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Home , Orbital mechanics, Orbit modeling, Perturbation (astronomy)

Flight and Orbital

Lecture slides

Challenge the future 1 and AE2-104, lecture hours 17+18: Perturbations Ron Noomen October 25, 2012

AE2104 Flight and Orbital Mechanics 1 | Example:

Questions: • what is the purpose of this mission? • where is the located? • why does it use a solar sail? • ….

[Wikipedia, 2010]

AE2104 Flight and Orbital Mechanics 2 | Overview

• Orbital mechanics (recap)

• Irregularities field • Third-body perturbations • Atmospheric • Solar • Thrust

• Relativistic effects • Tidal • Thermal forces  Special missions only; not treated here • Impacts debris/micrometeoroids

AE2104 Flight and Orbital Mechanics 3 | 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 ; • 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)

AE2104 Flight and Orbital Mechanics 4 | 2-dimensional Kepler

[Seligman, 2010]

AE2104 Flight and Orbital Mechanics 5 | 2-dimensional Kepler orbits

a: semi-major axis [m] e: eccentricity [-] [Cornellise, Schöyer and Wakker, 1979] θ: [deg] E: [deg]

AE2104 Flight and Orbital Mechanics 6 | 2-dimensional Kepler orbits: general equations

2 r  a (1  e )  p ; r  a (1  e) ; r  a (1  e) 1  e cos 1  e cos p a

2 E  E  E  V      tot kin pot 2 r 2a

2  2 1    2 V      ; Vcirc   ; Vesc   r a  r a r

3 T  2  a  AE2104 Flight and Orbital Mechanics 7 | 2-dimensional Kepler orbits: equations (cnt’d)

ellips: 0 ≤ e < 1 a > 0 Etot < 0  n  3 a satellite altitude specific energy  1 e E [km] [km2/s2/kg] tan  tan 2 1 e 2 launch 0 -62.4  M  E esin E platform M  n(t  t0) SpaceShipOne 100+ -61.4 () M  Ei esin Ei Ei1  Ei  imaginary sat 100 -30.8

1ecosEi Kepler’s Equation Kepler’s r  a(1ecosE) Envisat 800 -27.8 LAGEOS 5900 -16.2

AE2104GEO Flight and Orbital35900 Mechanics -4.78 | 2-dimensional Kepler orbits: equations (cnt’d)

ellips: 0 ≤ e < 1 a > 0 Etot < 0

AE2104 Flight and Orbital Mechanics 9 | 2-dimensional Kepler orbits: equations (cnt’d)

: e = 1 a = ∞ Etot = 0

p r  1  cos

1  1  M  tan  tan3 2 2 6 2

M  n (t  t0)  n  p3 2 2 2 V  Vesc  r AE2104 Flight and Orbital Mechanics 10 | 2-dimensional Kepler orbits: equations (cnt’d)

: e > 1 a < 0 Etot > 0

 e  1 F tan  tanh 2 e  1 2

M  e sinh F  F

M  n (t  t0)

 n  (a)3 r  a (1  e cosh F) 2 2 2 2 2 V  Vesc  V   V r AE2104 Flight and Orbital Mechanics 11 | 3-dimensional Kepler orbits

i: inclination [deg] [Cornelisse, Schöyer and Wakker, 1979] Ω: of the ascending node, or longitude of the ascending node [deg] ω: argument of pericenter [deg] u = ω + θ : argument of latitude [deg]

AE2104 Flight and Orbital Mechanics 12 | 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 )

2 2 rxy = sqrt ( x + y )

λ = atan2 ( y / rxy , x / rxy ) δ = asin ( z / r )

AE2104 Flight and Orbital Mechanics 13 | Orbital perturbations: introduction

Questions: • what forces? • magnitude of forces? • how to model/compute in satellite ? • analytically? numerically? • accuracy? • efficiency?

AE2104 Flight and Orbital Mechanics 14 | Inclusion in

Option 1: include directly in equation of motion

d x2  amain agrav adrag asolrad a3rdbody etcetera d t2 Option 2: express as variation of

da 2a2  p e.g.  Sesin  N dt p  r  (not further treated here; cf. ae4-874 and ae4-878)

AE2104 Flight and Orbital Mechanics 15 | Perturbations:

• Irregularities gravity field • Third body • Atmospheric drag • Solar radiation pressure • Thrust

AE2104 Flight and Orbital Mechanics 16 | Perturbations:

• Irregularities gravity field • Third body • Atmospheric drag • Solar radiation pressure • Thrust

AE2104 Flight and Orbital Mechanics 17 | Irregularities gravity field

Already treated in lecture hours 15+16

e.g., North-South due to J2: 2 4 acc  3 J2 Re r sin( )cos( )

e.g., East-West acceleration due to J2,2: 2 4 acc  6 J2,2 Re r 3cos( )sin(2(  2,2))

AE2104 Flight and Orbital Mechanics 18 | Perturbations:

• Irregularities gravity field • Third body • Atmospheric drag • Solar radiation pressure • Thrust

AE2104 Flight and Orbital Mechanics 19 | Third-body perturbations

Reference frame

  M  rsd rd  rs   G rs  G Md    3  3 3  rs  rsd rd 

Attractional forces (by definition): • satellite <-> • satellite <-> perturbing body • Earth <-> perturbing body

AE2104 Flight and Orbital Mechanics 20 | Third-body perturbations (cnt’d)

Attractional forces (practice): • Earth attracts satellite • perturbing body attracts satellite M << M , M • perturbing body attracts Earth sat E S • net effect counts M rr rr GGMmain ps  p s3 s p  3 3 rs r ps r p

AE2104 Flight and Orbital Mechanics 21 | Third-body perturbations (cnt’d)

perturbing acceleration ap:    rps rp  a    p p  3 3   r r   ps p 

maximum when on straight :

main sat 3rd body 3 amppr  2 s a m r mainmax main p

AE2104 Flight and Orbital Mechanics 22 | Third-body perturbations (cnt’d)

2 situations: 1. heliocentric (i.e., orbits around ) 2. planetocentric (i.e., orbits around Earth)

AE2104 Flight and Orbital Mechanics 23 | Third-body perturbations (cnt’d)

Conclusions: • influence increases with distance to Sun • O(10-7) m/s2 AE2104 Flight and Orbital Mechanics 24 | Third-body perturbations (cnt’d)

Conclusions: • influence Sun decreases with distance to Sun • influence increases with distance to Sun AE2104 Flight and Orbital Mechanics 25 | • acceleration from Sun O(10-2) m/s2; dominant (central body!!) • near : 3rd body becomes dominant Third-body perturbations (cnt’d)

Conclusions: • in : Sun dominant • near Earth: Earth itself dominant AE2104 Flight and Orbital Mechanics 26 | • central body vs. 3rd body perturbation ?? Third-body perturbations (cnt’d)

Sphere of Influence: • 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

Sun Earth sat AE2104 Flight and Orbital Mechanics 27 | Third-body perturbations (cnt’d)

accSun,3rd accEarth,3rd  accEarth,main accSun,main

without derivation :

0.4  Mmain  rSoI  r3rd    M3rd 

example : Earth main body, Sun 3rd body :

0.4  M Earth  rSoI,Earth  distEarthSun    930,000 km  MSun  AE2104 Flight and Orbital Mechanics 28 | is at 384,000 km…. Third-body perturbations (cnt’d)

planet Sphere of Influence [km] [% AU] [% distance planet-Sun] 1.1 × 105 0.08 0.2 6.2 × 105 0.4 0.6 Earth 9.3 × 105 0.6 0.6 5.8 × 105 0.4 0.3 4.8 × 107 32.2 6.2 5.5 × 107 36.5 3.8 5.2 × 107 34.6 1.8 8.7 × 107 57.9 1.9 3.2 × 106 2.1 0.1 AE2104 Flight and Orbital Mechanics 29 | Third-body perturbations (cnt’d)

2 situations: 1. heliocentric (i.e., orbits around Sun) 2. planetocentric (i.e., orbits around Earth)

AE2104 Flight and Orbital Mechanics 30 |

Third-body perturbations (cnt’d)

   center of ref. frame

AE2104 Flight and Orbital Mechanics 31 | 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 AE2104 Flight and Orbital Mechanics 32 | 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: 0.4  Mmain  rSoI  r3rd    M3rd  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 ). 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?

6 30 24 11 Data: 1 AU = 149.6 × 10 km, mSun = 2.0 × 10 kg, mEarth = 6.0 × 10 kg, μSun = 1.3271 × 10 3 2 3 2 km /s , μEarth = 398600 km /s . Answers: see footnotes below (BUT TRY YOURSELF FIRST!!) AE2104 Flight and Orbital Mechanics 33 | 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 AE2104 Flight and Orbital Mechanics 34 | Third-body perturbations (cnt’d)

Conclusions: • Earth dominant (central body; within SoI) • Moon most important 3rd body; Sun directly after AE2104 Flight and Orbital Mechanics 35 | • effect of planets about 4 orders of magnitude smaller Third-body perturbations (cnt’d)

[Wikimedia, 2010] celestial ΔV-budget body for GEO sat [m/s/yr]

Moon 36.93

Sun 14.45

AE2104 Flight and Orbital Mechanics 36 | Third-body perturbations (cnt’d)

Question 2:

Consider a hypothetical planet X with 5 × 1025 kg, orbiting the Sun in a with radius 3 AU. The orbital plane coincides with the (i.e., the orbital plane of the Earth). a) Make a sketch of the situation when the gravitational attraction of this planet X on 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).

-11 3 2 3 2 Data: G = 6.673 × 10 m /kg/s ; μEarth = 398600 km /s ; 1 AU = 149.6 × 106 km AE2104 Flight and Orbital Mechanics 37 | Answers: see footnotes below (BUT TRY YOURSELF FIRST!!) 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. 3 2 3 2 Consider the Earth-Moon system: μEarth = 398600 km /s , μMoon = 4903 km /s , average distance Earth-Moon = 384,000 km. What is the radius of the SoI of the Moon? 0.4  Mmain  rSoI  r3rd    M3rd  c) Suppose that the Earth and Moon have equal . 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! AE2104 Flight and Orbital Mechanics 38 | Perturbations:

• Irregularities gravity field • Third body • Atmospheric drag • Solar radiation pressure • Thrust

AE2104 Flight and Orbital Mechanics 39 | C S 1 V a   D  V 2 ;    exp (  h / H ) drag m 2 V 0

[Wertz, 2009]

[Wertz, 2009]

AE2104 Flight and Orbital Mechanics 40 | Atmosphere (cnt’d)

Altitude [km] Atmospheric [kg/m3]

minimum maximum after 200 1.8 × 10-10 3.5 × 10-10

[Wertz, 2009]: 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 AE21042.3 Flight × 10 and-16 Orbital Mechanics1.2 × 10-1541 | Atmosphere (cnt’d) [Wertz, 2009]:

AE2104 Flight and Orbital Mechanics 42 | Atmosphere (cnt’d)

altitudes of and GFZ-1:

AE2104 Flight and Orbital Mechanics 43 | Atmosphere (cnt’d) [Wertz, 2009]:

orbit 2

orbit 1

AE2104 Flight and Orbital Mechanics 44 | Atmosphere (cnt’d) circular orbits:

2 Δa2π = -2π (CDA/m) ρ a [m] 2 2 ΔT2π = -6π (CDA/m) ρ a / V [s]

ΔV2π = π (CDA/m) ρ a V [m/s]

Δe2π = 0 [-]

L = - H / Δa2π [rev] [Wertz, 2009]:

AE2104 Flight and Orbital Mechanics 45 | Atmosphere (cnt’d)

GOCE:

[ESA, 2010]

• launch date: March 17, 2009 • altitude: 250 km • phase solar cycle? • ….. AE2104 Flight and Orbital Mechanics 46 | Perturbations:

• Irregularities gravity field • Third body • Atmospheric drag • Solar radiation pressure • Thrust

AE2104 Flight and Orbital Mechanics 47 | Solar radiation pressure

• 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

1 SC A rSun-sat 6 A arad  (1 )  4.510 2 c m r m ( rSun-sat/AU ) Sun-sat

reflection coefficient unit vector Sun  sat

AE2104 Flight and Orbital Mechanics 48 | Solar radiation pressure (cnt’d)

SC = 1371 W/m2

AE2104 Flight and Orbital Mechanics 49 | Solar radiation pressure (cnt’d)

2 satellite arad [m/s ]

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

AE2104 Flight and Orbital Mechanics 50 | Perturbations:

• Irregularities gravity field • Third body • Atmospheric drag • Solar radiation pressure • Thrust

AE2104 Flight and Orbital Mechanics 51 | Thrust

2 2 r  r   / r  aT sin  1 d  2  r   aT cos  r dt   [Petropoulos 2001] (or : r  2r  aT cos )

AE2104 Flight and Orbital Mechanics 52 | Thrust (cnt’d)

Example 1: acceleration of 0.190 mm/s2 in along-track direction (i.e., α=0°)

AE2104 Flight and Orbital Mechanics 53 | Thrust (cnt’d)

Example 2: acceleration of 0.190 mm/s2 in radial direction (i.e., α=90°)

AE2104 Flight and Orbital Mechanics 54 | Thrust (cnt’d)

Example 3: orbit of SMART-1 [ESA, 2010]

AE2104 Flight and Orbital Mechanics 55 | Thrust (cnt’d)

High thrust: • can compete against central gravity (i.e., launch) • instantaneous changes (orbits around Earth + interplanetary orbits)

Low thrust:

• attractive since high Isp • for interplanetary missions • station-keeping AE2104 Flight and Orbital Mechanics 56 | • since 2012: transfer LEO  GEO Summary: (1)

AE2104 Flight and Orbital Mechanics 57 | Summary: Low Earth Orbit (2)

AE2104 Flight and Orbital Mechanics 58 | Summary: Low Earth Orbit (3)

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 • very good 1st-order approximation (2-4 orders of magnitude Δ)

AE2104 Flight and Orbital Mechanics 59 | Summary: interplanetary orbit (1)

AE2104 Flight and Orbital Mechanics 60 | Summary: interplanetary orbit (2)

Conclusions interplanetary orbits: • low thrust important • Solar radiation can be important • Kepler orbit very good 1st-order approximation (4-6 orders of magnitude Δ)

AE2104 Flight and Orbital Mechanics 61 | Summary perturbing forces

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.

AE2104 Flight and Orbital Mechanics 62 | Summary perturbing forces

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 exerted by the Earth’s gravity field. c) Compute the magnitude of the various perturbing sources.

3 2 3 2 11 Data: μEarth = 398600 km /s ; μMoon = 4903 km /s ; μSun = 1.327 × 10 3 2 -11 3 km /s ; Re = 6378 km; hGOCE = 250 km; ρatm,avg = 6.2 × 10 kg/m ; 6 2 5 AU = 149.6 × 10 km; SC = 1371 W/m ; c = 3 × 10 km/s; mGOCE = 2 1050 kg; SGOCE = 1.0 m ; CD,GOCE = 2.2; CR,GOCE = 1.2; TGOCE = 10 -6 -6 mN; J2 = 1082 × 10 ; J2,2 = 1.816 × 10 ; λ2,2 = -14.9°

AE2104 Flight and Orbital Mechanics 63 | Answers: see footnotes below (BUT TRY YOURSELF FIRST!!)