Calibration of atmospheric density model using two-line element data

Transcription

1 Astrodynamics Vol 2, No, 3 24, Calibration of atmospheric density model using two-line element data Yuan Ren ( ), Jinjun Shan Department of Earth and Space Science and Engineering, York University, Toronto, 3J P3, Canada ABSTRACT For satellites in orbits, most perturbations can be well modeled; however the inaccuracy of the atmospheric density model remains the biggest error source in orbit determination and prediction The commonly used empirical atmospheric density models, such as Jacchia, NRLSISE, DT, and Russian GOST, still have a relative error of about 0% 30% Because of the uncertainty in the atmospheric density distribution, high accuracy estimation of the atmospheric density cannot be achieved using a deterministic model A better way to improve the accuracy is to calibrate the model with updated measurements Twoline element (TLE) sets provide accessible orbital data, which can be used in the model calibration In this paper, an algorithm for calibrating the atmospheric density model is developed First, the density distribution of the atmosphere is represented by a power series expansion whose coefficients are denoted by the spherical harmonic expansions Then, the useful historical TLE data are selected The ballistic coefficients of the objects are estimated using the BSTAR data in TLEs, and the parameterized model is calibrated by solving a nonlinear least squares problem Simulation results show that the prediction error is reduced using the proposed calibration algorithm KEYWORDS atmospheric density model two-line element (TLE) Research Article Received: 05 December 206 Accepted: 3 June Tsinghua University Press Introduction Atmospheric drag is the main error source of orbit determination and prediction of low Earth orbit (LEO) satellites because the existing models are not accurate enough The simplest atmospheric density model is the exponential model, which is a static and spherically symmetrical model It cannot be used in high-accuracy studies In order to improve the accuracy, the empirical density models [], such as Jacchia [2], NRLSISE [3], DT [4], and Russian GOST [5] are found by combining theoretical computation with measurements of satellite drag [6] Their accuracies are much better than that of the exponential model, but still no better than 0% 30% [7,8] The atmospheric density is affected by some uncertain factors such as solar radiation and geomagnetic activity; hence it is very difficult to obtain a high-accuracy atmospheric density with such techniques There are two approaches to increase the accuracy of atmospheric density models The first is to derive the atmospheric density directly using measurements from a high-accuracy on-board accelerometer [9, 0] Unfortunately, only a very limited number of satellites have the capability of providing high-accuracy measurements of acceleration, and up-to-date data from these satellites are difficult to obtain The second approach is to calibrate the atmospheric density model using large numbers of orbital measurements with lower accuracy [] At present, historical and up-to-date orbital data of LEO satellites in two-line element (TLE) format can be obtained easily, and can be used for the calibration of atmospheric density models In this paper, an atmospheric density calibration approach is developed First, the density distribution of the atmosphere is parameterized with a power series expansion whose coefficients are denoted by spherical harmonic expansions Then, these coefficients are calibrated by using the historical TLE data of satellites and space debris in low Earth orbits yren@yorkuca

2 4 Y Ren, J Shan (LEOs) The ballistic coefficients, which are needed in the calibration, are estimated using the BSTAR values in TLEs In order to obtain a continuous time estimation of atmospheric density, a sliding analysis window technique is introduced Finally, the proposed calibration algorithm is verified by using the crossvalidation technique Results show that the prediction accuracy of the calibrated model is higher than that of the empirical model The rest of this paper is organized as follows In Section 2 the parameterization method is introduced The effect of air drag on the orbit is analyzed in Section 3 The TLE pretreatment technique is presented in Section 4 The calibration algorithm is presented in Section 5 and the numerical results can be found in Section 6 Section 7 concludes this paper 2 Atmospheric density model and parameterization The simplest atmospheric density model is the following exponential model: ( ) h h0 ρ = ρ 0 exp () H where ρ 0 is the base density at a nominal altitude h 0, h is the altitude, and H is a scale height that depends upon the selected average isothermal temperature In this model, there is a linear relationship between the altitude h and natural logarithm of atmospheric density ln ρ However, it is well known that the atmospheric density distribution highly depends on the solar flux and geomagnetic activity, which are not included in the exponential atmospheric model In order to improve the estimation accuracy, the empirical atmospheric models were developed The commonly used empirical atmospheric models, such as Jacchia, NRLSISE, and DT, rely on the correlation between variations in density, temperature, and composition with a few selected space weather proxies, such as F 0,7 and K p The achievable model accuracy is around 5% 30% Figure shows the atmospheric density distribution at 00:00 UTC, 06:00 UTC, 2:00 UTC, and 8:00 UTC on January, 205 obtained using the NRLSISE-00 model It can be found that the peak atmospheric density always appears near the point under the Sun The peak moves westward with time During the winter, the peak appears near the tropic of Capricorn, while during the summer, the peak lies near the tropic of Cancer (a) 00:00 UTC (b) 06:00 UTC (c) 2:00 UTC (d) 8:00 UTC Fig Atmospheric density distribution obtained by NRLSISE-00 model h = 400 km, date: Jan, 205, daily F 0,7 for previous day and 8-day average of F 0,7 flux are both 50, and daily magnetic index is 40 (A p) [2]

3 Calibration of atmospheric density model using two-line element data 5 The first step of the atmospheric density model calibration is to parameterize the distribution of the density Figure 2(a) shows the typical relationship between the natural logarithm of atmospheric density and altitude computed by using the NRLSISE-00 model It can be approximated by the following power series expansion: l max ln ρ(h) = l=0 ( ) l 2h hh h L a l (2) h H h L where a l (l = 0,, l max ) are the coefficients of the power series, h L and h H denote the minimal and the maximal altitude that can be expressed by this 2h h H h L expansion The expression,, is used to h H h L scale and centralize the input h Figure 2(b) shows the relative error of the power series approximations with different l max It can be seen that the relative error is smaller than 2% when l max = 2 This accuracy is acceptable, so the value of l max must be greater than or equal to 2 Note that, Eq (2) only represents the density on a plumb-line with fixed latitude and longitude To approximate the atmospheric density globally, coefficient a l should have different values for different locations In other words, a l (l = 0,, l max ) should be the functions of λ and φ, and can be represented by the spherical harmonics expansions Thus, Eq (2) can be modified to l max ln ρ(λ, φ, h) = l=0 [ nmax n n=0 m=0 P nm (sin φ) ( Clnm cos mλ + S lnm sin mλ ) ] ( 2h hh h L h H h L ) l (3) where Pnm ( ) are the fully normalized associated Legendre polynomials of degree n and order m Clnm and S lnm are the model coefficients to be determined by the measurements Note that, the value of 2h h H h L h H h L should lie in the range [, ]; otherwise the power series expansion will lose accuracy Correspondingly, the value of h should always lie in [h L, h H ] During calibration, extra attention should be paid to the original orbital data because the altitude of the perigee or apogee may fall outside the interval [h L, h H ] In order to decrease the variation of the coefficients with respect to the time, the longitude and latitude in the Earth-centered inertial frame are not used directly in this expansion The λ and φ in Eq (3) are defined in a Sun-fixed coordinate frame [3], whose three axes of coordinates are defined as follows: i = r s r s, k = r s v s r s v s, j = k i k i (4) where r s and v s are the position and velocity vectors of the Sun in the Earth-centered J2000 coordinate frame They can be obtained by using the JPL DE405 ephemeris For a spacecraft with a position vector r in the Earth-centered J2000 coordinate frame, λ and φ can be computed by λ = arcsin r k r, φ = arctan r j r i (5) (a) In Eq (3), the number of coefficients in the expansion is (n max + )(n max + 2)(l max + ) When m = 0, sin mλ = 0, so the coefficient S ln0 is useless Then Eq (3) becomes: ρ(λ, φ, h)=exp { lmax ( 2h hh h L l=0 h H h L [ P n0 (sin φ) C ln0 ) l n max n=0 n + P nm (sin φ) ( Clnm cos mλ+ S lnm sin mλ )]} (6) m= (b) Fig 2 (a) Typical h ln ρ relationship; (b) relative error of approximations with different l max In the modified expansion, the total number of coefficients is (n max + ) 2 (l max + ) Theoretically, this expansion can approximate any density distribution

4 6 Y Ren, J Shan near the Earth with suitable lmax and nmax, and in the Sun-fixed reference frame the peak density always appears in a relatively fixed region, which makes the coefficients of expansion more stable than in the Earth-fixed frame The approximation accuracy varies for different λ, φ, and h combinations The general approximation accuracy of Eq (6) is measured by averaging the approximation accuracies on a series of sampling points In this research, sampling points lie on the intersects of a grid defined in a spherical coordinate system In this grid, λ increases from 0 to 358 deg with a step size of 2 deg φ increases from 90 to 90 deg with a step size of deg, and h increases from 200 to 000 km with a step size of 50 km For nmax equal to 4, 5, and 6, the average relative error values are 5%, 079%, and 073%, respectively It can be found that the approximation error does not decrease much further if nmax is greater than 5 The expansion with nmax = 5 and lmax = 2 (468 coefficients) will be used in the model calibration The relative approximation error of this expansion is shown in Fig 3, where it can be seen that the maximum relative error is smaller than 4% 3 Orbital dynamical model and model calibration The acceleration due to atmospheric drag is given by ad = 2 CD Av (t) ms ρvr2 e v where CD is a drag coefficient, Av (t) is the crosssectional area of the satellite in the direction of velocity relative to the ambient atmosphere, ms is the mass of the spacecraft, vr is the velocity magnitude relative to the ambient atmosphere, and e v is the unit vector in the relative velocity direction [4] The velocity relative to the rotating atmosphere, vr e v, can be computed by vr e v = v ω E r (8) where ω E is the spin velocity of the Earth Av (t) For simplicity, CD can be denoted by B, and ms Eq (7) can be rewritten to a D = Bρvr2 e v 2 (9) where B is uncertain and time-varying because in most cases the cross-sectional area Av is unknown and changes with the attitude of the satellite, ρ is the uncertain variable to be estimated, and vr and e v also contain some uncertain factors, for example, the thermospheric wind, which is very strong at high latitudes under geomagnetic storm conditions [7] Theoretically, the semi-major axis of the satellite decreases continuously under the influence of air (a) h = 200 km (b) h = 300 km (c) h = 500 km (d) h = 700 km (e) h = 900 km (f) h = 000 km Fig 3 (7) Relative error of the expansion

5 Calibration of atmospheric density model using two-line element data 7 drag, while in reality, under the short term influence of the Earth non-spherical gravitation perturbation the variation of the semi-major axis is not monotonic Fortunately, the mean mean-motion n is given in the two-line element set, which only reflects the long term variation of the semi-major axis, and the short term perturbations are computed by simplified perturbations models (SGP4) Eliminating the influence of the short term perturbation, the variation of the mean mean-motion can be denoted by dn dt 3 2 µ 2 3 ρbv 3 r (0) Equation (0) can be modified into the following integral format: n 2 3 (t k ) n 2 3 (t i ) + µ 2 3 tk t i ρbv 3 r dt () Assuming that the density distribution of the atmosphere is fixed over a short time span, referred to here as the analysis window, ρ can be computed by Eq (6) Clnm and S lnm in Eq (6) are also fixed over this window If B is known, n 2 3 (t k ) can be predicted by Eq (), and the observation of n 2 3 (t i ) can be obtained from the TLE data The key problem of model calibration is to find the suitable values of Clnm and S lnm to minimize the error between the prediction and the observation of n 2 3 (t i ) To solve the estimation problem, the number of constraints must be greater than the number of unknown coefficients For a set of TLE data, if N data points can be found in an analysis window, the number of constraint is N Figure 4 shows the number of constraints with respect to time The time interval between adjacent TLEs can be seen to have increased remarkably since November 202 The cause of this increase is not known Nevertheless, in most cases, more than 000 constraints can be obtained in an arbitrary 4-day analysis window In some analysis windows, the number of the constraints is still less than that of the unknown parameters, and the system becomes underdetermined This problem is more serious when a shorter analysis window is used In this case, priori information about the density distribution is needed The detailed algorithm will be presented in Section 52 In practice, the atmospheric density distribution varies continuously A wider analysis window gives discrete outputs with a bigger step size In order to avoid this problem, a sliding analysis window is used As shown in Fig 5, the analysis window moves with a small step size dt, which is much smaller than the width of the analysis window In each analysis window, an estimation of the atmospheric density distribution can be obtained In the common region of multiple analysis windows, the density distribution can be obtained by computing the average value of these estimations with Eq (2): Q Aver = k k Q i (2) i= where Q i is an arbitrary parameter (C lnm or S lnm ) in the estimation of the ith analysis window and k is the number of concurrent analysis windows 4 Pretreatment of two-line elements Historical records of orbital element sets (TLE data) can be downloaded from the Space-Track website ( An Application Programming Interface (API) is provided by Space- Fig 4 Number of constraints vs time Fig 5 Sliding analysis window

6 8 Y Ren, J Shan Track to batch-download the TLE sets which satisfy given conditions Using this API, it can be found that the orbital data of 40,373 objects have been recorded in this database However, the orbits of most of these objects have since decayed Using the Recent TLEs option of the Space-Track website, 5,07 on-orbit objects can be obtained The distribution of their semimajor axes and eccentricities is shown in Fig 6(a) In the region above the solid line, the altitude of the periapsis is higher than 200 km, while in the region below the dashed line, the altitude of the apoapsis is lower than 000 km In this research, the atmospheric density from 200 to 000 km will be calibrated, hence we only choose those points that lie in the region enclosed by the solid line, dashed line, and y axis (see Fig 6(b)) 5375 points are found in this region The historical TLE data (from st Jan 200 to st Oct 204) of these 5375 objects were downloaded from the Space-Track website These TLEs cannot be used to calibrate the atmospheric model yet, without further filtering, for two reasons The first is that some TLE data contain the influence of an orbit maneuver These must be eliminated before the calibration: orbit maneuver information is not provided by TLEs If the orbit maneuver is used to raise the orbit, it is easy to detect because the mean semi-major axis will not increase under the effect of air drag However, the maneuver is difficult to detect when it is used to lower the orbit or change the inclination slightly The second reason is that for most spacecraft the ballistic coefficients are unknown The SGP4-type drag coefficient is provided in TLEs, and it can in theory be transformed into the commonly used ballistic coefficient However, previous work has reported that the ballistic coefficient obtained by this method has a very large error [5] In order to solve these problems, the original TLE data are filtered and the ballistic coefficients are calibrated It is assumed that there are two sets of adjacent TLEs whose epochs are t i and t k (t k > t i ) The orbit in the time interval [t i, t k ] can be computed by SGP4 Then, with an empirical atmospheric density model (NRLSISE-00 model), the prediction of n 2 3 (t k ) can be computed by Eq () ñ (t k ) is the value of the next TLE data, which is considered as the observation The error between the observation and prediction is regularized by Eq (3): Fig 6 (a) (b) a e distribution of the on-orbit objects ε = n 2 3 (t k) ñ 2 3 (t i ) ñ 2 3 (t k ) ñ 2 3 (t i ) (3) where ñ (t i ) and ñ (t k ) are the mean mean-motion read from these two sets of TLE data From Eq (), the value of B is determined by Ref [5]: B = BST AR (4) where BST AR is the SGP4-type drag coefficient, which can be obtained directly from the TLE B = BST AR is the original relationship between the ballistic coefficient B and BST AR given by Ref [6] However, many researchers have shown that this formula has poor accuracy, hence Sang proposed a modified formula as shown in Eq (4), where is a correction factor, and chosen to be 0 in Ref [5] The TLE data used in this paper may have different orbital properties (different altitude range, inclination, or time interval) from those used in Ref [5], so will be calibrated before the model calibration In this calibration, ε for all 5375 sets of chosen TLE are computed, and 6,58,76 sets of ε are obtained A suitable which makes the mean value of these ε zero will be computed and chosen as the

7 Calibration of atmospheric density model using two-line element data 9 modifier factor During the -calibration, a TLE will be used only if it satisfies the following four rules: () Its BST AR must be greater than 0 (2) The time interval between two TLE sets is longer than 48 h (3) The mean semi-major axis decreases, ie, n (t k ) < n (t i ) (4) ε is smaller than Rule () can remove those TLE without the useful BST AR information Rule (2) can remove TLEs which only cover a small region in the space Rule (3) removes orbit segments with orbit-raising maneuvers Rule (4) is used to remove orbit segments with other maneuver types Figure 7 shows the distribution of ε before and after the modification 5 Calibration algorithm The calibration algorithm can be divided into two steps The first step is to identify the parameters from the empirical model These parameters will be used as the initial guess The second step is the parameter calibration, ie, to correct the parameters until the error (a) Before modification ( =, ε = 0903) between the prediction and the measurement is smaller than the specified tolerance 5 Parameter identification from the empirical model The coefficients C lnm and S lnm can easily be identified from the empirical models such as NRLSISE-00 by solving two classes of linear least squares problems The first class is ρ(h ) ρ(h 2) ρ(h k) = )( 2h h ( H h L 2h )( 2h2 h ( H h L 2h2 ( 2hk )( 2hk ) 2 ( 2h ) 2 ( 2h2 ) 2 ( 2hk ) lmax ) lmax ) lmax a 0 a a lmax (5) where h is the altitude and ρ(h) is the density at h obtained from the empirical model There are k measurements and a l (l = 0,, l max ) are the coefficients to be determined If we solve this problem for different (λ, φ) combinations, l max + surfaces of a l denoted by a l (λ, φ), can be obtained Then, the coefficients C lnm and S lnm can be obtained by solving l max + second class linear least squares problems as shown in Eq (6): a l (λ, φ ) a l (λ 2, φ 2) a l (λ k, φ k ) = P 00(sφ ) P 0(sφ )cλ P0(sφ )sλ P nm(sφ )s(mλ ) P 00(sφ 2) P 0(sφ 2)cλ 2 P0(sφ 2)sλ 2 P nm(sφ 2)s(mλ 2) P 00(sφ k ) P 0(sφ k )cλ P0(sφ k )sλ P nm(sφ k )s(mλ k ) where s is sine function and c is cosine function C l00 C l0 S l0 S lnm (6) 52 Parameter calibration (b) After modification ( = 678, ε = 0) Fig 7 Distribution of ε The purpose of parameter calibration is to find the coefficients Slnm, Clnm which have the best agreement with the measurements (historical TLE data) From the historical TLE data, one can get the discrete mean mean-motion denoted by [n (t 0 ), n (t ),, n (t k )] and can also reconstruct the continuous orbit of the satellites The orbit can

8 20 Y Ren, J Shan be obtained by an SGP4 propagator for TLEs, or by numerical integration for classical orbital elements The prediction and the measurement will be denoted by n and ñ, respectively When the expansion shown in Eq (6) approximates the actual atmospheric density distribution closely, n ñ will be minimized The partial differential relations between n 2 3 ñ 2 3 and C lnm, S lnm can be written as [ ] [ δ(n 2 3 ñ 2 n ) = n 2 3, δ C ] lnm lnm S lnm δ S (7) lnm Using Eqs (6) and (), the analytic forms of the elements in the Jacobian can be written as 2 n 3 = µ 2 3 lnm n 2 3 S lnm = µ 2 3 tk ( 2h hh h L ρ t i h H h L tk t i ) l Pnm(sin φ) cos mλbv 3 r dt (8) ( ) 2h hh h l L ρ Pnm(sin φ) sin mλbv 3 h H h dt r L (9) In Eqs (8) and (9), λ, φ, h, and v r are functions of time, which can be obtained by numerical propagation or SGP4 ρ is a function of λ, φ, and h, and its value can be computed by Eq (6) In order to identify the coefficients C lnm and S lnm, a series of measurement are needed The number of the measurements should be greater than the number of coefficients The coefficients can be obtained by solving the nonlinear least squares problem shown in the following equation: ε ε ε 000 ε lnm S lnm δ C 000 ε ε 2 = 2 ε ε 2 lnm S lnm δ C lnm ε k 000 lnm S lnm δ S lnm (20) where ε is the relative error defined in Eq (3) and ñ (t k ) and ñ (t i ) are constants that can be read from TLE data, hence the elements in the Jacobian can be computed by lnm = S lnm = n 2 3 ñ 2 3 (t k ) ñ 2 3 (t i ) (2) lnm n 2 3 ñ 2 3 (t k ) ñ 2 3 (t i ) S (22) lnm Theoretically, the unknown coefficients Slnm and C lnm can be obtained by solving the nonlinear least squares problem shown in Eq (20) However, it is found that the system is not always overdetermined even though the number of constraints is greater than the number of the unknown coefficients For example, when a 3-day analysis window from 8th August, 204 0:00:00 UTC to th August, 204 0:00:00 UTC is used, 729 constraints can be obtained from the TLE data, and the number of unknown coefficients is 637 (l max = 2 and n max = 6) The dimension of the Jacobian is However, the rank of the Jacobian equals 534, ie, the system is underdetermined When the analysis window is extended to 4 days (from 8th August, 204 0:00:00 UTC to 2th August, 204 0:00:00 UTC), the number of constraints increases to 2348, while the rank of the Jacobian only increases slightly to 552 The reason for this phenomenon is that most of the objects in TLE data are satellites or space debris originally from satellites Their orbits are consequently close to commonly used satellite orbits, hence the distribution of these objects is not uniform There are some regions where no orbit passes through Consequently the density information in these regions cannot be obtained by TLE data The only way to patch such regions is to use the empirical models There are several ways to integrate the information from empirical models with TLE data In this research, the virtual satellite method is used This method contains the following four steps: () Using the algorithm introduced in Section 5 to parameterize the NRLSISE-00 model The center point epoch of the analysis window is used as the input of the NRLSISE-00 model F 07 solar flux, 8-day average F 07 solar flux, and magnetic index a p are chosen as 50, 50, and 9, respectively (2) Generating a series of virtual satellites, and propagating their orbits using the parameterized NRLSISE-00 model The orbits of these virtual satellites are circular The inclinations are uniformly distributed in 0 90 deg, and the altitudes are uniformly distributed in km Only the gravitational force of the Earth and the air drag are considered by way of perturbations Earth s non-spherical gravitation and the third-body perturbation are not considered because they do not generate the secular variation of the mean mean-motion (3) Recording the epoch and orbital elements of the virtual satellite orbits for a given time span The mean mean-motion is computed by n = µ/a 3, and considered as the measurement (4) Obtaining the unknown coefficients C lnm and S lnm by solving the nonlinear least squares problem shown in Eq (23)

9 Calibration of atmospheric density model using two-line element data 2 ε ε 000 ε k ε = 000 ε 000 ε ε p p 000 ε ε lnm S lnm lnm S lnm ε ε lnm S lnm ε p lnm ε p S lnm δ C 000 δ C lnm δ S lnm (23) In this system, k constraints are from TLE data, and p constraints are from the virtual satellites ε denotes the relative error from the virtual satellites, and is computed from Eq (3) If k > p, the observations (TLE data) are more believable, otherwise the empirical model is more believable Note that, in order to predict the mean mean-motion, the orbit between two measurements has to be generated The orbits of real objects are generated using an SGP4 propagator, and the orbits of the virtual satellites are propagated by a 7th and 8th order Runge Kutta method When p is big enough, the orbits of the virtual satellites can cover the region between 200 and 900 km altitude uniformly, and the Jacobian will be full rank In this research, the nonlinear least squares problem is solved by the subroutine LDER in the inpack- package, which is the FORTRAN code of the Levenberg arquardt algorithm (LA) Similar to other gradient-based algorithms, LA only finds local minima, and the risk of finding a local minimum is determined completely by the initial guess The initial guess here is given by the empirical model, which has 0% 30% relative error; therefore the initial guess may be quite far from the global minimum and the risk of being trapped in a local minimum is relatively high To avoid this, a multi-point start technique is used First, an initial guess is given by an empirical model; then, by adding random perturbations to each coefficient, a series of initial guesses can be generated in the neighborhood of the original initial guess The boundary of the neighborhood and the distribution of the initial guesses in the neighborhood can be controlled by the characteristics of the random perturbation The nonlinear least squares problem in Eq (20) is then solved by using these perturbed initial guesses and the result with the minimum 2 norm is chosen as the final result The flowchart of this calibration algorithm is shown in Fig 8 6 Numerical simulation Numerical simulations have been conducted to verify the proposed calibration algorithm In the first simulation, the model was calibrated using only the orbital data of the virtual satellites, with no measurement error To demonstrate that the calibration algorithm works, the calibration results need to be very close to the empirical model used to generate the virtual data This simulation is used to test the feasibility of the algorithm under ideal conditions, ie, no measurement error In the second simulation, the real TLE data and the orbital data from the virtual satellites will be used together to calibrate the atmospheric density model The calibration results contain the priori information of the empirical model and the observation from TLE data With an appropriate mixing ratio of this information, a high-fidelity atmospheric density model can be obtained 6 Calibration based on the orbital data of the virtual satellites In this simulation, a half-day analysis window is used The initial epoch is 8th August, 204 0:00:00 UTC, and the center point epoch is 8th August, 204 Fig 8 Flowchart of the calibration algorithm

10 22 Y Ren, J Shan 06:00:00 UTC Using these epochs, the mean solar flux and magnetic index, the NRLSISE-00 model is parameterized in the Sun-fixed reference frame 60 circular orbits of virtual satellites are then generated All these orbits start at the initial epoch and terminate half a day after the initial epoch The inclination and altitude are randomly generated between 0 and 90 deg, and 200 and 900 km, respectively Both satisfy the uniform distribution l max and n max are set to be 2 and 5, so the number of coefficients is 468 The dimension of the Jacobian is The iteration starts from 0 sets of randomly generated initial guesses 8 of them were seen to converge to the same minimum value During the computation, the Jacobian is checked in each iteration step and found always to be full rank Figure 9(a) shows the converging process It can be found that the modulus of the error vector between the prediction and the measurement decreases to 0 7 after 0 iterations Figure 9(b) shows the elements of the error vector after the 0th iteration The maximum error between the prediction and the measurement is less than In order to verify the accuracy of the calibrated model, the atmospheric density distributions at 200, 500, and 900 km altitude are computed by the series expansion, and compared with the NRLSISE-00 model It can be seen from Fig 0 that the relative error is always less than 4% This means that under ideal conditions the proposed algorithm can identify the atmospheric density model using orbital data 62 Calibration based on TLE data In this section, the atmospheric density model is calibrated using the real TLE data, and the calibrated model is verified via cross-validation For instance, if N constraints can be obtained from the TLE data, then P (P < N) sets of them will be used to calibrate the model, and the other N P sets will be used for validation The N P sets of validation constraints will be computed in the empirical model and calibrated model, respectively If the constraints obtained from the calibrated model have a smaller mean error than that obtained from the empirical model, the proposed calibration algorithm is effective Note that any arbitrary N P sets of constraints can be used for (a) (a) (b) (b) Fig 9 Constraint values with respect to the number of iterations (a) odulus of constraints vs number of iterations; (b) values of constraints after the 0th iteration (c) Fig 0 Relative error of the calibrated atmospheric density model at (a) 200 km, (b) 500 km, and (c) 900 km

11 Calibration of atmospheric density model using two-line element data 23 validation The selection of the validation constraints does not affect the final result In this simulation, a 4-day analysis window is used In this window, 2397 constraints can be obtained from the TLE data Among them, 500 constraints are chosen randomly and used in model calibration 300 virtual satellites are used to generate the orbital data For each satellite 3 data points are recorded The time span between the data points is 2 days So there are 600 constraints from the virtual satellites oreover, the virtual orbital data covers the spatial and time span required Figure shows the convergence history The iteration is terminated when the difference between the kth iteration and the (k +)th iteration is less than % The simulation stops after ten iterations The modulus of the error vector decreases from 99 to 002 Due to the observation error, the modulus cannot converge to zero The remaining 897 constraints from the TLEs are used to verify the calibrated model These constraints have been computed in the empirical model and the calibrated model separately, and the distributions of the errors are shown in Figs 2(a) and 2(b), respectively It can be seen that the variance decreases after the calibration, ie, the calibrated model is more congruent to the real TLE data ultiple simulation runs are conducted For each run, the 500 calibration constraints and the 897 verification constraints are chosen randomly It is found that the distribution of the error does not change much In other words, the calibration result is independent of the selection of observations 7 Conclusions In this paper, the atmospheric density model is Fig Constraint values with respect to the number of iteration (a) (b) Fig 2 Distribution of the constraints from TLE: (a) before calibration, (b) after calibration calibrated by using the TLE data First, the density distribution of the atmosphere is parameterized using a power series expansion whose coefficients are denoted by several spherical harmonic expansions The accuracy of the expansion approximation is validated through numerical simulation Second, useful TLE data are selected, and the ballistic coefficients are estimated using BST AR values in the TLEs Third, in the calibration algorithm, the information from TLE data and the information from the empirical model are integrated The coefficients of the series expansion can be obtained by solving a nonlinear least squares problem Finally, the proposed algorithm is verified by numerical simulation The results show that under ideal conditions, the relative error of the calibrated model is 4% Under the effect of the observation error contained in TLE data, the calibration algorithm can converge in 0 iterations, and the calibrated model can provide a

12 24 Y Ren, J Shan higher accuracy than that of the empirical model References [] Shanklin, R E, Lee, T, Samii,, allick, K, Cappellari, J O Comparative studies of atmospheric density models used for earth satellite orbit estimation Journal of Guidance, Control, and Dynamics, 984, 7(2): [2] Jacchia, L G Revised static models of the thermosphere and exosphere with empirical temperature profiles SAO Special Report No 332 Smithsonian Institution Astrophysical Observatory, 97 [3] Hedin, A E SIS-86 thermospheric model Journal of Geophysical Research: Space Physics, 987, 92(A5): [4] Bruinsma, S, Thuillier, G, Barlier, F The DT empirical thermosphere model with new data assimilation and constraints at lower boundary: Accuracy and properties Journal of Atmospheric and Solar-Terrestrial Physics, 2003, 65(9): [5] Cefola, P, Volkov, I I, Suevalov, V V Description of the Russian upper atmosphere density model GOST In: Proceedings of the 37th COSPAR Scientific Assembly, 2008: 476 [6] Vallado, D A Fundamentals of Astrodynamics and Applications Springer Science & Business edia, 200: [7] Doornbos, E, Klinkrad, H, Visser, P Atmospheric density calibration using satellite drag observations Advances in Space Research, 2005, 36(3): [8] Doornbos, E, Klinkrad, H, Visser, P Use of two-line element data for thermosphere neutral density model calibration Advances in Space Research, 2008, 4(7): 5 22 [9] Liu, H, Lühr, H, Henize, V, Köhler, W Global distribution of the thermospheric total mass density derived from CHAP Journal of Geophysical Research: Space Physics, 2005, 0(A4): A0430 [0] Bruinsma, S, Tamagnan, D, Biancale, R Atmospheric densities derived from CHAP/STAR accelerometer observations Planetary and Space Science, 2004, 52(4): [] Shoemaker, A, Wohlberg, B, Koller, J Atmospheric density reconstruction using satellite orbit tomography Journal of Guidance, Control, and Dynamics, 205, 38(4): [2] Picone, J, Hedin, A E, Drob, D P, Aikin, A C NRLSISE-00 empirical model of the atmosphere: Statistical comparisons and scientific issues Journal of Geophysical Research: Space Physics, 2002, 07(A2): SIA 5- SIA 5-6 [3] Hinks, J C, Psiaki, L Simultaneous orbit and atmospheric density estimation for a satellite constellation In: Proceedings of the AIAA/AAS Astrodynamics Specialist Conference, Guidance, Navigation, and Control and Co-located Conferences, 200: AIAA [4] King-Hele, D G Satellite Orbits in an Atmosphere: Theory and Application Springer Science & Business edia, 987: [5] Sang, J, Bennett, J C, Smith, C H Estimation of ballistic coefficients of low altitude debris objects from historical two line elements Advances in Space Research, 203, 52(): 7 24 [6] Hoots, F R, Roehrich, R L odels for propagation of NORAD element sets Spacetrack Report No 3 Defense Technical Information Center, 980 Yuan Ren joined the Department of Earth and Space Science and Engineering of York University as the research fellow in September 20 Prior to this, he was a arie- Curie experienced researcher at the Department of Applied athematics I, Polytechnic University of Catalonia, Spain, from 2008 to 200, and an assistant professor at the School of Aeronautics and Astronautics, Shanghai Jiao Tong University, China, from 2007 to 2008 He received his BEng, Eng, and PhD degrees all from Harbin Institute of Technology, China, in 2002, 2004, and 2007, respectively His current research interests are orbital dynamics, three-body and multi-body problem, trajectory optimization and navigation yren@yorkuca Jinjun Shan joined the Department of Earth and Space Science and Engineering of York University as an assistant professor in July 2006 He was promoted to associate professor in July 20, and full professor in July 206 Dr Shan was awarded the Alexander von Humboldt (AvH) Research Fellowship for Experienced Researchers and JSPS Invitation Fellowship in 202 He is a senior member of IEEE and AIAA, and a professional engineer in Ontario since 2007 His current research interests are dynamics, control and navigation, smart materials and structures, multi-agent system, and orbit dynamics He is the founding director of Spacecraft Dynamics Control and Navigation Laboratory (SDCNLab) of York University jjshan@yorkuca