OBLATE-EARTH EFFECTS ON THE CALCULATION OF EC DURING SPACECRAFT REENTRY John B. Bacon(1), Mark J. Matney(1), (1)NASA Johnson Space Center XI-4, 2101 NASA Pkwy. Houston TX 77058 USA [email protected] (1)NASA Johnson Space Center XI-4, 2101 NASA Pkwy. Houston TX 77058 USA [email protected] ABSTRACT decay. All values are plotted versus latitude, causing a rocking back and forth as time progresses. Clearly, the The bulge in the Earth at its equator has been shown [1] dominant nonlinear factor in the otherwise spiral decay is to lead to a clustering of natural decays biased to occur the physical uplift of the atmosphere into the orbit path. towards the equator and away from the orbit’s extreme latitudes. Such clustering must be considered when predicting the Expectation of Casualty (Ec) during a natural decay because of the clustering of the human population in the same lower latitudes. This study expands upon prior work [1, 2], and formalizes the correction that must be made to the calculation of the average exposed population density as a result of this effect. Although a generic equation can be derived from this work to approximate the effects of gravitational and atmospheric perturbations on a final decay, such an equation averages certain important subtleties in achieving a best fit over all conditions. The authors recommend that direct simulation be used to calculate the Figure 1. Altitude, radius, and Minus Log(Density) plotted true E for any specific entry as a more accurate method. c vs. latitude for a decaying 50o orbit at 100kg/m2 ballistic A generic equation is provided, represented as a function number. While the J2 gravitational effect raises the local of ballistic number and inclination of the entering radius at the extreme latitudes, the elevation of the spacecraft over the credible range of ballistic numbers. atmosphere above the physical bulge at the equator dominates the density variations encountered by the 1. BACKGROUND decaying spacecraft. Both the gravitational and atmospheric In the final stages of orbital decay, the density scale perturbations are evident in structure of the decay height shrinks to only a few kilometers. At the same concentration curves. time, the rapidly-circularizing orbit encounters two effects of the oblate Earth, each of which (at moderate to With such strong cyclic variation in the drag profile, it high inclinations) permute the “effective altitude” to a should be expected that the distribution of natural decays degree comparable to or exceeding such typical scale around the orbit path (measured in Argument of Latitude heights. The primary influence is the elevation of the [ArgLat with symbol Θ], which is the angle along an orbit atmosphere relative to Earth’s center in the equatorial in the orbit plane measured from the north-bound pass of regions, simply rising in lock step with the geoid. the equator; generally computed as the true anomaly plus Because the Earth’s radius at the equator is 21 kilometers the argument of perigee) should instead show a statistical (km) higher than at its poles, the effective density (and clustering that reflects this cyclic forcing function at one- thus, the local deceleration) could hypothetically increase half orbit period. Indeed, in earlier work, both simulation by a factor of nearly 50 at the equator relative to values and analysis of well-documented real, natural polar orbit at the poles under an idealized perfect circular Kepler decays showed a cyclic clustering towards the equator. orbit. Preliminary assessments with the initial model revealed that the statistical biasing of natural decays to the more Simultaneously, the J2 gravitational orbit perturbation heavily populated lower latitudes could have up to a 13% works in the opposite direction, raising the local radius of effect on the calculated value of Ec. When the authors the orbit at extreme latitudes, and dropping it at the presented the preliminary model they acknowledged that equator. There are additional J3 effects noted by further work would be necessary to completely define the Fremeaux, et al., [3] which emerge as a small correction statistical biases under different values of ballistic factor in the present study. Fig. 1 shows the progression number (BN), inclination, and perhaps solar flux of the local altitude and radius for a decaying polar orbit, conditions. This paper reports the conclusion of that and the progression in the Log10 of density during the work, covering a full range of credible ballistic numbers and all orbit inclinations. Solar flux was shown in bracket the 360° cycle.) A clustering of decays in one subsequent publication [2] to be inconsequential to the portion of the arc then yields a ratio less than one, and a decay latitude bias problem. In the improved model the rarefaction or sparseness of decays will lead to a number authors have reformulated the analysis in terms of ArgLat greater than one. Although this may seem instead of latitude, which has allowed better comparison counterintuitive, representing a variable angle per of the physics across cases. incremental decay scenario (vs. variable decays per unit angle) was the most easily immediately-normalized 2. APPROACH function when establishing common compression curves. When the function is inverted to a probability distribution Tens of thousands of natural decays were simulated in the function, each local point must be binomially expanded, NASA General Mission Analysis Tool (GMAT), in as discussed later. parametric studies every 5° of inclination for spacecraft with 50, 100, 150, and 200 kg/m2 BN decaying for Example: if 57.137 uniform increments of a slightly more than 50 orbits each. The orbits were satellite’s mass caused the decay’s ArgLat to vary propagated to 90 km altitude. (Past research [2] has by 360°, the average step size would be shown that at 90 km the residual time and distance to go (360°/57.137) = 6.2751°. If the argument of to the surface is approximately independent of ArgLat for decay latitude shifted only 5.72° from one any given ballistic number. Therefore, to reduce iteration to the next at a particular region, the local computation time from weeks to days, the decays were compression rate would be (5.72°/6.2751) = truncated at 90 km). This truncation must be accounted 0.9115° for when considering actual expectation of casualty, because ground intercept will be several degrees downrange of the calculated entry point. GMAT is a certified NASA mission planning tool with an adaptive step, ninth order Runge-Kutta integrator with eighth order error control that can numerically integrate the path of a decaying object through a selectable atmosphere. In this study, the MSIS-E 2000 atmosphere was modeled at F(10.7) identical daily and 90-day average values of 115 janskys, and the global geomagnetic index Kp constant was uniform at 3. Orbits were propagated with 4th-order spherical harmonics and full atmospheric responses to day-of-year and the orbit’s Right Ascension of the Ascending Node (RAAN). The mass of the modeled spacecraft was dithered in tiny steps to achieve a cumulative 2% spacecraft mass Figure 2. An example relative compression curve for variation in 101 uniform steps per run. The mass dither BN=200 decays at 50° inclination and RAAN=180°. The is considered by the authors to be the subtlest and most curve illustrates the excellent fit of the five-term Fourier linear way to induce a smooth variation in a very approximation. At the vernal equinox, this orbit has a beta nonlinear process. This method allows the initial angle of zero and passes directly under the subsolar point, conditions to be as nearly identical as possible, while creating a day-night asymmetry in the decay rate. The near allowing the forces being studied to integrate in the mirror of this curve occurs at 90° RAAN (see Fig 6). The nonlinear ways under study. The 2% total mass variation asymmetry is removed by plotting the rates at four in a decay with more than 50 remaining orbits at orthogonal RAANs and averaging the curves. (This is best inception is enough to observe one full cycle of the done after curve-fitting, because there is no way to align the ArgLat of decay. (One hundred and one runs are needed individual statistical points to achieve the same values of to generate 100 gaps. It is the gaps that drive the ArgLat in all four plots.) Note in the curve how sparse the compression data.). entry events are at the peaks relative to the valleys, in terms Once a full cycle of ArgLat was observed, the relative of spatial density of the points. compression (or “compression factor”) C(Θ) of the decays was recorded at each value of Θ. C(Θ) is The resulting plot of relative compression vs. ArgLat calculated by talking the ratio of the spacing between the shows a consistent cyclic behavior of concentrating ArgLat of any one iteration’s decay and the next entries as the decay location approaches the equator iteration’s value, divided by the averaged step size over (ArgLat= 0 and 180), receding rapidly after passing it, the full 360° cycle of ArgLat. (The non-integer number and becoming most rarefied approaching the extreme of steps is linearly interpolated between integer steps that latitudes (ArgLat = 90 and 270). An example plot is shown in Fig. 2. A 5-term Fourier analysis of the data is superimposed as the solid line, showing an excellent fit, 3. RESULTS with generally <0.7% average error of the fit from Each compression curve takes the general form of a recorded values. The universal model of the compression sinusoidal curve in (two times the) ArgLat, (a curve over all conditions is generated by characterizing 2Θ harmonic), perturbed by 1Θ, 3Θ, and 4Θ harmonic the trend of the first four best-fit Fourier coefficients terms of generally lower amplitude, each term affected under varying ballistic number, RAAN, and inclination.