# Flight and Orbital Mechanics

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1 Flight and Orbital Mechanics Lecture slides Challenge the future 1

2 Material for exam: this presentation (i.e., no material from text book). Sun-synchronous orbit: used for a variety of earth-observing missions.

3 Introduction picture.

4 In essence, this is about understanding an important component in the space environment, and dealing with the consequences (in a positive way!).

5 This topic relies to a certain extent on the theory of Kepler orbits as lectured in ae The material covered in this powerpoint presentation needs to be studied for the exam; more (background) information on all topics can be found in Spacecraft Systems Engineering by Fortescue, Stark and Swinerd.

6 Some examples of relevant questions that you should be able to answer after having mastered the topics of these lectures.

7 The values mentioned in this table hold for the GOCE satellite (altitude 250 km), and are meant for illustration purposes only; actual acceleration values depend on specific circumstances (position, level of solar activity,.). More on perturbations in one of the next lecture hours.

8 Selecting a proper reference system and a set of parameters that describe a position in 3 dimensions is crucial to quantify most of the phenomena treated in this chapter, and to determine what a satellite mission will experience. Option 1: cartesian coordinates, with components x, y and z. Option 2: spherical coordinates, with components r (radius, measured w.r.t. the center-of-mass of the central object; not to be confused with the altitude over its surface), δ (latitude) and λ (longitude).

9 Pericenter is the general expression for the position at closest distance to the focal center; when the object orbits the Earth, one generally speaks of the perigee, and when it orbits the Sun, it is named perihelion. Similar expressions for the point at farthest distance apocenter -> apogee, apohelion.

10 The time of passage of a well-defined point in the orbit (e.g., the pericenter) is indicated by t p or, equivalently, τ ( tau ). Knowing this value, one can relate the position in the orbit to absolute time (cf. following sheets). The inclination i is the angle between the orbital plane and a reference plane, such as the equatorial plane. It is measured at the ascending node, i.e., the location where the satellite transits from the Southern Hemisphere to the Northern Hemisphere, so by definition its value is between 0 and 180. The parameters Ω and ω can take any value between 0 and

11 Parameter G is the universal gravitational constant ( m 3 /kgs 2 ), m sat represents the mass of the satellite, r is the distance between a mass element of the Earth and the satellite (written in bold it is the vector, directed from the mass element to the satellite), r is the distance between the satellite and the center-of-mass of the Earth (equal to r sat in the picture apologies for difference in notation), ρ is the mass density of an element dv of the Earth [kg/m 3 ], V is the total volume of the Earth, M is the total mass of the Earth ( kg), U is the gravity potential (last equation: for a symmetric Earth), and µ is the gravitational parameter of the Earth (=G M earth = m 3 /s 2 ). The last equation shows how to compute the radial acceleration. Similar expressions can be used to derive the acceleration in x, y and z direction (that is: by taking the partial derivatives w.r.t. these parameters, and adding a minus sign).

12 The Earth is irregular in shape and mass distribution; this picture illustrates the geoid, i.e. an equipotential surface w.r.t. a 3D ellipsoid; offsets are about 80 m in either direction. 11

13 The first equation gives the classical description of the gravity field potential, with respectively the main term, the zonal terms (independent of longitude) and the sectorial and tesseral terms (cf. next sheet). Parameter R e is the Earth s equatorial radius ( m); the satellite position is described in an Earthfixed (as in: co-rotating) reference frame by the radius r (w.r.t. the center of Earth), the latitude δ (w.r.t. the equator) and the longitude λ (w.r.t. an Earthfixed reference meridian: the 0 meridian crossing Greenwich). The parameters J n, J n,m and λ n,m are scaling and orientation coefficients of the gravity field model, respectively. J 2 is about , whereas the other J values are O(10-6 ). When developing a specific term of the gravity field potential, it is strongly recommended to develop the Legendre function or potential as function of the general argument x first, and substitute the actual argument sinδ at the very end. The (infinite long) series expansion, with different frequencies as determined by the Legendre polynomial/function, is best compared to a Fourier series to describe an arbitrary signal. 12

14 See previous sheet for role of degree n, order m and coefficient J n,m. EGM is the abbreviation of Earth Gravity Model.

15 J 2 is about , whereas the other J values are O(10-6 ). It is related to the equatorial bulge, the ring of extra material around the equator of the Earth. U 2 is the contribution of the flattening of the Earth to the total potential U (so, directly related to the value of J 2 ). It can also be written as U 2,0.

16 J 2 is about , whereas the other J values are O(10-6 ). U 2 can also be written as U 2,0. First develop P 2 as a function of the general argument x, and then substitute this with the actual argument sinδ. In these equations, parameter x does NOT represent the x-position of our spacecraft, but it is a general argument (as in y=f(x) ).

17 J 2 is about , whereas the other J values are O(10-6 ). U 2 can also be written as U 2,0. It can be interpreted as potential energy (per unit mass). In this illustration, it is evaluated at 1000 km altitude. Clearly, U 2 is independent of longitude and symmetric in latitude (see equation).

18 Straightforward treatment of gravity field components. Parameters r, δ and λ represent radius (w.r.t. center of Earth), geocentric latitude (w.r.t. equatorial plane) and longitude (w.r.t. Earth-fixed reference), respectively. Accelerations are derived by taking the derivative w.r.t. a parameter that expresses length in the required direction. Since an infinitesimal distance in NS direction is equal to rdφ (angle φ in radians, measured along the great circle), the expression for the acceleration in NS direction contains a scaling factor 1/r. In a similar fashion, an infinitesimal distance in EW direction, at a certain latitude δ, is equal to r cos(δ)dφ (the total length of a latitude circle is equal to 2πrcos(δ)). Therefore, the expression for EW acceleration contains a scaling factor 1/(r cos(δ)). When deriving expressions for the accelerations due to a particular term, always develop the Legendre functions P n,m and P n as a function of the general parameter x first, and not until the very end substitute x with the real argument sin δ.

19 Application of recipe on previous sheet. Again, for altitude of 1000 km. Radial acceleration due to J 2 is independent of longitude, and symmetric in latitude.

20 Idem. The NS acceleration due to J 2 is also independent of longitude, and antisymmetric in latitude. Note: the direction North-South is generally different from the z-direction, since the former follows the curvature of the surface of the Earth, whereas the latter is always pointed along the rotation axis of the Earth.

21 Idem. The EW acceleration due to J 2 is zero, since U is independent of longitude (it is symmetric around the z-axis, cf. illustration on sheet 14).

22 This is the most complicated derivation that you can encounter in this course ae2-104 (or what you can expect during an exam.). When doing such a derivation in an exam, the full derivation should be given (and not with the.. text as shown here, for the sake of brevity). First develop P 3 and P 3,2 as function of the general argument x, and then substitute this by the actual argument sinδ.

23 This is the most complicated derivation that you can encounter in this course ae2-104 (or what you can expect during an exam.). When doing such a derivation in an exam, the full derivation should be given (and not with the.. text as shown here, for the sake of brevity).

24 The collection of these (and other) terms can be regarded as corrections to the 1 st order model of a spherical, radially symmetric Earth. The equatorial bulge is represented by the J 2 (or J 2,0, as depicted in this plot) term, which is the dominant correction term in the Earth s gravity field model. The real gravity field can be approximated by an (in principle infinite) series of individual terms, with different characteristics (cf. a Fourier analysis).

25 Changes in parameters after one complete revolution of the satellite around the central body (Earth). So, no secular J 2 -effect on semi-major axis, eccentricity nor inclination.

26 The net force acting on the satellite when traveling over the Southern hemisphere causes the trajectory to be bent in northern direction (i.e., the inclination of the orbit increases). When over the Northern hemisphere, the same thing happens but with reverse sign the net effect on the inclination is zero, but the orientation of the orbit has shifted in a western direction (for inclinations smaller than 90º, as sketched here).

27 ANSWERS (TRY YOURSELF FIRST!!): 1. a r = - U/ r 2. a r = 3 µ J 2 R e2 r -4 ( sin 2 δ) 3. a r = ( sin 2 δ) [m/s 2 ] 4. Give sketch. Note: sketch is symmetric w.r.t. δ=0.

28 ANSWERS (TRY YOURSELF FIRST!!): 1. a NS = -1/r U/ δ 2. a NS = -3 µ J 2 R e2 r -4 sinδ cosδ 3. a NS = sinδ cosδ [m/s 2 ] 4. Give sketch. Note: sketch is anti-symmetric w.r.t. δ=0.

29 ANSWERS (TRY YOURSELF FIRST!!): 1. a r = - U/ r 2. a r = 4 µ J 3 R e3 r -5 (2.5 sin 3 δ 1.5 sinδ) 3. a r = (2.5 sin 2 δ 1.5 sinδ) [m/s 2 ] 4. Give sketch. Note: sketch is anti-symmetric w.r.t. δ=0.

30 ANSWERS (TRY YOURSELF FIRST!!): 1. a NS = -1/r U/ δ 2. a NS = -µ J 3 R e3 r -5 (7.5 sin 2 δ cosδ cosδ) 3. a NS = (7.5 sin 2 δ cosδ cosδ) [m/s 2 ] 4. Make sketch. Note: sketch is symmetric w.r.t. δ=0.

31 The ground track of a satellite is the succession of sub-satellite points (i.e., the projection of the satellite position on the surface of the Earth; altitude drops out). Raw sea-level observations need to be corrected for many phenomena, such as local tides, effects of local sea-floor geometry, etcetera. When evaluated at a single, well-defined spot, systematic errors in such corrections will drop out (whereas they can introduce errors when comparing observations at e.g. 30 km distance).

32 Overview of satellites equipped with altimeter instruments. Note the families Seasat - Geosat - GFO, ERS-1 ERS-2 ENVISAT, and TOPEX/Poseidon Jason-1 Jason-2. Satellites in red are US, satellites in green are European. GFO is the abbreviation for Geosat Follow-On. Examples of other satellites in Earthrepeat orbits (but not necessarily equipped with an altimeter instrument): SMOS, Cartosat, Sentinel-3.

33 L is longitude: measured in co-rotating Earth-fixed reference frame, positive in Eastward direction. Verify direction of shift in plot and definition of signs in equations. Eccentricity ~0 holds for 95% of actual spacecraft. T is the orbital period of the satellite [s]. T E is the rotational period of the Earth (sidereal day, [s]). Effect L 2 : cf. expressions on sheet 24.

34

35 The value for k follows from the top-level requirements. The value for j is a reasonable guess, knowing that the real satellite is expected to fly at a a certain altitude. What are the options for (a,e,i)?

36 3 examples for (j,k) = (14,1). Here we follow the approach of assuming a value for the semi-major axis, and solving for inclination. Can we identify possible combinations (a,i) (eccentricity e is assumed to be 0) that satisfy the requirements?

37 Examples for zero-eccentricity orbits, with a 1-day repeat cycle. A higher altitude corresponds with a longer orbital period, which means less revolutions per day. Circular orbits: altitude is semi-major axis minus Earth radius ( km). The black stars indicate the two possible solutions as derived on the previous sheet.

38 Examples for zero-eccentricity orbits, with a 3-day repeat cycle. A higher altitude corresponds with a longer orbital period, which means less revolutions per day. Circular orbits: altitude is semi-major axis minus Earth radius ( km). Clearly, going from a 1-day repeat period to a 3-day repeat period increases the range of options. Compare some lines with the plot on the previous sheet: 45 revolutions in 3 days is exactly identical to 15 revolutions in 1 day.

39 ERS-1 is the abbreviation for European Remote-Sensing satellite. ESA was extremely proud of this one, it s a classical one!

40 In standard configuration, the repeat period is 35 days. The repeat interval is typically traded against geometric resolution (i.e., the distance between neighboring ground tracks).

41 Zooming in on plot on previous sheet. 1 st order estimate misses real value by about 2 km..

42

43 42

44 Zooming in on plot on previous sheet. 1 st order estimate also misses real value by about 2 km Agreement w.r.t. reality is comparable with that of ERS-1 (same orbit altitude more or less the same additional disturbances of perturbations, drag, ).

45 ANSWERS (TRY YOURSELF FIRST!!): a) i = 24.0 b) i = c) T = 6207 s = min. d) Question a): T repeat = min = hr = days. Question b): T repeat = min = hr = days. Earth s flattening has opposite effects on orbits with inclination < 90 (J 2 precession acts in Westward direction) and orbits with inclination > 90 (J 2 precession acts in Eastward direction).

46 Having a satellite in a Sun-synchronous orbit does NOT mean that it is in constant sunlight throughout its mission!

47 Inertially stable orbit: no perturbations, so no changes in orientation. Sunsynchronous orbit: effect is fully due to J 2.

48 Overview of satellites orbiting Earth in Sun-synchronous orbits. Note the families Seasat - Geosat - GFO, ERS-1 ERS-2 ENVISAT, and TOPEX/Poseidon Jason-1 Jason-2. Satellites in red are US, satellites in green are European. GFO is the abbreviation for Geosat Follow-On.

49 Direct, 1-on-1 relation between semi-major axis and inclination. T is the orbital period of the satellite [s]; T E is the length of a sidereal day [s]; T ES is the orbital period of the Earth around the Sun [s]. Again, compare with equations on sheet 24. Note: inclination is always larger than 90.

50 ERS-1 is the abbreviation for European Remote-Sensing satellite.

51 No significant difference.

52 ANSWERS (TRY YOURSELF FIRST): a) The orientation of the orbital plane w.r.t. the direction towards the Sun is constant over time. b) a 3.5 = -(3/4π) J 2 R e2 sqrt(µ) T ES cos(i) c) a = ( cos(i) ) -3.5 [km] make table with some numbers sketch

53 Direct, 1-on-1 relation between semi-major axis and (j,k) combination; inclination follows directly from solution for semi-major axis. T is orbital period of satellite [s]; T E is the length of a sidereal day [s]; T ES is the orbital period of the Earth around the Sun [s]. Again, compare with equations on sheet 24.

54 Direct, 1-on-1 relation between semi-major axis and (j,k) combination; inclination follows directly from solution for semi-major axis.

55 Direct, 1-on-1 relation between semi-major axis and (j,k) combination; inclination follows directly from solution for semi-major axis.

56 ANSWERS (TRY YOURSELF FIRST): a) See 3 rd -previous sheet. b) a = km. c) i = 98.53

57

58 Straightforward application of recipe.

59 The J 2 contribution is important for the exact determination of the altitude for which the orbital period is equal to the length of a sidereal day (compared to an unperturbed Kepler orbit, the difference is about 500 m ): the value of the semi-major axis a has to be chosen such that there is equilibrium between the centripetal acceleration ω 2 r and the physical accelerations g (due to the central gravity field of Earth) and a r,j2 (the radial acceleration due to J 2 ). J 2,2 is important for station keeping (see next sheets).

60 J 2,2 -effect is in many cases ignored for LEO satellites, but is elementary for GEO spacecraft (note opposite direction of acceleration and effect; why?). What defines equilibrium points? What defines stability/instability of equilibrium points? Can we ignore J 2? Here, the effect of J 2,2 in along-track direction is evaluated only; it of course has a full, 3-dimensional effect (r,δ,λ or x,y,z).

61 ANSWERS (TRY YOURSELF FIRST!!): 1. a EW = -1/(r cosδ) U/ λ 2. a EW = -6 µ J 2,2 R e2 r -4 cos δ sin (2(λ-λ 2,2 )) 3. r GEO = km 4. a EW = sin (2(λ+14.9)) [m/s 2 ] 5. Located at -14.9, 75.1, and (East longitude) 6. Stable: 75.1 and ; unstable: and

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