Orbital Mechanics : - Interplanetary Explorer Mission - Planetary Explorer Mission HOMEWORK ASSIGNMENTS Felin Aur ́ elien 10784723 963785 Mauriello Tommaso 10803889 968434 Selva Melvin 10786461 962017 Zagatti Tommaso 10581563 964800 Figure 1: ”Hello From Above”, Mark Vande Hei from aboard the International Space Station. 1 Professor: Colombo, C — Delivery date: 11/02/2021 Msc. in Space Engineering Politecnico di Milano Contents I Interplanetary Explorer Mission 2 1 Objective of the study 2 2 Design process 2 2.1 Preliminary considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Search for the first guesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2.1 Ideal Hohmann transfer options . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2.2 Refined Hohmann transfers options . . . . . . . . . . . . . . . . . . . . . . 4 2.2.3 Best single arc transfer options . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Choice of the search time spans . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4 Optimization algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.4.1 Grid search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.4.2 Fmincon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4.3 Genetic algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4.4 Global search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 Final results 8 3.1 Best transfer options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2 Interpretation of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 II Planetary Explorer Mission 11 4 Ground tracks 11 4.1 Initial orbital values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.2 Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.3 Unrepeated ground tracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.4 Repeated ground tracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 5 Orbit propagation 14 5.1 Assigned perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 5.2 Cartesian propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 5.3 Gauss propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5.4 Crash possibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5.5 Comparison and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 6 Filtering 17 6.1 Unfiltered results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 6.2 Filtering principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 6.3 Construction of the filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 6.4 Filtered results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 7 Comparison with real data 20 7.1 Real spacecraft in the same orbital region . . . . . . . . . . . . . . . . . . . . . . 20 7.2 TLE processing and propagation model . . . . . . . . . . . . . . . . . . . . . . . 20 7.3 Comparison between real data and Gauss propagation . . . . . . . . . . . . . . . 20 1 List of Symbols Physics Constants μ ⊕ Earth’s gravitational constant J 2 Second zonal harmonic of Earth R ⊕ Earth’s radius Acronyms 2 BP Two-body problem T ARR Orbital period of the arrival planet T DEP Orbital period of the departure planet T F LY Orbital period of the flyby planet ARR Arrival planet DEP Departure planet FLY Flyby planet RAAN Right Ascension of the Ascending Node RF Reference Frame RSW Orbital Frame SOI Sphere Of Influence SRP Solar Radiation Pressure TLE Two Line Element Other Symbols δ Turn angle μ f ly Planetary constant of the flyby planet Ω Right ascension of the ascending node ω Argument of perigee φ Phasing angle θ True anomaly e Eccentricity r p Radius of periapsis of the hyper- bolic perigee passage Part I Interplanetary Explorer Mission 1 Objective of the study The goal of this study is to find the best transfer options between two planets of the solar system exploiting a powered flyby of a third planet. The assigned data are: • DEP: Mercury • FLY: Venus • ARR: Mars Furthermore, a 40-year time window (earliest departure: 02/01/2027, latest arrival: 02/01/2067) has been provided as the temporal range in which to find the best transfer option. The figure of merit for the whole search is the total mission cost in terms of ∆ v . The MATLAB software has been used to perform the required computations. 2 Design process 2.1 Preliminary considerations The purpose of exploiting the Venus powered flyby is to reduce as much as possible the ∆ v required for the mission. Thus it can be insightful to compute the range of values of ∆ v 2 we expect to find. The minimum ∆ v would be obtained with an ideal Hohmann transfer DEP-to-FLY, and another ideal Hohmann transfer FLY-to-ARR (with an unpowered flyby between them) immediately afterwards. Notice that this would also imply a perfect phasing of the three planets to allow for such a transfer. The maximum ∆ v can be considered the one required to do a single arc transfer DEP-to-ARR, without flyby. For this purpose the function computeDvRange.m has been used, obtaining : ∆ v min = 11 538 km/s ∆ v max = 20 712 km/s (1) Therefore, any value below ∆ v min would be suspicious (i.e. incorrect), while any value over ∆ v max would mean that the mission layout itself (departure – powered flyby – arrival) is useless since a single arc transfer DEP-to-ARR would be less expensive both in terms of ∆ v and required transfer time. In addition to the assigned constraint (time window) a new one must be imposed: the minimum radius of flyby passage, due to the presence of the atmosphere. A minimum flyby altitude of 300 km from the surface of Venus has been considered, using [2] as a reference. Since the total search span is 40 years, and since there are three free variables (date of departure, flyby and arrival) it would be too computationally expensive to evaluate the cost function for all the possible combinations of departure, flyby and arrival dates. Clearly not all these combinations of dates would be suitable (e.g. it would be useless to evaluate a transfer option whose flyby date occurs before the departure date). This means that a preliminary analysis must be carried out in order to get some appropriate first guesses to be used as inputs of optimization algorithms. 2.2 Search for the first guesses 2.2.1 Ideal Hohmann transfer options As stated previously the theoretical best two-arc transfer would be the one made of two consecutive Hohmann transfers. Thus some simple first guesses can be obtained looking for the occasions that allow an ideal Hohmann transfer DEP-to-FLY and FLY-to-ARR. This is shown in the simplified pseudo algorithm for getBestTransferOptionsHohmann.m Algorithm 1: getBestTransferOptionsHohmann 1: Compute transf erOptionsDepT oF ly = array of transfer options for an ideal Hohmann transfer DEP-to-FLY, size N 2: for transf erOptionDepT oF ly = 1 , 2 , . . . , N do 3: Compute the phasing FLY-ARR at the flyby time 4: Store for each transfer option the difference ∆ φ between the actual and the desired phasing for an Hohmann transfer FLY-to-ARR 5: end for 6: Order the transf erOptionsDepT oF ly from ∆ φ min to ∆ φ max This function returns the best transfer options in terms of three-planet phasing for a double arc ideal Hohmann transfer. The absolute angular position φ of the planets has been computed from the gamma direction as: 3 φ = θ + Ω + ω (2) (this is actually a reasonable approximation given the small orbit inclination, otherwise a spherical triangle must be solved). The phasing has been computed as: ∆ φ depT oF ly = φ f ly − φ dep ∆ φ f lyT oArr = φ arr − φ f ly (3) 2.2.2 Refined Hohmann transfers options The actual values of eccentricity and inclination with respect to the ecliptic of the three planets are: Planet eccentricity (-) inclination ( ° ) Mercury 0.20563 7.005 Venus 0.006772 3.39458 Mars 0.0934 1.85061 Due to the quite high eccentricity of Mercury’s orbit it’s evident that the approximation of coplanar and circular orbit may not be very accurate especially for the first transfer arc. To evaluate the accuracy of the approximation a comparison between the phasing of the planets in the ideal and real case has been conducted for both transfers with the script comparisonRealIdeal.m . As expected, the script shows that the ideal phasing can differ quite a lot from the real one for the Mercury-Venus pair, up to 36 ° (i.e. the planet is 36 ° away from where it is expected to be in the ideal case). For this reason a new function called getBestTransferOptionsHohmannRefined.m has been built, following the same logic of the ideal Hohmann transfer options, but making use of real ephemerides ( uplanet.m ). 2.2.3 Best single arc transfer options Although the refined Hohmann transfer options could already be considered as good first guesses, in order to study more in detail the problem a step further has been taken, that is considering the actual best single arc transfers DEP-to-FLY and FLY- to-ARR (i.e. computing the single transfer porkchop plots). This is done by the function getBestTransferOptionsSingleArcTransfer.m . The refined Hohmann transfer options have been used as first guesses for this search, and the GlobalSearch algorithm has been used in place of Fminunc for computing the minimum ∆ v for the single arc transfer in order to avoid the convergence to a local minimum, as shown in Fig. 2, where Fminunc converged to the wrong minimum since the first guess happened to be in the wrong valley. Once the optimal dates for both the transfer arcs have been obtained, for each transfer option DEP-to-FLY the closest opportunity FLY-to-ARR is sought. Thus the transfer options are ordered in terms of increasing difference between the date of flyby associated to the first arc and the closest flyby date for the second arc, with a procedure similar to what has been shown in the pseudo algorithm for getBestTransferOptionsHohmann.m (but in this case temporal, and not angular, differences have been considered). This results in an array of transfer options, each of them consisting in: • Departure date for the single transfer arc DEP-to-FLY • Flyby date for the single transfer arc DEP-to-FLY 4 Figure 2: Porkchop plot for the first transfer option DEP-to-FLY, and the minimum ∆ v found with Fminunc and GlobalSearch • Closest flyby date for the single transfer arc FLY-to-ARR • Arrival date for the single transfer arc FLY-to-ARR (associated to the closest flyby date for the same arc) In the end, both the dates of flyby will be considered as first guesses of the optimization algorithm and, as consequence, two different dates of arrival will be taken into account. This require the computation of a first guess for the arrival date when considering the flyby date associated to the first transfer arc. This is done by the function bestArrivalTime.m , which executes a single arc transfer optimization with a fixed flyby date (single dof opti- mization). These will be the first guesses actually used in the optimization algorithms. 2.3 Choice of the search time spans It’s necessary to define the range of time around the first guesses where the minimum of the cost function should be sought, since this is required by some optimization algorithms. After some trials a base time span of 80 days has been chosen for the departure and flyby dates. In addition, for the flyby dates it’s reasonable to assume that the the best date will be located somewhere in the middle between the two dates under consideration. In fact the two dates represent a local minimum of the single arc transfer, which means that going further away from these dates will result in an increase in the single arc transfer ∆ v . Inside the time span defined by the 2 flyby dates going away from one date means getting closer to the other, while outside of this interval the distance from both the flyby dates is increased and, consequently, the required ∆ v , thus it’s reasonable to expect the 5 best flyby date to fall inside this range. The problem under study is still a complex one, with many variables influencing the optimal solution, so it could happen that in some cases the best flyby date will occur outside this range (e.g. due to the presence of the constraint). But since both the flyby dates will be considered as first guesses, with their respective guess on the arrival date, a time span equal to the difference between the two flyby dates will be selected, so to cover the whole time span between the two dates and also a large time range outside of it. If this distance is less than the base time span of 80 days, then the time span of 80 days is selected by default. Instead, for the arrival date a time span of 1.5 times the time span on the flyby date has been selected. 2.4 Optimization algorithms Once good first guesses and appropriate time range have been selected, it’s necessary to decide which optimization algorithm to use. They’ll have to minimize the cost function which is given by: ∆ v tot = ∆ v dep + ∆ v f lyByP ow + ∆ v arr (4) Where ∆ v dep and ∆ v arr are computed simply as vectorial differences between the velocities of the planet orbits (which are assumed to be the initial and final velocity of the transfer) and the velocity at the beginning of the transfer arc DEP-to-FLY and at the end of the transfer arc FLY-to-ARR. The ∆ v f lyByP ow is instead the ∆ v to be given at the perigee of the hyperbolic flyby orbit in order to obtain the desired change in heliocentric velocity. This is computed with the function FlyByPow.m , which also returns the minimum radius of perigee of the hyperbolic passage. Various optimization algorithms that have been tested for the problem under consideration are discussed in the following paragraphs. 2.4.1 Grid search A grid search is performed by evaluating the cost function in a set of points inside a suitable range. Once a minimum amidst these points is found, the search range is refined around this minimum. This is done recursively until the desired tolerance is achieved (i.e. the difference between the minimum obtained at two successive iterations is less than a certain threshold). This approach is rather simple, but presents the following shortcomings: • It may converge to a local minimum; • It needs many sample points in the selected range to achieve good accuracy. This in turns implies many cost function evaluations (i.e. a long computational time). • It deals poorly with constrained optimizations. For this problem, in order to obtain good accuracy and prevent the loss of some results due to the constrained optimization the grid search has been performed, at the first iteration, dividing the interval in slices of three days. This proved effective in terms of accuracy, less effective in terms of computational time. The function used to perform this minimization is minDeltaVgridSearch.m . The completion of the search required 36 hours. 6 2.4.2 Fmincon Fmincon is a built-in MATLAB function that works by following the gradient in the point where the cost function is evaluated, like a ball that rolls following the local slope until it reaches a stationary point, and that can also deal with constrained optimizations. It’s quite an efficient algorithm, but it will only converge to a local minimum, which is not guaranteed to be also a global minimum (as happened in Fig. 2). It proved to be the most computationally efficient algorithm among the ones tested, completing the whole search in only 10 minutes. Although it found many good transfer options, it was not able to find the actual global minimum, and also missed some other good transfer options that didn’t happen to be in the same valley as the first guess. The function used to perform the minimization with Fmincon is minDeltaVfmincon.m 2.4.3 Genetic algorithm A genetic algorithm exploits the idea at the basis of the natural selection by constructing a population whose individuals are the solutions to the problem to be optimized, and the fittest individuals (those who best minimize the function) are those selected for reproduc- tion. The advantage of this kind of algorithms is (in theory) the avoidance of local minima, though this is not guaranteed. Also, since they rely on random processes, they won’t con- verge to the same minimum when run multiple times with the same first guesses. For the problem under study the MATLAB function ga did not show any positive result with different options, both in terms of accuracy and computational time, thus has not been used. The function used to perform the minimization with ga is minDeltaVgenAlg.m 2.4.4 Global search The global search algorithm allows to find the actual global minimum of the cost function in a given evaluation range, at the expenses of a higher computational cost. It repeatedly runs a local solver, which was chosen to be Fmincon . In the end this has been the preferred algorithm because it allowed to obtain, in a reasonable time (ca. 8 hours), not only the best transfer options, but also many other good ones, with just slightly more required ∆ v tot , that had been missed by Fmincon . It was also preferred to the alternative of using Fmincon on an increased numbers of first guess for several reasons: • the computational time would have increased exponentially with the number of first guesses; • the convergence to a global minimum would still not be guaranteed; • the GlobalSearch algorithm is already optimized to execute the local solver on multiple points in order to find the global minimum. The definitive function for this search is minDeltaVGlobalSearchRefined.m , which also employs Fmincon in order to consider also the (rare) cases in which Fmincon would con- verge to a better result from the same initial guess. 7 3 Final results 3.1 Best transfer options The best transfer option in terms of ∆ v tot is characterized by: • ∆ v tot = 17.763 km/s • ∆ v dep = 11.505 km/s • ∆ v arr = 6.162 km/s • ∆ v f lyByP ow = 0.096 km/s • ∆ v f lyByP ow ∆ v tot = 0.54% • r p = 6.3518e3 km • h p = 300 km • ∆ t f lyBy = 25.14 h • Departure date = 8th Apr 2046 • FlyBy date = 5th Oct 2046 • Arrival date = 25th Apr 2047 • Transfer Time ≈ 382 days = 12.7 months As expected the best flyby date falls in between the best flyby date for the first arc transfer (4th May 2046) and the best flyby date for the second arc transfer (7th Dec 2046). Figs. 3 and 4 (planets and Sun not to scale) show the whole trajectory and the planet positions at the three mission times. Fig. 5 shows instead the 3D porkchop plot associated to the best transfer option. The combination of dates with no associated ∆ v are not respectful of the minimum radius constraint. It’s interesting to notice that the spacecraft trajectory is tangential to the planet trajectory at departure and arrival (so in these points the trajectory can be considered Hohmann-like). Instead at the flyby point the trajectory is secant to the flyby planet trajectory. Note also that the flyby occurs at the second intersection point, so this trajectory is associated to a long flight time. Another interesting transfer option is the following: • ∆ v tot = 18.108 km/s • ∆ v f lyByP ow = 1.298e-5 km/s • Departure date = 7th Mar 2044 • FlyBy date = 14th Apr 2044 • Arrival date = 19th Sep 2044 • Transfer Time ≈ 195 days = 6.5 months Figure 3: Best option: planets positions at departure and flyby and interplanetary trajectory DEP-to-FLY 8 Figure 4: Best option: interplanetary trajectory DEP-to-FLY-to-ARR with planets positions at arrival time, and hyperbolic trajectory of the Venus powered flyby Figure 5: 3D Porkchop plot of the best transfer option. This solution, even if it requires an additional 0.345 km/s of ∆ v tot with respect to the optimal one, can be considered a better one considering other parameters. The transfer time has been halved, and the ∆ v f lyByP ow is extremely small (the flyby could be considered unpowered). This means that smaller engines are required for the flyby maneuver; clearly the same engines must be used for the final maneuver (∆ v arr ), but considering the case of a final unpowered Mars flyby the whole ∆ v tot = ∆ v dep would be given simply by the launcher at Mercury departure, and the increased ∆ v tot cost would be compensated by the decreased complexity of the spacecraft and the increased mass available for scientific payloads (since smaller thrusters on the spacecraft can be used and less fuel must be brought by). 3.2 Interpretation of the results The best transfer options found are not actually associated with the best first guesses (the first elements of the array returned by getBestTransferOptionSingleArcTransfer.m ). Indeed the closer first guess to the optimal solution is the number 87 out of 100 (100 being the number of DEP-to-FLY transfer occasions in the given 40-year time span). To understand this result it must be recalled that the radius of periapsis of an hyperbolic passage is inversely proportional to the hyperbolic trajectory turning angle. This can be easily shown for unpowered trajectories, but still holds for powered ones. Indeed, 9 exploiting the following expressions from Curtis[1]: e = 1 + r p v 2 ∞ μ f ly δ = 2 arcsin 1 e (5) it’s easy to obtain: r p = ( 1 sin( δ/ 2) − 1) μ f ly v 2 ∞ (6) which is a function approaching ∞ for δ approaching 0 ° , and approaching zero for δ approaching 180 ° (note that the turning angle will always be less than 180 ° ). A flyby between two ideal Hohmann arcs would imply a heliocentric velocity of the spacecraft parallel to the planet velocity both before and after the flyby, thus a turning angle of 180 ° , and hence a zero radius of periapsis. This indicates that we cannot expect to find a DEP-to-FLY-to-ARR trajectory composed of two very Hohmann-like transfer arcs (which are those associated to the best first guesses) when enforcing the constraint on the minimum radius of perigee. And indeed, when the best first guesses are used with the optimization algorithms, the best transfer options show very inclined interplanetary trajectories with respect to the ecliptic. This is due to two reasons: • An out-of-plane heliocentric trajectory allows for a smaller required turning angle, and thus can be associated to a higher r p which is respectful of the constraint. • When the planets are aligned such to allow for a Hohmann transfer (i.e. the transfer arc is approximately 180 ° ) the Lambert solver ( lambertMR.m ) is able to find a very inclined transfer plane. Of course these solutions are associated with a large ∆ v tot (in the order of 40-80 km/s) since they require a double plane change. Instead, the in-plane heliocentric trajectories that will be associated to a smaller turning angle will be secant trajectories (i.e. the flyby planet is not met at the apoapsis of the first transfer arc). This case corresponds to the actual best options found, which are associated to ”worse” first guesses. Still, evaluating the difference between the first guesses on the departure, flyby and arrival dates and their associated optimal values, and taking the mean of these differences for the first 10 best transfer options, one would get: ∆ t dep = 41 1 days ∆ t f ly = 63 1 days ∆ t arr = 83 7 days (7) Computing their relative importance with respect to the associated planet period: ∆ t dep T DEP = 46 7% ∆ t f ly T F LY = 28 8% ∆ t arr T ARR = 12 2% (8) This shows that the first guesses are not very accurate for the departure date, while the accuracy is increased for the flyby date and it’s further improved for the arrival date. The poor accuracy in the departure date guess can be associated to the short orbital period of Mercury, and its consequent rapid change of phasing with respect to Venus. Finally, the best transfer option couldn’t be found using the chosen optimization algorithm with the discarded first guesses (ideal and refined Hohmann transfer options). This shows that the great focus that has been given to the search of good first guesses really helped in finding the best solution, and it has also given more insight on the physics behind the problem under study and allowed for a more detailed interpretation of the final results. 10 Part II Planetary Explorer Mission 4 Ground tracks 4.1 Initial orbital values The first task to accomplish is to choose the initial keplerian elements values. It has been decided to make the satellite trajectory begin just above the Eiffel tower in Paris. a (km) e i ( ° ) RAAN ( ° ) w ( ° ) f ( ° ) 38455 0.6122 74.9912 345 51 0 Table 1: Table of the initial keplerian elements 4.2 Perturbations In this part, the only perturbation that is taken into account is the J2 effect (second zonal harmonic). This perturbation is originated by the oblateness of the Earth (which is not a perfect sphere but it’s slightly flattened at its poles). It is recalled that J 2 = 0 1082626925638815 10 − 2 4.3 Unrepeated ground tracks It is observed from Fig-6 that the ground tracks advance westwards by an angle ∆ λ such that : ∆ λ = T · w E = 46 47 ◦ W (9) Where : • T = 75048 s : period of revolution of the satellite • w E = 7 2916 10 − 5 rad/s : angular velocity of the Earth Thus, taking into account only J2 perturbation, it is possible to determine the influence of the J2 effect on the spacecraft by computing the error between the position for perturbed and unperturbed cases. 11 (a) over an orbit (b) over a day (c) over ten days Figure 6: Unrepeated ground tracks with J2 effects 4.4 Repeated ground tracks In order to plot repeated groundtracks with a prescribed ratio between k revolutions of the satellite and m revolutions of the planet, it is mandatory to ensure that : m k = ̃ T ̃ T E (10) where : • ̃ T : satellite nodal period. • ̃ T E : Greenwich nodal period. • m = 5, k= 6: Respectively the desired number of revolutions of the Earth and the satellite after which the ground track shall be repeated. In the 2BP the numerator ̃ T is simply the satellite orbital period, while ̃ T E is the Earth sidereal day = 23 h 56 m 18 s . In the case of perturbed orbits we have to take into accounts new terms. Considering the J2 effect, the new value of the perturbed orbit semi-major axis that allows for a repeating ground track can be found by solving the following equation: m k = w E − ̇ Ω n + ̇ w + ̇ M (11) Where ̇ Ω, ̇ ω and ̇ M are respectively the first derivatives of the RAAN, of the argument of perigee and of the mean anomaly, that are functions of ( a, e, i, R ⊕ , J 2 , μ ⊕ ). 12 (a) Repeated ground track over 20 orbits (b) Repeated ground track over 72 orbits Figure 7: Repeated ground tracks in respectively 20 and 72 orbits Once the value of the new semi-major axis is found the (Cartesian) integration of the equations can be performed. The results have been plotted for both the perturbed and unperturbed cases and they are presented in Fig-7. What is notable is that the ground tracks do not exactly repeat themselves. To be convinced by this point, the maximum angular difference between the perturbed and un- perturbed ground tracks has been measured. This results are shown in Table 2 and 3. Null latitudes help determine the longitude difference at each passage between perturbed and unperturbed trajectories. According to the former repeated ground tracks, null latitudes match to points where the error angle (in longitude) is the greatest. Propagation Null-latitude First passage Last passage No J2 J2 Error No J2 J2 Error 15 days 115.76 ° 115.74 ° 0.02 ° -174.32 ° -175.23 ° 0.91 ° 60 days 115.82 ° 115.80 ° 0.02 ° -39.50 ° -51.94 ° 12.44 ° Table 2: Table of the longitudes for UNREPEATED ground track, computed at null latitude passage, with two propagation times of 15 and 60 days Propagation Null-latitude First passage Last passage No J2 J2 Error No J2 J2 Error 15 days 117.90 ° 117.89 ° 0.01 ° -6.02 ° -9.07 ° 3.05 ° 60 days 117.80 ° 117.85 ° 0.05 ° -5.99 ° -17.26 ° 11.27 ° Table 3: Table of the longitudes for REPEATED ground track, computed at null latitude passage, with two propagation times of 15 and 60 days Conclusion : 1. The longitude difference between perturbed and unperturbed, for both repeated and unrepeated ground tracks grows with time. This difference is called the accuracy error. 2. The repeated ground track does not repeat exactly as it can be noticed Fig-7. Rea- sons that could possibly explain this phenomenon are the tolerance of the integrator and the expressions for ̇ Ω, ̇ ω and ̇ M , which are by themselves approximations. 13 5 Orbit propagation 5.1 Assigned perturbations In addition to the J2 effect exposed in part-4 on the groundtracks, the SRP pertur- bation has been assigned for this study. The perturbing effect is caused by the photons emitted by the Sun hitting the spacecraft. The formula expressing the SRP acceleration, obtained from Curtis[1] (12.96), is: p = − ν S c C R A S m ˆ u = − p SR ˆ u (12) With: • ν : Coefficient equal to 1 if the sun is lighting the spacecraft, 0 otherwise. See Algorithm 12 3 from Curtis[1] to see how it is implemented. • S c : Energy flux from the sun. ( S c = 4 56 × 10 − 6 N/m 2 at a distance of 1AU). • C R : Drag coefficient. A value of 1.2 has been considered. • A S m : Spacecraft area-to-mass ratio relative to the Sun. A value of 4 m 2 /kg has been considered. • ˆ u: Unit vector pointing from the Earth to the Sun. The approximation of considering the unit vector Earth → Sun and not Spacecraft → Sun has been made considering that the two directions differs of only 0.02 degrees. 5.2 Cartesian propagation The Cartesian propagation is the method that has been used in part-4. It consists in solving the equations of motion while taking into account also the perturbing accelerations. To be consistent it has been decided to compute all the acceleration in the inertial (Earth- centered) RF. This choice is justified by the fact that at the end the keplerian elements will be computed with the function car2kep that has, as inputs, the position and velocity vector in the inertial RF. Hence it is obtained : ̈ r RF = − μ ⊕ r 3 r ︸ ︷︷ ︸ 2 BP + 3 2 J 2 μ ⊕ R 2 ⊕ r 4 x r (5( z r ) 2 − 1) y r (5( z r ) 2 − 1) z r (5( z r ) 2 − 3) ︸ ︷︷ ︸ J 2 ef f ect − p SR cos ( λ ) cos ( ) sin ( λ ) sin ( ) sin ( λ ) ︸ ︷︷ ︸ SRP ef f ect (13) with : • λ : Apparent solar ecliptic longitude (function of the JD2000 date). • : Obliquity angle (function of the JD2000 date). The parameters that are functions of the JD2000 date have been computed according to the ephemerides provided by Seidelmann[3], considering as initial date the ”25-dec-2020 00:13:13” (chosen arbitrarily). 14 5.3 Gauss propagation The Gauss propagation is a method that computes the time derivatives of the ke- plerian elements, and than the perturbing accelerations from those. As the Gauss prop- agation is a function of the perturbations, its form varies depending on the chosen RF. Consequently, as the SRP effect is given in the RSW frame in Curtis[1], it has been chosen to use the Gauss propagation in the RSW frame. da dt = 2 a 2 h ( e sin( θ ) a r + p r a s ) de dt = 1 h ( p sin( θ ) a r + (( p + r ) cos θ + re ) a s ) di dt = r cos( θ + ω ) h a w d Ω dt = r sin( θ + ω ) h sin i a w dω dt = 1 he ( − p cos( θ ) a r + ( p + r ) sin( θ ) a s ) − r sin( θ + ω ) cos i h sin i a w dθ dt = h r 2 + 1 eh ( p cos( θ ) a r − ( p + r ) sin( θ ) a s ) (14) where : a r a s a w = − 3 2 J 2 μ ⊕ R 2 ⊕ r 4 1 − 3 sin( i ) 2 sin( f + ω ) 2 sin( i ) 2 sin 2( f + ω ) sin(2 i ) sin( f + ω ) ︸ ︷︷ ︸ J 2 ef f ect − p SR u r u s u w ︸ ︷︷ ︸ SRP ef f ect Note : u r , u s , u w are function of λ, , i, Ω , ω, θ . See equations (12.105) from Curtis[1] for their complete definition. 5.4 Crash possibility In order to avoid any kind of singularities and no-sense results, it has been im- plemented an ”Impact Event”, which stops the propagation (whether it is Cartesian or Gauss) if the magnitude of the position vector of the spacecraft is smaller than R ⊕ + h , where h is the minimum allowed altitude, which has been set to 100 km (official limit of space). Its implementation is pretty straightforward, and served as quick check for the correctness of the results. 5.5 Comparison and Analysis Once both propagation methods have been implemented the comparison between them has been carried out and it is shown in Fig. 8. At first glance the two methods seem to match perfectly. Also, an oscillatory phenomenon, which is more detectable on the semi-major axis but still present on the other keplerian elements, can be noted. This aspect will be discussed in Section-6. In order to analyze more in detail the difference a plot of the relative error (with respect to Gauss propagation) between the keplerian elements obtained with the two propagation methods has been made, and it’s visible in Fig. 9. The order of magnitude of each relative error is shown in Table-4. ∆( a ) /a Gauss ∆( e ) /e Gauss ∆( i ) /i Gauss ∆(Ω) / Ω Gauss ∆( ω ) /ω Gauss ∆( θ ) /θ Gauss 10 − 8 10 − 9 10 − 11 10 − 10 10 − 9 10 − 6 Table 4: Order of magnitude of the relative error between Gauss and Cartesian propagation with respect to Gauss 15 Figure 8: Cartesian & Gauss propagation over a year with initial keplerian elements from section-1.1 and initial time as mentioned in section-5.2 Figure 9: Relative error, with respect to Gauss propagation, between Cartesian & Gauss propagation for each keplerian element 16 From Table-5 it can be seen that the computational time is not affected by the discretiza- tion, rather by the total duration of the propagation. Moreover, the Cartesian propagation is always slower than Gauss, and this difference increases as the total duration grows. T[years] N 1000 5000 10000 Gauss Cartesian Gauss Cartesian Gauss Cartesian 1/2 49.5s 64.5s 55.3s 64.4s 55.5s 65.9 1 105.3s 135.4s 104.7s 125.2s 108.8s 132.5s 2 212.7s 256.3s 213.9s 260.1s 190.6s 243.1s Table 5: Computational time as a function of discretization and total duration of propagation (a) Evolution of the orbit under J 2 and SRP over 10 years with Gauss propagation (b) Initial and final orbit under J 2 and SRP over 10 years with Gauss propagation Figure 10: 3D visualisation of the evolution of the orbit with Gauss propagation 6 Filtering 6.1 Unfiltered results In this part, only the Gauss propagation method will be used since it has been observed (see Tab-5) that it is the fastest method between the two, and also since the relative error between the two methods, as shown in Fig. 9, is small. The results will be computed for a one year duration. The unfiltered results have already been presented in Fig-8. 6.2 Filtering principle As it can be seen in Fig-11, only slight oscillations affect the eccentricity, RAAN, inclination, argument of perigee and true anomaly. Instead, more significant oscillations appear for the semi-major axis. Thus, the goal is to find a way to remove the undesired frequencies that generate this oscillatory behaviour. 17 6.3 Construction of the filter The decision has been to focus on MATLAB’s function movmean whose effect can be assimilated to a low pass filter. Its use requires uniformly spaced data in time. Its implementation is the following : A f iltered = movmean ( A, a ) (15) Where A is a vector and a a window. For each element of A , movmean will substitute it with the mean evaluated on the window with the selected length a and centered on this element. Three different methods have been studied in this part. Two are based on the inputs, and one on the output, in which the windows will be computed as: window = ceil ( size · T ∆ t ) (16) ∆ t = ceil ( n days N − 1 ) (17) Where T is the cut-off period, ∆ t the uniform time step, n days the number of days in seconds, size an integer that enlarge the basic window and N the number of points of the simulation. 1 st method: The cut-off period is selected as the solar cycle period. The corresponding frequency, and consequently the ∆ t , is than found. 2 nd method: The cut-off period is selected as the spacecraft nodal period. The corre- sponding frequency, and consequently the ∆ t , is than found. Parameter Method 1 st 2 nd Solar Cycle J2 T cutof f 11 years 75048s number of points 250000 55 size 2 4 Window 500000 220 Table 6: Parameter values for the filter in the two first methods for N = 500000 & T total = 22 years 3 rd method: Analysis of the plots in Fig-11 and graphical individuation of several dis- turbing periods As it can be seen, the 1 st and 2 nd methods are based only on the frequency of the pertur- bations. However,the ranges of the disturbing frequencies are not necessarily related to the frequency of the perturbations. Thus the third approach has been preferred. As seen on Tab-7, three major frequencies associated to the semi-major axis can be identified. As regards other keplerian elements, a common cut-off frequency, based on a mean of a one-day-period, is considered. 18 Element a (km) e (-) i ( ° ) RAAN ( ° ) w ( ° ) f ( ° ) Period (days) 0.75 9.7 31.7 0.91 0.94 1.17 1.02 1.14 Cut-off period (days) 1 9.7 31.7 1 1 1 1 1 window 110 1644 3474 110 110 110 110 110 Table 7: Selected parameters for the filter with size = 4 Figure 11: Unfiltered Keplerian elements, with a focus on the disturbing periods Note: For the semi-major axis, three different filters were used with respectively 1, 9.7 and 31.7 days as cut-off period. This superposition is due to the fact that several parasite frequencies affected the unfiltered plot. 6.4 Filtered results While operating the filtering of the keplerian elements the Gibbs phenomenon was managed and the oscillations at the boundaries avoided. Those boundaries effects - which occurred at the beginning and at the end of each vector of keplerian elements - have been avoided by multiplying the window of each element by a size factor. The filtered results are presented Fig-12. In conclusion, finding an accurate filter to remove the disturbing frequencies has led to the development of different methods that evolved throughout the study. In the end, the graphical approach on the unfiltered results turned out to be the most accurate one. 19