ISO 27852:2016

INTERNATIONAL ISO
STANDARD 27852
Second edition
2016-07-01
Space systems — Estimation of orbit
lifetime
Systèmes spatiaux — Estimation de la durée de vie en orbite
Reference number
©
ISO 2016
© ISO 2016, Published in Switzerland
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ii © ISO 2016 – All rights reserved

Contents Page
Foreword .iv
Introduction .v
1 Scope . 1
2 Normative References . 1
3 Terms, definitions, symbols and abbreviated terms . 1
3.1 Terms and definitions . 1
3.2 Symbols . 3
3.3 Abbreviated terms . 4
4 Orbit lifetime estimation . 4
4.1 General requirements . 4
4.2 Definition of orbit lifetime estimation process . 4
5 Orbit lifetime estimation methods and applicability . 5
5.1 General . 5
5.2 Method 1: High-precision numerical integration. 6
5.3 Method 2: Rapid semi-analytical orbit propagation . 7
5.4 Method 3: Numerical table look-up, analysis and fit formula evaluations . 7
5.5 Orbit lifetime sensitivity to sun-synchronous . . 7
5.6 Orbit lifetime statistical approach for high-eccentricity orbits (e.g. GTO) . 7
6 Drag modelling .13
6.1 General .13
6.2 Atmospheric density modelling .13
6.3 Long-duration solar flux and geomagnetic indices prediction .14
6.4 Approach 1: Monte Carlo random draw of solar flux and geomagnetic indices .15
6.5 Method 3: Equivalent constant solar flux and geomagnetic indices .19
6.6 Atmospheric density implications of thermospheric global cooling .23
7 Estimating ballistic coefficient (C A/m) .23
D
7.1 General .23
7.2 Estimating aerodynamic force and SRP coefficients .24
7.2.1 Aerodynamic and solar radiation pressure coefficient estimation via a
“panel model” .24
7.3 Estimating cross-sectional area with tumbling and stabilization modes .27
7.4 Estimating mass .28
Annex A (informative) Space population distribution .29
Annex B (informative) 25-year lifetime predictions using random draw approach .32
Annex C (informative) Solar radiation pressure and 3rd-body perturbations .37
Annex D (informative) Sample code for drag coefficient estimation via panel model .39
Bibliography .41
Foreword
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different types of ISO documents should be noted. This document was drafted in accordance with the
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Barriers to Trade (TBT) see the following URL: Foreword - Supplementary information
The committee responsible for this document is ISO/TC 20, Aircraft and space vehicles, Subcommittee
SC 14, Space systems and operations.
This second edition cancels and replaces the first edition (ISO 27852:2011), which has been technically
revised.
iv © ISO 2016 – All rights reserved

Introduction
This International Standard is a supporting document to ISO 24113 and the GEO and LEO disposal
standards that are derived from ISO 24113. The purpose of this International Standard is to provide
a common consensus approach to determining orbit lifetime, one that is sufficiently precise and
easily implemented for the purpose of demonstrating compliance with ISO 24113. This project offers
standardized guidance and analysis methods to estimate orbital lifetime for all LEO-crossing orbit
classes.
INTERNATIONAL STANDARD ISO 27852:2016(E)
Space systems — Estimation of orbit lifetime
1 Scope
This International Standard describes a process for the estimation of orbit lifetime for spacecraft,
launch vehicles, upper stages and associated debris in LEO-crossing orbits.
This International Standard also clarifies the following:
a) modelling approaches and resources for solar and geomagnetic activity modelling;
b) resources for atmosphere model selection;
c) approaches for spacecraft ballistic coefficient estimation.
2 Normative References
The following documents, in whole or in part, are normatively referenced in this document and are
indispensable for its application. For dated references, only the edition cited applies. For undated
references, the latest edition of the referenced document (including any amendments) applies.
ISO 24113, Space systems — Space debris mitigation requirements
3 Terms, definitions, symbols and abbreviated terms
3.1 Terms and definitions
For the purposes of this document, the terms and definitions given in ISO 24113 and the following apply.
3.1.1
orbit lifetime
elapsed time between the orbiting spacecraft’s initial or reference position and orbit demise/reentry
Note 1 to entry: An example of the orbiting spacecraft’s reference position is the post-mission orbit.
Note 2 to entry: The orbit’s decay is typically represented by the reduction in perigee and apogee altitudes (or
radii) as shown in Figure 1.
Figure 1 — Sample of orbit lifetime decay profile
3.1.2
earth equatorial radius
equatorial radius of the Earth
Note 1 to entry: The equatorial radius of the Earth is taken as 6 378,137 km and this radius is used as the
reference for the Earth’s surface from which the orbit regions are defined.
3.1.3
high area-to-mass
HAMR
space objects are considered to be high area-to-mass (or HAMR) objects if the ratio of area to mass
exceeds 0,1 m /kg
3.1.4
LEO-crossing orbit
low-earth orbit, defined as an orbit with perigee altitude of 2 000 km or less
Note 1 to entry: As can be seen in Figure A.1, orbits having this definition encompass the majority of the high
spatial density spike of spacecraft and space debris.
3.1.5
long-duration orbit lifetime prediction
orbit lifetime prediction spanning two solar cycles or more (e.g. 25-year orbit lifetime)
3.1.6
mission phase
period of a mission during which specified communications characteristics are fixed.
Note 1 to entry: The transition between two consecutive mission phases may cause an interruption of the
communications services.
3.1.7
post-mission orbit lifetime
duration of the orbit after completion of all mission phases
Note 1 to entry: The disposal phase duration is a component of post-mission duration.
2 © ISO 2016 – All rights reserved

3.1.8
space object
man-made object in outer space
3.1.9
orbit
path followed by a space object
3.1.10
solar cycle
≈11-year solar cycle based on the 13-month running mean for monthly sunspot number and is highly
correlated with the 13-month running mean for monthly solar radio flux measurements at the 10,7 cm
wavelength
Note 1 to entry: Historical records back to the earliest recorded data (1945) are shown in Figure 2.
Note 2 to entry: For reference, the 25-year post-mission orbit lifetime constraint specified in ISO 24113 is overlaid
onto the historical data; it can be seen that multiple solar cycles are encapsulated by this long time duration.
Figure 2 — Solar cycle (≈11-year duration)
3.2 Symbols
a orbit semi-major axis
A spacecraft cross-sectional area with respect to the relative wind
A earth daily geomagnetic index
p
β ballistic coefficient of spacecraft = C · A/m
D
C spacecraft drag coefficient
D
C spacecraft reflectivity coefficient
R
e orbit eccentricity
-22 -2 -1
F solar radio flux observed daily at 2 800 MHz (10,7 cm) in solar flux units (10 W m Hz )
10,7
F Bar solar radio flux at 2 800 MHz (10,7 cm), averaged over three solar rotations
10,7
H apogee altitude = a (1 + e) − R
a e
H perigee altitude = a (1 – e) − R
p e
m mass of spacecraft
R equatorial radius of the Earth
e
3.3 Abbreviated terms
3Bdy third-body (perturbations)
CAD computer-aided design
GEO geosynchronous earth orbit
GTO geosynchronous transfer orbit
HAMR high area-to-mass ratio
IADC Inter-Agency Space Debris Coordination Committee
ISO International Organization for Standardization
LEO low earth orbit
N/A not applicable
RAAN orbit right ascension of the ascending node (angle between vernal equinox and orbit ascending
node, measured CCW in equatorial plane, looking in–Z direction)
SRP solar radiation pressure
STSC Scientific and Technical Subcommittee of the Committee
UNCOPUOS United Nations Committee on the Peaceful Uses of Outer Space
4 Orbit lifetime estimation
4.1 General requirements
The orbital lifetime of LEO-crossing mission-related objects shall be estimated using the processes
specified in this International Standard. In addition to any user-imposed constraints, the post-mission
portion of the resulting orbit lifetime estimate shall then be constrained to a maximum of 25 years per
ISO 24113 using a combination of (a) initial orbit selection, (b) spacecraft vehicle design, (c) spacecraft
launch and early orbit concepts of operation which minimize LEO-crossing objects, (d) spacecraft
ballistic parameter modifications at EOL, and (e) spacecraft deorbit maneuvers.
4.2 Definition of orbit lifetime estimation process
The orbit lifetime estimation process is represented generically in Figure 3.
4 © ISO 2016 – All rights reserved

[4]
Figure 3 — Orbit lifetime estimation process
5 Orbit lifetime estimation methods and applicability
5.1 General
[3]
There are three basic analysis methods used to estimate orbit lifetime, as depicted in Figure 3.
Determination of the method used to estimate orbital lifetime for a specific space object shall be based
upon the orbit type and perturbations experienced by the spacecraft as shown in Table 1.
Table 1 — Applicable method with mandated conservative margins of error (in percent) and
required perturbation modelling
Special orbit Conservative margin applied to each method
Orbit apogee Sun- High Method 1: Method 2: Method 3: Method 3
altitude, km sync? area-
Numerical Semi- Table look-up Graph,
to-
integration analytic formula
mass?
fit
Apogee < 2 000 km No No No margin 5 % margin 10 % margin 25 % margin
req’d
Apogee < 2 000 km No Yes No margin; 5 % margin; 10 % margin IFF N/A
use SRP use SRP C ≈ 1,7
r
Apogee < 2 000 km Yes No No margin 5 % margin N/A N/A
req’d
Apogee < 2 000 km Yes Yes No margin 5 % margin; N/A N/A
req’d; use SRP
use SRP
Apogee > 2 000 km Either Either No margin 5 % margin; N/A N/A
req’d; use use 3Bdy+
3Bdy+SRP SRP
N/A = not applicable
3Bdy = third-body perturbations
SRP = solar radiation pressure
Method 1, certainly the highest fidelity model, utilizes a numerical integrator with a detailed gravity
model, third-body effects, solar radiation pressure, and a detailed spacecraft ballistic coefficient model.
[4] [5]
Method 2 utilizes a definition of mean orbital elements, semi-analytic orbit theory and average
spacecraft ballistic coefficient to permit the very rapid integration of the equations of motion while still
retaining reasonable accuracy. Method 3 is simply a table lookup, graphical analysis or evaluation of
formulae that have been fit to pre-computed orbit lifetime estimation data obtained via the extensive
and repetitive application of Methods 1 and/or 2. It is worth noting that all methods (1 through 3) shall
include at gravity zonals J and J at a minimum.
2 3
5.2 Method 1: High-precision numerical integration
Method 1 is the direct numerical integration of all accelerations in Cartesian space, with the ability
to incorporate a detailed gravity model (e.g. using a larger spherical harmonics model to address
resonance effects), third-body effects, solar radiation pressure, vehicle attitude rules or aero-torque-
driven attitude torques, and a detailed spacecraft ballistic coefficient model based on the variation of
the angle-of-attack, with respect to the relative wind. Atmospheric rotation at the Earth’s rotational
rate is also easily incorporated in this approach. The only negative aspects to such simulations are
(a) they run much slower than Method 2, (b) many of the detailed data inputs required to make this
method realize its full accuracy potential are simply unavailable, and (c) any gains in orbit lifetime
prediction accuracy are frequently overwhelmed by inherent inaccuracies of atmospheric modelling
and associated inaccuracies of long term solar activity predictions/estimates. However, to analyse a
few select cases where such detailed model inputs are known, this is undoubtedly the most accurate
method. At a minimum, Method 1 orbit lifetime estimations shall account for J and J perturbations
2 3
and drag using an accepted atmosphere model and an averaged ballistic coefficient. In the case of
high apogee orbits (e.g. geosynchronous transfer orbits) or other resonant orbits, sun and moon third-
body perturbations and solar radiation pressure effects shall also be modelled (see Reference [28] for
additional discussion).
6 © ISO 2016 – All rights reserved

5.3 Method 2: Rapid semi-analytical orbit propagation
[4] [5]
Method 2 analysis tools utilize semi-analytic propagation of mean orbit elements influenced by
gravity zonals J and J and selected atmosphere models. The primary advantage of this approach over
2 3
direct numerical integration of the equations of motion (Method 1) is that long-duration orbit lifetime
cases can be quickly analysed (e.g. 1 s versus 1 700 s CPU time for a 30-year orbit lifetime case). While
incorporation of an attitude-dependent ballistic coefficient is possible for this method, an average
ballistic coefficient is typically used. At a minimum, Method 2 orbit lifetime estimations shall account
for J and J perturbations and drag using an accepted atmosphere model and an average ballistic
2 3
coefficient. In the case of high apogee orbits (e.g. GTO), sun and moon third-body perturbations shall
also be modelled.
5.4 Method 3: Numerical table look-up, analysis and fit formula evaluations
In this final method, one uses tables, graphs and formulae representing data that was generated
by exhaustively using Methods 1 and 2 (see 5.2 and 5.3). The graphs and formulae provided in this
International Standard can help the analyst crudely estimate orbit lifetime for their particular case
of interest; the electronic access to tabular look-up provided via this International Standard (at www.
CelesTrak.com) permits the analyst to estimate orbit lifetime for their particular case of interest via
interpolation of Method 1 or Method 2 gridded data; all such Method 3 data in this International
Standard were generated using Method 2 approaches. At a minimum, Method 3 orbit lifetime products
shall be derived from Method 1 or Method 2 analysis products meeting the requirements stated above.
When using this method, the analyst shall impose at least a 10 % margin of error to account for table
look-up interpolation errors. When using graphs and formulae, the analyst shall impose a 25 % margin
of error.
5.5 Orbit lifetime sensitivity to sun-synchronous
For sun-synchronous orbits, orbit lifetime has some sensitivity to the initial value of RAAN due to the
density variations with the local sun angle. Results from numerous orbit lifetime estimations show that
orbits with 6:00 am local time have longer lifetime than orbits with 12:00 noon local time by about 5,5 %.
[3]
This maximum difference (500 d) translates into a 5 % error which can be corrected by knowing
the local time of the orbit. As a result, Methods 1 or 2 analyses of the actual sun-synchronous orbit
condition shall be used when estimating the lifetime of sun-synchronous orbits (see References [28]
and [38], where more details are given).
5.6 Orbit lifetime statistical approach for high-eccentricity orbits (e.g. GTO)
For high-eccentricity orbits (particularly geosynchronous transfer orbits or GTO), it can be difficult
to iterate to lifetime threshold constraints due to the coupling in eccentricity between the third-body
perturbations and the drag decay. Due to this convergence difficulty, only Method 1 or 2 analyses shall
be used when determining initial conditions which achieve a specified lifetime threshold for such orbits.
Sample analyses of GTO launcher stages (see References [29] and [30]) highlight this orbit lifetime
sensitivity to initial conditions (orbit, spacecraft characteristic and force model), leading to a wide
spectrum of orbital lifetimes.
Some theoretical considerations about the dynamical properties of GTO orbits are provided in
References [29] and [36].
The following test case illustrates the complex dynamical properties of GTO. Initial parameters are
provided in Table 2.
Table 2 — GTO initial conditions for the Monte Carlo simulation
Perigee altitude 200 km
Apogee altitude GEO altitude
Inclination 2°
Table 2 (continued)
Area to mass ratio 5e-3 m²/kg
Solar activity Constant (F10.7 = 140
sfu Ap = 15)
Drag coefficient Constant = 2,2
Reflectivity coefficient Constant = 2
Figure 4 shows lifetime results (years) when varying the initial date and the initial local time of perigee.
This latest parameter is defined as the angle in the equator between the sun direction and the orbit
perigee, measured in hours. The date was chosen from day 1 to 365 in year 1998 and the local time of
perigee was chosen by varying the right ascension of ascending node from 0π to 2π. A total of 2 500
different initial conditions were generated.
Figure 4 — Lifetime variations with respect to initial date and local time of perigee (year)
The shapes of the lifetime contours confirm that initial day of year and local time of perigee are initial
conditions that make sense to describe GTO evolution since strong patterns are visible. The amplitudes
of lifetimes variations are worth noting: from several months to more than 50 years. Previous results
(see References [30] and [37]) are illustrated here: the longest lifetimes are obtained for initial sun-
pointing (12 h local time) or anti sun-pointing (24 h local time) perigee with an initial date around the
solstices. Note that the dark red pixels drawn in dark blue areas, as seen for initial day 60 and local time
7 h, are an indication of the presence of strong resonance phenomena. We know that the year also has
an influence, to a lesser extent, through the moon perturbation.
Figure 5 shows semi-major axis evolution for several propagations of a typical low-inclined GTO. The
different curves correspond to changes of 0,1 % or 1 % in the area to mass ratio of the object (A/m),
which is far below the level of incertitude on this parameter. These dispersions lead to variations of
decades in the re-entry duration. Such a strong non-linear behaviour is explained by the aforementioned
resonances. One can see that semi-major axis evolutions are quite similar between all propagation
cases until the entrance in the coupling between J2 and sun perturbations, for a semi-major axis equal
to about 15 500 km. The duration of the resonance (period when the semi-major axis remains constant)
and, thus, the rest of the propagation are completely different. A similar figure can be plotted by keeping
the area to mass ratio constant and slightly changing the solar activity.
8 © ISO 2016 – All rights reserved

Figure 5 — SMA evolution sensitivity to slight A/m variations (from 0,1 to 2 %)
These examples show that resonance phenomena have substantial impacts on orbital elements
evolution that can neither be predicted nor managed. Cumulated uncertainties on drag force between
the extrapolation start (mission disposal manoeuvre, for example) and the instant when the resonance
occurs make the entry condition in this resonance prone to strong variations. As a consequence, trying
to estimate lifetime of GTOs using only one extrapolation may lead to erroneous conclusion since tiny
changes in the initial conditions, spacecraft characteristics or force models end in very different lifetime
results. Exceptions to that would be objects on a GTO whose semi major axis has already decreased
enough to avoid resonances or to be very close to them. However, since resonance conditions change
with regards to the possible resonant angles, one can see that performing several propagation cases
is advised to get robust results. As a conclusion, only statistical results are adequate to estimate the
strong variations of GTO lifetimes.
As a consequence, one should not say “this object’s lifetime is Y years” in GTO but rather “the lifetime of
this object is shorter than Y years with a probability p”, coming from a cumulative distribution function
(see example below).
The key parameter uncertainties to be taken into account in the lifetime estimation are
— initial conditions (date, orbit parameters),
— ballistic coefficient and drag coefficient, and
— solar activity.
The following test case (see Reference [32]) provides results of Monte Carlo simulations. Initial
parameters are described in Table 3. A total of 2 500 different initial conditions were generated.
Table 3 — Hypothesis for the Monte Carlo simulation
Parameter Nominal value Dispersions
Perigee altitude 180 km Small dispersions :
1sigma standard deviation about 1 km, cor-
related to other orbit parameters.
Apogee altitude GEO altitude Small dispersions:
1sigma standard deviation about 50 km,
correlated to other orbit parameters.
Inclination 6° Small dispersions:
1sigma standard deviation about 0,01°, cor-
related to other orbit parameters.
Area to mass 5e-3 m²/kg Uniform distribution
ratio
+/–20 %
wrt nom. value
Drag coefficient Function of geo- None
detic altitude
Reflectivity Constant = 1,5 None
coefficient
Solar activity Randomly chosen using data from the past
Date Uniform distribution between day 1 to day 365, for years
between 2015 and 2033 (the dispersion of the year enables to
cover the moon perturbation).
Local time of Gaussian distribution, mean value 22 h standard deviation
perigee 50 min
Figure 6 and Figure 7 provide a statistical histogram and cumulative distribution function of orbit
lifetime for this test case.
Figure 6 — Histogram of orbital lifetimes
10 © ISO 2016 – All rights reserved

Figure 7 — Cumulative distribution function of orbital lifetimes
The question of statistical convergence may be addressed by computing a confidence interval for the
Monte Carlo results, associated to a confidence level. The so-called “interval of Wilson with correction
for continuity” (see Reference [31]) has been well-adapted for this purpose.
In this approach, the upper p1 and lower p2 limits of this interval are given by Formula (1):
2 2
21nf +−uu−−un21−+41fn[( −+f )]1
aa//2 22a/
p1 =
2()nu+
a/2
(1)
2 2
2nf ++−uu12−−un−+14fn[(11−−f )]
aa//2 22a/
p2=
2()nu+
a/2
where
n is number of single runs (orbit propagations);
f is observed probability = number of lifetimes lower than a certain value divided by n;
-1
u = Φ (1- α/2) (= 1,96, for example, for a confidence interval of 95 %). Φ is the cumulative nor-
α/2
mal distribution function.
Figure 8 — Example of evolution of the observed probability (lifetimes lower than 25 years) and
95 % confidence interval
As shown in Figure 8 and Figure 9, after “N” Monte Carlo runs, one can compare the limit (upper or lower)
of the confidence interval with the targeted probability for the lifetime to be lower than a certain value.
Figure 9 — Example of cumulative distribution function of orbital lifetimes with a 95 %
confidence interval (500 runs)
12 © ISO 2016 – All rights reserved

6 Drag modelling
6.1 General
The three biggest factors in orbit lifetime estimation are (a) the selection of an appropriate atmosphere
model to incorporate into the orbit acceleration formulation, (b) the selection of appropriate atmosphere
model inputs, and (c) determination of a space object’s ballistic coefficient. We will now spend some
time discussing each of these three aspects.
6.2 Atmospheric density modelling
There are a wide variety of atmosphere models available to the orbit analyst. The background, technical
basis, utility and functionality of these atmosphere models are described in detail in References [6] to
[15]. This International Standard will not presume to dictate which atmosphere model the analyst shall
use. However, it is worth noting that in general, the heritage, expertise and especially the observational
data that went into creating each atmosphere model play a key role in that model’s ability to predict
atmospheric density, which is in turn, a key factor in estimating orbit lifetime. Many of the early
atmosphere models were low fidelity and were created on the basis of only one, or perhaps even just a
part of one, solar cycle’s worth of data.
The advantage of some of these early models is that they typically run much faster than the latest high-
fidelity models (see Table 4) without a significant loss of accuracy. However, the use of atmosphere
models that were designed to fit a select altitude range (e.g. the “exponential” atmosphere model
depicted below) or models that do not accommodate solar activity variations should be avoided as they
miss too much of the atmospheric density variations to be sufficiently accurate.
There are some early models (e.g. Jacchia 1971 shown below) which accommodate solar activity
variations and also run very fast; these models can work well for long-duration orbit lifetime studies
where numerous cases are to be examined. Conversely, use of the more recent atmosphere models
are encouraged because they have substantially more atmospheric drag data incorporated as the
foundation of their underlying assumptions. A crude comparison of a sampling of atmosphere models
for a single test case is shown in Figure 10 and Figure 11, illustrating the range of temperatures and
densities exhibited by the various models. Although this International Standard does not presume to
direct which atmosphere model the analyst should use, the reader is encouraged to seek atmosphere
model guidance from ISO 14222 to select proper atmosphere and associated indices. However, it is
also noted that the lengthy prediction timespan associated with this International Standard makes a
number of atmosphere models suitable for estimation of orbital lifetimes spanning 25 years, to include,
[10] [11] [12] [13] [14]
but not limited to, the NRLMSISE-00, JB2006, JB2008, GRAM-07, DTM-2000 and
[15]
GOST models.
Table 4 — Comparison of normalized density evaluation runtimes
Atmosphere model 0 < Alt < 5 000 km 0 < Alt < 1 000 km
Exponential 1,00 1,00
Atm1962 1,43 1,51
Atm1976 1,54 1,54
Jacchia 1971 13,68 17,31
MSIS 2000 141,08 222,81
JB2006 683,85 584,47
Figure 10 — Temperature comparison by atmosphere model
Figure 11 — Comparison of a small sampling of atmosphere models
6.3 Long-duration solar flux and geomagnetic indices prediction
Utilization of the higher-fidelity atmosphere models mentioned in 6.2 requires the orbit analyst to
specify the solar and geomagnetic indices required by such models. Care must be taken to obtain the
proper indices required by each model; subtle difference may exist in the interpretation of similarly
named indices when used by different atmosphere models (e.g. centrally-averaged vs. backward-
averaged F Bar).
10,7
14 © ISO 2016 – All rights reserved

Key issues associated with any prediction of solar and geomagnetic index modelling approach are as
follows.
a) F Bar predictions should reflect the estimated mean solar cycle as accurately as possible. One
10,7
such prediction is shown in Figure 12.
b) Large daily F and A index variations about the mean value induce non-linear variations in
10,7 p
atmospheric density and the selected prediction approach should account for this fact, i.e. one
should account for the highly non-linear aspects of solar storms versus quiet periods.
c) The frequency of occurrence across the day-to-day index values is highest near the lowest
prediction boundary (see Figure 13).
d) F cycle timing/phase is always imprecise and should be accounted for; the resultant time bias
10,7
that such a prediction error would introduce can yield large F prediction errors of 100 % or more.
10,7
e) The long-time duration orbit lifetime constraint specified in ISO 24113 (i.e. 25 years) would
require that the solar/geomagnetic modelling approach provide at least that many years (i.e. 25) of
predictive capability.
f) Predicted F values should be adjusted to correct for earth-sun distance variations.
10,7
g) Some atmosphere models (e.g. JB2006 and JB2008), due to the newly invented indices adopted
thereby, preclude the use of historical indices for long-term orbit lifetime studies while currently
also precluding use of any predictive forecasting model(s) for those indices until such time as those
become publicly available.
Accounting for these constraints, the user shall adopt one of the following three acceptable approaches.
[16] [17]
— Approach #1: Utilize Monte Carlo sampling of historical data mapped to a common solar
cycle period.
— Approach #2: Utilize a predicted F Bar solar activity profile generated by a model such as is
10,7
[19]
detailed in Figure 12, coupled with a stochastic or similar generation of corresponding F and
10,7
A values, e.g. Reference [19].
p
— Approach #3: Utilize a “Mean Equivalent Static” set of solar and geomagnetic activity. Note that
while such an approach produces equivalent solar and geomagnetic indices that are suitable for
efficient and equivalent orbit lifetime estimation, such static values are only valid for the cycles
fit, the selected orbit prediction span (i.e. 25 years) with an associated probability level and the
adopted atmosphere model. New sets of Mean Equivalent Static indices would likely need to be
generated for any changes in the above functional dependencies.
Since Approach #2 is a well-known and common approach, the focus of the remainder of this subclause
[3]
will be devoted to (Approach #1) the Monte Carlo “Random Draw” approach and the “Mean Equivalent
Static” approach (Approach #3).
6.4 Approach 1: Monte Carlo random draw of solar flux and geomagnetic indices
Note (see Figure 2) that we already have more than five solar cycles of observed solar and geomagnetic
data to choose from. Processing of this data maps each coupled and correlated triad of datum (F ,
10,7
F Bar, and A ) into a single solar cycle range of 10,825 46 years (3 954 d), with the “averaged” solar
10,7 p
minimum referenced to 25 February 2007.
By mapping this historical data into a single solar cycle (see Figure 14 through Figure 16), the user can
then sample coupled triads of (F , F Bar, and A ) data corresponding to the orbit lifetime simulation
10,7 10,7 p
day within the mapped single solar cycle. This solar/geomagnetic data can then be updated at a user-
selectable frequency (e.g. once per orbit or day), thereby simulating the drag effect resulting from solar
and geomagnetic variations consistent with historical trends for these data. Since we have accumulated
daily data since the February 14, 1947, on any given day within the 3 954-day solar cycle, we have at
least five data triads to choose from. It is important that the random draw retain the integrity of each
data triad since F , F Bar and A are interrelated.
10,7 10,7 p
In the Monte Carlo approach for modelling solar and geomagnetic data, coupled triads of (F , F Bar,
10,7 10,7
and A ) data are selected for each day (or alternately for each orbit rev) of the orbit lifetime simulation,
p
thereby simulating the drag effect resulting from solar and geomagnetic variations consistent with
historical trends for these data. The atmospheric density estimated from atmospheric models utilizing
a given (F , F Bar, and A ) triad can then be directly utilized by either Method 1 (numerical
10,7 10,7 p
integration) or Method 2 (semi-analytic) approaches. Due to the introduced step-function change in
atmospheric density, it may be beneficial to restart Method 1 integration at each parameter set change;
for semi-analytic (e.g. with orbital revolution time steps via Gaussian quadrature), a new parameter set
can be drawn at an orbit revolution time step; thus, no numerical difficulties will be introduced.
The starting epoch of the simulation can be selected to match the anticipated actual end-of-mission
epoch. Alternately, starting epochs can be sampled throughout the entire aggregate solar cycle and to
ensure that the median (50th-percentile) value meets the specified orbit lifetime criteria of ISO 24113.
The C++ code used to implement this atmospheric variation strategy is publicly available at www.
CelesTrak.com so that users of this orbit lifetime standard can adopt this standardized (F , F Bar
10,7 10,7
and A ) implementation Approach #1 if desired.
p
Figure 12 — Solar flux estimated upper, lower and representative trends
16 © ISO 2016 – All rights reserved

Figure 13 — Solar flux distribution in percentage of localized min/max variation
Figure 14 — F10.7 normalized to average solar cycle
Figure 15 — F10.7 bar normalized to average cycle
Figure 16 — A normalized to average cycle
p
18 © ISO 2016 – All rights reserved

It can be seen from Figure 16 that A is (a) unpredictable, (b) loosely correlated with the solar cycle,
p
and (c) volatile. Figure 17 demonstrates that density varies greatly (i.e. several orders of magnitude)
depending upon A ; thus, a geomagnetic storm can induce large decreases in orbital energy (orbit decay)
p
that the use of some average A value would miss. Correspondingly, the analyst should incorporate A
p p
variations into the geomagnetic index predictions.
Figure 17 — Log (density) variation as a function of A value
p
6.5 Method 3: Equivalent constant solar flux and geomagnetic indices
A third method for simulation of solar flux and geomagnetic indices is the use of “Equivalent Constant
Solar Flux and Geomagnetic Indices.” In this method, the user generates (or obtains from a qualified
third party) and incorporates pre-computed constant equivalent values of solar flux and geomagnetic
activity into the orbit lifetime estimation process which will yield the same orbital lifetime as would
the use of actual measured (dynamic) indices. Since the starting epoch with respect to solar cycles may
be not well known and is a sensitive parameter of orbit lifetime estimation (about ±4 years for a typical
LEO orbit), the starting epoch can be included in the Monte Carlo as a random parameter (the initial
day is a random realization within the first solar cycle). This pre-computation of equivalent constant
indices avoids the use of random draws each orbit and repeated Monte Carlos for the actual orbit of
interest.
If Method 3 is employed, the equivalent constant indices shall be carefully tuned by the analyst to match
historical solar and geomagnetic influences on orbit decay over a long-duration (i.e. 25-year) timespan
corresponding to the atmosphere model and orbit inclination of interest. This ensures that a space
object having a specific ballistic coefficient and starting orbit that has a 25-year lifetime computed by
using the constant equivalent solar activity yields an orbit lifetime estimate of less than 25 years with a
z % probability level (see Figure 22).
Figure 18 — Statistical equivalency between constant and variable solar activity
Equivalent constant solar flux and geomagnetic activity indices are obtained via the following
algorithm.
a) Select the initial orbit and a ballistic coefficient.
b) Select a representative geomagnetic A index (e.g. 15).
p
NOTE As there are two uncertain parameters (solar flux and geomagnetic index) with no evident
correlation between them, this method is faced with two degrees of freedom versus only one output
(estimated orbit lifetime). The method circumvents this by adopting a representative geomagnetic A index
p
value which is averaged over the timespan of interest (e.g. 25 years).
c) Using the random draw Monte Carlo approach previously outlined (see Method #1), generate
n possible future solar activities (including random draws of starting date which encompass all
possible launch dates, including launch slips).
d) Estimate orbit lifetime using the solar activity profiles, where n is appropriately sized using Dagum
(Chernoff – Hoeffding) bounding methods.
e) As shown in Figure 23 and adopting z % to be 50 % (the median value), determine OLT_50 % (the
median orbit lifetime of the trials).
Figure 19 — Lifetime cumulative distribution function
f) Iterate on eithe
...