relative acceleration of

moving (non-rotating) frame of reference in coord. components

21321

21321

21321

zr

z

yr

y

xr

x

+ 6 initial conditions


1m

2m

r

2R

1R

i

j

k

1i

1j

1k



relative angular momentum of body per unit mass

Angular momentum

25

1m

2m

r

rrrmrm

h

2

2

1

rrrrrrdt

hd

1 x time derivative

r

2m

trajectory




Angular momentum

26

rrrmrm

h

2

2

1

rrrrrrdt

hd

1 x time derivative

03

rr

r

1m

2m

r

r

trajectory

2m




Angular momentum

27

rrrmrm

h

2

2

1

rrrrrrdt

hd

1 x time derivative

03

rr

r

0

dt

hdangular momentum is conserved 1m

2m

r

r

trajectory

2m




Angular momentum

28

angular momentum can be expressed as

velocity vector can be expressed as

rvvvr

vrvrvrrrh r

1m

2m

r

r

v rv

trajectory

2m




Angular momentum

29

angular momentum can be expressed as

velocity vector can be expressed as

rvvvr

vrvrvrrrh r

11 khkrvvrh

1k

unit vector time invariant

h rvh

magnitude of angular momentum time invariant

1m

2m

r

r

v rv

trajectory

2m




Angular momentum

30

11 khkrvvrh

1k


h magnitude of angular momentum time invariant

1m

2m

r

r

trajectory

v rv

1i

1j

Cartesian coord. system

2m




Angular momentum

31

11 khkrvvrh

1k


h

rrdt

dv


1m

2m

r

r

trajectory

v rv

1i

1j

ri

j

Cartesian coord. system Polar coord. system

2m




Angular momentum

32

11 khkrvvrh

1k


h

rrdt

dv


1m

2m

r

r

trajectory

v rv

1i

1j

ri

j


1121 khkrkrrh

2m



Equation of orbit

Solution of problem

33

1m

2m

r

r

trajectory

v rv

1i

1j

ri

j


03

rr

r

cross product with h

hrr

hr

3



Equation of orbit

Solution of problem

34

1m

2m

r

r

trajectory

v rv

1i

1j

ri

j


03

rr

r


hrr

hr

3

1

2 krh

rirr

and expressed by polar coordinates

r

h



Equation of orbit

Solution of problem

35

1m

2m

r

r

trajectory

v rv

1i

1j

ri

j


03

rr

r


hrr

hr

3

1

2 krh

rirr

jhr


r

h


Equation of orbit

angular momentum is const. vector

1. The two body problem Solution of problem

36

1m

2m

r

r

trajectory

v rv

1i

1j

ri

j


03

rr

r


hrr

hr

3

1

2 krh

rirr

jhr

0

dt

hd


r

h



Equation of orbit

Solution of problem

37

1m

2m

r

r

trajectory

v rv

1i

1j

ri

j


03

rr

r


hrr

hr

3

1

2 krh

rirr

jhr

0

dt

hd

jdt

dhr

dt

d


r

h

angular momentum is const. vector



Equation of orbit

Solution of problem

38

1m

2m

r

r

trajectory

v rv

1i

1j

ri

j


11 sin cos jiir

jdt

dhr

dt

d

11 cos sin jij

Unit vectors of polar coord. system are not constant vectors



Equation of orbit

Solution of problem

39

1m

2m

r

r

trajectory

v rv

1i

1j

ri

j


11 sin cos jiir

jdt

dhr

dt

d

11 cos sin jid

id r

11 cos sin jij




Equation of orbit

Solution of problem

40

1m

2m

r

r

trajectory

v rv

1i

1j

ri

j


11 sin cos jiir

jdt

dhr

dt

d

11 cos sin jid

id r

11 cos sin jij

jd

id r




Equation of orbit

Solution of problem

41

1m

2m

r

r

trajectory

v rv

1i

1j

ri

j


11 sin cos jiir

jdt

dhr

dt

d

11 cos sin jid

id r

11 cos sin jij

jd

id r

ridt

dhr

dt

d


is scalar time invariant parameter



Equation of orbit

Solution of problem

42

1m

2m

r

r

trajectory

v rv

1i

1j

ri

j


jdt

dhr

dt

d

ridt

dhr

dt

d

eihr r


is integration constant, i.e. const. vector

e



Equation of orbit

Solution of problem

43

1m

2m

r

r

trajectory

v rv

1i

1j

ri

j


jdt

dhr

dt

d

ridt

dhr

dt

d

eihr r

eirhrr r


e

dot product with r




Equation of orbit

Solution of problem

44

1m

2m

r

r

trajectory

v rv

1i

1j

ri

j


jdt

dhr

dt

d

ridt

dhr

dt

d

eihr r

eirhrr r

cos1 erhrr

dot product with r


e

cbacba

using




Equation of orbit

Solution of problem

45

1m

2m

r

r

trajectory

v rv

1i

1j

ri

j


jdt

dhr

dt

d

ridt

dhr

dt

d

eihr r

eirhrr r

cos1 erhrr

cos1

1

2

e

hr

cbacba

using

dot product with r


e


Scalar equation of orbit is eccentricity

re



Equation of orbit

Solution of problem

46

1m

2m

r

r

trajectory

v rv

1i

1j

ri

j


cos1

1

2

e

hr

rvh

cos1 ehr

hv


re



Equation of orbit

Solution of problem

47

1m

2m

r

r

trajectory

v rv

1i

1j

ri

j


cos1

1

2

e

hr

rvh

cos1 ehr

hv

sin ehdt

drrvr


re



energy of system written in inertial frame of reference placed in center of mass

Energy law

48

1m

2mr


pkktot EEEE 21




Energy law

49

1m

2mr


pkktot EEEE 21

r

mmGvmvmE mmtot

212

22

2

112

1

2

1 expressed by inertial

motion




Energy law

50

1m

2mr


pkktot EEEE 21

r

mmGvmvmE mmtot

212

22

2

112

1

2

1

r

mmGv

mm

mmEtot

212

21

21

2

1

expressed by inertial motion

expressed by relative motion

21

21

mm

mm

reduced mass of system




Energy law

51

1m

2mr


pkktot EEEE 21

r

mmGvmvmE mmtot

212

22

2

112

1

2

1

r

mmGv

mm

mmEtot

212

21

21

2

1



21

21

mm

mm

reduced mass of system

r

v

2

2specific orbital energy (total energy per unit reduced mass) vis viva equation




Energy law

52

1m

2mr


pkktot EEEE 21

r

mmGvmvmE mmtot

212

22

2

112

1

2

1

r

mmGv

mm

mmEtot

212

21

21

2

1



r

v

2

2specific orbital energy (total energy per unit reduced mass) vis viva equation

specific energy expressed by

22

2

12

1e

h

e



Shape of trajectory depends on eccentricity

Equation of orbit is equation of conic sections:

circle

ellipse

parabola

hyperbola

Trajectories

53

e

0e

10 e

1e

1e

cos1

1

2

e

hr

equation of orbit

0e 10 e 1e 1e 0h



Circle: (bounded trajectory)

Trajectories

54

0e

cos1

1

2

e

hr

equation of orbit speed of motion

period specific energy

2hr

r

cos1 eh

vv r

v

r

rT

/

2

2/32 rT

22

2

12

1e

h

r

2

1




Trajectories

55

0e

2/32 rT

trajectories of satellite in different altitude passed in time EarthT

central body circ. velocity [km/s]

circ. period [min.]

Earth 7.90 84.48

Moon 1.68 108.36

Mars 3.55 100.19

Sun (surface) 436.7 166.91

Sun (Earths) 29.78 5.26x105

rv




Trajectories

56

0e

s 86164 GEOT

km42164GEOr

km/s07.3GEOv

trajectories of satellite in different altitude passed in in time .min48.84EarthT

2/32 rT

rv



Ellipse: (bounded trajectory)

Trajectories

57

10 e

a semimajor axis

empty focus

b semiminor axis

P -periapsis A - apoapsis

C - center

F - focus




Trajectories

58

10 e

cos1

1

2

e

hr

1

1

2

e

hrP

1

1

2

e

hrA

1

1

e

e

r

r

A

P

PA

PA

rr

rre

P -periapsis a semimajor axis

F - focus

empty focus

b semiminor axis

A - apoapsis

r

-true anomaly

C - center




Trajectories

59

10 e

cos1

1

2

e

hr

1

1

2

e

hrP

1

1

2

e

hrA

AP rra 2

2

2

-1

1

e

ha


F - focus

empty focus

b semiminor axis

A - apoapsis

r

-true anomaly

C - center




Trajectories

60

10 e


F - focus

empty focus

b semiminor axis

A - apoapsis

r

-true anomaly

AP rra 2 21 -eab

PA

PA

rr

rre

2 PA rrCF

C - center

eaCF 222 CFab




Trajectories

61

10 e eccentricity flattening

2

222

a

bae

a

baf

211 ef

1.0 e 1.0 f




Trajectories

62

10 e eccentricity flattening

1.0 e 1.0 f

usage: description of orbits

usage: description of planet shape

298.257 /1 f

flattening of Earth: 21.4 km diff. in radius




Trajectories

63

cos1

1

2

e

hr


period specific energy 2

2

2

12

1e

h

a

2

1

10 e

2

32

aT

h

abT

2

r

v

2

2

rav

2




Trajectories

64

10 e

ellipses with equal semimajor axis : a

a

2

1

2

32

aT

equal period and orbital energy

location of orbits

shape of ellipses




Trajectories

65

10 e

ellipses with equal semimajor axis : a

a

2

1

2

32

aT

equal period and orbital energy

rp[km] vp[km/s] ra[km] va[km/s]

42164 3.07 42164 3.07

29514.8 4.19 54813.2 2.25

16865.6 6.15 67462.4 1.53

8432.8 9.22 75895.2 1.02

rav

2


Parabola: (open trajectory)

- true anomaly

r

1. The two body problem Trajectories

66

1e

F - focus

directrix

P -periapsis

cos11

1

2

hr


specific energy

2

2

2

12

1e

h

0

r

v

2

2

rv

2




Trajectories

67

1e

central body esc. velocity [km/s]

Earth 11.18

Moon 2.37

Mars 5.02

Sun (surface) 617.5

Sun (Earths) 42.12

rv

2Earthp Rh

10

1




Trajectories

68

1etrajectories of satellite in different

Earthp Rh10

1

hp [km] (Earth) vp[km/s]

0 11.18

637.8 10.66

1275.6 10.20

1913.4 9.80

2551.2 9.44

3189.0 9.12

3826.8 8.83

Earthp Rh10

1




Trajectories

69

1etrajectories of satellite in different

Earthp Rh10

1

Earthp Rh10

1 Earthp Rh

10

1




Trajectories

70

1e

Earthp Rh10

1 Earthp Rh

10

1 Earthp Rh

10

1




Trajectories

71

1e

Earthp Rh10

1

Earthp Rh10

1

Earthp Rh10

1



Hyperbola: (open trajectory)

Trajectories

72

1e

F- focus empty focus

asymptotes

vertex

a- semimajor axis




Trajectories

73

1e

F- focus empty focus

asymptotes

vertex

a- semimajor axis

r


specific energy

22

2

12

1e

h

r

v

2

2

cos1

1

2

e

hr

a2

rav

22




Trajectories

74

1etrajectories of satellite with different periapsis:

1.0e

Earthp Rr

e vp[km/s]

1.1 11.45

1.2 11.72

1.3 11.98

1.4 12.24

1.5 12.49

1.6 12.74

1.7 12.99




Trajectories

75

1etrajectories of satellite with different periapsis:

1.0e

Earthp Rr

e vp[km/s]

1.1 11.45

1.2 11.72

1.3 11.98

1.4 12.24

1.5 12.49

1.6 12.74

1.7 12.99




Trajectories

76

1etrajectories of satellite with different alt. eccentricity: 1.1e

Earthp Rh 1.0

hp vp[km/s]

0.1xREarth 11.45

0.2xREarth 10.92

0.3xREarth 10.45

0.4xREarth 10.04

0.5xREarth 9.68

0.6xREarth 9.35

0.7xREarth 9.05




Trajectories

77

1etrajectories of satellite with different alt. eccentricity: 1.1e

Earthp Rh 1.0

hp vp[km/s]

0.1xREarth 11.45

0.2xREarth 10.92

0.3xREarth 10.45

0.4xREarth 10.04

0.5xREarth 9.68

0.6xREarth 9.35

0.7xREarth 9.05



Two cases can be investigated:

time as a function of position

position as a function of time

Only ellipse orbit is presented, but similar expressions can be derived for all trajectories

Time and position

78



True, Mean and Eccentric anomalies

Time and position

79

orbit

auxiliary circle




Time and position

80

orbit

auxiliary circle

location of satellite

true anomaly

focus




Time and position

81

orbit

auxiliary circle

eM

location of satellite

virtual location on circle with const. motion with the same period as satellite has

true anomaly

mean anomaly

eM

focus




Time and position

82

orbit

auxiliary circle

eE

eM

location of satellite projection of location on circle

true anomaly

eccentric anomaly eE

focus

virtual location on circle with const. motion with the same period as satellite has

mean anomaly

eM



Time as a function of position Time and position

83

22 rdt

drh

using mean anomaly




84

22 rdt

drh

drh

dt 21

using mean anomaly




85

22 rdt

drh

drh

dt 21

cos1

1

2

e

hr

using mean anomaly




86

22 rdt

drh

drh

dt 21

cos1

1

2

e

hr

223

cos1

e

dhdt

using mean anomaly




87

22 rdt

drh

drh

dt 21

cos1

1

2

e

hr

223

cos1

e

dhdt

0

22

3

cos1 e

dhtt p

using mean anomaly



Time as a function of position - ellipse Time and position

88

0pt

0

22

3

cos1 e

dhtt p

10 e

using mean anomaly




89

0pt

0

22

3

cos1 e

dhtt p

10 e

cos1

sin1

2tan

1

1tan2

1

1

cos1

21

2/320

2 e

ee

e

e

ee

d

using mean anomaly




90

0pt

0

22

3

cos1 e

dhtt p

10 e

cos1

sin1

2tan

1

1tan2

1

1

cos1

21

2/320

2 e

ee

e

e

ee

d

using mean anomaly

eM

ee

d2/32

0

21

1

cos1




91

10 e

using mean anomaly

]rad[

]rad[ eM

(circle) 0e

15.0e

eM

e

ht

2/322

3

1

1

cos1

sin1

2tan

1

1tan2

21

e

ee

e

eM e




92

10 e

using mean anomaly

]rad[

]rad[ eM

cos1

sin1

2tan

1

1tan2

21

e

ee

e

eM e

.5760e




93

10 e

cos1

sin1

2tan

1

1tan2

21

e

ee

e

eM e

eM

e

ht

2/322

3

1

1

using eccentric anomaly

eE eEsin




94

10 e

cos1

sin1

2tan

1

1tan2

21

e

ee

e

eM e

eM

e

ht

2/322

3

1

1


eE eEsin

EeEM e sin

Keplers equation




95

10 e

2tan

1

1tan2 1

e

eEe


]rad[

]rad[ eE

(circle) 0e

15.0e




96

10 e


]rad[

]rad[ E

2tan

1

1tan2 1

e

eEe

.5760e




97

10 e

2tan

1

1tan2 1

e

eEe

defined




98

10 e

2tan

1

1tan2 1

e

eEe

defined

EeEM e sin




99

10 e

2tan

1

1tan2 1

e

eEe

defined

EeEM e sin

eM

e

ht

2/322

3

1

1




100

10 e

2tan

1

1tan2 1

e

eEe

defined

EeEM e sin

eM

e

ht

2/322

3

1

1

eM

Tt

2




101

10 e

2tan

1

1tan2 1

e

eEe

defined

EeEM e sin

eM

e

ht

2/322

3

1

1

eM

Tt

2

ntM e

Tn

2

average angular velocity



Position as a function of time - ellipse Time and position

102

tT

M e2

t

10 e

defined




103

eee EeEM sin

tT

M e2

t

10 e

defined




104

must be computed numerically

eee EeEM sin

tT

M e2

t

10 e

defined

eE




105

must be computed numerically

eee EeEM sin

tT

M e2

t

10 e

defined

the problem of finding true anomaly for defined time is called Keplers problem

eE

2tan

1

1tan2 1 e

E

e

e


Frame of reference

Earth-based systems

Orbital elements

Calculation of elements


106


Frame of reference

107


to describe orbits in three dimensions, the coordinate system in frame of reference must be defined

Newton laws are valid in inertial frame of reference

practically only pseudoinertial frame of reference can be considered

coordinate system is formed in considered frame of reference


Frame of reference

108


coord. system is defined by:

origin, fundamental plane and preferred direction

choice of frame of reference and subsequently coordinate system depends on considered trajectory:

Interplanetary trajectory Interplanetary systems, e.g. Heliocentric coordinate system

Earth orbits Earth-based systems


Earth-based systems

109


Geocentric Equatorial System (GES) - the most common system in astrodynamics

the center of coord. system is at Earths center

not-rotating coord. system

fundamental plane Earths equator plane

axis X points towards the vernal equinox

axis Z extends through the North Pole


Earth-based systems

110


Geocentric Equatorial System (GES):

is often considered as Earth-Centered Inertial system (ECI)

ECI frame of reference is not fixed in space:

gravitational forces of planets planetary precession

gravitational forces of Moon and Sun luni-solar precession with period 26,000 years

combined effect general precession

inclination of Moon additional torque on Earths equatorial bulge nutation with period 18,6 years

due to precession and nutation equinox is moving


Earth-based systems

111


Geocentric Equatorial System (GES):

for all precise applications, ECI must by defined on specific date

J2000 - commonly used ECI frame is defined with the Earth's Mean Equator and Equinox at 12:00 Terrestrial Time on 1 January 2000

other Earth-based systems:

Earth-Centered, Earth-Fixed Coord. System rotate with Earth

Perifocal Coord. System


Orbital elements

112


Location of the satellite:

1. the location of the orbital plane in defined coord. system of chosen frame of reference

2. the position of the elliptical orbit in this plane

3. the characteristics of ellipse

4. the position of the moving satellite on the orbit

i ,

)(or , ahe

)(or M



113


the goal is to determine orbital elements from: position vector

velocity vector

both vectors are defined in GES at time

kvjvivv zyx

0t

krjrirr zyx



114


vr

and

kvjvivv zyx

krjrirr zyx

state vector



115


vr

and zyx

zyx

vvv

rrr

kji

vrh

kvjvivv zyx

krjrirr zyx

1st element

state vector

h



116


vr

and

2nd element

h

hi z1cos

kvjvivv zyx

krjrirr zyx

1st element

state vector

khjhihh zyx

h

i


kvjvivv zyx


117


h

vr

and

i

vvrr

rve

21

krjrirr zyx

khjhihh zyx

3th element

2nd element

1st element

state vector

e


khjhihh zyx


118


h

vr

and e

i vector of node line

zyx hhh

kji

hkn 100

kvjvivv zyx

krjrirr zyx

kejeiee zyx

3th element

2nd element

1st element

state vector

n



119


h

vr

and e

n

i 4th element

kvjvivv zyx

krjrirr zyx

khjhihh zyx

kejeiee zyx

kjninn yx

0

n

nx1cos

3th element

2nd element

1st element

state vector



120


h

vr

and e

n

i

5th element

kvjvivv zyx

krjrirr zyx

khjhihh zyx

kejeiee zyx

kjninn yx

0

e

e

n

n

1cos

4th element

3th element

2nd element

1st element

state vector



121


h

vr

and e

n

i

6th element

kvjvivv zyx

krjrirr zyx

khjhihh zyx

kejeiee zyx

kjninn yx

0

r

r

e

e

1cos

5th element

4th element

3th element

2nd element

1st element

state vector



122


input parameters:

Example:

km/s 3.435 1.581, 7.556,-

km 1567.56 6174.08, 2004.75,

p

p

v

r



123


input parameters:

0

30

45

196.0

28

s/km 56430.1 2

e

i

hExample:

km/s 3.435 1.581, 7.556,-

km 1567.56 6174.08, 2004.75,

p

p

v

r



124


input parameters:

0

30

45

196.0

28

s/km 56430.1 2

e

i

hExample:

Orbit in 2D view:

km/s 3.435 1.581, 7.556,-

km 1567.56 6174.08, 2004.75,

p

p

v

r



125


input parameters:

Example:

Orbit in 3D view:

km/s 3.435 1.581, 7.556,-

km 1567.56 6174.08, 2004.75,

p

p

v

r



126


input parameters:

km/s 3.435 1.581, 7.556,-

km 1567.56 6174.08, 2004.75,

p

p

v

r

Example:

Orbit in 2D map: Orbit in 3D view:



127


input parameters:

Example: GEO

Orbit in 2D map: Orbit in 3D view:

GEO, circular orbit,

2.5i



128


input parameters:

Example: GEO

Orbit in 2D map: Detail view:

GEO, circular orbit,

2.5i



129


input parameters:

Example: GEO


GEO,

0i

01575.0e



130


input parameters:

Example: GEO


GEO,

5.2i

01575.0e



131


input parameters:

Example: Molnija

Orbit in 2D map:

41.63i

75.0e

Orbit in 3D view:

km 40089ahkm260ph


Perturbing forces

Geopotential

Orbit propagation

Variation of parameters

Examples of orbits


132


Perturbing forces

133


Orbits of Earth satellites are influenced by 2 facts:

The Earth is not exactly spherical and the mass distribution is not exactly spherically symmetric

The satellite feels other forces apart from the Earths attraction:

attractive forces due to other heavenly bodies

forces that can be globally categorized as frictional

All these influences are called perturbations


Perturbing forces

134


Perturbing forces

Conservative forces can be derived from potential: flattening of the Earth Attraction of the Moon Attraction of the Sun Attraction by other planets

Non-conservative forces cannot be derived from potential dissipative forces: atmospheric drag radiation pressure


Perturbing forces

135


Influence of perturbing forces expressed by accelerations: GM attraction of Earth (sphere shape) J2 flattening of the Earth (Earth ellipsoid) J4, J6 potential of Earth expressed by higher orders Moon, Sun, Planets their attraction

sou

rce:

Cap

de

rou

: H

and

bo

ok

of

Sate

llite

Orb

its


Geopotential

136


Potential of single mass point:

position of masses

1m

r

r

mmGE p

21

potential energy of mass in gravitational field of mass

1m2m2

m


Geopotential

137



position of masses

1m

r

r

mmGE p

21


1m2m

rm

ErU

p

2

gravitational potential

2m


Geopotential

138



position of masses

1m

r

r

mmGE p

21


1m2m

rm

ErU

p

2


Ur grad

equation of motion expressed by potential

2m


Geopotential

139


Potential of Earth: 1. approximation - sphere

position of masses

M

2m

d

dMmGdE p

2

potential energy of mass in gravitational field of dM

2mdM

d

r


Geopotential

140



position of masses

M

2m

d

dMmGdE p

2


2m


dMd

r

d

GdM

m

dEdU

p

2


Geopotential

141



position of masses

M

2m

d

dMmGdE p

2


2m


MM

d

GdMdUU

dMd

r

d

GdM

m

dEdU

p

2


Geopotential

142



position of masses

M

2m

d

dMmGdE p

2


2m


MM

d

GdMdUU

integration over sphere boundary

equal potential as single mass potential

rr

GMrU

dMd

r

d

GdM

m

dEdU

p

2


Geopotential

143


Potential of Earth: 2. approximation - ellipsoid

position of masses

M

2mdM

d

r

Position of : longitude latitude radius r

2m


Geopotential

144



position of masses

M

2mdM

dM

Ellipsoid: longitude latitude radius

d

r


2m


Geopotential

145



position of masses

M

2mdM

dM


d

r


2m

2

cos21

rrrd

angle between and r

d


Geopotential

146



position of masses

M

2mdM

dM


d

r


2m

2

cos21

rrrd

angle between and r

d

M

d

GdMU


Geopotential

147



position of masses

M

2mdM

d

r

M

d

GdMU

using: expansion of 1/d in terms of Legendre polynomials symmetric properties of ellipsoid


Geopotential

148



position of masses

M

2mdM

d

r

M

d

GdMU

using: expansion of 1/d in terms of Legendre polynomials symmetric properties of ellipsoid

2

1sin31,,,

2

2

2

Jr

R

rrUrU

is equatorial radius R

dimensionless coefficient 2J zx II

MRJ

22

13

2 100826.1J


Geopotential

149


Potential of Earth: expansion to higher degrees

potential is function of all 3 coordinates, i.e. ,,rU


Geopotential

150




l

m

lmlmlm

l

l

PmSmCr

R

rrU

00

sinsincos,,


Geopotential

151


l

m

lmlmlm

l

l

PmSmCr

R

rrU

00

sinsincos,,

parameters are obtained from precise observation of the motion of satellites




Geopotential

152


l

m

lmlmlm

l

l

PmSmCr

R

rrU

00

sinsincos,,

Legendre functions





Geopotential

153


l

m

lmlmlm

l

l

PmSmCr

R

rrU

00

sinsincos,,

Legendre functions sinsin lmPm

sincos lmPm

products





Geopotential

154


l

m

lmlmlm

l

l

PmSmCr

R

rrU

00

sinsincos,,

Legendre functions sinsin lmPm

sincos lmPm imlmlm PH e sin,

products

Complex functions called Spherical Harmonics





Geopotential

155


l

m

lmlmlm

l

l

PmSmCr

R

rrU

00

sinsincos,,


for m=0: , , are called zonal harmonics, for m=l: are called sectoral harmonics all other functions are called tesseral harmonics

00 lS ,0lH

,llH

,lmH

ll JC 0




Geopotential

156



Zonal Harmonics

l=0

l=1 l=2 l=3

m=0 m=1 m=2 m=3

Spherical Harmonics (SH)


Geopotential

157



Sectoral Harmonics

l=0

l=1 l=2 l=3

m=0 m=1 m=2 m=3



Geopotential

158



Tesseral Harmonics

l=0

l=1 l=2 l=3

m=0 m=1 m=2 m=3



Geopotential

159



l=0

l=1 l=2 l=3

m=0 m=1 m=2 m=3


geopotential model of Earth is using coefficients in SH expansion, for example Goddard Earth Model 10b (GEM10b) is using 21x21 SH expansion


Orbit propagation

160


the goal is to solve equation of motion with initial conditions

potential U expresses influence of central acceleration and perturbative acceleration

for example, perturbative potential

00 )0( and )0(

grad

rtrrtr

Ur

RUU 0

rU

0

2

1sin3 2

23

2

J

r

RR


Orbit propagation

161


analytical methods: general perturbations

expresses modification of motion

enable to determine whether the eccentricity increases, the orbit begins to precess, and so on

numerical methods: special perturbations

one step methods purely mathematical approach: Runge-Kuta

multistep methods methods developed by astronomers to determine the motions of planets: Adams-Bashforth, Adams-

Moulton

special methods design specially for artificial satellites



162


Variation of parameters is analytical method to investigate influence of perturbation on planetary or satellite motion

Mathematical intro:

tgytfdt

dy

diff. equation with right hand side



163



Mathematical intro:

tgytfdt

dy


0 ytfdt

dy

homogenous equation



164



Mathematical intro:

tgytfdt

dy


0 ytfdt

dy

homogenous equation

dttfy

dy



165



Mathematical intro:

tgytfdt

dy


0 ytfdt

dy

homogenous equation

dttfy

dy

dttf

cy ehomogenous solution c int. constant



166



Mathematical intro:

tgytfdt

dy


0 ytfdt

dy

homogenous equation

dttfy

dy

dttf


to obtain solution of eq. with right hand side, we allow c to be function of t



167



Mathematical intro:

tgytfdt

dy


0 ytfdt

dy

homogenous equation

dttfy

dy

dttf



tg

dt

dc dttf

e



168



Mathematical intro:

tgytfdt

dy


0 ytfdt

dy

homogenous equation

dttfy

dy

dttf



tg

dt

dc dttf

e

dttgCtc

dttfe



169


Similar process can be applied to system of diff. eq.

diff. equation of motion can be written as system of equations

vdt

rd

Rrrdt

vdgrad

3



170




vdt

rd

Rrrdt

vdgrad

3

solution without right hand side

constants 6 ,trr

constants 6 ,tvv



171




vdt

rd

Rrrdt

vdgrad

3


constants 6 ,trr

constants 6 ,tvv

6 int. constants are 6 orbital elements

Meai ,,,, ,



172




vdt

rd

Rrrdt

vdgrad

3


constants 6 ,trr

constants 6 ,tvv


Meai ,,,, ,

variation of all 6 orbital elements

tMteta

ttit

, ,

, , ,



173




vdt

rd

Rrrdt

vdgrad

3


constants 6 ,trr

constants 6 ,tvv


Meai ,,,, ,

variation of all 6 orbital elements

tMteta

ttit

, ,

, , , calculation of parameters



174




i

R

inabdt

d

sin

1

R

inab

iR

inabdt

di

sin

cos

sin

1

e

R

ena

b

i

R

inab

i

dt

d

3sin

cos

M

R

nadt

da

2

M

R

ena

bR

ena

b

dt

de

4

2

3

e

R

ena

b

a

R

nadt

dM

4

22

Lagranges planetary equations



175


Perturbative potential must be expressed by orbital elements

2

1sin3 2

23

2

J

r

RR

Meai ,,,, ,



176



2

1sin3 2

23

2

J

r

RR

Meai ,,,, ,

cos1

1 2

e

ear

sinsinsin i

RR2. approximation - ellipsoid



177



average value of R in one period T

2

1sin3 2

23

2

J

r

RR

Meai ,,,, ,

cos1

1 2

e

ear

sinsinsin i


dMRdtRT

RT

2

00 2

11



178



average value of R in one period T

2

1sin3 2

23

2

J

r

RR

Meai ,,,, ,

cos1

1 2

e

ear

sinsinsin i


dMRdtRT

RT

2

00 2

11

2sin3

14

1 222/323

2

iJea

RR



179


Perturbative potential can be decomposed into average (secular) and periodic part

ps RRR



180



ps RRR

average value in one period is zero



181



ps RRR

RRs




182



Replacing perturbative potential by its secular part

ps RRR

RRs


RsR R



183




ps RRR

RRs


RsR

2sin314

1 222/323

2

iJea

RR

R



184




ps RRR

RRs


RsR

2sin314

1 222/323

2

iJea

RR

R

ieaRR ,,



185


Lagranges planetary equations ieaRR ,,

M

Ra

dt

da

M

RRe

dt

de,

RRi

dt

di,

i

R

dt

d

e

R

i

R

dt

d,

e

R

a

RM

dt

dM,



186



M

Ra

dt

da

aie , , are constants

M

RRe

dt

de,

RRi

dt

di,

i

R

dt

d

e

R

i

R

dt

d,

e

R

a

RM

dt

dM,



187



M

Ra

dt

da

aie , , are constants

M

RRe

dt

de,

RRi

dt

di,

i

R

dt

d

e

R

i

R

dt

d,

e

R

a

RM

dt

dM,

i

a

RnJ

edt

dcos

12

3 2

222

1cos3

14

3 22

22/32

i

a

RnJ

en

dt

dM

1cos5

14

3 22

222

i

a

RnJ

edt

d



188



Keplers orbit

input parameters:

km/s 3.435 1.581, 7.556,-

km 1567.56 6174.08, 2004.75,

p

p

v

r



189



Keplers orbit Perturbed orbit after 100 x T only J2 is considered

9.53

9.32

input parameters:

km/s 3.435 1.581, 7.556,-

km 1567.56 6174.08, 2004.75,

p

p

v

r



190


Numerical solution of orbital equations

Red color perturbed orbit in specific time range Blue color unperturbed Keplers orbit

Tt 40 ,0 TTt 80 ,40 TTt 012 ,80 TTt 016 ,120


Examples of orbits

191


Sun-synchronous orbits

Earth rotates counterclockwise around the Sun with angular velocity 0.986 per day

if satellite orbit rotates clockwise with the same angular velocity, position of orbit relative to the Sun will be still the same

i

a

RnJ

edt

dcos

12

3 2

222

ia

R

RJ

dt

dcos

2

3 2/7

32

i

Ra /


Examples of orbits

192



Earth rotates counterclockwise around the Sun with angular velocity 0.986 per day

if satellite orbit rotates clockwise with the same angular velocity, position of orbit relative to the Sun will be still the same

i

Ra /

Rai for 6.95min

180for km 12331max ia

Operating S-s satellites: orbit: circular or near circular

km 900700h


Examples of orbits

193


Sun-synchronous orbits Landsat 4:

km799.7285

07.99

a

i

Blue: Keplers orbit Red: sun-synchronous orbit Orange: Sun and sun beam

view from Sun


Examples of orbits

194



view from Earth

Landsat 4:

km799.7285

07.99

a

i


Examples of orbits

195



Landsat 4:

km799.7285

07.99

a

i


Impulsive maneuvers

Hohmann transfer

Non-Hohmann transfer

Plane change maneuvers


196


Impulsive maneuvers

197


brief firings of rocket motors change the magnitude and direction of the velocity vector instantaneously

during an impulsive maneuver, the position of the spacecraft is considered to be fixed, only the velocity changes impulsive maneuver is an idealization

velocity increment is related to consumed propellant

0

0Ln S

SSeS

m

mmuv


Impulsive maneuvers

198





gIu se

0

0Ln S

SSeS

m

mmuv


Impulsive maneuvers

199





gIu se

0

0Ln S

SSeS

m

mmuv

gI

v

S

S S

S

m

m

0

e1


Impulsive maneuvers

200


gIu se

0

0Ln S

SSeS

m

mmuv

gI

v

S

S S

S

m

m

0

e1

Propellant Specific impulse Is [s]

cold gas 50

Monopropellant hydrazine

230

LOX/LH2 455

Ion propulsion >3000

specific impulse characteristics

[-] / 0SS mm

[m/s]Sv

s 50sI

s 455sI

s 230sI


Impulsive maneuvers

201


impulse at periapsis

1, PP vr

Pv

1

23

Pv

P

0.019

km/s8.7

km300

1

1

Earth

e

v

rr

P

P

A


Impulsive maneuvers

202



1, PP vr

Pv

PPP vvv 12

1

23

Pv

P

0.019

km/s8.7

km300

1

1

Earth

e

v

rr

P

P

A


Impulsive maneuvers

203



22 PPvrh

1, PP vr

Pv

PPP vvv 12

1

23

Pv

P

0.019

km/s8.7

km300

1

1

Earth

e

v

rr

P

P

A


Impulsive maneuvers

204



22 PPvrh

1, PP vr

Pv

PPP vvv 12

1

23

Pv

P

0.019

km/s8.7

km300

1

1

Earth

e

v

rr

P

P

2e new orbit

A


Impulsive maneuvers

205



22 PPvrh

1, PP vr

Pv

PPP vvv 12

1

23

Pv

P

0.019

km/s8.7

km300

1

1

Earth

e

v

rr

P

P

2e

Documents

Orbital Mechanics - stuba.sk Mechanics Space for Education, Education for Space 1. The two body problem 2. Orbits in three dimensions 3. Orbital perturbations 4. Orbital maneuvers