Department of Mathematics - PowerPoint Presentation

Slide1

Department of MathematicsUniversity of DelhiBhavneet KaurLady Shri Ram College For WomenUniversity of Delhi

Robe's Restricted Problem of 2+2 BodiesSlide2

Three-Body ProblemSuppose m1, m2, m3 are three masses moving under their gravitational attraction, then the study of motion of these bodies under their mutual gravitational force is called the three-body problem.Slide3

Restricted Three-Body Problem

m

2

m

1

m

3

F

1

F

2

r

O

x

ySlide4

Robe’s Restricted Three-Body Problem

M

1

m

2

O

(Origin)

m

3

η

ξ

R

13

R

R

32

m

1Slide5

MotivationSlide6

MotivationSlide7

Robe's Restricted Problem of 2+2 BodiesSlide8

Various forces acting on m3Attraction of m2Attraction of m4Gravitational force exerted by fluid of density ρ1 on m3

Buoyancy forceSlide9

Equations of Motion of m3The equation of motion of m3 in the inertial systemThe equation of motion of m3 in the synodic systemSlide10

We haveEquations of Motion of m3Slide11

Equations of Motion of m3Slide12

Equilibrium SolutionsThe equilibrium solutions of m3 & m4 are given byVξ =0= Vη; Vξ''

=0= V

η

'

‘Slide13

Equilibrium SolutionsCollinear Equilibrium SolutionsSlide14

Equilibrium Solutions

The equations of equilibrium solutions can be written asSlide15

Equilibrium SolutionsSlide16

Equilibrium Solutions

The above equations can be combined asSlide17

Equilibrium Solutions

Location of Collinear Equilibrium Solutions of the Robe's restricted problem of 2+2 bodies. Circles denote the positions of m

3

and triangles denote the positions of m

4

±Slide18

Equilibrium SolutionsSlide19

Equilibrium SolutionsNon-Collinear Equilibrium SolutionsSlide20

Equilibrium SolutionsSlide21

Equilibrium SolutionsSlide22

Equilibrium SolutionsDetermination of the range of φ to evaluate the equilibrium solutions of m3 (and similarly m

4

)

O

η

φ

(m

2

)

d

1

ξ

θ

1

(Slide23

Equilibrium SolutionsSlide24

Equilibrium SolutionsSlide25

Equilibrium SolutionsSlide26

Equilibrium Solutions

Location of Non Collinear equilibrium solutions (when they exist) in the Robe's restricted problem of 2 + 2 bodiesSlide27

Stability of Equilibrium SolutionsSlide28

Stability of Equilibrium Solutions

If any of the roots of the characteristic equations is positive and real, the corresponding equilibrium solution will be unstable.Slide29

Stability of Equilibrium SolutionsStability of Collinear Equilibrium Solutions

whereSlide30

Stability of Equilibrium SolutionsSlide31

Stability of Equilibrium SolutionsSlide32

Stability of Equilibrium Solutions

Stability of Non Collinear Equilibrium SolutionsSlide33

Stability of Equilibrium SolutionsSlide34

ApplicationsWe consider the two primaries Earth (m1*) and Moon (m2) and the infinitesimal masses, two submarines m3 and m4.Slide35

ApplicationsSlide36

ApplicationsSlide37

ApplicationsSlide38

ApplicationsWe observe that K≠1 -µ and K ′ ≠1 -µ . Hence, there are no non collinear equilibrium solutions of the system in this particular case.We observe that the collinear equilibrium solutions m3 and m4 in this case are unstable.Slide39

ConclusionSlide40

Further Scope of the ProblemSlide41
Slide42

Robe's Restricted Problem of 2+2 Bodies with One of the Primaries an Oblate BodySlide43

Plastino in their paper “Robe’s Restricted three-body problem revisited” studied the Slide44

Robe's Restricted Problem of 2+2 Bodies when the Bigger Primary is a Roche EllipsoidGeometry of the Robe's restricted problem of 2 + 2 bodies when the bigger primary m

1

is considered as Roche EllipsoidSlide45

Robe's Restricted Problem of 2+2 Bodies when the Bigger Primary is a Roche Ellipsoid and the Smaller Primary is an Oblate BodySlide46

ReferencesSlide47

ReferencesSlide48

Bhavneet Kaurbhavneet.lsr(at)gmail.comLady Shri Ram College For WomenDelhi UniversityTHANK YOU