[1] Uncertainty in the atmospheric density is a crucial error source in calculating orbits of satellites in low Earth orbit. As a result, establishing accurate thermospheric neutral density models is important for predicting the motion of these satellites. Unfortunately, since density data in the altitude range between 140 and 200 km are sparse, predicting the neutral density to estimate atmospheric drag effects on the motion of satellites operating in this altitude region is subject to relatively large errors. A previous study found that the Jacchia-Bowman model (JB2006) is the most reliable thermospheric empirical neutral density model above 200 km and the Naval Research Laboratory's Mass Spectrometer Incoherent Scatter (NRLMSISE-00) model, whose core formulation is based on incoherent scatter radar data, can be considered a more reliable neutral density model below approximately 140 km. We have developed a simple bridging technique to blend the two models between these two regions. A two-body model with atmospheric drag was used to compare effects of various atmospheric density models. These preliminary tests are conducted by propagating the positions of satellites orbiting between 140 and 200 km, with various ballistic coefficients, using the JB2006, the NRLMSISE-00, and the bridging technique.

[2] With many satellites orbiting the Earth, it is essential to know their positions and to track their motions accurately in order to maintain the planned orbit and make sure they do not collide with other satellites or debris. Thermospheric density modeling is one of the essential factors for orbit prediction, since for low-Earth orbiting spacecraft atmospheric drag produces one of the largest perturbations to the orbit. In addition to orbit prediction, knowledge of atmospheric density is also important for reentry analysis, ground track maintenance, and precise orbit determination, since the implementation of these techniques also require atmospheric drag to be estimated. Several thermospheric neutral density models have been developed over the past several decades, and various ongoing efforts have focused on error reduction. However, models which have been developed do not represent all conditions. Each model has its own strengths and weaknesses, and is applied to different regions according to its validity.

[3] For the calculation of the thermospheric neutral density, Jacchia-class models and MSIS-class (Mass Spectrometer Incoherent Scatter) models are widely used. Since Jacchia-class models were formulated based on the density derived from the drag data measured by satellites at heights above approximately 200 km [Bowman et al., 2006], they are more reliable at relatively higher altitude. On the other hand, the bases of MSIS-class models' formulation are data from ground-based measurements with some sounding rockets and satellites measurements [Picone et al., 2002], which make them more reliable at lower altitudes but less reliable at higher altitudes when compared to Jacchia-class models. A simple mathematical (not physics-based) bridging technique [Kane et al., 2008] connecting the valid regions of MSIS-class models at lower altitudes and Jacchia-class models at higher altitudes was developed to provide a better estimate of the density data, when compared to using models outside of their region of validity.

[4] This paper presents preliminary comparisons obtained from simulations of the two-body motion of satellites with various ballistic coefficients using a simple atmospheric drag model with density data calculated from the MSIS model, the Jacchia model, and the blending technique. These atmospheric density models are applied to satellites flying through the region between 140 km and 200 km.

2. Model Descriptions

[5] The Jacchia model is a more operational model, and is used in tracking and predicting the orbital behavior of satellites. This empirical model is based primarily on drag data obtained by observing the orbital motion of numerous satellites. In contrast, the satellite mass spectrometer and ground-based incoherent scatter radar data are the major data sources of the MSIS model, which is widely used in the research community. Additionally, the incoherent scatter radar data is the core of the MSIS formulation which is directly related to the temperature in the model. The latest version of the Jacchia-class model is the Jacchia-Bowman 2006 [Bowman et al., 2006] (JB2006) model which uses new solar indices of extreme ultraviolet (EUV) and middle ultraviolet (MUV). The Naval Research Laboratory's MSIS Extension 2000 [Picone et al., 2002] (NRLMSISE-00) is an MSIS-class model that includes data of total mass density from satellite accelerometers, orbit determination, and molecular oxygen number density.

2.1. JB2006 Model

[6] Jacchia-65 [Jacchia, 1965] (J65) was the first Jacchia-class thermospheric empirical model. Its main drawback is the constant boundary conditions at 120 km. In 1970, Jacchia developed the Jacchia-70 [Jacchia, 1970] (J70) model, which lowered the height of the constant boundary conditions to 90 km from 120 km. Jacchia reformulated the J70 model using newer and more complete data in 1971 [Jacchia, 1971] (J71). A few years later, satellite mass spectrometer data were included and some equations were revised [Jacchia, 1977] (J77). In 2006, solar indices S_{10.7} (26 nm∼34 nm solar EUV emission) and M_{10.7} (279.56 and 280.27 nm MUV emission) were included to calculate global minimum nighttime exospheric temperature, and this model is designated the JB2006 model [Bowman et al., 2006]. New exospheric temperature and semiannual density equations for representing thermospheric density variations, temperature correction equations for diurnal and latitudinal effects, and density correction factors for model corrections required at high altitude were included at the JB2006 model.

2.2. NRLMSISE-00 Model

[7] The first version of MSIS-class models (MSIS-1) [Hedin et al., 1977] was introduced by Hedin et al. in 1977. Since the data for formulating the MSIS-1 model were mostly taken under low to moderate solar activity conditions while little data were taken under high solar activity conditions (10.7 cm solar radio flux, F_{10.7} > 170), it has weaknesses in representing high solar activity conditions. Furthermore, the lower boundary of the MSIS-1 model is 120 km. Hedin et al. developed the MSIS-83 [Hedin, 1983] model in 1983, which lowered the boundary to 90 km, and included data from rocket flights, seven satellites, and five incoherent scatter radars, including data obtained during high solar activity. The MSIS-83 model also adopted alternate representation of geomagnetic activity (three hourly a_{p} indices) in addition to the daily geomagnetic activity index, A_{p} (daily geomagnetic activity index). To improve the representation of polar region morphology, Hedin et al. developed the MSIS-86 model [Hedin, 1987] by adding or changing terms to better represent seasonal variations in the polar regions. The lower limitation of the MSIS-class model was extended to the Earth's surface in the MSISE-90 model [Hedin, 1991]. The Naval Research Laboratory (NRL) upgraded the MSISE-90 model in the thermosphere, including numerous drag and accelerometer data sets, and named it the NRLMSISE-00 model [Picone et al., 2002]. The NRLMSISE-00 model adopted a new component, named anomalous oxygen, which allows for O^{+} and hot atomic oxygen contributions to the total mass density above 500 km. As a result, the NRLMSISE-00 model can compute both total neutral mass density as in the previous MSIS models, and effective mass density, which denotes the sum of the total neutral mass density and the anomalous oxygen contribution above 500 km.

3. Analysis of Model Simulation

3.1. Comparison of Data From Simulation and Measurement in 2005

[8]Table 1 [Marcos et al., 2006] provides satellite number, “true” ballistic coefficient (B), inclination (degree), perigee and apogee height at the beginning of year 2000, for satellites used in this study. Densities are calculated from both models and compared with those obtained using the satellite tracking data sets from Table 1. Total mass densities obtained from the satellite observations are the daily average density values corresponding to the perigee point (height, local solar time, latitude) at 0h UT. All the densities for a day (−12h to +12h UT) are averaged to obtain the daily average density values. The reference perigee height is determined as the height that represents a midpoint of all perigee heights; 240 km, 200 km, 220 km, and 255 km for the satellites listed in the Table 1. The reference density values at the reference perigee heights are used to compare both models. The latitude coverage of satellites providing measured density data range between ±34.9°, ±38.7°, ±52.1°, and ±30.3°, and are listed in the Table 1.

[9] The density values of the reference perigee heights are shown in Figure 1 [Kim, 2008]. In order to compare both models quantitatively, the ratio, R, between measured density and model density is used [Marcos et al., 2006]. The mean ratios and standard deviations are, respectively,

where, R_{i} is the ratio of the ith density measurement to the model and N is the number of data points. Mean ratios and standard deviations of density values in Figure 1 are listed in Table 2. Both mean ratios and standard deviations of the JB2006 model are more accurate than those from the NRLMSISE-00 model in these altitude regions.

Table 2. Mean Ratio and Standard Deviation

Height (km)

Mean Ratio (Measured/Model)

Standard Deviation (%)

JB2006

NRLMSISE-00

JB2006

NRLMSISE-00

200

1.064

1.082

7.89

9.08

220

1.017

1.027

6.53

7.96

240

1.008

0.961

8.17

10.42

255

1.010

1.022

6.02

9.19

3.2. Comparison of Empirical Models

[10] As previously noted, density data in the altitude range between 140 km and 200 km are sparse. Therefore, it is impossible to draw an all-encompassing conclusion as to the accuracy of the calibration of the thermospheric density models in this altitude region. However, an extrapolation can be achieved in this region with density data from the satellites above and with sounding rocket data from below, thus connecting the two regions to establish a thermospheric density profile for the gap in question.

[11] In the altitude above 200 km, as shown in Figure 1, the JB2006 model is generally more reliable than the NRLMSISE-00 model. In this study, the primary altitude range of concern is 200 km ∼ 250 km. The fact that the JB2006 model is more reliable than the NRLMSISE-00 model is supported by a previous paper which shows the comparison of several empirical neutral density models between the height of 200 km and 600 km analyzed by Marcos et al in 2006 [Marcos et al., 2006]. The study showed that the standard deviations for the JB2006 model are significantly lower at all altitudes than those from other models.

[12] However, no density data below 200 km is included in the formulation of the JB2006 model, which is less reliable below 200 km. The core of the NRLMSISE-00 model is the incoherent scatter radar data but also includes lower thermospheric data measured by ground-based equipment. Therefore, it is reasonable to conclude that the NRLMSISE-00 model is more reliable in the lower thermosphere, below approximately 140 km.

4. Formulation for Bridging Two Models

[13] The calibrations are shown in Figures 2 and 3. Density and temperature values are plotted on days 30 and 300 in 2006. Between the boundary altitudes, the weighting portion of each model gradually shifts. 100% of the density and temperature values from the NRLMSISE-00 model and 0% of those from JB2006 model are used at the lower boundary altitude of 140 km. These percentages gradually decrease for the NRLMSISE-00 model and increase for the JB2006 model, so at the higher boundary altitude of 200 km, the density and temperature values are 0% from the NRLMSISE-00 model and 100% from the JB2006 model. Equation (3) shows the weighted blending relationship for the density or temperature values from the two models [Kim et al., 2008]. This linear fit was found to work as well as any higher order or nonlinear fit (in fact, often more stably), so in the absence of a full-physics based model, it was decided to keep it simple. Considering the data sets used in formulating the different models, it is reasonable to assume that the density values from the calibration are more accurate than either model.

where,

h

height of interest; altitude between lower and upper boundary;

h_{L}

lower boundary of height for blending (140 km for this study);

h_{U}

upper boundary of height for blending (200 km for this study);

J

density or temperature value from the JB2006 model;

M

density or temperature value from the NRLMSISE-00 model;

S

blended value of density or temperature.

5. Analysis of Model Simulation in Two-Body System

[14] Next, we compare the results of a numerical simulation using the blending technique, the JB2006 model, and the NRLMSISE-00 model by using the two-body equations of motion (equation (4)). Orbital drag acceleration due to the atmospheric density can be calculated by equation (5), and is always opposite to the velocity vector.

where,

_{D}

orbital drag acceleration due to atmospheric density;

position vector of satellite;

μ

standard gravitational parameter (398,600 km^{3}/s^{2} for the Earth);

ρ

atmospheric density;

V

velocity of satellite.

[15] For simplicity and better representation of the differences produced by each model, an equatorial orbit with perigee height of 140 km and apogee height of 200 km was simulated. An equatorial orbit was chosen to eliminate latitude-dependent effects. The reference coordinate system is shown in Figure 4. Densities were calculated every hour from each model, and then averaged for each day. The daily averaged density values were then used to calculate the orbital drag acceleration due to the changing atmospheric density. In order to compare the effect of atmospheric drag, two ballistic coefficients (from Table 1) were used; B = 0.02145 m^{2}/kg for satellite number 25935, representative of a large ballistic coefficient, and B = 0.00356 m^{2}/kg for satellite number 6073, representative of a small ballistic coefficient. The equations of motion were simulated for 5 days, using a fourth-order variable-step size Runge-Kutta numerical integrator; starting at 0000 UT on day 30 and ending at 2400 UT on day 34 in 2006.

[16] To examine the effects of atmospheric drag from the three models, we compared the range difference and true anomaly difference with a drag-free orbit propagation, and the results of this simulation are shown in Figure 5. Table 3 shows two representative differences, from Figure 5, after the end of the second and fifth days. The effects of the large ballistic coefficient are about 6 times greater than those of the small ballistic coefficient.

Table 3. Average Difference of Range With True Anomaly

Second Day

Fifth Day

B = 0.00356 (m^{2}/kg)

B = 0.02145 (m^{2}/kg)

B = 0.00356 (m^{2}/kg)

B = 0.02145 (m^{2}/kg)

MSISE-00

−26.5 m

−159.8 m

−67.8 m

−408.5 m

Blending

−27.2 m

−163.6 m

−69.4 m

−418.0 m

JB2006

−29.2 m

−175.7 m

−74.5 m

−449.0 m

[17] Next, we compare the propagation using the NRLMSIS and JB2006 models to the results obtained using the blending method. Figure 6 shows satellite position differences in the X-Y plane at the end of day 5. Figure 7 shows the trend in along-track differences of the two models when compared to the blending technique, as a function of time. At the end of the fifth day, the satellite with a large ballistic coefficient with density from the JB2006 model is located 11.98 km ahead of the position found using the blending method. At the same time, using the NRLNSISE-00 model, the satellite is located 3.77 km behind the satellite's position found using the blending technique.

6. Conclusions

[18] The work presented here focused on the development of bridging techniques to blend altitude variations of the Jacchia-Bowman model (JB2006) and the Naval Research Laboratory's Mass Spectrometer Incoherent Scatter model (NRLMSISE-00). Although it is difficult to compare the accuracy of these two models due to the lack of measurements of density, we found we could proceed using reasonable assumptions based on the data set of each model. In the altitude region above 200 km, the JB2006 model is the most reliable neutral density model compared to other models because the basis of its formulation it is the density derived from drag data measured using satellites above about 200 km. However, since the Jacchia-class model data sets do not include any data below 200 km, the JB2006 model is less reliable there. Below approximately 140 km, it is reasonable to conclude that the NRLMSISE-00 model is more reliable than JB2006 since the basis of formulating the MSIS-class models is rocket-borne mass spectrometry. It should be clarified that the bridging technique presented was mathematical in nature, and not physics-based; however, we anticipate this to evolve in future work, as discussed below.

[19] We found that the blending method produces reasonable results in the altitude range between the JB2006 and NRLMSISE-00 models, as expected; however, this approach might be limited in practice as real density values may not actually be between the values from JB2006 and NRLMSISE00. Before an all-encompassing conclusion can be made, a comparison with actual tracking data would be useful. Nevertheless, we believe that the results obtained from this method were more accurate than simply using either model outside their region of validity. This paper is not a final result, but an intermediate step toward our goal of comparing satellite tracking-derived atmospheric density data below 200 km in altitude. A continuing goal of this work is to obtain additional density measurements, then to make a physics-based density model on a diurnal timescale using the NCAR (National Center for Atmospheric Research) TIE-GCM (Thermosphere-Ionosphere-Electrodynamics General Circulation Model) to provide better density estimations even if measurements are not available between the values from the two models.