This section presents the characteristics of the orbital propagator and the reference models used.
2.4.1 Two-Body Problem
The classical problem of determining the motion of two bodies due solely to their own mutual gravitational attraction is resolved by the Newton’s Law of Universal Gravitation. In [14, 23] it is shown that by considering an inertial frame, such as the ECI frame represented in Figure 2.2, making some simplifying assumptions, and by applying the Newton’s second law, the path of one of the masses relative to the other is a conic section (circle, ellipse, parabola, or hyperbola) whose shape is determined by the eccentricity.
Through straightforward steps can be demonstrated that the vector differential equation of the relative motion for the two-body problem is given by
:¯r GpM mq
r3 r¯ (2.20)
where G is the universal gravitational constant, M is the mass of the main body, and m is the mass of the secondary body. In the case of interest, the main body is represented by the Earth and the secondary body by the spacecraft, therefore the mass m is much less than M . Hence it follows
GpM mq GM µ (2.21)
where µ is called the gravitational parameter which for the Earth is
µ 3.986004418 1014 m3{s2 (2.22)
Then Equation 2.20 can be written again as
:¯r µ
r3 ¯r (2.23)
It is possible to define the so-called state vector of the spacecraft which comprises its position ¯r and velocity ¯v. The time derivative of the state vector return the velocity and acceleration that is given by Equation 2.23. By integrating the six components of this vector, the instantaneous position and velocity of the spacecraft relative to the inertial frame FECI can be calculated to define the characteristics of the orbit. Hence, the six components of the state vector are sufficient to uniquely determine an orbit.
Line of nodes
Ascending node
P
Vernal equinox direction
Periapsis direction Satellite’s position
at epoch
Figure 2.6: Classical orbital elements, adapted from [14]
As an alternative to the three components of the position vector ¯r and the three components of the velocity vector ¯v, the six classical orbital parameters represented in Figure 2.6 can be used to describe the orbit and the position of the spacecraft. Among these six parameters, five are sufficient to define the shape, size and orientation of the orbit, while the sixth element is required to specify the position of the spacecraft along the orbit. In [14] the classical orbital elements are defined as
a, semi-major axis: a constant defining the size of the conic orbit;
e, eccentricity: a constant defining the shape of the conic orbit;
i, inclination: the angle between the ˆK unit vector and the angular momentum vector ¯h;
Ω, longitude of the ascending node: also known as RAAN angle, in the fundamental plane, between the ˆI unit vector and the point where the spacecraft crosses through the fundamental plane in a northerly direction measured counterclockwise when viewed from the north side of the fundamental plane;
ω, argument of periapsis: the angle, in the plan of the spacecraft’s orbit, between the ascending node and the periapsis point, measured in the direction of the satellite’s motion;
ν, true anomaly at epoch: the angle, in the plane of the spacecraft’s orbit, between periapsis and the position of the spacecraft at a particular time, t, called the ”epoch”.
An orbit that can be described by the six orbital parameters, considered constant over time (except for the true anomaly), is called a Keplerian orbit.
The orbit period is a direct consequence of the Kepler’s third law and can be calculated as
T 2 π d
a3
µ (2.24)
Further considerations and insights are made in [40, 14, 23], here it is only recalled that the con-servation of energy and angular momentum applies to Keplerian orbits.
In general, an orbit is characterised by several aspects
• energy of the orbit, which in turn determines the type of conic;
• altitude;
• inclination;
• direction of motion.
In the case of interest, a circular low Earth orbit (LEO) is considered.
2.4.2 Perturbations
Keplerian orbits are the closed-form solutions of the two-body equation of relative motion 2.23.
This equation is based on the folliwing simplifying assumptions
• there are only two bodies in space;
• their gravitational fields are spherically symmetric;
• the only source of interaction between the two bodies is the mutual gravitational attraction.
Any effect that causes the motion to deviate from a Keplerian trajectory is known as a perturbation.
To account for the perturbations, contributions to the acceleration of the spacecraft are added. It follows that Equation 2.23 becomes
:¯r µ
r3 ¯r ¯ap (2.25)
where the vector ¯apis the net perturbative acceleration from all sources other than the spherically symmetric gravitational attraction between the two bodies. Common perturbations of two-body motion include gravitational interactions with celestial objects like the moon and the sun, a non-spherical central body, solar radiation pressure, and atmospheric drag.
There are two main categories of perturbations techniques, these are described in Table 2.1. In this thesis work, attention was focused on the category of special perturbations, particularly in Cowell’s method.
Cowell’s method is the simplest and most straight forward of all perturbation methods. The ap-plication of the method consists on integrating the equations of motion, including all the pertur-bations, step-by-step numerically. Having the analytical formulation of the perturbation, the state vector can be integrated by applying a numerical integration scheme.
Type Description Examples
Special Perturbations are techniques which deal with the direct nu-merical integration of the equations of mo-tion including all necessary perturbing accel-erations
Cowell, Encke, Variation el-ements techniques
General Perturbations involve an analytic integration of series expan-sions of the perturbing accelerations
SGP, SGP4, BL
Table 2.1: Categories of perturbation techniques, from [14]
In [26] a comparative study is carried out where the performances of different orbital propagators are evaluated, computational time and root mean square errors are used as comparison metrics.
The results of the study show that special perturbation techniques are more accurate than general perturbation techniques; however, this accuracy comes at the cost of computational efficiency.
On the other hand, analytic theories (general perturbation techniques) perform poorly but are effi-cient. As computational capacity becomes increasingly available even on board spacecraft, accu-racy was prioritised over computational cost.
2.4.3 Orbit Propagator
The orbit propagator developed for the Attitude Determination and Control System implements a special perturbation solution of orbital motion using a fixed step size Runge-Kutta 4th integration method. The acceleration vector integrated at each step consists of the following contributions
¯
atot ¯:r ¯ag ¯asun ¯asrp ¯ad a¯moon (2.26) where
¯
ag is the acceleration due to gravity
¯
asun is the acceleration due to the Sun
¯
asrp is the acceleration due to solar radiation pressure
¯
ad is the acceleration due to atmospheric drag
¯
amoon is the acceleration due to the Moon
Figure 2.7 shows the schematic framework of the orbit propagator, implemented in this thesis work, which highlights the inputs required to calculate the different contributions.
Acceleration due to the Gravity Field of the Earth
The gradient of the potential of a central body yield the acceleration, if the body is spherically symmetric then the acceleration is that given in Equation 2.23. In this case, the non-ideal case is considered, therefore in order to derive the non-ideal acceleration, it is necessary to perform the gradient of a potential function that includes perturbations due to a nonspherical Earth. In [40] the procedure for deriving an aspherical-potential function U is shown in detail, here only the final results used in the Matlab code implementation are reported.
The aspherical-potential function is given by
U µ r
"
1
¸8 l2
Cl,0
RC r
l
Plpsin ϕq
¸8 l2
¸l m1
RC r
l
Pl,mpsin ϕq
Cl,mcospmλq Sl,msinpmλq
* (2.27)
where λ and ϕ are respectively the east longitude and the latitude of the spacecraft, µ is the gravi-tational parameter defined in 2.22, r a
x2 y2 z2is the geocentric radius of the spacecraft, S and C are unnormalized harmonic coefficients of the geopotential, and Pl are the Legendre polynomials of degree l while Pl,mare the Legendre polynomials of degree l and order m.
The analysis of satellite motion allows to empirically determine the C and S coefficients from observations. These harmonic coefficients are available for example in Earth Geopotential Model 96 (EGM96) gravity model which is used in the orbit propagator.
Figure 2.7: Orbit Propagator Framework
Coefficients Cl,0 Jlwhere Jl are the zonal harmonics of the planet. J2 is the strongest per-turbation due to the Earth’s shape, in particular reflects the Earth’s oblateness. There are other two type of spherical harmonics which are sectoral and tesseral harmonics. These are described in [40], however their effect is less important than that of the J2coefficient.
The conventional Legendre polynomials are defined as Plpsin ϕq 1
2ll!
dl dpsin ϕql
sin2ϕ 1l
(2.28) while the so-called associated Legendre functions are
Pl,mpsin ϕq p1 sin2ϕqm{2 dm
dpsin ϕqmPlpsin ϕq (2.29) Acceleration can be derived by taking the gradient of the aspherical-potential function in Equation 2.27.
¯
ag ∇U (2.30)
Breaking out individual components :x
"
1 r
BU
Br z
r2a
x2 y2 BU
Bϕ
* x
"
1 x2 y2
BU Bλ
* yµx
r3 (2.31)
:y
"
1 r
BU
Br z
r2a
x2 y2 BU
Bϕ
* y
"
1 x2 y2
BU Bλ
* xµy
r3 (2.32)
:z 1 r
BU Brz
ax2 y2 r2
BU Bϕ µz
r3 (2.33)
where
BU
Br µ r2
¸8 l2
¸l m0
RC r
l
pl 1qPl,mpsin ϕq
"
Cl,mcospmλq Sl,msinpmλq
*
(2.34)
BU Bϕ µ
r
¸8 l2
¸l m0
RC r
l"
Pl,m 1psin ϕq m tanpϕqPl,mpsin ϕq
*
Cl,mcospmλq Sl,msinpmλq
(2.35)
BU Bλ µ
r
¸8 l2
¸l m0
RC r
l
mPl,mpsin ϕq
Sl,mcospmλq Cl,msinpmλq
(2.36)
Acceleration due to Solar Gravity
The perturbing acceleration due to the presence of a third body is discussed in [23]. In particular, the acceleration due to solar gravity is given by
¯
asun µ@
r¯@{s r@{s3 r¯@
r3@
(2.37) where ¯r@{sis the vector from the spacecraft to the sun, ¯r@is the vector from the Earth to the Sun in FECI. These vectors are represented in Figure 2.8.
The sun’s geocentric position vector ¯r@ in its apparent motion around the earth can be found following the indications given in [23, 33].
According to The Astronomical Almanac [33] the apparent solar ecliptic longitude is given by the formula
λ L 1.915sin M 0.0200sin 2M (2.38)
where L and M are expressed both in degree and are respectively the mean longitude and mean anomaly of the sun. These can be calculated by
L 280.459 0.985 647 36n (2.39)
M 357.529 0.985 600 23n (2.40)
Earth
Spacecraft
i: Sun or Moon
Figure 2.8: Perturbation of a spacecraft’s earth orbit by solar or moon gravity.
Adapted from [23]. The scheme is not to scale.
λ, L, and M are angles in the range 0to 360. n is the number of day since J 200
n JD 2, 451, 545.0 (2.41)
The Julian day can be calculated as shown in Equations 2.2, 2.3, and 2.4.
The obliquity ε, which is the angle between earth’s equatorial plane and the ecliptic plane, can be found in terms of n
ε 23.439 3.56 107n (2.42)
Then, the distance from the Earth to the sun is
r@ p1.00014 0.01671 cos M 0.000140 cos 2Mq AU (2.43) This is only the norm of the vector, but it is necessary to know also the direction. To do so, the geocentric ecliptic frameis introduced. In this frame,the unit vector ˆu along the earth-sunline is provided by the solar ecliptic longitude λ given in Equation 2.38
ˆ
ueclipticsun cos λ ˆI1 sin λ ˆJ1 (2.44) In Equation 2.37 the vector ¯r@is measured in the FECI frame, therefore it is necessary to evaluate ˆ
u in FECI. The transformation from the geocentric-ecliptic frame to the ECI frame is a clockwise rotation through the obliquity ε around the positive X axis
ˆ
uECIsun A1pεqTuˆeclipticsun
1 0 0
0 cos ε sin ε 0 sin ε cos ε
cos λ sin λ
0
cos λ cos ε sin λ sin ε sin λ
(2.45)
Therefore, the sun vector ¯r@in ECI frame can be calculated as
¯
r@ r@uˆECIsun (2.46)
To avoid numerical problem, due to subtractions between two nearly equal numbers, Equation 2.37 is rewritten as suggested in [15]
¯
asun µ@ r@{s3
Fpqq¯r@ ¯r
(2.47) where Fpqq is given by
Fpqq q2 3q 3
1 p1 qq3{2q (2.48)
with
q r¯ p2¯r@ ¯rq
r2@ (2.49)
Acceleration due to Solar Radiation Pressure
The output from the sun contain momentum, which produce an effective pressure on spacecraft surfaces. The perturbing force on the satellite due to the solar radiation pressure is given by
F νS
cγAscuˆ@,s (2.50)
where
ν is the shadow function, which is equal to 0 when the spacecraft is in the Earth’s shadow while is equal to 1 when the spacecraft is in sunlight
S
c is the solar radiation pressure, where S is the solar constant and c the speed of light S
c 1367pN m{sq{m2
2.998 108 m{s 4.56 106N{m2 (2.51) γ is the radiation pressure coefficient, or reflectivity constant, which lies between 0 and 2 Ascis the surface area of the spacecraft normal to the incident radiation
ˆ
u@,sis the unit vector pointing from the satellite toward the Sun Denoting by m the mass of the spacecraft, the acceleration is given by
¯
asrp F
m (2.52)
During the integration process, the software must determine if the satellite is in Earth shadow or sunlight. As indicated in [40], to determine when a satellite is in the earth’s shadow it is possible to use the following procedure. The procedure is based on determining whether or not a line of sight exists between two given vectors. Considering Figure 2.10, assuming that body A is the spacecraft and body B is the sun then it is possible to check whether the spacecraft is in Earth shadow if the sum of θ1and θ2is greater than the angle θ.
The angle θ between the two position vectors may be found from the dot product operation θ cos1
r¯@ ¯r r@r
(2.53) while it is easy to see from Figure 2.10 that
θ1 cos1
RC r
θ2 cos1
RC r@
(2.54) Therefore, if θ1 θ2 θ then there is no line of sight which means that the spacecraft is in Earth shadow. In this case there is no perturbing force, hence ν 0. On the other hand if θ1 θ2 ¡ θ then there is line of sight and so the spacecraft is in sunlight ν 1.
Figure 2.9: Example of computed unit sun vector in orbital frame measured from an orbit with an altitude of 470km
T2
T1
A B A B
T
Figure 2.10: Eclipse model, adapted from [23]
Acceleration due to Atmospheric Drag
The acceleration of the spacecraft due to atmospheric drag is evaluated using the following ex-pression
¯
ad 1 2ρvrel
CDA m
¯
vrel (2.55)
where ρ is the atmospheric density and it is calculated considering the 1976 U.S. Standard atmo-sphere[36]. ¯vrel is the relative velocity of the spacecraft with respect to the atmosphere. In this implementation it is assumed that the atmosphere rotates with the earth, whose angular velocity is ωC, it follows that ¯vatm ¯ωC ¯r. Therefore, the relative velocity is given by
¯
vrel ¯v ¯vatm (2.56)
The negative sign in Equation 2.55 is due to the fact that the drag force on an object acts in the direction opposite to the relative velocity vector.
CD is the dimensionless drag coefficient, A is the spacecraft frontal area, and m is the space-craft mass.
Acceleration due to the Moon
The acceleration experienced by the spacecraft due to the Moon follow the same scheme presented for the perturbation of the solar gravity. In this case, the Moon replaces the sun as the third body, the acceleration become
¯
amoon µK
¯rK{s r3K{s r¯K
rK3
(2.57)
where ¯rK{s is the vector from the spacecraft to the moon, ¯rK is the vector from the Earth to the moon in FECI.
The unit vector ˆu from the center of the Earth to that of the moon is given in the geocentric ecliptic frame by an expression similar to Equation 2.44
ˆ
ueclipticmoon cos δ cos λ ˆI1 cos δ sin λ ˆJ1 sin δ ˆK1 (2.58) where λ is the lunar ecliptic longitude, and δ is the lunar ecliptic latitude. The components of ˆ
ueclipticmoon in the geocentric equatorial frame are found as in Equation 2.45
ˆ
uECImoon A1pεqTuˆeclipticmoon
1 0 0
0 cos ε sin ε 0 sin ε cos ε
cos δ cos λ cos δ sin λ
sin δ
cos δ cos λ
cos ε cos δ sin λ sin ε sin δ sin ε cos δ sin λ cos ε sin δ
(2.59)
Then, the geocentric equatorial position of the moon is ¯rm rmuˆECImoon. The distance to the moon rmcan be calculated as
rm RC
sin HP (2.60)
where HP is the horizontal parallax. The Astronomical Almanac [33] presents the following formulas for the time variation of lunar ecliptic longitude λ, lunar ecliptic latitude δ, and lunar horizontal parallax HP
λ b0 c0T0
¸6 i1
aisinpbi ciT0q
δ
¸4 i1
disinpei fiT0q
HP g0
¸4 i1
cosphi kiT0q
(2.61)
where T0is the number of Julian day centuries since J2000 for the current Julian day J D given in Equation 2.5. The necessary coefficients are listed in Table 2.2.
Longitude, λ Latitude, δ Horizontal Parallax, HP
i ai bi ci di ei fi gi hi ki
0 - 218.32 481267.881 - - - 0.9508 -
-1 6.29 135.0 477198.87 5.13 93.3 483202.03 0.0518 135.0 477198.87 2 -1.27 259.3 -413335.36 0.28 220.2 960400.89 0.0095 259.3 -413335.38 3 0.66 235.7 890534.22 -0.28 318.3 6003.15 0.0078 253.7 890534.22 4 0.21 269.9 954397.74 -0.17 217.6 -407332.21 0.0028 269.9 954397.70
5 -0.19 357.5 35999.05 - - -
-6 -0.11 106.5 966404.03 - - -
-Table 2.2: Coefficients for computing lunar position
2.4.4 International Geomagnetic Reference Field
As mentioned earlier, in order to verify the performance of the determination algorithms, it is nec-essary to have a model of the magnetic field to simulate the operation of the magnetometer whose measurements are used to determine the attitude. The Earth’s magnetic field model is also used to simulate the detumbling phase in which magnetic torquers are used.
In International System of Units the geomagnetic field is measured in Tesla, however given the very small values it is often used its submultiple nano-Tesla (nT). The Earth’s magnetic field is produced by internal sources primarly inside Eart’s core and is mainly that of a magnetic dipole.
It is not constant but subject to continuous variations and intensity due to external or local causes.
Further insights can be found in [44], however it should be noted that the lack of surface electric currents implies that outside the Earth, the magnetic field B has zero curl
∇ ¯B 0 (2.62)
therefore the field can be expressed as the gradient of a scalar potential, V
B¯ ∇V (2.63)
The absence of magnetic monopoles implies that the divergence of B is equal to zero which led to the Laplace’s equation
∇2V 0 (2.64)
Given the spherical nature of the boundary at the Earth’s surface, the Laplace’s equation has a solution suitably expressed as a finite series expansion in terms of spherical harmonic coefficients
Vpr, θ, ϕ, tq a
¸N n1
¸n m0
a r
n 1
gnmptq cos mϕ hmnptq sin mϕ
Pnmpcos θq (2.65)
where a is a reference radius of the Earth; gmn and hmn are the spherical harmonic coefficients called Gaussian coefficients; r, θ, and ϕ are respectively the geocentric distance, east longitude from Greenwich, and coelevation. The Pnmpcos θq are Schmidt semi-normalized associated Legen-dre functions of degree n and order m. The parameter N specify the maximum spherical harmonic degree.
To use Equation 2.65 in order to evaluate the magnetic field at any point, the Gaussian coeffi-cients must be known. The model used in this work is the International Geomagnetic Reference Field (IGRF): the thirteen generation. As reported in [11], the IGRF is a set of spherical har-monic coefficients which can be input into the mathematical model to describe the large-scale, time-varying portion of Earth’s internal magnetic field between epoch 1900 A.D. and the present.
The coefficients of this thirteen generation has been obtained from ground observatories, from observations recorded by satellites and magnetic surveys (in [11] it is reported a list of World Data System data centers and services). The IGRF is produced and maintained by an international task force of scientists under the auspices of the International Association of Geomagnetism and Aeronomy (IAGA) Working group V-MOD.
Given the continuous unpredictably variations of the Earth’s core field on timescales ranging from months to millions of years the IGRF must be regularly revised to account for temporal changes, typically every 5 years. It follows that Gauss coefficients gmnptq and hmnptq change in time at 5-years epoch intervals. Expressions for the time dependence of these parameters are given in [11].
The IGRF thirteen generation provides a Definitive Geomagnetic Reference Field (DGRF) model for epoch 2015, a Non-Definitive Geomagnetic Reference Field (basically labeled as IGRF) model for epoch 2020, and a predictive IGRF secular variation model for the 5-year time interval 2020 to 2025. To give an example, Figures 2.11a, and 2.11b shows respectively the global map of the IGRF-13 total field magnitude and its predicted secular variations. These Figures* can be found in [11]. While Figure 2.12 presents an example of magnetic field computed with the IGRF model.
*Included in the article’s Creative Common license http://creativecommons.org/licenses/by/4.0/
60000 50000
40000 30000
60000 50000
40000
00006
90°N
60°N
30°N 0°
30°S
60°S
90°S
180°
150°E 120°E 90°E 60°E 30°E 0°
30°W 60°W 90°W 120°W 150°W 180°
Total Field (F) in nT 2020
(a) Total field
80 60 40 20
0 -20
-40 -80-100
-60 -80
100 80
20 0
-100 90°N
60°N
30°N 0°
30°S
60°S
90°S
180°
150°E 120°E 90°E 60°E 30°E 0°
30°W 60°W 90°W 120°W 150°W 180°
Predicted average change in Total Field (F) for 2020-2025 (nT/year)
(b) Predicted secular variation in total field Figure 2.11: Maps of total field for epoch 2020 (a), and maps of predicted annual
secular variation in total field (b) over 2020 to 2025. Both at the WGS84 ellipsoid surface for epoch 2020. From [11]
Figure 2.12: Example of computed magnetic field vector in Centred Earth-Fixed frame measured from an orbit with an altitude of 470 km and51.6 inclination