Impact of tidal density variability on orbital and reentry predictions

Authors


Corresponding author: J. M. Leonard, Department of Aerospace Engineering Sciences, University of Colorado Boulder, Campus Box 431, Boulder, CO 80309-0429, USA. (jale4769@colorado.edu)

Abstract

[1] Since the first satellites entered Earth orbit in the late 1950's and early 1960's, the influences of solar and geomagnetic variability on the satellite drag environment have been studied, and parameterized in empirical density models with increasing sophistication. However, only within the past 5 years has the realization emerged that “troposphere weather” contributes significantly to the “space weather” of the thermosphere, especially during solar minimum conditions. Much of the attendant variability is attributable to upward-propagating solar tides excited by latent heating due to deep tropical convection, and solar radiation absorption primarily by water vapor and ozone in the stratosphere and mesosphere, respectively. We know that this tidal spectrum significantly modifies the orbital (>200 km) and reentry (60–150 km) drag environments, and that these tidal components induce longitude variability not yet emulated in empirical density models. Yet, current requirements for improvements in orbital prediction make clear that further refinements to density models are needed. In this paper, the operational consequences of longitude-dependent tides are quantitatively assessed through a series of orbital and reentry predictions. We find that in-track prediction differences incurred by tidal effects are typically of order 200 ± 100 m for satellites in 400-km circular orbits and 15 ± 10 km for satellites in 200-km circular orbits for a 24-hour prediction. For an initial 200-km circular orbit, surface impact differences of order 15° ± 15° latitude are incurred. For operational problems with similar accuracy needs, a density model that includes a climatological representation of longitude-dependent tides should significantly reduce errors due to this source.

1. Introduction

[2] Improvements in atmospheric density modeling over the past few decades have resulted in more accurate orbital predictions and conjunction analyses. Current requirements for improvements in orbital predictions have made clear the need for further refinements to density models. The LEO (low-earth orbit, ca. 200–800 km) regime is a highly variable “space weather” environment, and orbital drag remains a major uncertainty in orbital prediction and conjunction analyses. Unmodeled atmospheric density variations continue to greatly impact orbital predictions and conjunction analyses.

[3] The variability of density in the LEO regime is driven by 3 sources: absorption of highly variable UV and EUV solar radiation from the Sun's chromosphere and corona; energy from the solar wind, reprocessed by the magnetosphere and deposited into the upper atmosphere in the form of energetic particles, electric fields and currents; and waves propagating upward from the lower atmosphere, in particular solar thermal tides with periods of 12 and 24 hours. Thermosphere density responses to variable solar wind and solar radiation forcing are generally captured within empirical models [Picone et al., 2002; Bowman et al., 2008] through empirical relationships involving various magnetic and solar indices. Solar thermal tides generated in-situ in the thermosphere through absorption of EUV radiation are reasonably represented in such models, but “meteorological influences” originating below 100 km, which are generally manifested in the form of upward-propagating tides and other waves, are not. An important component of thermal tides originates in latent heating associated with deep tropical convection, and these waves carry the signal of troposphere weather, land-sea differences and other processes into the reentry and orbital regimes [Hagan and Forbes, 2002, 2003; Bruinsma and Forbes, 2010; Zhang et al., 2010a, 2010b; Oberheide and Forbes, 2008; Oberheide et al., 2009, 2011a, 2011b]. In particular, the longitude variability imposed by these processes on the orbital drag environment has only recently been discovered [Oberheide and Forbes, 2008; Oberheide et al., 2009, 2011a, 2011b; Forbes et al., 2009]. The question emerges: How important are these local-time and longitude-dependent density variations to accurate orbital and reentry predictions? It is the purpose of this study to answer this question.

[4] The leading error source for orbit determination in the LEO regime is that of mis-modeled atmospheric drag. Atmospheric drag perturbations remove energy from an orbit, producing secular and periodic decay in the semimajor axis, eccentricity, and inclination. Drag changes can also produce periodic variations in the true anomaly, mean anomaly, and Right Ascension of the Ascending Node. The decay in the semimajor axis and eccentricity lowers the orbit into higher density regions of the atmosphere, ultimately to the point where it causes the satellite to reenter and impact the surface. The specific density at any location in a satellite's orbit is influenced by the day-night cycle, location of the subsolar point, solar activity, geomagnetic activity, seasonal variations, and much more. McLaughlin [2005] gives an introduction to the time-varying effects of density in the neutral atmosphere. Sabol and Luu [2002] give a comprehensive summary of the atmospheric variability drivers and some of the problems with temporal resolution of certain parameters used in current empirical models. A detailed overview of satellite drag modeling research is given by Marcos et al. [2003]. McLaughlin and Bieber [2008], McLaughlin et al. [2008a, 2008b], and McLaughlin et al. [2011] describe attempts to characterize corrections to total density variations from precision orbit determination products. Another method known as dynamic calibration of the atmosphere has also been used to quantify variations and corrections in atmospheric modeling [Bowman, 2002; Bowman and Storz, 2002; Bowman et al., 2004a, 2004b; Storz et al., 2005; Cefola et al., 2003; Yurasov et al., 2004, 2008; Wilkins et al., 2007a, 2007b]. All of these methods rely heavily on altering existing empirical atmospheric density models such as Jacchia [1971] or NRLMSISE-00 [Picone et al., 2002]. These methods show that there is a generally accepted 10–15% error in most empirical models.

[5] In this work, the existing NRLMSISE-00 empirical model [Picone et al., 2002] is modified to include tidal variability, and numerical experiments are conducted to evaluate the impact that tidal density variations have on a series of orbital and reentry predictions that explore the parameter space of the problem. The basis for this empirical model modification is the recent work of Oberheide et al. [2011b] who provide an observation-based Climatological Tidal Model of the Thermosphere (CTMT). CTMT includes the 6 (8) most important diurnal (semidiurnal) tidal components for temperature, density, and zonal, meridional and vertical winds and is based on fitting a set of physics-based tidal basis functions (called Hough Mode Extensions, or HMEs) to 2002–2008 mean TIMED (Thermosphere-Ionosphere Energetics and Dynamics) satellite tidal diagnostics in the mesosphere/lower thermosphere (MLT). The model extends from 80 to 400 km, pole to pole, and applies to moderate (F10.7 ∼ 110 s.f.u.) solar conditions. Validation with independent tidal diagnostics from the CHAMP satellite around 400 km demonstrates the realism of the model [Oberheide et al., 2011a].

[6] In the following two sections a brief overview is provided of the tidal density structures present in the CTMT model as well as a defined set of orbital and reentry predictions that explore the effects of the density variations when added to a baseline empirical density model. Results for numerical experiments are presented in Section 4 and 5 with conclusions from the study presented in Section 6.

2. The Variable Drag Environment

[7] In this section some insight into the spatial and temporal variability in density that will be examined is provided. Figure 1 illustrates the relative density variability in percent over the equator versus longitude at four local solar times (0600, 1200, 1800, 2400 LST). The 60–150 km altitude regime (alternatively the reentry regime) is highly structured, and illustrates where a spectrum of waves have achieved large amplitude (∼10–20%) after having grown in amplitude after propagating from the less tenuous atmosphere below. Note also the quasi-wave-4 structure in longitude, which reflects the primary harmonic of the land-sea (and hence deep convection and latent heating) distribution in the lower atmosphere (see Section 1). The waves are strongly damped by molecular diffusion processes between about 120 and 150 km. As the wave spectrum propagates to higher altitudes, the shorter-scale waves are preferentially dissipated, leaving only the larger vertical scale waves to influence the orbital drag regime.

Figure 1.

Height versus longitude variability of relative total mass density (percent) over the equator at four local times, based on the Oberheide et al. [2011b] model.

[8] Figures 2 and 3 similarly illustrate the height versus local time variations at four different longitudes (Figure 2) and latitude versus longitude variability at four different local times at a single altitude (100 km) in the reentry regime (Figure 3). Significant density variations occur near the equator as seen in Figure 3. These variations of ∼±10% have a drastic impact on the behavior of a satellite that traverses this regime.

Figure 2.

Height versus local time variability of relative total mass density (percent) at 30° latitude at four longitudes (0°, 30°, 60°, 90°), based on the Oberheide et al. [2011b] model.

Figure 3.

Latitude versus longitude variability of relative total mass density (percent) at 100 km altitude at four local times (0600, 1200, 1800, 2400 LT) based on the Oberheide et al. [2011b] model.

3. Numerical Experiments

3.1. Models

[9] NASA's Program to Optimize Simulated Trajectories II (POST2) [Striepe, 2004] was used to simulate the satellite's trajectory. POST2 is a generalized point mass, discrete parameter targeting and optimization trajectory simulation program [Striepe, 2004] often used for mission and system development support, designing reference trajectories, Monte Carlo dispersion analyses, trade studies, as well as mission planning and operation support at NASA Langley Research Center. POST2 has the ability to simulate three-degree-of-freedom (3DOF) and 6DOF trajectories for multiple vehicles, simultaneously, in a wide variety of different flight regimes. POST2 also has the capability to utilize various models such as gravity, vehicle, propulsion, guidance, control, sensor and navigation system models [Striepe, 2004]. Our simulations utilize POST2's 3DOF integration of translational equations of motion along the trajectory.

[10] POST2 was used with an efficient seventh-order Runge-Kutta-Fehlberg RKF 7(8) algorithm [Fehlberg, 1968] to integrate the satellite's true equations of motion. The true equations of motion include the acceleration of Earth's gravitational field, as well as atmospheric drag. Striepe [2004] provides a detailed description of the dynamic models for orbit and reentry prediction used in this analysis. The GRACE gravity model (GGM02C) [Tapley et al., 2005] was added to POST2and is used for the evaluation of Earth's gravitational acceleration for spherical harmonic terms up to degree 100 and order 100 using the normalized Cartesian model [Gottlieb, 1993]. A simple drag force model with spherical body was used. To approximate a typical LEO satellite, a mass of 800 kg was assumed along with an area-to-mass ratio (AMR) of 0.0125 m2/kg. These values are based on the ICESat satellite [Schutz et al., 2005] that reentered in August 2010. The large area to mass ratio was chosen under the assumption that during an uncontrolled reentry, the tumbling spacecraft's average cross sectional area would remain high and not the minimum experienced during operations. For the simulations in this study, a drag coefficient of 2.0 was assumed. Note that the coefficients for other satellites may vary from the values used here, and the orbit differences given next would show corresponding increases or decreases for such satellites; however, the results would be consistent with the findings presented here. The US Naval Research Laboratory Mass Spectrometer and Incoherent Scatter Radar (NRLMSISE-00) model was used as the base atmospheric density model in all simulations [Picone et al., 2002]. The NRLMSISE-00 model was then modified by superimposing CTMT tidal variations previously described. The CTMT model provides a correction to the empirical density model in terms of a percent deviation of the density with respect to the mean value. The density deviation is then added to the model density to form the modified empirical model with CTMT corrections. A Solar flux F10.7 value of 110 sfu and a geomagnetic index, Ap, of 7 are assumed.

3.2. Implementation of Tidal Density Corrections

[11] The CTMT atmospheric density tidal variations are obtained for a four-dimensional grid based on latitude, longitude, altitude, and local solar time. The latitude varies from −90 to 90 degrees with a grid step size every 5 degrees. The longitude varies from 0 to 360 degrees with a grid step size every 10 degrees. The altitude varies from 0 to 400 km with a grid step size of 4 km. Local solar time varies from 0 to 24 hours with a grid step size of 1 hour. A four-dimensional grid is obtained for each month of the year and stored for use in POST II.

[12] For the determination of density variation due to the CTMT model during numerical integration of the equations of motion, a four-dimensional, nth degree piecewise Lagrange polynomial interpolation method is used. For this work, a 7th degree method was used in which the interpolated value lied in the midsection to reduce the likelihood of oscillatory behavior exhibited near the endpoints. The percent deviation of the density value obtained from the interpolation procedure is multiplied by the model density providing a density deviation that is then added to the model density to obtain the modified density model with CTMT corrections. The satellite's coordinates are then converted to latitude, longitude, and altitude for the tidal density variation to be determined. Once the interpolated value is obtained, it is then used to modify the NRLMSISE-00 model after an initial baseline density value is obtained from the model.

3.3. Method

[13] The approach used in our numerical experiments was to fix certain parameters in order to isolate the local time and longitudinal effects prescribed by the CTMT model and absent in the empirical NRLMSISE-00 atmospheric model. Orbit differences are examined over a period of a single orbit as well as 24 hours in order to identify small periodic timescale effects as well as a longer secular effects. Second, an analysis of reentry predictions was performed in which two identical orbits are integrated to impact, one with the altered CTMT model included and one using only an unaltered NRLMSISE-00 model. It was assumed that the reentry was uncontrolled and that no break-up occurs. This allows for a better understanding of the effects of the model on the reentry path without having to include Monte Carlo runs for reentry uncertainty.

4. Orbit Differences

[14] Two circular LEO altitude regimes are examined: 200 km and 400 km. For each orbit both single-orbit and 24-hour predictions are performed. Each case consists of four separate initial local solar times of 0, 6, 12 and 18 hours. For each altitude regime, a set of 24 orbits is rotated by 15 degrees in the longitude of the ascending node while keeping the local solar time at epoch the same. Each orbit has an inclination of 90 degrees with the ascending node located at the equator.

4.1. Single-Orbit Prediction Results

[15] In this section, the cases described previously are integrated over a single orbit for both 200-km and 400-km circular orbits. The results compare an orbit that is integrated with the CTMT-altered NRLMSISE-00 model and one that is not. The differences are quantified by the final absolute position difference that occurs at the end of the integration period as well as the mean absolute position difference of the 24 integrated orbits and the associated 3-sigma uncertainty.

[16] The absolute position difference of all 24 orbits with a specific local solar time were averaged together to determine the mean orbital position difference for each month and its 3-sigma bounds for the 200-km and 400-km circular orbits. It must be noted that there are some situations in which the in-track difference is positive (CTMT model removes less energy from the orbit) and negative (CTMT model removes more energy from the orbit). This analysis does not distinguish whether the satellite integrated with the CTMT model is either leading or trailing the satellite integrated with the standard atmospheric model; rather, the expected absolute position difference is the only concern. Therefore, the absolute position difference as we define it can never be negative and must always grow from zero. As expected, the addition of tidal density variations to the baseline atmospheric model significantly changes the behavior of the satellite over a single orbit. The majority of the difference occurs in the in-track direction with some radial loss. The means and standard deviations based on LST are relatively consistent over the single orbit interval for a 200-km orbit but vary for the 400-km orbit. For the 200-km orbit case, each of the orbits has a mean absolute position difference of about 45 m with a 3-sigma uncertainty of about 80 m, relatively independent of LST. The LST has a more significant effect on the 400-km predictions in which an LST of 0 or 6 hours has a mean absolute position difference of about 1.5 m (2.25 m 3-sigma), whereas an LST of 12 or 18 hours has a mean absolute position difference of about 0.7 m (1.2 m 3-sigma).

[17] The final absolute position difference means and 3-sigma standard deviations for all the months after a single 200-km orbit integration provide an understanding of how the CTMT tidal influence varies over the course of a year. There is roughly a 30–60 m position difference after a single orbit integration for each month. July is noticeably the month with the smallest differences (∼30 m) and smallest 3-sigma bounds. While the average position difference based on the various initial longitudes is consistent from month to month, the spread in the standard deviation is quite significant, ranging from 80 m to 170 m (3-sigma). This shows that the initial longitude of the ascending node is a significant factor when using these tidal variations in the atmospheric model.

[18] While this gives a good indication of the overall CTMT effects on a single-orbit propagation, Figure 4 shows the distribution of position differences based on the initial longitude of the orbit. Three of the twelve months are represented to provide a perspective on the distribution of differences based on the initial LST and location of the ascending node. Overall, after a single orbit integration there is not much structure to the differences, which are rather random. However, for the months of March, May, and November there exist a set of longitudes in which the differences are noticeably larger than the rest (May is shown in Figure 4). This is primarily due to the highly structured atmospheric region that these orbits are experiencing within certain longitudinal bands.

Figure 4.

Single-orbit prediction difference for 200-km circular orbit for (a) January, (b) May, and (c) July. Angular direction is longitude (deg) and radial direction is position difference (m) for local solar times of 0 (blue), 6 (green), 12 (red), and 18 (teal) hours.

[19] The absolute position differences based on local solar time for the single-orbit 200-km altitude orbit integration are summarized with histograms as in Figure 5. Each histogram represents the number of cases that fall within certain absolute position difference bins. The four LST values being investigated (0, 6, 12, and 18 hrs) are represented with their corresponding means and standard deviations. The distribution of absolute position differences after a single orbit integration are consistent for each of the LSTs. The means range from ∼38 m (LST of 6 and 12 hrs) to ∼47 m (LST of 0 hrs). The standard deviations for each of the LST distributions is consistent from one LST to the next and is ∼26 m. For the case with an LST of 12 hours, one notices a tighter distribution than the other LST cases. Though the distributions are similar for each LST, the 12-hour case is the one with the lowest mean (∼38 m) and standard deviation (∼25 m). Overall, the distribution of 200-km single-orbit absolute position differences are consistent from one LST to the next.

Figure 5.

Prediction difference histograms for single-orbit integrations at 200-km for different LSTs including all months.

[20] A similar analysis was conducted for the 400-km circular orbit. An interesting pattern occurs for the 400-km circular-orbit integrations that did not appear for the 200-km circular orbits. The orbits with an LST of 0 and 6 hours have similar means and 3-sigma bounds for every month but differ from orbits with LSTs of 12 and 18 hours as seen in Figure 6. Each histogram in Figure 6 represents the number of cases that fall within certain absolute position difference bins for the four LST values being investigated (0, 6, 12, and 18 hrs). For the cases with an LST of 0 and 6 hours, the mean is ∼1.3 m with a standard deviation of ∼0.7 m. The distributions of absolute position differences for an LST of 12 and 18 hours are shifted and condensed when compared to that of the cases with an LST of 0 and 6 hours. For the cases with an LST of 12 and 18 hours, the mean is ∼0.7 m with a standard deviation of ∼0.35 m. This shows that for the 400-km circular orbit case, the LST of the orbit plays an important role on the distribution of absolute position differences after a single orbit, which was not the case for the 200-km circular orbit case.

Figure 6.

Prediction difference histograms for single-orbit integrations at 400-km for different LSTs including all months.

4.2. Twenty-Four Hour Prediction Results

[21] Although the analysis thus far has given an indication of the effects of the inclusion of tidal density variations for short integration times, it is also relevant to look at longer timescales typically of concern for orbit predictions. In this section, the prediction differences that occur over a period of twenty-four hours are investigated.

[22] The absolute position differences for a 24-hour prediction are expected to grow with time. Within 24 hours, there are significant losses in orbital energy and the absolute position differences grow to a few kilometers for the 200-km orbit case and several hundred meters for the 400-km case. For the 200-km orbit case, the magnitudes of the 3-sigma bounds are rather consistent from month to month and have a value of ∼5 km; however, the mean absolute position difference varies significantly with July being the lowest (∼5 km) and November being the highest (∼20 km). This shows that the distribution of position differences based on the various initial longitudes and LSTs is consistent from month to month but seasonal variations in mean position difference can be significant. The fact that the standard deviation is significant for each month shows that the initial longitude of the ascending node does exert a significant influence when using these tidal variations in the atmospheric model.

[23] When the orbit is integrated over 24 hours instead of a single day, minute variations in the density models are removed and averaged trends begin to appear. Figure 7 shows the absolute position differences based on initial longitudinal variations of the orbits integrated over 24 hours. There is no noticeable difference between predictions that assume different LSTs or initial longitudes for most cases. The months of January and April show that there is an average difference expected in the orbit over 24 hours. The month of July is the one case in which there are significant variations based on the longitude of the orbit's node, although the total errors for July are small. Each LST has an average absolute position difference of ∼14 km and a standard deviation of ∼5 km. The similarities in the histograms given in Figure 8 confirms that there is no major dependence on LST but some dependence on the longitude of the orbit for 24-hour predictions.

Figure 7.

24-hour orbit prediction difference for 200-km circular orbit for (a) January, (b) April, and (c) July. Angular direction is longitude (deg) and radial direction is position difference (km) for local solar times of 0 (blue), 6 (green), 12 (red), and 18 (teal) hours.

Figure 8.

Prediction difference histograms for 24-hour orbit integrations at 200-km for different LSTs including all months.

[24] The 24-hour orbit position differences for the 400-km circular orbit case are also of importance for this study. The same pattern as before in the single-orbit 400-km circular-orbit predictions occurs again for the 400-km circular-orbit predictions for 24 hours; that is, LSTs of 0 and 6 hours have similar means and 3-sigma bounds for every month but differ from LSTs of 12 and 18 hours (not shown). The mean of the 0 and 6 hour LST 400-km orbits are ∼50 m less than the mean position difference for an LST of 12 or 18 hours. Another seasonal trend also appears when looking over the various months. There exists a set of months, June through September, that show a significant drop in the average position difference (mean of ∼125 m and ∼75 m 3-sigma) when compared to the other months (mean ∼275 m and ∼90 m 3-sigma). The 3-sigma variances for these months are similar to all of the other months, indicating that the atmospheric density variation for the months of June through September have a smaller average than the other months.

[25] Similar to the previous 24-hour 200-km case, the minute variations in the density models on a single orbit case at 400-km are averaged out and an overall trend appears. There is still the noticeable difference in LST but there is some dependence on the initial longitude of the ascending node of the 400-km orbit. The similarities in the histograms given in Figure 9 confirm that there is a slight dependence on LST and some dependence on the longitude of the orbit for 24-hour predictions. The difference in absolute position difference for the months of June through September are clearly visible in Figure 9, creating the bimodal distribution. The first mode which includes mostly the months of June through September has an average absolute position difference of about 125 m for each LST with a standard deviation of 25 m. The second mode of the bimodal distribution has an average of about 275 m and a standard deviation of about 30 m. There is a distinct difference in the absolute position difference due to seasonal variations in the CTMT model for the 400-km circular orbit case.

Figure 9.

Prediction difference histograms for 24-hour orbit integrations at 400-km for different LSTs including all months.

5. Reentry Predictions

[26] Within the last few years there have been several uncontrolled satellite reentries. In order to predict the impact location of these satellites, empirical density models are employed. In their current form, they cannot predict orbit lifetime very accurately until the final stages of reentry. It is vital to have the most accurate density model in order to perform accurate orbit lifetimes. Once a satellite drops below 150 km, it usually only has an orbit lifetime of a few revolutions. The following simulation results show that the highly structured tidal density perturbations that exist below 150 km dramatically affect the reentry trajectory of a satellite.

[27] In this section, variability in reentry prediction is investigated for an uncontrolled reentry. The primary purpose is to determine what type of overall variability in impact location can be attributable to density variations in the reentry regime that are of tidal origin. The simulations begin with an initially circular LEO orbit of 200 km with an inclination of 90 degrees. Reentry prediction orbits consist of four initial local solar times of 0, 6, 12, and 18 hours. Each of these initial local solar time orbits include a set of 24 circular 200-km orbits rotated by 15 degrees in the longitude of the ascending node while keeping the local solar time at the nodal epoch the same. These orbits are then integrated forward in time until impact. Each orbit is integrated with the tidal density variability included and excluded.

[28] Figure 10a illustrates the typical reentry altitude path based on the latitude of the satellite. The background contour is for a specific longitude of 0 degrees and is a representation of the typical density field encountered by the simulation. The reentry trajectories are within a few kilometers of each other when they approach the reentry regime of 0–150 km. The trajectories then begin to deviate due to the highly structured atmosphere within this region. Figure 10b shows the reentry trajectories for two orbits with the same longitude of the ascending node and a background contour with the actual CTMT density variation being experienced at their reentry longitude of approximately 53 degrees.

Figure 10.

Reentry trajectory profiles for (a) all trajectories with a modified density model for an LST of 0 hours and (b) two reentry trajectories with and without the density model modifications.

[29] Three sample months of the reentry impact latitude differences are shown in Figure 11. The impact latitude differences are given in degrees (as a reference, 1 degree latitude is roughly 110 km). A histogram of the impact latitude differences are given in Figure 12 for the four LST values. The values given in this histogram are for all impact trajectories for every month based on the LST given and give an idea of the expected variations in the impact latitude. The majority of the values fall within the range of 0–30 degrees difference in impact latitude with an average of 17.8 degrees. The standard deviations for each of the LSTs varies from ∼14 degrees to ∼15 degrees. The actual statistics of the distribution do not behave in a gaussian distribution and cannot be transformed into such; thus, higher order statistics are not given.

Figure 11.

Impact latitude difference for an initial 200-km circular orbit for (a) January, (b) February, and (c) March. Angular direction is longitude (deg) and radial direction is impact latitude difference (deg) for local solar times of 0 (blue), 6 (green), 12 (red), and 18 (teal) hours.

Figure 12.

Impact latitude difference histograms for an initial 200-km circular for different LSTs.

6. Summary and Conclusions

[30] In this paper we investigate the effects of known local-time and longitude-dependent density variations on LEO orbital prediction as well as reentry. The NRLMSISE-00 empirical density model is modified using the Climatological Tidal Model of the Thermosphere (CTMT) to include local-time and longitude-dependent variability. The sensitivities of orbit predictions and reentry variability to these effects are analyzed for various local solar times and months. Predicted position differences for both the 200-kmand 400-km altitude regimes are analyzed for single-orbit and 24-hour orbit integrations. Differences in impact latitude for a satellite at an initial orbit altitude of 200 km are also analyzed.

[31] Typical in-track prediction differences incurred by tidal effects are found to be of order 200 ± 100 m for satellites in 400-km circular orbits and 15 ± 10 km for satellites in 200-km circular orbits for a 24-hour prediction, and of order 40 ± 40 m for 200-km circular orbits and 0.25–2.5 m for 400-km circular orbits for the corresponding single-orbit integrations. Significant month-to-month variations are also identified. These differences are thought to exceed error magnitudes expected to be significant for U.S. Air Force applications [Anderson et al., 2009].

[32] The impacts of tidal variability on reentry predictions and impact locations are also analyzed. It is shown that a 200-km LEO orbit has a significant variation in the impact location (of order 15° ± 15° latitude) due to the upward propagating waves that the CTMT model describes. These large perturbation waves that achieve maximum amplitudes in the reentry regime below 150 km have the largest effect on the reentry path and impact location. A significant dependency on local solar time is also found.

[33] The CTMT model allows one to quantitatively prescribe thermospheric variability due to tides propagating upwards from the lower atmosphere. Since the model has been calibrated with TIMED satellite tidal diagnostics in the mesosphere/lower thermosphere and validated using CHAMP tidal diagnostics, the variations are reasonably well accounted for, at least in a climatological or average sense. The tidal spectrum can sometimes vary significantly over timescales ranging from days to weeks, however, and hence our previous reference to “space weather” in the context of atmospheric tides. Although we use a climatological model in the present study, this does not diminish the fact that the local time and longitude variabilities in the model are “typical” and thus can be used to evaluate impacts on orbital and reentry predictions. Moreover, on any given day, the salient features of the actual tidal density distribution are likely to be captured by the CTMT. For operational problems with accuracy needs similar to the prediction differences described above, a density model that includes a climatological representation of longitude-dependent tides should therefore significantly reduce errors due to this source. The CTMT data files are available upon request from co-author J. Forbes.

Acknowledgments

[34] This work was supported by AFOSR MURI award FA9550-07-377 1-0565 and NSF Space Weather grant ATM-0719480 to the University of Colorado.

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