Single Particle Closed Orbits in Yukawa Potential

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1 Single Particle Closed Orbits in Ykawa Potential Rpak Mkherjee 3, Sobhan Sonda 3 arxiv:75.444v [physics.plasm-ph] 6 May 7. Institte for Plasma Research, HBNI, Gandhinagar, Gjarat, India.. Ramakrishna Mission Vivekananda University, Belr Math, Howrah, Kolkata, India. 3. Ramakrishna Mission Residential College (Atonomos), Narendrapr, Kolkata, India. rpakmkherjee@gmail.com sonda6@gmail.com May 9, 7 Abstract: We stdy the orbit of a single particle moving nder the Ykawa potential and observe the precessing ellipse type orbits. The amont of precession can be tned throgh the copling parameter α. With a sitable choice of the copling parameter; we can get a closed bond orbit. In some cases we have observed some petals which can also have a closed bond natre with an appropriate choice of the copling constant. A threshold energy has also been calclated for the bondness of the orbits. Keywords: Ykawa Potential, Precessing Ellipse, Closed-Bond Orbit, Critical Point PACS No: 45..D, 45..dh, 45.5.Pk, 45..dc

2 Introdction: It is widely known that there are only two types of central potential r and r in which all finite motions take place in closed paths[]. Some exceptions of the statement above have been reported recently[, 3]. On the other hand Ykawa potential or screened colomb potential of the form V (r) = α r e r λ, ( λ is the screening parameter that determines the range of this interaction) has a long standing legacy to represent varios physical systems. Its application ranges from astronomy, high-energy physics, nclear physics, condensed matter physics to plasma physics and many other branches of physics. In the domain of plasma physics this potential is sed mostly in Strongly Copled Plasmas which has several applications in modelling the cores of white dwarfs, planetary rings, atmospheric lightning, molten salts and plasma technology. Hence qite natrally one may ask what is the orbit of a single particle nder this screened colomb potential. Can we expect some bond orbits for the non-zero vale of α? This stdy is a more general case of the previos stdies [, 3] that has been done till now regarding the closed bond orbits distinct from colomb or harmonic oscillator potentials becase by the trncation of the exponential series in the Ykawa potential one gets back the potentials assmed in the literatre so far. Hence this stdy is a sperset of the previos stdies performed so far to the best of or knowledge. The paper is organised as follows. Section discsses the theory of central force motion for any arbitrary potential. Following the theory we have calclated the trajectories for Ykawa potential in section 3 and the thrst has been given on the calclation for threshold energy for bond orbit in Ykawa potential in section 4. In section 5, some possible applications of this stdy has been mentioned. A Brief Discssion of Central Force Motion The total energy of a particle moving nder a central force is given by, mṙ + J + V (r) = E, mr J = mr θ where m is the mass of the particle, r is the radial distance of the particle from the force center, J is the anglar momentm of the particle, V is the potential of the particle and θ is the anglar coordinate of the particle with respect to some reference axis. For r = i.e. ṙ = J d; the above eqation looks like m dθ m J ( ) d + J m dθ m + V () = E For energy to be constant, i.e. de =, dθ d dθ = m d (V + J ) = m dv eff J d m J d

3 where dj = since, for central force the anglar period θ of the radial oscillators r(θ) dθ is independent of the anglar momentm J. In dimensionless coordinates, = and a V = m V the eqation of motion becomes, J a d dθ = dv eff d with V eff = V () +. Now if the form of the potential is known, we can find the trajectory of the particle. The threshold energy vale for a bond orbit in any central force can be calclated from the total energy (h) conservation relation, my + V eff () = h, where, V eff () = F ()d and y = d dθ y = ± m [h V eff()] () Now, if we differentiate the above eqation we get, y dy d = dv eff m d Here, for dv eff =, if dy is not ndefined, i.e. the crve plotted from eqation (), does d d not ct the axis perpendiclarly, then y = which in trn represents h = V eff i.e. the minimm threshold vale of energy for bond orbit. Now if dy is ndefined, i.e. the crve plotted from eqation, cts the axis perpendiclarly, then from the roots ( c, ) or the critical points of the eqation dv eff d = we find d dv (c) some eqilibrim points of the system. Now from the relations = and dv eff = d d we can conclde that V () has either a relative extremm or a horizontal inflection point at = c. If, I. the V eff () has a relative minimm at = c, then V eff ( c ) = h (threshold or minimm vale for energy) and the critical point is a centre and is stable. II. the V eff () has a relative maximm at = c, then the critical point is a saddle point and is nstable. III. the V eff () has a horizontal inflection point at = c, then the critical point is of a degenerate type called a csp and is nstable. 3 Motion of a Single Particle nder Ykawa Potential The Ykawa or screened colomb potential is given by: () V (r) = α r e r λ (3) V eff () = mα J a e (4) 3

4 where, λ is the range of the Ykawa potential. The eqation of motion is given by d = + αe dθ ( + ), where, α = mα J a (5) which can be rewritten in terms of energy as: my + αe = h y = ± h + αe (6) where we have chosen m =. The exact vales of α differ for different physical cases depending on the natre of application[,, 3, 6]. The Eqation (5) is difficlt to solve for an exact analytical soltion. For planetary type motion the eqation of motion with or parameters becomes, ( d dθ + = + αe λ + ) (7) λ In ref[] an approximate analytical soltion has been obtained by expanding the R.H.S. of (7) in a Taylor series and trncating it to the second order. Ths the eqation looks like, [ d dθ + α p = p [ + αe a p/λ ( a ) ] p e a p/λ = λ p ( + a )] p λ a p λ where a p p = / and is the npertrbed (α = ) soltion. The soltion of the above eqation given by, (θ) = p + e cos ω(θ θ ) ω = [ α(a p/λ )e a p/λ ] / is only valid for α << and can deviate significantly for particle orbits which lie away from the npertrbed (α = ) vale. The aim of or present stdy is to explore the fll soltion space of eqation (5) by retaining the complete nonlinear form of the force term. We obtain these soltions by a nmeical soltion of the eqation. We adopt another analytical tool viz. a linear stability analysis for a particle nder ykawa type potential. We proceed to some extent and then trn to the nmerical analysis for different α (copling constant). 4

5 4 Linear Stability Analysis[7] for Single Particle Motion in Ykawa Potential For the sake of analysis, we add a small viscos term proportional to velocity (µ d ), in the dθ eqation of motion and eventally pt µ = for the final calclation. Ths for Ykawa Potential the modified eqation of motion (5) can be written as, ( (θ) = f( (θ), (θ)) = µ + α + ) e (θ) = F ((θ), y(θ)) = y y (θ) = G((θ), y(θ)) = µy + α ( + ) e Hence, the fixed points f(y, ) = are: (y, ) = (,.4) for α =.5. (y, ) = (,.346) for α =.5. (y, ) = (,.475) for α =.5. (y, ) = (,.796) for α =.5. (y, ) = (,.894) for α =. The Jacobian for the above set of eqation will be given by[4], J(, y ) = ( F G F y G y ) =,y=y = ( ) + α e 3 µ =,y=y The Eigenvales of the above matrix will be, λ, = ( τ ± ) τ 4 = ( ( µ ± µ 4 α ) ) 3 e = It is checked that the vales of are positive for all vales of α given above. Hence, for the above parameter vales the fixed point can not be a saddle point. 8 Now, for > if τ < and τ 4 > then f(y, ) is 6 a stable node. 4 if τ < and τ 4 < then f(y, ) is a stable spiral. if τ > and τ 4 > then f(y, ) is an nstable node. if τ > and τ 4 > then f(y, ) is an nstable spiral. if τ = and τ 4 > then f(y, ) is a ntrally stable center. Delta Vale of Delta with alpha.... alpha "Delta" : 5

6 The existance of the limit cycle arond each of the fixed points (for different vales of α) has been checked and it is fond that for none of the cases there exists any limit cycle. Hence if we continosly change µ from positive to negative the fixed point changes from stable to nstable spiral. However at µ = we do not have a tre hopf bifrcation becase there are no limit cycles on either side of the bifrcation. This sitation is identical to the case of a damped pendlm or a dffing oscillator. Frther we analyse the case of µ = nmerically for the qantitative nderstanding of the parameter vales with closed orbits.. 5 Nmerical Analysis for Single Particle Motion in Ykawa Potential For nmerical analysis, we have sed a simple Rnge Ktta 4th Order solver and have sed standard accepted algorithm for avoiding the nmerical singlarities, if encontered. Interestingly we have shown that for some typical vales of α we get a closed bond orbit. In some cases the orbits have some petals and the nmber of petals can also be controlled throgh the jdicios choice of the copling constant. Here we present some reslts of or simlation that helps to nderstand the dependency on the copling constant easily. For α =.5; we observe a constant precesion of the ellipse. Bt the ellipse does not close itself when it completes one revoltion. We have tned the vale of the copling constant to.5 and have observed the expected closre..5 alpha=.5 "alpha_p5 :.5 alpha =.5 "alpha_p5" : Figre : Precessing Ellipse with α =.5 and Closed orbits with α =.5 For α =.; we observe almost same behavior bt the amont of precesion has increased. Still we had to tne the copling constant to. to get a closed bond orbit. 6

7 .5 alpha=. "alpha_p" :.5 alpha =. "alpha_p" : Figre : Precessing Ellipse with α =. and Closed orbits with α =. For α =.5; we observe a small groping of the precessed orbits. The precession occrs in a bnch of orbits. Hence we may observe that there are two different vales of precession within one complete revoltion. When we set the copling constant to.49 we sddenly observe a closed bond orbit maintaining the natre of groping..5 alpha =.5 "alpha_p5" :.5 alpha =.49 "alpha_p49" : Figre 3: Precessing Ellipse with α =.5 and Closed orbits with α =.49 For α = ; we observe two distinct types of orbits, which has been frther investigated in the next step. Here a 3-fold larger orbit as well as a 3-fold smaller orbit are fond. The tning of the copling constant has been done accordingly to obtain the closed bond orbit. 7

8 alpha =. "alpha_p" :.5 alpha =. "alpha_p" : Figre 4: Precessing Ellipse with α =. and Closed orbits with α =. For α = 5; we observe two distinct classes of petals, one within the other. Maintaining the two distinct petal strctres we have been able to find the closed bond orbits for some parameter vale (α). alpha = 5. "alpha_5p" : alpha = 5. "alpha_5p" : Figre 5: Precessing Ellipse with α = 5. and Closed orbits with α = 5. For α = also; we observe two distinct classes of petals bt the ratio of the bigger to smaller orbit has changed. Here also we have observed closed orbits. 8

9 alpha =. "alpha_p" : alpha = "alpha_9p9" : Figre 6: Precessing Ellipse with α =. and Closed orbits with α = Ths we observe that increasing the magnitde of copling constant increases the precession speed. In some cases it also changes the natre of the orbits. Ths if one can tne the copling constant in sch a way that after one complete rotation the precessed orbit matches with the first orbit, it traces back the same path and hence forms a closed bond orbit. The same is tre for the orbits with two distinct types of petals. Ths for Ykawa potential bondness criteria is jst a catios choice of the copling parameter that we have presented extensively for many parameter (copling constant; α) vales in the above figres. 6 Energy diagrams for ykawa potential: Now we calclate the threshold energy for the bond orbits. For a constant vale of copling constant, α =.5, we plot eqation 6 for different vales of energy. (h =,,.5,.5,.,.5,.5). Note that all the crves ct the negative axis perpendiclarly representing dv eff to be ndefined. This states that we have not reached the threshold d energy vale. For energy vales less than.5 we observe no crossing in the negative axis representing that we have gone frther below the threshold energy for copling constant α =.5 9

10 .5 alpha =.5 "h=" : "h=" : "h=.5" : "h=.5" : "h=." : "h=.5" : "h=.5" :.5 y Figre 7: Energy diagram for constant copling parameter α =.5 and varying energy h Again for a constant h =.5 we plot for different vales of copling constant (α =.5,.5,.5,.5, ).5 h =.5 "alpha=.5." : "alpha=.5." : "alpha=.5." : "alpha=.5." : "alpha=.." :.5 y Figre 8: Energy diagram with constant energy h =.5 and varying copling parameter α Now we calclate the minimm vales of energy and the -nllclines graphically. For a constant vale of α we start with a bond orbit in the negative axis and keep on decreasing energy ntill the circle redces to a point. Then if the vale of energy is frther decreased; sddenly we observe that the point in the negative axis disappears. This corresponds to the minimm vale of energy for a closed orbit in the θ diagram. In the figre below we have plotted the minimm vale of energy (h) pto which the point in

11 the negative axis was fond. The position of the point gives the vale of the -nllcline and we have measred that vale for different α..5.5 Plot for Minimm Energy "alpha=." : "alpha=.5" : "alpha=.5" : "alpha=.5" : "alpha=.5" :.5 y Figre 9: Threshold energy vales for different copling parameters α From the above graph we have made a table for copling constant and minimm threshold energy for a bond orbit (that we have obtained nmerically by varying the parameter h) and the -nllcline. Vale of α Minimm vale of Vale of h(+ve) for which closed crve occrs in ve axis Table : Minimm vales of energy for bond orbits and the -nllclines Now from the eqation dv eff d = we get: ( α + ) e = and solving this eqation analytically as well as graphically we get the table below. The graphical soltion is given for a comparison.

12 .5 nllcline x -.5*exp(-/x)*(+/x) x -.5*exp(-/x)*(+/x) x -.5*exp(-/x)*(+/x) x -.5*exp(-/x)*(+/x) x -.*exp(-/x)*(+/x) dv(eff)/d Figre : Graphical soltion of -nllclines Vale of α Vale of Table :the -nllclines vs the copling constant This is exactly in agreement with the vale provided above in the Table. We also note that d V eff >, hence V d eff has minima at those points given in the above table for different α. Hence the paths shown in the above figres are stable. 7 Conclsion: Ths from the above stdy we have fond a set of vales of copling constant for a fixed energy vale above a certain threshold limit. So, it is evident that a proper tning of the copling constant and energy may change the motion of a particle from closed to aperiodic or vice-versa. A possible application for this phenomena is the removal of otermost electrons of a heavy atom. The otermost electrons experience a screened colomb or ykawa type of interaction de to the presence of other inner electrons. Ths tning the strength of external magnetic

13 field, one can tne the anglar momentm (J)of the otermost electron which in trn affects the copling constant (α = mα ). For a fixed energy, a proper choice of magnetic J a field can case easy removal of otermost electrons from their shells. Very heavy elements or atoms in very excited states, can be exposed to external electric field that will alter the energy of the otermost electron, keeping the anglar momentm conserved, so that its orbit becomes aperiodic. One can also think of electrons jst below the condction band in a metal ndergoing some periodic orbits, can be taken p to the condction band by modlating the copling constant as mentioned above. In case of strongly-copled complex plasma the caging effect on the dst particles can give rise to some of these closed orbits. This in trn may lead to some oscillating collective behaviors in complex plasmas. Another important application of ykawa potential is in astronomy to explain anomaly in the period of orbits of planets[8], mean motion of planets in solar system[3] and in long time rn of satellites[9]. For astrophysical measrements based on radar signals near the sn, get also affected by the ykawa correction term in gravitational potential[6]. For all the above cases the periodicity and closre isses discssed above are qite relevant. Acknowledgements: RM and SS are thankfl to Mritynjay Knd, Sayantani Bhattacharyya (presently at IIT Kanpr) and Abhijit Sen for their valable sggestions and discssions. The athors also thank an anonymos referee of Indian Jornal of Physics for several insightfll sggestion. References:. L D Landa and E M Lifshitz Corse of Theoretical Physics (Mechanics) (Pergamon Press : Oxford) Vol Ch 3, Sec 4, p 3 (969). I Rodrigez and J L Brn Er. J. Phys. 9 4 (998). 3. J L Brn and A F Pacheco Celestial Mech. Dyn. Astr 96 3 (6) 4. M Malek Nonlinear Systems of Ordinary Differential Eqations (California State University, East Bay) 5. S L Ross Differential Eqations (John Willey & Sons) Ch 3, Sec 3.3, p 66 (984) 6. C Sparrow The Qalitative Theory of Ordinary Differential Eqations (Lectre Notes : MA37) p 5 (9) 7. S H Strogatz Nonlinear Dynamics and Chaos (Levant Books) Ch 8, Sec 8., p 53 and Ch 5, Sec 5., p I Haranas, O Ragos and V Mioc Astrophys Space Sci 33 7 () 3

14 9. I Haranas and O Ragos Astrophys Space Sci 33 5 (). E Fischbach and C L Talmadge The Search for Non-Newtonian Gravity (Springer) p 3 (999). J W Moffat and V T Toth ArXiv:7.796v5 [gr-qc] (9). L Iorio Scholarly Research Exchange 8 Article ID I Haranas, I Kotsireas, G Gomez, M J Fllana, I Gkigkitzis Astrophys Space Sci 36:365 (6) 4. I Haranas, I Gkigkitzis Astrophys Space Sci 337:693-7 () 5. L Iorio and M L Rggiero Int. J. Mod. Phys. A (No. 9) (7) 6. I Haranas, O Ragos Astrophys Space Sci 334:7-74 () 7. O Bertolami Int. J. Mod. Phys. A 6 (No. A) 3- (7) 4

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