Trajectory analysis of the comet 46p/Wirtanen

JOURNAL OF KUFA–PHYSICS A SPECIAL ISSUE FOR KUFA´S FIRST CONFERENCE FOR PHYSICS

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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)


709

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


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 )


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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 .


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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).


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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


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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


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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.


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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

Trajectory analysis of the comet 46p/Wirtanen

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