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Atmospheric Drag
Modeling the Space Environment
Manuel Ruiz Delgado
European Masters in Aeronautics and Space
E.T.S.I. Aeronáuticos
Universidad Politécnica de Madrid
April 2008
Atmospheric Drag – p. 1/29
Image courtesy NASA
Atmospheric Drag
Atmospheric Drag – p. 2/29
Image courtesy NASA
Atmospheric Drag
Effects of Air Drag: MIR Station Reentry March 22, 2001
Watch MIR deorbit video on Youtube (simulation by AGI)
Atmospheric Drag – p. 3/29
Aerodynamic Drag
Space Aerodynamics
Perturbations of Keplerian motion
Free molecular flow
Ballistic coefficient
Drag computation
High atmosphere
Structure of the atmosphere
Sun influence: F10.7
Geomagnetic activity influence: Kp
Atmospheric Models
Static: Exponential, Harris-Priester, US Standard
Dynamic: Jacchia, MSISE, COSMOS
Atmospheric Drag – p. 4/29
Perturbations of Keplerian Motion
Accelerations of the Satellite (BC=50)
1e+006
Kepler
J2
C22
Sun
Moon
Drag (low)
Drag (high)
Prad
Shuttle
10000
Acceleration (m/s2)
100
1
ISS
0.01
0.0001
1e−006
1e−008
0
100
200
300
400
500
Height (km)
600
700
800
900
Atmospheric Drag – p. 5/29
Perturbations of Keplerian Motion
Accelerations of the Satellite (BC=50)
1e+006
Kepler
J2
C22
Sun
Moon
Drag (low)
Drag (high)
Prad
10000
Acceleration (m/s2)
100
1
0.01
GPS
GEO
0.0001
1e−006
1e−008
0
5000
10000
15000 20000 25000
Height (km)
30000
35000
40000
Atmospheric Drag – p. 6/29
Space Aerodynamics
Free molecular flow:
Knudsen No.
≫1
Molecules interact one by one with the body: incident flow not disturbed
by the body.
Space Aerodynamics
Free molecular flow:
Knudsen No.
≫1
Molecules interact one by one with the body: incident flow not disturbed
by the body.
Ma
L
Kn = =
d
Re
L : Mean free path of the molecules
d : Characteristic longitude of satellite
Space Aerodynamics
Free molecular flow:
Knudsen No.
≫1
Molecules interact one by one with the body: incident flow not disturbed
by the body.
Ma
L
Kn = =
d
Re
Kn ≫ 1
Kn ∼ 1
Kn ≪ 1
L : Mean free path of the molecules
d : Characteristic longitude of satellite
Free molecular flow
Transition
Continuum flow
Space environment
(Complex: reentry)
Classical aerodynamics
ECSS-E-10-04A defines Kn > 3 as free molecular regime
Free molecular flow over 150 km (small satellites) or 250 km
(shuttle, ISS)
Atmospheric Drag – p. 7/29
Impact Types
p1
n
θ θ
p2
Elastic impact:
Drag coefficient:
p1
n
p2
p2 = p1 + 2p1 cos θn
CD = 4
p2 = p1 /2
Diffuse refflection:
Drag coefficient:
CD = 2 − 4
Impact Types
p1
n
θ θ
p2
Elastic impact:
Drag coefficient:
p1
n
p2
p2 = p1 + 2p1 cos θn
CD = 4
p2 = p1 /2
Diffuse refflection:
Drag coefficient:
CD = 2 − 4
p1
Absorption (diffuse emission later):
Drag coefficient:
p2 = 0
CD = 2
p1
Atmospheric Drag – p. 8/29
Atmospheric Drag
Force over the surface dA⊥ , incidence angle θ:
∆m = ρvdA⊥ ∆t
⇒
θ
v∆
t
z
dA⊥
∆p
x
dF =
= ρv 2 [1 + f (θ)]dA⊥
∆t
v
y
Atmospheric Drag
Force over the surface dA⊥ , incidence angle θ:
∆m = ρvdA⊥ ∆t
⇒
θ
v∆
t
z
dA⊥
∆p
x
dF =
= ρv 2 [1 + f (θ)]dA⊥
∆t
Integrating over the whole surface gives the drag acceleration:
1 CD A
D
=−
aD =
ρ |vrel | vrel
m
2 m
v
y
Atmospheric Drag
Force over the surface dA⊥ , incidence angle θ:
∆m = ρvdA⊥ ∆t
⇒
θ
v∆
t
z
dA⊥
v
∆p
x
dF =
= ρv 2 [1 + f (θ)]dA⊥
∆t
y
Integrating over the whole surface gives the drag acceleration:
1 CD A
D
=−
aD =
ρ |vrel | vrel
m
2 m
Lateral drag:
vrel
vt
CD =
Ak
CD⊥ + CDk A⊥
Orbital speed: vrel ∼ 8 km/s
Thermal speed: vt ∼ 1 km/s ( 12 mv 2 = 32 kT )
Important for light or svelte craft
Atmospheric Drag – p. 9/29
Atmospheric Drag
1 CD A
D
aD =
=−
ρ |vrel | vrel
m
2 m
vrel
Speed relative to the atmosphere
◮ Rotation, winds
Atmospheric Drag
1 CD A
D
aD =
=−
ρ |vrel | vrel
m
2 m
vrel
CD
Speed relative to the atmosphere
Drag Coefficient:
◮ Rotation, winds
◮ difficult to measure
◮ CD ∼ 2 − 2.4 (1-4)
Atmospheric Drag
D
1 CD A
aD =
ρ |vrel | vrel
=−
m
2 m
vrel
CD
A
Speed relative to the atmosphere
Drag Coefficient:
Frontal area
◮ Rotation, winds
◮ difficult to measure
◮ CD ∼ 2 − 2.4 (1-4)
◮ depends on attitude
Atmospheric Drag
1 CD A
D
aD =
=−
ρ |vrel | vrel
m
2 m
vrel
CD
A
ρ
Speed relative to the atmosphere
Drag Coefficient:
Frontal area
Atmospheric density:
◮ Rotation, winds
◮ difficult to measure
◮ CD ∼ 2 − 2.4 (1-4)
◮ depends on attitude
◮ ∼ 15% error
Atmospheric Drag
1 CD A
D
aD =
=−
ρ |vrel | vrel
m
2 m
vrel
CD
A
ρ
Speed relative to the atmosphere
Drag Coefficient:
◮ Rotation, winds
◮ difficult to measure
◮ CD ∼ 2 − 2.4 (1-4)
◮ depends on attitude
Frontal area
Atmospheric density:
◮ ∼ 15% error
m
β=
Ballistic coefficient: (β ↑, aD ↓)
CD A
CD A
Some authors use the opposite form: BC=
m
Atmospheric Drag – p. 10/29
Computing Drag
4 problems:
Calibrating CD or β : Differential Correction
MapleOD
Propagating orbits with drag: atmospheric model
Computing satellite lifetime: averaged equations
Atmospheric research
King-Hele
Computing Drag
4 problems:
Calibrating CD or β : Differential Correction
MapleOD
Propagating orbits with drag: atmospheric model
Computing satellite lifetime: averaged equations
Atmospheric research
King-Hele
Effects on the orbit
Seculars: a ↓, e ↓→ Reentry
Circularization phase
Spiral phase: reenty
Spiral Maplanim
Mir ISS
Mars
Periodic: Ω, ω, i (through atmospheric rotation)
Atmospheric Drag – p. 11/29
Structure of the Atmosphere
102
km
Ionosphere
103
Exosphere
Thermopause
Dominant constituent
He
Thermosphere
ISS, Shuttle
O
Mesopause
Mesosphere
Stratopause
Stratosphere
101
Clouds ↑
N2
Tropopause
Mt Everest
Troposphere
100
Sea Level
Atmospheric Drag – p. 12/29
Constituents - Solar low
Constituents: Low Solar Activity
1e+030
N2
O
O2
He
Ar
H
N
3
Density (molec/m )
1e+025
1e+020
1e+015
1e+010
100000
1
1e−005
1e−010
0
100 200 300 400 500 600 700 800 900
Height (km)
Atmospheric Drag – p. 13/29
Exospheric Temperature T∞ vs Solar Activity
◮
1800
1600
1400
High
Mean
Low
T (ºK)
1200
1000
800
600
400
200
0
0
100
200
300
400
500
600
700
800
900
Atmospheric Drag – p. 14/29
Density vs Solar Activity
◮◮
100
High
Mean
Low
1
3
Density (kg/m )
0.01
0.0001
1e−006
1e−008
1e−010
1e−012
1e−014
1e−016
0
100 200 300 400 500 600 700 800 900
Atmospheric Drag – p. 15/29
Location-Related Changes
In Static Models, properties change only with location:
Location-Related Changes
In Static Models, properties change only with location:
*** Height: Hydrostatic equilibrium ⇒ ρ = ρ0 e
h0 −h
H
◮
hell
Location-Related Changes
In Static Models, properties change only with location:
*** Height: Hydrostatic equilibrium ⇒ ρ = ρ0 e
h0 −h
H
hell
◮
φg
** Latitude: change of height through flattening
S
Height over the Ellipsoid
changes with longitude:
∆hell = 0 − 21 km ⇔ ∆ρ
hcir
h''ell
S
h'ell
φg
E
h ell
S
Location-Related Changes
In Static Models, properties change only with location:
*** Height: Hydrostatic equilibrium ⇒ ρ = ρ0 e
h0 −h
H
hell
◮
φg
** Latitude: change of height through flattening
S
Height over the Ellipsoid
changes with longitude:
∆hell = 0 − 21 km ⇔ ∆ρ
hcir
h''ell
S
h'ell
φg
h ell
S
E
* Longitude:
Temporal change (day/night)
λg
Subsolar hump
Small space variation (seas, mountains → atmosphere), mainly
at low heights.
Atmospheric Drag – p. 16/29
Causes of Time-Related Changes
In Time-varying Models, properties change with location and time:
Sun
spots
UV/EUV
radiation
Index
F10.7
Solar
activity
Density
ρ(t)
Solar
wind
Internal
geomagnetic
field
Geomagnetic
activity
Index
Kp / Ap
Atmospheric Drag – p. 17/29
Time Changes Due to the Sun
∼ 85%
◮
Image courtesy NASA
Sunspot 11 year cycle:
Sunspot Number ∼ EUV (10-120 nm) ⇒ T∞ ⇒ ρ
EUV not measurable: PROXY F10,7 , F10.7 81
Atmospheric Drag – p. 18/29
Time Changes: Sun and Geomagnetic Field
Diurnal variations:
∼ 15%
Solar UV radiation heats up the atmosphere: ρ ↑
Max: subsolar hump, delayed 2-2:30 pm.
Antipod Min
Density ρ depends on:
Apparent local solar time LHA⊙ of satellite
Solar declination δ⊙
Geodetic latitude φg of satellite
jach/hed
Time Changes: Sun and Geomagnetic Field
∼ 15%
Diurnal variations:
Solar UV radiation heats up the atmosphere: ρ ↑
Max: subsolar hump, delayed 2-2:30 pm.
Antipod Min
Density ρ depends on:
Apparent local solar time LHA⊙ of satellite
jach/hed
Solar declination δ⊙
Geodetic latitude φg of satellite
Magnetic storms:
Earth field’s fluctuations:
Solar storms: short but large effect:
small effect
Up to 30%
Influence ρ through the geomagnetic indices Kp or Ap
Atmospheric Drag – p. 19/29
Other Changes
Variable 0 − 10%
Solar rotation period of 27 days:
Visible sunspots change
EUV radiation changes
Affects ρ through F10.7 and F10.7
81
(81 day average)
Other Changes
Variable 0 − 10%
Solar rotation period of 27 days:
Visible sunspots change
EUV radiation changes
Affects ρ through F10.7 and F10.7
81
(81 day average)
Semi-annual variation: Sun distance changes.
Small
Cyclical variations: 11-year cycles are not regular. ESA’s standard
cycle.
Small
Atmospheric rotation: difficult to know. Decreases with height.
Co-rotation is a good estimate.
< 5%
Winds: Not well known. Models not mature. Low orbits.
Tides: The atmosphere also suffers tides. Models.
Small
Small
Atmospheric Drag – p. 20/29
Data Sources
Before Space Age: nothing known about the properties of the
atmosphere above 150 km
Early satellites: orbit tracking. Assume CD , compute ρ
Careful with NORAD TLE’s ṅ: may include other accelerations
On-board accelerometers: non-gravitational accelerations
On-board mass spectrometers: chemical composition, temperature
Incoherent scatter ground-based radar: electron and ion distribution,
which is related to neutral density and composition
Atmospheric Drag – p. 21/29
Static Models
Properties
Simple, low computation time, reasonable results
Good for theoretical or long-range studies (averaged)
Errors up to 40% (Mean Sun) or 60% (High Sun)
Time-varying models also have errors (∼15%)
Static Models
Properties
Simple, low computation time, reasonable results
Good for theoretical or long-range studies (averaged)
Errors up to 40% (Mean Sun) or 60% (High Sun)
Time-varying models also have errors (∼15%)
Exponential structure:
Spherical symmetry, co-rotating with Earth
Hydrostatic equilibrium + perfect gas: ρ = ρ0 e
Reference density and height, ρ0 , h0
Scale height H (changes with h!)
h0 −h
H
◮
Atmospheric Drag – p. 22/29
Static Models
US Standard Atmosphere 62, 76
(0-1000 km)
Tabulated
Ideal, stationary atmosphere, at 45o N, moderate solar activity
CIRA 65-90
(0-2500 km)
COSPAR-International Reference Atmosphere.
CIRA-72 and -86 incorporate dynamic models for h > 100km
Harris-Priester
(0-1000 km)
Static. Fast. Tabulated for T∞ ⇒ Interpolate
Includes subsolar hump (only LHA⊙ , equinoctial)
Atmospheric Drag – p. 23/29
Time-Varying Models
Comprehensive: include all the main effects
Inherent errors: unpredictable Sun, proxies, data fit
Better with past measured data. Reasonable predictions
Numerically intensive
∼ 15%
Time-Varying Models
Comprehensive: include all the main effects
Inherent errors: unpredictable Sun, proxies, data fit
Better with past measured data. Reasonable predictions
Numerically intensive
Jacchia-Roberts (65,71,77, 81)
∼ 15%
(70-700 km)
The first. Uses satellite data. Late, also ISR
Profile for T∞ (F10,7 , F 10,7 , Kp , φg , λ, δ⊙ , LHA⊙ , MJD, UT)
Numerical int. diffusion PDE of each constituent: ρ(h) .
Roberts: Integrate several profiles, tabulated polynomial fit
Computationally intensive
FORTRAN: MET/71, 77
Vallado and Montenbruck describe different modifications of the Jacchia model
Atmospheric Drag – p. 24/29
Time-Varying Models
MSIS 83, 86, MSISE 90, 2000
(0-2000 km)
Mass Spectrometer & Incoherent Scatter + satellite tracking
¯ i)
Profile T∞ (JD, hel , λg , φg , LST, F10.7 , F̄10.7 , Api , Ap
Diffusion PDE for each constituent:
1 dni
ni dh
+
1
Hi
+
1+αi dT
T dh
=0,
series integration (faster)
Add partial densities: ρ(h) =
P
ρi
More recent, faster, exact; J-R still better in some cases
ESA recommended standard / Mean cycle for predictions
FORTRAN code available / Indices data sources:
• ftp://ftp.ngdc.noaa.gov/STP/GEOMAGNETIC_DATA/INDICES/KP_AP/
• http://celestrak.com/SpaceData/
(Average F107 computed)
Atmospheric Drag – p. 25/29
Time-Varying Models
COSMOS
(160-600 km)
Tracking data fit of the COSMOS satellites
ρ = ρn k1 k2 k3 k4
ρn - Night density profile: exponential
k1 - Solar activity correction, F10.7 , 4 values
k2 - Day/Night correction
k3 - Semi-annual correction (small)
k4 - Geomagnetic correction, ap
Very simple, modular, fast, available (cf. Vallado)
Good for orbits similar to the COSMOS satellites
“Density Model for Satellite Orbit Predictions.” GOST 25645-84
Atmospheric Drag – p. 26/29
Comparison of Time-Varying Models
Model
Jacchia 71
Jacchia-Roberts
Jacchia-Lineberry
Jacchia-Gill
Jacchia 77
Jacchia-Lafontaine
MSIS 77
MSIS 86
TD88
DTM
CPU
1,00
0,22
0,43
0.11
10,69
0,86
0,06
0,32
0,01
0,03
∆ρ
0,01
0,13
0,02
0,13
0,13
0,18
0,21
0.91
0,40
∆ρmax
0,03
0,35
0,08
0,35
0,36
0,53
1,45
7,49
1,22
Data from Montenbruck, p. 100
Atmospheric Drag – p. 27/29
Conclusions
Atmospheric Drag is significant between 200-700 km
Uncertainties in CD , ρ, A
Static models have large erros
Time-varying models’ typical error is about 15%
Because of the model: indirect proxy
Because of the Sun’s uncertainty
Because of the fast solar storms
Density is the heaviest computation load of orbit propagation
Use the simplest model within the required precision
New models coming, error down to 5% : Solar-2000, HASDM
Space sensors allow direct measuring of EUV, without proxies
Atmospheric Drag – p. 28/29
COWELL with drag acceleration
ẏ = f (y, t)
Begin
Input data
KB/File
Initializations
Load Indices
Common block
ITRF / H, Lat-Long
rGCRF
ODE Integrator
Call Int step
Save Data
FILE
Call Derivs
Density
Hell , φg , λ
Drag Accel
Other Accel
End
Atmospheric Drag – p. 29/29