- Email: [email protected]
Anharmonicity effects in bounce motion of particles
ARTICLE IN PRESS
Planetary and Space Science 54 (2006) 113–116 www.elsevier.com/locate/pss
Anharmonicity effects in bounce motion of particles Jacob Sundara Rajaa, B. Binukumarb, a
10 (61/80 D), Athivilai, Neyyoor (P.O), 629802, India Division of Epidemiology & Clinical Research, Regional Cancer Centre, Medical College (P.O), Trivandrum-695011, Kerala, India
b
Received 23 September 2004; accepted 29 August 2005 Available online 28 November 2005
Abstract Equations for parallel motion for a particle trapped in a magnetic field have been considered and improved solutions of differential equations have been derived. The expressions for the change in energy of the particle ðDwÞ and diffusion coefficient ðDww Þ have been presented in a simple form using the improved solution. r 2005 Elsevier Ltd. All rights reserved. Keywords: Anharmonicity; Magnetic field; Diffusion coefficient
6 4 21 c3 ¼ 1919 20 cðLRE Þ 10 cðLRE Þ ,
1. Introduction Viswanathan and Renuka (1978), Raju (1996) studied the anharmonicity effects in the bounce motion of a particle. The equations for parallel motion obtained by them, respectively, are as follows: d2 s þ o20 s þ ds3 ¼ 0 dt2 and
(1)
2Mc1 ; m
d¼
4Mc2 ; m
d2 s þ o2 s þ ds3 ¼ 0 dt2
(3)
and
d2 s þ o20 s þ ds3 þ bs5 ¼ 0, (2) dt2 where ‘s’ is the distance from the magnetic equator measured along the line of force, ‘m’ is the mass of the particle, ‘M’ is the magnetic moment and o20 ¼
where RE is the radius of earth, L is the McIlwain parameter measured on the earth’s radius. Instead of solving Eqs. (1) and (2), they solved the following equations.
b¼
6Mc3 , m
d2 s þ o2 s þ ds3 þ bs5 ¼ 0, dt2
(4)
where o ¼ o0 þ o1 þ o2 þ and s ¼ s0 þ s1 þ s2 þ : They used the secular perturbation method1 to solve the equations. They have imposed a restriction on the integrating constant and in that case the solution is not general. The solution of Eq. (4) takes the form s ¼ a1 cosðo0 tÞ þ a3 cos 3ðo0 tÞ þ a5 cos 5ðo0 tÞ.
B0 0:31 104 c¼ 3¼ , L L3
For obtaining Dw and Dww they changed o0 to o and cosine to sine and used the form
c1 ¼ 92 cðLRE Þ2 ,
s ¼ a1 sinðot þ jÞ þ a3 sin 3ðot þ jÞ þ a5 sin 5ðot þ jÞ.
4 c2 ¼ 39 8 cðLRE Þ ,
We present a simple method to solve the basic equations (3) and (4) without imposing a restriction on the integrating constants and the series for s is in the sine series. The
Corresponding author. Tel.: +91 471 2522271; fax: +91 471 255 0782.
E-mail address: [email protected] (B. Binukumar). 0032-0633/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.pss.2005.08.006
1
The method is shown in the appendix.
ARTICLE IN PRESS 114
J. Sundara Raja, B. Binukumar / Planetary and Space Science 54 (2006) 113–116
expressions for change in energy, Dw and diffusion coefficient, Dww are derived in a simple form using a new solution.
series in the following form: f 11 ðs; tÞ ¼
1 X
2. Solution of equation for parallel motion Consider Eq. (4). The initial solution is taken as s ¼ s0 satisfying ðd2 s0 =dt2 Þ þ o2 s0 ¼ 0. The initial solution takes the form s0 ¼ b sinðot þ jÞ.
(5)
Applying (5) in (4) we obtain the first approximation s1 from d2 s1 ¼ ðo2 s0 þ ds30 þ bs50 Þ dt2 3db3 10bb5 ¼ o2 b þ þ sinðot þ jÞ 4 16 3 db 5bb5 bb5 þ sin 5ðot þ jÞ. þ sin 3ðot þ jÞ 4 16 16 Integrating the above equation, we obtain the approximation of s ¼ s1 as s ¼ s1 ¼ a1 sinðot þ jÞ þ a3 sin 3ðot þ jÞ þ a5 sin 5ðot þ jÞ þ ,
ð6Þ
1 db3 5bb5 þ a3 ¼ 2 , 9o 4 16 (7)
Viswanathan and Renuka (1978) used the following relation for estimating a and it is based on the assumption that the particle oscillates between the mirror points: B0 cot2 a ¼ 0, L3 where a is the equatorial pitch angle of the particle. Raju (1996) used c 2 a4 þ c 1 a2
on ¼
0otot,
(8)
B0 cot2 a ¼ 0. (9) L3 The coefficients c1 , c2 and c3 can be obtained from the formula given earlier and they can be applied in (8) or (9) to estimate a. This value of a can be applied in (7) to obtain b, a3 and a5 . c 3 a6 þ c 2 a4 þ c 1 a2
3. Diffusion coefficient For obtaining Dw, the change in energy of the particle, they expressed the oscillatory force f 11 ðs; tÞ as a Fourier
2pn , t
ð10Þ
where K n is the wave number corresponding to the frequency on and cn is the phase of the Fourier series. Z t ds f 11 ðs; tÞ dt Dw ¼ 0 dt Z t 1 X ds cosðK n s on t þ cn Þ dt. ¼ fn ð11Þ 0 dt n¼1 To evaluate integral (11) they used the summation method. We evaluate the integral (11) by exact integration to obtain Dw and Dww . Also the components of the Fourier series (10) have different periods. The terms K n s and ds=dt where s is given in (6) have a period 2p=o. The component on t has a period t when its cosine is taken. If t is a multiple N of 2p=o, on ¼
where a1 ; a3 and a5 are the amplitudes of the particle and are given by 1 3db3 10bb5 a1 ¼ a ¼ 2 o2 b þ þ , o 4 16
5 1 bb a5 ¼ . 2 25o 16
f n cosðK n s on t þ cn Þ;
n¼1
2pn 2pno no ¼ ¼ t 2pN N
which tends to zero for large N and for any finite n. That is another defect of Fourier series (10). Due to this defect they obtained non-zero values of Dw and Dww for n is a multiple of N. We correct this defect and obtain non-zero values for Dw and Dww for all values of n from one to infinity. We construct Fourier series in the intervals of width 2p=o ¼ t1 by changing on to vn ¼ 2pn=t1 , where vn is the frequency in the interval 0oto2p=o. In this case vn ¼ no and the corrected Fourier series becomes f 11 ðs; tÞ ¼
1 X
f n cosðK n s vn t þ cn Þ;
0oto2p=o,
n¼1 1 X
Z
t
ds cosðK n s vn t þ cn Þ dt 0 dt n¼1 Z t 1 X ds exp iðK n s vn t þ cn Þ dt ¼ Re fn 0 dt n¼1 Z t1 1 X ds ivn ¼ Re N f n expðicn Þ ðiK n Þ1 i Kn dt 0 n¼1
Dw ¼
fn
exp iðK n s vn tÞ dt Z vn t1 exp iðK n s vn tÞ dt þ Kn 0 [since t is a multiple of 2p=o ¼ t1 ]. Z t1 ds ivn exp iðK n s vn tÞ dt i Kn dt 0 ¼ ½exp iðK n s vn tÞt01 ¼ exp½iK n sðt1 Þ ivn t1 exp½iK n sð0Þ ¼ 0.
ð12Þ
ARTICLE IN PRESS J. Sundara Raja, B. Binukumar / Planetary and Space Science 54 (2006) 113–116
2p 2p o þ j þ a3 sin 3 oþj Since sðt1 Þ ¼ a1 sin o o 2p þ a5 sin 5 oþj o
Hence, Eq. (12) becomes Dw ¼ Re Nt1 ¼ to
¼ sð0Þ
0
2p ¼ expði2npÞ ¼ 1 o
t1
exp iðK n s vn tÞ dt Z t1 ¼ exp½iK n a1 sinðot þ jÞ þ iK n a3 sin 3ðot þ jÞ 0
þ iK n a5 sin 5ðot þ jÞ ivn t dt 1 1 1 X X X ¼ J‘ðK n a1 ÞJ m ðK n a3 ÞJ j ðK n a5 Þ ‘¼1 m¼1 j¼1
exp½ið‘ þ 3m þ 5jÞj Z t1 exp½ið‘ þ 3m þ 5jÞot ivn t dt, 0
where J‘ is the Bessel function defined by expðix sin yÞ ¼
1 X
J‘ðxÞ expði‘yÞ.
‘¼1
Taking ‘ þ 3m þ 5j ¼ r, Z t1 exp½ið‘ þ 3m þ 5jÞot ivn t dt 0 Z t1 ¼ expðirot ivn tÞ dt 0 Z t1 2p it1 v ðro vn Þ dv; ¼ exp 2p 2p 0 Z 2p t1 it1 ¼ ðro noÞ dv exp 2p 0 2p t1 ¼ 2p ( 2p t1 if r ¼ n; ¼ 0 if ran:
where t ¼
t1 v 2p
So, Z t1 0
exp iðK n s vn tÞ dt X X X J‘ðK n a1 ÞJ m ðK n a3 Þ ¼ t1 ‘
m ‘þ3mþ5j¼n
j
J j ðK n a5 Þ expðinjÞ ¼ t1 H n expðinjÞ, where Hn ¼
X
X
X
‘
m ‘þ3mþ5j¼n
j
f n exp½iðcn þ njÞ
f nH n
no Hn Kn
n cosðcn þ jn Þ. Kn
ð13Þ
Then the diffusion coefficient, Dww , is given by Z 1 2p Dww ¼ ðDwÞ2 dðc þ jÞ. 2p 0
expðivn t1 Þ ¼ exp ino Z
1 X n¼1
1 X n¼1
¼ a1 sin j þ a3 sin 3j þ a5 sin 5j and
115
J‘ðK n a1 ÞJ m ðK n a3 ÞJ j ðK n a5 Þ,
Since c and j are continuous, we use integration instead of ensemble average as follows: " #2 Z 1 1 2p 2 2 X n ¼ t o f nH n cosðcn þ jn Þ dðcn þ jn Þ; 2p 0 Kn 1 ðwhere jn ¼ njÞ 1 X 1 t2 o 2 X n n0 ¼ f n f n0 H n H n0 K n K n0 2p n¼1 n0 ¼1 Z 2p cosðcn þ jn Þ cosðcn0 þ jn0 Þ dðcn þ jn Þ 80 0 if n0 an; > > < 2 1 ¼ t2 o 2 X 2 n f H 2n p if n0 ¼ n > > : 2p n¼1 n K n 2 1 t2 o 2 X n 2 ¼ fn H 2n . Kn 2 n¼1
ð14Þ
Expressions (13) and (14) yield Dw and Dww in a simple form obtained by exact integration. The summation is true for all values of n from one to infinity, which is an important result in this work.
Appendix To solve the equation d2 s þ o2 s þ ds3 þ bs5 ¼ 0. dt2
(i)
Raju (1996) applied the secular perturbation method as illustrated below. Eq. (i) is written as d2 s þ o2 s þ ðds3 þ bs5 Þ ¼ 0, dt2 where ¼ 1 at the end o ¼ o0 þ o1 þ 2 o2 þ and s ¼ s0 þ s1 þ 2 s2 þ .
(ii)
ARTICLE IN PRESS J. Sundara Raja, B. Binukumar / Planetary and Space Science 54 (2006) 113–116
116
Applying these in (ii) 2 2 d s0 d s1 2 2 3 5 þ o s þ o s þ 2o o s þ ds þ bs þ 1 0 0 0 0 0 1 0 0 dt2 dt2 þ ¼ 0.
Since cosðo0 tÞ corresponds to resonant frequency, the term containing it is omitted in the expression for s1 . This is possible if the corresponding coefficient is zero, i.e., 2o0 o1 a þ
Neglecting higher powers of and equating the coefficients to zero, we obtained 2
d s0 þ o20 s0 ¼ 0, dt2
(iii)
d2 s1 þ o20 s1 þ 2o0 o1 s0 þ ds30 þ bs50 ¼ 0. dt2 From (iii) we obtain
(iv)
s0 ¼ a cosðo0 tÞ.
(v)
¼ 2o0 o1 a cosðo0 tÞ
(vii)
This restricts the arbitrary parameter a and the solution obtained by this method is therefore not a general one. Applying (vii) in (vi), we obtain 3 d2 s1 da 5ba5 ba5 2 þ cos 5ðo0 tÞ. þ o s ¼ tÞ cos 3ðo 1 0 0 dt2 4 16 16
On integrating, we obtain
Applying (v) in (iv) d2 s1 þ o20 s1 dt2 ¼ ½2o0 o1 a cosðo0 tÞ þ da3 cos3 ðo0 tÞ þ ba5 cos5 ðo0 tÞ
3da3 10ba5 þ ¼ 0. 4 16
s1 ¼
1 da3 5ba5 1 ba5 þ cos 5ðo0 tÞ. cos 3ðo0 tÞ þ 2 16 8o0 4 24o20 16
da3 ½cos 3ðo0 tÞ þ 3 cosðo0 tÞ 4
ba5 ½cos 5ðo0 tÞ þ 5 cos 3ðo0 tÞ þ 10 cosðo0 tÞ 16 3da3 10ba5 þ ¼ 2 o0 o1 a þ cosðo0 tÞ 4 16 3 da 5ba5 ba5 þ cos 5ðo0 tÞ. cos 3ðo0 tÞ 4 16 16
References
ðviÞ
Raju, K., 1996. Studies on anharmonicity of particle motion in geomagnetic field models, Ph.D. Thesis. University of Kerala, Trivandrum. Viswanathan, K.S., Renuka, R., 1978. Pitch angle diffusion by bounce resonance. Planet. Space Sci. 26, 75–80.
Planetary and Space Science 54 (2006) 113–116 www.elsevier.com/locate/pss
Anharmonicity effects in bounce motion of particles Jacob Sundara Rajaa, B. Binukumarb, a
10 (61/80 D), Athivilai, Neyyoor (P.O), 629802, India Division of Epidemiology & Clinical Research, Regional Cancer Centre, Medical College (P.O), Trivandrum-695011, Kerala, India
b
Received 23 September 2004; accepted 29 August 2005 Available online 28 November 2005
Abstract Equations for parallel motion for a particle trapped in a magnetic field have been considered and improved solutions of differential equations have been derived. The expressions for the change in energy of the particle ðDwÞ and diffusion coefficient ðDww Þ have been presented in a simple form using the improved solution. r 2005 Elsevier Ltd. All rights reserved. Keywords: Anharmonicity; Magnetic field; Diffusion coefficient
6 4 21 c3 ¼ 1919 20 cðLRE Þ 10 cðLRE Þ ,
1. Introduction Viswanathan and Renuka (1978), Raju (1996) studied the anharmonicity effects in the bounce motion of a particle. The equations for parallel motion obtained by them, respectively, are as follows: d2 s þ o20 s þ ds3 ¼ 0 dt2 and
(1)
2Mc1 ; m
d¼
4Mc2 ; m
d2 s þ o2 s þ ds3 ¼ 0 dt2
(3)
and
d2 s þ o20 s þ ds3 þ bs5 ¼ 0, (2) dt2 where ‘s’ is the distance from the magnetic equator measured along the line of force, ‘m’ is the mass of the particle, ‘M’ is the magnetic moment and o20 ¼
where RE is the radius of earth, L is the McIlwain parameter measured on the earth’s radius. Instead of solving Eqs. (1) and (2), they solved the following equations.
b¼
6Mc3 , m
d2 s þ o2 s þ ds3 þ bs5 ¼ 0, dt2
(4)
where o ¼ o0 þ o1 þ o2 þ and s ¼ s0 þ s1 þ s2 þ : They used the secular perturbation method1 to solve the equations. They have imposed a restriction on the integrating constant and in that case the solution is not general. The solution of Eq. (4) takes the form s ¼ a1 cosðo0 tÞ þ a3 cos 3ðo0 tÞ þ a5 cos 5ðo0 tÞ.
B0 0:31 104 c¼ 3¼ , L L3
For obtaining Dw and Dww they changed o0 to o and cosine to sine and used the form
c1 ¼ 92 cðLRE Þ2 ,
s ¼ a1 sinðot þ jÞ þ a3 sin 3ðot þ jÞ þ a5 sin 5ðot þ jÞ.
4 c2 ¼ 39 8 cðLRE Þ ,
We present a simple method to solve the basic equations (3) and (4) without imposing a restriction on the integrating constants and the series for s is in the sine series. The
Corresponding author. Tel.: +91 471 2522271; fax: +91 471 255 0782.
E-mail address: [email protected] (B. Binukumar). 0032-0633/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.pss.2005.08.006
1
The method is shown in the appendix.
ARTICLE IN PRESS 114
J. Sundara Raja, B. Binukumar / Planetary and Space Science 54 (2006) 113–116
expressions for change in energy, Dw and diffusion coefficient, Dww are derived in a simple form using a new solution.
series in the following form: f 11 ðs; tÞ ¼
1 X
2. Solution of equation for parallel motion Consider Eq. (4). The initial solution is taken as s ¼ s0 satisfying ðd2 s0 =dt2 Þ þ o2 s0 ¼ 0. The initial solution takes the form s0 ¼ b sinðot þ jÞ.
(5)
Applying (5) in (4) we obtain the first approximation s1 from d2 s1 ¼ ðo2 s0 þ ds30 þ bs50 Þ dt2 3db3 10bb5 ¼ o2 b þ þ sinðot þ jÞ 4 16 3 db 5bb5 bb5 þ sin 5ðot þ jÞ. þ sin 3ðot þ jÞ 4 16 16 Integrating the above equation, we obtain the approximation of s ¼ s1 as s ¼ s1 ¼ a1 sinðot þ jÞ þ a3 sin 3ðot þ jÞ þ a5 sin 5ðot þ jÞ þ ,
ð6Þ
1 db3 5bb5 þ a3 ¼ 2 , 9o 4 16 (7)
Viswanathan and Renuka (1978) used the following relation for estimating a and it is based on the assumption that the particle oscillates between the mirror points: B0 cot2 a ¼ 0, L3 where a is the equatorial pitch angle of the particle. Raju (1996) used c 2 a4 þ c 1 a2
on ¼
0otot,
(8)
B0 cot2 a ¼ 0. (9) L3 The coefficients c1 , c2 and c3 can be obtained from the formula given earlier and they can be applied in (8) or (9) to estimate a. This value of a can be applied in (7) to obtain b, a3 and a5 . c 3 a6 þ c 2 a4 þ c 1 a2
3. Diffusion coefficient For obtaining Dw, the change in energy of the particle, they expressed the oscillatory force f 11 ðs; tÞ as a Fourier
2pn , t
ð10Þ
where K n is the wave number corresponding to the frequency on and cn is the phase of the Fourier series. Z t ds f 11 ðs; tÞ dt Dw ¼ 0 dt Z t 1 X ds cosðK n s on t þ cn Þ dt. ¼ fn ð11Þ 0 dt n¼1 To evaluate integral (11) they used the summation method. We evaluate the integral (11) by exact integration to obtain Dw and Dww . Also the components of the Fourier series (10) have different periods. The terms K n s and ds=dt where s is given in (6) have a period 2p=o. The component on t has a period t when its cosine is taken. If t is a multiple N of 2p=o, on ¼
where a1 ; a3 and a5 are the amplitudes of the particle and are given by 1 3db3 10bb5 a1 ¼ a ¼ 2 o2 b þ þ , o 4 16
5 1 bb a5 ¼ . 2 25o 16
f n cosðK n s on t þ cn Þ;
n¼1
2pn 2pno no ¼ ¼ t 2pN N
which tends to zero for large N and for any finite n. That is another defect of Fourier series (10). Due to this defect they obtained non-zero values of Dw and Dww for n is a multiple of N. We correct this defect and obtain non-zero values for Dw and Dww for all values of n from one to infinity. We construct Fourier series in the intervals of width 2p=o ¼ t1 by changing on to vn ¼ 2pn=t1 , where vn is the frequency in the interval 0oto2p=o. In this case vn ¼ no and the corrected Fourier series becomes f 11 ðs; tÞ ¼
1 X
f n cosðK n s vn t þ cn Þ;
0oto2p=o,
n¼1 1 X
Z
t
ds cosðK n s vn t þ cn Þ dt 0 dt n¼1 Z t 1 X ds exp iðK n s vn t þ cn Þ dt ¼ Re fn 0 dt n¼1 Z t1 1 X ds ivn ¼ Re N f n expðicn Þ ðiK n Þ1 i Kn dt 0 n¼1
Dw ¼
fn
exp iðK n s vn tÞ dt Z vn t1 exp iðK n s vn tÞ dt þ Kn 0 [since t is a multiple of 2p=o ¼ t1 ]. Z t1 ds ivn exp iðK n s vn tÞ dt i Kn dt 0 ¼ ½exp iðK n s vn tÞt01 ¼ exp½iK n sðt1 Þ ivn t1 exp½iK n sð0Þ ¼ 0.
ð12Þ
ARTICLE IN PRESS J. Sundara Raja, B. Binukumar / Planetary and Space Science 54 (2006) 113–116
2p 2p o þ j þ a3 sin 3 oþj Since sðt1 Þ ¼ a1 sin o o 2p þ a5 sin 5 oþj o
Hence, Eq. (12) becomes Dw ¼ Re Nt1 ¼ to
¼ sð0Þ
0
2p ¼ expði2npÞ ¼ 1 o
t1
exp iðK n s vn tÞ dt Z t1 ¼ exp½iK n a1 sinðot þ jÞ þ iK n a3 sin 3ðot þ jÞ 0
þ iK n a5 sin 5ðot þ jÞ ivn t dt 1 1 1 X X X ¼ J‘ðK n a1 ÞJ m ðK n a3 ÞJ j ðK n a5 Þ ‘¼1 m¼1 j¼1
exp½ið‘ þ 3m þ 5jÞj Z t1 exp½ið‘ þ 3m þ 5jÞot ivn t dt, 0
where J‘ is the Bessel function defined by expðix sin yÞ ¼
1 X
J‘ðxÞ expði‘yÞ.
‘¼1
Taking ‘ þ 3m þ 5j ¼ r, Z t1 exp½ið‘ þ 3m þ 5jÞot ivn t dt 0 Z t1 ¼ expðirot ivn tÞ dt 0 Z t1 2p it1 v ðro vn Þ dv; ¼ exp 2p 2p 0 Z 2p t1 it1 ¼ ðro noÞ dv exp 2p 0 2p t1 ¼ 2p ( 2p t1 if r ¼ n; ¼ 0 if ran:
where t ¼
t1 v 2p
So, Z t1 0
exp iðK n s vn tÞ dt X X X J‘ðK n a1 ÞJ m ðK n a3 Þ ¼ t1 ‘
m ‘þ3mþ5j¼n
j
J j ðK n a5 Þ expðinjÞ ¼ t1 H n expðinjÞ, where Hn ¼
X
X
X
‘
m ‘þ3mþ5j¼n
j
f n exp½iðcn þ njÞ
f nH n
no Hn Kn
n cosðcn þ jn Þ. Kn
ð13Þ
Then the diffusion coefficient, Dww , is given by Z 1 2p Dww ¼ ðDwÞ2 dðc þ jÞ. 2p 0
expðivn t1 Þ ¼ exp ino Z
1 X n¼1
1 X n¼1
¼ a1 sin j þ a3 sin 3j þ a5 sin 5j and
115
J‘ðK n a1 ÞJ m ðK n a3 ÞJ j ðK n a5 Þ,
Since c and j are continuous, we use integration instead of ensemble average as follows: " #2 Z 1 1 2p 2 2 X n ¼ t o f nH n cosðcn þ jn Þ dðcn þ jn Þ; 2p 0 Kn 1 ðwhere jn ¼ njÞ 1 X 1 t2 o 2 X n n0 ¼ f n f n0 H n H n0 K n K n0 2p n¼1 n0 ¼1 Z 2p cosðcn þ jn Þ cosðcn0 þ jn0 Þ dðcn þ jn Þ 80 0 if n0 an; > > < 2 1 ¼ t2 o 2 X 2 n f H 2n p if n0 ¼ n > > : 2p n¼1 n K n 2 1 t2 o 2 X n 2 ¼ fn H 2n . Kn 2 n¼1
ð14Þ
Expressions (13) and (14) yield Dw and Dww in a simple form obtained by exact integration. The summation is true for all values of n from one to infinity, which is an important result in this work.
Appendix To solve the equation d2 s þ o2 s þ ds3 þ bs5 ¼ 0. dt2
(i)
Raju (1996) applied the secular perturbation method as illustrated below. Eq. (i) is written as d2 s þ o2 s þ ðds3 þ bs5 Þ ¼ 0, dt2 where ¼ 1 at the end o ¼ o0 þ o1 þ 2 o2 þ and s ¼ s0 þ s1 þ 2 s2 þ .
(ii)
ARTICLE IN PRESS J. Sundara Raja, B. Binukumar / Planetary and Space Science 54 (2006) 113–116
116
Applying these in (ii) 2 2 d s0 d s1 2 2 3 5 þ o s þ o s þ 2o o s þ ds þ bs þ 1 0 0 0 0 0 1 0 0 dt2 dt2 þ ¼ 0.
Since cosðo0 tÞ corresponds to resonant frequency, the term containing it is omitted in the expression for s1 . This is possible if the corresponding coefficient is zero, i.e., 2o0 o1 a þ
Neglecting higher powers of and equating the coefficients to zero, we obtained 2
d s0 þ o20 s0 ¼ 0, dt2
(iii)
d2 s1 þ o20 s1 þ 2o0 o1 s0 þ ds30 þ bs50 ¼ 0. dt2 From (iii) we obtain
(iv)
s0 ¼ a cosðo0 tÞ.
(v)
¼ 2o0 o1 a cosðo0 tÞ
(vii)
This restricts the arbitrary parameter a and the solution obtained by this method is therefore not a general one. Applying (vii) in (vi), we obtain 3 d2 s1 da 5ba5 ba5 2 þ cos 5ðo0 tÞ. þ o s ¼ tÞ cos 3ðo 1 0 0 dt2 4 16 16
On integrating, we obtain
Applying (v) in (iv) d2 s1 þ o20 s1 dt2 ¼ ½2o0 o1 a cosðo0 tÞ þ da3 cos3 ðo0 tÞ þ ba5 cos5 ðo0 tÞ
3da3 10ba5 þ ¼ 0. 4 16
s1 ¼
1 da3 5ba5 1 ba5 þ cos 5ðo0 tÞ. cos 3ðo0 tÞ þ 2 16 8o0 4 24o20 16
da3 ½cos 3ðo0 tÞ þ 3 cosðo0 tÞ 4
ba5 ½cos 5ðo0 tÞ þ 5 cos 3ðo0 tÞ þ 10 cosðo0 tÞ 16 3da3 10ba5 þ ¼ 2 o0 o1 a þ cosðo0 tÞ 4 16 3 da 5ba5 ba5 þ cos 5ðo0 tÞ. cos 3ðo0 tÞ 4 16 16
References
ðviÞ
Raju, K., 1996. Studies on anharmonicity of particle motion in geomagnetic field models, Ph.D. Thesis. University of Kerala, Trivandrum. Viswanathan, K.S., Renuka, R., 1978. Pitch angle diffusion by bounce resonance. Planet. Space Sci. 26, 75–80.