Two-Body Problem with Varying Mass in Case. of Isotropic Mass Loss

Transcription

1 Adv Theo Appl Mech Vol no 69-8 Two-Body Poblem with Vaying Ma in Cae of Iotopic Ma o W A Rahoma M K Ahmed and I A El-Tohamy Caio Univeity Faculty of Science Dept of Atonomy Caio 6 Egypt FA Abd El-Salam Taibah Univeity Faculty of Science Dept of Math El-Madinah Saudi Aabia M I El-Saftawy King Abdulaziz Univ Faculty of Science Atonomy and Space Science Dept Jeddah Saudi Aabia malaftawy@kauedua Abtact The peent wok i concened with the two-body poblem with vaying ma in cae of only one body eect ma The poblem i teated analytically in the Hamiltonian fame-wok and the equation of motion ae integated uing the ie eie developed and applied epaately by Delva [] and Hanlmeie [] The Hamiltonian i contucted up to econd ode uing time a independent vaiable The equation of motion ae integated uing the ie opeato and ie eie yielding the olution diectly in tem of the coodinate Keywod Vaying ma Petubation two-body Intoduction The two-body poblem with vaiable ma i one of old-tanding; it oigin going back to the middle of 9 th centuy Howeve ome confuion ha peited

2 7 W A Rahoma et al a to the dynamical equation which have to be ued Solution of celetial mechanic poblem fo eveal vaiable-ma bodie have been analytically tied olved fo two baic ituation: the geneal two-body poblem and the eticted thee-body poblem with many modification The mathematical tool applied to thee two baic poblem uually elie upon the claical equation of motion in gavitational field with additional fomal tem due to the vaiability of the gavitating mae The motion i tudied with the aleady claical pocedue that intepet thee tem a ditinct fictitiou petubation applied to the unpetubed motion of tationay mae Thee exit a elated invee pocedue intepetation of the actual petubed motion a a fomally unpetubed one in the gavitational field geneated by a vaiable effective ma In thi pape we limit ouelve to the fit baic poblem (that of two vaiable-ma bodie Diffeent Model of the Ma o Since both the elative ate of ma change and the time inteval fo thi change mut be included into the equation of motion hee below ome inteeting model of the ate of ma change ae addeed Metchekii' model Metchekii wa the fit to point out a pecific cae of the two-body poblem with vaying ma which i integable by intoducing pecial pace-time vaiable in which the poblem i educed to the claical poblem of two bodie Polyakhova [6] and Pieto and Docobo []-[] Thee integable cae coepond to thee celebated model by Metchekii fo the change in the total ma of the ytem μ ( t = ( a + αt μ ( t = ( a+ αt μ ( t = ( a+ αt + βt whee a α β ae cetain contant and μ i ( t ( = m ( t + m ( t i = ae diffeent model fo the ma change Matin' model Matin [5] fom hi wok on double ta ytem with vaying ma eached to thi tatement n αm m& = ( whee i the value of adiu vecto between the two component of the ytem

3 Two-body poblem with vaying ma 7 Hadidemetiou [7] addeed ueful comment on the Matin' model of vaying ma: thee ae: I The dependence of the ate of ma lo on the ditance between the two component mut be due to a tidal inteaction but ince μ i the total ma of the ytem thi law tate that the tidal inteaction i independent of the atio of the mae of the two component which doe not eem ealitic II The effect of one ta on it ma-loing companion would mot pobably eult in a non-iotopic lo of ma; and conequently the teatment of the poblem by uual method i not valid III The tidal inteaction i not likely to poduce lage velocitie of eection of ma o that the eected paticle may not ecape fom the ytem but intead full on the othe ta Fo thee eaon uch law mut be teated with geat cae and in cloe connection with the mechanim the ma lo take place Jean model Jean [8] wa the fit to poe law of vaying ma a an atophyical poblem He baed hi tudie on the theoy developed by Eddington (maluminoity elation by genealized law of ma lo; n & = α (5 m m whee α n ae eal numbe the fit one i +ve appoximate to zeo and n vaying between and Thi law i called Eddington-Jean law Taking n = we have Metchekii fit integable cae while taking n = we have the lat Metchekii cae Andade and Docobo Model Analyzing the dynamic of binay ytem with time-dependent ma lo and the inteaction between the two component mut be taken into account Andade and Docobo [9] could uppoe that cloe to peiaton thee i an appeciable enhancement of ma lo Thi phenomenon will be called the peiaton effect and it will be moe noticeable the geate the eccenticity and the malle the minimum ditance between the two ta Of the whole et of law that take into account peiaton effect by mean of it dependence on ditance only ome of them give ie to new behavio in the evolution of the obital element uch a ecula vaiation of eccenticity In thee model they tudied the following time-and ditance-dependent ma-lo law Pθ & μ t; ; P = & μ t β (6 ( θ ( whee the fit tem epeent time-dependent ma lo and the lat one intoduce the peiaton effect whee i the ditance between the two

4 7 W A Rahoma et al component P θ i the total angula momentum and β i anothe mall paamete cloe to zeo Petubation Technique In many cae in celetial mechanic the eie development of the ditubing function i not eaily teated and i complicated To avoid thi difficulty we ue an altenative appoach (Delva []; and Hanlmeie [] in which the pocedue can be pefomed with an opeato A pecial linea diffeential opeato the ie opeato poduce a ie eie The convegence of the eie i the ame a fo Taylo eie ince the eie i only anothe analytical fom of the Taylo eie In addition we can change the tep ize eaily (if neceay et Η ( x y Px Py t be the Hamiltonian function ( x y be the coodinate ( P x Py be the conugate momenta and t be the time Then the equation of motion ae: dx dp x& = = Η P & x x = = dt P x dt x Η dy dp y& = = Η P & y y = = dt P y dt y Η The linea ie opeato ha the geneal fom: dx dy dp dp x y D = dt x dt y dt P x dt P y (7 t and the olution x( x y Px Py t y( x y Px Py t Px ( x y Px Py t P x y P P t ae then given by the ie eie and ( y x y = = ( x y { exp( } x x y P P t t t D x D x x = x = = = ( x y { exp( } y x y P P t t t D y D y y = y = P ( x y P P t = { exp( t t D} P = D P x x y x P x x = Px = P ( x y P P t = { exp( t t D} P = D P y x y y P y y = Py = x y Px!! Py!! (8 (9 ( ( whee D x D y D P x and D P y ae to be evaluated fo the initial condition x x y P P t y x y P P t P x y P P t P x y P P t ( ( ( and ( x y x y x x y y x y

5 Two-body poblem with vaying ma 7 Hamiltonian The Hamiltonian i contucted in tem of Delaunay' vaiable a (Depit; []: μ μ Κ Κ( l G; t = + & eine ( μ We need fit to expand the Hamiltonian function ( in a mall paamete ε uing Jean' law of ma Fo ε = the Kepleian cae would be obtained It eem that the paamete ε hould be elated to the coefficient α which appea in the Jean law of ma vaiation We will imply chooe ε a nondimenional value of α Thi choice alo utifie the application of the method a fa a the econd ode only ince α i vey mall and highe ode would not contibute ignificantly Expanding the function m ( t in a Taylo eie yield:- ( p p m = m + m ( p = p! ( p th whee m i the value of ma in cetain initial intant t m i the p deivative of the function m with epect to t evaluated at t = t The Hamiltonian can now be ewitten in expandable fom a: m α Κ { ( G τ = + ine+ τ η ( } ( =! with ( n ( G = m n e ( ( ( ( ( n + + m ( n η = + and τ = t t ( 5 Solution of the Poblem To olve ou poblem a technique due to Delva [] and Hanlmeie [] i ued Defining: n A = m n ( ( ( ( ( n + ( + B = m n + then can be witten a η B = Ae η =

6 7 W A Rahoma et al The vaiation in the obital element can be witten a: U& α E = Κ τ = co E u =! u m u& = Κ = U α E τ U in E co E τη U =! U U whee ( X U denote the patial deivative of ( X with epect to U The nonvanihing final expeion of the vaiation in the obital element ae: m a G l& α τ = + in E + in E + τη (5 =! e α a G g τ & = G in E in E (6 =! e a & α τ = co E (7 =! Since only the mutual gavitational attaction i conideed and the ma lot by one body of the ytem i tanfeed to it companion the total ma of the ytem i kept contant Thi tun a contant mean motion n ~ The linea ie opeato D in tem of the Delaunay element ha the geneal fom: dl dg d dg D = dt l dt g dt dt G t Applying the opeato D to l g G and t yield: dl l Dl = + = l& + n% (8 dt t dg Dg = = g& (9 dt d D = = & ( dt dg DG = = G& = ( dt Dt = ( The olution l ( l g G t g ( l g G t ( l g G t ( l g G t ae then given in tem of the ie eie a:- l ( l g G t = D l G g l g G t ( = l = D g = g!! and ( (

7 Two-body poblem with vaying ma 75 whee l g G t ( G l g G t ( D l D g = = = = D D and D G G!! (5 (6 D G ae to be evaluated fo the initial condition l ( l g G t g ( l g G t ( l g G t G ( l g G t and To find the tem of the eie it will be neceay to calculate the multiple action of D to the vaiablel g G and t The ingle action to l g and G poduce: Dx = x x = ( l g G (7 and hence the multiple action give D x = D x 5 The Seie fo l The double action of the ie opeato D on the mean anomaly l can be computed a: dl dg d dg D l = l& + n dt l dt g dt dt G t % (8 Setting A e G = = A e e 6B η = Equation (8 can be witten a: 5 5 m m m a l a l l a l D l = α ϕnm co ne + α nm + γn in ne + τ ρnm co ne n= m= (9 whee the non-vanihing coefficient ae φ l m = + l mg & φ = 6 l mg l mg φ 6 & φ 6 e l γ & l G G ( = + + e e e l G l μg = & e

8 76 W A Rahoma et al l G + e l G 8 l G G = & = + 6 e e l μ G 8 8 l μ G = & 5 8 e 8 = e l G l G & = ( 6 e + e e e l G l G 5 6 & e 6 l μg l G 5 = 8 & 5 6 8e 8e l l m ρ = η & ρ = η η + + l G m G ρ = η 8 l G m l G m ρ = η + e & ρ = η + then the olution fo l i given by ( [ ] [ ] ( l t = l + Dl t t + D l l l l [ ] & % ( = l + l + n t t + D l l l l ( 5 The Seie fo g The double action of the ie opeato D on the mean anomaly g can be computed a: dl dg d dg D g = [ g& ] ( dt l dt g dt dt G t Setting G G = A e Equation ( can be witten a: 5 5 m m m a g a g g a g D g = α ϕnm co ne + α nm + γn in ne + τ ρnm co ne n= m= ( whee the non-vanihing coefficient ae

9 Two-body poblem with vaying ma 77 g m φ G mg = & φ g 5 g mg g mg φ = 5 & φ = 5 e g γ G & g G G + G e e g G G G ( 5 G + e + 5 e e e g G g μg 5 & 6 9 6e e g G G G e g G g G G = e e e g G g μg 5 & 5 = 7 6e e g G G = e g μg g G 5 = 7 & 5 = 5 e 6e g m g m G G ρ = η G + G & ρ = η 5 8 g G m g G m ρ = η + e & ρ = η + then the olution fo g i given by = & ( ( [ ] [ ] ( g t = g + Dg t t + D g g g g [ ] [ & ] ( = g + g t t + D g g g g ( 5 The Seie fo The double action of the ie opeato D on momenta can be computed a: dl dg d dg D = & dt l dt g dt dt G t ( which can be witten a:

10 78 W A Rahoma et al 5 5 m a = αϕnm + α nm + τρnm n= m= { } D in ne co ne in ne (5 whee the non-vanihing coefficient ae m φ m e = & φ G & 8 μg 5 = 5 & = e G G μg + 5 & 5 5 e e e μg & 5 = 5 e G G μg + 5 & 5 5 e e e G m = & ρ = 8 η + e m ρ = η + then the olution fo i given by ( [ ] [ ] ( t = + D t t + D [ ] [ & ] ( = g + g t t + D (6 5 The Seie fo G The double action of the ie opeato D on momenta G equal to zeo then: G t = G (7 ( [ ] G 6 Concluion The following concluding emak and note can be outlined:

11 Two-body poblem with vaying ma 79 I The eection of ma fom any body depend on many paamete amongt the mot impotant of which ae the cental condenation (which mean moe o le the degee of igidity of the body and the velocity of otation which povide the extenal laye with the angula momentum that activate the poce of eection of ma But to implify the model we aumed that the ate of ma eection depend explicitly olely on the ma of the body II Since the diffeent model of vaiable ma aume that the ma lo take place iotopically ie thee i no pefomed diection in the pace it i expected to find the Hamiltonian fee fom dependence on the inclination Thi eflect the abence of H in the Hamiltonian of the poblem Theefoe h & = h = cot III Alo a the ta ae aumed point mae the Hamiltonian i fee fom oientation angle ( g h Thi mean that G and H ae kept contant Refeence [] A Depit Celet Mech (98 [] A Hanlmeie Celet Mech (98 5 [] C Pieto and J A Docobo Aton and Atophy 8(997a 657 [] C Pieto and J A Docobo Celet Mech & Dyn Aton 68(997b 5 [5] E Matin Reale Stazione Aton e Geof di Calofote (Cagliai (9 [6] E N Polyakhova Aton Rep 8 (99 8 [7] J D Hadidemetiou Icau 5(965 [8] J H Jean Mon Not R Aton Soc 85(9 [9] M Andade and J A Docobo Wind Bubble Exploion: A Confeence to Honou John Dyon Pàtzcuao Michoacén Mexico 9- Septembe Edito: S J Athu and W J Henney Intituto de Atonomia Univeidad Nacional Autónoma de México (

12 8 W A Rahoma et al [] M Delva Celet Mech (98 5 Received: Apil