Spatial density approach for modelling of the space debris population

Abstract

This article proposes a continuum density approach for space debris modelling. The debris population in Low Earth Orbit (LEO) is represented through its density in semi-major axis, eccentricity and inclination. The time evolution of the density in orbital elements is modelled through the continuity equation. The perturbing effect of aerodynamic drag is included in the divergence term, while the effect of fragmentation can be seen as source term in the equation. The spatial density is then calculated from the orbital element density at each time. The proposed continuum method is used to analyse the evolution of the debris population in LEO; as initial condition the debris 2013 population is used. Then, the effect of a breakup event is superimposed onto the global population of space debris and its effect analysed; the fragment distribution caused by the breakup up of satellite DMSP-F13 is considered as test case scenario.

Full text

2749
AAS 16-465
SPATIAL DENSITY APPROACH FOR MODELLING OF
THE SPACE DEBRIS POPULATION
Camilla Colombo,* Francesca Letizia† and Hugh G. Lewis‡
This article proposes a continuum density approach for space debris modelling.
The debris population in Low Earth Orbit (LEO) is represented through its den-
sity in semi-major axis, eccentricity and inclination. The time evolution of the
density in orbital elements is modelled through the continuity equation. The per-
turbing effect of aerodynamic drag is included in the divergence term, while the
effect of fragmentation can be seen as source term in the equation. The spatial
density is then calculated from the orbital element density at each time. The pro-
posed continuum method is used to analyse the evolution of the debris popula-
tion in LEO; as initial condition the debris 2013 population is used. Then, the ef-
fect of a breakup event is superimposed onto the global population of space de-
bris and its effect analysed; the fragment distribution caused by the breakup up
of satellite DMSP-F13 is considered as test case scenario.
INTRODUCTION
The space surrounding our planet is densely populated by an increasing number of man-made
space debris, most of which have been generated from the break-up of operational satellites, aban-
doned spacecraft or upper stages of launchers. Space debris is internationally recognised as a hazard
to current and future space activities and space agencies are currently cooperating to identify appro-
priate and sustainable space debris mitigation measures.
The debris evolution in Low Earth Orbit (LEO) is dominated by the effects of the Earth’s oblate-
ness and the atmospheric drag, which is the only natural way debris objects are removed from their
orbits, to re-enter and burn in the atmosphere. Long-term studies of the debris environment per-
form simulation of the space debris populations over 100 to 200 years to observe the effects of
the growing space activities (e.g. launches), the uncertainty of the physical environment (e.g. at-
mosphere model, changes in the Earth’s atmosphere due to the solar activities) and the spacecraft
parameters (such as attitude, solar and drag coefficient, material deterioration), the consequences
of fragmentation and explosion of inoperative objects and active satellites[1]. From the other side,
these long-term studies aim at evaluating the efficacy of mitigation rules, such as passive disposal,
collision avoidance manoeuvres, end-of-life guidelines, active debris removal, to reduce the risk to
operating satellites and ensure the long term sustainability of space.
Surveys of the existing evolution models are available[2,3] . Most of these evolutionary debris
models[4,5,6,7] use semi-analytical methods to propagate the dynamics under orbit perturbations and
* Ph.D., Associate Professor, Astronautics Research Group, University of Southampton, SO17 1BJ, UK.
E-mail: c.colombo@soton.ac.uk.
† Ph.D., Senior Research Assistant, Astronautics Research Group, University of Southampton, SO17 1BJ, UK.
E-mail: f.letizia@soton.ac.uk.
‡ Ph.D., Senior Lecturer, Astronautics Research Group, University of Southampton, SO17 1BJ, UK.

2750
make some assumptions on natural phenomena, the future evolution space activities, compliance
with mitigation guidelines, and debris interaction (e.g. criteria for collision, number of fragmen-
tation events per year). To ensure the analyses are robust to these uncertainties and to overcome
the absence of a complete set of experimental data through observations several Monte Carlo runs
are used to consider an large number of evolution scenarios[8,9,10,4]. This dramatically increases the
computational time and limits the variety of the possible analyses. To overcome this limitation, sim-
plified were proposed based on a grid discretization of the debris population in altitude bins and a
variational approach to allow for a quick evaluation of the debris evolution[11,12] . In some cases the
computation of the collision risk for a target spacecraft is done starting from the number of objects
in each bin through a Poisson distribution. Some other models are instead based on a fitting process
of the deterministic high-fidelity models[13,14]. At the other end, a fully analytical model for LEO
was proposed by McInnes[15], borrowing the use of the continuity equation from fluid dynamics
and planetary science. In his work the evolution of debris is described through their spatial density.
Letizia et al.[16] extended the continuity equation method to more than a single variable and ap-
plied it to the fragment clouds generated by a single collision or fragmentation event in space. The
knowledge of the spatial density and the distribution of relative velocities (between the cloud and a
target spacecraft) within the cloud was used to compute the collision probability, via the kinetic gas
theory[17]. In this ways, maps of collision risk can be produced in a very short computational time;
these maps can be used for evaluating the risk on operative spacecraft.
In this paper, we extend our previous research on the modelling of clouds through a continuum
approach, which demonstrated to be an efficient way to propagate the density of particles in the
space of orbital elements. We use a semi-analytical continuum density approach for debris mod-
elling; the debris population in LEO is represented through its spatial density in orbital elements of
semi-major axis, eccentricity and inclination. The time evolution of the density in orbital elements
is modelled through the continuity equation that describes the debris flow evolution through a local
representation via the Jacobian of the dynamics equations. With respect to existing particle-in-a-box
approaches, where some representative objects are propagated to then rebuild the spatial density a
posteriori, here an additional equation is added to the system dn
/dt that describes the time history
of the density of space debris in the phase space, similarly to was was done by Nazarenko[18] and
Smirnov et al.[19]. The proposed continuum method is validated though comparison with the actual
debris evolution fully propagated element-wise by a semi-analytical propagator. As initial condition
the debris population in January 2013 is used. Then, a source term is added to the continuum equa-
tion, which represent a fragmentation. New fragments are thus added onto the population and their
effect is superimposed onto the whole debris population; the case of the breakup of DMSP-F13 is
considered. This paper will briefly describes, in the first Section, the approach developed for the
propagation of debris fragments. The second Section will detail the application of the density based
method to the description of the debris density evolution. The method will be applied in the third
Section to study the evolution of the debris population in 2013 provided by the European Space
Agency.
DENSITY-BASED PROPAGATION FOR A CLOUD OF DEBRIS FRAGMENTS
The propagation method CIELO (debris Cloud Evolution in Low Orbits)[20] was developed to
the aim of describing the evolution of space debris fragments resulting from breakup and collision
in space and to assess the risk that they pose to operative spacecraft. Indeed, even in case of low
intensity fragmentations, thousand of objects of dimension smaller than 5cm are created. The

2751
Figure 1: Schematics of the CiELO method.
inclusion of all these objects in long-term evolutionary studies would be prohibitive in terms of
computational time. Within our approach the fragmentation cloud is described in terms of its spatial
density, whose evolution in time under the effect of drag is obtained by applying the continuity
equation, following the approach proposed by McInnes[15] .
The breakup is modelled through the standard NASA breakup model[21,22] that gives the distribu-
tion of objects in terms of their relative velocity with respect to the nominal orbital velocity where
the fragmentation took place (which is function of the kinetic energy contribution from the event)
and the distribution of area-to-mass. The following evolution of the fragments in LEO is dominated
by Earth’s oblateness and atmospheric drag. In particular the effect of the Earth’s oblateness causes
the distribution of the anomaly of the ascending node and the anomaly of the perigee of the frag-
ments orbits. This phenomenon takes place over a period of time in the order of months, until the
objects form a band around the Earth with minimum and maximum latitude approximately equal
to the inclination where the initial fragmentation took place. For the following phase of the evo-
lution, the atmospheric drag can be considered as the main perturbation and it works as a natural
sink mechanics which removes fragments from their original orbits. In this regime, the continuous
method can be applied to find an analytical expression which describes the time evolution of the
spatial density. Compared to formulation by[15], where the debris density is function of the radial
distance from the Earth (r) only, the continuum method was extended to express the cloud density
as function of semi-major axis (a) and eccentricity (e)[23]. Apart of giving an insight into the evo-
lution of fragments as a whole, the proposed approach drastically reduces the computational time,
allowing the study of many fragmentation scenarios. In Fig. 1 a schematic the CIELO method is
shown.
Continuum approach
The evolution of the density is obtained through the continuity equation that describes the change
in the density of a dispersed set starting from the knowledge of the velocities of the particles. In
particular, if nrepresents the fragments density, the continuity equation can be written as
∂n
∂t +∇•f=˙n+−˙n−(1)
where ∇•fmodels the forces acting on the system and accounts for slow/ continuous phenomena

2752
(such as orbit perturbations) and ˙n+−˙n−represents the sources and the sinks of the system, so it
can models fast/discontinuous events (e.g., the injection of new fragments due to launches). Once
the initial condition for nis known, the continuity equation is used to obtain its evolution with time,
with very low computational effort. The method was previously applied to describe the evolution
of interplanetary dust[24,25], nano-satellites constellations [26] and high area-to-mass spacecraft [27] .
The multi-dimension extension of the continuity equation Eq. 1 was fully derived in[16], following
the approach by Gor’kavyi[28] . The idea is to work in the phase space of the orbital elements by
simply writing the divergence in rectangular coordinates; this simplifies by far the mathematical
formulation. In the last phase of the cloud evolution, when drag is the dominant factor, the semi-
major axis and the eccentricity can be chosen as phase space variables. The vector fin Eq. 1 can be
written as a vector field with two components, respectively, the rate of variation of the semi-major
axis aand eccentricity ecaused by drag:
f=n(a, e;t)va(a, e;t)
ve(a, e;t)(2)
The expressions of the velocities were further simplified by Letizia et al.[16] , to obtain an explicit
analytical solution:
va=−√μRHcDA
Mρ0exp −a−RH
Hf(RH,˜e(a),H)
ve=0 (3)
where ˜e(a)expresses a fixed reference value of the eccentricity for each value of the semi-major
axis. The value of ˜e(a)was set starting from the initial distribution n0(a, e). Given the expression in
Eq. 3, the continuity equation Eq. 1 can be solved adopting the method of characteristics obtaining
the following expression for the density:
n(a, e;t)=n0(ai,e
i)va(ai)
va(a)(4)
with n0is the initial density at the band formation and ai,eiare two functions obtained by inverting
the characteristics of the system at the initial time[16]:
ai(a, t)=−Hlog exp a−RH
H+ε(RH+˜e(a),H)√RH
Ht(5)
ei(e, t)=e. (6)
With Equation 4 the value of the density in the phase space at any time is known once the initial
condition is given. As an example, Fig. 2 shows the value of the density in the phase space at the
band formation and after 1000 days for a fragmentation at 700 km.
Once the phase space density is known at any time tin any point of the domain, the spatial density
can be retrieved from the phase space density by the transformations developed by Sykes[29] and
Kessler[30] .
DENSITY PROPAGATION FOR THE WHOLE DEBRIS POPULATION
This fully-analytical density propagation method described in the previous Section can be applied
between 700 and 1000 km [16], therefore it can also be employed to describe the whole LEO region.

2753
0 500 1000 1500 2000
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
a−RE[km]
e
TB+ 0 days
0 500 1000 1500 2000
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
a−RE[km]
e
TB+ 1000 days
0
5
10
15
20
25
30
NF
Figure 2: Visualisation of cloud density (in number of fragments) following a fragmentation at 700
km at the band formation (TB = 92 days) and after 1000 days[16].
The initial debris population at time t0is known. For each object in the population we know its
type (i.e. mission related object, payload, debris, rocket body), the area-to-mass A/M and the orbit
condition and orbital elements.
In time, the Earth’s oblateness causes the debris’ orbits to rotate with a precession rate that de-
pends on the object’s orbital parameters and that is, therefore, different among the objects in the
population. We can expect that, after a certain time, the right ascension of the ascending node
and the anomaly of the perigee will be equally distributed among objects of the same kind, due to
differences in launching time and conditions.
As a first attempt in applying the proposed continuous technique to the global population of space
debris, some simplifying assumptions will be made and they are justified here. The mean anomaly
of the objects in long-term propagation studies is usually randomised, while many Monte Carlo runs
are used to take into account differences in initial conditions, together with the uncertainties in the
models[8,9,10,4]. In the continuum approach this is equivalent to assume that the mean anomaly of the
object can be considered to be uniformly distributed across each orbit, therefore it can be removed as
a variable from the continuity equation. The argument of perigee and the longitude of the ascending
node are also randomised. Therefore, M,ωand Ωcan be excluded from the dependence of fin Eq.
3. With the hypothesis of a non-rotating atmosphere, the dependence on the inclination ican also
be removed. Under these assumption, fcan be written as a vector field with two components in a
and eas in Eq. 3.
Eq. 4 can be now used on each point of the initial grid of the aand edomain to compute how
the phase-space density evolve over time. Note that, in this work, we are assuming that no further

2754
launches are recorded after time t0, no collision among satellites are considered and no objects are
removed from the population due to active debris removal. Each one of these terms will be added in
a future extension of this work as the continuity equation can be also handle this cases though the
term ˙n+−˙n−in Eq. 1.
As said, the only perturbation on the debris population is due to the effect of drag. The effect
will be different depending on the object’s A/M. This is tackled by dividing the considered domain
in A/M bins. Note that in Ref. [23] the multi-dimensional extension of the continuity equation was
also applied to record the evolution of object with different area-to-mass; A/M, indeed can be
added as a further phase-space variable, with a zero variation in time (i.e. the area-to-mass does not
change over time). However, it was demonstrated that this did not result in an improvement in the
computational time; for this reason, the binning approach is used here for the A/M.
Fragmentation as superimposition of the effects
Now, let’s suppose that a fragmentation takes place at a given time tf. The NASA breakup
model[22,21] can be used to obtain the fragment distribution given the mass of the projectile an its
impact velocity vc. The method previously described can be now used to compute the evolution
of the density of the fragment cloud for any time t>t
f. Therefore, at time t+
fthe fragments
resulting from the fragmentation event add up to the whole debris population. From a mathematical
point of view, this means that the new objects are added to the a−egrid and used to compute the
new phase-space density at time t+
f. The propagation is then continued to evaluate the effect of the
fragmentation cloud on the whole debris population.
RESULTS
The debris population for January 2013 is used here as initial condition; this is limited to objects
larger than 10 cm. Only objects in LEO are considered with semi-major axis a≤2000 km. Figure
3 shows the initial distribution in semi-major axis and inclination. In Figure 4 instead, the initial
debris distribution is shown in semi-major axis and eccentricity, distinguishing among the object
types (MRO = Mission related Object, PL = payload, DEB = Debris, RB = Rocket Body). It can be
noted that the objects in LEO larger than 10 cm have much lower eccentricity values than for the case
of single breakups where the fragments have higher area-to-mass ratio. The grid considered for the
computation is discretized with bin sizes of 20 km for semi-major axis and 0.0002 for eccentricity.
The objects were divided in 15 bins of area to mass ratio in the rage of 0.001-13.35 m2/kg such
that each bin contains the same number of objects (the difference in number of objects in each bin
is less than 5%).
Debris long-term evolution
The evolution of the debris population can be computed with the method proposed, the density in
phase space is propagated with the continuity equation, then the spatial density is calculated. Fig-
ure 5 show the debris population evolution over 25 years. The results obtained with the analytical
method (continuous line) are compared with the results obtained by a numerical method (dashed
line) which integrates each single objects and reconstruct the spatial density though a binning ap-
proach. The continuous method is able to accurately follow the debris evolution. The aerodynamic
drag acts differently depending on the value of the area-to-mass ratio of the objects as visible from
Figure 6. This was already noted by McInnes[15], however here the real debris population is used

2755
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0
20
40
60
80
100
120
Semi-major axis - RE[km]
Inclination [deg]
0
50
100
150
200
250
Figure 3: Initial debris distribution in LEO in semi-major axis and inclination.
0 500 1000 1500 2000
0
0.5
1
1.5
2×10−2
Semi-major axis - RE[km]
Eccentricity
RB
0 500 1000 1500 2000
0
0.5
1
1.5
2×10−2
Semi-major axis - RE[km]
Eccentricity
PL
0 500 1000 1500 2000
0
0.5
1
1.5
2×10−2
Semi-major axis - RE[km]
Eccentricity
DEB
0 500 1000 1500 2000
0
0.5
1
1.5
2×10−2
Semi-major axis - RE[km]
Eccentricity
MRO
Figure 4: Initial debris distribution in LEO in semi-major axis and eccentricity, distinguishing
among the object types.

2756
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0
20
40
60
80
Altitude [km]
Density ×109[1/km3]
0
10
25
Figure 5: Debris population evolution over 25 years. Continuous line: analytical, dashed line:
numerical
0 500 1000 1500 2000
0
1
2
3
4
×10−9
Altitude [km]
Density ×109[1/km3]
A/M bin at 0.001 m2/kg
0 500 1000 1500 2000
0
1
2
3
4
×10−9
Altitude [km]
Density ×109[1/km3]
A/M bin at 0.23301 m2/kg
Figure 6: Debris population evolution over 25 years (analytical propagation). The evolution of two
different area-to-mass bins is shown.

2757
and the propagation is performed in aand e, not only in r. The method can also be applied to longer
term studied of 100 years as shown in Figures 7 and 8.
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0
20
40
60
80
Altitude [km]
Density ×109[1/km3]
0
100
Figure 7: Debris population evolution over 100 years. Continuous line: analytical, dashed line:
numerical.
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0
20
40
60
80
Altitude [km]
Density ×109[1/km3]
0
20
40
60
80
100
Figure 8: Debris population evolution over 100 years continuous line with analytical propagation.

2758
Effect of a fragmentation
A breakup on an 800 km Sun-synchronous orbit is now considered and the feedback of the event
on the whole debris population assessed. For the fragments distribution, the case of the fragmen-
tation of DMSP-F13 on 25 February 2015 is used a test case scenario. The object had an orbit of
perigee and apogee altitude of respectively 842 and 856 km and a mass of 830 kg. The proposed
method can be used for a quick estimation of the future development of space debris population also
under the effects of a fragmentation event. Figure 9 shows the spatial density of the space debris
at the moment the fragmentation takes place and a zoom on the altitude where the fragmentation
happens, where a distinct jump in spatial density can be recognised. The light blue line shows the
initial population (January 2013), the darker blue line the debris evolution without fragmentation
(in February 2015) and the red line adds up the fragmentation (in February 2015). The effect of
the fragmentation after 20 year since January 2013 can be seen in Figure 10, a density difference of
6.5% at the altitude of the fragmentation at time t+
fis diluted over time and in 2038 is decreased to
3.8% (visible in the peak at altitude of around 750 km). So, over time the effect of drag level out
the cloud peak to the envelope of the whole debris population.
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0
20
40
60
80
Altitude [km]
Density ×109[1/km3]
0
2.149 year - with fragmentation
2.149 year - without fragmentation
Figure 9: Debris population + DMSP-F13 fragmentation after 2.149 years.

2759
CONCLUSIONS
The modelling of the contribution of small debris fragments to the collision risk requires methods
that do not rely on the propagation of single objects; in this case, density-based models offer an
interesting alternative. A method based on the continuity equation, previously developed to describe
the evolution of the density of debris clouds produced by single fragmentations, is here extended to
the study of the global debris population. The results presented show the feasibility of this approach
for such applications with long term propagation. The accuracy of the method is demonstrated
over long time span of 100 years so it is suitable for environment evolution studies. The effect of
a fragmentation on the background population can be easily modelled though the superimposition
of the effects. Future work will be devoted to complete model the sources and sinks of the debris
population system and to measure the collision risk for spacecraft considering also background
population. Such an approach has a potential application to perform collision risk analysis for small
satellites.
ACKNOWLEDGEMENT
The authors acknowledge The European Space Agency for the permission to use the data for the
2013 debris population. The authors acknowledge the use of the IRIDIS High Performance Com-
puting Facility, and associated support services at the University of Southampton, in the completion
of this work.
REFERENCES
[1] J. Dolado Perez, B. Revelin, and R. Di Costanzo, “Introducing MEDEE,” Sixth European
Conference on Space Debris (L. Ouwehand, ed.), ESA Communications, Aug. 2013.
[2] V. Chobotov, Orbital mechanics. Reston: AIAA, 3rd ed., 2002.
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0
20
40
60
80
Altitude [km]
Density ×109[1/km3]
0
2.149 year - with fragmentation
2.149 year - without fragmentation
20 year - without fragmentation
20 year - with fragmentation
Figure 10: Debris population + DMSP-F13 fragmentation over 20 years from t0.

2760
[3] H. Sdunnus, P. Beltrami, H. Klinkrad, M. Matney, A. Nazarenko, and P. Wegener, “Compari-
son of debris flux models,” Advances in Space Research, Vol. 34, Jan. 2004, pp. 1000–1005,
10.1016/j.asr.2003.11.010.
[4] A. Rossi, A. Cordelli, C. Pardini, L. Anselmo, and P. Farinella, “Modelling the space debris
evolution: Two new computer codes,” Advances in the Astronautical Sciences - Space Flight
Mechanics, San Diego, Univelt, Apr. 1995, pp. 1217–1231.
[5] H. G. Lewis, A. E. White, R. Crowther, and H. Stokes, “Synergy of debris mitigation and re-
moval,” Acta Astronautica, Vol. 81, No. 1, 2012, pp. 62–68, 10.1016/j.actaastro.2012.06.012.
[6] J. Radtke, S. K. Flegel, V. Braun, C. Kebschull, C. Wiedemann, F. Sch¨
afer, M. Rudolph,
U. Johann, J. Utzmann, and M. H ¨
oss, “Fragmentation Consequence Analysis for LEO and
GEO Orbits,” GreenOps Workshop, Noordwijk, Nov. 2013. ESA AO 1/7121/12/F/MOS.
[7] J.-C. Liou, D. Hall, P. Krisko, and J. Opiela, “LEGEND–A three-dimensional LEO-to-GEO
debris evolutionary model,” Advances in Space Research, Vol. 34, No. 5, 2004, pp. 981–986.
[8] P. H. Krisko, “The predicted growth of the low-Earth orbit space debris environment an
assessment of future risk for spacecraft,” Proceedings of the Institution of Mechanical En-
gineers, Part G: Journal of Aerospace Engineering, Vol. 221, Jan. 2007, pp. 975–985,
10.1243/09544100JAERO192.
[9] L. Anselmo, A. Rossi, and C. Pardini, “Updated results on the long-term evolution of the
space debris environment,” Advances in Space Research, Vol. 23, Jan. 1999, pp. 201–211,
10.1016/S0273-1177(99)00005-8.
[10] A. Jenkin and R. Gick, “Dilution of disposal orbit collision risk for the medium Earth orbit
constellations,” Tech. Rep. 8506, The Aerospace Corporation, Los Angeles, 2005.
[11] A. Rossi, L. Anselmo, A. Cordelli, P. Farinella, and C. Pardini, “Modelling the evolution of the
space debris population,” Planetary and Space Science, Vol. 46, Nov. 1998, pp. 1583–1596,
10.1016/S0032-0633(98)00070-1.
[12] A. Nazarenko, “Modeling Technogenous Contamination of the Near-Earth Space,” Solar Sys-
tem Research, Vol. 36, No. 6, 2002, pp. 513–521, 10.1023/A:1022113421686.
[13] H. G. Lewis, G. G. Swinerd, R. J. Newland, and A. Saunders, “The fast debris
evolution model,” Advances in Space Research, Vol. 44, Sept. 2009, pp. 568–578,
10.1016/j.asr.2009.05.018.
[14] C. Kebschull, V. Braun, S. K. Flegel, J. Gelhaus, M. M ¨
ockel, J. Radtke, C. Wiedemann,
H. Krag, I. Carnelli, and P. Voersmann, “A Simplified Approach to Analyze the Space Debris
Evolution in the Low Earth Orbit,” 64th International Astronautical Congress, International
Astronautical Federation, Sept. 2013. IAC-13.A6.2.3.
[15] C. R. McInnes, “An analytical model for the catastrophic production of orbital debris,” ESA
Journal, Vol. 17, No. 4, 1993, pp. 293–305.
[16] F. Letizia, C. Colombo, and H. G. Lewis, “Multidimensional extension of the continu-
ity equation method for debris clouds evolution,” Advances in Space Research, 2015.
doi:10.1016/j.asr.2015.11.035. Accessed 8 December 2015, 10.1016/j.asr.2015.11.035.

2761
[17] F. Letizia, C. Colombo, and H. G. Lewis, “Collision probability due to space debris clouds
through a continuum approach,” Journal of Guidance, Control, and Dynamics, 2015. doi:
10.2514/1.G001382. Accessed 10 September 2015, 10.2514/1.G001382.
[18] A. Nazarenko, “The development of the statistical theory of a satellite ensemble motion and its
application to space debris modeling,” Second European Conference on Space Debris, 1997.
[19] N. Smirnov, A. Nazarenko, and A. Kiselev, “Modelling of the space debris evolution based on
continua mechanics,” Space Debris, 2001.
[20] F. Letizia, C. Colombo, and H. G. Lewis, “Analytical model for the propagation of small debris
objects clouds after fragmentations,” Journal of Guidance, Control, and Dynamics, Vol. 38,
No. 8, 2015, pp. 1478–1491, 10.2514/1.G000695.
[21] N. L. Johnson and P. H. Krisko, “NASA’s new breakup model of EVOLVE 4.0,” Advances in
Space Research, Vol. 28, No. 9, 2001, pp. 1377–1384, 10.1016/S0273-1177(01)00423-9.
[22] P. H. Krisko, “Proper Implementation of the 1998 NASA Breakup Model,” Orbital Debris
Quarterly News, Vol. 15, No. 4, 2011, pp. 1–10.
[23] F. Letizia, C. Colombo, and H. G. Lewis, “Multidimensional extension of the continu-
ity equation method for debris clouds evolution,” Advances in Space Research, 2015.
doi:10.1016/j.asr.2015.11.035. Accessed 8 December 2015, 10.1016/j.asr.2015.11.035.
[24] N. Gor’kavyi, “A new approach to dynamical evolution of interplanetary dust,” The Astrophys-
ical Journal, Vol. 474, No. 1, 1997, pp. 496–502, 10.1086/303440.
[25] N. Gor’kavyi, L. Ozernoy, J. Mather, and T. Taidakova, “Quasi-stationary states of dust flows
under Poynting-Robertson drag: New analytical and numerical solutions,” The Astrophysical
Journal, Vol. 488, No. 1, 1997, pp. 268–276, 10.1086/304702.
[26] C. R. McInnes, “Simple analytic model of the long term evolution of nanosatellite constel-
lations,” Journal of Guidance Control and Dynamics, Vol. 23, No. 2, 2000, pp. 332–338,
10.2514/2.4527.
[27] C. Colombo and C. R. McInnes, “Evolution of swarms of smart dust spacecraft,” New Trends
in Astrodynamics and Applications VI, New York, Courant Institute of Mathematical Sciences,
June 2011.
[28] N. Gor’kavyi and L. Ozernoy, “Quasi-stationary states of dust flows under poynting-robertson
drag: new analytical and numerical solutions,” The Astrophysical . . . , 1997, pp. 1–22,
10.1086/304702.
[29] M. Sykes, “Zodiacal dust bands: Their relation to asteroid families,” Icarus, Vol. 9, 1990.
[30] D. J. Kessler, “Collision probability at low altitudes resulting from elliptical orbits,” Advances
in Space Research, Vol. 10, No. 3, 1990, pp. 393–396.