Galactic Contraction and the Collinearity Principle

Transcription

1 TECHNISCHE MECHANIK, Band 23, Heft 1, (2003), Manuskipteingang: 12. August 2002 Galactic Contaction and the Collineaity Pinciple F.P.J. Rimott, FA. Salusti In a spial galaxy thee is not only a Keplefoce acting on an individual sta but also a tansvese pail oppo sing the motion. The elatively small tansvese pall is due to the atmospheic dag exeted by intestella gas (hydogen, cosmic dust). It is also shown that the ams of a spial galaxy consist of Wad spials, that thee is an obital enegy loss fo each individual sta in a contacting galaxy, and that the size ofthe Wad spial obseved can be used to pedict the speed of the galaxy s contacting. Fo inside the galaxy s cental sphee it is shown that the path of a sta descibes a logaithmic spial and that thee is an associated obital enegy loss. 1 Equations of Motion The collineaity pinciple equies that the stas obiting in a galaxy aange themselves into a flat disk (Rimott, 1998). Within the galactic disk (Figue 1) pola coodinates can be used fo Newton s second law. Fo an individual sta m K z = m('' 92) (1) D m<é+2é) (2) G : gavitional constant M : mass of galaxy (without m) 21

2 D = (constant) dag foce ' ' Accoding to Wad (2000) the tem f in equation (1) is often so small that it can pofitably be neglected. This is appaently also the case fo spial galaxies. Consequently fom equation (1) - GM e2 = 2m - GM o 9 = 2 (3). Diffeentiated with espect to time -- e = i G. 1:4 (4) 2 Equations (3) and (4) enteed into equation (2) give 1 GM. D = m 3 (5) 2 ' which elates the dag to the adius change ate. Solved fo the adius change ate equation (5) becomes = - - (6) mxl GM If D > 0 then " < 0, chaacteizing a contacting galaxy. 2 Shape of the Galactic Ams We begin with equations (5) which may also be witten as d 2D : dt (6) whee we have chosen to use 1 -. D = Em600 = constant (7) fo an initial inteval, The subscipt 0 efes to the sta consideed, Fo any subsequent inteval the dag may have a diffeent new value. On the othe hand equation (7) may be valid, at least appoximately, fo an extended potion of, and possibly the whole spial am. We assume the latte to be the case. Integated between = 0 fo t = 0 and : fo t : t we aive atapaamete equation 2 <8) 22

3 which was fist obtained by CA. Wad in connection with dag studies (see Rimott and Salusti, 2001). Making use of equations (3) and (7) we may also wite (9) instead of equation (8). By using (10) we obtain the Wad spial in pola coodinates _ 0 (1+c6)2 (11) with D c = K0 (12) and GMm K0 = 2 Vo (13) Fom these esults we conclude that the ams of a spial galaxy ae fomed by Wad spials. In Figue 2a Wad spial with c : 0.08 is depicted. Figue 2. A Wad Spial with c =

4 I 3 Obital Enegy The obital enegy of a single sta m on a cicula obit can be shown (Rimott, 1989) to be E = _ GMm 2 (l4) If subject to adius change ate ' the associated enegy change ate is -. M E:GMj": G D (15) Z" F In paticula fo a contacting galaxy ' < O and consequently E < 0, in othe wods thee is an enegy loss fo the sta m in a contacting galaxy, as equied by the collineaity pinciple. 4 Speed of Contacting Fo the physical appeaance of a spial am one can infe as to how fast a spial galaxy will contact on its way to become an elliptical galaxy. With the coefficient c known fom obsevation, the initial contacting speed #0 can be calculated. Fom equations (5), (11) and (12) 1 GM. D - CKO = m 30 2 d 01' 0 = 2c GM 0 (16) 5 Numeical Values fo Sun and Milky Way Fo ou own Milky Way and ou own Sun (Geine, 1989) which is located in one of the spial ams, we have m = 1.989(1030)kg fo the Sun 0 = 5(1020)m 90 = 6(10 16)s l v0 = 060 = 5(6)1o4 = 3(105)m/s M z 5.967(1041)kg fo the Milky Way G = 6.67(10 )m3kg ls'2 If the Sun moves close to the galactic cente at a ate of, say, 0 ate = 4.5(104)m/s then we have fo the enegy E : 6.67(10 )5.967(104 ) ) 1(52)104 ( 4.5(104)) = 7.125(1024)w fo the dag in tems of the adius change ate D = %m60#0 = %1.989(1030)6(10'16)4.5(104) = 2.685(10 9)N 24

5 fo the Keple foce magnitude K0 _ GMm _ 6.67(10' )5.967(1o4 )1.989(1030) 02 52(1040) =3.166(1020)N and fo the atio 0 = 2 = K0 A atio of C = 0.08 has been used fo the Wad spials of Figues 1 and 2. A contacting ate of 0 = 45 km/s : 4.5(104)m/s may seem to be athe lage. But conside this: Fo a galaxy of, say, 0 : 5(1020) m adius, shinking at a ate of 4.5 (104) m/s to half its size would mean that it takes 2 = limo 2 z 5.556(10 5)s = 1.762(108)a 2 4.5(10 ) i.e million yeas! 6 Inside the Cental Sphee Physical evidence has it that spial galaxies still have a nucleus that can be modeled by a small cental sphee (Figue 1) of unifom sta distibution. The gavitational field fo a point mass m inside a solid sphee M of uni fom mass (=sta) distibution and of oute adius R (Rimott, 1989) is 2 U = G ;m{3 %} (17) Leading to a Keple foce inside the sphee of K: -5 at] 2 GM R3 (18) Newton 3 second law equies then, that in adial diection GMm R 3 = m('' é)2) (19) With 5" = O as an appoximation we have Ö = Ä = constant (20) and The velocity of the sta m is {3:0 (21) ~2 v = J2+292 = M")wa :62 (22) 25

6 2 In View of ' << 292 we simplify and wite v=9= GM R3 (23) We also let the dag foce decease accodingly towads the galactic cente and wite fo it I, D R (24) and get fo Newton s second law, in tansvese diection, ED 2 m(é+2é) (25) With = 0 fom equation (21), we have. D =. ZmRG (26) which integated between the limits : R fot : 0 and = fo t = I gives = Rexp. I ZmRG (27) o with 6 f : T 6 (28) we have =Rexp D 6 2KR (29) a logaithmic spial, with a Keple foce of magnitude KR = mrö2 = GMm R2 (30) The kinetic enegy of a sta m inside the cental sphee is l - i T = mv2 2 imze2 2 m G : l R3 2 R3 (31) Togethe with the potential enegy fom equation (l7) we have fo the sta s obital enegy 2 1; = [j.t 7* :..EZQZÜZ Eä ll_ R 2 R2 (32) 26

7 and fo its time deivative ZGMm E: R3 i (33) We conclude that E < 0 if ' < o (34) i.c. contaction is associated with an obital enegy loss. We also have E = 2GM ' at = R (35) R2 and E = 0 at : O (36) The obital enegy change ate may also be expessed in tems of the dag, equation (26), as E = E2D = at : R the obital enegy change ate becomes R GM 20 (37) Rx/F E z _ EKD (38) R 7 The Inteface At the inteface : R of galactic oute disk and inne sphee, we find that all quantities, with one exception, ae continuous. Fo the dag foce we compae equations (7) and (24). Fo the obital enegy we compae equations (l4) and (32). Fo the obital enegy change ate we compae equations (15) and (35). Fo the magnitude of the Keple foce we compae equations (1) and (30). Fo the angula velocity we compae equations (3) and (20). The single exception is the adial velocity component which changes fom = 2D R 2 (39) mv GM as given by equation (6), when appoached fom outside the cental sphee, to x» = D R3/2 (40> 211wGM fom equation (26), when appoached fom inside the cental sphee. Moving inwads its magnitude deceases by a facto of fou and povides an explanation fo the sudden incease of the sta population fom disk to sphee at the inteface. 27

8 8 Conclusion By the simple expedient of taking into account the influence of a small atmospheic dag, exeted on each individual sta of a shinking spial galaxy, the spial natue of the galaxy, consisting of Wad spials outside the cen tal sphee and logaithmic spials inside the cental sphee, eveals itself. Anothe consequence is that, in acco dance with the collineaity pinciple, each individual sta gives up some of its obital enegy in the pocess. The shape of the Wad spials is also a measue of how quickly a spial galaxy shinks moving towads becoming an elliptical galaxy, the subsequent stage in the evolution pocess of galaxies. Refeences Geine, Walte: Theoetische Physik. Band 2, Teil 2, Velag Hai Deutsch, 466 S, (1989). Rimott, F.P.J.: Das Kollineaitätspinzip. Technische Mechanik, 18, l, (1998), Rimott, F,P,J.: Intoductoy Obit Dynamics, Vieweg, 193 p, (1989). Rimott, F.P.J.; Salusti, EA: The Wad Spial in Obit Dynamics. Poceedings, CANCAM O], , (2001). Wad, C.A,: Effects of the Dag Measued with a Fluid Cell on the Obit of the Space Shuttle. Pivate Com munication, (2000), 1 4. Addesses: Pofesso F P.J. Rimott, Depatment of Mechanical and Industial Engineeing, Univesity of To onto, Toonto, Ontaio, Canada MSS 3G8. Pofesso FA. Salusti, Depatment of Mechanical, Aeospace and Industial Engineeing, Ryeson Univesity, Toonto, Ontaio, Canada MSB 2K3. e mail: fimott@ha hinet.on.ca; salusti@yeson.ca 28