Orbit Analysis
Reference: ECSS-E-ST-10-04C (Space Environment), IADC Space Debris Mitigation Guidelines, CubeSat Design Specification Rev. 14
1. Orbital Elements
1.1 Keplerian Elements
| Element |
Symbol |
Value |
Notes |
| Semi-major axis |
a |
6921.14 km |
R_E + h = 6371.0 + 550.0 |
| Eccentricity |
e |
0.0001 |
Near-circular (launcher insertion) |
| Inclination |
i |
97.59 deg |
Sun-synchronous requirement |
| RAAN |
Omega |
TBD (launch-dependent) |
Determines LTAN |
| Argument of Perigee |
omega |
0 deg |
Circular orbit, undefined |
| True Anomaly |
nu |
0 deg (epoch) |
Arbitrary at epoch |
| LTAN |
- |
10:30 |
Local Time of Ascending Node |
1.2 Derived Parameters
Orbital Period: T = 2*pi * sqrt(a^3 / mu)
= 2*pi * sqrt(6921.14^3 / 398600.4418)
= 5742.4 s = 95.71 min
Mean Motion: n = 2*pi / T = 1.094e-3 rad/s = 15.04 rev/day
Orbital Velocity: v = sqrt(mu / a) = sqrt(398600.4 / 6921.14) = 7.59 km/s
Ground Track Speed: v_gt = v * cos(i) * (R_E / a) = 7.04 km/s (approx)
Where mu = 398600.4418 km³/s² (Earth gravitational parameter)
2. Sun-Synchronous Orbit (SSO) Derivation
2.1 SSO Condition
A sun-synchronous orbit requires the RAAN to precess at +0.9856 deg/day (matching Earth’s
orbital rate around the Sun). The J2 perturbation provides this precession:
dOmega/dt = -3/2 * n * J2 * (R_E/a)^2 * cos(i) / (1 - e^2)^2
Setting dOmega/dt = +0.9856 deg/day = 1.991e-7 rad/s:
cos(i) = - (dOmega/dt) * 2 * a^2 * (1-e^2)^2 / (3 * n * J2 * R_E^2)
Where:
J2 = 1.08263e-3 (Earth oblateness)
R_E = 6371.0 km
a = 6921.14 km
n = 1.094e-3 rad/s
cos(i) = -(1.991e-7) * 2 * 6921.14^2 / (3 * 1.094e-3 * 1.08263e-3 * 6371.0^2)
= -0.1323
i = arccos(-0.1323) = 97.59 deg
2.2 SSO Inclination vs. Altitude
| Altitude (km) |
Semi-major axis (km) |
Required Inclination (deg) |
| 400 |
6771 |
97.05 |
| 450 |
6821 |
97.20 |
| 500 |
6871 |
97.40 |
| 550 |
6921 |
97.59 |
| 600 |
6971 |
97.79 |
| 650 |
7021 |
97.99 |
| 700 |
7071 |
98.19 |
3. Perturbation Analysis
3.1 J2 Oblateness (Dominant Perturbation)
| Effect |
Formula |
Rate |
Impact |
| RAAN precession |
dOmega/dt = -3/2 * n * J2 * (R_E/a)^2 * cos(i) |
+0.9856 deg/day |
Maintains SSO |
| Argument of perigee drift |
domega/dt = 3/4 * n * J2 * (R_E/a)^2 * (5*sin^2(i) - 4) |
+2.17 deg/day |
Irrelevant (circular) |
| Mean anomaly secular drift |
dM/dt correction |
+0.003 deg/day |
Absorbed into mean motion |
Higher-order zonal harmonics (J3, J4) contribute < 0.01 deg/day and are neglected for mission planning.
3.2 Atmospheric Drag
Drag acceleration: a_drag = -1/2 * rho * v^2 * (C_D * A_ref / m)
Ballistic coefficient: BC = m / (C_D * A_ref)
Where:
C_D = 2.2 (typical for CubeSat)
A_ref = 0.03 m² (3U cross-section: 0.1m x 0.3m)
m = 2.18 kg (CBE mass)
BC = 2.18 / (2.2 * 0.03) = 33.0 kg/m²
Atmospheric density at 550 km (NRLMSISE-00 model):
| Solar Activity (F10.7) |
rho (kg/mÂł) |
Drag Accel (m/s²) |
Altitude Loss (km/yr) |
| Solar minimum (70 sfu) |
2.0e-13 |
5.3e-7 |
0.8 |
| Solar moderate (140 sfu) |
1.5e-12 |
3.9e-6 |
6.2 |
| Solar maximum (250 sfu) |
8.0e-12 |
2.1e-5 |
33.1 |
3.3 Solar Radiation Pressure (SRP)
SRP acceleration: a_srp = P_sr * (1 + q) * A_ref / m
Where:
P_sr = S/c = 1361 / 3e8 = 4.54e-6 N/m² (solar radiation pressure at 1 AU)
q = 0.6 (reflectivity, 0 = absorb, 1 = perfect mirror)
A_ref = 0.03 m² (assuming worst-case orientation)
m = 2.18 kg
a_srp = 4.54e-6 * 1.6 * 0.03 / 2.18 = 1.0e-7 m/s²
| Perturbation |
Acceleration (m/s²) |
Period Effect |
Relative Magnitude |
| J2 |
~1e-3 |
Secular precessions |
Dominant |
| Drag (moderate solar) |
~4e-6 |
Orbit decay |
Secondary |
| SRP |
~1e-7 |
Long-period oscillations |
Tertiary |
| Lunar/Solar gravity |
~5e-8 |
Long-period |
Negligible |
| Solid Earth tides |
~1e-9 |
Short-period |
Negligible |
3.4 Orbit Maintenance (Delta-V Budget)
No orbit maintenance is planned (no propulsion system). Expected natural evolution:
| Year |
Altitude Estimate (km) |
Notes |
| 0 (launch) |
550.0 |
Injection altitude |
| 0.5 |
547-549 |
Minimal decay |
| 1.0 |
543-548 |
Depends on solar cycle |
| 1.5 |
537-545 |
Mission requirement: h > 500 km |
| 2.0 (EOL) |
530-540 |
End of nominal mission |
4. Eclipse Analysis
4.1 Eclipse Geometry
Sun direction
<----
/--------\ Shadow cylinder
/ SUNLIT \ (cylindrical approximation)
------/ ORBIT \------
| / ___ \ |
| | / \ | | Eclipse region
| | | EARTH | | |
| | \_____/ | |
| \ / |
------\ /------
\ ECLIPSE /
\--------/
Eclipse condition: satellite enters Earth's shadow cone
Eclipse half-angle: alpha_e = arcsin(R_E / (R_E + h)) = 66.9 deg
4.2 Eclipse Duration vs. Beta Angle
The beta angle is the angle between the orbital plane and the Earth-Sun line.
For SSO at 10:30 LTAN, beta varies seasonally:
Eclipse fraction: f_e = (1/pi) * arccos(sqrt(h^2 + 2*R_E*h) / ((R_E+h)*cos(beta)))
This is valid when |beta| < beta_star, where:
beta_star = arcsin(R_E / (R_E + h)) = arcsin(6371/6921) = 66.9 deg
(Above beta_star: no eclipse, full sunlight)
| Beta (deg) |
Eclipse Duration (min) |
Eclipse Fraction (%) |
Season (10:30 LTAN) |
| 0 |
35.7 |
37.3 |
Equinox (worst) |
| 10 |
35.3 |
36.9 |
Near equinox |
| 20 |
34.1 |
35.6 |
Moderate |
| 30 |
31.9 |
33.3 |
Moderate |
| 40 |
28.4 |
29.7 |
Approaching solstice |
| 50 |
22.8 |
23.8 |
Near solstice |
| 60 |
12.7 |
13.3 |
Near full sun |
| 66.9 |
0.0 |
0.0 |
Full sun period |
For a 10:30 LTAN SSO at 550 km, the beta angle ranges approximately from
-23.4 + 10.5 = -12.9 deg to +23.4 + 10.5 = +33.9 deg over a year, meaning eclipses occur
year-round but vary in duration.
5. Ground Station Access Statistics
5.1 Tashkent Ground Station (41.3N, 69.2E)
Analysis performed using SGP4 propagator over 30-day simulation:
| Minimum Elevation |
Passes/Day |
Avg Duration (min) |
Max Duration (min) |
Total Contact/Day (min) |
| 0 deg |
8-10 |
9.2 |
14.1 |
78 |
| 5 deg |
6-8 |
8.0 |
12.8 |
54 |
| 10 deg |
5-7 |
7.2 |
11.5 |
43 |
| 20 deg |
3-5 |
5.8 |
9.2 |
22 |
| 30 deg |
2-3 |
4.5 |
7.1 |
11 |
5.2 Access Gap Analysis
| Metric |
Value |
| Average gap between passes |
2.5 hours |
| Maximum gap (worst case) |
8.2 hours |
| Minimum gap (consecutive high passes) |
92 min (1 orbit) |
| Passes with max elevation > 60 deg |
~2 per day |
| Passes with max elevation > 30 deg |
~4 per day |
5.3 Ground Station Pass Geometry
Elevation
(deg)
90 |
| * (max elevation)
60 | * *
| * *
30 | * *
| * *
10 | * * (AOS/LOS at 10 deg min elev)
5 | * AOS * LOS
0 |---*--+--+--+--+--+--+--+--+--*--> Time (min from AOS)
0 1 2 3 4 5 6 7 8 9 10
Typical high-elevation pass profile (max el = 75 deg)
Total duration: ~11 min above 10 deg elevation
5.4 Data Volume Per Pass
| Link |
Rate (effective) |
5-min pass |
8-min pass |
12-min pass |
| UHF Downlink |
4,080 bps |
150 KB |
240 KB |
360 KB |
| S-band Downlink |
117,760 bps |
4.3 MB |
6.9 MB |
10.4 MB |
| UHF Uplink |
4,080 bps |
150 KB |
240 KB |
360 KB |
6. Repeat Ground Track Analysis
6.1 Ground Track Repeat Condition
A repeat ground track occurs when:
N_orbits * T_orbit = M_days * T_sidereal_day
Where T_sidereal_day = 86164.1 s
For T_orbit = 5742.4 s:
Revs per sidereal day = 86164.1 / 5742.4 = 15.005
Near-repeat patterns:
15 revs in 1 day: drift = 0.005 * 360/15 = 0.12 deg/day longitude
211 revs in 14 days: near-exact repeat
422 revs in 28 days: even closer repeat
6.2 Ground Track Coverage
| Parameter |
Value |
| Inter-track spacing at equator |
360 / 15 = 24 deg = 2670 km |
| Track width (nadir imaging, FOV 30 deg) |
~290 km |
| Coverage gap at equator |
~2380 km |
| Days to fill equatorial gaps (no maneuver) |
~14 days |
| Latitude for daily coverage |
> 72 deg |
| Near-polar coverage |
Complete (overlapping tracks) |
6.3 Imaging Revisit Analysis
For a camera with 30 deg cross-track FOV at 550 km:
Swath width = 2 * h * tan(FOV/2) = 2 * 550 * tan(15) = 295 km
| Latitude |
Revisit Time (days) |
Overlap Fraction |
| 0 deg (equator) |
3-4 |
0% (gaps exist) |
| 20 deg |
2-3 |
~5% |
| 40 deg |
1-2 |
~25% |
| 60 deg |
< 1 |
~55% |
| 80 deg |
< 1 |
~90% |
7. Deorbit Timeline Analysis
7.1 Orbital Lifetime Estimation
Using the exponential decay model with atmospheric density from NRLMSISE-00:
dh/dt = -(rho * v * C_D * A) / (2 * m) * T / pi
Lifetime integration (numerical, from 550 km to 200 km reentry):
| Scenario |
F10.7 avg |
Initial Alt (km) |
Lifetime (years) |
Compliant (< 25 yr) |
| Solar minimum (quiet sun) |
70 |
550 |
18-22 |
YES |
| Solar moderate |
140 |
550 |
7-10 |
YES |
| Solar maximum (active) |
200 |
550 |
3-5 |
YES |
| Worst case (prolonged min) |
70 |
550 |
~22 |
YES (marginal) |
7.2 Altitude Decay Profile (Moderate Solar Activity)
Altitude
(km)
550 |*
525 | *
500 | *
475 | *
450 | *
425 | **
400 | **
375 | **
350 | ***
325 | ***
300 | ****
275 | ******
250 | ********
225 | *****
200 | ** (reentry)
+---+---+---+---+---+---+---+---+---+---+---> Years
0 1 2 3 4 5 6 7 8 9 10
Moderate solar activity (F10.7 = 140 sfu)
BC = 33 kg/m²
7.3 Compliance with Debris Mitigation Guidelines
| Guideline |
Requirement |
UniSat Status |
| IADC (25-year rule) |
Deorbit within 25 years of EOL |
COMPLIANT (7-22 years) |
| French Space Operations Act |
25-year limit |
COMPLIANT |
| US Government Orbital Debris Policy (2024) |
5-year limit for new missions |
NON-COMPLIANT in worst case |
| ESA Zero Debris Charter |
Best-effort minimization |
Under review |
Risk: The new US 5-year guideline may not be met under prolonged solar minimum.
Mitigation options: (1) deploy drag sail at EOL, (2) lower initial altitude to 500 km,
(3) accept risk if not launching under US jurisdiction.
8. Orbit Determination and Propagation
8.1 GNSS-Based OD
| Parameter |
Value |
| GNSS receiver |
u-blox MAX-M10S |
| Position accuracy |
< 2.5 m (3D RMS) |
| Velocity accuracy |
< 0.05 m/s (RMS) |
| Fix rate |
1 Hz (configurable to 0.1 Hz for power saving) |
| Cold start time |
< 30 s |
| TLE update frequency |
Every orbit (via GNSS fix) |
8.2 Propagation Model
For onboard orbit propagation (between GNSS fixes):
| Model |
Accuracy (24h propagation) |
CPU Load |
| SGP4/SDP4 (two-line elements) |
~1 km |
Minimal |
| J2 analytical |
~100 m |
Low |
| Numerical (RK4, J2+drag) |
~10 m |
Moderate |
Selected: SGP4 for coarse planning, J2 analytical for imaging predictions.
9. References
- Vallado, D.A., “Fundamentals of Astrodynamics and Applications”, 4th Ed., Chapter 9
- ECSS-E-ST-10-04C: Space Environment (2008)
- IADC-02-01: Space Debris Mitigation Guidelines, Rev. 2 (2020)
- US Government Orbital Debris Mitigation Standard Practices (2024 update)
- Picone, J.M., et al., “NRLMSISE-00 Empirical Model of the Atmosphere”, JGR, 2002
- Wertz, J.R., “Space Mission Engineering: The New SMAD”, Chapters 5-6