2.0 Schemes Considered for Constellation Deployment

2.1 Phase I Design: In-Plane Single Orbit configuration

Initially an ecliptic plane constellation with the 6 transponders being placed at the orbital radius from the sun of Jupiter but displaced in increments of 50º from the gas giant was envisioned.   This first option was dismissed immediately for concerns over transponder grouping near Lagrange points in the orbit of Jupiter and the lack of heliocentric Z-axis displacement.   The entire concept of passive navigation is founded in the maintenance of distinct positions for each transponder.   And, operations over long periods of time would place prohibitively large fuel requirements on the transponders for station keeping away from Jovian Lagrange points.   Also, without any significant out of plane displacement, the entire system could be hamstrung in attempts to position coplanar spacecraft.

2.2 Phase II: Two Orbit Primary/Secondary Configuration

Primary Transponder Ring and Gravitational Effects of Jupiter

Certain concerns were felt about the long term, approximately 100 years, effects of Jupiter's gravity upon a zero inclination to the ecliptic P-STAR transponder ring of three spacecraft.   A massive object placed 90º forward or rearward or Jupiter's orbit would only experience approximately 0.05% of the Sun's gravitational pull from Jupiter.   Intuition indicated that motion toward the Lagrange points from 90º would be a noncritical factor since displacement between transponders (not the actual positions) are the main concern; but the integrated effects could not be clearly predicted by inspection.

Therefore, a test mass was simulated in a position 90º forward of Jupiter's orbit for a period of approximately 96 years (8 Jovian orbital periods).   The interest was not in accuracy of orbital determination, but rather in the general effect of third-body Jovian gravity on the displacement of the mass from Jupiter.   The simulation was carried out numerically by integrating the mass trajectory as a two body system about the Sun with Jovian gravity as a perturbation using a 7-8 order Runge-Kutte integrator with a step size of .001*TJ (TJ=Jovian orbital period), approximately 4 days.   During this integration step, the Jovian gravity effect was considered to be a constant value.   After 8*TJ the forward orbital displacement of the mass has deteriorated to approximately 65º from 90º and was much nearer the Jovian Lagrange point.

Although this represented a downtrack motion of 25º relative to Jupiter, the mass was still in a stable orbit with a semi-major axis of approximately 5.5 AU.   The interpretation of this motion was that for a period of 100 years a primary P-STAR transponder ring of three spacecraft operating at 90º, 180º, and -90º displaced from Jupiter's orbit at 5.2 AU would maintain stability and substantial displacement from each other.

Secondary Transponder Ring and Extra-Ecliptic Displacement

With three stations placed in a 'four corners' arrangement with Jupiter, the Heliocentric X-Y position of a spacecraft can be passively determined.   However, P-STAR is required to:

  1. Achieve accurate Z axis positioning.
  2. Provide a fourth transponder for time variable estimation.
  3. Provide alternate transponders in the event that the line of sight between a spacecraft and a transponder passes through the Solar blind spot.
With these points in mind, a secondary transponder ring in a slightly inclined orbit was desired.   Unlike the primary transponder ring, where delivery was assumed via Hohmann transfer orbit, the secondary ring would utilize the gravity of Jupiter in a hyperbolic flyby to create the transponders' inclination.   The secondary transponder formation would also consist of three transponders at the points an equilateral triangle inscribed within the inclined elliptical orbit at 5+ AU.   Inclined orbits would ensure that at least one of the secondary ring transponders would be out of plane at all times.   It was also desirable that the orbital period of the secondary transponder ring be such that Jovian gravitational interaction would be at a minimum for the duration of the P-STAR mission.  

The establishment of such a system can be nominally achieved if one assumes certain convenient parameters.   Plausibly, one can estimate circular non-inclined orbits for Earth and Jupiter.   Also, one can assume frequent Hohmann transfer launch opportunities given the 13 month synodic period of the Earth-Jupiter system.   One can assume:

  1. existing motors with unspecified masses and Specific Impulses (Isp's) of 400 seconds and mass fractions of 0.1 for tailor made staging - very unrealistic
  2. placement in non-inclined Earth orbit by a large delivery system (the Titan IV was approximated) without penalty to spacecraft mass - only simplifies launch window timing
  3. zero inclination of the Earth's rotational axis to the ecliptic - only simplifies launch window timing
  4. maximum payload delivery for the approximated systems
However, the results of such an analysis were as follows:

Primary Transponder Ring - TK Model ptr.tk

Mission Time Event Mass
T+0:00 Titan IV Launch
860,000 kg
T+45:00 min LEO Delivery of Upper Stage
21,640 kg
T+2:00:00 hr Interplanetary Injection DV = 6.31 km/s
2048 kg
T+2.73 year 5 AU Circularization Burn DV = 5.647 km/s
275 kg

Secondary Transponder Ring - TK Model str.tk

Mission Time Event Mass
T+0:00 Titan IV Launch
860,000 kg
T+45:00 min LEO Delivery of Upper Stage
21,640 kg
T+2:00:00 hr Interplanetary Injection DV = 6.31 km/s
2048 kg
T+2.73 year 6 AU Inclined Elliptical Burn DV = 3 km/s
792 kg

The circularization DV for the primary transponder ring (calculated from 35flyby.tk) required such a large mass that, given a Titan IV mass limit for LEO, only 275 kg could be placed in a 5 AU heliocentric circular orbit.   While the technology and mass involved with each transponder will not be analytically explored, it is assumed that a minimum mass of 500 kg would be required for an operational unit.  

However, it was felt that the inclined orbits of the secondary transponder ring (calculated in rctooe.tk):

Secondary Transponder Ring Orbital Elements

Semi-Major Axis
5.73 AU
Long. Asc. Node
Arg. Periapse
13.73 year

corresponded to a promising enough payload to become the basis for a more realistic final launch and delivery scheme.  

2.3 Final Selection: Six Inclined Elliptical Orbits

The final design selection was driven by the apparent success (in terms of orbital elements and final payload mass) of the Phase II Secondary Transponder Ring orbit and by the need to instill more rigourous assumptions on the heliocentric environment.   The new scheme abandons the primary ring and instead calls for six separate transponder orbits with the following commonalities:

Final Constellation Scheme Orbital Elements

Semi-Major Axis
5.4022 AU
Arg. Periapse
12.576 year

Given the synodic period of the Earth-Jupiter system it will be possible to launch toward Jupiter approximately every 13 months.   The Hohmann transfer orbit time from Earth to Jupiter is 2.73 years.   During this time Jupiter moves through 83º of its orbit.   Since the Hohmann orbit's argument of ascending node is aligned with the Earth-Sun line at time of launch, Jupiter must be 97º ahead of Earth in the ecliptic plane at time of interplanetary injection.   Therefore, the hohangle.f and the position.f F77 programs were used to determine potential launch dates.   A note should be made here that the positions calculated from these routines were based upon the 1984 Astronomical Almanac ephemeris tables which contains the 'actual' inclinations and eccentricies of Earth and Jupiter rather than the noninclined circular approximations used here.   However, since these eccentricies and inclinations are small, a tolerance value is included in the hohangle.f launch criterion, and booster fuel calculations include contingency burns, the launch dates provided should suffice for the scope of this study.

Thus, in hohangle.f and position.f the orbits of Earth and Jupiter were both assumed to be circular and noninclined, which is more or less true, and the Space Shuttle and Titan IV were again used as LEO delivery systems.   However, the following assumptions were adapted from Phase II:

  1. only existing upper stage motors would be used with fuel restrictions and launch system fairing in place
  2. placement in Earth orbit at approximately 28º inclination to the equator
  3. 23º inclination of the Earth's rotational axis to the ecliptic
  4. maximum payload delivery for the approximated systems (Titan IV and Shuttle)

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Tim Crain
Graduate Aerospace Engineering
The University of Texas at Austin
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