International Journal of Astronomy and Astrophysics, 2012, 2, 129-155
http://dx.doi.org/10.4236/ijaa.2012.23018 Published Online September 2012 (http://www.SciRP.org/journal/ijaa)
Dynamic Problems of the Planets and Asteroids, and
Their Discussion
Joseph J. Smulsky1*, Yaroslav J. Smulsky2
1Institute of the Earth’s Cryosphere, Siberian Branch of Russian Academy of Sciences, Tyumen, Russia
2Institute of Thermophysics of Russian Academy of Sciences, Siberian Branch, Novosibirsk, Russia
Received February 9, 2012; revised March 15, 2012; accepted April 18, 2012
ABSTRACT
The problems of dynamics of celestial bodies are considered which in the literature are explained by instability and
randomness of movements. The dynamics of planets orbits on an interval 100 million years was investigated by new
numerical method and its stability is established. The same method is used for computing movements of two asteroids
Apophis and 1950 DA. The evolution of their movement on an interval of 1000 is investigated. The moments of their
closest passages at the Earth are defined. The different ways of transformation of asteroids trajectories into orbits of the
Earth’s satellites are considered. The problems of interest are discussed from the different points of view.
Keywords: Dynamics; Planets; Asteroids; Satellites; Stability; Discussion
1. Introduction
In the last decades a number of problems associated with
the precision of calculating movements have accumu-
lated in the celestial and the space dynamics. It was
found that there were discrepancies between the calcu-
lated and the observed movements. These differences
were the cause of the conclusions about chaotic move-
ments and the impossibility of accurately calculating
them. In addition to Newton’s gravitational forces were
introduced weaker forces of other nature, as well as new
substances “dark energy” and “dark matter”. In this paper
we consider the problems that are associated only with
the dynamics of the two groups of celestial objects: With
planets and asteroids.
In the study of the evolution of the Solar system over
geological time scales the number of researchers has
come to the conclusion that there are the instability of the
planetary orbits and chaotic motions in the Solar system.
For example, in paper [1], it is noted that the eccentricity
of the orbit of Mars can be greater than 0.2, and chaotic
diffusion of Mercury is so great that his eccentricity can
potentially reach values close to 1 and the planet could
be thrown out of the Solar system. Already at the time
intervals of 10 Myr (million years) there is the weak di-
vergence of the Earth’s orbit [2], which, according to
these authors, is caused by multiple resonances in the
Solar system. Due to them the movement of Solar system
is chaotic. Therefore the motion of the Earth [2] and
Mars [3] with an acceptable accuracy cannot be calcu-
lated for a time greater than 20 Myr.
The same problems of dynamics are occurred in the
study of motion of asteroids. Unlike the planets, their
movement is considered on a smaller time intervals.
However, the higher accuracy of determination of their
movement is required. In connection with the urgency of
the tasks of asteroids motion the problems of their dy-
namics is considered in more detail.
Over the past decade, the asteroids of prime interest
have been two asteroids, Apophis and 1950 DA, the first
predicted to approach the Earth in 2029, and the second,
in 2880. Reported calculations revealed some probability
of an impact of the asteroids on the Earth. Yet, by the
end of the decade refined orbital-element values of the
asteroids were obtained, and more precise algorithms for
calculating the interactions among solar-system bodies
were developed. Following this, in the present paper we
consider the motion evolution of both asteroids. In addi-
tion, we discuss available possibilities for making the
asteroids into the Earth-bound satellites. Initially, the
analysis is applied to Apophis and, then, numerical data
for 1950 DA obtained by the same method will be pre-
sented.
The background behind the problem we treat in the
present study was recently outlined in [4]. On June 19-20,
2004, asteroid Apophis was discovered by astronomers at
the Kitt Peak Observatory [5], and on December 20,
2004 this asteroid was observed for the second time by
astronomers from the Siding Spring Survey Observato-
*Corresponding author.
C
opyright © 2012 SciRes. IJAA

J. J. SMULSKY, Y. J. SMULSKY 130
ry [6]. Since then, the new asteroid has command inter-
national attention. First gained data on identification of
Apophis’ orbital elements were employed to predict the
Apophis path. Following the first estimates, it was re-
ported in [7] that on April 13, 2029 Apophis will ap-
proach the Earth center to a minimum distance of 38,000
km. As a result of the Earth gravity, the Apophis orbit
will alter appreciably. Unfortunately, presently available
methods for predicting the travel path of extraterrestrial
objects lack sufficient accuracy, and some authors have
therefore delivered an opinion that the Apophis trajectory
will for long remain unknown, indeterministic, and even
chaotic (see [4,7,8]). Different statistical predictions
points to some probability of Apophis’ collision with the
Earth on April 13, 2036. It is this aspect, the impact risk,
which has attracted primary attention of workers dealing
with the problem.
Rykhlova et al. [7] have attempted an investigation
into the possibility of an event that the Apophis will
closely approach the Earth. They also tried to evaluate
possible threats stemming from this event. Various means
to resist the fall of the asteroid onto Earth were put for-
ward, and proposals for tracking Apophis missions, made.
Finally, the need for prognostication studies of the Apo-
phis path accurate to a one-kilometer distance for a pe-
riod till 2029 was pointed out.
Many points concerning the prospects for tracking the
Apophis motion with ground- and space-based observing
means were discussed in [4,7-9]. Since the orbits of the
asteroid and Earth pass close to each other, then over a
considerable portion of the Apophis orbit the asteroid
disc will only be partially shined or even hidden from
view. That is why it seems highly desirable to identify
those periods during which the asteroid will appear ac-
cessible for observations with ground means. In using
space-based observation means, a most efficient orbital al-
location of such means needs to be identified.
Prediction of an asteroid motion presents a most chal-
lenging problem in astrodynamics. In paper [10], the
differential equations for the perturbed motion of the
asteroid were integrated by the Everhart method [11]; in
those calculations, for the coordinates of perturbing bo-
dies were used the JPL planetary ephemeris DE403 and
DE405 issued by the Jet Propulsion Laboratory, USA.
Sufficient attention was paid to resonance phenomena
that might cause the hypothetical 2036 Earth impact.
Bykova and Galushina [12,13] used 933 observations
to improve the identification accuracy for initial Apophis
orbital parameters. Yet, the routine analysis has showed
that, as a result of the pass of the asteroid through several
resonances with Earth and Mars, the motion of the aste-
roid will probably become chaotic. With the aim to eva-
luate the probability of an event that Apophis will impact
the Earth in 2036, Bykova et al. [12] have made about 10
thousand variations of initial conditions, 13 of which proved
to inflict a fall of Apophis onto Earth.
Smirnov [14] has attempted a test of various integra-
tion methods for evaluating their capabilities in predict-
ing the motion of an asteroid that might impact the Earth.
The Everhart method, the Runge-Kutta method of fourth
order, the Yoshida methods of sixth and eighth orders,
the Hermit method of fourth and sixth orders, the Mul-
tistep Predictor-Corrector (MS-PC) method of sixth and
eighth orders, and the Parker-Sochacki method were
analyzed. The Everhart and MS-PC methods proved to
be less appropriate than the other methods. For example,
at close Apophis-to-Earth distances Smirnov [14] used,
instead of the Everhart method, the Runge-Kutta method.
He came to the fact that, in the problems with singular
points, finite-difference methods normally fail to accu-
rately approximate higher-order derivatives. This conclu-
sion is quite significant since below we will report on an
integration method for motion equations free of such
deficiencies.
In paper [15] the mathematical problems on asteroid
orbit prediction and modification were considered. Pos-
sibilities offered by the impact-kinetic and thermonuclear
methods in correcting the Apophis trajectory were evalu-
ated.
An in-depth study of the asteroid was reported in paper
[4]. A chronologically arranged outline of observational
history was given, and the trend with progressively re-
duced uncertainty region for Apophis’ orbit-element
values was traced. Much attention was paid to discussing
the orbit prediction accuracy and the bias of various fac-
tors affecting this accuracy. The influence of uncertainty
in planet coordinates and in the physical characteristics
of the asteroid, and also the perturbing action of other
asteroids, were analyzed. The effects on integration ac-
curacy of digital length, non-spherical shape of Earth and
Moon, solar-radiation-induced perturbations, non-uni-
form thermal heating, and other factors, were examined.
The equations of perturbed motion of the asteroid were
integrated with the help of the Standard Dynamic Model
(SDM), with the coordinates of other bodies taken from
the JPL planetary ephemeris DE405. It is a well-known
fact that the DE405 ephemerid was compiled as an ap-
proximation to some hundred thousand observations that
were made till 1998. Following the passage to the ephe-
meris DE414, that approximates observational data till
2006, the error in predicting the Apophis trajectory on
2036 has decreased by 140,000 km. According to Gior-
gini et al. [4], this error proved to be ten times greater
than the errors induced by minor perturbations. Note that
this result points to the necessity of employing a more
accurate method for predicting the asteroid path.
In paper [4], prospects for further refinement of Apo-
phis’ trajectory were discussed at length. Time periods

J. J. SMULSKY, Y. J. SMULSKY 131
suitable for optical and radar measurements, and also
observational programs for oppositions with Earth in
2021 and 2029 and spacecraft missions for 2018 and
2027 were scheduled. Future advances in error minimi-
zation for asteroid trajectory due to the above activities
were evaluated.
It should be noted that the ephemerides generated as
an approximation to observational data enable rather ac-
curate determination of a body’s coordinates in space
within the approximation time interval. The prediction
accuracy for the coordinates on a moment remote from
this interval worsens, the worsening being the greater the
more the moment is distant from the approximation in-
terval. Therefore, the observations and the missions sche-
duled in paper [4] will be used in refining future ephe-
merides.
In view of the afore-said, in calculating the Apophis
trajectory the equation of perturbed motion were inte-
grated [4,10,15], while the coordinates of other bodies
were borrowed from the ephemerid. Difference integra-
tion methods were employed, which for closely spaced
bodies yield considerable inaccuracies in calculating
higher-order derivatives. Addition of minor interactions
to the basic Newtonian gravitational action complicates
the problem and enlarges the uncertainty region in pre-
dicting the asteroid trajectory. Many of the weak interac-
tions lack sufficient quantitative substantiation. More-
over, the physical characteristics of the asteroid and the
interaction constants are known to some accuracy. That
is why in making allowance for minor interactions expert
judgments were used. And, which is most significant, the
error in solving the problem on asteroid motion with
Newtonian interaction is several orders greater than the
corrections due to weak additional interactions.
The researches, for example, Bykova and Galushina
[12,13] apply a technique in Giorgini et al. 2008 to study
of influence of the initial conditions on probability of
collision Apophis with Earth. The initial conditions for
asteroid are defined from elements of its orbit, which are
known with some uncertainty. For example, eccentricity
value e = en ± σe, where en is nominal value of eccentric-
ity, and σe is root-mean-square deviation at processing of
several hundred observation of asteroid. The collision
parameters are searched in the field of possible motions
of asteroid, for example for eccentricity, 3σe, the initial
conditions are calculated in area e = en ± σe. From this
area the 10 thousand, and in some works, the 100 thou-
sand sets of the initial conditions are chosen by an acci-
dental manner, i.e. instead of one asteroid it is consid-
ered movement 10 or 100 thousand asteroids. Some of
them can come in collision with Earth. The probability of
collision asteroid with the Earth is defined by their
amount.
Such statistical direction is incorrect. If many mea-
surement data for a parameter are available, then the
nominal value of the parameter, say, eccentricity en, pre-
sents a most reliable value for it. That is why a trajectory
calculated from nominal initial conditions can be re-
garded as a most reliable trajectory. A trajectory calcu-
lated with a small deviation from the nominal initial con-
ditions is a less probable trajectory, whereas the prob-
ability of a trajectory calculated from the parameters
taken at the boundary of the probability region (i.e. from
e = en ±

e) tends to zero. Next, a trajectory with initial
conditions determined using parameter values trice
greater than the probable deviations (i.e. e = en ± 3

e) has
an even lower, negative, probability. Since initial condi-
tions are defined by six orbital elements, then simulta-
neous realization of extreme (boundary) values (± 3
 ) for
all elements is even a less probable event, i.e. the prob-
ability becomes of smaller zero.
That is why it seems that a reasonable strategy could
consist in examining the effect due to initial conditions
using such datasets that were obtained as a result of suc-
cessive accumulation of observation data. Provided that
the difference between the asteroid motions in the last
two datasets is insignificant over some interval before
some date, it can be concluded that until this date the
asteroid motion with the initial conditions was deter-
mined quite reliably.
As it was shown in paper [4], some additional active-
ties are required, aimed at further refinement of Apophis’
trajectory. In this connection, more accurate determina-
tion of Apophis’ trajectory is of obvious interest since,
following such a determination, the range of possible al-
ternatives would diminish.
For integration of differential motion equations of so-
lar-system bodies over an extended time interval, a pro-
gram Galactica was developed [16-19]. In this program,
only the Newtonian gravity force was taken into account,
and no differences for calculating derivatives were used.
In the problems for the compound model of Earth rota-
tion [20] and for the gravity maneuver near Venus [21],
motion equations with small body-to-body distances, the
order of planet radius, were integrated. Following the
solution of those problems and subsequent numerous
checks of numerical data, we have established that, with
the program Galactica, we were able to rather accurately
predict the Apophis motion over its travel path prior to
and after the approach to the Earth. In view of this, in the
present study we have attempted an investigation into
orbit evolution of asteroids Apophis and 1950 DA; as a
result of this investigation, some fresh prospects toward
possible use of these asteroids have opened.
2. Problem Statement
For the asteroid, the Sun, the planets, and the Moon, all
interacting with one another by the Newton law of gra-

J. J. SMULSKY, Y. J. SMULSKY 132
vity, the differential motion equations have the form [16]:
2
23
d, 1,2,,
d
n
ikik
ki ik
m
Gi
tr

 

rr n (1)
where i is radius-vector of a body with mass i rela-
tively Solar System barycenter; G is gravitational con-
stant; is vector and rik is its module; n = 12.
r
r
m
ik ik
As a result of numerical experiments and their analysis
we came to a conclusion, that finite-difference methods
of integration do not provide necessary accuracy. For the
integration of Equation (1) we have developed algorithm
and program Galactica. The meaning of function at the
following moment of time t = t0 + t is determined with
the help of Taylor series, which, for example, for coor-
dinate x looks like:
rr


00
1
1
!
Kk
k
k
x
xx
k

 
t (2)
where

0
k
x
is derivative of k order at the initial moment
t0.
The meaning of velocity
x

is defined by the similar
formula, and acceleration 0
x

 by the Equation (1).
Higher derivatives

0
k
x
we are determined analytically
as a result of differentiation of the Equation (1). The cal-
culation algorithm of the sixth order is now used, i.e.
with K = 6.
A few words about the method used and the program
Galactica. The algorithms of finite-difference methods
are derived from the Taylor series (2). In this case the
higher derivatives are determined by the difference of the
second derivatives at different steps. This leads to errors
of integration. There are methods [22,23], in which the
derivatives are determined by recurrence formulas. In the
program Galactica the derivatives are calculated under
the exact analytical formulas, which we have deduced.
This provides greater accuracy than other methods.
Besides in the program Galactica there are many other
details, which allow to not reducing the achieved in such
way accuracy. They were found in the process of creat-
ing the program Galactica. During its development more
than 10 different ways to control errors has been used. In
our book [19] the following checks are made mention of:
1) Checking the stability of the angular momentum M
of the Solar System;
2) Checking the magnitude of the momentum Р of the
Solar System;
4) Integrating backwards and forwards in time;
5) Integrating into a remote epoch and subsequent re-
turn to the initial epoch;
6) Picking persistent changes (orbital major semiaxis,
period, or precession axis etc.) and their checking;
7) Checking against test problems with exact analytic-
cal solutions, for example, n-axisymmetrical problem
[24];
8) Comparison with observations;
9) Comparison with other reported data.
The high accuracy of the program Galactica is firstly
allowed to integrate Equations (1) for the motion of the
Solar System for 100 million years [19]. The error was
less on some orders in comparison with work [25], in which
this problem has been solved for 3 million years by
Stórmer method.
The program Galactica allows solving the problem of
interaction of any number of bodies, which motion is
described by Equations (1). For example, the problem of
the evolution of Earth’s rotation axis [20] was solved. In
this task, the rotational motion of the Earth’s has been
replaced by a compound model of the Earth’s rotation.
The compound model of the Sun’s rotation was used in
another task [26], in which the influence of the oblate
rotating Sun on the motion of planets was established. In
all these problems, this method of integration and pro-
gram Galactica had no failures and we successfully used
them.
As noted above, the methods, which use Standard Dy-
namic Model, are approximation of the data of Solar Sys-
tem body’s observations. To calculate the future move-
ment of the asteroid the bodies’ positions are used out-
side the framework of observation. Therefore the calcu-
lation error increases with distance in time from the base
of observations. The base of observations is not used in
program Galactica, so this error is missing in it. A high
accuracy of the integration method of Galactica allows
for a smaller error compute motion of asteroids in their
rapprochement with the celestial bodies.
The free-access Galactica system, user version for per-
sonal computer, can be found at:
http://www.ikz.ru/~smulski/GalactcW. The system of Ga-
lactica, except the program Galactica, includes several
components, which are described in the User’s Guide
htt p://www.ikz.ru/~smulsk i/GalactcW /GalD i s cr E . pd f . The
Guide also provides detailed instructions for all stages of
solving the problem. After the free-access system is
created based on a supercomputer, we will post the in-
formation at the above site.
3. The Dynamics of the Planets’ Orbits
To study the dynamics of the Solar system we used the
initial conditions (IC) on the 30.12.1949 with the Julian
day JD0 = 2433280.5 in two versions. The first version of
the IC was based on ephemerid DE19 of Jet Propulsion
Laboratory, USA (JPL), a second version is based on the
ephemerid DE406. The first IC was in a coordinate sys-
tem of 1950.0, while the second—in the system of
2000.0. These initial conditions are given in our paper
[19] and on the website http://www.ikz.ru/~smulski/Data/
OrbtData. As already mentioned, the differential Equa-

J. J. SMULSKY, Y. J. SMULSKY 133
tions (1) were integrated for 11 bodies: 9 planets, the Sun
and the Moon. The motion of bodies is considered in
barycentric equatorial coordinates. The parameters of the
orbits of the planets are defined in the heliocentric coor-
dinate system and the orbit of the Moon is in geocentric
coordinates.
The main results obtained with the integration step t =
10–4 year and the number of double length (17 decimal).
Checking and clarifying the calculations were performed
with a smaller step, as well as with the extended length
of the numbers (up to 34 decimal places).
Methods for validation of the solutions and their errors
are investigated in [19]. The computed with the help of
the program Galactica change of the parameters of the
orbit of Mars at time interval of 7 thousand years is
shown in Figure 1 points 1. The eccentricity e and peri-
helion angle

p increase, and the longitude of the as-
cending node

Ω decreases in this span of time. The an-
gle of inclination of the Mars orbit i does not change
monotonically. In the epoch at T = 1400 years from 30
December 1949, it has a maximum value. The changing
these parameters for a century are called secular pertur-
bations. In contrast, the semi-major axis a and orbital
period P on the average remains unchanged, so the
graphs are given their deviations from the mean values.
These fluctuations relative to the average am = 2.279 ×
1011 m and Pm = 1.881 years have a small value. The
parameters e, i,

Ω and

p also vary with the same rela-
tive amplitudes as parameters a and P.
In Figure 1 the lines 2 and 3 show the approximation
of the observation data. As one can see, the eccentricity e
and the angles i,

Ω and

p coincide with observations in
the interval 1000 years, i.e. within the validity of ap-
proximations by S. Newcomb [27] and J. Simon et al.
[28]. Calculations for the semi-major axis a and the pe-
riods P also coincide with the observations and their fluc-
tuations comparable in magnitude with the difference
between the approximations of different authors. Similar
studies of secular changes in orbital parameters are made
for other planets. They are also compared with the ap-
proximations of the observational data, except for Pluto.
Its approximation of the observations is missing.
In Figure 2 there are the variations of the calculated
elements of the orbit of Mars in 3 million years into the
past. The eccentricity of the orbit e has short-period os-
cillations of amplitude 0.019 with the main period equal
to Te1 = 95.2 thousand years. It oscillates relatively the
average for 50 million years value of em = 0.066. The
longitude of ascending node 
Ω oscillates with the aver-
age period of ТΩ = 73.1 thousand years around the mean
value

Ωm = 0.068 radians. The angle of inclination of the
orbital plane to the equatorial plane i fluctuates with the
same period Ti = 73.1 thousand years around the mean
value іm = 0.405 radians. Longitude of perihelion φр al-
Figure 1. Secular changes of the Mars orbit 1 compared
with approximated observations by Newcomb [27] and
Simon et al. [28], 2 and 3, respectively: Eccentricity e; in-
clination i to equator plane for epoch J2000.0; longitude of
ascending node


relative to the x axis at J2000.0; longi-
tude of perihelion 
p; semi-major axis deviation Δa from the
average of 7 thousand years, in meters; orbital period de-
viation ΔP from average of 7 thousand years, in centuries.
Angles are in radians and time Т is in centuries from 30
December 1949; data points spaced at 200 yr.
Figure 2. Evolution of the orbit parameters of Mars for 3
Myr.

p is the angular velocity of the perihelion rotation in
the “century for the time interval of 20 thousand years: 
pm =
1687” century is the average angular velocity of perihelion
rotation during 50 million years; Те1, ТΩ and Тi are the
shortest periods of eccentricity, ascending node, and incli-
nation, respectively, in kyr (thousand years); em, im and 
 m
are the average values of appropriate parameters; Тр is a
rotation perihelion period averaged over 50 Myr. Other
notations see Figure 1.

J. J. SMULSKY, Y. J. SMULSKY 134
most linearly increases with time, i.e. the perihelion
moves in the direction of circulation of Mars around the
Sun, making on the average for a –50 million years, one
revolution for the time Tp = 76.8 thousand years. At the
same time its motion is uneven. As one can see from the
graph, the angular velocity of rotation of the perihelion

p fluctuates around the average value of

pm = 1687"
(arc-seconds) per century, while at time T = –1.35 mil-
lion years, it takes a large negative value, i.e. in this mo-
ment there is the return motion of the perihelion with
great velocity.
As a result of research for several million years the pe-
riods and amplitudes of all fluctuations of the orbits pa-
rameters of all planets received. The system of Equation
(1) with the help of the program Galactica was integrated
for 100 million years in the past and the evolution of the
orbits of all planets and the Moon was studied. Figure 3
shows changing parameters of the Mars orbit on an in-
terval from –50 million years to –100 million years. The
eccentricity e, angles of inclination i, ascending node 
Ω
monotonically oscillate. The fluctuations have several
periods, and duration of most of them is much smaller
than the interval in 50 million years. For example, the
greatest period eccentricity Te2 = 2.31 million years. The
angular velocity

p of perihelion rotation fluctuate
around the average value of

pm. In comparison with the
Figure 3. The evolution of the Mars orbit in the second half
of the period of 100 million years: T is the time in millions
of years into the past from the epoch 30.12.1949; other no-
tations see Figures 1 and 2.
eccentricity e one can see that negative values of 
p, i.e.
reciprocal motion of the perihelion occurs when the ec-
centricity of the orbit is close to zero.
On the interval from 0 to –50 million years the graph-
ics have the same form [19], as shown in Figure 3, i.e.
the orbit of Mars is both steady and stable and there is no
tendency to its change. Similar results were obtained for
other planets, i.e. these studies established the stability of
the orbits and the Solar system as a whole. The result is
important, since in the abovementioned papers [1-3] at
solving the problem of other methods after 20 million
years, the orbit begins to change, what led to the destruc-
tion of the Solar system in the future. Based on these
solutions their authors came to the conclusion about the
instability of the Solar system and chaotic motions in it.
During these researches we have established, that the
evolution of the planets orbits is the result of four move-
ments: 1) The precession of the orbital axis
S
around a
fixed in space vector of angular momentumof the
Solar system; 2) Nutational oscillations of the orbit axis
M
S
; 3) The oscillations of orbit; 4) the rotation of the orbit
in its own plane (rotation of the perihelion). The behavior
of the perihelion of the orbits of all planets in the span of
50 million years is shown in Figure 4 as a function of
angles perihelion φр versus time. The orbits of the eight
planets from Mercury to Neptune are orbiting counter-
clockwise, i.e. in the direction of orbital motion. The
orbit of Pluto, the only one, that rotates clockwise. Two
Figure 4. Variations of perihelion longitudes 
p and the
precession angle 
S for 50 Myr in orbits of nine planets
from Mercury to Pluto (numbered from 1 to 9), with the
respective perihelion periods (Tp) and precession cycles (TS)
in Kyr, which averaged over 50 Myr.

J. J. SMULSKY, Y. J. SMULSKY 135
groups of planets (Venus and Earth, Jupiter and Uranus),
as seen from the values of the periods Tp, have almost the
same velocity of perihelion rotation. The orbit of Saturn
has the highest velocity, and the orbit of Pluto has the
smallest one.
should also be emphasized that reported in literature
resonances and instabilities appear in the simplified equ-
ations of motion when they are solved by approximate
analytical methods. These phenomena do not arise at the
integrating not simplified Equation (1) by method (2).
The precession of the orbit occurs clockwise, i.e.
against the orbital motion of the planets The angle of
precession 
S of the orbit axis is defined in the plane
perpendicular to the vector of the momentum . This
plane crosses the equator plane on the line directed at an
angle φΩm = 0.068 to the x axis. Precession angle 
S is
the angle between the line of intersection and the node of
the planet orbit. The angle

S, as the angle of the peri-
helion φр, varies irregularly, but a large time interval of
50 million years, as shown in Figure 4, these irregular-
rities are not visible. As can be seen from the periods of
TS, the orbits axis of Jupiter and Saturn precess with
maximum velocity and Pluto—with the lowest one. For
the two groups of planets: Venus and Earth, Jupiter and
Saturn, the velocity of precession is practically the same.
M
4. Preparation of Initial Data of Asteroids
We consider the motion of asteroids in the barycentric
coordinate system on epoch J2000.0, Julian day JDs =
2451545. The orbital elements asteroids Apophis and
1950 DA, such as the eccentricity e, the semi-major axis
a, the ecliptic obliquity ie, the ascending node angle  ,
the ascending node-perihelion angle ωe, etc., and aste-
roids position elements, such as the mean anomaly M,
were borrowed from the JPL Small-Body database 2008
as specified on November 30, 2008. The data, represent-
ed to 16 decimal digits, are given in Table 1.
For Apophis in Table 1 the three variants are given.
The first variant is now considered. These elements cor-
respond to the solution with number JPL sol. 140, which
is received Otto Mattic at April 4, 2008. In Table 1 the
uncertainties of these data are too given. The relative un-
certainty value

is in the range from 2.4 × 10–8 to 8 ×
10–7. The same data are in the asteroid database by Ed-
ward Bowell [29], although these data are represented
only to 8 decimal digits, and they differ from the former
data in the 7-th digit, i.e., within value  . Giorgini et al. [4]
used the orbital elements of Apophis on epoch JD =
2453979.5 (September 01, 2006), which correspond to the
solution JPL sol.142. On publicly accessible JPL-system
It should be emphasized that the secular changes of
angles of inclination i and ascending node

Ω in Figure 1
and their variations, are presented by the graphics in Fig-
ures 2 and 3, are due to the precessional and nutational
motion of the orbit axis
S
.
The studies testify that the evolution of the planets or-
bits in the investigated range is unchanged and stable.
This allows us to conclude that the manifestations of in-
stability and chaos in the motion of the planets, as de-
scribed by other authors, most likely due to the methods
of solution which they have been used. Furthermore, it
Table 1. Three variants of orbital elements of asteroids Apophis on two epochs and 1950 DA on one epoch in the heliocentric
ecliptic coordinate system of 2000.0 with JDS = 2451545 (see JPL Small-Body Database [30]).
Apophis 1950 DA
1-st variant
November 30, 2008
JD01 = 2454800.5
JPL sol.140
Uncertainties
±

1-st var.
2-nd variant
January 04, 2010
JD02 = 2455200.5
JPL sol.144
3-rd variant
November 30, 2008
JD01 = 2454800.5
JPL sol.144.
November 30, 2008
JD0 = 2454800.5
JPL sol.51
Units
Elements
Magnitude
e 0.1912119299890948 7.6088e–08 0.1912110604804485 0.1912119566344382 0.507531465407232
a 0.9224221637574083 2.3583e–08 0.9224192977379344 0.9224221602386669 1.698749639795436 AU
q 0.7460440415606373 8.6487e–08 0.7460425256098334 0.7460440141364661 0.836580745750051 AU
ie 3.331425002325445 2.024e–06 3.331517779979046 3.331430909298658 12.18197361251942 deg
 204.4451349657969 0.00010721 204.4393039605681 204.4453098275707 356.782588306221 deg
ωe 126.4064496795719 0.00010632 126.4244705298442 126.4062862564680 224.5335527346193 deg
M 254.9635275775066 5.7035e–05 339.9486156711335 254.9635223452623 161.0594270670401 deg
tp 2454894.912750123770
(2009-Mar-04.41275013) 5.4824e–05 2455218.523239657948 (2010-Jan-22.02323966)
2454894.912754286546
(2009-Mar-04. 41275429)
2.454438.693685309
(2007-Dec-12.0419368531
JD
d
P 323.5884570441701
0.89
1.2409e–05
3.397e–08
323.5869489330219
0.89
323.5884551925927
0.89
808.7094041052905
2.21
D
yr
n 1.112524233059586 4.2665e–08 1.112529418096263 1.112524239425464 0.445153720449539 deg/d
Q 1.098800285954179 2.8092e–08 1.098796069866035 1.098800306340868 2.560918533840822 AU

J. J. SMULSKY, Y. J. SMULSKY 136
Horizons the solution sol.142 can be prolonged till No-
vember 30.0, 2008. In this case it is seen, that difference
of orbital elements of the solution 142 from the solution
140 does not exceed 0.5
 uncertainties of the orbit ele-
ments.
The element values in Table 1 were used to calculate
the Cartesian coordinates of Apophis and the Apophis
velocity in the barycentric equatorial system by the fol-
lowing algorithm (see [19,20,31,32]).
From the Kepler equation
sinEeE M (3)
we calculate the eccentric anomaly E and, then, from E,
the true anomaly 
0:

02 arctg11tg0.5ee E


 



(4)
In subsequent calculations, we used results for the
two-body interaction (the Sun and the asteroid) [21,32].
The trajectory equation of the body in a polar coordinate
system with origin at the Sun has the form:

11
1cos
p
R
r




 (5)
where the polar angle φ, or, in astronomy, the true ano-
maly, is reckoned from the perihelion position r = Rp;


111 e


 is the trajectory parameter; and Rp =


11
r
υ
21aa a is the perihelion radius. The expressions
for the radial and transversal velocities are
t
υ

22
11
11,
rp tp
υυ rυυr   , (6)
where for φ > π we have
 ; 0
r
υ
r
= r/Rp is the dimen-
sionless radius, and the velocity at perihelion is
 
1pSAs
υGm mR  
p
, (7)
where mS = m11 is the Sun mass (the value of m11 is given
in Table 2), and mAs = m12 is the Apophis mass.
Table 2. The masses mbj of the planets from Mercury to Pluto, the Moon, the Sun (1 - 11) and asteroids: Apophis (12a) and
1950 DA (12b), and the initial condition on epoch JD0 = 2454800.5 (November 30, 2008) in the heliocentric equatorial coor-
dinate system on epoch 2000.0 JDS = 2451545. G = 6.67259E–11 m3·s–2·kg–1.
Bodies masses in kg, their coordinates in m and velocities in m·s–1
Bodies,
j mbj xaj, vxaj, yaj, vyaj zaj, vzaj
–17405931955.9539 –60363374194.7243 –30439758390.4783
1 3.30187842779737E+23 37391.7107852059 –7234.98671125365 –7741.83625612424
108403264168.357 –2376790191.8979 –7929035215.64079
2 4.86855338156022E+24 1566.99276862423 31791.7241663148 14204.3084779893
55202505242.89 125531983622.895 54422116239.8628
3 5.97369899544255E+24
–28122.5041342966 10123.4145376039 4387.99294255716
–73610014623.8562 –193252991786.298 –86651102485.4373
4 6.4185444055007E+23 23801.7499674501 –5108.24106287744 –2985.97021694235
377656482631.376 –609966433011.489 –270644689692.231
5 1.89900429500553E+27 11218.8059775149 6590.8440254003 2551.89467211952
–1350347198932.98 317157114908.705 189132963561.519
6 5.68604198798257E+26
–3037.18405985381 –8681.05223681593 –3454.56564456648
2972478173505.71 –397521136876.741 –216133653111.407
7 8.68410787490547E+25 979.784896813787 5886.28982058747 2564.10192504801
3605461581823.41 –2448747002812.46 –1092050644334.28
8 1.02456980223201E+26 3217.00932811768 4100.99137103454 1598.60907148943
53511484421.7929 –4502082550790.57 –1421068197167.72
9 1.65085753263927E+22 5543.83894965145 –290.586427181992 –1757.70127979299
55223150629.6233 125168933272.726 54240546975.7587
10 7.34767263035645E+22 –27156.1163326908 10140.7572420768 4468.97456956941
0 0 0
11 1.98891948976803E+30 0 0 0
–133726467471.667 –60670683449.3631 –26002486763.62
12a 30917984100.3039 16908.9331065445 –21759.6060221801 –7660.90393288287
314388505090.346 171358408804.935 127272183810.191
12b 1570796326794.9 –5995.33838888362 9672.35319009371 6838.06006342785

J. J. SMULSKY, Y. J. SMULSKY 137
The time during which the body moves along an ellip-
tic orbit from the point of perihelion to an orbital position
with radius r is given by
 



1
111
32
1
21
π2arcsin 211
21
pr
p
p
p
Rrυ
tυ
r
R
υ




 


1
 
(8)
where rrp
At the initial time t0 = 0, which corresponds to epoch
JD0 (see Table 1), the polar radius of the asteroid r0 as
dependent on the initial polar angle, or the true anomaly

, can be calculated by Equation (5)The initial radial
and initial transversal velocities as functions of r0 can be
found using Equation (6).
υυυ
0
is the dimensionless radial velocity.
The Cartesian coordinates and velocities in the orbit
plane of the asteroid (the axis xo goes through the perihe-
lion) can be calculated by the formulas
00 0
cos ;sin
oo
xr yr

  (9)
0
00
cos sin
sin cos
xo rt
yo rt
υυ υ
υυ υ
0




 
  (10)
The coordinates of the asteroid in the heliocentric
ecliptic coordinate system can be calculated as



coscosΩsin sinΩcos
–sincosΩcos sinΩcos ;
eo eee
oee e

x
x
y


 

i
i


i
e
(11)


cossinΩsin cosΩcos
–sinsinΩcos cosΩcos ;
eo eee
oee e
yx i
y


 
 (12)
sinsincossin.
eoe eoe
zx iyi

  (13)
The velocity components of the asteroid ,
x
eye
υυ and
ze in this coordinate system can be calculated by Equa-
tions analogous to (11)-(13).
υ
Since Equation (1) are considered in a motionless equ-
atorial coordinate system, then elliptic coordinates (11)-
(13) can be transformed into equatorial ones by the Equ-
ations
0
00
;
cossin;
sinsin
ae
ae e
ae e
xx
yy z
zy z
0





 
 
(14)
where

0 is the angle between the ecliptic and the equator
in epoch JDS.
The velocity components ,
x
eye
υυ and
z
e
υ can be
transformed into the equatorial ones ,
x
aya
υυ and
z
a
by Equations analogous to (14). With known heliocentric
equatorial coordinates of the Solar system n bodies xai, yai,
zai i = 1, 2, … n, the coordinates of Solar system bary-
centre, for example, along axis x will be:
υ
1
n
ci aiSs
i
X
mx M




, where
1
n
Ss i
i
M
m

 is mass of
solar system bodies.
Then barycentric equatorial coordinates xi of asteroid
and other bodies will be
–
iai c
x
xX

.
Other coordinates yi and zi and components of velocity
,
x
iyi
υυ and
z
i
υ in barycentric equatorial system of co-
ordinates are calculated by analogous equations.
In the calculations, six orbital elements from Table 1,
namely, e, a ie,  ,  ωe, and M, were used. Other orbital
elements were used for testing the calculated data. The
perihelion radius Rp and the aphelion radius Ra =


1
21
p
Ra

 were compared to q and Q, respectively.
The orbital period was calculated by Equation (8) as
twice the time of motion from perihelion to aphelion (r =
Ra). The same Equation was used to calculate the mo-
ment at which the asteroid passes the perihelion (r = r0).
The calculated values of those quantities were compared
to the values of P and tp given in Table 1. The largest
relative difference in terms of q and Q was within 1.9 ×
10–16, and in terms of P and tp, within 8 × 10–9.
The coordinates and velocities of the planets and the
Moon on epoch JD0 were calculated by the DE406/
LE406 JPL-theory [33,34]. The masses of those bodies
were modified by us [18], and the Apophis mass was
calculated assuming the asteroid to be a ball of diameter
d = 270 m and density  = 3000 kg/m3. The masses of all
bodies and the initial conditions are given in Table 2.
The starting-data preparation and testing algorithm (3)-
(14) was embodied as a MathCad worksheet (program
AstCoor2.mcd).
5. Apophis’ Encounter with the Planets and
the Moon
In the program Galactica, a possibility to determine the
minimum distance Rmin to which the asteroid approaches
a celestial body over a given interval T was provided.
Here, we integrated Equation (1) with the initial condi-
tions indicated in Table 2. The integration was per-
formed on the NKS-160 supercomputer at the Computing
Center SB RAS, Novosibirsk. In the program Galactica,
an extended digit length (34 decimal digits) was used,
and for the time step a value dT = 10–5 year was adopted.
The computations were performed over three time inter-
vals, 0 - 100 years (Figure 5(a)), 0 - –100 years (Figure
5(b)), and 0 - 1000 years (Figure 5(c)).
In the graphs of Figure 5 the points connected with
the heavy broken line show the minimal distances Rmin to
which the asteroid approaches the bodies indicated by
points embraced by the horizontal line. In other words, a
point in the broken line denotes a minimal distance to

J. J. SMULSKY, Y. J. SMULSKY 138
Figure 5. Apophis’ encounters with celestial bodies during
the time T to a minimum distance Rmin, km: Mars (Ma),
Earth (Ea), Moon (Mo), Venus (Ve) and Mercury (Me); a, b—
T = 1 year; c—T = 10 years. T, cyr (1 cyr = 100 yr) is the
time in Julian centuries from epoch JD0 (November 30,
2008). Calendar dates of approach in points: A—13 April
2029; B—13 April 2067; C—5 September 2037; E—10 Oc-
tober 2586.
which, over the time T = 1 year, the asteroid will ap-
proach a body denoted by the point in the horizontal line
at the same moment. It is seen from Figure 5(a) that,
starting from November 30, 2008, over the period of 100
years there will be only one Apophis’ approach to the
Earth (point A) at the moment TA = 0.203693547133403
century to a minimum distance RminA = 38907 km. A next
approach (point B) will be to the Earth as well, but at the
moment TB = 0.583679164042455 century to a minimum
distance RminB = 622,231 km, which is 16 times greater
than the minimum distance at the first approach. Among
all the other bodies, a closest approach with be to the
Moon (point D) (see Figure 5(b)) at TD =
–0.106280550824626 century to a minimum distance
RminD = 3,545,163 km.
In the graphs of Figures 5(a) and (b) considered above,
the closest approaches of the asteroid to the bodies over
time intervals T = 1 year are shown. In integrating
Equation (1) over the 1000-year interval (see Figure
5(c)), we considered the closest approaches of the aste-
roid to the bodies over time intervals T = 10 years. Over
those time intervals, no approaches to Mercury and Mars
were identified; in other words, over the 10-year intervals
the asteroid closes with other bodies. Like in Figure 5(a),
there is an approach to the Earth at the moment TA. A
second closest approach is also an approach to the Earth
at the point Е at TE = 5.778503 century to a minimum
distance RminE = 74002.9 km. During the latter approach,
the asteroid will pass the Earth at a minimum distance
almost twice that at the moment TA.
With the aim to check the results, Equation (1) were
integrated over a period of 100 years with double digit
length (17 decimal digits) and the same time step, and
also with extended digit length and a time step dT = 10–6
year. The integration accuracy (see Table 3) is defined
[19] by the relative change of  Mz, the z-projection of the
angular momentum of the whole solar system for the
100-year period. As it is seen from Table 3, the quantity
 Mz varies from –4.5 × 10–14 to 1.47 × 10–26, i.e., by 12
orders of magnitude. In the last two columns of Table 3,
the difference between the moments at which the asteroid
most closely approaches the Earth at point A (see Figure
5(a)) and the difference between the approach distances
relative to solution 1 are indicated. In solution 2, ob-
tained with the short digit length, the approach moment
has not changed, whereas the minimum distance has re-
duced by 2.7 m. In solution 3, obtained with ten times
reduced integration step, the approach moment has
changed by –2 × 10–6 year, or by –1.052 minutes. This
change being smaller than the step dT = 1 × 10–5 for solu-
tion 1 and being equal twice the step for solution 3, the
value of this change provides a refinement for the ap-
proach moment. Here, the refinement for the closest ap-
proach distance by –1.487 km is also obtained. On the
refined calculations the Apophis approach to the Earth
occurs at 21 hours 44 minutes 45 sec on distance of
38905 km. We emphasize here that the graphical data of
Figure 5, a for solutions 1 and 3 are perfectly coincident.
The slight differences of solution 2 from solutions 1 and
3 are observed for Т > 0.87 century. Since all test calcu-
lations were performed considering the parameters of
solution 1, it follows from here that the data that will be
presented below are accurate in terms of time within 1’,
and in terms of distance, within 1.5 km.
At integration on an interval of 1000 years the relative
change of the angular momentum is Mz = 1.45 × 10–20.
How is seen from the solution 1 of Table 3 this value
exceeds Mz at integration on an interval of 100 years in
10 times, i.e. the error at extended length of number is
proportional to time. It allows to estimate the error of the
Table 3. Comparison between the data on Apophis’ en-
counter with the Earth obtained with different integration
accuracies: Lnb is the digit number in decimal digits.
No.
solution Lnb dT, yr
 Mz TAi–TA1, yr RminAi–Rmin A 1,
km
1 34 1 × 10–5 1.47 × 10–21 0 0
2 17 1 × 10–5 –4.5 × 10–14 0 –2.7 × 10–3
3 34 1 × 10–6 1.47 × 10–26 –2 × 10–6 –1.487

J. J. SMULSKY, Y. J. SMULSKY 139
second approach Apophis with the Earth in TE = 578
years by results of integrations on an interval of 100
years of the solution with steps dT = 1 × 10–5 years and 1 ×
10–6 years. After 88 years from beginning of integration
the relative difference of distances between Apophis and
Earth has become  R88 = 1 × 10–4, that results in an error
in distance of 48.7 km in TE = 578 years.
So, during the forthcoming one-thousand-year period
the asteroid Apophis will most closely approach the
Earth only. This event will occur at the time TA counted
from epoch JD0. The approach refers to the Julian day
JDA = 2462240.406075 and calendar date April 13, 2029,
21 hour 44'45" GMT. The asteroid will pass at a mini-
mum distance of 38905 km from the Earth center, i.e., at
a distance of 6.1 of Earth radii. A next approach of Apo-
phis to the Earth will be on the 578-th year from epoch
JD0; at that time, the asteroid will pass the Earth at an
almost twice greater distance.
The calculated time at which Apophis will close with
the Earth, April 13, 2029, coincides with the approach
times that were obtained in other reported studies. For
instance, in the recent publication [4] this moment is
given accurate to one minute: 21 hour 45' UTC, and the
geocentric distance was reported to be in the range from
5.62 to 6.3 Earth radii, the distance of 6.1 Earth radii
falling into the latter range. The good agreement between
the data obtained by different methods proves the ob-
tained data to be quite reliable.
As for the possible approach of Apophis to the Earth
in 2036, there will be no such an approach (see Figure
5(a)). A time-closest Apophis’ approach at the point C to
a minimum distance of 7.26 million km will be to the
Moon, September 5, 2037.
6. Apophis Orbit Evolution
In integrating motion Equation (1) over the interval –1
century ≤ T ≤ 1 century the coordinates and velocities of
the bodies after a lapse of each one year were recorded in
a file, so that a total of 200 files for a one-year time in-
terval were obtained. Then, the data contained in each
file were used to integrate Equation (1) again over a time
interval equal to the orbital period of Apophis and, fol-
lowing this, the coordinates and velocities of the asteroid,
and those of Sun, were also saved in a new file. These
data were used in the program DefTra to determine the
parameters of Apophis’ orbit relative to the Sun in the
equatorial coordinate system. Such calculations were
performed hands off for each of the 200 files under the
control of the program PaOrb. Afterwards, the angular
orbit parameters were recalculated into the ecliptic coor-
dinate system (see Figure 6).
As it is seen from Figure 6, the eccentricity е of the
Apophis orbit varies non-uniformly. It shows jumps or
Figure 6. Evolution of Apophis’ orbital parameters under
the action of the planets, the Moon and the Sun over the
time interval −100 years - +100 years from epoch November
30, 2008: 1—as revealed through integration of motion
Equation (1); 2—initial values according to Table 1. The
angular quantities:  , i
e, and ωe are given in degrees; the
major semi-axis a in AU; and the orbital period P in days.
breaks. A most pronounced break is observed at the mo-
ment TA, at which Apophis most closely approaches the
Earth. A second most pronounced break is observed
when Apophis approaches the Earth at the moment TB.
The longitude of ascending node  shows less breaks,
exhibiting instead rather monotonic a decrease (see Fig-
ure 6). Other orbital elements, namely, ie, ωe, a, and P,
exhibit pronounced breaks at the moment of Apophis’
closest pass near the Earth (at the moment TA).
The dashed line in Figure 6 indicates the orbit-ele-
ment values at the initial time, also indicated in Table 1.
As it is seen from the graphs, those values coincide with
the values obtained by integration of Equation (1), the
relative difference of e, , ie, ωe, a, and P from the initial
values at the moment T = 0 (see Table 1) being respec-
tively 9.4 × 10–6, –1.1 × 10–6, 3.7 × 10–6, –8.5 × 10–6, 1.7 ×
10–5, and 3.1 × 10–5. This coincidence testifies the reli-
ability of computed data at all calculation stages, includ-
ing the determination of initial conditions, integration of
equations, determination of orbital parameters, and trans-
formations between the different coordinate systems.
As it was mentioned in Introduction, apart from non-
simplified differential Equation (1) for the motion of
celestial bodies, other equations were also used. It is a
well-known fact (see Duboshin, 1976) that in perturbed-
motion equations orbit-element values are used. For this
reason, such equations will yield appreciable errors in
determination of orbital-parameter breaks similar to

J. J. SMULSKY, Y. J. SMULSKY 140
those shown in Figure 6. Also, other solution methods
for differential equations exist, including those in which
expansions with respect to orbital elements or difference
quotients are used. As it was already mentioned in Intro-
duction, these methods proved to be sensitive to various
resonance phenomena and sudden orbit changes ob-
served on the approaches between bodies. Equation (1)
and method (2) used in the present study are free of such
shortcomings. This suggests that the results reported in
the present paper will receive no notable corrections in
the future.
7. Influence of Initial Conditions
With the purpose of check of influence of the initial con-
ditions (IC) on Apophis trajectory the Equation (1) were
else integrated on an interval 100 years with two variants
of the initial conditions. The second of variant IC is
given on January 04, 2010 (see Table 1). They are taken
from the JPL Small-Body database [29] and correspond
to the solution with number JPL sol.144, received Steven
R. Chesley on October 23, 2009. In Figure 7 the results
of two solutions with various IC are submitted. The line
1 shows the change in time of distance R between Apo-
phis and Earth for 100 years at the first variant IC. As it
is seen from the graphs, the distance R changes with os-
cillations, thus it is possible to determine two periods: the
short period TR1 = 0.87 years and long period TR2. The
amplitude of the short period Ra1 = 29.3 million km, and
long is Ra2 = 117.6 million km. The value of the long
oscillation period up to T ~ 70 years is equal TR20 = 7.8
years, and further it is slightly increased. After ap-
proach of April 13, 2029 (point A in Figure 7) the am-
plitude of the second oscillations is slightly increased.
Both short and the long oscillations are not regular;
therefore their average characteristics are above given.
Let’s note also on the second minimal distance of
Apophis approach with the Earth on interval 100 years. It
occurs at the time TF1 = 58.37 years (point F1 in Figure 7)
on distance RF1 = 622 thousand km. In April 13, 2036
(point H in Figure 7) Apophis passes at the Earth on
distance RH1 = 86 million km. The above-mentioned
characteristics of the solution are submitted in Table 4.
The line 2 in Figure 7 gives the solution with the se-
cond of variant IC with step of integration dT = 1 × 10–5
years. The time of approach has coincided to within 1
minutes, and distance of approach with the second of IC
became RA2 = 37,886 km, i.e. has decreased on 1021 km.
To determine more accurate these parameters the Equa-
tion (1) near to point of approach were integrated with a
step dT = 1 × 10–6 years. On the refined calculations
Apophis approaches with the Earth at 21 hours 44 min-
utes 53 second on distance RA2 = 37,880 km. As it is seen
from Table 4, this moment of approach differs from the
moment of approach at the first of IC on 8 second. As at a
Figure 7. Evolution of distance R between Apophis and
Earth for 100 years. Influence of the initial conditions (IC):
1—IC from November 30, 2008; 2—IC from January 04,
2010. Calendar dates of approach in points: A—13 April
2029; F1—13 April 2067; F2—14 April 2080.
step dT = 1 × 10–6 years the accuracy of determination of
time is 16 second, it is follows, that the moments of ap-
proach coincide within the bounds of accuracy of their
calculation.
The short and long oscillations at two variants IC also
have coincided up to the moment of approach. After ap-
proach in point A the period of long oscillations has de-
creased up to TR22 = 7.15 years, i.e. became less than
period TR20 at the first variant IC. The second approach
on an interval 100 years occurs at the moment TF2 =
70.28 years on distance RF2 = 1.663 million km. In 2036
(point H) Apophis passes on distance RH2 = 43.8 million
km.
At the second variant of the initial conditions on Janu-
ary 04, 2010 in comparison with the first of variant the
initial conditions of Apophis and of acting bodies are
changed. To reveal only errors influence of Apophis IC,
the third variant of IC is given (see Table 1) as first of IC
on November 30, 2008, but the Apophis IC are calcu-
lated in system Horizons according to JPL sol.144. How
follows from Table 1, from six elements of an orbit e, a,
ie,  , ωe and M the differences of three ones: ie,  и ωe
from similar elements of the first variant of IC are 2.9,
1.6 and 1.5 appropriate uncertainties. The difference of
other elements does not exceed their uncertainties.
At the third variant of IC with step of integration dT =
1 × 10–5 year the moment of approach has coincided with
that at the first variant of IC. The distance of approach
became RA3 = 38,814 km, i.e. has decreased on 93 km.
For more accurate determination of these parameters the
Equation (1) near to a point of approach were also inte-
grated with a step dT = 1 × 10–6 year. On the refined cal-
culations at the third variant of IC Apophis approaches
with the Earth at 21 hours 44 minutes 45 second on dis-
tance RA3 = 38,813 km. These and other characteristics of
the solution are given in Table 4. In comparison with the
first variant IC it is seen, that distance of approach in
2036 and parameters of the second approach in point F1
are slightly changed. The evolution of distance R in a
Figure 7 up to T = 0.6 centuries practically coincides
with the first variant (line 1).

J. J. SMULSKY, Y. J. SMULSKY 141
Table 4. Influence of the initial conditions on results of integration of the Equation (1) by program Galactica and of the
equations of Apophis motion by system Horizons: TimeA and Rmin A are time and distance of Apophis approach with the
Earth in April 13, 2029, accordingly; RH is distance of passage Apophis with the Earth in April 13, 2036; TF and RF are time
and distance of the second approach (point F on Figure 7).
Solutions at different variants of initial conditions
Galactica Horizons
1 2 3 1 2 3
Parameters
30.11.2008
JPL sol.140
04.01.2010
JPL sol.144
30.11.2008
JPL sol.144
18.07.2006
JPL sol.144
30.11.2008
JPL sol.140
04.01.2010
JPL sol.144
TimeA 21:44:45 21:44:53 21:44:45 21:46:47 21:45:47 21:44:45
RminA , km 38905 37880 38813 38068 38161 38068
RH, 106 km 86.0 43.8 81.9 51.9 55.9 51.8
TF, cyr
from 30.11.08 0.5837 0.7138 0.6537 0.4237 0.9437 0.4238
RF, 103 km 622 1663 585 1515 684 1541
It is seen (Table 4) that the results of the third variant
differ from the first one much less than from the second
variant. In the second variant the change of positions and
velocities of acting bodies since November 30, 2008 for
04.01.2010 is computed under DE406, and in the third
variant it does under the program Galactica. The initial
conditions for Apophis in two variants are determined
according to alike JPL sol.144, i.e. in these solutions the
IC differ for acting bodies. As it is seen from Table 4,
the moment of approach in solutions 2 and 3 differs on 8
seconds, and the approach distance differs on 933 km.
Other results of the third solution also differ in the
greater degree with second ones, in comparison of the
third solution with first one. It testifies that the diffe-
rences IC for Apophis are less essential in comparison
with differences of results of calculations under two pro-
grams: Galactica and DE406 (or Horizons).
So, the above-mentioned difference of the initial con-
ditions (variants 1 and 3 tab. 4) do not change the time of
approach of April 13, 2029, and the distance of approach
in these solutions differ on 102 km. Other characteristics:
RH, TF and RF also change a little. Therefore it is possible
to make a conclusion, that the further refinement of Apo-
phis IC will not essentially change its trajectory.
The same researches on influence of the initial condi-
tions we have carried out with the integrator of NASA. In
system Horizons (the JPL Horizons On-Line Ephemeris
System, manual look on a site
http://ssd.jpl.nasa.gov/?horizons_doc) there is opportu-
nity to calculate asteroid motion on the same standard
dynamic model (SDM), on which the calculations in pa-
per [4] are executed. Except considered two IC we used
one more IC for Apophis at date of July 12, 2006, which
is close to date of September 01, 2006 in paper [4]. The
characteristics and basic results of all solutions are given
in Table 4. In these solutions the similar results are re-
ceived. For example, for 3-rd variant of Horizons the
graphs R in a Figure 7 up to T = 0.45 centuries practi-
cally has coincided with 2-nd variant of Galactica. The
time of approach in April 13, 2029 changes within the
bounds of 2 minutes, and the distance is close to 38,000
km. The distance of approach in April 13, 2036 changes
from 52 up to 56 million km. The characteristics of se-
cond approach for 100 years changes in the same bounds,
as for the solutions on the program Galactica. The above-
mentioned other relations about IC influence have also
repeated for the NASA integrator.
So, the calculations at the different initial conditions
have shown that Apophis in 2029 will be approached
with the Earth on distance 38 - 39 thousand km, and in
nearest 100 years it once again will approach with the
Earth on distance not closer 600 thousand km.
8. Examination of Apophis’ Trajectory in
the Vicinity of Earth
In order to examine the Apophis trajectory in the vicinity
of Earth, we integrated Equation (1) over a two-year pe-
riod starting from T1 = 0.19 century. Following each 50
integration steps, the coordinate and velocity values of
Apophis and Earth were recorded in a file.
The moment TA at which Apophis will most closely
approach the Earth falls into this two-year period. The
ellipse E0E1 in Figure 8 shows the projection of the
two-year Earth’s trajectory onto the equatorial plane xOy.
Along this trajectory, starting from the point E0, the Earth
will make two turns. The two-year Apophis trajectory in
the same coordinates is indicated by points denoted with
the letters Ap. Starting from the point Ap0, Apophis will
travel the way Ap0Ap1ApeAp2Ap0Ap 1 to most closely ap-
proach the Earth at the point Ape at the time TA. After that,
the asteroid will follow another path, namely, the path
ApeAp3Apf.
Figure 9(a) shows the trajectory of Apophis relative to

J. J. SMULSKY, Y. J. SMULSKY 142
Figure 8. The trajectories of Apophis (Ap) and Earth (E) in
the barycentric equatorial coordinate system xOy over a
two-year period: Ap0 and E0 are the initial position of Apo-
phis and Earth; Apf is the end point of the Apophis trajec-
tory; Ape is the point at which Apophis most closely ap-
proaches the Earth; the coordinates x and y are given in
AU.
Figure 9. Apophis’ trajectory (1) in the geocentric equato-
rial coordinate system xrOyr: a—on the normal scale, b—on
magnified scale on the moment of Apophis’ closest appro-
aches to the Earth (2); 3—Apophis’ position at the moment
of its closest approach to the Earth following the correction
of its trajectory with factor k = 0.9992 at the point Ap1; the
coordinates xr and yr are given in AU.
the Earth. Here, the relative coordinates are determined
as the difference between the Apophis (Ap) and Earth (E)
coordinates:
;
rApErAp
yyyxx x 
Along trajectory 1, starting from the point Ap 0, Apo-
phis will travel to the Earth-closest point Ape, the trajec-
tory ending at the point Apf. The loops in the Apophis
trajectory represent a reverse motion of Apophis with
respect to Earth. Such loops are made by all planets when
observed from the Earth (Smulsky 2007).
At the Earth-closest point Ape the Apophis trajectory
shows a break. In Figure 9(b) this break is shown on a
larger scale. Here, the Earth is located at the origin, point
2. The Sun (see Figure 8) is located in the vicinity of the
barycenter O, i.e., in the upper right quadrant of the
Earth-closest point Ape. Hence, the Earth-closest point
will be passed by Apophis as the latter will move in be-
tween the Earth and the Sun (see Figure 9(b)). As it will
be shown below, this circumstance will present certain
difficulties for possible use of the asteroid.
9. Possible Use of Asteroid Apophis
So, on April 13, 2029, we will become witnesses of a
unique phenomenon, the pass of a body 31 million tons
in mass near the Earth at a minimum distance of 6 Earth
radii from the center of Earth. Over subsequent 1000
years, Apophis will never approach our planet closer.
Many pioneers of cosmonautics, for instance, K. E. Tsi-
olkovsky, Yu. A. Kondratyuk, D. V. Cole, etc. believed
that the near-Earth space will be explored using large
manned orbital stations. Yet, delivering heavy masses
from Earth into orbit presents a difficult engineering and
ecological problem. For this reason, the lucky chance to
turn the asteroid Apophis into an Earth bound satellite
and, then, into a habited station presents obvious interest.
Among the possible applications of a satellite, the fol-
lowing two will be discussed here. First, a satellite can be
used to create a space lift. It is known that a space lift
consists of a cable tied with one of its ends to a point at
the Earth equator and, with the other end, to a massive
body turning round the Earth in the equatorial plane in a
24-hour period, Pd = 24 × 3600 sec. The radius of the sate-
llite geostationary orbit is

22
34π
42241 km 6.62
gsdA E
E
e
RPGmm
R


(16)
In order to provide for a sufficient cable tension, the
massive body needs to be spaced from the Earth center a
distance greater than Rgs. The cable, or several such ca-
bles, can be used to convey various goods into space
while other goods can be transported back to the Earth
out of space.
If the mankind will become able to make Apophis an
Earth bound satellite and, then, deflect the Apophis orbit
into the equatorial plane, then the new satellite would
suit the purpose of creating a space lift.
E
. (15) A second application of an asteroid implies its use as a

J. J. SMULSKY, Y. J. SMULSKY 143
“shuttle” for transporting goods to the Moon. Here, the
asteroid is to have an elongated orbit with a perihelion
radius close to that of a geostationary orbit and an apogee
radius approaching the perigee radius of the lunar orbit.
In the latter case, at the geostationary-orbit perigee goods
would be transferred onto the satellite Apophis and then,
at the apogee, those goods would arrive at the Moon.
The two applications will entail the necessity of solv-
ing many difficult problems which now can seem even
unsolvable. On the other hand, none of those problems
will be solved at all without making asteroid an Earth
satellite. Consider now the possibilities available here.
The velocity of the asteroid relative to the Earth at the
Earth-closest point Ape is AE = 7.39 km·s–1. The velo-
city of an Earth bound satellite orbiting at a fixed dis-
tance RminA from the Earth (circular orbit) is
v

1
min A3.2 kms
CEA E
vGmmR

 
(17)
For the asteroid to be made an Earth-bound satellite,
its velocity AE should be brought close to CE
v. We
performed integration of Equation (1) assuming the
Apophis velocity at the moment TA to be reduced by a
factor of 1.9, i.e., the velocity AE = 7.39 km·s–1 at the
moment TA was decreased to 3.89 km·s–1. In the later
case, Apophis becomes an Earth bound satellite with the
following orbit characteristics: eccentricity es1 = 0.476,
equator-plane inclination angle is1 = 39.2˚, major semi-
axis as1 = 74540 km, and sidereal orbital period Ps1 =
2.344 days.
v
v
We examined the path evolution of the satellite for a
period of 100 years. In spite of more pronounced oscilla-
tions of the orbital elements of the satellite in comparison
with those of planetary orbit elements, the satellite’s ma-
jor semi-axis and orbital period proved to fall close to the
indicated values. For the relative variations of the two
quantities, the following estimates were obtained: |  a| <
±2.75 × 10–4 and |  P| < ±4.46 × 10–4. Yet, the satellite
orbits in a direction opposite both to the Earth rotation
direction and the direction of Moon’s orbital motion.
That is why the two discussed applications of such a sat-
ellite turn to be impossible.
Thus, the satellite has to orbit in the same direction in
which the Earth rotates. Provided that Apophis (see Fig-
ure 9(b)) will round the Earth from the night-side (see
point 3) and not from the day-side (see line 1), then, on a
decrease of its velocity the satellite will be made a satel-
lite orbiting in the required direction.
For this matter to be clarified, we have integrated
Equation (1) assuming different values of the asteroid ve-
locity at the point Аp1 (see Figure 9). This point, located
at half the turn from the Earth-closest point Ape, will be
passed by Apophis at the time TAp1 = 0.149263369488169
century. At the point Аp1 the projections of the Apophis
velocity in the barycentric equatorial coordinate system
are 1Apx = –25.6136689 km·s–1, 1Apy = 17.75185451
km·s–1, and 1Apz = 5.95159206 km·s–1. In the numeri-
cal experiments, the component values of the satellite
velocity were varied to one and the same proportion by
multiplying all them by a single factor k, and then Equa-
tion (1) were integrated to determine the trajectory of the
asteroid. Figure 10 shows the minimum distance to
which Apophis will approach the Earth versus the value
of k by which the satellite velocity at the point Аp1 was
reduced.
v v
v
We found that, on decreasing the value of k (see Fig-
ure 10), the asteroid will more closely approach the
Earth, and at k = 0.9999564 Apophis will collide with the
Earth. On further decrease of asteroid velocity the aste-
roid will close with the Earth on the Sun-opposite side,
and at k = 0.9992 the asteroid will approach the Earth
center (point 3 in Figure 9(b)) to a minimum distance
Rmin3 = 39,157 km at the time T3 = 0.2036882 century.
This distance Rmin3 roughly equals the distance RminA to
which the asteroid was found to approach the Earth cen-
ter while moving in between the Earth and the Sun.
In this case, the asteroid velocity relative to the Earth
is also AE = 7.39 km·s–1. On further decrease of this
velocity by a factor of 1.9, i.e., down to 3.89 km·s–1
Apophis will become an Earth bound satellite with the
following orbit parameters: eccentricity es2 = 0.486, equ-
ator plane inclination angle is2 = 36˚, major semi-axis as2 =
76,480 km, and sidereal period Ps2 = 2.436 day. In addi-
tion, we investigated into the path evolution of the Earth
bound satellite over a 100-year period. The orbit of the
satellite proved to be stable, the satellite orbiting in the
same direction as the Moon does.
v
Figure 10. The minimum distance Rmin to which Apophis
will approach the Earth center versus the value of k (k is the
velocity reduction factor at the point Ap1 (see Figure 8)).
The positive values of Rmin refer to the day-side: The values
of Rmi n are given in km; 1—the minimum distance to which
Apophis will approach the Earth center on April 13, 2029
(day-side); 2—the minimum distance to which Apophis will
approach the Earth center after the orbit correction (night-
side); 3—geostationary orbit radius Rgs.

J. J. SMULSKY, Y. J. SMULSKY 144
Thus, for Apophis to be made a near-Earth satellite or-
biting in the required direction, two decelerations of its
velocity need to be implemented. The first deceleration is
to be effected prior to the Apophis approach to the Earth,
for instance, at the point Ap1 (see Figure 8), 0.443 year
before the Apophis approach to the Earth. Here, the
Apophis velocity needs to be decreased by 2.54 m/s. A
second deceleration is to be effected at the moment the
asteroid closes with the Earth. In the case under consid-
eration, in which the asteroid moves in an elliptic orbit,
the asteroid velocity needs to be decreased by 3.5 km·s–1.
Slowing down a body weighing 30 million tons by 3.5
km·s–1 is presently a difficult scientific and engineering
problem. For instance, in paper [7] imparting Apophis
with a velocity of 10–6 m/s was believed to be a problem
solvable with presently available engineering means. On
the other hand, Rykhlova et al. [7] consider increasing
the velocity of such a body by about 1 - 2 cm/s a difficult
problem. Yet, with Apophis being on its way to the Earth,
we still have a twenty-year leeway. After the World War
II, even more difficult a problem, that on injection of the
first artificial satellite in near-Earth orbit and, later, the
launch of manned space vehicles, was successfully solved
in a period of ten years. That is why we believe that, with
consolidated efforts of mankind, the objective under dis-
cussion will definitely be achieved.
It should be emphasized that the authors of Giorgini et
al. 2008 considered the possibility of modifying the Apo-
phis orbit for organizing its impact onto asteroid (144898)
2004 VD17. There exists a small probability of the aster-
oid’s impact onto the Earth in 2102. Yet, the problem on
reaching a required degree of coordination between the
motions of the two satellites presently seems to be hardly
solvable. This and some other examples show that many
workers share an opinion that substantial actions on the
asteroid are necessary for making the solution of the va-
rious space tasks a realistic program.
10. Asteroid 1950 DA Approaches to the
Earth
The distances to which the asteroid 1950 DA will ap-
proach solar-system bodies are shown versus time in
Figure 11. It is seen from Figure 11(a), that, following
November 30, 2008, during the subsequent 100-year pe-
riod the asteroid will most closely approach the Moon: at
the point A (TA = 0.232532 cyr and Rmin = 11.09 million
km) and at the point B (TB = 0.962689 cyr and Rmin =
5.42 million km). The encounters with solar-system bod-
ies the asteroid had over the period of 100 past years are
shown in Figure 11(b). The asteroid most closely ap-
proached the Earth twice: at the point C (TC = –0.077395
cyr and Rmin = 7.79 million km), and at the point D (TD =
–0.58716 cyr and Rmin = 8.87 million km).
Figure 11. Approach of the asteroid 1950 DA to solar-sys-
tem bodies. The approach distances are calculated with
time interval ∆T: a, b—∆T = 1 year; c—∆T = 10 years. Rmin,
km is the closest approach distance. Calendar dates of ap-
proach in points see Table 5. For other designations, see
Figure 5.
Over the interval of forthcoming 1000 years, the mi-
nimal distances to which the asteroid will approach so-
lar-system bodies on time span ∆T = 10 years are indi-
cated in Figure 11(c). The closest approach of 1950 DA
will be to the Earth: at the point E (TE = 6.322500 cyr
and Rmin = 2.254 million km), and at the point F (TF =
9.532484 cyr and Rmin = 2.248 million km).
To summarize, over the 1000-year time interval the
asteroid 1950 DA will most closely approach the Earth
twice, at the times TE and TF, to a minimum distance of
2.25 million km in both cases. The time TE refers to the
date March 6, 2641, and the time TF, to the date March 7,
2962.
Giorgini et al. [35] calculated the nominal 1950 DA
trajectory using earlier estimates for the orbit-element
values of the asteroid, namely, the values by the epoch of
March 10, 2001 (JPL sol.37). In paper [35], as the varia-
tion of initial conditions for the asteroid, ranges were set
three times wider than the uncertainty in element values.
For the extreme points of the adopted ranges, in the cal-
culations 33 collision events were registered. In this
connection, Giorgini et al. 2002 have entitled their pub-
lication “Asteroid 1950 DA Encounter with Earth in
2880”.
We made our calculations using the orbit-element va-

J. J. SMULSKY, Y. J. SMULSKY 145
lues of 1950 DA by the epoch of November 30, 2008
(JPL sol.51) (see Table 1). By system Horizons the JPL
sol.37 can be prolonged till November 30, 2008. As it is
seen in this case, the difference of orbital elements of the
solution 37 from the solution 51 on two - three order is
less, than uncertainties of orbit elements, i.e. the orbital
elements practically coincide.
With the aim to trace how the difference methods of
calculation has affected the 1950 DA motion, in Table 5
we give a comparison of the approach times of Figure 11
with the time-closest approaches predicted in paper [35].
According to Table 5, the shorter the separation between
the approach times (see points C and A) and the start time
of calculation (2008-11-30), the better is the coincidence
in terms of approach dates and minimal approach dis-
tances Rmin. For more remote times (see points D and B)
the approach times differ already by 1 day. At the point E,
remote from the start time of calculation by 680 year, the
approach times differ already by eight days, the approach
distances still differing little. At the most remote point F,
according to our calculations, the asteroid will approach
the Earth in 2962 to a distance of 0.015 AU, whereas,
according to the data of Giorgini et al. [35], a most close
approach to the Earth, to a shorter distance, will be in
2880.
So, our calculations show that the asteroid 1950 DA
will not closely approach the Earth. It should be noted
that our calculation algorithm for predicting the motion
of the asteroid differs substantially from that of Giorgini
et al. [35]. We solve non-simplified Equation (1) by a
high-precision numerical method. In doing so, we take
into account the Newtonian gravitational interaction only.
In paper [35], additional weak actions on the asteroid
were taken into account. Yet, the position of celestial
bodies acting on the asteroid is calculated from the
ephemerides of DE-series. Those ephemeredes approxi-
mate observational data and, hence, they describe those
data to good precision. Yet, the extent to which the pre-
dicted motion of celestial bodies deviates from the actual
motion of these bodies is the greater the farther the mo-
ment of interest is remote from the time interval during
which the observations were made. We therefore believe
that the difference between the present calculation data
for the times 600 and 900 years (points E and F in Table
5) and the data of Giorgini et al. [35] results from the
indicated circumstance.
11. Evolution of the 1950 DA Orbit
Figure 12 shows the evolution of 1950 DA orbital ele-
ments over a 1000-year time interval as revealed in cal-
culations made with time span ∆T = 10 years. With the
passage of time, the orbit eccentricity e non-monoto-
nically increases. The angle of longitude of ascending
node  , the angle of inclination ie to the ecliptic plane,
and the angle of perihelion argument ωe show more
monotonic variations. The semi-axis a and the orbital
period P both oscillate about some mean values. As it is
seen from Figure 12, at the moments of encounter with
the Earth, TE and TF, the semi-axis a and the period P
show jumps. At the same moments, all the other orbit
elements exhibit less pronounced jumps.
The dashed line in Figure 12 indicates the initial-time
values of orbital elements presented in Table 1. As it is
seen from the graphs, these values are perfectly coinci-
dent with the values for T = 0 obtained by integration of
Equation (1). The relative differences between the values
of e,
 , ie, ωe, a, and P and the initial values of these
parameters given in Table 1 are –3.1 × 10–4, –1.6 × 10–5,
–6.2 × 10–5, –1.5 × 10–5, –1.5 × 10–5, –1.0 × 10–4, and
–3.0 × 10–4, respectively. Such a coincidence validates
the calculations at all stages, including the determination
of initial conditions, integration of Equation (1), deter-
mination of orbital-parameter values, and the transforma-
tion between different coordinate systems.
It should be noted that the relative difference for the
Table 5. Comparison between the data on asteroid 1950 DA encounters with the Earth and Moon: Our data are denoted with
characters A, B, C, D, E, F, as in Figure 11, and the data by Giorgini et al. [35] are denoted as Giorg.
Source JD, days Date Time, days Body Rmin, AU
D
Giorg.
2433354
-
1950-03-13
1950-03-12
0.730
0.983
Earth
Earth
0.059273
0.059286
C
Giorg.
2451973
-
2001-03-05
2001-03-05
0.157
0.058
Earth
Earth
0.052075
0.052073
A
Giorg.
2463293
-
2032-03-02
2032-03-02
0.222
0.281
Moon
Earth
0.074158
0.075751
B
Giorg.
2489962
-
2105-03-09
2105-03-10
0.224
0.069
Moon
Earth
0.036260
0.036316
E
Giorg.
2685729
-
2641-03-06
2641-03-14
0.338
0.330
Earth
Earth
0.015070
0.015634
F
Giorg.
2802974
-
2962-03-07
2880-03-16
0.985
0.836
Earth
Earth
0.015030
0.001954

J. J. SMULSKY, Y. J. SMULSKY 146
Figure 12. Evolution of 1950 DA orbital parameters under
the action of the planets, the Moon, and the Sun over the
time interval 0 - 1000 from the epoch November 30, 2008:
1—As revealed through integration of motion Equation (1)
obtained with the time interval ∆T = 10 years: 2—Initial
values according to Table 1. The angular quantities,  , ie,
and

e, are given in degrees, the major semi-axis a—in AU,
and the orbital period P, in days.
same elements of Apophis is one order of magnitude
smaller. The cause for the latter can be explained as fol-
lows. Using the data obtained by integrating Equation (1),
we determine the orbit elements at the time equal to half
the orbital period. Hence, our elements are remote from
the time of determination of the initial conditions by that
time interval. Since the orbital period of Apophis is
shorter than that of 1950 DA, the time of determination
of Apophis’ elements is 0.66 year closer in time to the
time of determination of initial conditions than the same
time for 1950 DA.
12. Study of the 1950 DA Trajectory in the
Encounter Epoch of March 6, 2641
Since the distances to which the asteroid will approach
the Earth at the times TE and TF differ little, consider the
trajectories of the asteroid and the Earth at the nearest
approach time TE, March 6, 2641. The ellipse E0Ef in
Figure 13 shows the projection of the Earth trajectory
over a 2.5-year period onto the equatorial plane xOy.
This projection shows that, moving from the point E0 the
Earth will make 2.5 orbital turns. The trajectory of 1950
DA starts at the point A0. At the point Ae the asteroid will
approach the Earth in 2641 to a distance of 0.01507 AU.
The post-encounter trajectory of the asteroid remains
roughly unchanged. Then, the asteroid will pass through
Figure 13. The trajectories of Earth (1) and 1950 DA (2) in
the barycentric equatorial coordinate system xOy over 2.5
years in the encounter epoch of March 6, 2641 (point Ae): A0
and E0 are the starting points of the 1950 DA and Earth
trajectories; Af and Ef are the end points of the 1950 DA
and Earth trajectories; 3—1950 DA trajectory after the
correction applied at the point Aa is shown arbitrarily; the
coordinates x and y are given in AU.
the perihelion point Ap and aphelion point Aa, and the
trajectory finally ends at the point Af.
Figure 14(a) shows the trajectory of the asteroid rela-
tive to the Earth. The relative coordinates xr and yr were
calculated by a Equation analogous to (15). Starting at
the point A0, the asteroid 1950 DA will move to the point
Ae, where it will most closely approach the Earth, the end
point of the trajectory being the point Af. The loop in the
1950 DA trajectory represents a reverse motion of the
asteroid relative to the Earth.
On an enlarged scale, the encounter of the asteroid
with the Earth is illustrated by Figure 14(b). The Sun is
in the right upper quadrant. The velocity of the asteroid
relative to the Earth at the closing point Ae is vAE = 14.3
km·s–1.
13. Making the Asteroid 1950 DA an
Earth-Bound Satellite
Following a deceleration at the point Ae (see Figure
14(b)), the asteroid 1950 DA can become a satellite or-
biting around the Earth in the same direction as the Moon
does. At this point E (see Table 5) the distance from the
asteroid to the Earth’s center is RminE = 2.25 million km,
the mass of the asteroid being mA = 1.57 milliard ton.
According to (17), the velocity of a satellite moving in a
circular orbit of radius RminE is vCE = 0.421 km·s–1. For
the asteroid 1950 DA to be made a satellite, its velocity
needs to be brought close to the value vCE or, in other
words, the velocity of the asteroid has to be decreased by
∆V ≈ 13.9 km·s–1. In this situation, the asteroid’s mo-
mentum will become decreased by a value ma∆V =

J. J. SMULSKY, Y. J. SMULSKY 147
Figure 14. The 1950 DA trajectory in the geocentric equato-
rial coordinate system xrOyr: a—On ordinary scale; b—On
an enlarged scale by the moment of 1950 DA encounter
with the Earth: point O—The Earth, point Ae—The aster-
oid at the moment of its closest approach to the Earth; the
coordinates xr and yr are given in AU.
2.18 × 1016 kg·m/s, for Apophis the same decrease amounts
to ma·∆V = 1.08 × 1014 kg·m·s–1, a 200 times greater
value. Very probably, satellites with an orbital radius of
2.25 million km will not find a wide use. In this connec-
tion, consider another strategy for making the asteroid an
Earth-bound satellite. Suppose that the velocity of the
asteroid at the aphelion of its orbit (point Aa in Figure 13)
was increased so that the asteroid at the orbit perihelion
has rounded the Earth orbit on the outside of it passing
by the orbit at a distance R1. To simplify calculations, we
assume the Earth’s orbit to be a circular one with a radius
equals the semi-axis of the Earth orbit aE = 1 AU. So, in
the corrected orbit of the asteroid the perihelion radius
will be
1pcE
RaR (18)
Then, let us decrease the velocity of the asteroid at the
perihelion of the corrected orbit to a value such that to
make the asteroid an Earth-bound satellite. To check ef-
ficiency of this strategy, perform required calculations
based on the two-body interaction model for the asteroid
and the Sun (Smulsky 2007, Smulsky 2008). We write
the expression for the parameter of trajectory in three
forms:

1
1
122
0.5 1p
pa 2
p
pa
R
RR Rv Rv
a


  

, (19)
where

1
is the interaction parameter of the Sun and the asteroid,
ms is the Sun mass, mAs is the asteroid mass, and 
=
–0.6625 is the 1950 DA trajectory parameter.
Then, using (19), for the corrected orbit of the asteroid
with parameters Rpc and vac we obtain:

1
22
0.5 1pc
pc a
aac
R
RR Rv

 (21)
From (21), we obtain the corrected velocity of the as-
teroid at aphelion:


1
2
2pc
ac
aa pc
R
vRR R


 (22)
Using (19), we express 
 in terms of

1 and va, and
after substitution of this expression into (22) we obtain
the corrected velocity at aphelion:


1
2
p
ca
ac a
apcp
RR
vvRR R
 
 . (23)
From the second Kepler law, Ra·vac = Rpc·vpc, we de-
termine the velocity at the perihelion of the corrected
orbit:
p
cacap
vvRR
c

. (24)
As a numerical example, consider the problem on
making the asteroid 1950 DA an Earth-bound satellite
with a perihelion radius equal to the geostationary orbit
radius R1 = Rgs = 42,241 km. Prior to the correction, the
aphelion velocity of the asteroid is va = 13.001 km·s–1,
whereas the post-correction velocity calculated by Equa-
tion (23) is vac = 13.912 km·s–1. Thus, for making the
asteroid a body rounding the Earth orbit it is required to
increase its velocity at the point Aa in Figure 13 by 0.911
km·s–1. The corrected orbit is shown in Figure 13 with
line 3.
According to (24), the velocity of the asteroid at the
perihelion of the corrected orbit is vpc = 35.622 km·s–1.
Using Equation (7), for a circular Earth orbit with

 =
–1 and Rp = aE, and with the asteroid mass mAs replaced
with the Earth mass mE, for the orbital velocity of the
Earth we obtain a value vOE = 29.785 km·s–1. According
to (17), the velocity of the satellite in the geostationary
orbit is vgs = 3.072 km·s–1. Since those velocities add up,
for the asteroid to be made an Earth satellite, its velocity
has to be decreased to the value vOE + vCE = 32.857
km·s–1. Thus, the asteroid 1950 DA will become a geo-
stationary satellite following a decrease of its velocity at
the perihelion of the corrected orbit by vpc – (vOE + vCE) =
2.765 km·s–1.
We have performed the calculations for the epoch of
2641. Those calculations are, however, valid for any epoch.
Our only concern is to choose the time of 1950 DA orbit

s
As
Gm m   (20)

J. J. SMULSKY, Y. J. SMULSKY 148
correction such that at the perihelion of the corrected
orbit the asteroid would approach the Earth. Such a
problem was previously considered in Smulsky 2008,
where a launch time of a space vehicle intended to pass
near the Venus was calculated. The calculations by Equ-
ations (18)-(24) were carried out on the assumption that
the orbit planes of the asteroid and the Earth, and the
Earth equator plane, are coincident. The calculation me-
thod of Smulsky 2008 allows the calculations to be per-
formed at an arbitrary orientation of the planes. In the
same publication it was shown that, following the deter-
mination of the nearest time suitable for correction, such
moments in subsequent epochs can also be calculated.
They follow at a certain period.
In the latter strategy for making the asteroid 1950 DA
a near-Earth satellite, a total momentum ma·∆V =
ma·(0.911 + 2.765) × 103 = 5.77 × 1015 kg·m/s needs to
be applied. This value is 4.8 times smaller than that in the
former strategy and 53 times greater than the momentum
required for making Apophis an Earth satellite. It seems
more appropriate to start the creation of such Earth satel-
lites with Apophis. In book [36], page 189, it is reported
that an American astronaut Dandridge Cole and his
co-author Cox [37] advanced a proposal to capture pla-
netoids in between the Mars and Jupiter and bring them
close to the Earth. Following this, mankind will be able
to excavate rock from the interior of the planetoids and,
in this way, produce in the cavities thus formed artificial
conditions suitable for habitation. Note that another pos-
sible use of such satellites mentioned in [37] is the use of
ores taken from them at the Earth.
Although the problem on making an asteroid an Earth
satellite is a problem much easier to solve than the prob-
lem on planetoid capture, this former problem is none-
theless also a problem unprecedented in its difficulty. Yet,
with this problem solved, our potential in preventing the
serious asteroid danger will become many times en-
hanced. That is why, mankind getting down to tackling
the problem, this will show that we have definitely
passed from pure theoretical speculations in this field to
practical activities on Earth protection of the asteroid
hazard.
14. Discussion
In the 20th century, in the science of motion, namely, in
the mechanics the changes have been introduced that led
to the opinion of the movements’ indeterminacy. The Ge-
neral Theory of Relativity is beginning of the changes.
We have studied all aspects of this theory and established
its wrong reasons in the works [16,17,26,38]. Unlike
existing methods, our method does not use the distortion
of mechanics. Besides for the solution of differential
equations we have developed a new method of high ac-
curacy. Therefore, our results are more accurate and re-
liable. But our work causes sharp objections of support-
ers of the existing methods. Below they are presented in
the form of objections.
The success of scientific research often depends on the
worldview of a scientist. He can have a misconception in
the field, seemingly distant from the sphere of research.
However, it is not to permit it to set scientific truth. The
above applies to certain objections. We considered it
necessary bring them and give them an answer, because
this judgments interfere with the comprehension of sci-
entific truth.
1. Objection. It is stated, paragraph 1. INTRODUC-
TION, that presently available methods for predicting the
travel path of extraterrestrial objects lack sufficient ac-
curacy..., but this pronouncement is not justified in any
meaningful way. In fact, it is generally regarded that the
limitation on prediction is set by observational uncer-
tainties, not computational abilities. As is noted, the ra-
diation pressure forces set a limit on prediction of Apo-
phis and 1950 DA over very long periods of time, but
again, the limitation is on our ability to measure or esti-
mate these forces, not on computational limitations.
Answer. Apart from observational errors and the ra-
diation-pressure force, there exist many other factors
causing the difference between the calculated trajectory
and the actual motion of an asteroid. In our paper, vari-
ous approaches proposed by different authors are ana-
lyzed, and a method, free of many drawbacks, is used to
solve the problem.
The referee expresses an opinion that in our paper we
do not prove that methods capable of predicting the mo-
tion of asteroids with satisfactory accuracy are presently
lacking. However, the absence of such methods immedi-
ately follows from the publications under consideration.
It was not the point to prove that.
2. Objection. The suggestion to alter the orbits of these
two objects to put them in orbit about the Earth seems
absurd, and without justification. As noted, the change in
velocity required to accomplish this is in the several
km/sec range. It is barely conceivable with present tech-
nology to make a change of a few cm/sec, five orders of
magnitude less than would be required to place either
object in Earth orbit. The authors make the cavalier
statement that it might be possible to accomplish this,
making reference to the advance from bare orbiting of
instruments around the Earth to landing men on the
moon in only a bit more than a decade. But they ignore
the fact that the physics of how to do the latter was al-
ready known before the former was done, whereas in
moving asteroids around by km/sec increments of veloc-
ity is far beyond any currently understood technology.
It’s a bit like asking Christopher Columbus to plan a
vessel to transport 400 people across the Atlantic in six

J. J. SMULSKY, Y. J. SMULSKY 149
hours—he wouldn’t even know where to begin.
Answer. In our paper, we put forward an idea of cap-
turing an asteroid in Earth orbit, analyze available possi-
bilities in implementing this project, and calculate nec-
essary parameter values.
We do not consider the engineering approaches that
can be used in implementing the idea. That is a different
field of knowledge, and this matter is to be analyzed in a
separate publication.
As for the referee’s remark on Christopher Columbus,
the history saw how in 1485 the Columbus’ proposal
about an expedition to be send through the West Ocean
to India was rejected by the Mathematician Council in
Portugal. Later, in 1486, the project by Columbus was
also rejected by the Academic Senate in the University of
Salamanca, Spain, a famous university in the Middle
Ages alongside with the Montpellier, Sorbonne and Ox-
ford, because the project had incurred ridicule as resting
on the “very doubtful” postulate of Earth’s sphericity.
The objection expresses an opinion that the idea of
man’s exit into the outer space was implemented rather
fast because the physics necessary for solving the prob-
lems behind that project was known, whereas the physics
of how to implement our project presently remains an
obscure matter.
The problem of man’s exit into space was solved by
engineers rather than scientists. When engineers had
solved all problems, then established scientists had be-
come able to catch the physical essence of the matter.
Presently, academic scientists are in captivity of rela-
tivistic fantasies about micro- and macro-world. As a
result, they failed to properly understand the entire phy-
sical picture of our world, including the space travel phy-
sics. The best thing such physicists could do is not to
interfere into the projects actually important for mankind
like the project we discuss in our paper.
3. Objection. In considering the motion of the aster-
oids the paper describes only the asteroids are integrated,
the other perturbations are derived from planetary ephe-
merides.
Answer. In our study, we integrated not only the mo-
tion of the asteroid; we also integrated the motion of
other celestial bodies.
4. Objection. The integration method the authors pre-
sent is not new (though the implementation in software
may be). They present a simple, fixed-step Newtonian
integrator that models only gravitational point masses.
Far more sophisticated methods and physics have been
published before precisely because the approach the au-
thors go on to describe is inadequate.
Answer. Whether our integration method is new or not,
the definition here is rather relative. Formula (2) in our
paper gives a specialist the general idea behind our me-
thod and some details of its implementation is stated in
the paper. Since none of the already existing methods
was used in treating the problem we deal with in our
study, we qualify our method as an original one. In our
opinion, our method is akin to the method of Taylor-
Steffensen series rather than to the Newton method. In
this approach the derivatives are determined by recur-
rence formulas. In our method (program Galactica) the
derivatives are calculated under the exact analytical for-
mulas, which we have deduced. This provides greater
accuracy than other methods.
5. Objection. The presently available methods for pre-
dicting the travel path of extraterrestrial objects are fine.
They are the same ones used to deliver spacecraft to
planets and fit measurement data-arcs hundreds of years
long.
Answer. One of the deficiencies of presently available
methods for integrating the motion of celestial bodies is
that those methods were constructed so that to provide a
best fit to observational data. Within the period of avail-
able observations, those methods proved to yield rather
good results. On the other hand, calculations of the mo-
tion of a previously unobserved object will obviously
yield worse results. Also, calculations of the motion of a
body observed during some observation period per-
formed far outside this period will also yield less accu-
rate results.
6. Objection. It is the limited knowledge of the physic-
cal properties of the objects that is the problem. Given
measurements of those properties (spin, reflectivity, etc.),
proper prediction is possible within computable error
bounds.
Answer. The opinion that physical properties of an
asteroid such as reflectivity or spin may notably affect
the asteroid’s motion is an erroneous opinion. This opin-
ion is the consequence of deficiencies inherent to the
methods mentioned in Answer 5. The actual motion of
celestial bodies and spacecraft having been found differ-
ent from their calculated motion, the people dealing with
celestial mechanics undertook introducing additional
fictitious forces into motion equations, such as the Yar-
kovsky force, whose magnitude was assumed to be de-
pendent on the physical properties of a particular body
under study.
7. Objection. It is stated (p. 1. INTRODUCTION), that
“… the Apophis trajectory will for long remain... cha-
otic”. No. Error growth is almost entirely in the along-
track direction. It is not chaotic over relevant time-scales
and measurements likely in 3 years will radically reduce
those prediction uncertainties about 97%. This is de-
scribed in the papers the authors reference, so seems to
be a misunderstanding.
Answer. It was the authors of cited publications rather
than us that have qualified the motion of Apophis as a
chaotic motion.

J. J. SMULSKY, Y. J. SMULSKY 150
8. Objectio n. The planetary ephemeris error it is far
less than radiation related effects like solar pressure and
Yarkovsky thermal re-radiation. The Giorgini et al. 2008
paper referenced shows what is required for better pre-
diction is PHYSICAL KNOWLEDGE of the object (mea-
surement), not METHOD.
Answer. Indeed, Giorgini et al. [4] have demonstrated
that the solar pressure and Yarkovsky thermal re-radia-
tion may have a considerable influence on the motion of
celestial bodies. We, however, believe that the results by
Giorgini et al. are erroneous. First, the forces of Giorgini
et al. dealt with are fictitious nonexistent forces. Second,
the interaction constants of those forces were artificially
overestimated by Giorgini et al. [4].
We would like to deliver here some additional remarks
concerning the fictitious nature of some forces. More
than half a century ago it was shown by some physicists
that no light pressure is observed in nature. Unfortunately,
those results have been forgotten by many physicists.
The Yarkovsky force was introduced so that to com-
pensate for the difference between the observed motion
of celestial bodies, spacecrafts and their motion as pre-
dicted by contemporary theories. As it was already men-
tioned, the presently available methods for predicting the
motion of celestial bodies suffer from serious deficien-
cies. Those deficiencies need to be overcome, and we
believe that, following this, the difference between the
actual motion of bodies and their motion as predicted
assuming only the Newtonian gravity force to be opera-
tive will be made negligible and even exiled from final
results. Then, additional fictitious forces will no longer
be needed.
Let us give here some direct arguments proving that
the forces under discussion are in fact fictitious forces.
When in mechanics someone says that a force acts on a
body, this does not mean that the force presents a mate-
rial object. The sentence “a force acts on a body” is just
slang. In mechanics, we imply that some body acts on
another body. The influence is manifested in the changed
motion of the second body. A change in motion is de-
fined by body’s acceleration. Hence, the action exerted
by the first body consists in an acceleration experienced
by the second body.
Man has invented mechanics in which actions are de-
fined by an auxiliary quantity called the force. The force
was defined as a quantity proportional to acceleration
accurate to a factor (for details, see our books [16,17]).
So, the term “force” is not a name for an object in our
world. When somebody says that a body on an inclined
board experiences the actions due to the friction force
and due to the gravity force, we imply that the body is
acted upon by the board and, through the gravity interac-
tion, by the Earth. When somebody says that the Moon is
acted upon by the gravity force due to Earth, this means
that it is the Earth that acts on the Moon.
On the other hand, in the case of light-pressure and
Yarkovsky forces the acting bodies are missing. If one
thinks of light considering it as a photon flux, he has to
remember that photons have no mass, and they are there-
fore no physical bodies. Yarkovsky had invented his
force as a force due to either particles, which are also
nonexistent objects. Thus, both the light pressure and Yar-
kovsky thermal re-radiation are not actions due to bodies;
such forces therefore bear no relation to mechanics. The
only application fields of such forces are extra-sensory
perception and Hollywood movies. Those forces “can be
used” in ephemerid approximation models, such as SDM,
because they all the same need to be fitted to many hun-
dred thousand observations.
9. Objection. In p. 2. PROBLEM STATEMENT the in-
sufficient information was provided to determine what
integration algorithm was used by the authors. This is
unacceptable given the rest of the paper.
The previously published literature on this subject is
vast and highly developed and should be drawn upon and
referenced.
Answer. Formulas (1) and (2) in our paper give the
general idea behind our method, and they also define the
form of master equations used in it. Details of the algo-
rithm, and those of the method and equations, are too
numerous to be outlined in the paper. We exploited our
method over a period of more than ten years, and during
that period, using the method, we have solved many pro-
blems. Some of our results were reported in publications
[18-21,26,38]. In those publications, some details of the
algorithm were described, and ample data on the ade-
quacy of our method and credibility of solutions obtained,
given. Below we list some of the problems that were
tackled with the help of the Galactica software.
9.1. Evolution of planetary orbits and the orbit of
Moon over the period of one hundred million years [18,
19]. It was for the first time that non-simplified differen-
tial equations of motion were integrated. The periods and
amplitudes of planetary-orbit oscillations were evaluated,
and stability of the Solar System was demonstrated (see
Figures 3 and 4).
9.2. Optimal flight of a spacecraft to the Sun [21]. The
spacecraft was proposed to use the gravitational maneu-
ver near to Venus. The launch regime of the spacecraft
allowing minimization of its starting velocity was identi-
fied.
9.3. Compound model of Earth rotation and the evolu-
tion of Earth rotation axis [20]. The Earth is considered
as a system of several bodies located in the equatorial
plane of a central body. The motion of one of the periph-
eral bodies models the motion of Earth rotation axis. The
evolution of Earth rotation axis was calculated over a
period of 110 thousand years. It was found that the Earth

J. J. SMULSKY, Y. J. SMULSKY 151
rotation axis precesses relative to the non-stationary axis
of Earth orbital motion.
9.4. Compound model of Sun rotation and its out-
comes for the planets [26] and [38]. The Sun rotation
period is 25.38 days. The Galactica software was used to
predict the outcomes of the compound model of Sun ro-
tation on nearest planets. As a result of the calculations,
an excessive revolution of Mercury perihelion was iden-
tified, which was previously explained assuming other
mechanisms to be operative.
9.5. Multilayer ring structures [39]. The structure of
interest comprises several rings, each of the rings involv-
ing several bodies. Evolution of several such ring struc-
tures was calculated, and stable and unstable configure-
tions were identified.
10. Objectio n. In p. 4. PREPARATION OF INITIAL
DATA OF ASTEROIDS three pages of discussion and
Equations (5)-(14) on the transformation of orbital ele-
ments to Cartesian coordinates could be deleted. This
material is found in every introductory celestial mechan-
ics course and need not be belabored.
Answer. Orbital elements can be transformed into
Cartesian coordinates in different ways that yield diffe-
rent results. We have chosen the best transformation; we
have deduced them and therefore give it in our paper. In
addition, our consideration involves some formulas not
be found in standard courses on celestial mechanics.
11. Objection. Further, the authors state their goal is
to compute barycentric Cartesian coordinates, but then
describe only heliocentric transformations. No informa-
tion on if or how transformation from heliocentric to the
barycentric needed by their code is given leads the
reader to wonder if heliocentric coordinates were im-
properly used in the barycentric code.
Answer. The transformation of heliocentric coordi-
nates to barycentric ones are firstly omitted from our
paper as presenting a matter of common knowledge.
Guided by the referee’s remark, now we discuss it in our
paper after Equation (14).
12. Objection. In end of p. 4. PREPARATION OF INI-
TIAL DATA OF ASTREOIDS the authors have written:
“… the masses of those bodies were modified by us…”
This would introduce a dynamical inconsistency within
the planetary ephemeris used to compute perturbations
in the integration. Was the magnitude of this inconsis-
tency computed?
The coordinates from DE405/406 said to be used are
derived from the original planetary masses. Change
those masses and the positions will change, hence per-
turbations on the object being integrated, hence the re-
sult of the integration.
Answer. We integrate Equations (1) for a total of
twelve bodies, including the planets, Moon, Earth, and
Apophis. We did not use planet and Moon coordinates
taken from ephemerides; hence, any mass values can be
adopted. The closer are the mass values to real masses,
the better is the consistency between the calculated and
observational data. We have checked this fact. In Galac-
tica, the mass values and the initial data are specified in a
separate file, which can easily be replaced with another
file. Now, the relative mass values are taken from the
DE405 system, whereas the absolute values have been
recalculated as G·MEarth, where MEarth is the Earth mass
from the IERS system. The mass values adopted in our
calculations are indicated in Table 2.
13. Objection. Studying Figure 5 at length, it is unable
to interpret it. It seems to show two dots for Earth at
point A; the text says there is only one.
Time scale would be better in calendar years instead
of fractional centuries.
A figure is used if it shows relationships or trends
clearly. This figure does not. Why not a useful table of
numerical values?
Answer. The first dot in the horizontal line Ea refers
to time А. The second dot after interval Т = 1 year re-
fers to the Earth, too. As it is seen from the graph, here
the distance to which the asteroid closes the Earth is
greater than 4.25 × 107 km.
Figure 5 is indeed an uncommon representation. How-
ever, in case this uncommonness is overcome, Figure 5
gives a clear picture of the asteroid’s approach to all the
bodies over the whole considered time interval. No such
picture can be grasped from a table.
14. Objection. It is stated in end of p. 5 APOPHIS’
ENCOUNTER WITH THE PLANETS AND THE MOON
that “As for the possible approach of Apophis to the
Earth in 2036, there will be no such approach…”
This is another fundamental misunderstanding of the
paper resulting from an incorrect analysis.
The authors integrate a nominal orbit solution only
and find it does not closely approach the Earth in 2036.
However, it is necessary to examine not just the single
nominal orbit, but the set of statistically possible orbit
variations, defined by the orbit solution covariance ma-
trix, as well as physical uncertainties (uncertainites).
The papers the authors cite go into such statistical
approaches extensively.
Why does this fundamental issue of modern orbit de-
termination not exist in this paper?
The analysis the authors provide does not recognize
the statistical nature of the problem. The authors ap-
proach is not acceptable for analyzing such problems
because it ignores the statistical distribution of orbit
variations defined by the measurement dataset.
This alone renders the paper and its conclusions ir-
relevant to readers.
Answer. We regard such a statistical study a vain un-
dertaking.

J. J. SMULSKY, Y. J. SMULSKY 152
If many measurement data for a parameter are avail-
able, then the nominal value of the parameter, say, ec-
centricity en, presents a most reliable value for it. That is
why a trajectory calculated from nominal initial condi-
tions can be regarded as a most reliable trajectory. A tra-
jectory calculated with a small deviation from the nomi-
nal initial conditions is a less probable trajectory, where-
as the probability of a trajectory calculated from the pa-
rameters taken at the boundary of the probability region
(i.e. from e = en ± 
e) tends to zero. Next, a trajectory
with initial conditions determined using parameter values
trice greater than the probable deviations (i.e. e = en ±
3

e) has an even lower, negative, probability. Since ini-
tial conditions are defined by six orbital elements, then
simultaneous realization of extreme (boundary) values (±
3  ) for all elements is even a less probable event, i.e. the
probability becomes of smaller zero.
That is why it seems that a reasonable strategy could
consist in examining the effect due to initial conditions
using such datasets that were obtained as a result of suc-
cessive accumulation of observation data. Provided that
the difference between the asteroid motions in the last
two datasets is insignificant over some interval before
some date, it can be concluded that until this date the
asteroid motion with the initial conditions was deter-
mined quite reliably.
Such computations were carried out and described in
the additional Section 6. Influence of initial conditions.
15. Objection. In p. 6. APOPHIS ORBIT EVOLUTION
the authors describe integrating the orbit of Apophis
over 200 years, writing out a file of coordinates each
year. They then go back and, starting from each file, in-
tegrate one Apophis orbit period and save that to a file.
Why? 201 integrations are being done when one would
suffice. Is not going back and integrating from the start-
ing point of each yearly file the same as integrating con-
tinuously over the span?
Answer. On integration of Equation (1), we obtain
coordinates of each body in the barycentric system. For
determining a body’s orbital elements, it is required to
consider the coordinates of the body with respect to a
parent body (for an asteroid, with respect to the Sun)
during one orbital period. To avoid a complex logic in
choosing coordinate values, in integration over the whole
time interval of interest, we chose to adhere to the strat-
egy described in the paper.
16. Objection. In p. 9. POSSIBLE USE OF ASTEROID
APOPHIS the argument made for capturing Apophis into
Earth orbit is at a level suitable for sketching on a nap-
kin. No discussion of material properties, or mechanics.
The composition of Apophis is unknown and the discus-
sion amounts to speculation for personal entertainment.
Answer. In the paper, we describe available strategies
for making the asteroid an Earth-bound satellite and cal-
culate parameter values necessary for realization of such
a project. An analytical background behind those strate-
gies is developed. The motion of the asteroid after tra-
jectory correction and the motion of formed satellites
were determined by integrating Equations (1). We do not
describe all the obtained results in our paper; however,
those results were used to substantiate the proposed
strategies in capturing the asteroid in Earth orbit. Those
strategies are unobvious, and it should be remembered
that one can propose strategies that never can be imple-
mented. We propose realizable strategies. We have cal-
culated the orbit evolution of the satellites and proved
that those orbits can be made stationary for a long time.
The computations for satellites were made taking into
account the action exerted on them by all bodies. We
believe our calculations to be original. Following our
publication, other workers will move farther in this di-
rection.
How can those strategies be implemented? This matter
will be discussed after the present results are reported in
the literature. For the time being, we raise the issue of
making an asteroid an Earth-bound satellite. This issue is
given rather a deep analysis. All computations are per-
formed at a good scientific level. That is why our results
are not to be ignored, and the work, regarded as a sketch
on napkin, to be one day thrown away. It is more prob-
able that it is the statistical data on the asteroid’s en-
counter with the Earth rather than our paper that will be
one good day thrown away.
17. Objection. Ii is stated in the beginning p. 9 that
“Over subsequent 1000 years, Apophis will never ap-
proach our planet closer”. The analyses given cannot
support the statement. All uncertainties physical and
measurement are ignored by the authors. Only the single
nominal orbit is considered. This is unacceptable and the
results of no interest to readers.
Answer. Indeed, our calculations show that the aster-
oids will not hit the Earth. On conscientious analysis,
statistical data on such collisions in the cited publications
are also indicative of this fact. Only undisguised trick-
sters, reasoning from such statistics, can frighten the so-
ciety with the threat of Apophis danger. With passage of
time, people usually become aware of scientific trickery,
and this deteriorates their trustfulness to science. The
way we propose in our paper will allow mankind to de-
velop in the future a good method for preventing the po-
tential threat of asteroid’s collisions with Earth. Note that
such method can only be implemented if we find a way
for making asteroids Earth-bound satellites.
The dynamics of Solar system is not linear. Thus, there
can be orbits with initial conditions intermediate to those
that authors used, which can lead to closer approaches.
Actually, the theory predicts that there are KEYHOLES,
associated to RESONANT RETURNS which can lead to

J. J. SMULSKY, Y. J. SMULSKY 153
collisions. This aspect is missing in this work.
In this paper no new results. This looks more like the
report of a beginner entering in this field for the first
time, and just setting up the software tools and the con-
ceptual know-how to be able, in the future, to perform
research in this field. In particular some conceptual
building blocks are still missing, such as the notion of
chaos (mentioned just once as dreaded possibility, while
it is a well established fact that all the asteroids which
can impact the Earth are on chaotic orbits), and the ef-
fect of nonlinearity in the orbit determination and in the
propagation of the uncertainty to a future time.
Answer. There are three aspects in this objection:
1) Within the uncertainty ellipsoid of the orbital ele-
ments do not found a collision with Earth;
2) Do not shown the KEYHOLES of resonant and cha-
otic trajectories;
3) There are no new results.
Our paper shows that the search for collisions within
the uncertainty ellipsoid of the orbital elements is mean-
ingless work. We have also shown that the conclusion
about chaotic motion is caused by imperfection of meth-
ods of integrating the equations.
By another method and in another way we are solving
this problem. We have got new results: Asteroids Apo-
phis and 1950 DA do not impact the Earth. In addition
we are putting forward and are based the new idea: The
transformation of the asteroids in the satellites.
So, in our paper the new methods are used, the new
results are received and the new ideas are putted forward.
In contrast to our paper the published papers, which
we cited above, prove the false idea about collisions in
2036 and in 2880. These papers are misleading readers.
When the scientists’ errors become clear, the society has
intensified distrust of science.
In published papers the imaginary constructs are in-
vestigated: Chaos, resonances, keyholes, etc. Their au-
thors use methods with the imaginary precision by which
supposedly can determine the motion of the planets up to
mm and up to marcsec. We emphasize the imaginary
precision that arises when comparing the methods on
those observations, to which they are fitted. If someone is
using them to calculate the outside of this area, the mo-
tions of bodies differ significantly from the calculated
movements. The authors of published papers believe that
there are fictitious forces (Yarkovsky force, etc.), reso-
nances and keyholes, which make the body motion cha-
otic. That is, rather than to doubt the accuracy of the
methods they put forward the reasons for their justifica-
tion.
The same methods found that the Solar system after 20
million years ago is starting to change, and in the future
because of the chaos it begins to collapse. The reason for
these phenomena lies in the imperfection of methods for
calculating the motions. In contrast, our method allowed
us to integrate the equations of the Solar system motion
for 100 million years: The Solar system is stable and no
signs of change. So the keyholes, resonances, chaos and
the fictitious forces appear due to imperfect methods of
calculating the motions.
Our paper cannot be viewed superficially, it must be
deeply studied. It gives a lot of new knowledge about the
evolution of the asteroids motion, the accuracy of inte-
gration methods and on the ways in which to develop
these methods.
The modern celestial mechanics dominates by ideas of
indeterminacy, of unpredictable resonances and of cha-
otic motions. Our paper provides the mathematical tools
and techniques that allow us to calculate the movement
with known accuracies, and then to implement them. The
paper presents a path that each can go through and check
out our results. This is the science.
But the chaos, the resonances, the keyholes are not the
science, those are Extrasensory.
It is need return to the classical celestial mechanics,
the creators of which are not doubted the determinacy of
movements.
15. Conclusions
1) The instability and randomness in dynamics of
planets and asteroids is caused by imperfection of meth-
ods of account of movements;
2) The parameters of planets orbits are steady chang-
ing with the certain periods and amplitudes;
3) On 21 hour 45' GMT, April 13, 2029 Apophis will
pass close to the Earth, at a minimum distance of 6 Earth
radii from Earth’s center. This will be the closest pass of
Apophis near the Earth in the forthcoming one thousand
years;
4) Calculations on making Apophis an Earth bound
satellite appropriate for solving various space exploration
tasks were performed;
5) The asteroid 1950 DA will twice approach the Earth
to a minimal distance of 2.25 million km, in 2641 and in
2962;
6) At any epoch, the asteroid 1950 DA can be made an
Earth-bound satellite by increasing its aphelion velocity
by ~1 km·s–1 and by decreasing its perihelion velocity by
~2.5 km·s–1.
16. Acknowledgements
The authors express their gratitude to T. Yu. Galushina
and V. G. Pol, who provided them with necessary data on
asteroid Apophis. They are also grateful to the staff of
the Jet Propulsion Laboratory, USA, whose sites were
used as a data source from which initial data for integra-
tion of motion equations were borrowed. The site by

J. J. SMULSKY, Y. J. SMULSKY 154
Edward Bowell (ftp://ftp.lowell.edu/pub/elgb/) was help-
ful in grasping the specific features of asteroid data rep-
resentation and in avoiding possible errors in their use.
Krotov O. I. took part in calculations of the Apophis mo-
tion on the system Horizons. The calculations were car-
ried out on the supercomputer of the Siberian Super-
computer Centre of Siberian Branch RAS.
The study was carried out as part of Integration Pro-
grams 13 (2008-2011) of the Presidium of the Russian
Academy of Sciences.
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