027T1177190
$0.00+ .50
CopyrightO 1989COSPAR
1990
Adv. SpaceRes.Vol. 10,No. H, pp. (3)393-(3)396,
Printedin GreatBritain.All rishtsreserved.
COLLISION PROBABILITY AT LOW
ALTITUDES RESULTING FROM ELLIPTICAL
ORBITS
DonaldJ. Kessler
NASAlJohnson SpuceCenter, Houston, TX 77058, U.S.A.
ABSTRACT
probability
between
a spacecraft
and another
object
in Earth
The collision
orbit
can
perigee,
be expressed
as a function
of the orbital
apogee,
and inclination
of the object.
rr^,.^1 1,,
*L^
LLrs y,obability
is not a sensitive
function
of inclination.
v-ualrJ,
Collision
can only
when the
occur
spacecraft
is
located
at
an altitude
which
between
perigee
is
the
and
per unit
apogee altitudes
of the object.
The probability
time is largest
when the perigee
(i.e.,
are
and apogee
nearly
equal
the
orbit
is
nearly
circular).
Therefore,
it
is
usually
concluded
that
objects
in eircular
orbits
represent
the greatest
hazard
to other
altitudes,
atmospheric
drag
perigee
spacecraft.
However,
at
low
causes
and apogee to
change with
cime,
such that
circular
orbits
have a much shorter
lifetime
than many of the
which
pass
orbits
through
the
lower
altitudes.
Consequently,
when the
elliptical
probability
is
integrated
over
the
lifetime
of
the
orbiting
object,
some
collision
probability
to
have
a much higher
total
collision
than
elliptical
orbits
are
found
could represent
orbits.
Objects
in these elliptical
orbits
the greater
source of
circular
hazardous
objects
to
spacecraft
operating
in
1ow Earth
orbit.
Some common objects
in
payloads
are rocket
bodies
used to boost
from low Earth
orbit
to
these
elliptical
orbits
geosynchronous
orbit.
INTRODUCTION
probabilities
between orbiting
objects
when calculating
collision
A common assumption
the
circular
orbits.
This
assumption
is
common because
that
the
objects
are
in
is
most
orbiting
are in
near
circular
calculation
is
easier,
and in
Earth
orbit,
objects
of
their
time
at a particular
alticude,
their
Since
circular
orbits
spend all
orbits.
probability
that
altitude
is
much higher
than
an
to
the
collision
at
contribution
passing
through
the same altitude.
elliptical
orbit
short;
orbital
lifetimes
are relatively
in Earth
orbit,
at
low altitudes
However,
probability,
a constant
source
of
or
flux,
collision
a constant
therefore,
to maintain
or it
can
into
these altitudes,
source can be from new launches
is required.
This
objects
Because
this
altitudes.
down from
higher
dragging
of
older
objects
result
be as the
from the source
resulting
of the objects
lifetime
the orbital
source
is required,
constant
can
altitude
which
pass through
a particular
elliptical
orbits
Since
important.
is also
these
at
that
altitude,
orbits
than
circular
lifetime
orbital
a much
longer
have
flux.
a particular
to maintain
a smaller
source of objects
orbits
may require
elliptical
probabilities
the cc.rllision
used to calculate
paper will
discuss
the equations
This
to explain
of objects
production
rate
orbits,
and the required
from elliptical
resulting
and the STS-7 window
Solar-Max
surfaces
of
the returned
analysis
from
measured
the flux
'-"+
Tl l L
^^^
^-cduction
the
if
especially
and reasonable,
are
shown to be small
rates
l s ) s
P ! (
P r L .
source
is
in
certain
elliptical
orbits.
CALCULATING COLLISION PROBAB]LITIES
density
the spatial
calculating
by first
probabilities
can be calculated
Collision
flux
and collision
to calculate
density
that
spatial
then using
from an object,
resulting
orbital
with
an object
of finding
S, is the probability
density,
probabililuy
Spatial
/I/.
R
element
located
a volume
g,
and apogee q'within
perigee
i,
of
inclination
elements
by
B, and is given
and at latitude
of the Earth,
from the center
distance
S(R,B):
where
s(R)
is
the
spatial
density
averaged
(1)
s(R)f(B)
over
all
latitudes,
and
given
by
1
(2)
s(R) :
2 z r 2 R ( q + q 't) ( R - q ) ( q ' - R ) l t / '
(3)3e3
(3)3e4
D. J. Kessler
and
f(B)
latitudes,
is
the ratio
and given
of
by
che
spatial
density
at
B to
the
spatial
density
averaged
over
all
2
f(B) :
(3)
r ( s i n 2 i - " L n B ) ' /,
The orbital
inclination
of objects in low Earth orbit varies from near zero to 145
degrees.
This distribution
of inclination
produces a spatial
density distribution
with
latitude
which is nearly
constant
(within
a factor
of 2).
Consequently,
for most
calculations
the approximation that f(B):1 is appropriate,
and will be used here.
Flux, I', or the number of impacts per unit cross-sectional
area per unit time, is
given by
F:
where v
collision
is
the
relative
probability,
if
Sv
t/,\
collision
velocity.
N is small)
is given
The
by
average
number
of
impacts,
N,
N : Jraar
where
Ar
A
is
the
collision
souRcE STRENGTHS
Assume that
at some distance
atmospheric
drag
increment
of time
frorn the definition
cross-sectional
area
(or
the
(5)
exposed
to
the
flux
for
an
increment
of
time
REQUIRED FROM CTRCUI/,R ORBITS
G particles
per unit
time are produced. from an object
in circular
orbit,
above R.
Assume also
that
as the orbits
of these
particles
decay from
and pass
through
R, remaining
in
the
increment
of
altitude
dR for
an
dt,
and an equilibrium
spatial
density,
S, is established
at R.
Then
of G and S
G :
4zrRzSdR/dr
:
4nRzFvo/v
(6)
vo - dR/dt,
or the rate
of change of the orbital
semi-major
axis
due to atmospheric
and F is the measured flux
at R from particles
having
relative
velocity
v.
As an application
of this
equation,
analysis
of Solar-Max
louver
surfaces
gives
an
orbital
debris
flux
of 1X10-6 gm (about
0.1 mm diameter)
debris
of about 0.3/m2-yr.
Th"""
impacts
were
predominancly
paint
from
particles
The average
cross-section
of
/2/.
a
randomly
tumbling
surface
is I/4
its
surface
area,
so that
F in equation
5 is L.Z/mz-yr.
At 500 km altitude
(R : 6878 km),
and during
average
solar
activity,
a 1X10-6 gm parritle
will
decay
at
a rate
of
about
26 kn/day.
Assuming
an average
collision
vellcity
of g
km/sec.
gives
a required
production
rate
of
2.4X1010 particles
per
year,
or
about
24
kgm/yeat
of
0.1
mm particles.
other
size
impacts
were
also
measured.
The
st-ze
distribution
of
these
impacts
would
suggest
that
a slightly
larger
production
rate
is
required
at
smaller
sizes.
Over the
entire
size
distribution
measured
by Solar-Max,
a
particle
production
raLe,
assuming
circular
orbits,
of about
100 kgm/yr
is required.
The
effects
of
atomic
oxygen
on several
hundred
painted
just
spacecraft
above
Solar-Max
altitude
would give
this
this
production
rate.
where
drag,
SOURCE STRENGTH REQUIRED FROM ELLIPTICAL
ORBITS
T1^ ^
rrrE dvErd6e number of collisions
from an object
by combining
equations
4 and 5.
Since
collision
inclination,
v can be
taken
out
of
the
integral,
Consequently,
for elliptical
orbits,
in an elliptical
orbit
can be found
velocity
is
mostly
a function
of
along
with
the
collision
area A.
N, : evJsdt
(1)
The value
of
S is
a function
of
time
since
perigee
both
and apogee wiII
vary
due to
atmospheric
decay;
when the
apogee
is
above
about
5000 km altitude,
lunar
and solar
perturbations
will
also
cause the perigee
and apogee to vary.
Numerical
techniques
must
be used to determine
how these
orbital
parameters
change with
time.
The same program
used tn /3/,
which
was developed
by Alan Mueller,
will
also
be used here.
This
progt"t
combines gravitational
perturbations
with
equations
developed
by King-HeLe
/4/.
The average
number of collisions
from an object
in a circular
orbit
can be expressed
the same way, except
that
a singularity
occurs
when R
q:
q'.
This
singularity
is
integrable,
so that
the collision
frequency
for
a circular
orbit
is given by
N" :
The ratio
N"/N" is then the
source
required
from
circular
given
by
ratio
orbits
Av/(4rRzva)
of the source
to maintain
w : Jsar (4zrR2v6)
(8)
required
a given
from
f1ux.
elliptical
orbits
This
ratio,
W,
to the
is
then
(e)
(3)3e5
Collisions from Elliptical Orbits
the decay
equation
9 using
integrating
by numerical
of W was determined
The value
of time for
as a function
and apogee varied
1 shows how the perigee
Figure
model Ln /3/.
was 400 km, and initial
perigee
was 1 cm, initial
diameter
the particle
the case where
that
the amount of time
Note that
shown.
is also
orbit
A circular
apogee was 4000 km.
compared to
or 15 years,
6000 days,
400 km is about
through
passes
orbit
the elliptical
of
to drop more than 10 km, and have no possibility
orbit
a circular
than 10 days for
less
at 400 km.
an object
with
collision
2 and 3 represent
Figures
2 through
7.
in figures
of W is given
value
The calculated
aPogee
of
interest;
altitude
the
to
equal
is
height
perigee
initial
the
where
cases
and hence
gravitational
5000 km, which means that
Perturbations,
below
were kept
heights
for W.
values
give
the largest
of conditions
These types
are not important.
inclination,
in
more effective
2 and 3, apogees above 2000 km are more than 10 times
As seen in figures
a
v/ou1d require
measurements
The Solar-Max
orbits.
circular
than
" gi.rr.rt flux
producing
of
type
in
this
were
source
if
the
10 kgn/yr
than
less
of
rate
produccion
particle
500 km and
perigees
near
with
orbits
in elliptical
spacecraft
However,
orbit.
Lttipti""l
in
can be found
fragments
explosion
common, although
2000 km are not
than
larger
"pogl."
of orbits.
these types
25
r0 0 0 0
' ' - l
I
t-#
E
v
APOGEE HEIGHT
I
'\..
\\ i l
I
F
1 0 0 0r
f
H
l
\
r00 U
Fig. 1.
15
,/ru*r"uts
H
F
= 4oo KM
l0
&
HT
P E R I G E E H E I G-*-----1
i
20
3
t
C I R C U L A RO R B I T
-.ioo
1000
10
TIME (DAYS)
10000
for 1 cm
Decay profile
d i a . , 2 g m / c m zP a r t i c l e :
and circular
Elllptical
orbit.
0
r000
2000
3000
4000
5000
A P O G E EH E I G H T ( K ] ' 1 )
F i g . 2.
Required production rate
to maintain a given flux
400 km altitude
at
to a source in
ratioed
orbit.
circular
geosynchronous
into
payloads
from placing
results
orbit
A more common elliptical
with
a
in an orbit
(IUS or PAM) is left
stage
an upper
is used,
When the Shurtle
orbit.
of 28.5 degrees.
i
n
c
l
i
n
a
t
i
o
n
a
n
d
k
m
,
3
5
0
0
0
n
e
a
r
a
p
o
g
e
e
a
n
a
l
t
i
t
u
d
e
,
3
0
0
k
m
near
perigee
near 200 km, apogee near 36000,
a perigee
with
in an orbit
3d stage
its
leaves
The Ariane
of
types
these
sun angle,
initial
on the
Depending
7 degrees.
of about
and inclination
spacecraft
for
particles
of
source
significant
a potentially
represent
can
orbits
1 .
4 throu1|'
as seen in figures
500 km altitude
below
operating
L J
I
i
l
201
3
|
1 5 .
o
F
l n
DEPENDS ON
i
INITIAL
SUN ANGLE
5 1
'
0 '
0
1000 2000
3000 4000 5c00
0
J
rate
production
Required
flux
a given
to maintain
altitude
km
500
at
in
a source
to
ratioed
orbit.
circular
20000
30000
APOGEE HEIGHT (KM)
APOGEE HEIGHT (KM)
Irg.
r0000
Fig. 4.
rate
production
Required
flux
a given
to maintain
altitude
km
400
at
in
to
a source
ratioed
orbit.
circular
+oooo
(3)3e6
D. J. Kessler
50
P E R I G E E= 2 0 0 K M
= 7"
INCLINATTON
3
r
r 3 0
DEPE
N D S ON I N I T I A L
S U N AN G L E
\l
H
H
t 2 0
F
d
.\
<i
0
---
l0
suN ANGLEI
---- --]-=_*
,
r0000
20000
30000
A P O G E EH E I G H T ( K M )
Fig. 5.
,tI
- --<{;rtAL
=
7"
Fig. 6.
.l
,/
/l
DEPENDS
O/rllrrreri
I
/SUN
ts
l
ANGLE]
AVERAGEDOV{P. Nr
,..,.5UN ANGLE
./'_/
0
10000
20000
APOGEE HEIGHT
Fig.
l.
30000
zoooo
30000
+ o o0 0
APOGEE HEIGHT (KM)
= 2oo *-----n
PERTcEE
l0
roooo
40000
Required
production
rate
to maintain
a given
flux
at
400
km
altitude
ratioed
to
a source
in
circular
orbit.
INCLINATION
A V E R A G E DO V E R
-ALL SUN ANGLEI
-._.-l__
..'.1
. 5
-fnpni.us oll
B
INCLINATI
B
I
l
40
i
I
PERIGEE =
40000
(KM)
Required production rate
to maintain a given flux
at
300 km alrirude
ratioed
to a source in
circular
orbit.
CONCLUSIONS
At low altitudes,
objects
in elliptical
of
orbital
debris
than
objects
in circular
stages
currently
being
left
in
orbit
after
could
represent
the dominant
source of debris
Required
production
rate
to maintain
a given
flux
at
300
km
alritude
ratioed
to
a source
in
circular
orbit.
The initial
(an angle
sun angle
which
is
a
function
of
the
position
of
the
sun
and
orientation
of the orbital
plane
at the time
of
launch)
determines
whether
solar
perturbations
will
increase
or
decrease
the
orbital
lifetime
of
elliprical
orbirs
with
apogees
greater
rhan
5000 km altitude.
As
seen in figure
6, at 300 km, some sun angles
can be more than
50 times
more efficient
in
maintaining
a
given
flux
than
circular
orbits.
Since
the value
of v,
is
about
50
times
larger
ar
300 km than
"i
500 km, rhe
same flux
measured
on Solar-Max
at
500 km
could
be maintained
at
300 km from
a source
of about
L00 kgn/yr,
if
the
sources
lrere
in
elliptical
orbits
which
gives
rhese
high
values
for
W.
This
could
explain
the STS-7
window
pit
Thar
is,
rhe
flux
/5/.
represented
by
this
impact
on
the
STS-7
window
at
300 km would
require
a very
large
source
of
O.2 mm paint
particles,
if
the
impact
originated
from an object
in circular
orbit.
However,
a much more
reasonable
source
is
required
if
the
impact
originated
from an object
in an elliptical
orbit
having
a high value
of W.
orbits
can represent
a more important
source
orbits.
The long
orbital
lifetime
of upper
placing
payloads
into
geosynchronous
orbit
at altitudes
below 500 km.
ACKNOWLEDGEMENT
I would
like
to thank
Ron Madler
from the University
of Colorado
at
assistance
in setting
up and performing
the necessary
integration,
using
drag program,
to produce
the plots
used in the texc.
Boulder
for
his
the atmospheric
REFERENCES
1.
Kessler,
D.J.,
Derivation
of the collision
probability
between orbiting
objects:
The
lifetimes
of Jupiter's
outer moons, Icarus,
Vol.
48,1981,
pp 39-4g.
2.
Barrect,
R.A.,
Bernhard,
R.P.,
and McKay, D.S.,
Impact holes
and impact
flux
on
returned
Solar
Max louver
material,
Lunar and Planetary
Science XIX,
1988, pp 39-40.
3.
Su, S.-Y.,
and Kessler,
D.J.
Contribution
of explosion
and future
collision
fragments
to the orbital
debris
environment,
Adv. Space Res., Vol 5, No. 2,1985,
pp 25-34.
4.
King-Hele,D.G.,
Methods
predicting
for
satellite
orbital
lifetimes,
Journal
of rhe
British
Interplanetary
Society,
Vol.
31, 19j8, pp 181-L96.
5.
Kessler,
D.J. Orbital
debris issues, Adv. Space Res., Vol 5, No. 2,1985,
pp 3-10.