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Extension of the King-Hele orbit contraction method for accurate,
semi-analytical propagation of non-circular orbits
Stefan Frey
a,⇑
, Camilla Colombo
a
, Stijn Lemmens
b
a
Department of Aerospace Science and Technology, Politecnico di Milano, Via La Masa, 34, 20156 Milan, Italy
b
ESA/ESOC Space Debris Office, Robert-Bosch-Str. 5, 64293 Darmstadt, Germany
Received 16 October 2018; received in revised form 13 February 2019; accepted 11 March 2019
Available online 27 March 2019
Abstract
Numerical integration of orbit trajectories for a large number of initial conditions and for long time spans is computationally expen-
sive. Semi-analytical methods were developed to reduce the computational burden. An elegant and widely used method of semi-
analytically integrating trajectories of objects subject to atmospheric drag was proposed by King-Hele (KH). However, the analytical
KH contraction method relies on the assumption that the atmosphere density decays strictly exponentially with altitude. If the actual
density profile does not satisfy the assumption of a fixed scale height, as is the case for Earth’s atmosphere, the KH method introduces
potentially large errors for non-circular orbit configurations.
In this work, the KH method is extended to account for such errors by using a newly introduced atmosphere model derivative. By
superimposing exponentially decaying partial atmospheres, the superimposed KH method can be applied accurately while considering
more complex density profiles. The KH method is further refined by deriving higher order terms during the series expansion. A variable
boundary condition to choose the appropriate eccentricity regime, based on the series truncation errors, is introduced. The accuracy of
the extended analytical contraction method is shown to be comparable to numerical Gauss-Legendre quadrature. Propagation using the
proposed method compares well against non-averaged integration of the dynamics, while the computational load remains very low.
Ó2019 COSPAR. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/
by-nc-nd/4.0/).
Keywords: Orbit decay; Atmospheric drag; Semi-analytical propagation; King-Hele
1. Introduction
Numerical integration of the full orbital dynamics,
including short-periodic variations, can be demanding
from a computational point of view. For this reason,
Semi-Analytical (SA)
1
methods were developed to perform
this task in a less demanding manner (e.g. Liu, 1974). Such
methods remove the short-term periodic effects by averag-
ing the variational equations, thereby reducing the stiffness
of the problem. This is especially desired when orbits are to
be propagated for many initial conditions and over long
lifetimes, e.g. for estimating the future space debris
environment.
The calculation of the orbit contraction – i.e. the reduc-
tion in semi-major axis and eccentricity – induced by atmo-
spheric drag requires the integration of the atmosphere
https://doi.org/10.1016/j.asr.2019.03.016
0273-1177/Ó2019 COSPAR. Published by Elsevier Ltd.
This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
⇑
Corresponding author.
E-mail addresses: stefan.frey@polimi.it (S. Frey), camilla.colombo@polimi.it (C. Colombo), stijn.lemmens@esa.int (S. Lemmens).
1
The abbreviations used herein are, in alphabetical order; CIRA: COSPAR International Reference Atmosphere, CNES: Centre National d’E
´tudes
Spatiales, COSPAR: Committee on Space Research, DTM: Drag Temperature Model, GL: Gauss-Legendre, KH: King-Hele, NA: Non-Averaged,
NRLMSISE: Naval Research Laboratory Mass Spectrometer, Incoherent Scatter Radar Extended, SA: Semi-Analytical, SI-KH: SuperImposed King-
Hele
www.elsevier.com/locate/asr
Available online at www.sciencedirect.com
ScienceDirect
Advances in Space Research 64 (2019) 1–17
density along the orbit. Half a century ago, King-Hele
(KH) derived analytical approximations to these integrals
(King-Hele, 1964). Depending on the eccentricity of the
orbit, e.g. circular, near-circular, low eccentric and highly
eccentric, different series expansions were derived. Recom-
mendations are given, found empirically, on when to use
which formulation. Vinh et al. (1979) improved the theory
by removing the ambiguity arising from the regions of
validity in eccentricity and by applying the more mathe-
matically rigorous Poincare
´method for integration. The
classical theory was adapted to non-singular elements, mit-
igating the problems that theories formulated in Keplerian
elements have with vanishing eccentricities (Sharma, 1999;
Xavier James Raj and Sharma, 2006).
The advantage of these methods is that the averaged
contraction can be computed analytically using only a sin-
gle density evaluation at the perigee. However, the analyt-
ical methods assume exponential decay of the atmosphere
density above the perigee height. This fixed scale height
assumption potentially introduces large errors, especially
for highly eccentric orbits, if compared to propagation
using quadrature.
Averaging methods based on quadrature solve the inte-
gral numerically. No assumption on the shape of the den-
sity profile is required, however, the density needs to be
evaluated at many nodes along the orbit, slowing down
the integration of the trajectory.
This work proposes modelling the atmosphere density
by superimposing exponential functions, each with a fixed
scale height. The KH formulation is then used for the cal-
culation of the contraction of each individual component.
As the assumption of a fixed scale height is satisfied for
each component, the resulting decay rate is estimated with
great accuracy. Finally, each individual contribution is
summed up, resulting in the global contraction of the over-
all not strictly exponentially decaying atmosphere density.
This superimposed approach is not limited to the KH
method and can also be applied to the other analytical
methods described above.
The proposed method is applied during propagation of
different initial conditions from circular to highly elliptical
orbits and compared against propagations using numerical
quadrature of the contraction as well as against Non-
Averaged (NA) integration. The smooth atmosphere
derivative introduced here is independent of the underlying
atmosphere model and can be extended to include time-
variations, as shown here for the case of solar activity.
2. Background on atmospheric models
The atmosphere models discussed here can be divided
into reference models and the derivatives thereof. The
reference models commonly give the temperature, T,
2
and – more importantly for calculating the drag force –
the density, q, of Earth’s atmosphere as a function of the
altitude, h, and other input parameters. Examples are, in
increasing degree of complexity, the COSPAR Interna-
tional Reference Atmosphere (CIRA), the Jacchia atmo-
sphere (Jacchia, 1977), the Drag Temperature Model
(DTM) (Bruinsma, 2015) and the Naval Research Labora-
tory Mass Spectrometer, Incoherent Scatter Radar
Extended (NRLMSISE) model (Picone et al., 2002), all
of which are (semi-) empirical models.
Of these reference models, derivatives can be obtained
through fitting for two purposes: appropriate simplification
of the mathematical formulation can lead to significant
speed increases for a density evaluation; and adequate
reformulation of the model improves the accuracy of ana-
lytical SA contraction methods as will become apparent in
Section 3.3.
Sections 2.1 and 2.2 briefly introduce the Jacchia-77
atmosphere model and a derived non-smooth exponential
atmosphere model.
2.1. Jacchia-77 reference atmosphere model
The Jacchia-77 reference atmosphere (Jacchia, 1977)
estimates the temperature and density profiles of the rele-
vant atmospheric constituents as a function of the exo-
spheric temperature, T1. The density profile, qJ, is based
on the barometric equation and an empirically derived
temperature profile in order to comply with observations
of satellite decay. The static model is valid for altitudes
90 <h<2500 km and exospheric temperatures
500 <T1<2500 K.
The computation of qJcannot be performed analytically
and requires numerical integration for each of the 4 con-
stituents, nitrogen, oxygen, argon and helium, plus integra-
tion of atomic nitrogen and oxygen. A fast, closed-form
approximation is available (De Lafontaine and Hughes,
1983), but it was not considered here, as its modelled atmo-
sphere does not purely decay exponentially.
The scale height, H, is defined as
H¼ q
dq=dhð1Þ
and numerically approximated for the Jacchia model scale
height, HJ,as
HJhðÞ¼ qhðÞDh
qhþDhðÞqhðÞ Dh¼1m ð2Þ
Several thermospheric variations can be taken into
account, such as solar cycle, solar activity, seasonal or daily
variations. Generally, the objects of interest for SA propa-
gation dwell on-orbit for several months to hundreds of
years. Thus, only the variation with the 11-year solar cycle
is of interest here. The Jacchia reference uses the solar
radio flux at 10:7 cm, F, as an index for the solar activity
(see Fig. 1, source for data: Goddard Space Flight
Center, 2018). From F;T1can be inferred as (Jacchia,
1977)
2
The nomenclature of all the variables can be found in Appendix A.
2S. Frey et al. / Advances in Space Research 64 (2019) 1–17
T1¼5:48F4
5þ101:8F2
5ð3Þ
where Fis a smoothed F, commonly centred over an inter-
val of several solar rotations. Jacchia recommended to use
a smooth Gaussian mean based on weights which decay
exponentially with time. Fig. 1 shows the solar flux, F,
and the Gaussian mean with a standard deviation of
r¼3 solar rotations, i.e. 81 days, considering a window
of 3r. More recent models such as NRLMSISE or
DTM require Fto be a moving mean of 3 solar rotations
(ISO, 2013).
2.2. Non-smooth exponential atmosphere model
One very simple representation of the atmosphere den-
sity is using a piece-wise exponentially decaying model,
by dividing the altitude range into bins. Each bin is defined
by a lower altitude (base) and an upper altitude (base of the
next bin), hiand hiþ1, respectively, the base density, ^
qi,athi
and a scale height, Hi, chosen such that the density is con-
tinuous over the limits of each bin. Then, within each alti-
tude bin, the density, qNS , can be evaluated at each altitude
has follows
qNS hðÞ¼^
qiexp hhi
Hi
hi<h<hiþ1ð4Þ
Such a model can be derived from any atmospheric model.
Herein, the values given in Vallado (2013, Chapter 8.6) –
fitting the CIRA-72 model at T1¼1000 K – are used
for a comparison of models.
A problem with the non-smooth atmosphere model is
that it is non-physical, with discontinuities in H. At each
change of altitude bin, Hjumps from Hito Hiþ1. This
non-smooth behaviour poses a problem to the (variable-
step size) integrator, as the step size needs to be reduced
to accurately describe the sudden change in contraction
rate of the orbit. Thus, the number of function evaluations
and the total time to propagate the orbit increases. An
example is given in Fig. 2b, comparing the number of steps
required for propagation of an object subject to the non-
smooth qNS to one using the smooth qJas a function of alti-
tude. Evidently, each change of bin forces the integrator to
reduce the step size.
The equally simple parametric model introduced in Sec-
tion 4.1 does not suffer from these discontinuities.
3. Background on semi-analytical orbit contraction methods
During SA propagation of an object trajectory subject
to air-drag forces, the integrated change in the orbital ele-
ment space, i.e. the contraction of the orbit, over a full rev-
olution is of interest. This requires the integration of the
(weighted) density along the orbit, which can either be
done numerically using quadrature, or analytically.
Many quadrature rules exist (e.g. see Abramowitz et al.,
1972, p. 885–895) and they are independent of the underly-
ing function, making them versatile. However, they require
the evaluation of the density at multiple nodes along the
orbit, increasing the computational load of the function
evaluations during integration.
Analytical formulations, such as the one derived by D.
King-Hele more than half a century ago (King-Hele,
1964) require the density to be evaluated only once per iter-
ation in correspondence of the perigee altitude. Other
examples of analytical formulations are the ones derived
by Vinh et al. (1979), Sharma (1999) and Xavier James
Raj and Sharma (2006). While offering improvements to
the classical formulation of KH, such as being mathemat-
ically more rigorous and non-singular, they still suffer from
the same assumption of a fixed scale height. The method
proposed in Section 4addresses this problem for any of
the analytical formulations. For the sake of brevity, it is
only applied to the KH method.
Section 3.1 introduces the system dynamics used
throughout this work and discusses its averaging. Sections
3.2 and 3.3 introduce two averaging methods; the numeri-
cal Gauss-Legendre (GL) quadrature and the analytical
KH method.
3.1. Dynamical system and averaging
The main focus of this work is on correcting the errors
arising from the fixed scale height assumption. Important
Fig. 1. Daily 10.7 cm solar flux, and a Gaussian mean with r¼81 days and a window of w¼486 days, since beginning of 1970. The dashed lines
correspond to T1¼750;1000 and 1250 K, respectively, assuming F¼F.
S. Frey et al. / Advances in Space Research 64 (2019) 1–17 3
effects of an oblate Earth, such as a non-spherical atmo-
sphere or gravitational coupling (e.g. see Brower and
Hori, 1961), are not considered here. The superimposed
approach does not replace the averaging method, rather
it transforms one of the inputs, i.e. the atmosphere density,
to fit its assumptions. Hence, it is also applicable to more
elaborate theories.
The dynamical system used here is based on Lagrange’s
planetary equations, given in Keplerian elements, stating
the changes in the elements as a function of the applied
forces from any small perturbations (see King-Hele, 1964,
for more information). Only the tangential force induced
by the aerodynamic drag is considered, i.e.
fT¼1
2qv2dð5Þ
with the density, q, the inertial velocity, v, and the effective
area-to-mass ratio (i.e. the inverse of the ballistic coeffi-
cient), d, defined as d¼cDA=m, where cDis the drag coef-
ficient, Ais the surface normal to v, and mis the mass.
Atmospheric rotation is ignored here, but could be taken
into account by multiplying the right hand side of Eq. (5)
with the appropriate factor.
The variations of the semi-major axis, a, the eccentric-
ity, e, and the eccentric anomaly, E, with respect to time,
t, are
da
dt¼a2qdv3
lð6aÞ
de
dt¼aqdv
r1e2
cos Eð6bÞ
dE
dt¼1
r
l
a
1
2ð6cÞ
with Earth’s gravitational parameter, l, the radius, r, and v
given as
r¼a1ecos EðÞ ð7aÞ
v¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2l
rl
a
rð7bÞ
In order to reduce the stiffness of the problem, Eq. (6) is
averaged over a full orbit revolution, under the assumption
that aand eremain constant. The resulting contractions,
Daand De, for aand erespectively are
Da¼a2dZ2p
0
qhðÞ1þecos EðÞ
3
2
1ecos EðÞ
1
2
dEð8aÞ
De¼adZ2p
0
qhðÞ 1þecos E
1ecos E
1
2
cos E1e2
dEð8bÞ
with the altitude, h¼rR, given the mean Earth radius,
R.
For SA propagation of the orbit, the derivatives of the
variables with respect to time are approximated by the
change over one revolution divided by the time required
to cover the revolution
Fx¼dx
dtDx
Px2a;e½ ð9Þ
with the orbit period, P, defined as
P¼2pffiffiffiffiffi
a3
l
sð10Þ
3.2. Numerical approximation
The integrals in Eq. (8) can be approximated numeri-
cally using quadrature, e.g. GL quadrature (Abramowitz
et al., 1972, p. 887)
Fig. 2. Trajectories propagated for two different atmosphere models, qNS (orange) and qJ(blue), and two different contraction methods, KH (light) and
GL (dark). The initial state is hpha¼750 2000 km. (For interpretation of the references to colour in this figure legend, the reader is referred to the web
version of this article.)
4S. Frey et al. / Advances in Space Research 64 (2019) 1–17
Z2p
0
fEðÞdE pX
i
wifE
i
ðÞ;Ei¼xiþ1ðÞpð11Þ
where the node xiis the ith root of the Legendre Polynomial
PnxðÞ. The weights wiare given as
wi¼2
1x2
i
ðÞP0
nxi
ðÞ
2ð12Þ
and P0
nis the derivative of PnxðÞwith respect to x. The
nodes and weights remain constant during the propaga-
tion, so they are calculated (or read from a table) only once
upon initialisation. Routines to calculate (xi;wi) are avail-
able for various scientific programming tools, such as
MATLAB (MathWorks, 2018) and NUMPY(Oliphant, 2006).
Advantages of a numerical approximation of the inte-
grals in Eq. (8) is that it can be found for any atmospheric
model and that no series expansions are required. Disadvan-
tages are the need of multiple density evaluations and the
loss of an analytic formulation. E.g. the Jacobian cannot
be inferred analytically, but requires another quadrature.
3.3. Classical King-Hele approximation
Here, only a brief summary of the formulation is given.
The treatment of the full theory behind the KH formula-
tion can be found in King-Hele (1964). The integrals in
Eq. (8) can be approximated analytically by expanding
the integrands as a power series in efor low eccentric
orbits, and in the inverted auxiliary variable, z
1
z¼H
ae ð13Þ
for highly eccentric orbits, and cutting off at the appropri-
ate degree.
With the assumption that the density, q, decreases
strictly exponentially with altitude, i.e. with a fixed H, each
expanded integrand can be represented by the modified
Bessel function of the first kind, In, which for n2N0is
given as (Abramowitz et al., 1972, p. 376)
InxðÞ¼1
pZp
0
exp xcos hðÞcos nhðÞdhð14Þ
In Appendix B, the resulting equations are given up
to 5th order, higher than the 2nd order given originally
by KH.
The KH formulation is fast as it can be evaluated ana-
lytically and requires only a single density evaluation for
each computation of the contraction. The main problem
with the fixed Hassumption is the underestimation of q
at altitudes above the perigee altitude, hp, which for eccen-
tric orbits can induce large errors. Fig. 2a shows the trajec-
tories of an object in an initially eccentric orbit with perigee
and apogee height of hpha¼750 2000 km. They were
propagated with two different atmosphere models, qNS and
qJ, and using two different contraction methods, GL
quadrature and the KH formulation. For both atmosphere
models, the KH method overestimates the density decay
above perigee along the orbit, leading to an overestimation
of the lifetime of up to 40%, compared to the propagation
with the GL method. This is true – albeit sometimes less
pronounced – for any object in a non-circular orbit subject
to a non-strictly exponentially decaying atmosphere.
It has to be noted here that KH was aware of this prob-
lem and suggested a way to calculate the contraction of an
orbit with a varying scale height (see King-Hele, 1964,
Chapter 6). To keep the equations analytically integrable,
he approximates the varying Hlinearly, with a constant
slope parameter. Linear approximation of the true His
valid only locally. For low eccentric orbit configurations
this might be sufficient, but high eccentricities will re-
introduce the errors. Using a constant slope parameter will
thus lead to a new over- or underestimation of the drag
depending on e.
Another issue of the KH formulation is that it relies on
series expansion. As the eccentricity grows, the formulation
to calculate the contraction needs to switch from low to
high eccentric orbits. This introduces discontinuities, at a
classically fixed boundary eccentricity, eb.
4. Proposed new model for the semi-analytical computation
of the orbit contraction due to atmospheric drag
The proposed method of taking into account atmo-
spheric drag for SA integration of trajectories consists of
two parts: an atmosphere model based on constant scale
heights, introduced in Section 4.1; and the extension of
the KH formulation to reduce the errors induced by an
atmosphere which in its sum does not decay exponentially,
described in Section 4.2.
Table 1 shows an overview of how the proposed exten-
sion fits into the existing scheme of atmosphere models and
SA orbit contraction methods. As mentioned earlier, the
technique presented here is not limited to the KH method,
but could be applied to any averaging method which is
based on the fixed scale height assumption.
4.1. Smooth exponential atmosphere model
The smooth atmosphere model proposed here does not
in any way attempt to replace existing atmosphere density
Table 1
Non-exhaustive list of existing and newly proposed atmospheric models
and contraction methods.
S. Frey et al. / Advances in Space Research 64 (2019) 1–17 5
models. Instead, it is a derivation of those models. Nor is
the idea of modelling the atmosphere as a sum of exponen-
tials new: the Jacchia-77 reference model reduces – for each
atmospheric constituent – to such a mathematical formula-
tion if the vertical flux terms are neglected (Bass, 1980).
The novelty of this work is the combination of the atmo-
sphere model with the extended, superimposed KH formu-
lation. Sections 4.1.1 and 4.1.2 introduce the static and
variable atmosphere model, respectively.
4.1.1. Static model
The smooth exponential atmosphere model, qS, is mod-
elled by superimposing exponentials functions as
qShðÞ¼
X
np
p¼1
qphðÞ¼
X
np
p¼1
^
qpeh=Hpð15Þ
where the number of partial atmospheres, np, the partial
base densities, ^
qp, and the partial scale heights, Hp, are fit-
ting parameters. Note that the subscript pdoes not stand
for altitude bins, but for one of the partial atmospheres,
each of which is valid for the whole altitude range. While
it potentially could stand for a single atmosphere con-
stituent, it is not restricted as such. The superimposed scale
height, HS,is
HShðÞ¼qShðÞ
dqS=dh¼Pnp
p¼1qphðÞ
Pnp
p¼1qphðÞ=Hpð16Þ
The derivative of HSwith respect to his monotonically
increasing, as Hpis enforced to be larger than 0 for all p.
Hence, the smooth atmosphere model can only be fitted
to atmosphere models in altitude ranges where dH
dh>0.
Above h¼100 km, this is the case for qJfor a wide range
of T1. Even if the underlying model shows slightly negative
Hat the lower boundary h0, a partial atmosphere with a
small positive Hpcan still be fitted accurately.
To find the parameters, Hpand ^
qp, the model in Eq. (15)
is fitted to qJfor three different T1: in accordance to a low
solar activity, T1¼750 K; mean solar activity, T1¼1000
K; and high solar activity, T1¼1250 K (see Fig. 1). The fit
is performed in the logarithmic space as not to neglect
lower densities at higher altitudes, using least squares min-
imisation at heights between h0¼100 km and the upper
boundary, h1¼2500 km. To put more weights on the edges
of the fit interval, the densities are evaluated at N¼100
heights, hi, distributed as Chebyshev nodes (Abramowitz
et al., 1972, p. 889)
hi¼h0þh1
2þh1h0
2cos 2i1
2Np
i¼1;...;Nð17Þ
The number of partial atmospheres, np, is chosen to be 8, as
the cost function
Fig. 3. Cost function depending on number of partial atmospheres.
Table 2
Relative density fitting errors 8h2100;2500½km.
gqT1¼750 K T1¼1000 K T1¼1250 K
<0:1%8h>239 km 8h>308 km 8h>306 km
<0:5%8h>134 km 8h>153 km 8h>154 km
<1%8h>119 km 8h>119 km 8h>130 km
gq;max 1:6%(h¼115 km) 1:8%(h¼115 km) 1:9%(h¼115 km)
Table 3
Smooth atmosphere model parameters resulting from a fit to the Jacchia-77 model, valid for altitudes h2100;2500½km.
T1¼750 K T1¼1000 K T1¼1250 K
pHp^
qpHp^
qpHp^
qp
[km] [kg/m
3
] [km] [kg/m
3
] [km] [kg/m
3
]
14:9948 2:4955e þ02 4:9363 3:1632e þ02 4:9027 3:6396e þ02
210:471 8:4647e 04 11:046 5:2697e 04 11:437 3:8184e 04
321:613 9:1882e 07 24:850 3:7354e 07 25:567 2:8928e 07
437:805 1:2530e 08 46:462 1:0839e 08 44:916 1:2459e 08
549:967 1:3746e 09 64:435 1:0880e 09 76:080 9:2530e 10
6 174:23 1:5930e 13 147:46 3:8122e 13 111:09 1:6667e 11
7 315:15 1:1290e 14 314:53 4:8431e 14 354:23 5:9225e 14
8 1318:13:8065e 16 1214:64:2334e 16 892:19 1:7378e 15
6S. Frey et al. / Advances in Space Research 64 (2019) 1–17
C¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
NXN
i¼1ln qShi
ðÞ
qJhi
ðÞ
2
sð18Þ
which is the root mean square of the logarithmic density fit
residuals, stops improving (see Fig. 3). For
T12750;1000;1250½K, the relative error, gq, calculated
as
gqhðÞ¼
jqShðÞqJhðÞj
qJhðÞ ð19Þ
always remains below 0:1%and 1%for all h>308 km and
h>130 km, respectively, and the maximum relative error,
gq;max, does not exceed 2%, as can be seen in Table 2.
Hence, the density fit accurately represents the underlying
model. The model parameters can be found in Table 3.
Fig. 4 shows a comparison between the underlying and fit-
ted model, for T1¼1000 K.
A speed test for 2401 density and scale height evalua-
tions over the range 100 6h62500 km shows a near 60-
fold decrease in evaluation time for qScompared to qJ.
The implementation of the Jacchia-77 model used herein
is written in the coding language C (taken from Instituto
Nacional De Pesquisas Espaciais, 2018), and called from
within MATLAB, while the routine to calculate qSis imple-
mented and called directly in MATLAB. Thus, a further
decrease of computational time could be expected if also
the latter was implemented in C. The speed tests were per-
formed using the same processor architecture.
4.1.2. Variable model
Possible extensions to the smooth exponential atmo-
sphere model are the inclusion of a temporal dependence,
such as the solar cycle, annual or daily variations. Here,
the model is extended to incorporate the variability in the
atmosphere density due to a variable T1. To conserve
the mathematical formulation of the static model, the tem-
perature dependence is introduced in the fitting parameters,
^
qp¼^
qpT1
ðÞand Hp¼HpT1
ðÞ.
T1is a function of the solar proxy F(see Eq. (3)), so the
fitting range is defined by F. Generally, the long-term pre-
dictions for F– based on various numbers of previous solar
cycles – remain between F260;230½sfu (Vallado and
Finkleman, 2014; Dolado-Perez et al., 2015; Radtke and
Stoll, 2016). This translates into T12669;1321½K, as F
per definition remains in the same range as F. The param-
eters for the variable smooth exponential atmosphere
model derived below, and listed in Appendix D, are valid
for any T12T0¼650;T1¼1350½K. They should not
be used for T1outside this range, as polynomial fits tend
to oscillate strongly outside the fitting interval.
The dependence on T1is incorporated using a polyno-
mial least squares fit. Each partial atmosphere is fitted sep-
arately. The static parameters, fitted to the i¼1;2;...;M
static atmospheres with different T1, are converted
ai
p¼1=Hi
pð20aÞ
bi
p¼ln ^
qi
p
ð20bÞ
and each time-variable partial atmosphere is fitted to two
independent polynomials of order land mrespectively
ape
T1
¼X
l
k¼0
apk e
Tk
1ð21aÞ
bpe
T1
¼X
m
k¼0
bpk e
Tk
1ð21bÞ
Fig. 4. Fit of qSto qJfor T1¼1000 K. Additionally, the different
contributions of each partial atmosphere are shown (dotted) from p¼1
(dark) to p¼8 (light).
S. Frey et al. / Advances in Space Research 64 (2019) 1–17 7
using a normalised and unit-less e
T1, defined as
e
T1¼T1T0
T1T0ð22Þ
In vector notation, Eq. (21) can be written as
a¼
a1
.
.
.
anp
2
6
6
43
7
7
5¼
a10 ... a1l
.
.
...
.
anp0... anpl
2
6
6
43
7
7
5e
T0
1
.
.
.
e
Tl
1
2
6
6
43
7
7
5ð23aÞ
b¼
b1
.
.
.
bnp
2
6
6
43
7
7
5¼
b10 ... b1m
.
.
...
.
bnp0... bnpm
2
6
6
43
7
7
5e
T0
1
.
.
.
e
Tm
1
2
6
6
43
7
7
5ð23bÞ
To prevent over-fitting, the order of the polynomials
should remain well below the number of fitted static atmo-
spheres. Here, the model in Eq. (21) is fitted to M¼50 stat-
ically fitted models, distributed again as Chebyshev nodes
between T0and T1
Ti¼T0þT1
2þT1T0
2cos 2i1
2Np
i¼1;...;N
ð24Þ
The orders are chosen to be l¼m¼8 such that the error
remains below 0:5%for all h>155 km and
T12650;1350½K. If more accuracy is needed, the poly-
nomial order can be increased and/or spline polynomial
interpolation applied. Finally, the time-dependent atmo-
sphere is recovered by inverting Eq. (20)
HpT1
ðÞ¼1=ape
T1
ð25aÞ
^
qpT1
ðÞ¼exp bpe
T1
ð25bÞ
Fig. 5 compares the accuracy of the T1-variable smooth
exponential atmosphere model against the original Jacchia-
77 model. It shows the ratio between qST1
ðÞ=qJT1
ðÞfor
T1in the range from 650 K to 1350 K (left), and the cor-
responding parameters, ^
qpand Hpas a function of T1,
including the underlying parameters of the static fits
(right). Towards the lower edge of the temperature range
(i.e. T1!650 K), the polynomial fits for components
p¼57 do not well represent the underlying data. This
leads to increased but still tolerable errors in the altitude
range between 500 and 1500 km.
The advantage of this approach is, that the original
structure of the model is maintained, so it can be used with
the contraction model introduced in the next section.
4.2. Superimposed King-Hele approximation
The extension of the KH contraction formulation into
the SuperImposed King-Hele (SI-KH) formulation with a
superimposed atmosphere is straightforward. Replacing q
from Eq. (8) with the one defined in Eq. (15) leads to
Da¼X
np
p¼1
Dap¼a2dX
np
p¼1Z2p
0
qp
1þecos EðÞ
3
2
1ecos EðÞ
1
2
dEð26aÞ
De¼X
np
p¼1
Dep¼adX
np
p¼1Z2p
0
qp
1þecosE
1ecosE
1
2
cosE1e2
dE
ð26bÞ
i.e. each partial contraction reduces to the classical KH for-
mulation with the partial exponential atmosphere qp. The
important difference is that now Hpis constant over the
whole altitude range. The classical KH approximations –
extended up to 5th order – can be found in Appendix B
(dropping the subscript p). Finally, the rate of change is
Fig. 5. Quality of temperature dependent fit. Left: comparison for different T1. Right: evolution of Hp(top) and ^
qp(bottom) as a function of T1. The dots
show the parameters of the static fits, which were used to fit the variable model.
8S. Frey et al. / Advances in Space Research 64 (2019) 1–17
Fx¼dx
dt¼X
np
p¼1
Fx
ðÞ
p1
PX
np
p¼1
Dxpx2a;e½ ð27Þ
KH introduced the simple fixed boundary condition
eb¼0:2 to select between the approximation method for
low eccentric and high eccentric orbits, given in Appen-
dices B.2 and B.3, respectively. However, as Hpcan be
large, this condition is not always sufficient. Recall from
Eq. (13) that
z¼ae
Hð28Þ
For low aand high H;zcan approach unity at e¼0:2,
making the series expansion in 1=zinaccurate. Instead, it
is proposed to define ebbased on the truncation errors
found in the formulations for the low and high eccentric
orbits. The series truncation errors for the low eccentric
orbit approximation (Eq. (B.4)), using the order notation,
O, are of the order of
Olow
ae6
¼a2qexp zðÞI0e6ð29aÞ
Olow
ee6
¼aqexp zðÞI1e6ð29bÞ
If zis large (see justification below), I0=1zðÞ!exp zðÞ=ffiffiffiffiffiffiffi
2pz
p
and Eq. (29) becomes
Olow
ae6
¼a2qe6
ffiffiz
pð30aÞ
Olow
ee6
¼aqe6
ffiffiz
pð30bÞ
For the high eccentric orbit approximation (Eq. (B.6)), the
truncation errors are in the order of
Ohigh
a
1
z6
¼a2q1ffiffiz
p1þeðÞ
3
2
1eðÞ
1
2
1
z61e2
ðÞ
6ð31aÞ
Ohigh
e
1
z6
¼aq1ffiffiz
p1þe
1e
1
21
z61e2
ðÞ
5ð31bÞ
Assuming that the terms 1þeðÞ
3
2
1eðÞ
1
2
1
1e2
ðÞ
6and 1þe
1e
1
21
1e2
ðÞ
5are
dominated by 1=z6(see again below for a justification),
Eq. (31) simplifies to
Ohigh
a
1
z6
¼a2q1ffiffiz
p1
z6ð32aÞ
Ohigh
e
1
z6
¼aq1ffiffiz
p1
z6ð32bÞ
Equating the truncation errors from Eqs. (30) and (32),
using Eq. (13) and solving for eresults in the following
condition
eb¼ffiffiffiffi
H
a
rð33Þ
Note that this boundary is most exact if the series expan-
sions in both the low and high eccentric regimes are of
the same order.
The derivation of the boundary condition required the
assumptions of zto be large, such that
I0=1zðÞ!exp zðÞ=ffiffiffiffiffiffiffi
2pz
pand such that 1=z6dominates the
other eterms in Eq. (31). To validate the assumptions,
replace ain Eq. (33) with a¼hpþRE
=1eb
ðÞand solve
for eb, neglecting the negative solution
eb¼1
2yþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
y2þ4y
p
hi
where y¼H
hpþREð34Þ
Given Hmin=max ¼4:9=1320 km (see Table 3) and the valid
range for hp2h2100;2500½km, the extrema in eband
zb¼1=eb, are found to be
eb;min=max ¼0:023=0:361
zb;min=max ¼2:77=43
For any z>zb;min;I0=1remains close to exp zðÞ=ffiffiffiffiffiffiffi
2pz
p, being
off only þ6%and 16%, respectively, at zb;min . At the same
time, 1=z6dominates the terms dependent on ein Eq. (31)
by two to three orders of magnitude 8e<eb;max . Thus, the
assumptions made to derive ebare valid.
An advantage of an analytical expression of the dynam-
ics is that the Jacobian of the dynamics can be derived ana-
lytically too, which can be used for uncertainty
propagation. For a comprehensive discussion of the SI-
KH method, the partial derivatives of the dynamics as
derived by KH, with respect to aand e, are given in Appen-
dix C (again dropping the subscript p). As the SI-KH
method is simply a summation of the individual contribu-
tions of the partial atmosphere, the derivatives can equally
be summed up as
@Fx
@y¼X
np
p¼1
@Fx
@y
p
x;yðÞ2a;e½ ð35Þ
5. Validation
The validation section is split into two parts: Section 5.1
validates the smooth exponential atmosphere, qS,bycom-
paring it to the Jacchia-77 model, qJ, during SA propaga-
tion using the GL contraction method; Section 5.2
validates the proposed SI-KH approach by comparing
the contraction approximation along a single orbit, i.e.
Daand De, to numerical quadrature. For completeness,
propagations of a grid of initial conditions are performed
using the GL and SI-KH methods and NA integration.
The latter does not resort to any averaging technique, inte-
grating the full dynamics of Eq. (6), including E.
5.1. Validation of the smooth exponential atmosphere model
To validate qSagainst qJfor T1¼750;1000 and
1250 K and at the same time distinguish it from the effects
introduced by the SI-KH method on the resulting lifetime,
tL, the following orbits are propagated using the GL
method only for the computation of the orbit contraction.
S. Frey et al. / Advances in Space Research 64 (2019) 1–17 9
All physically feasible initial orbit configurations on a
46 46 grid from 250 6hp62500 km and
250 6ha<2500 km are propagated, using d¼1m
2
/kg.
The lower limit, 250 km, is selected as an object with such
a large don a circular orbit at this altitude survives for a
fraction of a day only at which point SA propagation
becomes inaccurate. The upper limit, 2500 km, is being
imposed by definition of qJ, but can be overcome by fitting
to another model. The chosen dis large, but does not limit
the validity of this validation, as inaccuracies from the SA
approach affect the propagation equally for both atmo-
sphere models.
The SA propagation is performed using MATLAB’s
ODE113 – a variable-step, variable-order Adams-Bashforth-
Moulton integrator (Shampine and Reichelt, 1997) – and
a relative error tolerance, crel ¼106, which is shown to
be sufficient for different orbital scenarios in Section 5.2.
Fig. 6 shows tLfor the initial orbit grid, for propagation
subject to qS, and the relative error, gtL, defined as
Fig. 6. Lifetimes and comparison of accuracy for lifetime estimation for objects being subject to qJand qS.
10 S. Frey et al. / Advances in Space Research 64 (2019) 1–17
gtL¼tLqS
ðÞtLqJ
ðÞ
tLqJ
ðÞ ð36Þ
comparing the propagations for each grid point using qS
and qJ, respectively. Table 4 contains information about
the maximum error and the workload. Over the whole
specified domain and for all T12750;1000;1250½K, gtL
remains within 0:1%;0:1%½, which considering the uncer-
tainties in atmospheric density modelling is more than
accurate enough (Sagnieres and Sharf, 2017). Towards
low perigees (hp<500 km), the fitted qSstarts to wobble
around the underlying model (see Fig. 4a), which is also
apparent for the propagated orbits. A 6-fold speed
improvement can be observed, as no numerical integration
is required when calculating the density with qS.
The reduction in function evaluations and computa-
tional time observable with an increasing T1is a conse-
quence of the different density profiles. Increasing T1
leads to an increased q, which increases the drag force
and thus decreases the lifetime. However, the variable-
step size integration method can compensate this by
increasing the step size. Two possible explanations are: as
the integrator is initialised with the same properties for
all three cases, the initially set (small) step size favours
shorter lifetimes; and the shape of the density profiles with
high T1are more smooth, decreasing the number of failed
function evaluation attempts.
5.2. Validation of the superimposed King-Hele method
The SA propagation relies on an accurate approxima-
tion of Daand De.Fig. 7 shows – for different orbital
Table 4
Comparison of 1081 propagations being subject to qJor qS, in total
number of function evaluations, Ntot
f, total integration evaluation time,
ttot
CPU , and the minimum and maximum lifetime estimation error, gtL;min=max.
T1[K] qNtot
f[–] ttot
CPU [s] gtL;min=max [%]
750 qJ593,255 1086:9
qS592,124 169:50:060=0:051
1000 qJ568,140 986:3
qS568,140 153:60:077=0:056
1250 qJ550,021 789:9
qS549,063 149:30:074=0:048
Fig. 7. Comparison for accuracy in Da(left) and De(right) for different approximation methods. The underlying atmosphere model is qSat T1¼1000 K.
Note that the colour bar range of the lower figure is 3 orders of magnitudes smaller than the one of the upper figure.
S. Frey et al. / Advances in Space Research 64 (2019) 1–17 11
configurations – the relative integral approximation error,
gDx, defined as
gDx¼DxC2
ðÞDxC1
ðÞ
DxC1
ðÞ x2a;e½ ð37Þ
where Cis the selected contraction method: C1is the
numerical GL method computed using 65 nodes; and C2
describes the analytical formulation, KH or SI-KH, using
series expansion up to 5th order.
Fig. 7a reveals why orbits are predicted to re-enter much
later using the classical KH contraction method: the den-
sity is underestimated at altitudes above hp. The largest
errors occur around hp¼125 km and 800 km, where the
rate of change in Hwith respect to his large. Around these
two altitudes, the contraction rate in ais underestimated by
more than 10%and 20%, respectively, if e>0:03. Using
the SI-KH the relative error remains well below 0:1%
8hp2100;2500½km and 8ha2100;100;000½km (see
Fig. 7b), a range that includes the vast majority of all Earth
orbiting objects. Discontinuities can be found whenever e
passes through eb¼ebHp
. The biggest step occurs for
the largest Hp. Those discontinuities slightly increase the
number of steps required during the integration. However,
given the averaged dynamics, crel can be chosen large
enough during integration mitigating the effects of the
discontinuities.
To see how the SI-KH compares against GL in SA
propagation and against NA propagation in terms of accu-
racy and computational power, the results from different
initial orbit conditions are compared, for two scenarios:
(a) Short-term re-entry duration: tL¼30 days
(b) Mid-term re-entry duration: tL¼360 days
The reasons why long-term re-entry cases are not dis-
cussed here are twofold: First, for long time spans, the
NA integration requires small relative tolerances. If they
are not met, the result cannot be trusted; Secondly, the
longer the time spans, i.e. the smaller d, the more accurate
the assumptions made for the SA propagation.
The initial conditions are spaced in hp2250;2500½km
and ha2250;100;000½km and consist of all the 1558 fea-
sible solutions on a 46 46 grid, where the grid spacing in
hais chosen to be logarithmic, as opposed to the equidis-
tant grid in hp. Two preliminary runs were performed using
the SI-KH method to calculate the lifetimes. This way, the
drequired to re-enter within the given time-span can be
estimated. Fig. 8 shows the grids of the resulting dfor both
scenarios. Note that dvaries by almost 13 orders of
magnitude.
The accuracy is described again as the relative lifetime,
gtL, this time defined as
gij
tLM1;M2;hp;i;ha;j
¼tLM1;hp;i;ha;j
tLM2;hp;i;ha;j
tLM2;hp;i;ha;j
ð38Þ
where Mis the selected contraction and integration
method, combined with a given relative integrator toler-
ance, crel, during integration. To give a feeling for the accu-
racy across all the different initial conditions, the 50%- and
100%-quantiles, i.e. the median and maximum denoted as
gtL;50%and gtL;100%, respectively, over all the jgij
tLjare given.
The computation effort is compared via the total number
of function calls, Ntot
f, and time required for the integration
itself, ttot
CPU
NfM1;M2
ðÞ¼
Ntot
fM1
ðÞ
Ntot
fM2
ðÞ ð39aÞ
tCPU M1;M2
ðÞ¼
ttot
CPU M1
ðÞ
ttot
CPU M2
ðÞ ð39bÞ
Table 5 contains these figures comparing the different
integration methods against each other. For both SI-KH
and GL, the absolute maximum error over the whole grid
and over both scenarios remains below 0:01%, when
decreasing crel from 106to 1012. Given this force model,
Fig. 8. The minimum effective area-to-mass ratio is dmin ¼1:5104in order to remain in orbit for 360 days from a circular orbit at hp¼ha¼250 km.
The maximum, in order to re-enter in 30 days from hp=ha¼250=100;000 km, is dmax ¼3:0106.
12 S. Frey et al. / Advances in Space Research 64 (2019) 1–17
it is therefore sufficient to use crel ¼106. For NA integra-
tion, this is not the case. While the maximum error remains
modest (0:18%) in the short-term case, it becomes large
when the re-entry span is increased to one year (26%),
when decreasing crel. Decreasing crel ¼109and comparing
to integration with crel ¼1012, reduces the maximum error
for the NA propagation in the mid-term case to 0:032%.
For the comparison of the SA techniques against NA
propagation, the tolerance of the latter is set to
crel ¼1012. Again, SI-KH and GL fare very similar. For
the short-term case, the boundaries of the SA propagation
can be recognised for very high d, leading to still small
maximum errors of 0:18%and 0:17%, respectively.
Fig. 9a shows the resulting lifetime comparison for SI-
KH and tL¼30 days. As dincreases to values above 104
m
2
/kg, the assumption of constant aand eover one orbit
starts to break down and small errors are introduced. This
might be an issue for small debris such as multi-layer insu-
lation fragments and paint flakes. For the mid-term scenar-
io, the maximum error reduces by one order of magnitude
for both SA methods tested. For high ha>10;000 km, the
series expansion applied in the SI-KH method introduces
small errors (see Fig. 9b).
6. Conclusion
The classical KH orbit contraction method allows to
analytically calculate the effects of drag on the orbit evolu-
tion averaged over an orbital period. However, it inaccu-
rately estimates the orbital decay for eccentric orbits
subject to a non-exponentially decaying atmosphere model.
To improve the accuracy, a smooth exponential atmo-
sphere model was proposed to be used in tandem with
the new SI-KH orbit contraction method.
The classical KH method was extended to the SI-KH
contraction method, making use of a superimposed atmo-
sphere model to satisfy the assumption of a strictly decay-
ing density for each component of the model. This greatly
reduces the errors in the estimated decay rates of objects in
eccentric orbits and subject to atmospheric density profiles
with variable scale height. The analytical method was val-
idated against an averaging technique based on numerical
quadrature. Further, the semi-analytical propagation of
orbits using the SI-KH method was validated against full
numerical integration of the dynamics. The approach is
applicable to any averaging techniques considering drag
and based on the fixed scale height assumption above peri-
Fig. 9. Relative error gtLwhen comparing SA propagation using SI-KH with crel ¼106against NA integration with crel ¼1012.
Table 5
Performance of the different propagation and contraction methods, for (a) tL¼30 days and (b) tL¼360 days and various relative integration tolerances,
crel. All figures are unit less.
M1M2gtL;50%gtL;100%NftCPU
(a) SI-KH/106SI-KH/1012 3:2e 68:4e 53:0e 12:9e 1
GL/106GL/1012 3:3e 67:0e 53:7e 13:7e 1
NA/106NA/1012 1:3e 32:5e 23:4e 13:4e 1
NA/109NA/1012 1:6e 63:1e 56:0e 16:2e 1
SI-KH/106NA/1012 8:7e 41:8e 31:1e 22:2e 2
GL/106NA/1012 8:7e 41:7e 31:0e 23:6e 2
(b) SI-KH/106SI-KH/1012 3:2e 66:9e 53:1e 13:2e 1
GL/106GL/1012 3:7e 66:9e 53:9e 14:0e 1
NA/106NA/1012 1:6e 22:6e 13:4e 13:7e 1
NA/109NA/1012 1:9e 54:1e 46:1e 16:4e 1
SI-KH/106NA/1012 7:0e 53:2e 45:8e 41:1e 3
GL/106NA/1012 7:2e 54:9e 45:8e 42:1e 3
S. Frey et al. / Advances in Space Research 64 (2019) 1–17 13
gee. Finally, the Jacobian of the dynamics governed by the
SI-KH method is given to be used for future applications
such as uncertainty propagation.
Acknowledgements
The research leading to these results has received fund-
ing from the European Research Council (ERC) under
the European Union’s Horizon 2020 research and innova-
tion programme as part of project COMPASS (Grant
agreement No 679086). The authors acknowledge the use
of the Milkyway High Performance Computing Facility,
and associated support services at the Politecnico di
Milano, in the completion of this work. The datasets gen-
erated for this study can be found in the repository at the
link www.compass.polimi.it/publications.
Appendix A. Nomenclature
EEccentric anomaly [rad or deg]
F10.7 cm solar flux [sfu]
FSmoothed 10.7 cm solar flux [sfu]
HAtmosphere density scale height [m or km]
InModified Bessel function of the first kind of order
n[–]
POrbit period [s]
RMean Earth radius [m or km]
TTemperature [K]
T1Exospheric temperature [K]
DaContraction over a full orbit period in a [m or km]
DeContraction over a full orbit period in e [–]
aSemi-major axis [m or km]
eEccentricity [–]
ebBoundary in e for selection of integral approxima-
tion method [–]
hHeight above Earth surface [m or km]
haApogee altitude [m or km]
hpPerigee altitude [m or km]
npNumber of partial atmospheres [–]
rRadial distance from Earth’s center [m or km]
tTime [seconds, days or years]
tLLifetime [seconds, days or years]
vIntertial velocity [m/s or km/s]
zAuxilary variable for integration of the decay rate
of highly eccentric orbits [–]
dInverse ballistic coefficient [m
2
/kg]
gqRelative atmospheric density error [– or %]
gDxRelative integral approximation error [– or %]
gtLRelative lifetime error [– or %]
crel Relative integration tolerance [– or %]
lEarth gravitational parameter [m
3
/s
2
or km
3
/s
2
]
qAtmosphere density [kg/m
3
or kg/km
3
]
^
qAtmosphere base density [kg/m
3
or kg/km
3
]
CContraction method
MContraction and integration method with given
crel
OOrder of series truncation error
JIndex corresponding to the Jacchia-77 atmosphere
model
NS Index corresponding to the non-smooth atmo-
sphere model
SIndex corresponding to the smooth atmosphere
model
pIndex corresponding to the partial smooth atmo-
sphere model
Appendix B. King-Hele Formulation
All the formulas presented here are explained and
derived in the work of King-Hele (1964). The analytical
formulas describe, for different eccentricities, the change
in the semi-major axis, a, and the eccentricity, e, over one
orbit as an approximation of Eq. (8). Please note that
one of the four cases was dropped, as it was introduced
only due to the Bessel functions becoming inaccurate for
small arguments. Today, the relevant mathematical soft-
ware packages are accurate and fast enough to overcome
this limitation.
Adaptations to the original formulation were made to
find the change directly in aand e, rather than aand
x¼ae, to calculate the change in the variables of
interest;
find a more appropriate boundary condition, eb, for the
selection of the phase (see Section 4.2);
increase the accuracy within each phase by taking into
account more terms after the series expansion.
The two functions, kaand ke, are introduced here for
later use when describing the rate of change in all the eccen-
tricity regimes described below, as
ka¼dffiffiffiffiffiffi
la
pqhp
ðB:1aÞ
ke¼ka=aðB:1bÞ
with the effective area-to-mass ratio, d, the gravitational
parameter, l, the atmospheric density, q, evaluated at the
perigee altitude, hp.
B.1. Circular orbit
For circular orbits, no integration needs to be approxi-
mated, as the integral can be solved analytically as
Da¼2pda2qhp
ðB:2aÞ
De¼0ðB:2bÞ
where hpreduces to the circular altitude. Dividing by the
orbital period, P, according to Eq. (9) and using the func-
tions defined in Eq. (B.1), the rate of change for circular
orbits is
14 S. Frey et al. / Advances in Space Research 64 (2019) 1–17
Fa¼da
dt¼kaðB:3aÞ
Fe¼de
dt¼0ðB:3bÞ
B.2. Low eccentric orbit
For small e<eba;HðÞ, a series expansion in eis per-
formed and then integrated using the modified Bessel func-
tion of the first kind, In(z), as
eT¼1ee
2e3e4e5
IT¼I0I1I2I3I4I5I6
ðÞ
Da¼2pdq hp
exp zðÞa2eTKl
aIþOe6
ðÞ
ðB:4aÞ
De¼2pdq hp
exp zðÞaeTKl
eIþOe6
ðÞ
ðB:4bÞ
with the auxiliary variable z¼ae=H, the scale height, H,a
single evaluation of the density at the perigee height, hp,
and the order of the series truncation error, O,ofe6. The
constant matrices are given as
Kl
a¼
1000000
0200000
3
403
40000
03
401
4000
21
64 028
64 07
64 00
030
64 015
64 03
64 0
2
6
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
7
5
Kl
e¼
0100000
1
201
20 000
05
801
8000
5
16 04
16 01
16 00
018
128 01
128 03
128 0
18
256 019
256 02
256 03
256
2
6
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
7
5
Dividing by Paccording to Eq. (9) and using the functions
defined in Eq. (B.1), the rate of change for low eccentric
orbits is
Fa¼da
dt¼kaexp zðÞeTKl
aIþOe6
ðB:5aÞ
Fe¼de
dt¼keexp zðÞeTKl
eIþOe6
ðB:5bÞ
B.3. High eccentric orbit
Instead of performing the series expansion in e, which is
infeasible for large values of e>eba;HðÞ, the expansion is
performed for the substitute variable, k2=z¼1cos E.
KH truncated the series already after two powers. Here,
instead, as Hcan be large and as the formulation should
be readily available for any hp<2500 km, it is extended
up to 5th power. The contractions over one orbit period
are
rT¼11
z1e2
ðÞ
1
z21e2
ðÞ
21
z31e2
ðÞ
31
z41e2
ðÞ
41
z51e2
ðÞ
5
eT¼1ee
2e3e4e5e6e7e8e9e10
Da¼2dffiffiffiffi
2p
z
qqhp
a21þeðÞ
3
2
1eðÞ
1
2
rTKh
aeþO1
z6
ðB:6aÞ
De¼2dffiffiffiffiffiffi
2p
z
rqhp
a1þe
1e
1
2
1e2
rTKh
eeþO1
z6
ðB:6bÞ
with the constant matrices
Kh
a¼
1
2
1
16
9
256
75
2048
3675
65;536
59;535
524;288
01
23
16 45
256 525
2048 33;075
65;536
03
16
75
128
675
2048
5985
16;384
288;225
524;288
00 3
16 75
128 105
2048
10;395
16;384
0015
256 3735
2048
21;945
32;768 344;925
262;144
00 0 45
256
13;545
2048 129;465
32;768
00 0 105
2048
110;985
16;384 7;687;575
262;144
00 0 0 525
2048 836;325
16;384
00 0 0 4725
65;536 16;288;965
524;288
00 0 0 0 33;075
65;536
00 0 0 0 72;765
524;288
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
Kh
e¼
1
23
16 15
256 105
2048 4725
65;536 72;765
524;288
01
4
9
32
75
512
735
4096
42;525
131;072
03
16
39
128 405
2048
525
16;384
152;145
524;288
00 3
32 375
256
735
4096 31;185
32;768
0015
256 1515
2048
123;585
32;768 530;145
262;144
00 0 45
512
31;605
4096 1;165;185
65;536
00 0 105
2048
40;845
16;384 10;235;295
262;144
00 0 0 525
4096 1;505;385
32;768
00 0 0 4725
65;536 5;716;305
524;288
00 0 0 0 33;075
131;072
00 0 0 0 72;765
524;288
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
Plugging Eq. (B.6) into Eq. (9), using the functions defined
in Eq. (B.1), and introducing the new functions
ca¼ffiffiffiffiffi
2
pz
r1þeðÞ
3
2
1eðÞ
1
2ðB:7aÞ
ce¼ffiffiffiffiffi
2
pz
r1þe
1e
1
2
1e2
ðB:7bÞ
the rate of change for highly eccentric orbits is
S. Frey et al. / Advances in Space Research 64 (2019) 1–17 15
Fa¼da
dt¼kacarTKh
aeþO1
z6
ðB:8aÞ
Fe¼de
dt¼kecerTKh
eeþO1
z6
ðB:8bÞ
Appendix C. Jacobian of Dynamics in aand e
The partial derivatives of the dynamics with respect to a
and eare given here for the three different regimes dis-
cussed in Appendix B. The partial derivatives of a partial
atmosphere defined in Eq. (15) (dropping the subscript
p), and given hp¼a1eðÞR, can be found as
@qhp
@a¼1e
Hqhp
ðC:1aÞ
@qhp
@e¼a
Hqhp
ðC:1bÞ
Thus, the partial derivatives of kaand ke(see Eq. (B.1))
with respect to aand eare
@ka
@a¼dffiffiffi
l
pqhp
2ffiffiffi
a
pþffiffiffi
a
p@qhp
@a
¼ka
1
2a1e
H
ðC:2aÞ
@ka
@e¼dffiffiffiffiffiffi
la
p@qhp
@e¼ka
a
HðC:2bÞ
@ke
@a¼dffiffiffi
l
pqhp
2a3
2þ1ffiffiffi
a
p@qhp
@a
¼ke1
2a1e
H
ðC:2cÞ
@ke
@e¼dffiffiffi
l
a
r@qhp
@e¼ke
a
HðC:2dÞ
C.1. Circular orbit
For circular orbits, the rate and derivative in evanishes
and the partial derivative of Fawith respect to a, combin-
ing Eqs. (B.3) and (C.2),is
@Fa
@a¼1
2a1
H
FaðC:3Þ
C.2. Low eccentric orbit
For low eccentric orbits, with e6eb, the partial deriva-
tive of Faand Fewith respect to aand e, combining Eqs.
(B.5) and (C.2) and using the product rule, are
@Fa
@a¼1
2a1
H
Fakaexp zðÞeTKl
a
e
H
@I
@zðC:4aÞ
@Fa
@e¼kaexp zðÞ
@eT
@eKl
aIþeTKl
a
a
H
@I
@z
ðC:4bÞ
@Fe
@a¼
1
2a1
H
Fekeexp zðÞeTKl
e
e
H
@I
@zðC:4cÞ
@Fe
@e¼keexp zðÞ
@eT
@eKl
eIþeTKl
e
a
H
@I
@z
ðC:4dÞ
where
@InzðÞ
@z¼1
2In1zðÞþInþ1zðÞðÞ
@I0zðÞ
@z¼I1zðÞ ðC:5Þ
and
@en
ðÞ
@e¼nen1ðC:6Þ
C.3. High eccentric orbit
Using the partial derivatives of caand cefrom Eq. (B.7)
with respect to aand e
@ca
@a¼ca1
2a
ðC:7aÞ
@ca
@e¼ca14eþe2
2e1e2
ðÞ
ðC:7bÞ
@ce
@a¼ce1
2a
ðC:7cÞ
@ce
@e¼ce12eþ3e2
2e1e2
ðÞ
ðC:7dÞ
it follows that
@
@akaca
ðÞ¼kaca1e
H
ðC:8aÞ
@
@ekaca
ðÞ¼kaca
a
H14eþe2
2e1e2
ðÞ
ðC:8bÞ
@
@akece
ðÞ¼kece1
a1e
H
ðC:8cÞ
@
@ekece
ðÞ¼kece
a
H12eþ3e2
2e1e2
ðÞ
ðC:8dÞ
and the partial derivatives of Faand Fefor high eccentric
orbits (see Eq. (B.8)), with ePeb, with respect to aand
ebecome
@Fa
@a¼
1e
H
Fakaca
@rT
@aKh
aeðC:9aÞ
@Fa
@e¼a
H14eþe2
2e1e2
ðÞ
Fakaca
@rT
@eKh
aeþrTKh
a
@e
@e
ðC:9bÞ
@Fe
@a¼
1
a1e
H
Fekece
@rT
@aKh
eeðC:9cÞ
@Fe
@e¼a
H12eþ3e2
2e1e2
ðÞ
Fekece
@rT
@eKh
eeþrTKh
e
@e
@e
ðC:9dÞ
where
rn¼ln¼ae
H1e2
n
ðC:10aÞ
@rn
@a¼nlnþ1ðÞ
e
H1e2
¼n
arnðC:10bÞ
@rn
@e¼nlnþ1ðÞ
a
H13e2
¼n13e2
ðÞ
e1e2
ðÞ
rnðC:10cÞ
16 S. Frey et al. / Advances in Space Research 64 (2019) 1–17
Appendix D. Variable atmosphere model parameters
Tables D.7 and D.8 list the parameters to calculate a
and baccording to Eq. (23) as a function of the normalised
e
T1. The two vectors are needed to recover ^
qpand Hp8p,
according to Eq. (25). Note that the model should only
be used for T12T0¼650;T1¼1350½K.
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Table D.8
Parameters to calculate bas a function of e
T1. The factors are of unit [ln
(kg/m
3
)].
pbp0bp1bp2
15:35674e þ01:36142e þ01:71993e þ0
26:96022e þ01:71534e 16:26282e þ0
31:33334e þ14:29240e þ01:12545e þ0
41:78792e þ12:89047e þ03:93500e þ0
52:09320e þ18:52674e þ05:08863e þ1
62:93700e þ15:68339e 22:61029e þ1
73:29807e þ14:90080e þ01:78391e þ1
83:51561e þ12:66659e þ01:73783e þ0
pbp3bp4bp5
11:48408e þ02:43815e þ09:19988e þ0
21:70218e þ13:66333e þ17:26606e þ1
31:41418e þ16:27283e þ11:53398e þ2
41:67754e þ11:15289e þ23:24667e þ2
51:56893e þ23:21951e þ24:61948e þ2
62:90804e þ21:47321e þ33:87334e þ3
79:35850e þ12:24591e þ23:60868e þ2
84:98942e þ02:71676e þ14:15537e þ1
pbp6bp7bp8
11:64492e þ11:28147e þ13:67526e þ0
29:47544e þ16:43396e þ11:72245e þ1
32:00134e þ21:29740e þ23:30267e þ1
44:59063e þ23:15704e þ28:42405e þ1
54:34126e þ22:32404e þ25:27733e þ1
65:21125e þ33:43718e þ38:85649e þ2
73:73065e þ22:15221e þ25:18052e þ1
81:88208e þ21:86631e þ25:96266e þ1
Table D.7
Parameters to calculate aas a function of e
T1. The factors are of unit
[km
1
].
pap0ap1ap2
11:98541e 11:40701e 21:87647e 2
29:71648e 27:16062e 34:77822e 2
35:05069e 23:33725e 21:85987e 2
42:83356e 21:64584e 23:32683e 2
52:18893e 28:84693e 35:46460e 2
66:24488e 34:90041e 36:03999e 3
72:82771e 33:17505e 31:93697e 3
88:53512e 47:92640e 41:24063e 3
pap3ap4ap5
11:72925e 22:77798e 29:95750e 2
21:51184e 13:51432e 17:02642e 1
31:03728e 15:51289e 11:41638e þ0
48:69501e 26:20406e 23:36952e 1
52:34999e 15:47095e 18:27779e 1
67:24190e 25:32824e 11:79828e þ0
74:29619e 21:78919e 13:53528e 1
84:65874e 31:87465e 28:70408e 3
pap6ap7ap8
11:76679e 11:37542e 13:94618e 2
29:01640e 16:03103e 11:59691e 1
31:87770e þ01:22379e þ03:11852e 1
48:28293e 16:99209e 12:06734e 1
57:76841e 14:02671e 18:74533e 2
62:85818e þ02:11311e þ05:91400e 1
73:82857e 12:16923e 15:02721e 2
83:62357e 24:73838e 21:66805e 2
S. Frey et al. / Advances in Space Research 64 (2019) 1–17 17
