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Publisher: Research Council of Science and Technology, Biratnagar, Nepal  p. 53 

 
 

BIBECHANA 
A Multidisciplinary Journal of Science, Technology and Mathematics 

ISSN 2091-0762 (print), 2382-5340 (online) 

Journal homepage: http://nepjol.info/index.php/BIBECHANA 

 

Energy of Sitnikov's restricted three body problem if the 

primaries are source of radiation and triaxial rigid bodies 
 

R.R.Thapa
1,2*
 

1
University Department of Mathematics., T.M. Bhagalpur University, Bhagalpur, India 

2P.G. Department of Mathematics., P.G. Campus, Biratnagar, T.U., Nepal. 

*E-mail: thaparajuram@yahoo.com  
Accepted for publication: December 13, 2014 

 

Abstract 

In this paper the joint effect of source of radiation and triaxial rigid body has been studied. The energy 

of Sitnikov's restricted three body problem when primaries are sources of radiation and energy of 

Sitnikov's restricted problem of three bodies when primaries are triaxial rigid bodies have been 

studied to calculate the joint effect. Equation of motion of the third body of infitesimal mass, if 

primaries are sources of radiation and triaxial rigid bodies,  are calculated.     

DOI: http://dx.doi.org/10.3126/bibechana.v12i0.11707      © 2014 RCOST:  All rights reserved.         
 

Keywords:  Dimensional distance; oscillatory motion; periodicity; radiation parameter. 

 
 

1. Introduction 

Sitnikov [1] problem can be integrated in case of restricted problem if primaries are taken to be equal 

masses. If primaries are triaxial rigid bodies then distance between them will be changed. Then third 

body can move along z axis. But if primaries are sources of radiation then the third body moves along 

the line perpendicular through the centre of mass of the primaries and is perpendicular to the plane of 

motion.     

Thapa et al. [2] obtained the periodicity of Sitnikov's restricted three body problem when the 

primaries are sources of radiation. They found the equations of motion of the third body and total 

energy of the system. R.R. Thapa et  al. [3] also obtained the equations of motion of the Sitnikov's 

restricted problem of three bodies when Primaries are triaxial rigid bodies. They got the equations of 

motion in dimensionless variables and Cartesian form as  

 

2

2

x

y

z

x ny

y nx

z

− = Ω

+ = Ω

= Ω

ɺɺ ɺ

ɺɺ ɺ

ɺɺ

 

 



 

  

Thapa/ BIBECHANA 12 (2015) 53-58:  p. 54 

 

    

2
2 2 1 2

3

1 2 1

2

1 2 1 2

3 5

2 1

2 2 2

1 2 1 1

5 5 5

2 1 2

(1 )(2 )1
where  ( )

2 2

(2 ) 3(1 )( )

2 2

3 ( ) 3(1 ) 3

2 2 2

− −−
Ω = + + + +

′ ′− − −
+ −

′ ′ ′− −
− − −

n
x y

r r r

y

r r

y z z

r r r

µ σ σµ µ

µ σ σ µ σ σ

µ σ σ µ σ µσ

 

In present work we have proposed to get the energy of restricted three body problem if the primaries 

are source of radiation and triaxial rigid bodies. 

 

2. Equation of motion  

We will adopt the notation and terminology of szehebely [4]. 

We have the equation of motion of the third body of infinitesimal mass is 

       

3

1 2 1 2 1 1

3 5 7
2 2 22 2 2

2 2 2

3(2 2 ) 15( )(1 )

4 4
4 4 4

σ σ σ σ σ σ′ ′ ′− + − +∂Ω − −
= = − +

∂
     

+ + +     
     

ɺɺ
z zp z

z
z

a a a
z z z

 

32

1 1

3 5 72
2 2 22 2 2

2 2 2

( )(1 ) 3 15

4

4
4 4 4

σ σα ′+− −
= − +
     

+ + +     
     

zd z p z z

dt
a a a

z z z

    (1) 

where 1 2 1 22 2α σ σ σ σ′ ′= − + −  

Equation (1) is non-linear equation. It can be integrated by linearizing it. 

 

 3. Energy of the third body 

  If we multiply (1) by2
dz

dt
 we get,  

32

1 1

3 5 72
2 2 22 2 2

2 2 2

15( )2(1 ) 3 2
2 2

4 4
4 4 4

′+−
= − − × +

     
+ + +     

     

zdz d z p z dz dz dz
z

dt dt dt dt dt
a a a

z z z

σ σα

              

Or, 

2 2
2

2

1 1

3 5 7
2 2 22 2 2

2 2 2

4 415( )1 (1 ) 3

2 4

4
4 4 4

 
+ − ′+−   = − − × + 

       
+ + +     

     

a a
z z

d dz p z dz z dz dz

dt dt dt dt dt
a a a

z z z

σ σα       

    



 

  

Thapa/ BIBECHANA 12 (2015) 53-58:  p. 55 

                                                                                                    

2

1 1

3 5 5 7
2 2 2 22 2 2 2

2 2 2 2

15( )(1 ) 3

4 4

4
4 4 4 4

 
 
 ′+−

− − + − 
        

+ + + +        
         

p z dz z dz z dz a z dz

dt dt dt dt
a a a a

z z z z

σ σα      (2) 

If we substitute  
du dz
u z
dt dt

=  and 
2

2 2

4

a
u z= +  we get,  

( ) ( ) ( ) ( )

2 2

1 1

3 5 5 7
2 2 2 22 2 2 2

21 1

2 4 4 6

1 1 1 12 4 4

15( )1 (1 ) 3 1 1

2 4 4
4

15( )(1 ) 3 1 1 1

4 4 4

1 3 15 1 15
( ) (

4 4 4

 
′+−    = − − + −       

′+−  = − − + −  

′ ′= − − + + − +

d dz p du du du a du
u u u u

dt dt dt dt dt dt
u u u u

p du du du du
a

u dt u dt u dt u dt

du du du

u dt u dt u dt

σ σα

σ σα

α
σ σ σ σ

( )

2

6

21 3 3
51 1 1 1

1
)
4

5( ) 3( )(1 )

4 4 16

− − −
−

⋅

′ ′+ +−
= + − +

a du

u dt

ap du du du d
u

dt dt dt dt

σ σ σ σα

             

2 21 3 3 5

1 1 1 1

2 2
1 3 3 51 1

1 1

2

1 1 1 1

3 3

5( ) 3( )1
(1 )

2 4 4 16

3( )1 5
(1 ) ( )

2 4 4 16

5( ) 3( )1 (1 )

2 4 4

− − − −

− − − −

′ ′+ +  = − + − 
 

′+  ′⇒ = − + − + + + 
 

′ ′+ +− ⇒ = + − + 
 

∫ ∫ ∫ ∫ ∫
ad dz du du du du

dt p dt dt dt dt
dt dt dt dt dt dt

adz
p u u u u c

dt

adz p

dt u u u

σ σ σ σα

σ σα
σ σ

σ σ σ σα 2

5

2 2

1 1 1 1

3 3 52
2 2 22 2 22

2 2 2

2 2

1 1 1 1

3 32
2 2 22 22

2 2 2

1

16

5( ) 3( )1 (1 )

2

4 4 164 4 4 4

5( ) 3( )1 (1 )

2

4 4 164 4 4 4

+

′ ′+ +− ⇒ = + − + + 
       + + + +     

     

′ ′+ +− ⇒ − − + − 
       + + + +     

     

c
u

adz p
c

dt a a a az z z z

adz p

dt a a a az z z z

σ σ σ σα

σ σ σ σα
5

2

= c

     

( )2 2
1 1 1 2 1 2 1 1

3 52
2 22 22

2 2

5 5 2 2 3( )1 (1 )

2

4 164 4 4

′ ′ ′+ − + − + ′+− ⇒ − + − = 
     + + +   

   

adz p
c

dt a a az z z

σ σ σ σ σ σ σ σ
 

 

 



 

  

Thapa/ BIBECHANA 12 (2015) 53-58:  p. 56 
 

 

( )2 2
1 2 1 2 1 1

3 52
2 22 22

2 2

3 3 3( )1 (1 )

2

4 164 4 4

′ ′+ + + ′+− ⇒ − + − = 
     + + +   

   

adz p
c

dt a a az z z

σ σ σ σ σ σ
 

2 2

1 1

1 3 5
2 2 22 2 2

2 2 2

3( )1 (1 )
 constant

2

4 16
4 4 4

σ σ ′+− ⇒ − + − = 
       

+ + +     
     

adz p p

dt
a a a

z z z

 

2 2

1 1

1 3 5
2 2 22 2 2

2 2 2

2

1 1

3 52
2 22 22

2 2

3( )1 (1 )
constant.

2

4 16
4 4 4

3( )1 (1 )
constant.

2

4 164 4 4

σ σ

σ σ

′+− = − + − = 
       

+ + +     
     

′+− = − + − = 
     + + +   

   

adz p p
E

dt
a a a

z z z

dz p p
E

dt a a az z z

 

Now from equation (1)     

2
31 1

3 5 72
2 2 22 2 2

2 2 2

2
31 1

3 5 72
3 5 72 2 22 2 2

2 2 2

2

2 3

15( )(1 ) 3
0

4 4
4 4 4

15( )(1 ) 3
0

4 4 4
1 4 1 4 1

2 2 2

(1 )8

σ σα

σ σα

′+−
+ + − =
     

+ + +     
     

′+−
⇒ + + − =

          + + +          
          

−
⇒ +

d z p z z
z

dt
a a a

z z z

d z p z z
z

dt
a z a z a z

a a a

d z p z

dt a

3 5
2 22 2

2 5 2

7
3 2 2

1 1

7 2

4 24 4
1 1

15( ) 32 4
1 0

(as    so 1)

α

σ σ

− −

−

   
+ + +   

   

′  + ×
− + = 

 

 << <<< 
 

z z z

a a a

z z

a a

z
z a

a

 

2 2 22
3

1 12 3 5 7

(1 )8 3 2 24 5 2 15 32 7 2
1 ... 1 ... ( ) 1 ... 0
2 2 2

     − ×     ′⇒ + − + + − + − + − + =                         

d z p z z z z z
z

dt a a a a a a

α
σ σ

 

Neglecting higher order term of (z/a) above the third order,   we get, 

 

 



 

  

Thapa/ BIBECHANA 12 (2015) 53-58:  p. 57 

 

( )

22 2
31 1

2 3 3 5 5 2 7

2
31 1

7 2

2
31 1

2 3 5 5 7 7

2
2 2

2 5 7

480( )(1 )8 8 12 24 24 5 4

2 2

480( ) 7 4
0

2

480( )(1 )8 24 48 240
... 0

8(1 ) 48
3 5 1

d z p z z z z z z
z

dt a a a a a a a

z
z

a a

d z p
z z

dt a a a a a

d z p
a z a

dt a a

σ σα α

σ σ

σ σα α

α α

′+− × + − ⋅ + − × − 
 

′+
+ × =

′+−   + + − + + + =   
   

−
+ + − + +( ) 3

1 1
0( ) 0zσ σ ′+ =

  

2
2 3

0 0
d z

z z
dt

η ε+ − =                      (4) 

( )
2

2 2

0 5 5

2

1 17

8(1 ) 8 ( 3 )
where 3

48
and 5 10( )

p q a
a

a a

a
a

α
η α

ε α σ σ

− +
= + = 


′ = + + +  

 

The equations (3), (4) represent energy of Sitnikov's restricted three body problem if the primaries are 

source of radiation and triaxial rigid bodies. 

 
 

4.  Result and Discussion  
 

The energy of third body depends on variable z, radiation parameter p, (parameter) α  

= 1
2

1
121 22 σ−σ+σ−σ , distance between the primaries a; where 

2 2
1 3

1 2

a a
,

5R

−
σ =  

2 2 2 2 2 2
2 12 3 1 3 2 3

2 1 22 2 2

a a b b b b
, ,

5R 5R 5R

− − −
σ = σ = σ = ; R is dimensional distance between the 

primaries and 321 a,a,a ; 321 b,b,b are lengths of semi-axes of two primaries parallel to axes 

respectively. The kinetic energy of the third body depends on variable z but potential energy depends 

on 0 andη ε . 

where 
5

2
2
0

a

)3a(
q8

α+
=η   and )](105a[(

a

48 12

7
σ+σ+α+=ε . 

Hence the third body will perform oscillatory motion. 

 

 

Acknowledgement  
 

I want to acknowledge with thanks my supervisor Dr. M.R. Hassan, Department of Mathematics, 

S.M. College, Bhagalpur, T.M.B.U. Bghagalpur, 812007, India for his meticulous guidance during the 

work.  

 

(3) 



 

  

Thapa/ BIBECHANA 12 (2015) 53-58:  p. 58 

 

References 
 

[1]  K.A. Sitnikov, Dokl. Akad, Nauk, USSR 132(2) (1960) 303. 

[2] R.R. Thapa, and M.R. Hassan, Journal of the Bihar Mathematical society 27(2013) 99-

105. 

[3] R.R. Thapa and M.R. Hassan, A research journal in mathematics "M0" 1(2014) 37-39.  

[4] V. Szebehely, Theory of orbits, Academic Press, New York  (1967).