LEO OR.A.SI Presentation Version No.17

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
Illumination(%)
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.

2.1. 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.

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
 Sun synchronous
 Local Solar time at node crossings = 6 h
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
LocalSolarTime(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
0,4
DistancefromMeanGridNode(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
GroundTrackShifts
[Km]
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
DistancefromPreviousNode(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-SatelliteLongitude(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
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
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
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

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
Eccentricity
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
112,4ArgumentofPerigee(deg)
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.

 Sun synchronous
 Overfly through Earth point with longitude
λ = 390 East and latitude φ = 220 North.
Overfly while ascending.
eccentricity)

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
GeodeticLatitude(deg)
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
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
Longitude (deg)
Shifted Ground Track
Original Ground Track
20
o
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.

 Sun synchronous
eccentricity)

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
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
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
Cross
Longitude
earth
2
cos1
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
di
l
dt
da
lsign
dt
di
l
dt
da
llilal iaiaMaxia 04







dt
di
h
dt
da
hsign
dt
di
h
dt
da
hHihah iaiaMaxia 4
(4.1)
(4.2)
The period for symmetric evolution for each case is given from the formulas:
dt
di
l
dt
da
l
l
T
ia
Max
lo


 0
4
dt
di
h
dt
da
h
H
T
ia
Max
H


 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
di
l
dt
da
lsign
dt
di
l
dt
da
llilal
dt
ld
iaiaMaxia 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.
     
 
 
 
2
1
2
2
cos
cos
1
sin
sin
cos








L
i
i
L
iaisensLll earthoL
(4.7)








dt
di
h
dt
da
hsign
dt
di
h
dt
da
hHihah
dt
Hd
iaiaMaxia 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

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.

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.

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.

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

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.

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.
)cos(
sec
t
dt
d
LRaah
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

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.

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.
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.

4.5.5.1 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.

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.

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

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.

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.

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.

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

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.

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 r
ue
ea
rrh 



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:
0
dt
ed
0
0 



dt
d
dt
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
ee
k






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)

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
kTT
ktRe
kTT
ktRe
ooy
oox
















(I.7)
)
2
sin()
2
sin()sin()
2
cos()cos(
)
2
sin()
2
sin()cos()
2
cos()sin(
kTkT
kt
kT
ktRe
kTkT
kt
kT
ktRe
oooy
ooox
















)
2
(sin)sin(2)sin()cos(
)
2
(sin)cos(2)sin()sin(
2
2














kT
ktkTktRe
kT
ktkTktRe
oooy
ooox


(I.8)
(I.9)

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
e
B
kT
AkT
R
e
BkTA
kT




(I.11)
(I.12)
(I.10)
Whose solution is:
cos(kT)-1
)sin(
2
1
)sin(
cos(kT)-1
)sin(
2
1
)cos(





 






 

kTe
e
R
ktB
kTe
e
R
ktA
x
y
o
o
y
x
o
o


Now we use the trigonometric identity:
cos(x)-1
)sin(
)
2
cot(
xx


So we have that:
(I.14)
(I.13)
)
2
cot(
2
1
)sin(
)
2
cot(
2
1
)cos(














kT
ee
R
ktB
kT
ee
R
ktA
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(
,
,
kT
ee
ktReee
kT
ee
ktReee
xy
oyfyy
yx
oxfxx






And from (5):
2
)
2
cot(
2
)
2
cot(
o
xy
y
o
yx
x
R
Te
ee
e
R
Te
ee
e








(I.15)

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(3
2
223
3
33









 i
ea
iRJ
ae
E

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(3
2
3
3
33








 i
a
iRJ
ae
E

So from (I.3), (I.5) and (I.17) we have since ex = e cos(ω) :
)(sin
4
5
-1
2
)sin(3
)cos( 2
3
3
33






 i
ea
iRJ
a
e
A
eR
Ae
R
A
k
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-tB
m
1j
j
i
o
1
i  








 j
j
n
i
o
T
t
ptp  (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:
periodlong:TT
periodmedium: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:
 
 
ninclinatiomeantheiandaxissemimajormeanthearadius,Earththeisaand
1sin43
2
3
1
2sin
2
1
sin2
2
3
2sincos
4
3
2cos2sin
8
3
3sinsin
12
7
sinsin
4
7
1
2
3
3cossin
12
7
cossin
4
5
1
2
3
2cossin
2
3
e
1
2
2
2
2
2
2
2
2
2
2
22
2
2
22
2
2
2
2
2
























































































































vu
i
a
a
J
where
ui
a
a
Ju
ui
a
a
J
ui
a
a
Ji
uiui
a
a
Je
uiui
a
a
Je
ui
a
a
Ja
e
e
e
e
e
y
e
x
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 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
OsculatingMajorSemiAxis(Km)
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*sin(ArgumentofPerigee)
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
OsculatingInclination(deg)
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
AscendingNode(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
Inclination(deg)
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
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

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
DistanceBetweenConsecutiveinTimeNodes(Km)
MJD (ddddd.ddd)
Distance Between Consecutive in Time Nodes on the Equator
Distance Between Consecutive in Time Nodes on 70
o
Latitude
Difference due to zonal harmonics

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
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
a
TT
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(ArgumentofPerigee)
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(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
Altitude(km)
ArgumentofLatitude(deg)
D O Y ( d d d .d d d )
50
100
150
200
250
300
350
700
702
704
706
708
710
712
714
716
718
720
722
724
726
728
730
732
50
1 0 0
1 5 0
2 0 0
2 5 0
3 0 0
3 5 0
Altitude(km)
DOY(ddd.ddd)
Detail
In accordance to [2], the simplified expression for altitude deviation Δh is given as follows :
 
latitudeofargumenttheisvωandaxissemimajortheisa
)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:
 
 
 
 
 
periodcanomalistiKepleriantheis
a
nwhere
1
cos(i)
2
3
1
1)(cos5
4
3
1
1)(cos3
4
3
3
mean
22
2
2
22
22
2
2
3
2
22
2


































ea
a
J
n
e
i
a
a
J
n
e
i
a
a
J
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
RAAN(deg)
DOY (ddd.ddd)
RAAN
Sun's Right Ascension
90
o
= 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

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.
7223iandyear1Twhere
2
1
T
H
4sin
2
i
sin(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
Inclination(deg)
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]:
   
   



































i
a
a
Jiaand
i
a
a
J
a
a
dt
l
dt
l
dt
ld
alno
e
alnoe
e
alno
alno
alno
e
o
2sin6)tan(l
)(cos4
2
7
3
7
1
2
3
lwhere
dida
minsidereal
2
2min2
2
2
2
minsidereal
min
minsidereal1
21







dt
di
h
dt
da
h
dt
da
dt
di
i
T
T
dt
Hd
Sun
Earth
21
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
MJD (ddddd.ddd)
Distance from Mean Grid Node for LTA = 6 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
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
MJD (ddddd.ddd)
Phase Difference for Equatorial Nodes and LTA = 6h
51690 51695 51700 51705 51710 51715 51720
1,0
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
- -
- -
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 -






 


2
frozen
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:
 
 
   
 
ratiomasstoareaeffectivetheisA/m
andAU1ofdistanceatpressuresolartheisNm65.4
)cos(
1
2
λ
cos
orbit withtheofpartangulardilluminatetheisλand
2
1
2πq
)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
H
where
qq
a
m
A
P
dt
vd
qq
a
m
A
P
dt
di
q
a
m
A
P
dt
de
AU
meane
Earth
AN
e
mean
AU
e
mean
AU
eemean
AU
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
2
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
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
- -
spacecrafttheofrunit vectopositiontherˆand
tcoefficienpressuresolartheCwithratiomastoareaeffectivetheis
ˆ
p
m
AC
where
r
m
AC
adt
ed
a
m
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 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
MeanMajorSemiAxis(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
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
Altitude(km)
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
696
698
700
702
704
706
708
710
712
714
716
718
720
722
724
726
728
730
732
5 0
1 0 0
1 5 0
2 0 0
2 5 0
3 0 0
3 5 0
Altitude(km)
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
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 3
J
naa
P
P 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
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/).

 a = 7077.701 Km
 i = 98.203o (Sun Synchronous)
 Period = 98.88 min
LANDSAT

Aside 1.1 LANDSAT
Figure A.1: Ground track – Time span shown: 1 day - Elevation aspect angle = 33o

Aside 1.2 LANDSAT

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

 a = 6728.217 Km
 i = 34.99o
 Period = 91.31min
TTRM – Tropical Rainfall Measuring Mission

Aside 2.1 TRMM

Aside 2.2 TRMM
Figure A.6: Ground track – Time span shown: 1 day - Elevation aspect angle = 90o (North Pole)

Aside 2.3 TRMM

 a = 25507.6 Km
 i = 64.8o
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

 a = 26560.906 Km
 i = 55.0o
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)

Aside 4.2 NAVSTAR/GPS

 a = 29993.689 Km
 i = 56o
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

Aside 5.3 Galileo

 a = 26553.629 Km
 i = i = 63.2o
 e = 0.736
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

Aside 6.3 Molniya

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

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