Reaction to the empty set

5.4.4.1 Introduction

In this module, two-body propagation is used over one day and orbit maintenance is assumed. J2 and J4 perturbations are not used. The module propagates a Walker constellation of swarms.

There is one sub-plane per swarm and logarithmic spacing is used between the sub-orbits (or

“rings”) of the swarm. The swarms are configured to project a circle in the horizontal plane.

The swarm configuration consists of one center-satellite and three sub-satellites per sub-orbit.

This configuration was used to create the baselines necessary to measure beacon angle of arrival data and to create the fill factor to eliminate ambiguity. The orbit code is contained within the orbitprop.m and swarmorbits.m files.

5.4.4.2 Required Inputs

The orbit module takes inputs from the following modules:

DESIGN CONSTANTS TIME

The inputs are as follows:

DESIGN.perigee_altitude DSEIGN.apogee_altitude DESIGN.number_of_planes DESIGN.sats_per_swarm DESIGN.radius_of_swarm DESIGN.swarms_per_plane

CONSTANTS.subplanes_per_swarm (=1 for B-TOS) CONSTANTS.arg_perigee (=0)

CONSTANTS.inclination CONSTANTS.earth_radius CONSTANTS.earth_mu

CONSTANTS.propagation_time_secs CONSTANTS.propagation_steptime_secs CONSTANTS.walker_interplane_spacing CONSTANTS.walker_raan_spread

CONSTANTS.propagate_only_centersat (0 or 1) TIME.time_resolution

TIME.no_edp_sats

5.4.4.3 Output Descriptions

The outputs of swarmorbits.m are as follows:

SWARMORBITS.swarmsat:

A matrix of the orbital elements for each satellite, including apogee and perigee altitudes, inclination, argument of perigee, RAAN, and true anomaly.

ERROR.swarmorbits.anyerror:

Equals 1 if there are any errors in swarmorbits.m, otherwise zero.

ERROR.swarmorbits.satsperswarm_is_1:

Equals 1 if true, otherwise zero.

ERROR.swarmorbits.subplanes_lessthan_satsperswarm:

Equals 1 if true, otherwise zero.

The outputs of orbitprop.m are as follows:

ORBIT.average_revisit_time:

Average revisit time for a grid of points; the grid is based upon the spatial resolution.

ORBIT.spatial_resolution:

The nadir angle swept out by the center satellite during t = time resolution.

ORBIT.instant_percent_global_cov:

Percentage of grid covered in t = time resolution; does not include polar regions north and south of latitude 65 degrees because grid currently does not extend to those regions.

ERROR.orbitprop.error_from_swarmorbits:

Equals 1 if an error is output from swarmorbits.m, otherwise zero.

ERROR.orbitprop.satsperswarm_morethan_26:

Equals 1 if true, otherwise zero.

ERROR.orbitprop.no_edp_sats:

Equals 1 if there are no working EDP satellites, otherwise zero.

5.4.4.4 Key Assumptions Fundamental equations

The orbital parameters for each of the satellites in the swarm must be determined in order to provide the proper inputs to STK. The original swarm projects a vertical 2:1 ellipse along the global orbit. The ionospheric mapping mission requires distinct measurements distributed horizontally over a segment of the ionosphere. We decided to project a circle in the horizontal plane with a radius equal to the semi-major axis of the ellipse. The individual satellites must be given cross-track elements relative to the reference orbit at the center of the swarm. These incremental differences in orbital parameters are derived from the geometry of the swarm and uniquely describe the orbit for each satellite. These parameters include the following:

• Delta RAAN (Right Ascension of the Ascending Node)

• Delta Inclination

• Delta perigee

• Delta apogee

• Delta argument of perigee

• Delta true anomaly

The spatial resolution is defined as a conical angle originating at the center of the Earth and is determined by the time resolution (time between data sets) and the orbital velocity. The spatial resolution projects a circle on the surface of the Earth. The effective field of view (FOV) is a conical angle that originates at the center of the swarm and projects the same size circle on the Earth’s surface. The FOV is used in STK to calculate revisit time and global coverage statistics.

Rationale for any simplifications

The average delta V’s for station-keeping due to atmospheric drag were found to be small at the altitudes considered, so a constant was used in the spacecraft module. It was later determined that for large swarm radii, the delta V requirements for formation-keeping in the outer sub-orbits can be large due to J2 effects. This could be alleviated by not projecting a horizontal circle, at least for the outer sub-orbits. This sensitivity analysis has been done for some frontier architectures, but not for the entire tradespace.

The effective field of view was utilized to emulate an optical system so that the coverage and revisit statistics could be calculated by STK.

Evolution of calculations

The module was developed using the A-TOS code as a baseline. Since the number of sub-orbits per swarm was not a design variable in B-TOS, the logarithmic spacing calculation was not used in the same manner. The number of satellites per swarm constrained the number of sub-orbits by placing one satellite at the center and three in each succeeding sub-orbit. Discrete sets of satellite numbers were then considered.

5.4.4.5 Fidelity Assessment

The module used STK to ensure high fidelity orbit trajectories. This required a Matlab-STK interface.

5.4.4.6 Verification

Extreme cases were tested in order to test the assumptions. Visual inspections of the swarm geometry in three-dimensional STK animations were also used to verify the configuration.