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JOURNAL OF KUFA–PHYSICS A SPECIAL ISSUE FOR KUFA´S FIRST CONFERENCE FOR PHYSICS
707
Trajectory analysis of the comet 46p/Wirtanen
M.J.F. Al Bermani
H.H.Jawad
Physics department
College of Science
University of Kufa
Abstract:
The main objectives are to investigate the trajectories of 46p/Wirtanen as short
periodic comets under the effects of sun gravitational attractions and perturbations
caused by the nine planets of the solar system. Perturbations due to the major planets,
Jupiter and Saturn are considered for their dominance. .Numerical integration of the
equation of motion of the comet through the solar system was performed by using
Adams-Bashforth method.
الخالصة
انشس واالضطشاباث انناجت عن انشئست ه انبحذ ف يساساث انزنباث انذوست ححج حأرش جاربت األهذاف
القت يزم صحم وانشخشي حأرشاث جاربت انكىاكب انساسة انخسعت . وحى انخشكض عهى حأرشاث انكىاكب انع
_باشفىسد .آديضطشقت بسبب كخهها انكبشة . حى حم يعادالث حشكت انزنب خالل انجىعت انشست باسخخذاو
Trajectory analysis of the comet 46p/Wirtanen:M.J.F. Al Bermani , H.H.Jawad
708
1- Introduction
The orbits of comets can be classified in to three major types ellipse ,parabola and
hyperbola ,while comet in the elliptical orbits are periodic making repeated returns,
those in parabolic or hyperbolic orbits are close to sun only for a single short period of
time (Montenbrouck,1994).
The comet 46p/wirtanen is a good example for the short period comets with elliptical
orbits it was discovered in 1948 by C.A.Wirtanen as an object of 16th
magnitude
a month and half after a comet perihelion time
The two approaches of comet to Jupiter in 1972 and 1984 are responsible for
significant change of orbital elements. (Malgorzata&Grzegorz,1996)
The orbit of 46p/wirtanen has been investigated by (Vaghi & Rickman,1982) ,a
major perturbation in 1971 reducing the orbital period from 6.7 to 5.9 years and
perihelion distance from 1.61 to 1.26 AU and again in 1977 which reduce both period
and perihelion distance still further to 5.5 years and 1.08 A U (Belyaev,1986),
(Malgorzata&Grzegorz,1999),slight further reduction in the perihelion distance
occurred with 1.064 AU at 1997 return and 1.059 AU in 2002 where the period in
2002 is 5.44 years(kidger,2004),further studied was carried by
(Grav,2001),(Aksnes&Grav,2005).
2- Orbit model The position of comet in its orbits is defined by the radius vector r and the true
anomaly .
In two body problem (where we neglect the perturbations from the other bodies) we
have
cos.e
pr
1 (1)
Where p is the is semi-latus rectum (some time known as the orbital parameter) and
it defines the size of the orbit, where equation (1) correspond to a conic in which the
sun is at one of the foci, so in order to determine the position of the comet in its orbit
at any given instant, we must solve the Kepler equation for eccentric anomaly E
)t(M)t(Esine.)t(E
180 (2)
Where )(360)(T
tttM , T , t , t is the mean anomaly, orbital period ,the time of
perihelion and the time after perihelion passage respectively .
The Kepler equation cannot be solved analytically with respect to time so we need to
use iteration method.
The Newton iteration method is sufficient to elliptical and hyperbolic orbits where the
iterative expression for ellipse and hyperbolic orbits are
1
1
180
Hcosh.e
)t(h
MHHsinh.eHH
Ecos.e
)t(MEsin.eE
EE
(3)
JOURNAL OF KUFA–PHYSICS A SPECIAL ISSUE FOR KUFA´S FIRST CONFERENCE FOR PHYSICS
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Where H corresponding to the eccentric anomaly ,the equations of position and
velocity of comet at elliptic orbits are given
Eeary
eEarx
sin1sin.
)(coscos.
2
(4)
Ee
E
a
eGMy
Ee
E
a
GMx
cos.1
cos)1(
cos.1
sin
2
.
(5)
and those of hyperbolic orbit are
Hsinheasin.ry
)eH(coshacos.rx
21
(6)
1
12
1
Hcosh.e
Hcosh
a
)e(GMy
Hcos.e
Hsinh
.
a
GMx
(7)
With parabolic orbits x and y are first expressed in term of )2/tan( :
)2
tan(.2sin.
))2
(tan1(cos. 2
qry
qrx
(8)
The true anomaly is expressed in term of time by Barker equation
)(2
)2
(tan3
1)
2tan(
3
3
tt
q
GM
(9)
which is third equation of )2
tan(
and could be solved directly for for a given time
t if we set )(22
33 tt
q
GMA and
3 2 1 AAB then
BB /1)2
tan(
, )/1arctan(.2 BB
Where 2320 .10.32712.1 smGM , q are the solar gravitational constant and solar
mass and the perihelion distance respectively (Montenbrouck, 1994).
The Stumpff method is more suitable for parabolic and near parabolic orbits, from
Kepler equation
)tt(a
GM)t(Esin.e)t(E
3 (10)
As )1/( eqa and 2
3 /)sin( EEEc we have first
Trajectory analysis of the comet 46p/Wirtanen:M.J.F. Al Bermani , H.H.Jawad
710
)tt()q
e(GME).E(c.eE).e(
312
31
and introducing new variable Ee
EceU .
1
)(.3 3
we obtain the modified form
)tt.(q
GM.)E(c.eUU
33
3
26
3
1
This equation has the same general form as Barker equation and can be solved for U
with EEEc /)sin()(1 , 2
2 /))cos(1()( EEEc we further obtain
2
36
22
1
13
6
1
2
1221
1
2
36
22
11
U.c
c.qr
U.
c.ce
e.qEsine
e
qsin.ry
Uec
c.q)eE(cos
e
qcos.rx
So the equation of velocity of a comet will be
)er
x(
)e(q
GMy
)r
y(
.
)e(q
GMx
1
1
(11)
These arranged don’t alter the fact that the Kepler equation must still be solved
iteratively we begin with the assumption that 0E 6/1)(3 Ec and determine the
appropriate value for U by solving Kepler/Barker equation
UBU
1 , )tt.(
q
GM)E(ecA
323
63
2
from this we obtain improved values
)E(c.e
eUE
33
1 and
23E
EsinE)E(c
With which we obtain amore accurate value ( u ) these steps are repeated until U no
longer varies beyond the desired degree of accuracy.
In the N-body problem we took in to account the perturbations from other bodies
in this case the nine planet which will cause the comet's motion to depart from a
purely Keplarian one so the orbital elements will slightly change with time.
The equation of motion in this case could be given by
9
1333
ii
r
ir
irr
irr
iGM
r
rGMr
(12 )
JOURNAL OF KUFA–PHYSICS A SPECIAL ISSUE FOR KUFA´S FIRST CONFERENCE FOR PHYSICS
711
Where r , ii M,r,r the comet's acceleration ,distance from the sun, distance from the
planet i and the mass of the planet i to calculate the acceleration at any given time
the heliocentric coordinates r of the comet and the corresponding coordinates ir for
the nine planets from Mercury to Pluto has been required ,the position r and
velocity of the comet concerned may be combined as a single , six-dimensional
vector known as a state vector
)t(z
)t(y
)t(x
)t(z
)t(y
)t(x
)t(
)t(r)t(y
(13)
The change of y of the position and velocity with time may therefore be expressed
as differential equation of the form
)y,t(fy where the function combine the velocity and acceleration which are
themselves depend on time and position ,to obtain usable results with larger step
values the procedure adopted is so called multi step method with improved
approximation which are based on several earlier values of f to illustrate the
principal behind this procedure let us assume that we have approximations i
y for the
state vector )i
t(y at times jhtt j where ij ,......1,0 if we then integrate
both sides of )y,t(fy with respect to time t from ( it to 1it ) we then obtain
the equivalent equation
hi
t
it
dt)).t(y,t(f)i
t(y)i
t(y1
To avoid the problem that the integral depend on the unknown state vector )t(y the
integrand is replaced by a polynomial )(tp that interpolates some of the values of the
function ),( jji ytff at earlier times jt which in accordance with assumption that
we made above are already known .if the polynomial of 3rd order written in form
3
3
2
21)( aaaatp with )(1
itth
and appropriate coefficients
611
32
33
13
661
152
123
32
6111
182
93
21
66
/)i
fi
fi
fi
f(a
/)i
fi
fi
fi
f(a
/)i
fi
fi
fi
f(a
/)i
f(a
Are entered we obtain the Adams-Bashforth 4th –order equation
iiiii
ht
t
ii ffffh
ydttpyyi
i
555937924
).( 1231
Which give good approximation of the state vector for larger step sizes, repeating the
process enables us to obtain corresponding values for subsequent times jhti .
Trajectory analysis of the comet 46p/Wirtanen:M.J.F. Al Bermani , H.H.Jawad
712
3- Results
The orbits model was simulated by implementing computer code written by
(Montenbrouck,1994) and using the orbital elements from (Aksnes&Grav,2005) to
calculate the equatorial coordinates ,geocentric distance ,and heliocentric distance
covering the time from 2009 to 2025 ,the results has been calculated for perturbations
under the effect of the nine planets .
The relations between the time in days with the heliocentric distance and the
geocentric distance in astronomical units have been shown in figures (1), (2).
Also the right ascension and declination versus time was illustrated in figures (3), (4).
JOURNAL OF KUFA–PHYSICS A SPECIAL ISSUE FOR KUFA´S FIRST CONFERENCE FOR PHYSICS
713
Figure (1) the relation between the heliocentric distance in AU and time in days for
perturbed and unperturbed orbit
Figure (2) the relation between the geocentric distance in AU and time in days for
perturbed and unperturbed orbits
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
1 720 1439 2158 2877 3596 4315 5034 5753 6472 7191
Time(days)
Hel
oce
ntr
ic d
ista
nce
(A
U)
متسلسلة1
perturbed orbitمتسلسلة2
unperturbed orbit
0
1
2
3
4
5
6
7
1 720 1439 2158 2877 3596 4315 5034 5753 6472 7191
Time(days)
Geo
cen
tric
dis
tan
ce (
AU
)
متسلسلة1
متسلسلة2
unperturbed orbit
perturbed orbit
Trajectory analysis of the comet 46p/Wirtanen:M.J.F. Al Bermani , H.H.Jawad
714
Figure (3) the relation between the Right ascension in hours and time in days for
perturbed and unperturbed orbits
Figure(4) the relation between the Declination in hours and time in days for perturbed
and unperturbed orbits
-40
-20
0
20
40
60
80
1 720 1439 2158 2877 3596 4315 5034 5753 6472 7191
Time (days)
DE
C(d
egre
es)f
or
per
turb
ed a
nd
un
per
turb
ed o
rbit
s
متسلسلة1
متسلسلة2un perturbed orbit
perturbed orbit
0
5
10
15
20
25
30
1 720 1439 2158 2877 3596 4315 5034 5753 6472 7191
TIME(dayes)
RA
(ho
urs)
fo
r p
eru
rb
ed
an
un
pertu
rb
ed
orb
it
مت
سل
سلة
1مت
سل
سلة
2
un perturbed orbit
perturbed orbit
JOURNAL OF KUFA–PHYSICS A SPECIAL ISSUE FOR KUFA´S FIRST CONFERENCE FOR PHYSICS
715
4- Discussion Computer simulation of the trajectory of 46p/wirtanen comet has been performed by
using two computers programs namely COMET and NUMINT written by
(Montenbrouck,1994) . The gravitational attraction of the sun was taken as two body
problem by the program COMET and the perturbation caused by the gravitational
attraction of the nine planets was treated by the second program NUMIT. The time
interval was taken from (01-01-2009 to 30-12-2025). The results are shown in the
flowing figures. Figure (1) shows the variation of heliocentric distance (AU) against
time (days).it can be seen from this figure that there is a noticeable deviation from the
unperturbed motion due to the nine planets. the geocentric distance variations against
time is shown in figure (2).the slight deviation of the geocentric distance can be
attributed to the gravitational attraction of the giant planets (Jupiter and
Saturn).similar variations in the right ascension and declination are shown in figures
(1,2) also change in perihelion distance proving there is a significant effect of the nine
planets
In other hand we can determine the period of comet until the year 2025 in the
predicting ephemeris for this comet can be appliance for other comets.
Trajectory analysis of the comet 46p/Wirtanen:M.J.F. Al Bermani , H.H.Jawad
716
5- References
1-Astronomy on personal computer, Oliver Montenbruck, Thomas Pfleger, Springer
Verlag, 1994
2-Malgorzata Królikowska and Grzegorz Sitarski, A&A, 310,992-998(1996)
3-Belyaev,N,A.,Kresak,L.,Pittich,E,M.,&Pushkarev,A,N.1986,in catalogue of short-period
comets .Bratislava: Slovak Academy of Sciences, Astronomical Institute .
4- Vaghi, S., & Rickman, H. 1982, in Sun and planetary system, Proceedings of the Sixth
European Regional Meeting in Astronomy, Dubrovnik, Yugoslavia, A82-47740 24-89
(Dordrecht: D. Reidel Publishing Co.), 391
5- Malógorzata Kr´olikowska and Sl´awomira Szutowicz, A&A. 343, 997–1000 (1999)
6- M. R. Kidger, A&A 420, 389–395 (2004).
7- K. Aksnes1 and T. Grav, A&A 441, 815–818 (2005).
8- Grav, T. 2001. The Orbital Motion of Periodic Comet 46P/Wirtanen, Cand. Scient.
(MS) thesis, University of Oslo