Mathematical Modelling of Gravitational Control of
Flights of Meteorites by Artificial Satellites
A.S. Alekseev and Yu.A. Vedernikov
Institute of Computational Mathematics and Mathematical Geophysics
Novosibirsk, Russia
Abstract
Among the methods proposed for anti-asteroid protection of the Earth, gravitational
control of the flights of meteorites by a group of artificial satellites is the most secure
and environmentally safe. In this method, a threatening asteroid or meteorite is escorted by a system of artificial satellites controlled from the Earth. Calculations have
shown that five such satellites forming a chain can shift considerably the center of the
asteroid mass and thereby change its trajectory. For multiple corrections of the flight
of an asteroid, several “parades” of its artificial satellites are necessary.
Introduction
We consider three techniques of a controlled deviation of the dangerous space body
path from the original path. These techniques are environmentally clean and safe in
contrast with the technique based on using the nuclear explosion devices (NEDs).
Figure 1: The critical cone of the dangerous space body.
Consider a dangerous meteorite moving towards the Earth (see Figure 1). Let us draw
a line connecting the mass center of the meteorite with the center of the Earth. We
now construct a right circular cone whose symmetry axis coincides with the above
line, the apex lies in the center of mass of the meteorite, and the base diameter coincides with the Earth diameter. Denote by αc the semi-apex angle of the cone. If
the flight trajectory of the meteorite is inclined to the symmetry line of the cone by
an angle α0 < αc , then the meteorite will inevitably collide with the Earth. Thus,
in order to avoid the meteorite impact with the Earth we can formulate the following
problem: deviate the trajectory of the dangerous meteorite by an angle α > αc . As we
will see below, there are at least three techniques for the solution of this task, which
make no use of nuclear explosion devices.
The satellite technique of asteroid deflection
In this technique, several artificial satellites are delivered to the neighborhood of a
dangerous space body. The x- and y-coordinates of the center of mass (inertia) of the
mechanical system “the space body + the satellites” are determined by well-known
formulas
1 Ns
1 Ns
xc =
mk xk , yc =
∑
∑ mk yk ,
MΣ k=1
MΣ k=1
where Ns is the number of artificial satellites, MΣ is the total mass of the system, mk
is the mass of the kth satellite located at the distances xk and yk from the center of
mass, see Figure 2.
x
m3
m2
m1
y
The center
The of mass Σ
asteroid
x1
x2 x3
x4
m4
xc
The artificial
satellites
Figure 2: The satellite technique of the asteroid deflection.
Let us now make some estimations involving the masses of the Earth, moon, and the
asteroids. The moon mass amounts to 1/10 of the Earth mass, and 40 thousand of
asteroids observed in telescopes amount to 1/1000 of the mass of our planet, which
equals 6 · 1023 kg. We now take the same aspect ratio of the masses of the satellite and
of the asteroid as for the asteroids belt and the Earth and calculate the deviation of the
center of mass of the linear system (see Figure 2) with ten satellites for the distances
xk = 10kD, where D is the asteroid diameter. In this case the value of xc measured
from the asteroid center equals 0.54D, that is, it goes beyond the asteroid perimeter.
In the case of four satellites the position of the center of mass xc of the system equals
approximately 0.1D.
Retaining the ratio of the masses of satellites and the asteroid let us now halve the
distances xk . As a result, the value of xc also nearly halves in the case of ten satellites.
For four satellites, the position of the center of masses (inertia) becomes to be equal
to 0.05D. If we follow the way of a ten-fold reduction of the satellite masses, then in
the case of distances xk = 10kD the position of the center of masses of the system of
ten satellites amounts to xc = 0.06D, and in the case of four satellites it amounts to
xc ≈ 0.001D.
With regard for the weight and dimensional constraints it appears to be hardly possible to deliver an artificial satellite with the mass of more than 1,000 kg. Then there
remains the only possibility for realization of the given elegant method, which consists of the use of natural satellites near the dangerous asteroid. 10 % of small planets
possess their satellites of natural origin [1]. It is advisable to augment right these
natural satellites with a control system consisting of several artificial satellites.
While distributing the “parade” of artificial satellites in the xOy plane one must take
into account the shift of the center of masses (inertia) along the Oy axis and account
for its averaged value. This relatively simple mathematical operation is advisable
in the cases of multiple “parades” of the asteroid satellites, which are necessary for
controlling its flight in the near-Earth space. The third coordinate z is used in the
regime of the verification of the process of the deviation of the dangerous asteroid
from the Earth.
Considerable efforts of the mathematicians will of course be needed to ensure the
delivery of satellites to the space body to be deflected and to position the satellite
orbits in the same plane. This is, however, already the scope of the conventional
rocket dynamics, in which the Newton’s law of universal gravitation is widely used
[2, 3, 4].
Thus, besides the constraints for the weight of the relatively small asteroid to be deflected it is necessary to apply the gravitational flight control in the presence of natural satellites near the celestial body. Only in such case it is possible to guarantee the
realization of environmental and technical advantages of the proposed gravitational
method in the simplest case.
Asteroid deflection with the aid of gravitational riders
This technique for the deflection of dangerous asteroids was first proposed by [3].
It may be considered as a development of the foregoing technique and it eliminates
some of the physical shortcomings of the above presented first technique.
Figure 3: The forces acting on
the gravitational rider.
Figure 4: The motion diagram of the
satellite.
Consider an artificial satellite in the form of Indian clubs (Figure 3) with two equal
spheres each of which has the mass m/2. We will neglect the mass of the connecting
rope (the bar). Let the rope be perpendicular to the line connecting the Earth center
with the rope center, and let r be the distance between the center of satellites
√ masses
and the Earth center, l be the half-length of the Indian clubs. Then R = l 2 + r2 is
the distance of a sphere from the Earth.
The Newton’s force
¯
¯
Mm ¯¯ ∂UN ¯¯
FN = 2 = ¯
2R
∂R ¯
acts on each sphere, where UN is the integral force function of each sphere. Each of
the forces F1 and F2 is directed to the Earth center. The total force is determined with
the aid of the parallelogram of forces (Figure 3) and is equal to
¯ ¯
¯ ∂U ¯ Mm
1
F = ¯¯ ¯¯ = 2 ·
.
∂r
r
(1 + d 2 )3/2
It is seen that the total force F is smaller than the Newton’s force FN . The effect of the
body extension introduces an additional pushing force. Although this force is nearly
vanishing for small satellites, it is available. This effect was proposed to be the basis
for a resonance maneuvering in space [3].
The motion takes place along a planar orbit and is determined by the integrals of areas
and energy:
V 2 U(r)
dϕ
= c,
−
= h.
r2
dt
2
m
Introducing the osculatory focal parameter p(t) and the osculatory eccentricity of the
orbit e(t) we can present the radius vector r and the velocity V of the osculatory orbit
as
s
p
r=
,
1 + e cos ν
V=
M
(1 + e2 + 2e cos ν) .
p
Here ν is the true anomaly in perturbed motion. By virtue of the constancy of p the
area and energy integrals may be written as
·
¸
2p M 1
2
P = P0 ;
e +
− U(r) = h
M r
m
where h is a new constant quantity. With regard for the fact that the integral force
function is equal to
Mm
Mm
U=√
= √
,
2
2
e +r
r 1 + α2
where α = e/r, the last expression may be rewritten as
e2 +
2p
α2
√
·
=h
2
r 1 + α + 1 + α2
The actual motion can, however occur in some part of the plane (e2 , r = r/p) rather
than in the entire plane:
³p
´2
e2 ≥
− 1 ≡ e2∗ (r),
r
where
³p
´2
e2∗ (r) =
−1 .
r
The diagram of the actual motion of the pulsating Indian clubs is presented in Figure
4 for fixed p and h. The satellite moves in a bounded neighborhood of the Earth so
that it does not move away from the Earth and retains a similarity with the Kepler’s
ellipse. When the satellite is at the least distance from the Earth its eccentricity is
minimal, and at a point farthest from the Earth the eccentricity is maximum. These
additions to the forces may be taken into account in a refined method of gravitational
rider with inert spheres (the upper half of Figure 5).
An incorporation of the formation of ball lightnings represents a very exotic variant
of the gravitational technique of the asteroid flight control, see the lower half of Figure 5. These artificial ball lightnings can be generated as high-temperature plasmoids
by a micro-plasma galvano setup “Bayasite” [5]. Under the appropriate voltage and
current it is possible to generate a ball lightning including a group of micro-plasmoids
from the aluminum cathodes. After an additional improvement the above experimental scheme is delivered to the orbit of a dangerous space object and forms several ball
lightnings. This set of lightnings is shifted in several stages along spiral curves to the
center of masses (inertia) of the mechanical system “asteroid + satellites”.
Upon achieving a certain energy concentration the cloud of ball lightnings explodes
and ensures a shock deflection of the asteroid. Thus, the controlling effect of the gravitational technique can be enhanced in this way. This may also lead to an alteration
of the pressure center.
The conventional
gravitational
riders
The
asteroid
The
gravitational
riders
with ball lightnings
Figure 5: The gravitational technique of the asteroid deflection.
The shift of the asteroid pressure center by sun sails
The sun wind represents the stream of particles and fine dust. The solar radiation
pressure has a vanishingly small value at the Earth’s orbit: p = 4.5 · 10−6 Pa [3]. If
the spacecraft is, however, supplied with a sail, which is sufficiently light and has
a large size, then the total force of the sun wind pressure on the sail may give the
spacecraft an acceleration necessary for the maneuvering in the solar system. This
can result in an alteration of the center of pressure developed by the sun parachute
similar to the Ural prototype shown in Figure 6.
The testing of “sun sails” intended for transport operations in the near-Earth space
and for interplanetary flights can be carried out both on a ballistic trajectory and in
the satellite orbit (Figure 7).
To shift the pressure center of the asteroid the parachutes manufactured of special
texture are delivered together with the satellites by a single carrier rocket or by a
single artillery missile, see Figure 8.
While performing the experiment in a near-Earth orbit in 1999 the carrier rocket
“Volna” (the “Wave”) has brought onto the ballistic flight trajectory an orbital acceleration booster with spacecraft, which accommodated a small-size rocket engine.
The engine has come into operation at the time when the ballistic trajectory apogee
was reached, thus providing entry onto a circular orbit. The “sun sail” was then set
up, and a further flight of the spacecraft continued.
Gritzner [6] has proposed to enhance the “sun wind” effect with the aid of hemispherical reflectors rotating near the controlled asteroid. These reflectors may be used to
redirect the laser beams emitted from the earth.
Figure 6: Accommodation of “sun sail” spacecraft on the missile: 1 3rd stage of missile; 2
apogee rocket engine; 3 adapter with spinning system; 4 container; 5 spacecraft; 6 container
cover; 7 “sun sail” in transportation state; 8 “sun sail” in flight configuration.
Figure 7: Scheme showing the launch of “sun sail” spacecraft.
References
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asteroids. Reviews of topical problems. Uspekhi Fizicheskih Nauk, 165, 661 (in
Figure 8: The asteroid with several parachutes mounted on it and several gravitational
riders in the asteroid orbits.
Russian).
[2] Gorbatenko, C.A., E.M. Makashov, Yu.F. Kolushkin and L.V. Sheftel, 1969,
Flight Mechanics. Engineers’ Handbook. Mashinostroenie, Moscow (in Russian).
[3] Beletsky, 1977, Essays about the Motion of Space Bodies, Nauka, Moscow (in
Russian).
[4] Alekseev, A.S., and Vedernikov, Yu.A., Ricochet and shock-cumulative methods of protecting the Earth against asteroids”, pp 14–18 in Ecology, Planetary
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[5] Vedernikov, Yu.A., B.S. Gisatullin and Yu.S. Khudyakov, 2001, A historical
experience and prospects for the application of the micro-plasmoid technique
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Protection and Safety, The SM Scientific and Production Union, St. Petersburg,
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[6] Gritzner, Ch., 1996, Analysis of Alternative Systems for Orbit Alteration of
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