The Photogravitational Circular Restricted Four-body Problem with Variable Masses

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1 The Photogravtatonal Crcular Restrcted Four-body Problem wth Varable Masses Abdullah A. Ansar Department of Mathematcs College of Scence Majmaah Unversty Al-Zulf Kngdom of Saud Araba Abstract In ths paper we examne the stablty of the equlbrum ponts n the photogravtatonal crcular restrcted four-body problem by consderng all the masses are varyng wth tme one of the three masses are taken as source of radaton pressure and all the masses are placed at the vertces of an equlateral trangle. We derve the equatons of moton under the effect of solar radaton pressure by usng the Meshchersk transformatons. We have plotted the equlbrum ponts tme seres surfaces of moton of nfntesmal body Poncare surface of sectons and Newton-Raphson basn of attracton. Here fve equlbrum ponts are found but t s ten n the classcal case. From the tme seres t s observed that the orbt wll not be perodc. The surfaces of moton of the nfntesmal body are studed n the space. The Poncare surface of secton s n the dscrete type of pattern. The convergence of the equlbrum ponts s studed by the Newton-Raphson basn of attracton. Fnally we have examned the stablty of the equlbrum ponts and found that all the equlbrum ponts are unstable. Keywords: Radaton Pressure; varable masses; equlateral trangle; Poncare surface of sectons; Newton-Raphson basn of attracton.. Introducton The restrcted problem s studed by many scentsts and mathematcans n the two-body threebody four-body and N-body models. Chernkov [8] nvestgated the stablty of equlbrum ponts by Lyapunov s methods n the restrcted three body problem (Sun-Planet-Partcle) wth the effects of solar radaton pressure. Perezhogn [7] studed the stablty of the sxth and seventh lbraton ponts n the photogravtatonal crcular restrcted three body problem. Bhatnagar et al. [7] studed about the lagrangan ponts n the photogravtatonal restrcted three body problem. They examned the stablty of the equlbrum ponts and observed that the equlbrum ponts are stable n the lnear sense and unstable around the trangular ponts. Mgnard [4] studed the restrcted three-body problem wth the ncluson of solar radaton pressure. And observed that trangular equlbrum ponts are no longer exst. Smmons [0] nvestgated the restrcted 3-body problem wth radaton pressure and observed that nne equlbrum ponts exsts fve n the plane of moton and four n the out of plane. Sharma [9] nvestgated the varaton of the solar radaton pressure and oblateness factor on the statonary solutons of the planar restrcted three-body problem when the more massve prmary s a source of radaton and smaller prmary s an oblate spherod wth ts equatoral plane concdent wth the plane of moton. AbdulRaheem et al. [4] nvestgated the stablty of equlbrum ponts under the nfluence of the Corols and centrfugal forces together wth the effects of oblateness and radaton pressure of the prmares. It s observed that the collnear ponts are unstable and the trangular ponts are condtonally stable dependng on the radaton factor and oblateness. It s also observed that the Corols force has a stablzng tendency whle the centrfugal force radaton and oblateness of the prmares have destablzng effects. Kalvourds et al. [0] nvestgated the dynamcal propertes and the parametrc evoluton of perodc orbts n the restrcted four body problem wth radaton pressure. They llustrated the zero-velocty curves and surfaces and also they examned the stablty of the equlbrum ponts. Abouelmagd [] nvestgated the stablty and perodc orbts n the restrcted three body problem under the effects of oblateness and radaton pressure. He observed that the collnear ponts are unstable whle trangular ponts are condtonally stable dependng on the radaton pressure and oblateness. And also the elements of perodc orbts around equlbrum ponts are affected by oblateness. Sngh et al. [5] studed numercally the restrcted four-body problem by consderng all the prmares as source of radaton pressure placed at the vertces of an equlateral trangle. They observed that the equlbrum ponts are unstable. Pushparaj et al. [8] studed the nteror resonance perodc orbts around the Sun n the photogravtatonal restrcted three body problem by the method of Poncare surface of secton. They observed that the perod of tme decrease wth the ncrease of radaton pressure. Many researchers have explored about the varable masses and the Newton-Raphson basns of attracton as Meshchersk [3] Lchtenegger[] Sngh [ 3 4] Douskos [9] Zhang [6] Kumar [] Asss [5] Abouelmagd [3] Mttal [5] Zotos [ ]. 30

2 We have studed the crcular restrcted fourbody problem n whch the masses of the prmares as well as the mass of the nfntesmal body vary wth tme and one of the prmares s taken as source of radaton pressure. We have dscussed our problem n varous sectons. In the frst secton we have revewed the lterature. In the second secton we have derved the equatons of moton of the nfntesmal varable mass under the effects of the radaton pressure. In the thrd secton we have llustrated the numercal analyss (equlbrum ponts tme seres surface of moton of the nfntesmal body Poncare surface of secton and Newton-Raphson basn of attracton). In the fourth secton we have examned the stablty of the equlbrum ponts. Fnally n the ffth secton we have concluded the problem. Our problem has great applcatons n the Astronomy and Astrophyscs.. Equatons of moton Let m m and m 3 be the masses of the three prmares placed at the vertces of an equlateral trangle of sde. The fourth nfntesmal body havng mass m movng under the nfluence of the prmares but not nfluencng them. The one of the prmares as m s consdered as source of radaton pressure wth factor q and all the masses are varyng wth tme. The prmares are revolvng n the crcular orbts around ther center of mass whch s consdered as orgn. The prmary m s placed at x-axs the lne passng through the orgn and perpendcular to the x-axs s taken as y -axs and the lne passng through the orgn and perpendcular to the plane of moton of the prmares s taken as z-axs (Fg. ). Intally we supposed that the synodc plane concdes wth the nertal plane and revolvng wth mean moton w about z-axs. Usng the procedure of Abdullah [] we can wrte the equatons of moton of the varable nfntesmal body n the restrcted four body problem as Fg. : Confguraton of the crcular restrcted four-body problem wth solar radaton pressure at B. m (x w y) + (x w y w y w x) = m mg(x x ) mg(x x ) q mg(x x ) r r r3 m (y + wx) + (y + wx + w x w y) = m mg( y y ) m G( y y ) q mg( y y ) r r r3 m m Gz m Gzq m Gz z + z = m r r r where r = (x x ) + ( y y ) + z ( = 3) are the dstances from the prmares to the nfntesmal body G s the gravtatonal constant. Let µ ( t) = mg ( = 3). Usng Meshchersk transformaton [3] x= ξrt ( ) y= hrt ( ) z = ζrt ( ) dt dτ = R = (t) r r Rt ( ) w0 w( t) = x (t) (t) = ξr y = hr R () t 3

3 µ µ µ µ R(t) R(t) 0 0 (t) = (t) = m R at bt c R (t) ac b = k ( = 3) 0 m = (t) = + + where k abc µ 0 µ 0 µ 0 µ 30m0 are constants. Consderng unt of mass dstance and tme at ntal tme t 0 such that µ = = w = G = a t + b = a (constant) Introducng the mass parameter as µ 0 = µ µ 0 = ( µ aµ ) µ 30 = aµ a << Fnally the equatons of moton become ξ'' h' aξ' =Ω h'' + ξ' ah' =Ω ξ h ζ'' aζ' =Ω. () where a ξh r r r ( ) ( ) 3 = + + ζ Ω= ( a + k )( ξ + h + ζ ) ζ µ ( µ aµ ) q aµ r ξ ξ h h ζ µξ ( ξ ) ( a + k ) ξ ah 3 r ( µ aµ )( ξ ξ) q aµξ ( ξ ) = 0 () µh ( a + k ) h aξ r r3 3 r (3) ( µ aµ )( h h) q aµh ( h ) = r r3 µζ ( µ aµζ ) q ( a + k ) ζ 3 3 r r a µζ = 0. 3 r3 (4) We have shown the equlbrum ponts graphcally n ( ξh ) plane (Fg. ) ( ξ ζ ) plane (Fg. 3) and ( hζ ) plane (Fg. 4). In the ( ξh ) plane we found fve equlbrum ponts. The green dots denote the locatons of the equlbrum ponts and the purple dots denote the locatons of the prmares. But Baltaganns [6] have gotten ten equlbrum ponts wth unequal masses n the classcal case. In the ( ξ ζ ) plane we found fve equlbrum ponts. The green dots denote the locatons of the equlbrum ponts. In the ( hζ ) plane we found three equlbrum ponts. The black dots denote the locatons of the equlbrum ponts. ( ξ h ) = ( 0)( ξ h ) = ( ) 3 3 ( ξ h ) = ( ) Prme ( ) s the dfferentaton w.r.t. τ. 3. Numercal Analyss 3. Equlbrum ponts The equlbrum ponts can found by takng ξ'' = ξ' = h'' = h' = ζ '' = ζ ' = 0n equaton ().e. 3

4 Fg. : Locatons of equlbrum ponts at a = 0. k = 0.4 a = 0.0 µ = 0.09 q = 0.8. Fg. 4. Locatons of equlbrum ponts at a = 0. k = 0.4 a = 0.0 µ = 0.09 q = Tme seres We have plotted the tme seres n between ( τξ ) (Fg. 5.) and ( τh )(Fg. 6). These tme seres show that the orbts wll not be perodc. Fg. 3. Locatons of equlbrum ponts at a = 0. k = 0.4 a = 0.0 µ = 0.09 q = 0.8. Fg. 5. Tme seres n between ( τξ ) at a = 0. k = 0.4 a = 0.0 µ = 0.09 q =

5 Fg. 6. Tme seres n between ( τh ) at a = 0. k = 0.4 a = 0.0 µ = 0.09 q = Surfaces.Surfaces of the moton of the nfntesmal body In ths secton we have drawn the surface of moton of the nfntesmal body by consderng the equatons ( and 3) (Fg. 7) the equatons ( and 4) (Fg. 8) the equatons (3 and 4) (Fg. 9). Fg. 8. The surface of moton of the nfntesmal body n ( ξh = 0 ζ) -plane Fg. 7. The surface of moton of the nfntesmal body n ( ξhζ= 0) -plane Fg. 9. The surface of moton of the nfntesmal body n ( ξ= 0 hζ ) -plane 34

6 . Poncare surface of secton We have drawn the Poncare surface of secton and got a dscrete type of graph (Fg. 0). Fg. 0: Poncare surface of secton 3.4 Newton-Raphson basn of attracton We have drawn the basns of attracton by usng the smple and accurate Newton-Raphson teratve method for solvng systems of equaton. Ths method s also applcable for systems of multvarate functons. The teratve algorthm of our problem s gven by the system Fg. : Newton-Raphson basn of attracton under the effects of solar radaton pressure where orange dots ndcate the locatons of the equlbrum ponts and black dots ndcate the locatons of the prmares. ξ h n n ΩΩ ΩΩ ξ hh h ξh = ξn ΩξξΩhh ΩξhΩhξ ξ h ΩΩ ΩΩ ( n n ) h ξξ ξ hξ = hn ΩξξΩhh ΩξhΩhξ ξ h ( n n ). (5) ξ h Where n n are the values of the ξ andh coordnates of the (n-) th step of the Newton-Raphson teratve process. The ntal pont ( ξh ) s a member of the basn of attracton of the root f ths pont converges rapdly to one of the equlbrum ponts. Ths process stops when the successve approxmaton converges to an attractor wth some predefned accuracy. For the classfcaton of the equlbrum ponts on the ( ξh ) plane we wll use color code. In ths way a complete vew of the basn structures created by the attractors (Fg. ). Also we have shown the zoomed part of the basns of attractons near the prmares (Fg. ). We used Mathematca software for fndng basn of attracton. Fg. : The zoomed part of Fg. near the prmares. 4. Stablty Usng the procedure gven by Mccuskey [6] we can examne the stablty of the equlbrum ponts n the photogravtatonal crcular restrcted four-body problem. Let us suppose 35

7 ξ= ξ0 + ah = h0 + βζ = ζ0 + γ puttng these values n equaton () we get a β aa a β γ '' ' ' = Ω ξξ + Ω ξh + Ωξζ Table : Egen Values Coordnate of Equlbrum Ponts Egen Values ( ) β a aβ a β γ '' + ' ' = Ω hξ + Ω hh + Ωhζ L γ aγ = aω + βω + γ Ω (6) '' ' ζξ ζh ζζ Where aβ and γ are the small dsplacements of the nfntesmal body from the equlbrum ponts. Suffx zero denotes the value at the equlbrum pont. To solve equaton (6) let λτ λτ λτ a = Ae β = Be γ = Ce where A B and C are parameters. After substtutng these values n equaton (6) we get L L A λ aλ ξξ λ ξh ξζ ( Ω ) B( +Ω ) CΩ = 0 A( λ Ω ) + B ( λ aλ Ω ) CΩ = hξ hh hζ AΩ B Ω + C( λ aλ Ω ) = 0 (7) ζξ ζh ζζ The equaton (7) wll have a non-trval soluton for A B and C f λ aλ Ω ( λ+ω ) Ω ξξ ξh ξζ λ hξ λ aλ hh hζ Ωζξ Ωζh λ aλ Ωζζ +Ω Ω Ω = 0 L 4 L 5 λ 3 aλ + λ (4 + 3 a Ω Ω Ω ) ξξ hh ζζ + λ ( 4a a + a Ω + a Ω + a Ω ξξ hh ζζ +Ω 4 ) + λ (( Ω ) Ω ( ) +Ω Ω ξh ξh ξζ ξξ hh ( Ω ) 4Ω +Ω Ω +Ω Ω hζ ζζ ξξ ζζ ζζ hh + 4 a Ω a ( Ω +Ω +Ω )) ξh ξξ hh ζζ + λ( Ω 4 Ω a ( Ω ) + a ( Ω ) ξh ζζ ξh ξζ a Ω Ω + a ( Ω ) a Ω Ω ξξ hh hζ ξξ ζζ a Ω Ω ) + (( Ω ) Ω +Ω ( Ω ) hh ζζ ξζ hh ξξ hζ Ω Ω Ω Ω Ω = ( ξh ) ζζ ξξ hh ζζ ) 0 (8) We solved the equaton (8) for all the values of the equlbrum ponts and for each equlbrum pont we found sx egenvalues n whch at least one s ether postve real value or postve real part of the egenvalues (Table ). Therefore all the equlbrum ponts are unstable. 5. Conclusons In ths paper we explore the locatons and stablty of the crcular restrcted four-body problem by supposng all the masses are varyng wth tme and one of the masses s source of radaton pressure. The three prmares are placed at the vertces of an equlateral trangle and revolvng around ther center of mass whch s taken as orgn. We derve the equatons of 36

8 moton whch are dfferent from the classcal case by a and k the factors. We have plotted the graphs for the locatons of the equlbrum ponts n three planes the tme seres the surfaces and the Newton-Raphson basn of attracton. In the ( ξh ) plane we found fve equlbrum ponts (Fg. ) whch are ten n the classcal case where the green dots denote the locatons of the equlbrum ponts and the purple dots denote the locatons of the prmares. In the ( ξ ζ ) plane we found fve equlbrum ponts (Fg. 3) where the green dots denote the locatons of the equlbrum ponts. In the ( hζ ) plane we found three equlbrum ponts (Fg. 4) where the black dots denote the locatons of the equlbrum ponts. From the tme seres we observed that the orbts are not perodc (Fg. 5 & 6). We have drawn the surfaces of the moton of the nfntesmal body n three spaces (Fgs ) and the Poncare surface of secton where we got dscrete type graph (Fg. 0). We also have drawn the Newton-Raphson basn of attracton (Fg. ) where we have shown the convergence of the equlbrum ponts and the locatons of equlbrum ponts are ndcated by orange dots and the locatons of the prmares are ndcated by black dots. The Fg. s the zoomed part of the Fg. near the prmares. Zotos [ ] are llustrated basns of attracton n the classcal case and they found scorpon type graph whch s dfferent from my case. Fnally we examne the stablty of the equlbrum ponts for the crcular restrcted four-body problem wth radaton pressure and found at least one postve real value or one postve real part of the egenvalues (Table ). Hence all the equlbrum ponts are unstable. Acknowledgements We are thankful to the Deanshp of scentfc research and the department of Mathematcs College of Scence Al-Zulf Majmaah Unversty Kngdom of Saud Araba for provdng all the facltes n the completon of ths research work. References []. Abdullah (07). Dynamcs n the crcular restrcted three body problem wth perturbatons. Internatonal Journal of Advanced Astronomy. vol. 5 no. pp []. Abouelmagd E. I. Sharaf M.A. (03). The moton around the lbraton ponts n the restrcted three-body problem wth the effect of radaton and oblateness. Astrophys. Space sc. vol. 344 pp [3]. Abouelmagd E. I. Mostafa A. (05). Out of plane equlbrum ponts locatons and the forbdden movement regons n the restrcted three-body problem wth varable mass. Astrophyscs Space Sc DOI 0.007/s [4]. AbdulRaheem A. Sngh J. (006). Combned effects of perturbatons radaton and oblateness on the stablty of equlbrum ponts n the restrcted three-body problem. The Astronomcal Journal [5]. Asss S. C. et al. (04). Escape dynamcs and fractal basn boundares n the planar earth-moon system Celest. Mech. Dyn. Astr [6]. Baltaganns A.N. Papadaks K.E. (0). Equlbrum ponts and ther stablty n the restrcted four-body problem. Int. J. Bfurc. Chaos 79. [7]. Bhatnagar K. B. Chawla J. M. (979). A study of the Lagrangan ponts n the photogravtatonal restrcted three body problem. Indan journal of pure and appled mathematcs. Vol. 0 pp [8]. Chernkov Yu. A. (970). The Photogravtatonal restrcted three body problem Sovet Astronomy 4() pp [9]. Douskos C. N. (00). Collnear equlbrum ponts of Hll s problem wth radaton pressure and oblateness and ther fractal basns of attracton. Astrophys. Space Sc [0]. Kalvourds T. Arrbus M. Elpe A. (007). Parametrc evoluton of perodc orbts n the restrcted four body problem wth radaton pressure. Planetary and space scence. Vol. 55 pp []. Kumar R. Kushvah B. S. (04). Stablty regons of equlbrum ponts n the restrcted four body problem wth oblateness effects. Astrophyscs Space Sc []. Lchtenegger H. (984) The dynamcs of bodes wth varable masses. Celest. Mech [3]. Meshchersk I.V. (949). Studes on the Mechancs of Bodes of Varable Mass. GITTL Moscow. [4]. Mgnard F. (984) Stablty of L 4 and L 5 aganst radaton pressure. Celest. Mech [5]. Mttal A. et. al. (06) Stablty of lbraton ponts n the restrcted four-body problem wth varable mass. Astrophyscs and space sc DOI 0.007/s [6]. Mccuskey S. W. (963). Introducton to Celestal Mechancs. Addson Wesley USA. [7]. Perezhogn A. A. (976). Stablty of the sxth and seventh lbraton ponts n the photogravtatonal restrcted crcular three body problem. Sovet Astronomy letters. Vol pp. 7. [8]. Pushparaj N. Sharma R. K. (07). Interor resonance perodc orbts n the photogravtatonal restrcted three-body problem. Advances n astrophyscs. Vol. () pp [9]. Sharma R. K. (987). The lnear stablty of lbraton ponts of the photogravtatonal restrcted three body problem when the smaller prmary s an oblate spherod. Astrophys. Space Sc. vol.35 pp [0]. Smmons J. F. L. McDonald A. J. C. Brown J. C. (985). The restrcted three body problem wth radaton pressure. Celestal Mechancs. Vol. 35 pp []. Sngh J. et al. (984). Effect of perturbatons on the locaton of equlbrum ponts n the restrcted problem of three bodes wth varable mass. Celest. Mech. 3 (4) []. Sngh J. Ishwar B. (985). Effect of perturbatons on the 37

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