OR.A.SI © Explores LEO Region

Antonios Arkas

Flight Dynamics Engineer

Table of Contents

Preface

1. Geometric and Celestial Events

2. Reference Orbit Module

3. Ground Track Shift Module

4. Preliminary On Station Mission Analysis

Appendix I Frozen Eccentricity Calculation

Appendix II Impact of Perturbations on LEO Orbits

II.1 Perturbations and Evolution of Orbital Elements

II.2 Effects of non-spherical geopotential

II.3 Effects of Luni-Solar Gravitation

II.4 Effects of Solar Radiation Pressure

II.5 Effects of Air Drag

Aside Ground Track Plotting Capabilities of OR.A.SI

Preface

OR.A.SI software was initially developed for the support of Flight Dynamics operations of

Hellas Sat 2 geostationary spacecraft and thus its modules were specialized for GEO

missions. The LEO regime not only poses new challenges from software development point

of view but it is also very interesting on its own since it’s characterized from more complicate

Flight Dynamics operations and control, with respect to the GEO ones.

The lack of specific modules for LEO control poses a formidable opportunity for OR.A.SI to

start a new interesting exploration to the territory of LEO orbital control. Even though the

core of OR.A.SI, which comprise the two Runge-Kutta integrators and the Adams-Bashforth-

Moulton integrator, are fully capable of accurately integrating such kind of orbits, it lacks the

modeling of solid Earth and ocean tides as well as the perturbations from Earth albedo and

infra-red radiation pressure. Prior to the development of software modules for the modeling of

these perturbations, which are vital for the accuracy of LEO propagation, the exploration of

LEO will start from the analysis of its most basic characteristics.

The evolution of this LEO code will be presented with the gradual implementation of various

modules and the presentation of their functionality and the corresponding code validation

with the addition of new chapters to the existing ones.

Both GEO and LEO OR.A.SI code is developed with C++ Builder 6 IDE.

1. Geometric and Celestial

Events

1.1. Geometric and Celestial Events for LEO

At the time being the celestial and geometric events for GEO orbits are enhanced with the

following events for LEO:

Detection of both ascending and descending node crossings for whatever geodetic

latitude.

Acquisition (AOS) and loss of signal (LOS) for whatever Earth station and desired

cut-off elevation.

Detection of eclipse transitions (penumbra entrance, penumbra-umbra transition,

umbra-penumbra transition and penumbra exit).

1.2. Node Crossings Detection

The module which detects the node crossings and their sub-satellite longitude, is absolutely

necessary for the accurate calculation of the semi major axis needed for the accomplishment

of a repeat ground track orbit with desired characteristics (Number of days per cycle and

number of revolutions per cycle). The node crossings module has the following

characteristics:

Detection of both ascending and descending node crossings for whatever

geodetic latitude.

UTC and local mean solar time of each crossing.

Sub-satellite longitude of each crossing.

Angular and linear distance between consecutive in time or consecutive in

space node crossings of the same type.

Calculation of consecutive number of repeat cycle (phasing) for repeat ground

track orbits.

Absolute cardinal number of each type of crossing.

Relative cardinal number of each type of crossing with respect to the

contemporary repeat cycle to which it belongs.

Difference between the position of a node and the corresponding node in the

mean grid for phased orbits.

Scenario to be used for the demonstration of node crossings

detection, AOS/LOS detection and eclipse transition detection.

Orbit Characteristics

Revolutions/Day = 14.5

Cycle Duration = 16 days

Equator Crossing Points = 233 per cycle

Sun synchronous

Local Solar time at node crossings = 3 h

Minimum altitude variance (frozen

eccentricity)

Earth Station for AOS/LOS Detection

Longitude (East) = 22.6860o

Geodetic Latitude = 38.8224o

Geodetic Height = 70 m

Cut-Off Elevation = 8o

Temperature = 15o C

Pressure = 1020 mbar

Humidity = 5%

1.3. Node Crossings Detection GUI and Consecutive in Time Node Crossings Example

The detection of node crossings is done with the utilization of 8th order Lagrange interpolation on

osculating ephemeris produced for the desired satellite.

Far apart from the GUI output, the software produces an extensive ASCI file containing all the details for

each node crossing.

This snapshot shows the results for contiguous node crossing (consecutive in space) detection.

1.4. Node Crossings Detection GUI and Consecutive in Space Node Crossings Example

1.5. Earth Station AOS/LOS Detection

• The detection of Earth station AOS/LOS is done with the utilization of 8th order Lagrange

interpolation on antenna pointing data (APD) produced for the desired satellite.

• The AOS/LOS events are generated for the specific value of cut-off elevation for each Earth station

which can be changed from the Earth Station database.

• The detection of AOS/LOS takes account of tropospheric refraction based on Hopfield model.

1.6. Eclipse Transition Detection

The detection of eclipse transitions is done with the utilization of 8th order Lagrange interpolation on a

continuous function of the apparent radius of Earth and Sun and the angular Earth-Sun separation as seen

from the satellite.

1.7. Spacecraft Illumination During Transition from Umbra to Penumbra and Light

1,00065 1,00068 1,00071 1,00074 1,00077 1,00080 1,00083

0

20

40

60

80

100

Illu

min

ation

(%

)

DOY (ddd.ddd)

Illumination

Figure 1.1: Spacecraft illumination while passing from umbra to penumbra and finally to light. The calculation

is done for a Sun synchronous phased orbit with local solar time at ascending node LTA = 3 h and mean major

semi axis a= 7077.469 Km. Penumbra transit duration is 11.787 sec.

2. Reference Orbit

Module

2.1. Reference LEO Module Characteristics

The purpose of this module is the calculation of the initial osculating state corresponding to a

Sun synchronous, repeat ground track, minimum altitude variance orbit with either a desired

local time at node crossings or overfly a desired Earth point. The initial state is calculated

based on the following input parameters:

State epoch.

Number of days per repeat cycle K.

Number of revolutions per repeat cycle M or alternatively the desired spacing

between contiguous crossings.

Desired eccentricity or automatic calculation of eccentricity for minimum

altitude variance orbit (frozen eccentricity).

Desired inclination or automatic calculation of inclination needed for Sun

synchronous nodal regression.

Local time at node crossings or geodetic coordinates of an Earth point for an

overfly.

For the case of frozen eccentricity, the argument of perigee is set automatically to ω = 90ο.

The derivation of the formula for the algorithm which computes the frozen eccentricity, can

be found in Appendix I.

2.2. Reference LEO Module GUI

The scenario to be used for the demonstration of the new module

capabilities is the LANDSAT polar, Sun synchronous, frozen

eccentricity, repeat ground track type of orbit.

Desired Reference Orbit Characteristics

Revolutions/Day = 14.5

Cycle Duration = 16 days

Equator Crossing Points = 233 per cycle

Sun synchronous

Local Solar time at node crossings = 6 h

Minimum altitude variance (frozen

eccentricity)

The calculation of the initial state is done under

the assumption of no air drag and for a

geopotential of 9th degree and order (OR.A.SI is

currently capable of simulating a geopotential up

to 36th degree and order).

Validation of Sun-Synchronicity

Method

1. Exhibition of Sun synchronicity

regularity through the stability of

local Solar time on ascending node

crossings for a period of two

months.

2. Achievement of the desired local

Solar time on ascending node

crossings.

2.3 Stability of Local Solar Time at Ascending Node

Figure 2.1: Evolution of local Solar time at ascending node. Shift of local Solar time amounts to less than 0.4 sec for a

1 month period comprising 466 orbits i.e. two repeat cycles. Equation of time hasn’t been taken account.

51545 51550 51555 51560 51565 51570

6,0513

6,0514

6,0515

6,0516

6,0517

6,0518

Loca

l S

ola

r T

ime

(h)

MJD (ddddd.ddd)

Local Solar Time

Validation of Repeat Ground Track

Regularity

Method

Exhibition of repeat ground track

stability through the following

characteristics:

1. Stability of ground track shift

(distance between consecutive in

time equatorial node crossings).

2. Stability of distance between

consecutive in space equatorial

node crossings.

3. Stability of difference between

actual track and mean grid

measured along the equator.

4. Coincidence of equatorial node

crossings during consecutive

passes.

2.4 Repeat Ground Track Pattern (16 days Period of Propagation)

2.5.1 Repeat Ground Track Definition of Terms

Phasing grid or repeat ground track is the network of ground tracks which cover the Earth in a quasi-

symmetrical way, during a specific time period called cycle duration, in relation to the Equator, and at the

end of which the satellite track nominally returns to the same position in relation to an Earth fixed

reference frame [2] pp.752).

Mean grid (ground track) is revolution symmetric around the polar axis, with an orbit of constant

inclination (i). If M is the number of orbits completed during a cycle, a constant angular interval may then

be defined, measured between two orbits which are contiguous in space, in any plane parallel to the

equatorial plane, and is know as the cycle track interval (having a value of: 2π/Μ) ([2] pp.752).

True grid (ground track) differs from the mean grid because it includes the effects of perturbations to the

satellite orbit: if only the irregularities in the geopotential are considered, then the influence of the tesseral

and zonal terms may be established: the latter give rise to a slight “warping”, depending on the latitude,

of the true grid in relation to the mean grid ; the former will prevent the cycle track interval from

remaining constant along the entire parallel ([2] pp.752).

Ground track shift λs is the distance between two consecutive in time equator crossings:

Ground track shift λs = Distance of noden+1 from noden

Phase difference at the Equator (Δlo) and at latitude L (ΔlL) is the difference between the actual track

and the reference grid, measured along the Equator or a parallel of latitude L, correspondingly. The

reference grid can be either the true or the mean grid ([2] pp.787).

2.5.2 Evolution of Difference Between Actual and Mean Ground Track Equatorial Nodes

The impact of perturbations on the regularity of the repeat ground track is readily demonstrated from the

difference between the actual equatorial node crossings and the corresponding crossings of the mean grid.

The daily fluctuation of few hundred meters is due to the tesseral harmonics of the geopotential [2] pp.806.

Figure 2.2: Evolution of difference between actual and mean grid equatorial node position.

51544 51545 51546 51547 51548 51549 51550 51551 51552

-0,2

-0,1

0,0

0,1

0,2

0,3

0,4D

ista

nce fro

m M

ean G

rid N

ode (

Km

)

MJD (ddddd.ddd)

Distance from Mean Grid Node

2.5.3 Impact of Tesseral Harmonics on Repeat Ground Track Regularity

The impact of tesseral harmonics on the regularity of the repeat ground track is readily demonstrated from

the fluctuation of the distance between two consecutive in time equator crossings which is called:

ground track shift λs = Distance of noden+1 from noden

The negative value of the distance in the diagram, indicates the westward motion of the ascending node

crossings (each node crossing resides westward with respect to the previous one).

51540 51560 51580 51600

-2752,1

-2752,0

-2751,9

-2751,8

Gro

und T

rack S

hift

s [K

m]

MJD (ddddd.ddd)

Ground Track Shift s for No Air Drag

Figure 2.3: Evolution of ground track shift with 1σ = 68 m for a 2 month period.

2.5.4 Impact of Tesseral Harmonics on Distance Between Consecutive in Space Node Crossings

The impact of the tesseral harmonics from the Earth geopotential prevents the distance between two

consecutive in space node crossings from remaining constant along the entire parallel [2] pp.753

Figure 2.4: Variation of distance between consecutive in space node crossings (1σ = 38.51 m ).

-50 0 50 100 150 200 250 300 350 400

171,92

171,94

171,96

171,98

172,00

172,02

172,04

172,06

172,08

Dis

tan

ce f

rom

Pre

vio

us N

od

e (

Km

)

Sub-Satellite Longitude (deg)

Distance from Previous Node

2.6.1 Deviation of the Ground Track from the Reference Repeat Pattern for 2 Month Period

Accurate Iterative Computation of a = 7068.5834Km

The deviation of the actual ground track from the reference repeat pattern, can be seen from the values of

geographical longitude corresponding to the same relative number of crossing in a repeat cycle but

belonging to different passes. The ascending node is moving from top to bottom and from left to the right.

Figure 2.5: Movement of ascending node sub-satellite point for a 2 month period.

Iterative computation of major semi axis.

0 50 100 150 200 250

-50

0

50

100

150

200

250

300

350

400 Sub-Satellite Longitude

Sub-S

ate

llite

Longitude (

deg)

Node Number in Cycle

Ascending Node Movement

2.6.2 Detail of the Ascending Node Sub-Satellite Longitude vs the Relative Number of Node Crossing

Accurate Computation of a = 7068.637 Km

The following diagram depicts a detail from Figure 2.5, of the sub-satellite longitude pertaining to the

same cardinal number of node, for 4 consecutive passes. The ground track’s accuracy of repeating itself

after each pass is evident from the close proximity of the ascending node longitude pertaining to the same

consecutive number in different passes.

This diagram corresponds to a 64 days

propagation period. Since the major

semi axis corresponds to a 16 days

repeat cycle, there are 4 different passes

during this time period of propagation.

Clarification:

A pass is different from a revolution and

occurs whenever the satellite passes over

the same area on the Earth again.

Figure 2.6: Sub-satellite longitude of ascending node crossings for four

consecutive passes. Iterative calculation of major semi axis.

43,999998 43,999999 44,000000 44,000001 44,000002

89,236

89,238

89,240

89,242

89,244

89,246

89,248

89,250

89,252

89,254

89,256

89,258

89,260

Sub-Satellite Longitude

Sub-S

ate

llite

Longitude (

deg)

Node Number in Cycle

First Pass

Second Pass

Third Pass

Fourth Pass

2.6.3 Deviation of the Ground Track from a Strict Repeat Pattern for 2 Month Period

Approximate Analytical Calculation of a = 7083 Km

The following diagram is the same with Figure 2.5 but the major semi axis has been approximated by an

analytical expression and not an iterative process. The jittering behavior of consecutive node crossings is a

indication of deviation from the nominal repeat pattern.

Figure 2.7: Movement of ascending node sub-satellite point for a 2 month period.

Analytic approximation of major semi axis.

0 50 100 150 200 250

-50

0

50

100

150

200

250

300

350

400 Sub-Satellite Longitude

Su

b-S

ate

llite

Lon

gitu

de

(d

eg

)

Node Number in Cycle

Ascending Node Movement

2.6.4 Detail of the Ascending Node Sub-Satellite Longitude vs the Relative Number of Node Crossing

Approximate Analytical Calculation of a = 7083 Km

The following diagram depicts a detail from Figure 2.7, of the sub-satellite longitude pertaining to the

same cardinal number of node, for 4 consecutive passes. If the ground track was absolutely repeating itself

then the ascending node longitude pertaining to the same consecutive number in different passes, should

have been the same but the approximate calculation of major semi axis results in a substantial movement

of the node crossing during consecutive passes.

Figure 2.8: Sub-satellite longitude of ascending node crossings for four consecutive passes.

Analytic approximation of major semi axis.

43,7 43,8 43,9 44,0 44,1 44,2 44,3

25

30

35

40

45

50

55

60

65

70

75

80

85

90

95

Sub-Satellite Longitude

Su

b-S

ate

llite

Lon

gitu

de

(d

eg

)

Node Number in Cycle

First Pass

Second Pass

Third Pass

Fourth Pass

2.6.5 Ideal Repeat Ground Track

In the case of an ideal repeat ground track, the longitude of ascending node crossings pertaining to the

same relative number of crossing, should coincide for all passes. This is the case of the repeat ground track

calculated with the new module which calculates the perturbed semi axis iteratively.

Figure 2.9: Zoom in of Figure 2.3 depicts the regularity of the repeat pattern when the

perturbed major semi axis is calculated iteratively.

0 10 20 30 40 50 60 70

-50

0

50

100

150

200

250

300

350

Sub-Satellite Longitude

Su

b-S

ate

llite

Lon

gitu

de

(d

eg

)

Node Number in Cycle

2.7 Difference Between Consecutive in Space and Consecutive in Time Crossing Points

In repeat ground tracks there is a significant difference between two consecutive node crossings separated

from the ground shift distance λs = 2751 Km = 24.71o (3.1) and the distance between two contiguous

node crossings which for our case is equal to 2*π*Rearth/233 = 172 Km = 1.545o. The difference is evident

from the following diagram.

Figure 2.10: Difference between consecutive in space and consecutive in time crossing points.

Validation of Minimum Altitude

Variance Characteristic Method

Exhibition of minimum altitude

variance and accurate computation of

frozen eccentricity through the

following characteristics:

1. Close curve evolution of mean

eccentricity tip, around

equilibrium point with small

radius.

2. Small variation of mean

eccentricity modulus.

3. Small variation of mean argument

of perigee.

4. Small variation of altitude for the

same argument of latitude.

2.8.1 Evolution of Mean Eccentricity Vector Around Frozen Eccentricity

Figure 2.10: Evolution of mean eccentricity vector for a period of 116 days, around the frozen eccentricity center,

with radius 0.000013.

-0,000024 -0,000016 -0,000008 0,000000 0,000008 0,000016

0,00118

0,00119

0,00120

0,00121

0,00122

0,00123

esin

()

ecos()

e*sin()

2.8.2 Evolution of Mean Eccentricity Modulus for a Period of 116 Days

Figure 2.11: Evolution of mean eccentricity modulus for a period of 116 days.

0 20 40 60 80 100 120

0,001425

0,001430

0,001435

0,001440

0,001445

0,001450

0,001455

0,001460

Eccentr

icity

DOY (ddd.ddd)

Eccentricity

2.8.3 Evolution of Mean Argument of Perigee for a Period of 116 Days

Figure 2.12: Evolution of mean argument of perigee for a period of 116 days.

0 20 40 60 80 100 120

110,8

111,0

111,2

111,4

111,6

111,8

112,0

112,2

112,4A

rgum

en

t o

f P

erig

ee

(d

eg

)

DOY (ddd.ddd)

Argument of Perigee

2.8.4 Evolution of Osculating and Mean Eccentricity for a Period of 116 Days

Figure 2.13: Evolution of mean and osculating eccentricity for a period of 116 days.

2.8.5 Altitude Evolution for a Period of 116 Days

Figure 2.14: Altitude evolution for a period of 116 days. Maximum variation for the same value of

argument of latitude is 52 m

Validation of Overfly Accuracy

Method

Verification of sub-satellite point

passing over a desired location on

Earth through propagation of the

calculated initial state and direct

confirmation of the overfly.

Desired Reference Orbit Characteristics

Revolutions/Day = 14.5

Cycle Duration = 16 days

Equator Crossing Points = 233 per cycle

Sun synchronous

Overfly through Earth point with longitude

λ = 390 East and latitude φ = 220 North.

Overfly while ascending.

Minimum altitude variance (frozen

eccentricity)

The calculation of the initial state is done under

the assumption of no air drag and for a

geopotential of 9th degree and order (OR.A.SI is

currently capable of simulating a geopotential up

to 36th degree and order).

38,0 38,2 38,4 38,6 38,8 39,0 39,2 39,4 39,6 39,8 40,0

21,0

21,2

21,4

21,6

21,8

22,0

22,2

22,4

22,6

22,8

23,0

Ge

od

etic L

atitu

de

(d

eg

)

Longitude (deg)

Geodetic Latitude

2.9 Overfly from Earth Point With λ = 39ο East and 22o North

-50 0 50 100 150 200 250 300 350 400

-100

-80

-60

-40

-20

0

20

40

60

80

100

Geodetic L

atitu

de (

deg)

Longitude (deg)

Geodetic Latitude

Figure 2.15: Verification of overfly a desired Earth point for a repeat ground track, Sun-synchronous,

minimum altitude variance orbit.

3. Ground Track Shift

Module

0 50 100 150 200 250 300 350

-100

-80

-60

-40

-20

0

20

40

60

80

100

Geodetic L

atitu

de (

deg)

Longitude (deg)

Shifted Ground Track

Original Ground Track

20o longitude shift

3.1. Ground Track Shift Module Characteristics

The purpose of this module is to shift the reference orbit in time, longitude of the ascending

node crossing points or synchronize it with the desired characteristics (crossing point

longitude or epoch) of a specific ascending node. Essentially these rotational transformations

of the ground track are necessary for the production of the orbit which will serve as a

reference for subsequent calculation of the needed orbital maneuvers based on the deviation

of the actual ground track from the reference one. The characteristics and capabilities of this

module are the following:

Shift ground track in time (definition of shift in days, repeat cycles or shift to a specific

date).

Cross track shift or shifting the ground track in longitude.

Synchronize the reference ground track with the characteristics of an ascending node

(epoch and node crossing longitude).

For all the methods of ground track shift, an appropriate shifting in time can be calculated

in order for the shifted ground track to maintain the same local solar time at the ascending

node crossings with the reference ground track.

Definition of the reference ground tracks either with the initial state or with an ephemeris

file.

4.2. Ground Track Shift Module Interface

Both GEO and LEO OR.A.SI software is implemented to be multiplatform. The choice of the desired

spacecraft from OR.A.SI database is done through the menu Spacecraft. The results of every module are

automatically saved in the appropriate directories of the chosen spacecraft. The interface has the flexibility

to accept as an input and produce as an output the orbits in the form of orbital elements of ephemeris files

whose names and paths are chosen from Input/Output Files menu.

Validation of Ground Track Shifting

Methods

Method

Direct comparison of the results with

the selected shifting method and the

desired shifting parameters.

Desired Reference Orbit Characteristics

Revolutions/Day = 14.5

Cycle Duration = 16 days

Equator Crossing Points = 233 per cycle

Sun synchronous

Minimum altitude variance (frozen

eccentricity)

The calculation of the initial state is done under

the assumption of no air drag and for a

geopotential of 9th degree and order (OR.A.SI is

currently capable of simulating a geopotential up

to 36th degree and order).

3.3.1 Ground Track Shift In Time (1 day shift) – Shifted Ground Track State

Calculation of 1 day time shifted ground track state.

3.3.2 Ground Track Shift In Time (1 day shift)

Reference Ground Track One Day Time Shifted Ground Track

Comparison between the node crossings of the reference ground track and the one which has been time

shifted one day in the future.

3.4.1 Ground Track Shift In Longitude (20o shift) - Shifted Ground Track State

Calculation of 20o longitude shifted ground track state.

3.4.2 Ground Track Shift In Longitude (20o shift) – Shifted Ground Track

Comparison between the reference ground track and the one which has been shifted in longitude for 20o.

Figure 3.1: Reference ground track and 20o longitude shifted ground track

3.4.3. Ground Track Shift in Longitude with Maintenance of LTAN (20o shift)

Reference Ground Track 20o Longitude Shifted Ground Track

Comparison between the node crossings of the reference ground track and the one which has been shifted

in longitude for 20o while maintaining the local solar time at node crossings. The maintenance of LTAN

dictates that the total time offset to be applied, should be an integral number of days so the nodes of the

shifted ground track are time shifted almost 1 day in the future.

3.5.1 Cross Track Shift (800 Km) – Calculation of Shifted Ground Track State

Calculation of 800 Km cross shifted ground track state.

3.5.2 Cross Track Shift (800 Km) – Comparison Between Ground Tracks

A cross track shift of 800 Km corresponds to an equivalent longitude shift of 7.2601o.

0 50 100 150 200 250 300 350

-100

-80

-60

-40

-20

0

20

40

60

80

100

Geodetic L

atitu

de (

deg)

Longitude (deg)

Geodetic Latitude

Figure 3.2: Cross track shift of 800 Km

210 215 220 225 230

-8

-6

-4

-2

0

2

4

6

800 Km

Geodetic L

atitu

de (

deg)

Longitude (deg)

Geodetic Latitude

7.2601o

3.5.3 Cross Track Shift (800 Km) – Validation with Equivalent Longitude Shift

Reference Ground Track 800 Km Cross Track Shift = 7.2601o Longitude Shift

The difference between the longitude of the first

descending node crossings is 7.2601o as expected.

The correspondence between cross track and

longitude shift is given in [2] pp.755 formula (5):

ia

CrossLongitude

earth

2cos1

ShiftTrack Shift

3.6.1 Ground Track Synchronization with AN Longitude and Maintenance of LTAN

Calculation of state corresponding to synchronization with ascending node (AN) at 72o with

maintenance of local solar time and 1 month time shifting.

3.6.2 Ground Track Synchronization with AN Longitude and Maintenance of LTAN

Reference Ground Track Synchronization with AN at 72o

In this case the reference ground track is synchronized with an ascending node (AN) of longitude 18o and

enforce the same local solar time. The AN of the reference ground track with the closest longitude is the

one with longitude 72.239o and Node Number 1. In the shifted ground track the ascending node with

number 1 has the desire longitude of 72o and the same local solar time with the one in the reference ground

track.

4. Preliminary On Station

Mission Analysis

4.1 Characteristics of On Station Preliminary Mission Analysis Tool

The mission analysis tool is a design tool whose purpose is the determination of the orbit

control strategy which satisfies simultaneously the constrains expressed by the desired values

of the operational parameters. The operational parameters are defined as the difference

between the mission parameters and their nominal value. The four operational parameters

which are taken account from this tool, are the following:

Δlo : phase difference at the equator

Δl : phase difference at a desired latitude L

ΔhL : altitude difference at desired latitude L

ΔH : deviation of local time at ascending node

The orbit control strategy determined from this tool, is defined by the following:

1. Orbit control deadband by means of maximum allowable deviation of mean major semi

axis Δamax and mean inclination Δimax.

2. The Δa and Δi biases needed for the initialization of the strategy.

3. Effective values of operational parameters. These are the values which are actually

achieved by the strategy and have equal or smaller dynamic ranges than the desired ones.

4. All the maneuvers needed for the control of the orbit. These maneuver are characterized

by their type (Δa or Δi correction), their ΔV magnitude and the thrust dates with respect

to the beginning of the mission (ETC).

4.2 Working Hypothesis for Validity of Mission Analysis Tool

The mission analysis tool is based on approximate analytic formulas and thus its validity is

based on certain assumptions with out violating the generality of its results. These

approximations mainly concern the modeling of the impact of perturbing forces on the orbit

elements evolution and consequently the modeling of operational parameters evolution. The

aforementioned hypothesis are as follows:

Every mission corresponds to a Sun-synchronous, frozen eccentricity, repeat ground track

(phased) orbit.

The equations of evolution are linearized around their nominal orbit values.

Only the geopotential term J2 is used for the modeling of non-spherical Earth.

The secular evolution of Δlo and ΔH are considered near parabolic and are given by

formulas (II.9) and (II.10).

The secular drift of inclination for Sun-synchronous orbits is given by (II.7) and it is

consider constant during the whole mission, thus the behavior seen in Figure II.15 is not

simulated.

The solar flux F10.7 and the geomagnetic activity index Kp are considered constant during

the whole mission.

The impact of the solar radiation pressure and air drag on mean eccentricity evolution are

not modeled thus no eccentricity control maneuvers are calculated by the tool.

The calculated maneuvers are impulsive.

4.3.1 Mission Analysis Tool GUI – Input Section

The input section is very simple and comprises the

following data:

1. The osculating state containing all the

characteristics of the orbit. This state is produced

from the Reference Orbit Module.

2. The nominal values of the operational parameters

describing the orbit control specifications.

3. The eccentricity radius Ro which is the distance of

mean eccentricity from the frozen eccentricity (see

Figure I.2).

4. The mission duration expressed in years.

5. The characteristics of the chosen spacecraft

are automatically retrieved from the database.

4.3.2 Mission Analysis Tool GUI – Output Section: Strategy Parameters

The output section is divided to the strategy

parameters and the maneuver parameters. The former

comprises the following data:

1. The secular drift of mean major semi axis and

mean inclination.

2. The nominal maneuver cycle pertaining to each

operational parameter (no coupling is considered).

3. The biased of the orbital elements and the

operational parameters for the initialization of the

strategy.

4. The Δa-Δi deadband.

5. The values of the operational parameters achieved

by the implementation of the strategy.

5. The actual control cycles pertaining to the

maneuvers predicted by the tool. These cycles

take account the coupling between the

simultaneous validity of all four constrains

concerning the operational parameters.

4.3.3 Mission Analysis Tool GUI – Output Section: Maneuver Parameters

4.3.4 Mission Analysis Tool GUI – Output Section: Maneuver Parameters

Every control strategy is realized with at most two different type of maneuvers which are

distinguished from their cycle duration and the operational parameter that they control. Each

maneuver corresponds to either a Δa or Δi correction or a combination of both. The Maneuver

and Consumption Characteristics of the output section, presents all the details concerning

the maneuvers pertaining to the strategy. The output fields of this section contain the

following information:

1. Type of the implemented strategy in accordance to the primary (smaller) maneuver cycle

and the operational parameter maintained form this maneuver.

2. The type of each of the maneuvers (or maneuver) computed by the strategy, in accordance

to the operational parameters that is controlled by the maneuver.

3. The maneuver cycle for each maneuver type.

4. The Δa and Δi increments of the maneuver.

5. The ΔV increments and mass consumptions corresponding to Δa and Δi corrections.

6. The total ΔV increment and mass consumption, corresponding to Δa and Δi corrections,

for the same type of maneuver.

7. The number of maneuver corresponding to each different type of maneuver.

8. The total number of both type of maneuvers.

9. The total ΔV increment and mass consumption for both type of maneuvers for the whole

mission duration.

4.3.5 Mission Analysis Tool GUI – Overview of Output Section

4.3.6 Mission Analysis Tool GUI – Output Section: Δa-Δi Plot

Figure 4.1: Diagram of Δa – Δi deviations from nominal orbit (current case corresponds to Δlo maintenance for di/dt > 0)

Mission Beginning

Natural Evolution

Inclination Correction Major Semi Axis Correction

Since the strategy is realized by Δa and Δi corrections , the evolution of these two deviations from their

nominal values are very essential for the understanding of the philosophy of the implemented strategy. The

red line corresponds to the slope of null dΔlo/dt and the blue one to the slope of null dΔΗ/dt/

4.3.7 Mission Analysis Tool GUI – Output Section: Δlo and – Δl Evolution Plots

Figure 4.2: Diagram of Δlo and Δl deviations from nominal orbit.

These two diagrams depict the evolution of the phase at the Equator deviation and of the phase at non-

zero latitude deviation. The x-axis of each diagram corresponds to the elapsed time counter since the

beginning of the mission, expressed in days.

4.3.8 Mission Analysis Tool GUI – Output Section: ΔH and – Δh Evolution Plots

Figure 4.3: Diagram of ΔH and Δh deviations from nominal orbit.

These two diagrams depict the evolution of the local time deviation at the ascending node and the altitude

deviation for the desired latitude. The x-axis of each diagram corresponds to the elapsed time counter since

the beginning of the mission, expressed in days.

4.4.1 Mathematical Background of the Model Used for Analytic Propagation

These evolution of Δlo and ΔH is supposed to be parabolic as described in II.3.3.1. This can be seen in the

leftmost plots of Figures 4.2. and 4.3.

The impact of Δa and Δi corrections on these two operational parameters are given, in accordance to [2],

as follows :

dt

dil

dt

dalsign

dt

dil

dt

dallilal iaiaMaxia 04

dt

dih

dt

dahsign

dt

dih

dt

dahHihah iaiaMaxia 4

(4.1)

(4.2)

The period for symmetric evolution for each case is given from the formulas:

dt

dil

dt

dal

lT

ia

Maxlo

04

dt

dih

dt

dah

HT

ia

MaxH

4

(4.3)

(4.4)

The secular drift of inclination is considered constant and it is given from (II.7).

4.4.2 Mathematical Background of the Model Used for Analytic Propagation

The initial time derivatives of of Δlo and ΔH for symmetric parabolic evolution, are given in accordance

to [2] as:

dt

dil

dt

dalsign

dt

dil

dt

dallilal

dt

ldiaiaMaxia 000

0

0 2 (4.5)

(4.6)

where Δa0 and Δi0 are the initial values of the corresponding deviations.

The evolution of phase difference ΔlL at non-zero latitude L, depends on the latitude, the deviation on

Equator Δlo and the inclination deviation.

21

2

2

cos

cos1

sin

sincos

L

i

i

LiaisensLll earthoL

(4.7)

dt

dih

dt

dahsign

dt

dih

dt

dahHihah

dt

HdiaiaMaxia 200

0

where isens has the value of +1 if the latitude corresponds to the ascending part of the orbit and -1 in the

opposite case.

From equation (4.7) it is evident that the evolution of ΔlL will be a linear one, depending on the sign of

di/dt, with a superimposed periodic term with parabolic evolution due to the evolution of phasing

deviation at the Equator. This behavior is evident in the rightmost plot of Figure 4.2.

The altitude deviation evolution is given from (II.5).

Method

The validity of the produced orbital control strategy is directly validated from the evolution

of all four operational parameters under the effect of the designed orbital maneuvers and

their graphical representation for the whole mission duration.

This method is considered valid at the extent of the accepted assumptions described in 4.2.

The states used for all cases, correspond to a repeat ground track, Sun-synchronous, frozen

eccentricity orbits and the spacecraft has ballistic coefficient BC = 0.02 m2/Kgr.

4.5.1 Mission Analysis Tool Validation Method

Test Cases

1. Primary correction is for phasing at the equator Δlo - State LTA = 3 h with di/dt > 0 :

Δlo = 10 Km, Δl = 10 Km at L = 78o, ΔΗ = 10 min and Δh = 400 m.

2. Primary correction is for altitude deviation Δh - State LTA = 3 h with di/dt > 0 :

Δlo = 10 Km, Δl = 10 Km at L = 78o, ΔΗ = 10 min and Δh = 200 m

3. Primary correction is for LTA deviation ΔH - State LTA = 3 h with di/dt > 0 :

Δlo = 200 Km, Δl = 42 Km at L = 78o, ΔΗ = 0.001 min and Δh = 800 m

4. Correction cycles of Δlo and ΔH are very close - State LTA = 3 h with di/dt > 0 :

Δlo = 10 Km, Δl = 42 Km at L = 78o, ΔΗ = 0.01 min and Δh = 400 m

5. Very tight control of Δlo - State LTA = 9 h with di/dt < 0 :

Δlo = 0.1 Km, Δl = 0.1 Km at L = 45o, ΔΗ = 10 min and Δh = 400 m

6. Primary correction is for phasing at the equator Δlo - State LTA = 6 h with di/dt ≈ 0 :

Δlo = 10 Km, Δl = 10 Km at L = 78o, ΔΗ = 10 min and Δh = 400 m.

4.5.2.1 Test Case 1: Primary Correction is Phasing Δlo at the Equator

The state in this case corresponds to local time at ascending node LTA = 3 h so di/dt > 0. The smallest

maneuver cycle corresponds to the maintenance of phase Δlo at the equator so all the other maneuver

cycles are condition in accordance to the one of Δlo. The first maneuver of this strategy is a Δa correction

for Δlo maintenance while the second one, with a cycle equal to the smallest one of ΔΗ and Δl, is a Δi

correction essentially imposed from Δh maintenance. This maneuver indirectly controls ΔΗ and Δl and is

accompanied by a Δa correction which compensates the disturbance of Δlo from the Δi correction

4.5.2.2 Test Case 1: Primary Correction is Phasing Δlo at the Equator

Figure 4.4: Diagram of Δa – Δi deviations from nominal orbit. The vertical lines correspond to Δa corrections while the

lines almost parallel with the Δi axis, are the inclination corrections. The natural evolution, since di/dt > 0, is done from the

left to the right part of the diagram. The control of Δa is done symmetrical with the slope of null dΔlo/dt. The Δα deadband is

imposed from the altitude constrain. Plot time span is covers the whole mission.

4.5.2.3 Test Case 1: Primary Correction is Phasing Δlo at the Equator

Figure 4.5: Plots of evolutions of Δlo and – Δl deviations from nominal orbit. The correction cycle of Δlo is the smallest one

with period of 44.67 days while the correction of Δl is done every 249.47 days. The superimposed periodic parabolic

evolution on the linear decay of Δl is evident in the rightmost plot.

4.5.2.4 Test Case 1: Primary Correction is Phasing Δlo at the Equator

Figure 4.6: Plots of evolutions of ΔH and – Δh deviations from nominal orbit. The correction cycle of ΔH is equal to the

one of Δl while the correction cycle of Δh is the same with Δlo.

The limits of ΔH has been readjusted from the strategy in order to be conformal with the desired

maneuver cycle. The new limits are always smaller than the desired ones given in the input section.

4.5.3.2 Test Case 1: Primary Correction is Altitude Deviation Δh

The state in this case corresponds to local time at ascending node LTA = 3 h so di/dt > 0. The

smallest maneuver cycle corresponds to the maintenance of altitude deviation Δh so all the other

maneuver cycles are condition in accordance to the one of Δh. The first maneuver of the strategy is

a Δa correction which controls the evolution of Δh while the second one, with a cycle equal to that

of Δlo, is a Δi correction controlling Δlo , ΔΗ and Δl.

Figure 4.7: Diagram of Δa – Δi deviations from nominal orbit. The vertical lines correspond to Δa corrections while the

lines almost parallel with the Δi axis, are the inclination corrections. The natural evolution, since di/dt > 0, is done from the

left to the right part of the diagram. The Δα deadband is imposed from the altitude constrain. The plots time span is 300

days.

4.5.3.2 Test Case 2: Primary Correction is Altitude Deviation Δh

Figure 4.8: Evolutions plots for Δlo and – Δl deviations from nominal orbit. The correction cycle of Δlo and Δl are the same

and equal to 297.03 days. The small perturbations from the parabolic evolution is due to Δa corrections for Δh

maintenance.

4.5.3.3 Test Case 2: Primary Correction is Altitude Deviation Δh

Figure 4.9: Evolutions plots for deviations ΔH and – Δh from nominal orbit. The correction cycle of ΔH is equal to the

one of Δl and Δlo while the correction cycle of Δh is the smallest one with period of 8.078 days.

The interesting feature at the rightmost plot is the harmonic evolution of Δh due to the rotation of

eccentricity around the frozen eccentricity.

4.5.3.4 Test Case 2: Primary Correction is Altitude Deviation Δh

)cos(sec

tdt

dLRaah

ular

emean

4.5.4.1 Test Case 3: Primary Correction is LTA Deviation ΔH

The state in this case corresponds to local time at ascending node LTA = 3 h so di/dt > 0. The

smallest maneuver cycle corresponds to the maintenance of LTA ΔH so all the other maneuver

cycles are condition in accordance to the one of ΔH. The first maneuver of the strategy is a Δa

correction which controls the evolution of ΔH while the second one, with a cycle equal to the one

of Δlo and Δl, is a Δi correction controlling Δlo and Δl. This maneuver is accompanied by a Δa

correction which compensates the disturbance of ΔH from the Δi correction

Figure 4.9: Diagram of Δa – Δi deviations from nominal orbit. The vertical lines correspond to Δa corrections while the

lines almost parallel with the Δi axis, are the inclination corrections. The natural evolution, since di/dt > 0, is done from the

left to the right part of the diagram. The control of Δa is done symmetrical with the slope of null dΔΗ/dt. The Δα deadband is

imposed from the altitude constrain. The plots time span is 200 days.

4.5.4.2 Test Case 3: Primary Correction is LTA Deviation ΔH

Figure 4.10: Evolutions plots for Δlo and – Δl deviations from nominal orbit. The correction cycle of Δlo and Δl are the

same and equal to 35.53 days.

4.5.4.3 Test Case 3: Primary Correction is LTA Deviation ΔH

The operational limits of Δlo and Δl has been readjusted from the strategy in order to be conformal

with the desired maneuver cycle. The new limits are always smaller than the desired ones given in

the input section.

Figure 4.11: Evolutions plots for deviations ΔH and – Δh from nominal orbit. The correction cycle of ΔH is equal to the

one of Δh and is the smallest one with period of 14.28 days.

4.5.4.4 Test Case 3: Primary Correction is LTA Deviation ΔH

4.5.5.1 Test Case 4: Δlo and ΔΗ Have Similar Control Cycles

The state in this case corresponds to local time at ascending node LTA = 3h so di/dt > 0. The

maneuver cycle corresponds to the simultaneous maintenance of phase Δlo at the equator and local

time deviation ΔΗ. The only maneuver of the strategy is a Δa correction accompanied by a Δi

correction for the simultaneous control of Δlo and ΔΗ .

Figure 4.12: Diagram of Δa – Δi deviations from nominal orbit. Simultaneous control of Δa and Δi axis. The natural

evolution, since di/dt > 0, is done from the left to the right part of the diagram. The Δα deadband is imposed from the

altitude constrain. The plots time span cover the whole mission duration.

4.5.5.2 Test Case 4: Δlo and ΔΗ Have Similar Control Cycles

Figure 4.13: Evolutions plots for Δlo and – Δl deviations from nominal orbit. The control cycles of all operational

parameters are the same and equal to 44.69 days.

4.5.5.3 Test Case 4: Δlo and ΔΗ Have Similar Control Cycles

Figure 4.14: Evolutions plots for deviations ΔH and – Δh from nominal orbit. The control cycles of all operational

parameters are the same and equal to 44.69 days.

4.5.5.4 Test Case 4: Δlo and ΔΗ Have Similar Control Cycles

4.5.6.1 Test Case 5: Very Tight Control of Δlo and Δl with di/dt < 0

This is the same strategy with case one and the substantial differences are the very tight control of

100 m of Δlo and Δl and the state which corresponds to local time at ascending node LTA = 9 h so

di/dt < 0. The interesting characteristic of this case is that despite the control of Δlo and Δl being

tighter with respect to Case1, the mass consumption is almost the same. This happens because

despite the fact the there are much more maneuvers in Case 2, their magnitude is much smaller

since the Δlo and Δl deadbands are smaller.

Figure 4.15: Diagram of Δa – Δi deviations from nominal orbit. Simultaneous control of Δa and Δi axis. The natural

evolution, since di/dt < 0, is done from the right to the left part of the diagram. The Δα deadband is imposed from the tight

Δlo constrain. The plots time span cover is 60 days.

4.5.6.2 Test Case 5: Very Tight Control of Δlo and Δl with di/dt < 0

4.5.6.3 Test Case 5: Very Tight Control of Δlo and Δl with di/dt < 0

Figure 4.16: Plots of evolutions of Δlo and – Δl deviations from nominal orbit. The correction cycle of Δlo is equal to that

of Δlo with a period of 6.09 days.

Figure 4.17: Evolutions plots for deviations ΔH and – Δh from nominal orbit. The control cycle of ΔΗ is 5.07 days while

Δh is controlled simultaneously with Δl0 and Δl with a control cycle of 6.9 days.

4.5.6.4 Test Case 5: Very Tight Control of Δlo and Δl with di/dt < 0

The state in this case corresponds to local time at ascending node LTA = 6h so di/dt ≈ 0. The

maneuver cycle corresponds to the simultaneous maintenance of phase Δlo at the equator and local

time deviation ΔΗ. The only maneuver of the strategy is a Δa correction accompanied by a Δi

correction for the simultaneous control of Δlo and ΔΗ .

4.5.7.1 Test Case 6: Primary Correction is Phasing Δlo at the Equator With di/dt ≈ 𝟎

Figure 4.18: Diagram of Δa – Δi deviations from nominal orbit. Simultaneous control of Δa and Δi axis. Since di/dt ≈ 0, no

inclination maneuvers are needed for Δl maintenance. The Δα deadband is imposed from the tight Δlo constrain. The plots

time span cover the whole mission duration.

4.5.7.2 Test Case 6: Primary Correction is Phasing Δlo at the Equator With di/dt ≈ 𝟎

Figure 4.19: Evolutions plots for Δlo and – Δl deviations from nominal orbit. The control cycles of all operational

parameters are the same and equal to 50.2 days.

4.5.7.3 Test Case 6: Primary Correction is Phasing Δlo at the Equator With di/dt ≈ 𝟎

Figure 4.20: Evolutions plots for deviations ΔH and – Δh from nominal orbit. No control is needed for ΔΗ while Δh is

controlled simultaneously with Δl0 and Δl with a control cycle of 50.2 days.

4.5.7.4 Test Case 6: Primary Correction is Phasing Δlo at the Equator With di/dt ≈ 𝟎

Appendix I

Frozen Eccentricity Calculation

I.1 Frozen Orbits Concept and their Significance in LEO Spacecraft Control

Frozen orbits are specialized orbits that try to fix one or more orbital elements in the presence of

perturbations [1]. Many Earth imaging or reconnaissance and surveillance missions, necessitates the

minimization of the altitude variation by fixing the eccentricity and the argument of the perigee. The

dependence of the altitude hell from eccentricity e and argument of perigee ω is obvious from the conical

section representation of the Keplerian orbit:

sitesiteellp rue

earrh

cos1

1 2

where rsite is the Earth’s radius at the sub-satellite point and u is the Position in Orbit or Argument of

Latitude v + ω.

The zonal harmonics of the geopotential cause long and short periodic variations in the eccentricity. The

passive control of the eccentricity, argument of perigee and thus altitude, consists in choosing the

appropriate values for e and ω so that the long periodic motion of these two Keplerian elements, is

reduced.

(I.1)

Conditions for Frozen Orbit

Nulling the long-periodic variations of eccentricity and argument of

the perigee:

0dt

ed

0

0

dt

ddt

de

(I.2)

I.2.1 Long-periodic Evolution of Eccentricity Vector

In the case of frozen eccentricity what we are interested in is the evolution of its mean value emean which is

the outcome of a single averaging of the osculating eccentricity for a period of one repeat cycle. The

conditions (I.2) for the frozen eccentricity, impose that the solution of the eccentricity evolution close to

the equilibrium value of the frozen eccentricity ef, should be closed curves as seen from the following

diagram:

ex

ey

fe

oR

e

1e

2e

Where ef is the frozen eccentricity, e1

the mean eccentricity at time to, e2 the

mean eccentricity at time to+T.

Movement around the circle will be

CW if the inclination is within the

range [ic, π- ic], where ic = 63.4o is the

critical inclination, otherwise it will be

CCW.

Figure I.2: Evolution of mean eccentricity close to the equilibrium point of frozen eccentricity

)sin(

)cos(

ee

ee

y

x

(I.3)

I.2.2.1 Calculation of Frozen Eccentricity from Two Values of the Mean Eccentricity

In order to implement an iterative process for the calculation of the frozen eccentricity, there must be a

closed form expression of the frozen eccentricity calculation from two values of the mean eccentricity.

What follows is the derivation of this formula.

In accordance to Figure I.2:

)sin(

)cos(

,

,

ktRee

ktRee

oyfy

oxfx(I.4)

where ef,x and ef,y are the components of the frozen eccentricity, Ro is the radius of the circle around which

the mean eccentricity evolves, k is the angular frequency of this rotation and φ the initial phase. If we take

the time derivative of both equation in (I.4), square them and sum them, we get the value of k to be:

ooo

yx

R

e

R

ee

R

eek

22222

(I.5)

Let the values of mean eccentricities e1 and e2 which are calculated with a time difference T of more than

one repeat cycle. Then in accordance to (I.4):

)sin()(sin)()(

)cos()(cos)()(

,1,2

,1,2

oooooyoyyyy

oooooxoxxxx

ktRtkRteteeee

ktRtkRteteeee

(I.6)

I.2.2.2 Calculation of Frozen Eccentricity from Two Values of the Mean Eccentricity

If we use the trigonometrical identities which give the difference of cos(x) and sin(x) functions for two

different arguments x, we get:

now we expand the trigonometric functions whose arguments is the sum of terms kto+T/2+φ:

Which after the multiplication of terms becomes:

)2

cos(2

cos2

)2

sin(2

sin2

kTTktRe

kTTktRe

ooy

oox

(I.7)

)2

sin()2

sin()sin()2

cos()cos(

)2

sin()2

sin()cos()2

cos()sin(

kTkTkt

kTktRe

kTkTkt

kTktRe

oooy

ooox

)2

(sin)sin(2)sin()cos(

)2

(sin)cos(2)sin()sin(

2

2

kTktkTktRe

kTktkTktRe

oooy

ooox

(I.8)

(I.9)

I.2.2.3 Calculation of Frozen Eccentricity from Two Values of the Mean Eccentricity

The two equations of (I.9) constitute a linear system of equations with respect to the unknowns:

From (I.9) and (I.10) and we get the linear system:

)sin(

)cos(

o

o

ktB

ktA

)2

(sin2)sin(

)sin()2

(sin2

2

2

o

y

o

x

R

eB

kTAkT

R

eBkTA

kT

(I.11)

(I.12)

(I.10)

Whose solution is:

cos(kT)-1

)sin(

2

1)sin(

cos(kT)-1

)sin(

2

1)cos(

kTee

RktB

kTee

RktA

xy

o

o

y

x

o

o

Now we use the trigonometric identity:

cos(x)-1

)sin()

2cot(

xx

I.2.2.4 Calculation of Frozen Eccentricity from Two Values of the Mean Eccentricity

So we have that:

(I.14)

(I.13)

)2

cot(2

1)sin(

)2

cot(2

1)cos(

kTee

RktB

kTee

RktA

xy

o

o

yx

o

o

Now we can compute the difference between the initial mean eccentricity and the frozen eccentricity from

(I.4) and (I.13):

2

)2

cot(

)sin(

2

)2

cot(

)cos(

,

,

kTee

ktReee

kTee

ktReee

xy

oyfyy

yx

oxfxx

And from (5):

2

)2

cot(

2

)2

cot(

o

xy

y

o

yx

x

R

Teee

e

R

Teee

e

(I.15)

I.2.2.5 Calculation of Frozen Eccentricity from Two Values of the Mean Eccentricity

From [1] to the order of J3 we get for the derivative of eccentricity modulus close to the frozen

eccentricity:

(I.16) cos)(sin

4

5-1

)1(2

)sin(32

223

3

33

i

ea

iRJa

eE

Frozen eccentricity values are very small (e = 0.00115 for ω = 90o) so we can approximate (I.16) with:

(I.17) Acoscos)(sin

4

5-1

2

)sin(32

3

3

33

i

a

iRJa

eE

So from (I.3), (I.5) and (I.17) we have since ex = e cos(ω) :

)(sin4

5-1

2

)sin(3)cos( 2

3

3

33

i

ea

iRJa

e

A

eR

Ae

R

Ak

E

o

x

o

(I.18)

Final result described from (I.14) and (I.18) can be verified from [4] pp.101.

Appendix II

Impact of Perturbations on LEO Orbits and

Mission Operational Parameters

II.1 Perturbations and Evolution of Orbital Elements

This chapter presents the effects of perturbing forces on LEO orbits. These effects will be described for

the following perturbations:

• Earth gravity (non spherical geopotential)

• Luni-Solar gravity

• Atmospheric drag

• Solar radiation pressure

In general the variation of each orbital element p, under the influence of perturbing forces, can take the

form ([2] pp.798):

2sinAt-tBm

1j

j

i

o

1

i

j

j

n

i

oT

tptp (II.1)

This three terms of the equation above, are identified as follows:

• po represents the mean parameter associated with the parameter p, and theoretically it equal to the

nominal orbit parameter.

• The second term represents the secular evolution of the orbital parameter.

• The third terms represents the periodic evolution of the orbital parameter.

The spectral classification of each perturbation is given in accordance to the value of its fundamental

period T:

period long : TT

period medium : T

periodshort :

Earth

orbital

Earth

orbital

TT

TT

(II.2)

Appendix II

Chapter 2

Effects of Non-Spherical Geopotential

II.2 Perturbations Due to Non-spherical Geopotential

The non-spherical geopotential produces perturbations which cover all spectral types of short, medium

and long periodic as well as secular ones.

The following table presents the effect of the geopotential on the various orbital elements along with their

accompanying spectrum i.e. the different harmonics of the fundamental period if the evolution is periodic

and the first derivative with respect to time if the evolution is secular [2].

Spectrum/Orbital element Major Semi Axis Eccentricity Vector Inclination Right Ascension of

Ascending Node

Short Period Source: Zonal Terms Jn

T = Torbital/n n even ≥ 2

J2 – predominant

T = Torbital/n n = 1 and 3

Same as Major Semi Axis

Same as Major Semi Axis

Medium Period Source : Tesseral Terms Jn,m

T=2π/m(dΩ/dt - ωsidereal) m ≥ 1

T = Tearth/m for Sun-Synchronous

Same as Major Semi Axis

- Same as Major Semi

Axis

Long Period Source: Odd Zonal Terms

- T=2π/(dω/dt)secular - -

Secular Source: Even Zonal Terms

- (dω/dt)secular - (dΩ/dt)secular

Table II.1: Non-spherical geopotential effects on orbital elements and their spectrum.

II.2.1 Short-period (1 orbital revolution) Components

The short-period evolution of the orbital parameters is governed principally from the zonal term J2 = -C2,0

and in accordance to [2] pp.800, may be formulated as follows:

ninclinatiomean thei and axis semimajor mean thea radius,Earth theis a and

1sin432

31

2sin2

1sin2

2

3

2sincos4

3

2cos2sin8

3

3sinsin12

7sinsin

4

71

2

3

3cossin12

7cossin

4

51

2

3

2cossin2

3

e

1

2

2

2

2

2

2

2

2

2

2

22

2

2

22

2

2

22

2

vu

ia

aJ

where

uia

aJu

uia

aJ

uia

aJi

uiuia

aJe

uiuia

aJe

uia

aJa

e

e

e

e

ey

ex

e

As seen from these formulas, to the order of J2, the short-periodic variation depends on the argument of

latitude u which means that it depends on latitude.

(II.3)

Scenario Characteristics

The scenario to be used for the demonstration of the effects from

the non-spherical geopotential on orbital elements assumes that

there in no air drag and the nominal orbit is to be phased, Sun-

synchronous and frozen (minimum altitude variance).

Nominal Orbit Characteristics

Revolutions/Day = 14.5

Cycle Duration = 16 days

Revolutions per Cycle = 233

Sun synchronous

Local Solar time at Node Crossings = 6 h

Minimum Altitude Variance (Frozen Eccentricity)

Perturbations that Where Taken Account

Geopotential (9th degree and order)

Luni-Solar gravity

Solar radiation pressure

For very long period propagations, the 8th order adaptive step

size Runge-Kutta-Dormant-Prince integrator was used.

II.2.1.1 Major Semi Axis Short-period (1 orbital revolution) Component

Figure II.1: Short-periodic component of major semi axis. First equation from (II.3) gives for the amplitude the value of

δαmax = 9.155 Km. Mean major semi axis a= 7077.469 Km and mean inclination i = 98.202o.

-50 0 50 100 150 200 250 300 350 400

7065

7070

7075

7080

7085

7090

Oscula

ting M

ajo

r S

em

i A

xis

(K

m)

Argument of Latitude (deg)

Osculating Major Semi Axis

Mean Major Semi Axis

9,1 Km

II.2.1.2 Eccentricity Short-period (1 orbital revolution) Components

Figure II.2: Short-periodic component of eccentricity. Second and third equations from (II.3) give for the amplitudes the

values: δex = 4.5e-4 for u = 0ο and δey = 1.7e-3 for u = 3π/2ο. Mean major semi axis a= 7077.469 Km and mean

inclination i = 98.202o.

-0,0010 -0,0005 0,0000 0,0005 0,0010

-0,0010

-0,0005

0,0000

0,0005

0,0010

0,0015

0,0020

0,0025

0,0030

0.0017

0.00041(u=)

(u=)e*s

in(A

rgum

ent of P

erigee)

e*cos(Argument of Perigee)

e*sin(Argument of Perigee)

(u=)

(u=3)

emean

II.2.1.3 Inclination Short-period (1 orbital revolution) Component

Figure II.3: Short-periodic component of inclination. Fourth equation from (II.3) gives for the amplitude the value of

δi = 5e-3 deg. Mean major semi axis a= 7077.469 Km and mean inclination i = 98.202o.

0 50 100 150 200 250 300 350 400

98,196

98,198

98,200

98,202

98,204

98,206

98,208

98,210

Modulation from medium-periodic components

Oscu

latin

g I

nclin

atio

n (

de

g)

Argument of Latitude (deg)

Osculating Inclination

5e-3 deg

II.2.1.4 Right Ascension of the Ascending Node Short-period (1 orbital revolution) Component

Figure II.4: Short-periodic plus secular evolution of right ascension of the ascending node. Fifth equation from (II.3)

gives for the amplitude the value of δΩ = 5.4e-4 deg. Mean major semi axis a= 7077.469 Km and

mean inclination i = 98.202o.

0,99 1,00 1,01 1,02 1,03 1,04 1,05 1,06 1,07 1,08

190,736

190,743

190,750

190,757

190,764

190,771

190,778

190,785

190,792

190,799

190,806

190,813

190,820

Ascendin

g N

ode (

deg)

DOY (ddd.ddd)

Ascending Node

Secular Component of Ascending Node

II.2.2.1 Inclination Medium Period (1 day) Component

Figure II.5: Inclination medium period component of Tearth/2 = 0.5 days period, superimposed on the short-periodic

component with period of one orbital revolution.

1,0 1,5 2,0 2,5 3,0 3,5 4,0

98,196

98,200

98,204

98,208

98,212

Inclin

atio

n (

de

g)

DOY (ddd.ddd)

Osculating Inclination

Mean Inclination

Upper Envelope of "Inclination"

Lower Envelope of "Inclination"

II.2.2.2.1 Impact of Short and Medium Period Components on Repeat Ground Track Regularity

Figure II.6: Dependence of phase difference from the longitude and the latitude of the node crossing due to the tesseral

and zonal harmonics respectively.

The most prominent effect of short and medium-periodic components on LEO orbits, is the deformation

of the repeat ground track. Due to the tesseral harmonics, which depend on the longitude, the phase

difference between two consecutive in space nodes for the same parallel, is varying with a period of one

day. The zonal harmonics are responsible for the dependence of the phase difference from the latitude.

The phase difference is of the order of magnitude of 100 m at the Equator while it can reach 1 Km at

higher latitudes.

51544 51545 51546

-1,2

-1,0

-0,8

-0,6

-0,4

-0,2

0,0

0,2

0,4

0,6

0,8

1,0

Dis

tance fro

m M

ean G

rid N

ode (

Km

)

MJD (ddddd.ddd)

Distance from Mean Grid Node for Nodes on the Equator

Distance from Mean Grid Node for Nodes with Latitude 70o

Distance from Mean Grid Node for Nodes with Latitude 45o

II.2.2.2.2 Impact of Short and Medium Period Components on Repeat Ground Track Regularity

Figure II.7: Dependence of distance between consecutive in time nodes from the longitude and the latitude of the node

crossing due to the tesseral and zonal harmonics respectively.

The deviation of the true phasing grid from the mean grid due to tesseral and zonal harmonics, is more

evident in the following diagram which depicts dependence of the distance between consecutive in time

nodes, from the longitude and the latitude.

51544 51545 51546

-2752,6

-2752,5

-2752,4

-2752,3

-2752,2

-2752,1

-2752,0

-2751,9

-2751,8

-2751,7

-2751,6

-2751,5

-2751,4

-2751,3

-2751,2

-2751,1

-2751,0

Variation due to tesseral harmonics

Dis

tance B

etw

een C

onsecutive in T

ime N

odes (

Km

)

MJD (ddddd.ddd)

Distance Between Consecutive in Time Nodes on the Equator

Distance Between Consecutive in Time Nodes on 70o

Latitude

Difference due to zonal harmonics

II.2.2.2.3 Impact of Short and Medium Period Components on Repeat Ground Track Regularity

Figure II.8: Dependence of distance between consecutive in time nodes from the longitude and the latitude of the node

crossing due to the tesseral and zonal harmonics respectively.

-50 0 50 100 150 200 250 300 350 400

-2752,6

-2752,4

-2752,2

-2752,0

-2751,8

-2751,6

-2751,4

-2751,2

-2751,0

Difference due to zonal harmonics

Dis

tan

ce f

rom

Pre

vio

us N

od

e (

Km

)

Sub-Satellite Longitude (deg)

Distance Between Consecutive in Time Nodes on the Equator

Distance Between Consecutive in Time Nodes on 70o Latitude

Variation due to tesseral harmonics

II.2.3.1 Long-period Evolution of Mean Eccentricity Due to Non-spherical Geopotential

Figure II.9: Manifestation of long-periodic effects through the 115.9 days evolution of mean eccentricity close to the

equilibrium point of frozen eccentricity with ω = π/2.

The long-period effect of non-spherical geopotential is on the mean eccentricity vector which in

accordance to [2] pp.805, to the first order of J2 has a period of:

1)(cos5J 3

4 1-2

2

2

i

a

aTT

e

orbital (II.4)

For the case of the scenario with mean major semi axis a = 7077.469 Km and mean inclination

i = 98.202o , (II.4) gives T = 115.785 days.

-0,0010 -0,0005 0,0000 0,0005 0,0010

0,0000

0,0002

0,0004

0,0006

0,0008

0,0010

0,0012

esin

(Arg

um

en

t o

f P

erig

ee

)

ecos(Argument of Perigee)

esin(Argument of Perigee)

115.9 days evolution

around frozen eccentricity

-0,00003 -0,00002 -0,00001 0,00000 0,00001 0,00002

0,00118

0,00120

0,00122

0,00124

esin

(Arg

um

ent of P

erigee)

ecos(Argument of Perigee)

esin(Argument of Perigee)Detail

II.2.3.2 Long Period Evolution of Altitude Deviation Δh Due to Eccentricity Evolution

Figure II.10: The long periodic evolution of altitude for various position in orbit (argument of latitude).The variation of

latitude, for a specific position in orbit, exhibits periodic behavior with period of approximately 116 as expected by II.4 and II.5

50

100

150

200

250

300

350

728

730

732

2 5 0

2 5 5

2 6 0

2 6 5

2 7 0

2 7 5

2 8 0

2 8 5

2 9 0

Alt

itu

de

(k

m)

Argu

ment

of La

titude

(deg

)

D O Y ( d d d .d d d )

50

100

150

200

250

300

350

700702704706708710712714716718720722724726728730732

50

1 0 0

1 5 0

2 0 0

2 5 0

3 0 0

3 5 0

Altitu

de

(km

)

Argum

ent of

Latitud

e (deg)

DOY (ddd.ddd)

Detail

In accordance to [2], the simplified expression for altitude deviation Δh is given as follows :

latitude ofargument theis vω and axis semimajor theis a

)sin()cos(

where

evevaah yx (II.5)

In the absence of air drag the altitude deviation, for a specific argument of latitude, should variate in

accordance to the eccentricity and thus have a periodic evolution with period given by II.4.

II.2.4.1 Secular Components Due to Non-spherical Geopotential

The only orbital elements affected by secular variation from the geopotential, are the angular ones i.e. Ω,

ω and M. The secular evolution of Ω is the well known nodal regression, the secular evolution of ω is

the apsidal rotation and the secular component of M changes the anomalistic period with respect to the

non-perturbed one. To the first order of J2, in accordance to [2] pp.804, the time derivatives quantifying

the secular effects on the angular orbital elements, are the following:

period canomalistiKeplerian theis a

n where

1

cos(i)

2

3

1

1)(cos5

4

3

1

1)(cos3

4

3

3

mean

22

2

2

22

22

2

23

2

22

2

ea

aJ

n

e

i

a

aJ

n

e

i

a

aJ

n

nM

e

e

e

(II.6)

For the case of the scenario with amean = 7077.469 Km (Tanomalistic = 1.646 h) and imean = 98.202o, (II.4)

gives:

Tperturbed = 1.647 h Relative increase of anomalistic period = 0.062 %

dω/dt = - 3.092 deg/day Corresponds to a period of 115.785 days (see formula II.4)

dΩ/dt = 0.9876 deg/day Sun-synchronous orbit

II.2.4.2 Secular Evolution of Right Ascension of the Ascending Node

-50 0 50 100 150 200 250 300 350 400

-50

0

50

100

150

200

250

300

350

400

RA

AN

(de

g)

DOY (ddd.ddd)

RAAN

Sun's Right Ascension

90o

= 6h

Figure II.10: Secular evolution of the right ascension of the ascending node (RAAN) and right ascension of the Sun for a

period of 1 year. RAAN has 6h local solar time.

II.2.4.3 Secular Evolution of Argument of the Perigee

Figure II.11: The secular evolution of ω is evident from the CW rotation of mean eccentricity vector around the

equilibrium point of frozen eccentricity. Period of propagation is 1 year. Solar radiation pressure was not taken account.

-0,00003 -0,00002 -0,00001 0,00000 0,00001 0,00002

0,00118

0,00120

0,00122

0,00124

esin

(Arg

um

ent of P

erigee)

ecos(Argument of Perigee)

esin(Argument of Perigee)

Appendix II

Chapter 3

Effects of Luni-Solar Gravitation

II.3.1 Perturbations Due to Luni-Solar Gravitation

The Luni-Solar gravitation produces long period components for I, Ω, ω and M but in the special case of

Sun-synchronous orbits it is responsible for the secular evolution of phase difference (Δlo, ΔlL ) and local

solar time (ΔH) of ascending node. In the latter case the secular evolution of the aforementioned

operational parameters is imposed from the secular evolution of inclination whose time derivative, to the

first order, is give as follows ([2] pp.807):

Spectrum/Orbital element Major Semi Axis Eccentricity

Vector Inclination

Right Ascension of Ascending Node

Short Period - - -

-

Medium Period - - - -

Long Period - - T = Tmoon/n (n=2)

T = Tsun/m (m=1 and 2) Same as Inclination

Secular - - If Sun-synchronous Same as Inclination

Table II.2: Effects of Luni-Solar gravity on orbital elements and their spectrum.

72 23i andyear 1 T where

2

1

T

H4sin

2

isin(i)cos

2

3

o

eclipticSun

Earth

ecliptic4

2

Sun

orbital

T

T

dt

di(II.7)

The sign and magnitude of this time derivative, depend on the local solar time H of the ascending node.

II.3.2 Long Period Component of Inclination Due to Luni-Solar Gravitation

1 year period

180 190 200 210 220 230 240

98,2080

98,2083

Inclin

atio

n (

de

g)

DOY (ddd.ddd)

Inclination

~ 13.5 days

Detail

Figure II.12: Demonstration of long period component of inclination with period of 1 year = TSun and sub-period

of 13.5 days = Tmoon/2

II.3.3.1 Secular Effects on Phase Difference Δlo and Local Time Difference ΔΗ

The secular effects on phase difference at the equator Δlo and the local solar time (ΔH) of ascending node,

is described by their second time derivatives [2]:

ia

aJiaand

ia

aJ

a

a

dtl

dtl

dt

ld

alnoe

alnoe

e

alno

alnoalno

e

o

2sin6)tan(l

)(cos42

7

3

71

2

3l where

dida

minsidereal

2

2min2

2

2

2

minsidereal

minminsidereal1

21

dt

dih

dt

dah

dt

da

dt

dii

T

T

dt

Hd

Sun

Earth21

2

7)tan(

(II.9)

(II.10)

If the secular drifts of a and i are taken as constant, then the double integration of the formulas above

results in a parabolic time variation. The Luni-Solar perturbation is imposed by di/dt given from (II.7)

and the impact of the atmospheric drag is given by da/dt.

Under the consideration of no air drag, the parabolic evolution of these two operational parameters

(especially its direction) strongly depends on the nominal local solar time of the ascending node H

through the time derivative of the inclination and its dependence from H (last term of II.7).

II.3.3.2 Secular Effects on Phase Difference Δlo

51540 51560 51580 51600 51620 51640 51660 51680

-15

-10

-5

0

5

10

Dis

tan

ce f

rom

Mea

n G

rid

No

de

(K

m)

MJD (ddddd.ddd)

Distance from Mean Grid Node for LTA = 6 h

Distance from Mean Grid Node for LTA = 3 h

Distance from Mean Grid Node for LTA = 9 h

Figure II.13: Secular evolution of phase difference on the equator. For LTA = 6h the time derivative di/dt is almost zero,

for LTA = 3h it is positive and for LTA = 9h it is negative. Since for the case of a Sun-synchronous orbit with

imean = 98.202o, the term K2 is positive, the sign of the phase difference second derivative with respect to time,

will be the same with the sign of di/dt.

II.3.3.3 Secular Effects on Local Time Difference ΔΗ

51540 51560 51580 51600 51620 51640 51660 51680

0,046

0,047

0,048

0,049

0,050

0,051

0,052

0,053

0,054

0,055

0,056

0,057

0,058

h

MJD (ddddd.ddd)

for for LTA = 6 h

for for LTA = 3 h

for for LTA = 9 h

Figure II.14: Secular evolution of phase difference on the equator. For LTA = 6h the time derivative di/dt is almost zero,

for LTA = 3h it is positive and for LTA = 9h it is negative. Since for the case of a Sun-synchronous orbit with

imean = 98.202o, the multiplicative term of di/dt is positive, the sign of the phase difference second derivative

with respect to time,will be the same with the sign of di/dt.

II.3.3.4 Impact of Equation of Time and Lunar Gravity on Phase Difference Δlo

Figure II.15: The figure at the left sight depicts the 1-year and 6-month periodic components of phase difference at the

equator, as well as its dependence on the actual position of the Sun (Equation of time), for LTA = 6h.

Equation (II.7) is approximate because it doesn’t take account the lunar variation and the equation of

time. If these two are taken account then di/dt is expected to have also a 1-year and 6-month periodic

component which will manifest themselves in the evolution of phase difference for LTA = 6h.

51500 51550 51600 51650 51700 51750 51800 51850 51900 51950

-0,5

0,0

0,5

1,0

1,5

2,0

Dis

tance fro

m M

ean G

rid N

ode (

Km

)

MJD (ddddd.ddd)

Phase Difference for Equatorial Nodes and LTA = 6h

51690 51695 51700 51705 51710 51715 51720

1,0

Dis

tan

ce f

rom

Mea

n G

rid

No

de

(K

m)

MJD (ddddd.ddd)

Phase Difference for Equatorial Nodes and LTA = 6h

14 Days

Detail

II.3.3.5 Osculating States Used for Figures II.13 and II.14

LTA = 6h LTA = 3h

LTA = 9h

Appendix II

Chapter 4

Effects of Solar Radiation Pressure

II.4.1 Perturbations Due to Solar Radiation Pressure

The orbital parameters a and Ω are not affected from solar radiation pressure but the rest of the orbital

elements have a long periodic behavior which for near polar orbits has an angular frequency of

(dΩ/dt – ωsun) [2] pp.812. In Sun-synchronous orbits this periodic variation becomes secular.

Also in the case of solar radiation pressure the actual position of the frozen eccentricity point is slightly

different from ±π/2 which is theoretically calculated when the only perturbation is the spherical

geopotential.

Table II.3: Effects of solar radiation pressure on orbital elements and their spectrum.

Spectrum/Orbital element Major Semi

Axis Eccentricity Vector Inclination

Right Ascension of Ascending Node

Short Period Source: Zonal Terms Jn

- - -

-

Medium Period - - - -

Long Period - Very long period

T=2π/(dΩ/dt - ωsun) Same as for eccentricity -

Secular - For Sun-Synchronous

orbits Same as for eccentricity -

2frozen

The sign of ε depend on the relative values of the atmospheric drag and solar radiation pressure while the

value of this parameter depends essentially on the solar radiation pressure [2] pp.814.

II.4.2 Effects of Solar Pressure on Eccentricity and Inclination

The aforementioned secular variation for Sun-synchronous is given by the following drifts:

ratio mass toarea effective theis A/m

and AU 1 of distanceat pressuresolar theis Nm 65.4

)cos(

1

2

λcos

orbit with theofpart angular dilluminate theis λ and 2

12π q

)cos(sgn)cos(2

sin

2

)cos(sgn)sin(2

sin

)cos(4

sin3

2

1

2

e

e

1

1

1

eP

q

aa

T

Hwhere

qqa

m

AP

dt

vd

qqa

m

AP

dt

di

qa

m

AP

dt

de

AU

meane

Earth

AN

e

meanAU

e

meanAU

eemeanAU

y

(II.11)

II.4.3 Effects of Solar Pressure on Eccentricity and Inclination

For the case of the scenario with no air drag, effective area to mass ratio A/m = 0.01527, amean =

7077.469 Km and LTA = 6h the first two equations from II.10, give:

deydt = 0.000115 /day

di/dt = 1.8e-21 deg/day

-0,00002 -0,00001 0,00000 0,00001 0,00002

0,00118

0,00119

0,00120

0,00121

0,00122

0,00123

End

7

5

6

4

3

2esin

(Arg

um

en

t o

f P

erig

ee

)

ecos(Argument of Perigee)

esin(Argument of Perigee)

1

Start

Figure II.16: One year evolution of mean eccentricity when solar radiation pressure is taken account along with the rest

of the perturbations except the air drag. Compare with Figure II.11 depicting the evolution of mean eccentricity in the

absence of solar pressure.

Appendix II

Chapter 5

Effects of Atmospheric Drag

The atmospheric model used for the computation of the air drag is

Jacchia’s 1971 density model.

F10.7 and Kp are considered constant during the whole period of propagation

II.5.1 Perturbations Due to Atmospheric Drag

The effects of atmospheric drag on orbital elements are the following:

Decrease of the major semi axis due to decrease of the total energy of the orbit.

Progressive circularization of the orbit, i.e. decrease of eccentricity, because air drag is more

prominent at the perigee which has the lowest altitude and thus resides in denser parts of the

atmosphere.

The time derivatives of major semi axis and eccentricity are given as follows [2] pp.811:

Table II.4: Effects of air drag on orbital elements and their spectrum.

Spectrum/Orbital element Major Semi Axis Eccentricity Vector Inclination Right Ascension of

Ascending Node

Short Period Source: Zonal Terms Jn

Short period T=Torbital (solar bulge)

Same as major semi axis

-

-

Medium Period - - - -

Long Period T =27 days (Sun rotation) - - -

Secular Principal term related to

mean atmospheric density Same as major semi

axis - -

spacecraft theofr unit vectoposition ther̂ and

tcoefficien pressuresolar theC with ratio mas toarea effective theis

ˆ

pm

ACwhere

rm

AC

adt

ed

am

AC

dt

da

P

P

P

(II.12)

Scenario Characteristics

The scenario to be used for the demonstration of the effects from

atmospheric drag orbital elements assumes that there in no solar

radiation pressure and the nominal orbit is to be phased, Sun-

synchronous and frozen (minimum altitude variance).

Nominal Orbit Characteristics

Revolutions/Day = 14.5

Cycle Duration = 16 days

Revolutions per Cycle = 233

Sun synchronous

Local Solar time at Node Crossings = 6 h

Minimum Altitude Variance (Frozen Eccentricity)

Perturbations that Where Taken Account

Geopotential (9th degree and order)

Luni-Solar gravity

Atmospheric drag with the following characteristics:

• Spacecraft Ballistic Coefficient BC = 0.02 m2/Kg

• F10.7 = 200 SFU and F10.7ave = 155 SFU

• Kp = 4 (Quiet Sun)

II.5.2 Atmospheric Density and Decay Rate for BC = 0.02 m2/Kg (SPOT)

Figure II.17: Decay of mean major semi axis due to air drag, for a period of one year.

Maximum decay rate (da/dt)max = -13.14 m/day ρ = 1.43x10-13 Kg/m3 21/03/2000

Minimum decay rate (da/dt)min = -6.42 m/day ρ = 7x10-14 Kg/m3 21/06/2000

-50 0 50 100 150 200 250 300 350 400

7074,0

7074,5

7075,0

7075,5

7076,0

7076,5

7077,0

7077,5

7078,0

7078,5

Mea

n M

ajo

r S

em

i A

xis

(km

)

DOY (ddd.ddd)

Mean Major Semi Axis

II.5.3 Effect of Air Drag on Mean Eccentricity Vector

Figure II.18: Mean eccentricity evolution around the equilibrium point of frozen eccentricity, for a period of 1 year.

-0,00004 -0,00002 0,00000 0,00002

0,00116

0,00118

0,00120

0,00122

End

esin

(Arg

um

ent of P

erigee)

ecos(Argument of Perigee)

esin(Argument of Perigee)

Begin

50

100

150

200

250

300

350

728

730

732

5 0

1 0 0

1 5 0

2 0 0

2 5 0

3 0 0

3 5 0

Alt

itu

de

(k

m)

Argume

nt of La

titude (d

eg)DOY (ddd.ddd)

II.5.4.1 Effect of Air Drag on Altitude

Figure II.19: Altitude decay due to air drag, for a period of 1 year.

In accordance to II.5, the altitude for a specific argument of latitude, will retain its periodic evolution with

period given by II.4 but the most prominent effect will be a linear drift due to the decay of the mean

major semi axis Δa.

50

100

150

200

250

300

350

696698700702704706708710712714716718720722724726728730732

5 0

1 0 0

1 5 0

2 0 0

2 5 0

3 0 0

3 5 0

Alt

itu

de

(k

m)

Argume

nt of La

titude (d

eg)DOY (ddd.ddd)

Detail

II.5.4.2 Effect of Air Drag on Altitude

Figure II.20: Decay of mean altitude due to air drag, for a period of 1 year.

Decrease of mean altitude

01/01/2000

01/01/2001

II.5.5 Secular Effects of Air Drag on Phase Difference Δlo

Figure II.21: Comparison of the impact of air drag on phasing at the Equator and the corresponding impact of

Luni-Solar gravitation for a period of 5 months.

51540 51560 51580 51600 51620 51640 51660

0

200

400

600

Dis

tan

ce f

rom

Mea

n G

rid

No

de

(K

m)

MJD (ddddd.ddd)

Phase difference for Equatorial nodes due to air drag- LTA = 6h

Phase difference for Equatorial nodes without air drag- LTA = 6h

Phase difference for Equatorial nodes without air drag- LTA = 3h

In accordance to II.9, the secular evolution of phase difference on the equator will be parabolic. In the

following diagram there is a comparison between the impact of the Luni-Solar perturbation (Maximum

di/dt for LTA = 3h) on the equatorial phasing and the corresponding impact of the air drag. The curve

with di/dt = 00/day (LTA = 6h) is used as a null curve for reasons of comparison.

II.5.6.1 Secular Effects of Local Time Difference ΔH

Figure II.21: Comparison of the impact of air drag on the difference ΔΗ of the local solar time of the ascending node

and the corresponding impact of Luni-Solar gravitation for a period of 5 months.

51540 51560 51580 51600 51620 51640 51660 51680

0,0510

0,0515

0,0520

0,0525

0,0530

0,0535

0,0540

0,0545

0,0550

0,0555

0,0560

0,0565

0,0570

0,0575

0,0580

h

MJD (ddddd.ddd)

H for the case of air drag and LTA = 6h

H for the case of no air drag and LTA = 6h

H for the case of no air drag and LTA = 3h

The ground track shift λs is defined as the spacing between consecutive in time equator

crossing points. This figure is calculated from how far the Earth rotates during one nodal

period, relative to the ascending node located in inertial space [1]:

Psidereals

Where ωsidereal is the sidereal rate of Earth, dΩ/dt the nodal regression rate and PΩ the nodal

period.

The air drag affect the ground track shift through the change of nodal period PΩ which when

ignoring the higher order effects, becomes:

2

2 3J

naa

PP siderealsiderealsidereals

The accumulated effect on the ground track shift parameter after j revolutions is given in [1]

by the formula:

Pj sidereal

j

k

sk 1

2

1 2

1

From the last formula it is evident that the ground track shift grows quadratic ally with the

number of revolutions which has a severe impact on the repeat ground track.

(II.13)

(II.14)

(II.15)

II.5.7 Analysis of the Effect of the Air Drag on the Distance Between Consecutive in Time Nodes

II.5.6.2 Effect of Air Drag on Ground Track Shift λs

The quadratic effect of air drag on ground track shift, which is given by equation (II.14), it is evident in the

following diagram. The nodal period of the spacecraft is continuously decreasing since major semi axis is

decreasing due to air drag. The decrease of nodal period causes an unwanted eastward shifting of

consecutive ascending node crossings. The combination of the eastward shifting, due to the air drag, and

the natural westward movement of ascending node crossings results in a continuous quadratic decrease of

the ground shift

Figure II.22: Deterioration of ground track shift stability under the influence of air drag for the case of repeat ground track.

The propagation has a duration of two months and the ballistic coefficient used, for the case of air drag, is BC = 0.02 m2/Kg

The medium period fluctuation is due to tesseral harmonics while the secular one is due to air drag.

51540 51560 51580 51600 51620 51640 51660 51680

-2752,2

-2752,0

-2751,8

-2751,6

-2751,4

-2751,2

-2751,0

Dis

tance fro

m P

revio

us N

ode (

Km

)

MJD (ddddd.ddd)

Distance between consecutive in time equatorial nodes for the case of air drag

Distance between consecutive in time equatorial nodes for the case of no air drag

Second order polynomial fit for the case of air drag

Aside

Ground Track Plotting Capabilities of

OR.A.SI

All graphics presented in this Aside as well as all graphics of OR.A.SI, are produced from a

custom made class build for production of graphics and plots. This OR.A.SI class is based on

PNGwriter Open Source pngwriter C++ class (http://pngwriter.sourceforge.net/).

Orbit Characteristics

a = 7077.701 Km

i = 98.203o (Sun Synchronous)

Period = 98.88 min

Revolutions/Day = 14.5

LANDSAT

Aside 1.1 LANDSAT

Figure A.1: Ground track – Time span shown: 1 day - Elevation aspect angle = 33o

Aside 1.2 LANDSAT

Figure A.2: Ground track – Time span shown: 1 day - Elevation aspect angle = 0o

Aside 1.3 LANDSAT

Figure A.3: Ground track – Time span shown: 1 day - Elevation aspect angle = 90o (North Pole)

Aside 1.4 LANDSAT

Figure A.4: Beherman projection - Time span shown: 1 day

Orbit Characteristics

a = 6728.217 Km

i = 34.99o

Period = 91.31min

Revolutions/Day = 15.77

TTRM – Tropical Rainfall Measuring Mission

Aside 2.1 TRMM

Figure A.5: Ground track – Time span shown: 1 day - Elevation aspect angle = 15o

Aside 2.2 TRMM

Figure A.6: Ground track – Time span shown: 1 day - Elevation aspect angle = 90o (North Pole)

Aside 2.3 TRMM

Figure A.7: Beherman projection - Time span shown: 1 day

Orbit Characteristics

a = 25507.6 Km

i = 64.8o

Period = 675.73 min

Revolutions/Day = 2.13

GLONASS

Aside 3.1 GLONASS

Figure A.8: Ground track – Time span shown: 8 days - Elevation aspect angle = 5o

Aside 3.2 GLONASS

Figure A.9: Beherman projection - Time span shown: 8 days

Orbit Characteristics

a = 26560.906 Km

i = 55.0o

Period = 717.98 min

Revolutions/Day = 2.01

NAVSTAR/GPS

Aside 4.1 NAVSTAR/GPS

Figure A.10: Ground track – Time span shown: 1 day

Elevation aspect angle = 90o (North Pole) Elevation aspect angle = -90o (South Pole)

Figure A.11: Beherman projection - Time span shown: 8 days

Aside 4.2 NAVSTAR/GPS

Orbit Characteristics

a = 29993.689 Km

i = 56o

Period = 861.58 min

Revolutions/Day = 1.67

GALILEO

Aside 5.1 Galileo

Figure A.12: Ground track – Time span shown: 3 days – Elevation aspect Angle = 0o

Aside 5.2 Galileo

Figure A.13: Ground track – Time span shown: 3 days

Elevation aspect angle = 90o (North Pole) Elevation aspect angle = -90o (South Pole)

Aside 5.3 Galileo

Figure A.14: Beherman projection - Time span shown: 3 days

Orbit Characteristics

a = 26553.629 Km

i = i = 63.2o

e = 0.736

Period = 717.5 min

Revolutions/Day = 2.01

MOLNIYA

Aside 6.1 Molniya

Figure A.15: Ground track – Time span shown: 1 day – Elevation Aspect Angle = 60

Aside 6.2 Molniya

Figure A.16: Ground track – Time span shown: 1 day

Elevation aspect angle = 90o (North Pole) Elevation aspect angle = -90o (South Pole)

Aside 6.3 Molniya

Figure A.17: Beherman projection - Time span shown: 1 day

Bibliography

1. David A.Vallado, Second Edition 2004. Fundamentals of Astrodynamics and Applications.

2. CNES, Edited by Jean-Pierre Carrou, 1995. Spaceflight Dynamics Part I.

3. CNES, Edited by Jean-Pierre Carrou, 1995. Spaceflight Dynamics Part II.

4. ESA, NAPEOS Mathematical Models and Algorithms DOPS-SYS-TN-0100-OPS-GN