Hazard of collisions in geostationary ring
A probability of catalogue satellites collision in the geostationary orbit is calculated. The direct method is applied: the dangerous rendezvouses of the satellites are determined, and a probability of collisions under the dangerous rendezvouses is calculated. A density of distribution of the uncont...
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Головна астрономічна обсерваторія НАН України
2005
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irk-123456789-796852015-04-04T03:02:27Z Hazard of collisions in geostationary ring Klimyk, V.U. Kizyun, L.M. MS4: Positional Astronomy and Global Geodynamics A probability of catalogue satellites collision in the geostationary orbit is calculated. The direct method is applied: the dangerous rendezvouses of the satellites are determined, and a probability of collisions under the dangerous rendezvouses is calculated. A density of distribution of the uncontrolled satellites coordinates and their orbital elements is obtained. A density of distribution of the directions and values of the relative velocities under dangerous rendezvouses of the geosynchronous satellites is also calculated. 2005 Article Hazard of collisions in geostationary ring / V.U. Klimyk, L.M. Kizyun // Кинематика и физика небесных тел. — 2005. — Т. 21, № 5-додаток. — С. 393-397. — Бібліогр.: 3 назв. — англ. 0233-7665 http://dspace.nbuv.gov.ua/handle/123456789/79685 en Кинематика и физика небесных тел Головна астрономічна обсерваторія НАН України |
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MS4: Positional Astronomy and Global Geodynamics MS4: Positional Astronomy and Global Geodynamics |
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MS4: Positional Astronomy and Global Geodynamics MS4: Positional Astronomy and Global Geodynamics Klimyk, V.U. Kizyun, L.M. Hazard of collisions in geostationary ring Кинематика и физика небесных тел |
| description |
A probability of catalogue satellites collision in the geostationary orbit is calculated. The direct method is applied: the dangerous rendezvouses of the satellites are determined, and a probability of collisions under the dangerous rendezvouses is calculated. A density of distribution of the uncontrolled satellites coordinates and their orbital elements is obtained. A density of distribution of the directions and values of the relative velocities under dangerous rendezvouses of the geosynchronous satellites is also calculated. |
| format |
Article |
| author |
Klimyk, V.U. Kizyun, L.M. |
| author_facet |
Klimyk, V.U. Kizyun, L.M. |
| author_sort |
Klimyk, V.U. |
| title |
Hazard of collisions in geostationary ring |
| title_short |
Hazard of collisions in geostationary ring |
| title_full |
Hazard of collisions in geostationary ring |
| title_fullStr |
Hazard of collisions in geostationary ring |
| title_full_unstemmed |
Hazard of collisions in geostationary ring |
| title_sort |
hazard of collisions in geostationary ring |
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Головна астрономічна обсерваторія НАН України |
| publishDate |
2005 |
| topic_facet |
MS4: Positional Astronomy and Global Geodynamics |
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http://dspace.nbuv.gov.ua/handle/123456789/79685 |
| citation_txt |
Hazard of collisions in geostationary ring / V.U. Klimyk, L.M. Kizyun // Кинематика и физика небесных тел. — 2005. — Т. 21, № 5-додаток. — С. 393-397. — Бібліогр.: 3 назв. — англ. |
| series |
Кинематика и физика небесных тел |
| work_keys_str_mv |
AT klimykvu hazardofcollisionsingeostationaryring AT kizyunlm hazardofcollisionsingeostationaryring |
| first_indexed |
2025-07-06T03:42:06Z |
| last_indexed |
2025-07-06T03:42:06Z |
| _version_ |
1836867467794710528 |
| fulltext |
HAZARD OF COLLISIONS IN GEOSTATIONARY RING
V. U. Klimyk1, L. M. Kizyun2
1Space Research Laboratory, Uzhhorod National University
2a Daleka Str., 88000 Uzhhorod, Ukraine
e-mail: space@univ.uzhgorod.ua
2Main Astronomical Observatory, NAS of Ukraine
27 Akademika Zabolotnoho Str., 03680 Kyiv, Ukraine
e-mail: kisyun@mao.kiev.ua
A probability of catalogue satellites collision in the geostationary orbit is calculated. The direct
method is applied: the dangerous rendezvouses of the satellites are determined, and a probability
of collisions under the dangerous rendezvouses is calculated. A density of distribution of the un-
controlled satellites coordinates and their orbital elements is obtained. A density of distribution
of the directions and values of the relative velocities under dangerous rendezvouses of the geosyn-
chronous satellites is also calculated.
DISTRIBUTION OF UNCONTROLLED OBJECTS IN GEOSTATIONARY RING
To obtain a distribution of uncontrolled geostationary objects (UGO) we used the catalogue of orbital ele-
ments [3]. The orbital elements at 0.01 day interval were obtained using the analytical theory of UGO mo-
tion [1], taking into account the Earth’s nonsphericity, the Moon and the Sun attraction. Based on the UGO
orbital elements the coordinates λ, ϕ, r0 (geographic longitude λ, latitude ϕ (or declination δ0), and geocentric
distance r0), determined for all UGO positions in the Earth coordinate system were calculated at 0.01 day time
interval. In a such way the instantaneous distribution of all UGO was obtained at 1990–2004 time interval.
Then the mean values of these coordinates at a long time interval were calculated. The accuracy of the UGO
calculating position is 1◦ at the 15 years time interval.
Figure 1. Density function of δ for UGO (1), nor-
mal density function (2) and ϕ density function (isolated
points)
Figure 2. Density function of r for UGO (1) and for
geostationary objects in a libration orbit (L) and density
of normal distribution (2)
The density function of declination δ calculated using the data obtained by above-mentioned method is
shown in Fig. 1 (the curve 1), δ = δ0 − δs, where δ0 is UGO declination, δs is its mean value (for Uzhhorod
latitude δs = −6.2◦). The function is calculated for Uzhhorod using the points (δi
mi
n · Δδ ), where Δδ is the length
of splitting interval along δ axis (Δδ = 1◦, that is about 736 km in orbit), δi is the central point of an i-interval,
c© V. U. Klimyk, L. M. Kizyun, 2004
393
mi is the number of δ values falling in i-interval, n is the number of all δ values. The density function of δ is
symmetric relatively δs, but this distribution is not Gaussian curve (dotted curve 2 in Fig. 1).
A distribution of UGO geographical latitude ϕ doesn’t almost differ from δ distribution. In Fig. 1 the value
of ϕ density function for UGO is submitted by isolated points.
The density function of r is shown in Fig. 2 (the curve 1), r = r0 − rs, where rs = 42164 km. The curve is
obtained using the points from 10 km splitting interval. The density function of r is not Gaussian curve too.
The objects in a libration orbit change the form of the distribution curve. Almost all satellites of this type are
situated at rs± 30 km interval. A density function of r for such UGO is shown in Fig. 2 (the curve L). PL axis
for this curve is on the right.
Figure 3. Density function for UGO subsatellite longi-
tude λ (1) and for geostationary objects in a drift or-
bit (2)
Figure 4. Distribution of relative velocities of two UGO
(1) and relative velocities of UGO and CGS (2) at dan-
gerous rendezvouses
The density function of UGO subsatellite longitude λ is shown in Fig. 3 (the curve 1). A splitting interval
is 10◦. 55 000 points were used to determine this distribution. The curve maximum is founded near the libration
point with a longitude λ = 75◦. It depends greatly on many objects of �1 type in a libration orbit among this
distribution. By dashed line in Fig. 3 (the curve 2) is shown a density function of subsatellite longitude only
for geostationary objects (GO) in a drift orbit. 60 000 points are used.
Let us consider a spatial (in the coordinate plane ϕOr) frequency function of distribution P (ϕ, r) for UGO
taken from the catalogue. This function may be presented as an implicit function by ellipse equation:
(736 · ϕ)2
[−29.83 · P (ϕ, r) · k + 10953]2
+
(r − 42164)2
[0.001659 · P (ϕ, r)2 · k2 − 0.8621 · P (ϕ, r) · k + 121.1]2
= 1, (1)
where the value P (ϕ, r) in denominator is the density function obtained as: P (ϕ, r) =
mϕr
Δϕ · Δr · n . Here mϕr
is the number of object positions along all zone of UGO in Δϕ · Δr splitting field (Δϕ · Δr = 5000 km2), n is
the total number of objects positions, using for calculations (n = 92207). In (1) k = Δϕ · Δr · n ≈ 46 · 107.
If k = 1, the value P (ϕ, r) transforms to the number of objects positions along all GO zone in Δϕ · Δr area.
The dependence of ellipse semi-major of axis and P (ϕ, r) in (1), is linear with a high correlation coefficient
ρ = −0.995. The dependence of ellipse semi-minor of axis and P (ϕ, r) was estimated by parabola with a range
of semi-minor of axis from 10 km up to 120 km. An implicit function (1) is convenient to use as an inverse
function: to set a probability P (ϕ, r) or one of ellipse semi-axes; to receive an ellipse of identical probability
P (ϕ, r), that is the coordinates ϕ, r with a probability of collision P (ϕ, r).
DISTRIBUTION OF UGO RELATIVE MOTION PARAMETERS
AT DANGEROUS RENDEZVOUSES
A research problem of GO collisions hazard is in fact the task of the calculation of the probability of their
collision at some time interval. The probability of collision and appropriate cumulative distribution function
of the Earth’s satellites at low heights (up to 3000 km) depending on different conditions and parameters
(altitude above the Earth, the satellite size, and the object geographical latitude and longitude) is rather deeply
394
investigated [2]. We used this direct method for investigation of the probability of UGO collisions with a some
simplification. For this aim all UGO positions at the certain moments of time with a rather small step Δt at
two-year interval were calculated. It is enough to select Δt = 0.01 of day as the UGO position in the Earth
coordinate system slowly changes.
For each instantaneous UGO distribution we fix the moments of dangerous rendezvouses every two objects.
Under the dangerous rendezvous we understand the rendezvous of two objects at a distance does not exceeding
100 km. The calculated rendezvouses parameters are transferred by appropriate program in a special archives.
These parameters are: the moment of minimum objects rendezvous, the distance between the objects in this
moment, the international number and type of two approaching objects, their coordinates (geographical lon-
gitude and latitude, geocentric distance) and the rates of their change. The distances between the objects at
the moment of rendezvous can be used only as statistical estimations, because the errors of orbits determination
exceed the UGO distances at dangerous rendezvouses.
To determine the parameters of UGO relative motion at the dangerous rendezvouses we used the moving
rectangular coordinate system and called it a local system. The origin of this coordinate system coincides with
O position one of the two approaching objects, for example with a first object. The axis z is directed from
the Earth’s center along a geocentric radius vector �r of the first UGO, the x axis is perpendicular to z and
directed towards the west, the y axis is perpendicular to the coordinate plane xOz and directed to North Pole
of the Earth.
Such coordinate system is connected with geographical coordinates λ, ϕ, and r by approximate formulas:
x = (λ − λ1) · r, y = (ϕ− ϕ1) · r, z = r − r1, where ϕ1, λ1, r1 are geographical coordinates and absolute value
of radius vector of the first object. Vector of the relative speed Δ�ν = �ν2 − �ν1 is a main motion parameter at
dangerous rendezvous.
Let us consider two scalar parameters: a value Δν of relative velocity and a slope angle ψ between the x
axis and the velocity direction Δ�r. The value of a slope angle is determined as:
ψ =
∣
∣
∣
∣
∣
arctan
√
(Δy)2 + (Δz)2
Δx
∣
∣
∣
∣
∣
, (2)
where Δx, Δy, Δz are the coordinates of the second object in a local coordinate system. It is obviously that
0◦ ≤ ψ ≤ 90◦.
Figure 5. Distribution of ψ angles for two UGO (1), for
UGO and CGS (2) at dangerous rendezvouses
Figure 6. Density function of the distances between two
UGO at dangerous rendezvouses (1) and between UGO
and CGS (2)
The distribution of the relative velocities Δν and the angles ψ at the dangerous rendezvouses of two UGO
using the archives of dangerous rendezvouses for 1990 is shown in Fig. 4 and Fig. 5 (the curve 1). N1 in Fig. 4
and Fig. 5 is the number of rendezvouses at 20 m/s velocity interval and at 5◦-angle interval, respectively. It is
seen that the values of relative velocities at dangerous rendezvouses are situated always in the limits mainly of
zero up to 300 m/s. It is more than half of all velocity values (Fig. 4, the curve 1) are taken place in the area
from 0 up to 120 m/s with a 20–80 m/s maximum. Some small maximums are near 160 m/s and 240 m/s.
The directions of the approaching UGO are almost parallel: the angles ψ between the directions of the velocity
395
vectors of two UGO in the most cases do not exceed 20◦ (Fig. 5). The distribution curve of ψ angles changes
smoothly.
PROBABILITY OF UGO COLLISIONS
The value of a density function P (r) of the minimum distances at dangerous rendezvouses of UGO is calculated
(Fig. 6, shown by circles) as P (r) =
m
n · Δr . For this aim the data of archives of dangerous rendezvouses
are used. Here Δr is the grouping interval, m is the number of dangerous UGO rendezvouses, the minimum
distances between which are situated in the given interval Δr, n is the number of all dangerous rendezvouses.
Figure 7. Change of probability of UGO collisions dur-
ing one day at the one year interval
Figure 8. Change of probability of collision CGS with
UGO during one day at 5.5 year interval beginning
from 1990
There are about 2500–3000 dangerous rendezvouses of two UGO. Using the two years archives data of
5753 dangerous rendezvouses, during one year the approximate frequency function of the distances distribution
between UGO at dangerous rendezvouses is calculated (curve 1 in Fig. 6) as:
P (r) = 2.30923 · 10−4 r − 4.099228 · 10−7 r2. (3)
The mean GO size according to catalogue of 1996 is 8.15 m. Rms residual is about ±8.30 m.
We received in our calculations that the mean size of GO is 8 m. A probability of UGO collisions at dangerous
rendezvouses presented in the catalogue is Pz =
∫ 0.008
0 P (r) dr = 7.5 · 10−9, where the frequency function from
(3) is used. Under collision we understand such rendezvous of two UGO, that the distance between their centers
is less than 8 m.
In Fig. 7 the change of probability of UGO collisions during one day at one year interval is shown. Two curves
show the change of probability of collision for 1990 and 1997. The probability was calculated using the mean
values of number dangerous rendezvouses during 10 days. The mean probability of collision has increased during
one year from 2.0 · 10−5 in 1990 up to 2.5 · 10−5 in 1997. The cause of such change is the increasing of UGO
number in 1997.
In Table 1 a frequency P =
m
n
of participation of different types UGO in dangerous rendezvouses is
submitted. The archives data for 1990 are used. Here m is the number of UGO selected type from the archives
of dangerous rendezvouses, n is the number of all UGO from archives. It is seen that UGO in a drift orbit is
much more often approach at dangerous distances than UGO in a libration orbit. The reason of this is much
more number of UGO in a drift orbit in catalogue.
Let us give an estimation of frequency of UGO rendezvouses, which does not depend on number of selected
type objects in the catalogue. Enter the value Q =
P
P ′ , where P ′ =
r
k
is the frequency of selected type UGO
in catalogue (k is the number of all UGO in catalogue, r is a number of selected type UGO in catalogue). It is
seen from Table 1 that for UGO of different types Q is near upon a size, that is the participation of UGO in
rendezvouses is a little depends on objects type. Using the archives data one may be seen that the rendezvouses
of different types objects are taken place the most frequency (with frequency 0.87). Only 0.13 of all cases are
the objects of the same type.
396
Table 1. Distribution a number of rendezvouses of UGO different types using the data of dangerous rendezvouses
archives for 1990
UGO type �1 �2 �3 d1 d2 d3
Frequency P = m/n in archives 0.134 0.056 0.018 0.407 0.191 0.193
The ratio of frequencies P/P ′ 0.90 0.96 0.73 1.17 0.72 1.25
HAZARD OF COLLISION OF UNCONTROLLED OBJECTS WITH CONTROLLED SATELLITES
On finishing its resource a controlled geostationary satellite (CGS) is transferred outside CGS zone to geosyn-
chronous orbit and goes as UGO. Therefore, the hazard of UGO collision with CGS is considerably less than
in the case of two UGO rendezvouses. About 90 dangerous rendezvouses UGO with CGS is taken place during
one year; it is equal only about 3 per cent of all dangerous UGO rendezvouses.
The density function of distances between UGO and CGS at dangerous rendezvouses is shown in Fig. 6
(the curve 2). It is obtained from the least squares adjustment by points P (r) =
m
n · Δr (in Fig. 6 is designed
by crosses). The grouping interval is Δr = 5 m, m is the number of dangerous rendezvouses of UGO with CGS
a minimum distance between which is situated in the selected interval Δr, n is the number of all dangerous
rendezvouses UGO with CGS. Density function (the curve 2 in Fig. 6) may be presented as polynomial:
P ′(r) = 1.01003 · 10−4 r + 15.858612 · 10−7 r2. (4)
The curves 1 and 2 in Fig. 6 and so the functions (3) and (4) is differ so, that at the same number of dangerous
rendezvouses the probability of two UGO collision is much more than probability of collision UGO with CGS.
Really, P ′
z =
∫ 0.008
0 P ′(r) dr = 3.2 · 10−9, where P ′(r) is a function (4). The probability of collision UGO
with CGS at dangerous rendezvous P ′
z is 2.34 times less than probability Pz in (3). As a result it is obtained
that the probability of UGO collision with CGS in geosynchronous orbit is 75 time less a probability of two
UGO collision.
The change of probability of UGO collision with CGS during one day at 5.5 year interval is shown in Fig. 8.
The probability was calculated as a mean value at 50 day interval. The change of probability is very nonuniform.
The next UGO are the most dangerous: 78113D(d1) (121 rendezvouses with CGS during 1990–1995),
67066G(d3) (45 rendezvouses for six years), 86007A(d1) (27 rendezvouses), 87109A(d1) (27 rendezvouses),
75011F(d1) (25 rendezvouses), 77034C(d1) (23 rendezvouses), 70055A(d3) (19 rendezvouses). Only 121 uncon-
trolled geostationary objects can approach at dangerous distance to CGS.
The origin of local rectangular coordinate system x = (λ − λ1) · r, y = (ϕ − ϕ1) · r, z = r − r1 is placed
in the center of CGS. The slope angle ψ between the directions of UGO velocities and CGS ones at dangerous
rendezvous is calculated as (2).
In Figs. 4 and 5 (curve 2, with axes for them on the right) the distributions of relative velocities of Δν and
angles ψ is shown using the data of archives dangerous rendezvouses of UGO with CGS for 1990–1995. N2 is
the number of rendezvouses at 10 m/s velocity interval in Fig. 4 , N2 is the number of rendezvouses at 2◦ angle
interval in Fig. 5. It is seen from Fig. 4 that the values of relative velocities at dangerous rendezvouses are
situated within the limits of zero up to 300 m/s as well as in case of two UGO rendezvouses. A distribution is
very nonuniform. The largest maximum are for velocities: 200 m/s, 260 m/s, 45 m/s.
The ψ angles between velocities directions (Fig. 5) are situated mainly at the same area, as for two UGO
rendezvouses and do not exceed 20◦. But the distributions are very different. The maximum of curve (2) is in
the point of ψ = 8◦. Sometimes, ψ = 50◦ and may be higher.
[1] Bagrov A. V., Erpyliov N. P., Mikisha A. M., et al. Experimental geodynamic: observations of geostationary
satellites // Sci. Inform.–1991.–Issue 69.–P. 35–51.
[2] Khutorovsky Z. N., Kamensky S. Yu., Bojkob V. F., Smelov V. L. Risk of collisions of the space objects at low
height // Collisions in near Earth space (space debris).–Kosmoinform, IA RAN, 1995.–P. 19–91.
[3] Sochilina A. S., Kiladze R. I., Grigoriev K. V., Vershkov A. N. Catalogue of orbits of geostationary satellites.–
St.-Petersburg: ITA RAN, 1996.–104 p.
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