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Transcript of Ae2104 orbital-mechanics-slides 9
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1Challenge the future
Flight and Orbital Mechanics
Lecture slides
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1AE2104 Flight and Orbital Mechanics |
Flight and Orbital Mechanics
AE2-104, lecture hours 17+18: Perturbations
Ron Noomen
October 25, 2012
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2AE2104 Flight and Orbital Mechanics |
Example: solar sail spacecraft
[Wikipedia, 2010]
Questions:• what is the
purpose of this mission?
• where is the satellite located?
• why does it use a solar sail?
• ….
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3AE2104 Flight and Orbital Mechanics |
Overview
• Orbital mechanics (recap)
• Irregularities gravity field• Third-body perturbations• Atmospheric drag• Solar radiation pressure• Thrust
• Relativistic effects• Tidal forces• Thermal forces• Impacts debris/micrometeoroids
Special missions only; not treated here
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4AE2104 Flight and Orbital Mechanics |
Learning goals
The student should be able to:• mention and describe the various perturbing forces that
may act on an arbitrary spacecraft;• quantify the resulting accelerations;• make an assessment of the importance of the different
perturbing forces, depending on the specifics of the mission (phase) at hand.
Lecture material:• these slides (incl. footnotes)
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5AE2104 Flight and Orbital Mechanics |
2-dimensional Kepler orbits
[Seligman, 2010]
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6AE2104 Flight and Orbital Mechanics |
2-dimensional Kepler orbits
a: semi-major axis [m]
e: eccentricity [-]
θ: true anomaly [deg]
E: eccentric anomaly [deg]
[Cornellise, Schöyer and Wakker, 1979]
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7AE2104 Flight and Orbital Mechanics |
2-dimensional Kepler orbits: general equations
)1(;)1(;cos1cos1
)1( 2earear
ep
eea
r ap
arVEEE potkintot
22
2
rV
arV
arV esccirc
2;;
122
32
aT
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8AE2104 Flight and Orbital Mechanics |
2-dimensional Kepler orbits: equations (cnt’d)
ellips: 0 ≤ e < 1 a > 0 Etot < 0
)cos1(
cos1
sin1
)0(
sin
2tan
1
1
2tan
3
Eear
iEe
iEeiEMiEiE
ttnM
EeEM
E
e
e
an
satellite altitude
[km]
specific energy
[km2/s2/kg]
launch
platform
0 -62.4
SpaceShipOne 100+
(culmination)
-61.4
imaginary sat 100 -30.8
Envisat 800 -27.8
LAGEOS 5900 -16.2
GEO 35900 -4.7
Keple
r’s
Equation
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9AE2104 Flight and Orbital Mechanics |
2-dimensional Kepler orbits: equations (cnt’d)
ellips: 0 ≤ e < 1 a > 0 Etot < 0
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10AE2104 Flight and Orbital Mechanics |
2-dimensional Kepler orbits: equations (cnt’d)
parabola: e = 1 a = ∞ Etot = 0
rescVV
pn
222
3
2
3tan6
1
2tan
2
1 M
cos1
pr
)0( ttnM
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11AE2104 Flight and Orbital Mechanics |
2-dimensional Kepler orbits: equations (cnt’d)
hyperbola: e > 1 a < 0 Etot > 0
3)( an
2tanh
1
1
2tan
F
e
e
)0(
sinh
ttnM
FFeM
22222
)cosh1(
Vr
VescVV
Fear
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12AE2104 Flight and Orbital Mechanics |
3-dimensional Kepler orbits
i: inclination [deg]
Ω: right ascension of the ascending node,
or longitude of the ascending node [deg]
ω: argument of pericenter [deg]
u = ω + θ : argument of latitude [deg]
[Cornelisse, Schöyer and Wakker, 1979]
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13AE2104 Flight and Orbital Mechanics |
coordinates transformations
1) from spherical (r,λ,δ) (λ in X-Y plane; δ w.r.t. X-Y
plane) to cartesian (x,y,z):
x = r cos δ cos λ
y = r cos δ sin λ
z = r sin δ
2) from cartesian (x,y,z) to spherical (r,λ,δ):
r = sqrt ( x2 + y2 + z2 )
rxy = sqrt ( x2 + y2 )
λ = atan2 ( y / rxy , x / rxy )
δ = asin ( z / r )
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14AE2104 Flight and Orbital Mechanics |
Orbital perturbations: introduction
Questions:
• what forces?
• magnitude of forces?
• how to model/compute in satellite orbit?
• analytically? numerically?
• accuracy?
• efficiency?
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15AE2104 Flight and Orbital Mechanics |
Inclusion in orbit modeling
Option 1: include directly in equation of motion
Option 2: express as variation of orbital elements
(not further treated here; cf. ae4-874 and ae4-878)
etceterardbodysolraddraggravmaintd
d 32
2aaaaa
x
r
pNSe
p
a
dt
dae.g.
sin
2 2
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16AE2104 Flight and Orbital Mechanics |
Perturbations:
• Irregularities gravity field
• Third body
• Atmospheric drag
• Solar radiation pressure
• Thrust
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17AE2104 Flight and Orbital Mechanics |
Perturbations:
• Irregularities gravity field
• Third body
• Atmospheric drag
• Solar radiation pressure
• Thrust
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18AE2104 Flight and Orbital Mechanics |
Already treated in lecture hours 15+16
e.g., North-South acceleration due to J2:
e.g., East-West acceleration due to J2,2:
Irregularities gravity field
)cos()sin(4223 reRJacc
))2,2(2sin()cos(3422,26 reRJacc
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19AE2104 Flight and Orbital Mechanics |
Perturbations:
• Irregularities gravity field
• Third body
• Atmospheric drag
• Solar radiation pressure
• Thrust
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20AE2104 Flight and Orbital Mechanics |
Third-body perturbations
333d
r
d
sdr
sddMGs
sr
MGs
rrrr
Attractional forces (by definition):
• satellite <-> Earth
• satellite <-> perturbing body
• Earth <-> perturbing body
Reference frame
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21AE2104 Flight and Orbital Mechanics |
Third-body perturbations (cnt’d)
3 3 3
ps pmains s p
s ps p
MG G M
r r r
r rr r
Attractional forces (practice):
• Earth attracts satellite
• perturbing body attracts satellite
• perturbing body attracts Earth
• net effect counts
Msat << ME, MS
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22AE2104 Flight and Orbital Mechanics |
Third-body perturbations (cnt’d)
33
p
p
ps
psp
r
r
r
rpa
perturbing acceleration ap:
maximum when on straight line:
3
max
2p p s
main main p
a m r
a m r
main sat3rd body
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23AE2104 Flight and Orbital Mechanics |
Third-body perturbations (cnt’d)
2 situations:
1. heliocentric (i.e., orbits around Sun)
2. planetocentric (i.e., orbits around Earth)
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24AE2104 Flight and Orbital Mechanics |
Third-body perturbations (cnt’d)
Conclusions:
• influence increases with distance to Sun
• perturbation O(10-7) m/s2
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25AE2104 Flight and Orbital Mechanics |
Third-body perturbations (cnt’d)
Conclusions:
• influence Sun decreases with distance to Sun
• influence planets increases with distance to Sun
• acceleration from Sun O(10-2) m/s2; dominant (central body!!)
• near planet: 3rd body becomes dominant
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26AE2104 Flight and Orbital Mechanics |
Third-body perturbations (cnt’d)
Conclusions:
• in Solar System: Sun dominant
• near Earth: Earth itself dominant
• central body vs. 3rd body perturbation ??
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27AE2104 Flight and Orbital Mechanics |
Third-body perturbations (cnt’d)
Sphere of Influence:
• area around planet where gravity from planet is dominant (compared with gravity of other celestial bodies)
• 3-dimensional shape
• boundary
• 1st-order approximation: sphere with constant radius
• definition for determination location:
relative acceleration w.r.t. system 1 = relative acceleration w.r.t. system 2
EarthSun
sat
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28AE2104 Flight and Orbital Mechanics |
Third-body perturbations (cnt’d)
kmSunM
EarthMSunEarthdistEarthSoIr
rdM
mainMrdrSoIr
mainSunacc
rdEarthacc
mainEarthacc
rdSunacc
000,930
4.0
,
:body rd3Sun body,main Earth :example
4.0
33
:derivationwithout
,
3,
,
3,
Moon is at 384,000 km….
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29AE2104 Flight and Orbital Mechanics |
Third-body perturbations (cnt’d)
planet Sphere of Influence
[km] [% AU] [% distance
planet-Sun]
Mercury 1.1 × 105 0.08 0.2
Venus 6.2 × 105 0.4 0.6
Earth 9.3 × 105 0.6 0.6
Mars 5.8 × 105 0.4 0.3
Jupiter 4.8 × 107 32.2 6.2
Saturn 5.5 × 107 36.5 3.8
Uranus 5.2 × 107 34.6 1.8
Neptune 8.7 × 107 57.9 1.9
Pluto 3.2 × 106 2.1 0.1
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30AE2104 Flight and Orbital Mechanics |
Third-body perturbations (cnt’d)
2 situations:
1. heliocentric (i.e., orbits around Sun)
2. planetocentric (i.e., orbits around Earth)
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31AE2104 Flight and Orbital Mechanics |
Third-body perturbations (cnt’d)
center of ref. frame
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32AE2104 Flight and Orbital Mechanics |
Third-body perturbations (cnt’d)
Conclusions:
• influence of Sun as 3rd body increases with distance from Earth
• effective 3rd body acceleration by Sun O(10-6) m/s2
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33AE2104 Flight and Orbital Mechanics |
Third-body perturbations (cnt’d)
Question 1:
a) Compute the dimension of the Sphere of Influence of the Earth (when the Sun is considered as the perturbing body). The SoI is given by the following general equation:
b) What is the value of the radial attraction exerted by the Earth at this distance? (if you were unable to make the question a, use a value of 1 × 106 km for this position).
c) What is the effective gravitational acceleration by the Sun at this position? Assume that Earth, Sun and satellite are on a straight line.
d) What is the relative perturbation of the solar attraction, compared to that of the main attraction of the Earth?
Data: 1 AU = 149.6 × 106 km, mSun = 2.0 × 1030 kg, mEarth = 6.0 × 1024 kg, μSun = 1.3271 × 1011
km3/s2, μEarth = 398600 km3/s2.
Answers: see footnotes below (BUT TRY YOURSELF FIRST!!)
4.0
33
rdM
mainMrdrSoIr
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34AE2104 Flight and Orbital Mechanics |
Third-body perturbations (cnt’d)
Conclusions:
• influence of Moon as 3rd body increases with distance from Earth
• effective 3rd body acceleration by Moon O(10-5) m/s2 at GEO
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35AE2104 Flight and Orbital Mechanics |
Third-body perturbations (cnt’d)
Conclusions:
• Earth dominant (central body; within SoI)
• Moon most important 3rd body; Sun directly after
• effect of planets about 4 orders of magnitude smaller
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36AE2104 Flight and Orbital Mechanics |
Third-body perturbations (cnt’d)
celestial
body
ΔV-budget
for GEO sat [m/s/yr]
Moon 36.93
Sun 14.45
[Wikimedia, 2010]
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37AE2104 Flight and Orbital Mechanics |
Third-body perturbations (cnt’d)
Question 2:
Consider a hypothetical planet X with mass 5 × 1025 kg, orbiting the Sun in a circular orbit with radius 3 AU. The orbital plane coincides with the ecliptic (i.e., the orbital plane of the Earth).
a) Make a sketch of the situation when the gravitational attraction of this planet X on satellites around the Earth is largest.
b) Idem for the case when this would be smallest.
c) Compute the maximum and minimum perturbing acceleration due to this planet X, on a geostationary satellite (radius orbit is 42200 km).
Data: G = 6.673 × 10-11 m3/kg/s2; μEarth = 398600 km3/s2; 1 AU = 149.6 ×106 km
Answers: see footnotes below (BUT TRY YOURSELF FIRST!!)
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38AE2104 Flight and Orbital Mechanics |
Third-body perturbations (cnt’d)
Question 3:
The treatment of the motion of a satellite is driven by the fact whether the vehicle is inside the Sphere of Influence (SoI) or not.
a) Describe the concept of the SoI, and give its mathematical (underlying) definition.
b) The dimension of the SoI can be approximated by the equation given below. Consider the Earth-Moon system: μEarth = 398600 km3/s2, μMoon = 4903 km3/s2, average distance Earth-Moon = 384,000 km. What is the radius of the SoI of the Moon?
c) Suppose that the Earth and Moon have equal masses. What would now be the radius of the SoI? What would it be from a physical point of view? Discuss the results.
ANSWERS: see footnotes below. TRY YOURSELF FIRST!
4.0
33
rdM
mainMrdrSoIr
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39AE2104 Flight and Orbital Mechanics |
Perturbations:
• Irregularities gravity field
• Third body
• Atmospheric drag
• Solar radiation pressure
• Thrust
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40AE2104 Flight and Orbital Mechanics |
Atmosphere
[Wertz, 2009]
[Wertz, 2009]
)/(exp;2
10
2 HhV
Vm
SCD V
adrag
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41AE2104 Flight and Orbital Mechanics |
Atmosphere (cnt’d)
after
[Wertz, 2009]:
Altitude [km] Atmospheric density [kg/m3]
minimum maximum
200 1.8 × 10-10 3.5 × 10-10
300 8.2 × 10-12 4.0 × 10-11
400 7.3 × 10-13 7.6 × 10-12
500 9.0 × 10-14 1.8 × 10-12
600 1.7 × 10-14 4.9 × 10-13
700 5.7 × 10-15 1.5 × 10-13
800 3.0 × 10-15 4.4 × 10-14
900 1.8 × 10-15 1.9 × 10-14
1000 1.2 × 10-15 8.8 × 10-15
1250 4.7 × 10-16 2.6 × 10-15
1500 2.3 × 10-16 1.2 × 10-15
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42AE2104 Flight and Orbital Mechanics |
Atmosphere (cnt’d)
[Wertz, 2009]:
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43AE2104 Flight and Orbital Mechanics |
Atmosphere (cnt’d)
altitudes of Mir and GFZ-1:
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44AE2104 Flight and Orbital Mechanics |
Atmosphere (cnt’d)
[Wertz, 2009]:
orbit 2
orbit 1
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45AE2104 Flight and Orbital Mechanics |
Atmosphere (cnt’d)
circular orbits:
Δa2π = -2π (CDA/m) ρ a2 [m]
ΔT2π = -6π2 (CDA/m) ρ a2 / V [s]
ΔV2π = π (CDA/m) ρ a V [m/s]
Δe2π = 0 [-]
L = - H / Δa2π [rev]
[Wertz, 2009]:
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46AE2104 Flight and Orbital Mechanics |
Atmosphere (cnt’d)
GOCE:
• launch date: March 17, 2009
• altitude: 250 km
• phase solar cycle?
• …..
[ESA, 2010]
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47AE2104 Flight and Orbital Mechanics |
Perturbations:
• Irregularities gravity field
• Third body
• Atmospheric drag
• Solar radiation pressure
• Thrust
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48AE2104 Flight and Orbital Mechanics |
Solar radiation pressure
m
A
m
A
c
SC 6105.4sat-Sun
sat-Sun2)sat/AU-Sun(
1)1(rad
r
r
ra
• amount of energy emitted by Sun at 1 AU distance: ≈1371 W/m2
• value hardly dependent on solar activity ”Solar Constant” (SC)
• Solar radiation pressure [N/m2]: SC / c
• energy reduces with distance w.r.t. Sun: energy(r) = SC/r2
reflection coefficient
unit vector Sun sat
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49AE2104 Flight and Orbital Mechanics |
Solar radiation pressure (cnt’d)
SC = 1371 W/m2
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50AE2104 Flight and Orbital Mechanics |
Solar radiation pressure (cnt’d)
satellite arad [m/s2]
ISS (100-1000 m2; 500 ton) 9 – 90 × 10-10
ENVISAT 8.3 – 83 × 10-9
LAGEOS 3 × 10-9
Echo-1 4.4 × 10-5
solar sail (100x100 m; 300 kg) 1.5 × 10-4
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51AE2104 Flight and Orbital Mechanics |
Perturbations:
• Irregularities gravity field
• Third body
• Atmospheric drag
• Solar radiation pressure
• Thrust
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52AE2104 Flight and Orbital Mechanics |
[Petropoulos 2001]
)cos2:(
cos21
sin2/2
Tarror
Tardt
d
r
Tarrr
Thrust
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53AE2104 Flight and Orbital Mechanics |
Example 1:
acceleration of 0.190 mm/s2 in along-track direction (i.e., α=0°)
Thrust (cnt’d)
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54AE2104 Flight and Orbital Mechanics |
Example 2:
acceleration of 0.190 mm/s2 in radial direction (i.e., α=90°)
Thrust (cnt’d)
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55AE2104 Flight and Orbital Mechanics |
Example 3:
orbit of SMART-1 [ESA, 2010]
Thrust (cnt’d)
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56AE2104 Flight and Orbital Mechanics |
High thrust:
• can compete against central gravity (i.e., launch)
• instantaneous velocity changes (orbits around Earth + interplanetary orbits)
Low thrust:
• attractive since high Isp
• primary propulsion for interplanetary missions
• station-keeping
• since 2012: transfer LEO GEO
Thrust (cnt’d)
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57AE2104 Flight and Orbital Mechanics |
Summary: Low Earth Orbit (1)
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58AE2104 Flight and Orbital Mechanics |
Summary: Low Earth Orbit (2)
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59AE2104 Flight and Orbital Mechanics |
Conclusions near-Earth orbits:
• J2 acceleration is dominant perturbation for all LEO
• low thrust already important
• atmospheric drag dominant perturbation at very low altitudes
• Solar, lunar and J2,2 accelerations very small, but build up for GEO
• Kepler orbit very good 1st-order approximation (2-4 orders of magnitude Δ)
Summary: Low Earth Orbit (3)
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60AE2104 Flight and Orbital Mechanics |
Summary: interplanetary orbit (1)
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61AE2104 Flight and Orbital Mechanics |
Conclusions interplanetary orbits:
• low thrust important
• Solar radiation can be important
• Kepler orbit very good 1st-order approximation (4-6 orders of magnitude Δ)
Summary: interplanetary orbit (2)
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62AE2104 Flight and Orbital Mechanics |
Question 4:
a) Mention the 5 different categories of perturbing forces that may act on an arbitrary vehicle. Describe each category briefly (about 5 lines per item).
b) Give a brief description of the (relative) importance of these categories.
Summary perturbing forces
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63AE2104 Flight and Orbital Mechanics |
Question 5:
a) Mention the 5 different categories of perturbing forces that may act on an arbitrary vehicle.
b) Compute the magnitude of the main gravitational force exerted by the Earth’s gravity field.
c) Compute the magnitude of the various perturbing sources.
Data: μEarth = 398600 km3/s2; μMoon = 4903 km3/s2; μSun = 1.327 × 1011
km3/s2; Re = 6378 km; hGOCE = 250 km; ρatm,avg = 6.2 × 10-11 kg/m3; AU = 149.6 × 106 km; SC = 1371 W/m2; c = 3 × 105 km/s; mGOCE = 1050 kg; SGOCE = 1.0 m2; CD,GOCE = 2.2; CR,GOCE = 1.2; TGOCE = 10 mN; J2 = 1082 × 10-6; J2,2 = 1.816 × 10-6 ; λ2,2 = -14.9°
Answers: see footnotes below (BUT TRY YOURSELF FIRST!!)
Summary perturbing forces