Outline
MEO Satellite Dynamics
Alessandro Rossi
ISTI–CNR,
Spaceflight Dynamics Section
Via Moruzzi 1, 56124 Pisa, Italy
24th IADC Meeting
Tsukuba, April 10 - 13, 2006
Alessandro Rossi
MEO Satellite Dynamics
Long term evolution of objects in GLONASS-like orbits
Conclusions
Acknowledgments
This work was done in the framework of the ESA/ESOC
Contract No. 18423/04/D/HK: Analysis of Mitigation Measures
based on the Semi-Deterministic Model, for the upgrade of
SDM to SDM 4.0
Alessandro Rossi
MEO Satellite Dynamics
Long term evolution of objects in GLONASS-like orbits
Conclusions
Orbital evolution
Orbital propagation
Orbital region survey
Agenda
1
Long term evolution of objects in GLONASS-like orbits
Orbital evolution
Orbital propagation
Orbital region survey
2
Conclusions
Alessandro Rossi
MEO Satellite Dynamics
Long term evolution of objects in GLONASS-like orbits
Conclusions
Orbital evolution
Orbital propagation
Orbital region survey
Orbital evolution MEO - ETALON
Etalon 1 and 2 are identical
geodetic passive satellites of
Russia (former USSR), for
satellite laser ranging.
They are spheres with diameter
= 1.294 m, mass = 1415 kg.
Etalon-1 and Etalon-2 were
launched in 1989 in
GLONASS-like orbits:
1
E-1: a = 25 498 km,
e = 0.00061, i = 64.9◦
2
E-2: a = 25 498 km,
e = 0.00066, i = 65.5◦
Alessandro Rossi
MEO Satellite Dynamics
Long term evolution of objects in GLONASS-like orbits
Conclusions
Orbital evolution
Orbital propagation
Orbital region survey
Orbital evolution MEO - ETALON
The Etalon satellites are ideal test bodies to study the
perturbations affecting the long term evolution of the
navigation satellite orbits
The Etalon satellites are ideal bodies to test the accuracy
of the orbital propagators.
The Tuned Inter-Range Vectors (TIRV) are state vectors of
the Etalon satellites used to track them by the Laser
stations.
The accuracy of the TIRVs is about 1 m.
ASI (Matera Station) provided also one year (2004) of fitted
orbits with weighted RMS of a few cm.
Alessandro Rossi
MEO Satellite Dynamics
Long term evolution of objects in GLONASS-like orbits
Conclusions
Orbital evolution
Orbital propagation
Orbital region survey
Orbital evolution MEO - ETALON
The Etalon satellites are ideal test bodies to study the
perturbations affecting the long term evolution of the
navigation satellite orbits
The Etalon satellites are ideal bodies to test the accuracy
of the orbital propagators.
The Tuned Inter-Range Vectors (TIRV) are state vectors of
the Etalon satellites used to track them by the Laser
stations.
The accuracy of the TIRVs is about 1 m.
ASI (Matera Station) provided also one year (2004) of fitted
orbits with weighted RMS of a few cm.
Alessandro Rossi
MEO Satellite Dynamics
Long term evolution of objects in GLONASS-like orbits
Conclusions
Orbital evolution
Orbital propagation
Orbital region survey
Orbital evolution MEO - ETALON
The Etalon satellites are ideal test bodies to study the
perturbations affecting the long term evolution of the
navigation satellite orbits
The Etalon satellites are ideal bodies to test the accuracy
of the orbital propagators.
The Tuned Inter-Range Vectors (TIRV) are state vectors of
the Etalon satellites used to track them by the Laser
stations.
The accuracy of the TIRVs is about 1 m.
ASI (Matera Station) provided also one year (2004) of fitted
orbits with weighted RMS of a few cm.
Alessandro Rossi
MEO Satellite Dynamics
Long term evolution of objects in GLONASS-like orbits
Conclusions
Orbital evolution
Orbital propagation
Orbital region survey
Orbital evolution MEO - ETALON
The Etalon satellites are ideal test bodies to study the
perturbations affecting the long term evolution of the
navigation satellite orbits
The Etalon satellites are ideal bodies to test the accuracy
of the orbital propagators.
The Tuned Inter-Range Vectors (TIRV) are state vectors of
the Etalon satellites used to track them by the Laser
stations.
The accuracy of the TIRVs is about 1 m.
ASI (Matera Station) provided also one year (2004) of fitted
orbits with weighted RMS of a few cm.
Alessandro Rossi
MEO Satellite Dynamics
Long term evolution of objects in GLONASS-like orbits
Conclusions
Orbital evolution
Orbital propagation
Orbital region survey
Orbital evolution MEO - ETALON
The Etalon satellites are ideal test bodies to study the
perturbations affecting the long term evolution of the
navigation satellite orbits
The Etalon satellites are ideal bodies to test the accuracy
of the orbital propagators.
The Tuned Inter-Range Vectors (TIRV) are state vectors of
the Etalon satellites used to track them by the Laser
stations.
The accuracy of the TIRVs is about 1 m.
ASI (Matera Station) provided also one year (2004) of fitted
orbits with weighted RMS of a few cm.
Alessandro Rossi
MEO Satellite Dynamics
Long term evolution of objects in GLONASS-like orbits
Conclusions
Orbital evolution
Orbital propagation
Orbital region survey
ASI vs. TIRV
The difference between the
precise orbits and the TIRV
elements is within a few tens of
meters in a and a few parts in
10−6 in e.
Alessandro Rossi
MEO Satellite Dynamics
Long term evolution of objects in GLONASS-like orbits
Conclusions
Orbital evolution
Orbital propagation
Orbital region survey
ASI vs. TIRV
The difference between the
precise orbits and the TIRV
elements is within a few tens of
meters in a and a few parts in
10−6 in e =⇒ use of TIRV
correct for the study.
Alessandro Rossi
MEO Satellite Dynamics
Long term evolution of objects in GLONASS-like orbits
Conclusions
Orbital evolution
Orbital propagation
Orbital region survey
ETALONs orbital evolution from TIRVs- Semiaxis
Alessandro Rossi
MEO Satellite Dynamics
Long term evolution of objects in GLONASS-like orbits
Conclusions
Orbital evolution
Orbital propagation
Orbital region survey
ETALONs orbital evolution from TIRVs - Eccentricity
Alessandro Rossi
MEO Satellite Dynamics
Long term evolution of objects in GLONASS-like orbits
Conclusions
Orbital evolution
Orbital propagation
Orbital region survey
ETALONs orbital evolution from TIRVs - Inclination
Alessandro Rossi
MEO Satellite Dynamics
Long term evolution of objects in GLONASS-like orbits
Conclusions
Orbital evolution
Orbital propagation
Orbital region survey
ETALONs orbital evolution from TIRVs - Node
Alessandro Rossi
MEO Satellite Dynamics
Long term evolution of objects in GLONASS-like orbits
Conclusions
Orbital evolution
Orbital propagation
Orbital region survey
ETALONs orbital evolution from TIRVs - Node
Alessandro Rossi
MEO Satellite Dynamics
Long term evolution of objects in GLONASS-like orbits
Conclusions
Orbital evolution
Orbital propagation
Orbital region survey
TIRV vs. FOP (ETALON 1 - 63 y)
Alessandro Rossi
MEO Satellite Dynamics
Long term evolution of objects in GLONASS-like orbits
Conclusions
Orbital evolution
Orbital propagation
Orbital region survey
TIRV vs. SATRAP (ETALON 2 - 200 y)
4
2.9
x 10
2.8
Apogee/Perigee [km]
2.7
GPS
2.6
2.5
2.4
2.3
2.2
0
1
2
3
4
Time [days]
Alessandro Rossi
5
6
7
4
x 10
MEO Satellite Dynamics
Orbital evolution
Orbital propagation
Orbital region survey
Long term evolution of objects in GLONASS-like orbits
Conclusions
ETALON 2 Ecc. evolution 1990–2004 w. SATRAP
−4
10
ETALON 2
x 10
ALL
9
There is a secular
growth of the
eccentricity
caused mainly by
the third body
perturbations.
Eccentricity
SUN MOON NOPRES
SUN MOON NOPRES J22
8
SUN − NO MOON
J88 + RAD. PRESS
J88 − NO RAD. PRESS
J44
7
J33
J22
J2
6
0
Alessandro Rossi
1000
2000
3000
Time [days]
MEO Satellite Dynamics
4000
5000
6000
Long term evolution of objects in GLONASS-like orbits
Conclusions
Orbital evolution
Orbital propagation
Orbital region survey
Spectrum of the ETALON 2 Eccentricity perturbations
Alessandro Rossi
MEO Satellite Dynamics
Long term evolution of objects in GLONASS-like orbits
Conclusions
Orbital evolution
Orbital propagation
Orbital region survey
TIRV vs. FOP - Eccentricity Spectrum
Alessandro Rossi
MEO Satellite Dynamics
Long term evolution of objects in GLONASS-like orbits
Conclusions
Orbital evolution
Orbital propagation
Orbital region survey
SATRAP - Eccentricity Spectrum
Alessandro Rossi
MEO Satellite Dynamics
Long term evolution of objects in GLONASS-like orbits
Conclusions
Orbital evolution
Orbital propagation
Orbital region survey
SATRAP - Eccentricity Spectrum (long period)
Alessandro Rossi
MEO Satellite Dynamics
Long term evolution of objects in GLONASS-like orbits
Conclusions
Orbital evolution
Orbital propagation
Orbital region survey
Exploration of the MEO region
The influence of the node and semimajor axis on the
eccentricity growth of the orbits is explored by propagating, for
100 years with FOP and SATRAP, a large set of ETALON
clones.
Alessandro Rossi
MEO Satellite Dynamics
Long term evolution of objects in GLONASS-like orbits
Conclusions
Orbital evolution
Orbital propagation
Orbital region survey
Node clones
36 Etalon 2 like orbits, with
Ωi = 36.69◦ + (i × 10◦ ) i =
0, 35
Maximum eccentricity
growth after 100 years.
Alessandro Rossi
MEO Satellite Dynamics
Orbital evolution
Orbital propagation
Orbital region survey
Long term evolution of objects in GLONASS-like orbits
Conclusions
Semimajor axis clones
0.09
GLONASS
5:3
2:1
0.08
500 Etalon 2 like orbits
ai = 25400 + (i × 10) km,
i = 0, 459
(25400 < a < 29990).
Maximum eccentricity
0.07
0.06
GPS
0.05
0.04
0.03
0.02
Maximum eccentricity
growth after 100 years.
GALILEO
0.01
0
2.5
2.6
2.7
2.8
2.9
Semimajor axis [km]
Alessandro Rossi
MEO Satellite Dynamics
3
3.1
4
x 10
Long term evolution of objects in GLONASS-like orbits
Conclusions
Orbital evolution
Orbital propagation
Orbital region survey
Semiaxis clones - Ecc. evolution near resonance
0.07
ainiz = 26560
0.06
Eccentricity
0.05
0.04
0.03
0.02
ainiz = 26540
26550
0.01
26580
0
ainiz = 26570
0
0.5
1
1.5
2
2.5
3
Time [days]
Alessandro Rossi
MEO Satellite Dynamics
3.5
4
4
x 10
Orbital evolution
Orbital propagation
Orbital region survey
Long term evolution of objects in GLONASS-like orbits
Conclusions
Resonance onset
0.12
ONLY SUN AND MOON
0.1
0.08
Eccentricity
Only Luni-Solar
perturbations (no
gravity
harmonics)
0.06
0.04
0.02
0
0
0.5
1
1.5
2
2.5
Time [days]
Alessandro Rossi
MEO Satellite Dynamics
3
3.5
4
x 10
Orbital evolution
Orbital propagation
Orbital region survey
Long term evolution of objects in GLONASS-like orbits
Conclusions
Resonance onset
0.12
ONLY SUN AND MOON
ONLY GRAV. FIELD 2 x 2
0.1
Eccentricity
0.08
Only gravity
harmonics up to
degree ` = 2 and
order m = 2.
0.06
0.04
0.02
0
0
0.5
1
1.5
2
2.5
Time [days]
Alessandro Rossi
MEO Satellite Dynamics
3
3.5
4
x 10
Orbital evolution
Orbital propagation
Orbital region survey
Long term evolution of objects in GLONASS-like orbits
Conclusions
Resonance onset
0.12
ONLY SUN AND MOON
ONLY GRAV. FIELD 2 x 2
GRAV. FIELD 2 x 2 + SUN + MOON
0.1
0.08
Eccentricity
Gravity
harmonics up to
degree ` = 2 and
order m = 2 +
Luni-Solar
perturbations
0.06
0.04
0.02
0
0
0.5
1
1.5
2
2.5
Time [days]
Alessandro Rossi
MEO Satellite Dynamics
3
3.5
4
x 10
Orbital evolution
Orbital propagation
Orbital region survey
Long term evolution of objects in GLONASS-like orbits
Conclusions
Resonance onset
0.12
ONLY SUN AND MOON
ONLY GRAV. FIELD 2 x 2
GRAV. FIELD 2 x 2 + SUN + MOON
ONLY GRAV. FIELD 3 x 3
0.1
0.08
Eccentricity
Only gravity
harmonics up to
degree ` = 3 and
order m = 3 (No
Luni-Solar
perturbations).
0.06
0.04
0.02
0
0
0.5
1
1.5
2
2.5
Time [days]
Alessandro Rossi
MEO Satellite Dynamics
3
3.5
4
x 10
Long term evolution of objects in GLONASS-like orbits
Conclusions
Orbital evolution
Orbital propagation
Orbital region survey
Resonance theory
Given the expansion of the gravity field in terms of spherical
harmonics, the resonance argument for a J`m coefficient is:
ψlmpq = [(` − 2p + q)n − mθ̇]t + (l − 2p)ω + m(Ω − θ0 )
where: 0 ≤ p ≤ `, −∞ < q < +∞
n = mean motion
Ω, ω: argument of node and pericenter
θ = θ0 + θ̇ = right ascension of the Greenwich meridian
Large perturbations occur when:
ψ̇lmpq = [(` − 2p + q)n − mθ̇]t + (l − 2p)ω̇ + mΩ̇ ' 0
Alessandro Rossi
MEO Satellite Dynamics
Long term evolution of objects in GLONASS-like orbits
Conclusions
Orbital evolution
Orbital propagation
Orbital region survey
Resonance theory
For MEOs we have:
ω̇ ' 10−5 , Ω̇ ' −10−4 , n ' 2, θ̇ ' 1
So that:
2'
n
m
'
`
−
2p
+q
θ̇
Therefore the main resonant coefficients are:
` = 3, m = 2 for p = 1 and q = 0
` = 4, m = 4 for p = 1 and q = 0
` = 2, m = 2 for p = 0, 1 and q = ±1
Alessandro Rossi
MEO Satellite Dynamics
Orbital evolution
Orbital propagation
Orbital region survey
Long term evolution of objects in GLONASS-like orbits
Conclusions
Resonance onset
0.12
ONLY SUN AND MOON
ONLY GRAV. FIELD 2 x 2
GRAV. FIELD 2 x 2 + SUN + MOON
ONLY GRAV. FIELD 3 x 3
ONLY GRAV. FIELD 4 x 4
0.1
0.08
Eccentricity
Only gravity
harmonics up to
degree ` = 4 and
order m = 4 (No
Luni-Solar
perturbations).
0.06
0.04
0.02
0
0
0.5
1
1.5
2
2.5
Time [days]
Alessandro Rossi
MEO Satellite Dynamics
3
3.5
4
x 10
Orbital evolution
Orbital propagation
Orbital region survey
Long term evolution of objects in GLONASS-like orbits
Conclusions
Resonance onset
0.12
ONLY SUN AND MOON
ONLY GRAV. FIELD 2 x 2
GRAV. FIELD 2 x 2 + SUN + MOON
ONLY GRAV. FIELD 3 x 3
ONLY GRAV. FIELD 4 x 4
ONLY GRAV. FIELD 10 x 10
0.1
0.08
Eccentricity
Only gravity
harmonics up to
degree ` = 10
and order
m = 10 (No
Luni-Solar
perturbations).
0.06
0.04
0.02
0
0
0.5
1
1.5
2
2.5
Time [days]
Alessandro Rossi
MEO Satellite Dynamics
3
3.5
4
x 10
Orbital evolution
Orbital propagation
Orbital region survey
Long term evolution of objects in GLONASS-like orbits
Conclusions
Resonance onset
0.12
ONLY SUN AND MOON
ONLY GRAV. FIELD 2 x 2
GRAV. FIELD 2 x 2 + SUN + MOON
ONLY GRAV. FIELD 3 x 3
ONLY GRAV. FIELD 4 x 4
ONLY GRAV. FIELD 10 x 10
GRAV. FIELD 10 x 10 + SUN
0.1
0.08
Eccentricity
Gravity
harmonics up to
degree ` = 10
and order
m = 10 + Solar
perturbations
(i.e., no Moon).
0.06
0.04
0.02
0
0
0.5
1
1.5
2
2.5
Time [days]
Alessandro Rossi
MEO Satellite Dynamics
3
3.5
4
x 10
Orbital evolution
Orbital propagation
Orbital region survey
Long term evolution of objects in GLONASS-like orbits
Conclusions
Resonance onset
0.12
ONLY SUN AND MOON
ONLY GRAV. FIELD 2 x 2
GRAV. FIELD 2 x 2 + SUN + MOON
ONLY GRAV. FIELD 3 x 3
ONLY GRAV. FIELD 4 x 4
ONLY GRAV. FIELD 10 x 10
GRAV. FIELD 10 x 10 + SUN
GRAV. FIELD 10 x 10 + MOON
0.1
0.08
Eccentricity
Gravity
harmonics up to
degree ` = 10
and order
m = 10 + Lunar
perturbations
(i.e., no Sun).
0.06
0.04
0.02
0
0
0.5
1
1.5
2
2.5
Time [days]
Alessandro Rossi
MEO Satellite Dynamics
3
3.5
4
x 10
Orbital evolution
Orbital propagation
Orbital region survey
Long term evolution of objects in GLONASS-like orbits
Conclusions
Resonance onset
0.12
ONLY SUN AND MOON
ONLY GRAV. FIELD 2 x 2
GRAV. FIELD 2 x 2 + SUN + MOON
ONLY GRAV. FIELD 3 x 3
ONLY GRAV. FIELD 4 x 4
ONLY GRAV. FIELD 10 x 10
GRAV. FIELD 10 x 10 + SUN
GRAV. FIELD 10 x 10 + MOON
GRAV. FIELD 3 x 3 + SUN + MOON
0.1
0.08
Eccentricity
Gravity
harmonics up to
degree ` = 3 and
order m = 3 +
Lunar and Solar
perturbations
0.06
0.04
0.02
0
0
0.5
1
1.5
2
2.5
Time [days]
Alessandro Rossi
MEO Satellite Dynamics
3
3.5
4
x 10
Orbital evolution
Orbital propagation
Orbital region survey
Long term evolution of objects in GLONASS-like orbits
Conclusions
Resonance onset
0.12
ONLY SUN AND MOON
ONLY GRAV. FIELD 2 x 2
GRAV. FIELD 2 x 2 + SUN + MOON
ONLY GRAV. FIELD 3 x 3
ONLY GRAV. FIELD 4 x 4
ONLY GRAV. FIELD 10 x 10
GRAV. FIELD 10 x 10 + SUN
GRAV. FIELD 10 x 10 + MOON
GRAV. FIELD 3 x 3 + SUN + MOON
GRAV. FIELD 4 x 4 + SUN + MOON
0.1
0.08
Eccentricity
Gravity
harmonics up to
degree ` = 4 and
order m = 4 +
Lunar and Solar
perturbations
0.06
0.04
0.02
0
0
0.5
1
1.5
2
2.5
Time [days]
Alessandro Rossi
MEO Satellite Dynamics
3
3.5
4
x 10
Orbital evolution
Orbital propagation
Orbital region survey
Long term evolution of objects in GLONASS-like orbits
Conclusions
Resonance onset
0.12
ONLY SUN AND MOON
ONLY GRAV. FIELD 2 x 2
GRAV. FIELD 2 x 2 + SUN + MOON
ONLY GRAV. FIELD 3 x 3
ONLY GRAV. FIELD 4 x 4
ONLY GRAV. FIELD 10 x 10
GRAV. FIELD 10 x 10 + SUN
GRAV. FIELD 10 x 10 + MOON
GRAV. FIELD 3 x 3 + SUN + MOON
GRAV. FIELD 4 x 4 + SUN + MOON
GRAV. FIELD 6 x 6 + SUN + MOON
0.1
0.08
Eccentricity
Gravity
harmonics up to
degree ` = 6 and
order m = 6 +
Lunar and Solar
perturbations
0.06
0.04
0.02
0
0
0.5
1
1.5
2
2.5
Time [days]
Alessandro Rossi
MEO Satellite Dynamics
3
3.5
4
x 10
Orbital evolution
Orbital propagation
Orbital region survey
Long term evolution of objects in GLONASS-like orbits
Conclusions
Resonance onset
0.12
ONLY SUN AND MOON
ONLY GRAV. FIELD 2 x 2
GRAV. FIELD 2 x 2 + SUN + MOON
ONLY GRAV. FIELD 3 x 3
ONLY GRAV. FIELD 4 x 4
ONLY GRAV. FIELD 10 x 10
GRAV. FIELD 10 x 10 + SUN
GRAV. FIELD 10 x 10 + MOON
GRAV. FIELD 3 x 3 + SUN + MOON
GRAV. FIELD 4 x 4 + SUN + MOON
GRAV. FIELD 6 x 6 + SUN + MOON
GRAV. FIELD 8 x 8 + SUN + MOON
0.1
0.08
Eccentricity
Gravity
harmonics up to
degree ` = 8 and
order m = 8 +
Lunar and Solar
perturbations.
0.06
0.04
0.02
0
0
0.5
1
1.5
2
2.5
Time [days]
Alessandro Rossi
MEO Satellite Dynamics
3
3.5
4
x 10
Orbital evolution
Orbital propagation
Orbital region survey
Long term evolution of objects in GLONASS-like orbits
Conclusions
Resonance onset
0.12
ONLY SUN AND MOON
ONLY GRAV. FIELD 2 x 2
GRAV. FIELD 2 x 2 + SUN + MOON
ONLY GRAV. FIELD 3 x 3
ONLY GRAV. FIELD 4 x 4
ONLY GRAV. FIELD 10 x 10
GRAV. FIELD 10 x 10 + SUN
GRAV. FIELD 10 x 10 + MOON
GRAV. FIELD 3 x 3 + SUN + MOON
GRAV. FIELD 4 x 4 + SUN + MOON
GRAV. FIELD 6 x 6 + SUN + MOON
GRAV. FIELD 8 x 8 + SUN + MOON
GRAV. FIELD 10 x 10 + SUN + MOON
0.1
0.08
Eccentricity
Gravity
harmonics up to
degree ` = 10
and order
m = 10 + Lunar
and Solar
perturbations.
0.06
0.04
0.02
0
0
0.5
1
1.5
2
2.5
Time [days]
Alessandro Rossi
MEO Satellite Dynamics
3
3.5
4
x 10
Orbital evolution
Orbital propagation
Orbital region survey
Long term evolution of objects in GLONASS-like orbits
Conclusions
Node dependence at resonance
Initial semiaxis = 26560 km (2:1 resonance)
0.25
0.2
Eccentricity
Maximum
eccentricity
growth as a
function of initial
longitude of the
node, at the
resonance
semiaxis.
0.15
0.1
0.05
0
0
50
100
150
200
250
Initial longitude of the node [degrees]
Alessandro Rossi
MEO Satellite Dynamics
300
350
Long term evolution of objects in GLONASS-like orbits
Conclusions
Orbital evolution
Orbital propagation
Orbital region survey
Node dependence at resonance
Initial semiaxis = 26560 km (2:1 resonance)
0.25
0.2
Eccentricity
Maximum
eccentricity
growth as a
function of initial
Moon angle
(difference
between initial
longitude of the
node and initial
longitude of the
Moon), at the
resonance
semiaxis.
0.15
0.1
0.05
0
−50
0
50
100
150
200
Initial angular separation of the Moon [degrees]
Alessandro Rossi
MEO Satellite Dynamics
250
Long term evolution of objects in GLONASS-like orbits
Conclusions
Agenda
1
Long term evolution of objects in GLONASS-like orbits
Orbital evolution
Orbital propagation
Orbital region survey
2
Conclusions
Alessandro Rossi
MEO Satellite Dynamics
Long term evolution of objects in GLONASS-like orbits
Conclusions
(No) conclusions and Future work
Analytical theory of the resonance (from Kozai et al.)
Disposal of spacecraft
Collision risk
Alessandro Rossi
MEO Satellite Dynamics