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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/m³) 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