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An Approach for Nonlinear UncertaintyPropagation: Application to Orbital

Mechanics

Daniel Giza∗ , Puneet Singla† , Moriba Jah‡

An approach for nonlinear propagation of orbit uncertainties is discussedwhile making use of the Fokker-Planck-Kolmogorov Equation (FPKE). Thecentral idea is to replace evolution of initial conditions for a dynamical sys-tem with the evolution of a probability density function (pdf) for statevariables. The transition pdf corresponding to dynamical system state vec-tor is approximated by using a finite Gaussian mixture model. The meanand covariance of different components of the Gaussian mixture model arepropagated through the use of an Unscented Kalman Filter (UKF). Fur-thermore, the unknown amplitudes corresponding to different componentsof the Gaussian mixture model are found by minimizing the FPKE errorover the entire volume of interest. This leads to a convex quadratic min-imization problem guaranteed to have a unique solution. The two-bodyproblem model with non-conservative atmospheric drag forces and initialuncertainty will be used to show the efficacy of the ideas developed in thispaper.

I. Introduction

In July, 1994, the comet Shoemaker-Levy 9 collided with Jupiter at over 133,000 mph.

In March, 2004, a group of about large 30 meteors passed by the Earth at only one tenth

the distance to the moon; these space rocks were discovered only three days prior to the

event. A spectacular fireball lit up the sky above Northern Sudan on October 7, 2008. This

explosion was caused by the atmospheric entry of a small near-Earth asteroid, estimated to

∗Graduate Student, Department of Mechanical & Aerospace Engineering, University at Buffalo, Buffalo,NY-14260, Email: [email protected].†Assistant Professor, Senior AIAA, AAS Member, Department of Mechanical & Aerospace Engineering,

University at Buffalo, Buffalo, NY-14260, Email: [email protected].‡Director, Advanced Sciences and Technology Research Institute for Astrodynamics (ASTRIA), Air Force

Research Laboratory, 535 Lipoa Parkway, Suite 200, Kihei, HI-96753

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American Institute of Aeronautics and Astronautics

be no more than a few meters in diameter. Asteroid 2004 MN4 is predicted to make a very

close encounter in April of 2029, followed by other approaches in 2034 and 2036. The uncer-

tainty of the 2029 approach (± 1 Earth radii) is quite large compared to the predicted miss

distance of 5.6 Earth radii. This close encounter is not presently considered a high risk for

collision, however the subsequent encounters have larger uncertainties and cannot be accu-

rately predicted due to the nonlinearity of the dynamics and the large uncertainty associated

with the 2029 approach. According to a NASA catalog, there are a total of 1015 potentially

hazardous asteroids. After cataloging, the next challenge toward ensuring our safety is to

determine the path of these near Earth objects (NEOs) for threat analysis. Similarly, the

increasing amount of debris and inactive resident space objects (RSOs), particularly those

with high area-to-mass ratios, in both the low-earth-orbit (LEO) and geosynchronous-earth-

orbit (GEO) regime pose a threat to active RSOs and must be accurately tracked for threat

analysis.

A mechanism to represent the uncertainty is necessary before analysis can be done in an

efficient and consistent manner. Probabilistic means of representing uncertainty has been ex-

plored extensively and provides the greatest wealth of knowledge. The most common method

for representing orbital uncertainty is to approximate the initial distribution using a Gaus-

sian model derived from the orbit determination, and use linear error theory to propagate

the mean and covariance of the Gaussian model forward in time. This can lead to significant

errors when propagating uncertain orbits for large amounts of times.1–4 In Refs.,5,6 judicious

choices for coordinate systems is advocated to decrease the uncertainty propagation errors

while making use of linear error theory. In addition to this, several approximate techniques

exist in the literature to approximate the initial condition uncertainty evolution, the most

popular being Monte Carlo (MC) methods,7 Gaussian closure,8 Equivalent Linearization,9

and Stochastic Averaging.10,11 All of these algorithms except Monte Carlo methods are

similar in several respects, and are suitable only for linear or moderately nonlinear systems,

because the effect of neglected higher order terms can lead to significant errors. Monte Carlo

methods require extensive computational resources, and become increasingly infeasible for

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high-dimensional dynamic systems.12 Furthermore, all these approaches provide only an

approximate description of the uncertainty propagation problem by restricting the solution

to a small number of parameters - for instance, the first N moments of the sought pdf.

For stochastic continuous dynamic systems the exact evolution of state pdf is given by the

Fokker-Planck-Kolmogorov Equation (FPKE).13 Park et al.14 has discussed the use of FPKE

to analyze the spacecraft trajectory statistics by incorporating higher order Taylor series

term in the spacecraft dynamics. Analytical solutions for the Fokker-Planck Kolmogorov

equation exist only for a stationary probability density function and are restricted to a limited

class of dynamical systems.13,15 In this paper, we explore the use of recently developed

Adaptive Gaussian Mixture Model approach16,17 for accurate orbit uncertainty propagation

while incorporating the solution to the FPKE.

II. Equations of Motion

The classical differential equations governing the perturbed two-body problem1 are given

by

x+ µxr3

= adx(t, x, y, z, x, y, z), r2 = x2 + y2 + z2

y + µyr3

= ady(t, x, y, z, x, y, z), () ≡ d()/dt

z + µzr3

= adz(t, x, y, z, x, y, z), µ ≡ G(m1 +m2)

(1)

or in compact form:

r +µr

r3= ad (2)

Given the initial conditions{r(t0), r(t0)}, and prescription of the perturbing acceleration

function ad, Eqs. (1), (2) can be solved for the Cartesian coordinate trajectory. However,

the exact initial conditions and perturbing acceleration profiles are typically not known.

Throughout most of the modern history of physics, it has been assumed that it is possible to

shrink the uncertainty in the final dynamical prediction by measuring the initial conditions to

greater and greater accuracy. But Poincare noticed that certain astronomical systems did not

seem to obey the rule that increasing the accuracy of the initial conditions would effect the

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final prediction in a corresponding way.18 Poincare showed that a very tiny imprecision in the

initial conditions would grow in time at an enormous rate. Thus, two nearly-indistinguishable

sets of initial conditions for the same system would result in two final predictions that differ

vastly from each other. Similar observations were made by Lorenz in 1963 while studying

a primitive model of atmospheric current. He showed that even the smallest imaginable

discrepancy between two sets of initial conditions would always result in a large discrepancy

at later or earlier times.19,20 Thus, the presence of chaotic systems in nature seems to place a

limit on our ability to apply deterministic physical laws to predict motions with any degree of

certainty. This means that in order to make long-term forecasts with any degree of accuracy,

it is necessary to understand how the initial conditions and model uncertainty propagate

through the nonlinear mathematical models.

III. Uncertainty Propagation

In conventional deterministic systems, the system state assumes a fixed value at any

given instant of time. However, in stochastic dynamics it is a random variable and its time

evolution is given by a stochastic differential or difference equation:

x(t) = f(t,x) + g(t,x)Γ(t) (3)

where, g(t,x)Γ(t) is a stochastic term representing the combined effect of un-modeled dy-

namics, and external disturbances acting on the system. In the orbit uncertainty propagation

problem, this terms can constitute un-modeled external disturbances such as solar radiation

pressure, atmosphere drag for low-earth-satellite orbit etc. We further assume Γ(t) to be a

zero mean white-noise process with the correlation matrix Q. The uncertainty associated

with the state vector x(t) is usually characterized by a time dependent state pdf p(t,x). In

essence, the study of stochastic systems reduces to finding the nature of the time-evolution of

the system-state pdf. For stochastic continuous-time dynamic systems, the exact evolution

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of state pdf is given by the Fokker-Planck-Kolmogorov equation (FPKE)13

∂

∂tp(t,x) = fT (x, t)

∂p(t,x)

∂x+

1

2Tr

(g(t,x(t))QgT (t,x(t))

∂2p

∂x∂xT

)(4)

The FPKE is a formidable equation to solve, because of the following issues:

1. Positivity of the pdf: p(t,x)dx ≥ 0.

2. Normalization constraint of the pdf:∫

Rn p(t,x)dx = 1.

3. No fixed Solution Domain: how to impose boundary conditions in a finite region and

restrict numerical computations to the volume of interest.

Analytical solutions for the FPKE exist only for a stationary pdf and are restricted to a

limited class of dynamical systems.13,15 Thus researchers are actively looking at numerical

approximations to solve the FPKE,21–26 generally using the variational formulation of the

problem. However, these methods suffer from the “curse of dimensionality” since one needs

to discretize the space over which the pdf mass lies and this discretization comes at an

expense of overwhelmingly cost of computation.26 In this paper, we will make use of the

recently developed adaptive Gaussian mixture model approach to solve the FPKE efficiently.

The main idea of this approach is to approximate the joint probability density function by

a weighted sum of independent Gaussian pdfs. The weights associated with each Gaussian

elements are selected so that the FPKE residual error is minimized over an area of inter-

est.16,17,27 An advantage of this approach is that the problem of solving FPKE is posed as a

convex optimization problem with a unique solution. In Refs. [16,17,27–29], the effectiveness

of this approach has been demonstrated in solving the FPKE for higher dimension problems

including the 6-D spacecraft attitude estimation problem.29

IV. Gaussian Mixture Model

The main idea behind the Adaptive Gaussian Mixture model approach, is to solve the

FPKE by approximating the forecast pdf using a finite sum of Gaussian components and

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tools from convex optimization. Here, we briefly discuss the main idea and details can be

found in Refs. [16,17,27].

Let us consider the following equation depicting the Gaussian mixture model approxima-

tion for the forecast density function, p(t,x)

p(t,x) =N∑i=1

wipgiwhere pgi

= N (x(t) | µi,Pi) (5)

where µi and Pi represent the mean and covariance of the ith component of the Gaussian

mixture, N (x(t) | µi,Pi), respectively and wi denotes the amplitude of ith Gaussian in the

mixture. The positivity and normalization constraint on the mixture pdf, p(t,x), leads to

following constraints on the amplitude vector:

N∑i=1

wi = 1, wi ≥ 0, ∀i (6)

In Ref. [30], it is shown that since all the components of the mixture pdf of Eq. (5) are

Gaussian and thus, only estimates of their mean and covariance need to be maintained in

order to obtain the optimal state estimates. These can be propagated by using a continuous-

time derivation of the Unscented Kalman Filter,31 which uses a set of deterministically

chosen sigma-points that capture the mean and covariance of the initial distribution. The

propagation equations for mean and covariance are given as

Xi = [µi . . . µi] +√c [0 A −A] , c = α2(n+ κ) (7)

µi = f(Xi)wm (8)

Pi = XiWfT (Xi) + f(Xi)WXiT + g(t,µi)QgT (t,µi) (9)

where Xi is the n × 2n + 1 matrix of sigma-points and the weight vector, wm, and weight

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matrix, W , are given by

λ = α2(n+ κ)− n

W(mean)0 =

λ

n+ λ

W(cov)0 =

λ

(n+ λ) + (1− α2 + β)

W(mean)j =

1

(2(n+ λ)), j = 1, . . . , 2n

W(cov)j =

1

(2(n+ λ)), j = 1, . . . , 2n

wm =[W

(mean)0 . . . W

(mean)2n

]T(10)

W = (I− [wm . . . wm])× diag(W

(cov)0 . . . W

(cov)2n

)× (I− [wm . . . wm])T (11)

The constants α, β, and κ in the above equations are constant parameters of the method.

The spread of sigma points is determined by α and is typically a small positive value, i.e.

1×10−4 ≤ α ≤ 1. The parameter β is used to incorporate prior knowledge of the distribution,

and for is optimally chosen as 2 for a normal distribution. The parameter κ can be used to

exploit knowledge of the distributions higher moments, and for higher order systems choosing

κ = 3− n minimizes the mean-squared-error up to the fourth order.2

Notice that the weights wi corresponding to each Gaussian component are also unknown.

Hence, the idea is to use FPKE error as a feedback to update the amplitude of different

Gaussian components in the mixture pdf. In other words, we seek to minimize the FPKE

error under the assumption of Eqs. (5), (6), (8) and (9). Substituting Eq. (5) in Eq. (4)

leads to

e(t,x) =∂p(t,x)

∂t− LFP(p(t,x)) (12)

where we denote LFP(p(t,x)) as the Fokker-Planck operator. The application of this operator

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is given by,

LFP(p(t,x)) =N∑i=1

wtiLFP(pgi)

= lFPTw (13)

where w is the vector of weights at time t and the elements of lFP are given by the application

of the Fokker-Planck operator to the individual Gaussian components:

LFP(pgi) = −

∂pTgi

∂xf(t,x)− pgi

Tr

[∂f(t,x)

∂x

]+

1

2Tr

[Q

∂2pgi

∂x∂xT

](14)

The first term in Eq.(12) is the time derivative of the probability density function approxi-

mated by the Gaussian mixture, and is given by,

∂p(t,x)

∂t=

N∑i=1

(wtipgi

+ wti∂pTgi

∂µiµi + wtiTr

[∂pgi

∂Pi

Pi

])(15)

The derivatives of the first two moments, µ and P, are given by Eq.(8)-(9), and the derivative

of the weights, wi, is obtained by the following finite difference approach:

wti =1

∆t(wt

′

i − wti) where t′ = t+ ∆t (16)

Substituting Eq.(16) into Eq.(15), we obtain

∂p(t,x)

∂t=

N∑i=1

1

∆tpgiwt′

i +N∑i=1

(∂pTgi

∂µi

µi + Tr

[∂pgi

∂Pi

Pi

]− 1

∆tpgi

)︸︷︷︸

mDTi

wti (17)

=1

∆tpg

Twt′ + mDTTwt (18)

where wt′ is the vector of new weights to be found, pg is the vector of Gaussian mixture

components, and the elements of mDT are shown in the above equation. Furthermore, the

different derivatives of the Gaussian components in the above equations are computed using

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these analytical expressions,

∂pgi

∂µi

= P−1i (x− µi) pgi

∂pgi

∂Pi

=1

2

[P−1i (x− µi) (x− µi)

T P−1i −P−1

i

]pgi

∂pgi

∂x= −P−1

i (x− µi) pgi

∂2pgi

∂x∂xT=[(Pi)

−1 (x− µi) (x− µi)T P−1

i −P−1i

]pgi

Substituting Eq.(18) and Eq.(13) into the FPKE error equation, Eq.(12), gives

e(t,x) =1

∆tpg

Twt′ + (mDT − lFP)T wt (19)

Now, at a given time instant, after propagating the mean, µi, and the covariance, Pi, of

individual Gaussian elements using Eqs. (8) and (9), we seek to update weights by minimizing

the FPKE error over some volume of interest V .

minwt′

i

1

2

∫V

e2(t,x)dx (20)

s.tN∑i=1

wt′

i = 1

wt′

i ≥ 0, i = 1, · · · , N

Because the FPKE error, Eq.(19), is linear in the new Gaussian component weights, wt′i , the

problem can be written as a quadratic programming problem.

minwt′

i

1

2wTt′Mcwt′ + wT

t′Ncwt (21)

s.t 1TN×1wt′ = 1

wt′ ≥ 0N×1

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Where 1N×1 ∈ RN×1 is a vector of ones, 0N×1 ∈ RN×1 is a vector of zeros and the matrices

Mc ∈ RN×N and Nc ∈ RN×N are given by

Mc =1

∆t2

∫V

pgpgTdx (22)

Nc =1

∆t

∫V

pg (mDT − lFP)T dx (23)

The individual components of these two matrices are given by:

mcij =1

∆t2|2π(Pi + Pj)|−1/2 exp

[−1

2(µi − µj)

T

×(Pi + Pj)−1(µi − µj)

]for i 6= j (24)

mcii =1

∆t2|4πPi|−1/2 (25)

ncij =1

∆tpgi

∫V

(∂pTgj

∂µj

µj + Tr

[∂pgj

∂Pj

Pj

]− 1

∆tpgj

+∂pTgj

∂xf(t,x) + pgj

Tr

[∂f(t,x)

∂x

]− 1

2Tr

[Q

∂2pgj

∂x∂xT

])dx (26)

A major challenge in solving the convex minimization problem of Eq. (21) is the need

to evaluate integrals involving Gaussian pdfs over volume V . This volume V can be defined

by making use of the fact that the mass of a Gaussian pdf is concentrated in a finite vol-

ume about its mean. This is one of the major advantages of using the Gaussian mixture

model approximation, as it automatically defines the space over which probability mass lies.

Also, these integrals can be computed exactly for polynomial nonlinearity and in general

can be approximated by using Gaussian Quadrature, Monte Carlo integration or Unscented

Transformation methods.32 While in lower dimensions the Unscented Transformation and

Gaussian Quadrature methods are mostly equivalent, the Unscented Transformation is com-

putationally more appealing in evaluating integrals for higher dimensions as the number of

points taken to compute the integral grows only linearly with the number of dimensions.

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V. Low Earth Orbit with Atmospheric Drag

Non-conservative forces in orbital mechanics very quickly lead to the evolution of non-

Gaussian state-pdfs, even when the initial distribution is Gaussian. This is especially true

when the object of interest has a high area-to-mass ratio (HAMR). These HAMR objects

were first discovered in the geosynchronous regime and are hypothesized to be part of the

rapidly increasing population of space debris.33 After the recent collision of a US Iridium

satellite and the defunct Russian Cosmos 2251 satellite, the debris had to be accurately

tracked in order to analyze threat to other satellites in similar orbits. Similarly, some initial

speculation about HAMR objects indicates that several may be multi-layered insulation

material that have become separated from its spacecraft of origin by collision or some other

means. In the geosynchronous regime HAMRs are largely effected by solar radiation pressure

as well as third-body effects. In low-earth orbit the effect of atmospheric drag dominates,

and this is the problem focused on below.

Consider the dynamics of an object in low-earth-orbit (LEO) that is affected by noncon-

servative atmospheric drag forces, whose planar equations of motion are given by34

x+µx

r3= aDx(t, x, y, x, y), aD =

1

2

CdA

mρv2

rel

vrel|vrel|

y +µy

r3= aDy(t, x, y, x, y), ρ = ρ0e

−(r−R⊕)

h

where Cd is the coefficient of drag, A is the cross-sectional area, m is the mass of the object,

and ρ is the atmospheric density at a given altitude. The atmospheric density model is

assumed to be an exponential model with reference density ρ0. It is also worth noting that

the vrel is not the velocity state vector, but rather the velocity relative to Earth’s atmosphere.

For simulation, we assume that we have an initial state-pdf that is Gaussian with mean

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and covariance shown below,

µ0 =

6.6032× 106

0

0

7.7695× 103

P0 =

1.78× 106 0 0 0

0 2.50× 105 0 0

0 0 6.25 0

0 0 0 25

which correspond to a starting altitude of 225 km. The covariance matrix reflects a larger

uncertainty in the radial position than in the in-track position as well as a larger uncertainty

in the tangential velocity than in radial velocity. The ballistic coefficient is chosen as B = 1.4,

where B = CdAm

, which is consistent with a HAMR object. Other assumptions include a

circular orbit and the absence of process noise under the assumption that the perturbing

atmospheric drag forces are being accurately modeled. A Monte Carlo simulation is run

in which 50,000 samples are taken from the initial distribution and propagated for 2 orbit

periods. The resulting histograms are taken to be the true conditional pdf at a given time

to which the Adaptive Gaussian Mixture results can be compared.

Several other mixture components are chosen as the sigma points from −3σ to 3σ of

the initial distribution along the radial position and the tangential velocity, in this case x

and y respectively, as those are the two states with the highest uncertainty and have the

greatest effect on the final state-pdf. This leads to a total of 50 mixture components, as

all combinations of the sigma points for these two states are taken. The initial Gaussian

distribution is assigned a weight of 1, and the mixture components placed along the sigma

points are assigned a weight of 0. In Fig.(1), the histograms created from the Monte Carlo

simulation and the Adaptive Gaussian Mixture pdfs are shown. It is easily seen that the

atmospheric drag is beginning to curve and skew the distribution and that the Adaptive

Gaussian mixture accurately captures this behavior. In fact, after 1 orbital period the initial

distribution is assigned a weight of 0, evidencing that the initial Gaussian distribution no

longer accurately captures the behavior of the state-pdf. The time-evolution of the weights

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for the initial distribution and the dominant Gaussian components after two orbital periods,

T , are shown in Table 1. The coherence of the Adaptive Gaussian Mixture results to the

Monte Carlo results is best seen from the conditional pdf contours. In Fig.(2), the contours

of the Monte Carlo simulation, Adaptive Gaussian Mixture pdfs, as well as the Unscented

Kalman Filter propagation of the initial Gaussian distribution are shown. These contours

are generated at 1% of the pdf’s peak value done because although the contours of constant

probability are easily generated for the Unscented Kalman Filter, the Adaptive Gaussian

mixture and the Monte Carlo samples do not have an equivalent 3σ contour, and plotting

the 1% contour is done for consistency. It is clear that the Unscented Kalman Filter and

the Adaptive Gaussian mixture pdf are initially identical and remain similar for some time.

Eventually, however, the Unscented Kalman Filter no longer accurately represents the area of

uncertainty given by the Monte Carlo samples contour while the Adaptive Gaussian Mixture

contour is able to very accurately capture the non-Gaussian behavior.

VI. Conclusions

Orbit uncertainty propagation is an important problem to study while designing the orbit

for man-launched spacecraft or doing the risk-analysis for NEO collision with Earth. This

paper presents the formulation for accurate uncertainty propagation through orbital me-

chanics while making use of the Fokker-Planck-Kolmogorov Equation (FPKE). A Gaussian

mixture model is used to approximate the state transition pdf and furthermore, a convex

optimization problem is posed to update unknown parameters of the mixture model. This

approach is shown to provide much more accurate uncertainty propagation than traditional

methods that simply maintain a mean and covariance of an initial distribution approximated

by a single Gaussian, and is very attractive for orbital problems involving high area-to-mass

ratio objects under the non-conservative perturbing forces of atmospheric drag and solar

radiation pressure.

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(a) MC Histogram - 0.5 Orbits (b) AGM PDF - 0.5 Orbits

(c) MC Histogram - 1 Orbit (d) AGM PDF - 1 Orbit

(e) MC Histogram - 1.5 Orbits (f) AGM PDF - 1.5 Orbits

(g) MC Histogram - 2 Orbits (h) AGM PDF - 2 Orbits

Figure 1. Conditional PDFs

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(a) Initial PDF Contours (b) 0.25 Orbits (c) 0.5 Orbits

(d) 0.75 Orbits (e) 1 Orbit (f) 1.25 Orbits

(g) 1.5 Orbits (h) 1.75 Orbits (i) 2 Orbits

Figure 2. Conditional PDF Contours

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Table 1. Time-Evolution of Dominant Gaussian Components Weights

Time w1 w19 w20 w26 w27

0 1.0000 0.0000 0.0000 0.0000 0.0000

0.1T 0.9995 0.0000 0.0000 0.0000 0.0000

0.2T 0.9961 0.0005 0.0003 0.0006 0.0004

0.3T 0.9654 0.0011 0.0005 0.0060 0.0046

0.4T 0.3349 0.0593 0.0352 0.1230 0.0697

0.5T 0.0126 0.1003 0.0570 0.1877 0.1063

0.6T 0.0000 0.1022 0.0581 0.1952 0.1117

0.7T 0.0000 0.1089 0.0612 0.1999 0.1140

0.8T 0.0000 0.1156 0.0648 0.2022 0.1150

0.9T 0.0000 0.1202 0.0673 0.2030 0.1153

1.0T 0.0000 0.1220 0.0682 0.2031 0.1153

1.1T 0.0000 0.1210 0.0676 0.2034 0.1154

1.2T 0.0000 0.1177 0.0657 0.2039 0.1156

1.3T 0.0000 0.1135 0.0632 0.2047 0.1159

1.4T 0.0000 0.1105 0.0612 0.2055 0.1162

1.5T 0.0000 0.1129 0.0620 0.2068 0.1167

1.6T 0.0000 0.1230 0.0669 0.2082 0.1172

1.7T 0.0000 0.1393 0.0754 0.2072 0.1167

1.8T 0.0000 0.1560 0.0850 0.2025 0.1148

1.9T 0.0000 0.1674 0.0924 0.1961 0.1126

2.0T 0.0000 0.1712 0.0953 0.1928 0.1114

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