Single Particle Closed Orbit in Yukawa Potential
SSingle Particle Closed Orbits in Yukawa Potential
Rupak Mukherjee † , Sobhan Sounda ‡
1. Institute for Plasma Research, HBNI, Gandhinagar, Gujarat, India.2. Ramakrishna Mission Vivekananda University, Belur Math, Howrah, Kolkata, India.3. Ramakrishna Mission Residential College (Autonomous), Narendrapur, Kolkata, In-dia. † [email protected] ‡ [email protected] May 9, 2017
Abstract:
We study the orbit of a single particle moving under the Yukawa potential and observethe precessing ellipse type orbits. The amount of precession can be tuned through the cou-pling parameter α . With a suitable choice of the coupling parameter; we can get a closedbound orbit. In some cases we have observed some petals which can also have a closedbound nature with an appropriate choice of the coupling constant. A threshold energy hasalso been calculated for the boundness of the orbits. Keywords:
Yukawa Potential, Precessing Ellipse, Closed-Bound Orbit, Critical Point
PACS No: a r X i v : . [ phy s i c s . p l a s m - ph ] M a y Introduction:
It is widely known that there are only two types of central potential r and r in which allfinite motions take place in closed paths[1]. Some exceptions of the statement above havebeen reported recently[2 , V ( r ) = − αr e − rλ , ( λ is the screening parameter that determines the range of this interaction)has a long standing legacy to represent various physical systems. Its application rangesfrom astronomy, high-energy physics, nuclear physics, condensed matter physics to plasmaphysics and many other branches of physics. In the domain of plasma physics this potentialis used mostly in Strongly Coupled Plasmas which has several applications in modellingthe cores of white dwarfs, planetary rings, atmospheric lightning, molten salts and plasmatechnology. Hence quite naturally one may ask what is the orbit of a single particle underthis screened coulomb potential. Can we expect some bound orbits for the non-zero valueof α ?This study is a more general case of the previous studies [2 ,
3] that has been done till nowregarding the closed bound orbits distinct from coulomb or harmonic oscillator potentialsbecause by the truncation of the exponential series in the Yukawa potential one gets backthe potentials assumed in the literature so far. Hence this study is a superset of the previousstudies performed so far to the best of our knowledge.The paper is organised as follows. Section 2 discusses the theory of central force motion forany arbitrary potential. Following the theory we have calculated the trajectories for Yukawapotential in section 3 and the thrust has been given on the calculation for threshold energyfor bound orbit in Yukawa potential in section 4. In section 5, some possible applicationsof this study has been mentioned.
The total energy of a particle moving under a central force is given by,12 m ˙ r + J mr + V ( r ) = E, J = mr ˙ θ where m is the mass of the particle, r is the radial distance of the particle from the forcecenter, J is the angular momentum of the particle, V is the potential of the particle and θ is the angular coordinate of the particle with respect to some reference axis. For r = u i.e.˙ r = − Jm dudθ ; the above equation looks like12 m J m (cid:18) dudθ (cid:19) + J u m + V ( u ) = E For energy to be constant, i.e. dEdθ = 0, d udθ = − mJ ddu (cid:18) V + J u m (cid:19) = − mJ dV eff du dJdθ = 0 since, for central force the angular period ∆ θ of the radial oscillators r ( θ )is independent of the angular momentum J . In dimensionless coordinates, u = ua and V = mJ a V the equation of motion becomes, d udθ = − dV eff du with V eff = V ( u ) + u . Now if the form of the potential is known, we can find thetrajectory of the particle. The threshold energy value for a bound orbit in any centralforce can be calculated from the total energy ( h ) conservation relation,12 my + V eff ( u ) = h, where, V eff ( u ) = − u (cid:90) F ( u ) du and y = dudθ ⇒ y = ± (cid:114) m [ h − V eff ( u )] (1)Now, if we differentiate the above equation we get, y dydu = 1 m dV eff du (2)Here, for dV eff du = 0, if dydu is not undefined, i.e. the curve plotted from equation (1), doesnot cut the u axis perpendicularly, then y = 0 which in turn represents h = V eff i.e. theminimum threshold value of energy for bound orbit.Now if dydu is undefined, i.e. the curve plotted from equation 1, cuts the u axis perpendic-ularly, then from the roots ( u c ,
0) or the critical points of the equation dV eff du = 0 we findsome equilibrium points of the system. Now from the relations dV ( u c ) du = 0 and dV eff du = 0we can conclude that V ( u ) has either a relative extremum or a horizontal inflection pointat u = u c .If,I. the V eff ( u ) has a relative minimum at u = u c , then V eff ( u c ) = h (threshold or minimumvalue for energy) and the critical point is a centre and is stable.II. the V eff ( u ) has a relative maximum at u = u c , then the critical point is a saddle pointand is unstable.III. the V eff ( u ) has a horizontal inflection point at u = u c , then the critical point is of adegenerate type called a cusp and is unstable. The Yukawa or screened coulomb potential is given by: V ( r ) = − αr e − rλ (3) ⇒ V eff ( u ) = 12 u − mαJ a ue − u (4)3here, λ is the range of the Yukawa potential.The equation of motion is given by d udθ = − u + αe − u (cid:18) u (cid:19) , where, α = mαJ a (5)which can be rewritten in terms of energy as:12 my + 12 u − αue − u = h ⇒ y = ± (cid:113) h − u + 2 αue − u (6)where we have chosen m = 1. The exact values of α differ for different physical casesdepending on the nature of application[11 , , , d udθ + u = 1 + αe − λu (cid:18) λu (cid:19) (7)In ref[10] an approximate analytical solution has been obtained by expanding the R.H.S.of (7) in a Taylor series and truncating it to the second order. Thus the equation lookslike, d udθ + u (cid:20) − α (cid:18) a p λ (cid:19) e − a p /λ (cid:21) = u p u p = 1 p (cid:20) αe − a p /λ (cid:18) a p λ − a p λ (cid:19)(cid:21) where a p ∼ p = 1 /u and u is the unperturbed ( α = 0) solution. The solution of theabove equation given by, u ( θ ) = u p + u e cos ω ( θ − θ ) ω = (cid:2) − α ( a p /λ ) e − a p /λ (cid:3) / is only valid for α << α = 0) value. The aim of our present study is to explore the fullsolution space of equation (5) by retaining the complete nonlinear form of the force term.We obtain these solutions by a numeical solution of the equation.We adopt another analytical tool viz. a linear stability analysis for a particle underyukawa type potential. We proceed to some extent and then turn to the numerical analysisfor different α (coupling constant). 4 Linear Stability Analysis [7] for Single Particle Mo-tion in Yukawa Potential
For the sake of analysis, we add a small viscous term proportional to velocity ( µ dudθ ), in theequation of motion and eventually put µ = 0 for the final calculation.Thus for Yukawa Potential the modified equation of motion (5) can be written as, u (cid:48)(cid:48) ( θ ) = f ( u (cid:48) ( θ ) , u ( θ )) = − µu (cid:48) − u + α (cid:18) u (cid:19) e − u ⇒ u (cid:48) ( θ ) = F ( u ( θ ) , y ( θ )) = yy (cid:48) ( θ ) = G ( u ( θ ) , y ( θ )) = − µy − u + α (cid:18) u (cid:19) e − u Hence, the fixed points f ( y ∗ , u ∗ ) = 0 are:( y ∗ , u ∗ ) = (0 , − . α = 0 . y ∗ , u ∗ ) = (0 , − . α = 0 . y ∗ , u ∗ ) = (0 , − . α = 0 . y ∗ , u ∗ ) = (0 , − . α = 0 . y ∗ , u ∗ ) = (0 , − . α = 1.The Jacobian for the above set of equation will be given by[4], J ( u ∗ , y ∗ ) = (cid:18) ∂F∂u ∂F∂y∂G∂u ∂G∂y (cid:19) u = u ∗ ,y = y ∗ = (cid:18) − αu e − u − µ (cid:19) u = u ∗ ,y = y ∗ The Eigenvalues of the above matrix will be, λ , = 12 (cid:16) τ ± √ τ − (cid:17) = 12 (cid:18) − µ ± (cid:114) µ − (cid:16) − αu e − u (cid:17) u = u ∗ (cid:19) D e l t a a l pha Value of Delta with alpha " D elta" u 1:2 It is checked that the values of ∆ are posi-tive for all values of α given above. Hence,for the above parameter values the fixedpoint can not be a saddle point.Now, for ∆ > τ < τ − > f ( y ∗ , u ∗ ) isa stable node.if τ < τ − < f ( y ∗ , u ∗ ) isa stable spiral.if τ > τ − > f ( y ∗ , u ∗ ) isan unstable node.if τ > τ − > f ( y ∗ , u ∗ ) isan unstable spiral.if τ = 0 and τ − > f ( y ∗ , u ∗ ) is a nutrally stable center.5he existance of the limit cycle around each of the fixed points (for different values of α )has been checked and it is found that for none of the cases there exists any limit cycle.Hence if we continuously change µ from positive to negative the fixed point changes fromstable to unstable spiral. However at µ = 0 we do not have a true hopf bifurcation becausethere are no limit cycles on either side of the bifurcation. This situation is identical to thecase of a damped pendulum or a duffing oscillator.Further we analyse the case of µ = 0 numerically for the quantitative understanding of theparameter values with closed orbits.. For numerical analysis, we have used a simple Runge Kutta 4th Order solver and haveused standard accepted algorithm for avoiding the numerical singularities, if encountered.Interestingly we have shown that for some typical values of α we get a closed bound orbit.In some cases the orbits have some petals and the number of petals can also be controlledthrough the judicious choice of the coupling constant. Here we present some results of oursimulation that helps to understand the dependency on the coupling constant easily.For α = 0.05; we observe a constant precesion of the ellipse. But the ellipse does notclose itself when it completes one revolution. We have tuned the value of the couplingconstant to 0.0501 and have observed the expected closure. -1.5-1-0.5 0 0.5 1 1.5-1.5 -1 -0.5 0 0.5 1 1.5 u * s i n ( t he t a ) u*cos(theta) alpha = 0.05 " alpha_0p05 u 1:2 -1.5-1-0.5 0 0.5 1 1.5-1.5 -1 -0.5 0 0.5 1 1.5 u * s i n ( t he t a ) u*cos(theta) alpha = 0.0501 " alpha_0p0501 " u 1:2 Figure 1: Precessing Ellipse with α = 0 .
05 and Closed orbits with α = 0 . α = 0.1; we observe almost same behavior but the amount of precesion has increased.Still we had to tune the coupling constant to 0.11 to get a closed bound orbit.6 u * s i n ( t he t a ) u*cos(theta) alpha = 0.1 " alpha_0p1 " u 1:2 -1.5-1-0.5 0 0.5 1 1.5-1.5 -1 -0.5 0 0.5 1 1.5 u * s i n ( t he t a ) u*cos(theta) alpha = 0.11 " alpha_0p11 " u 1:2 Figure 2: Precessing Ellipse with α = 0 . α = 0 . α = 0.5; we observe a small grouping of the precessed orbits. The precessionoccurs in a bunch of 2 orbits. Hence we may observe that there are two different values ofprecession within one complete revolution. When we set the coupling constant to 0.49 wesuddenly observe a closed bound orbit maintaining the nature of grouping. -1.5-1-0.5 0 0.5 1 1.5-1.5 -1 -0.5 0 0.5 1 1.5 u * s i n ( t he t a ) u*cos(theta) alpha = 0.5 " alpha_0p5 " u 1:2 -1.5-1-0.5 0 0.5 1 1.5-1.5 -1 -0.5 0 0.5 1 1.5 u * s i n ( t he t a ) u*cos(theta) alpha = 0.49 " alpha_0p49 " u 1:2 Figure 3: Precessing Ellipse with α = 0 . α = 0 . α = 1; we observe two distinct types of orbits, which has been further investigatedin the next step. Here a 3-fold larger orbit as well as a 3-fold smaller orbit are found.The tuning of the coupling constant has been done accordingly to obtain the closed boundorbit. 7 u * s i n ( t he t a ) u*cos(theta) alpha = 1.0 " alpha_1p0 " u 1:2 -2-1.5-1-0.5 0 0.5 1 1.5-1.5 -1 -0.5 0 0.5 1 1.5 2 u * s i n ( t he t a ) u*cos(theta) alpha = 1.011 " alpha_1p011 " u 1:2 Figure 4: Precessing Ellipse with α = 1 . α = 1 . α = 5; we observe two distinct classes of petals, one within the other. Maintainingthe two distinct petal structures we have been able to find the closed bound orbits for someparameter value ( α ). -10-8-6-4-2 0 2 4 6 8 10-10 -8 -6 -4 -2 0 2 4 6 8 10 u * s i n ( t he t a ) u*cos(theta) alpha = 5.0 " alpha_5p0 " u 1:2 -10-8-6-4-2 0 2 4 6 8 10-10 -8 -6 -4 -2 0 2 4 6 8 10 u * s i n ( t he t a ) u*cos(theta) alpha = 5.1002 " alpha_5p1002 " u 1:2 Figure 5: Precessing Ellipse with α = 5 . α = 5 . α = 10 also; we observe two distinct classes of petals but the ratio of the bigger tosmaller orbit has changed. Here also we have observed closed orbits.8 u * s i n ( t he t a ) u*cos(theta) alpha = alpha_10p0 " u 1:2 -20-15-10-5 0 5 10 15 20-20 -15 -10 -5 0 5 10 15 20 u * s i n ( t he t a ) u*cos(theta) alpha = 9.91000045528 " alpha_9p91 " u 1:2 Figure 6: Precessing Ellipse with α = 10 . α = 9 . α ) values in the abovefigures. Now we calculate the threshold energy for the bound orbits. For a constant value ofcoupling constant, α = 0 .
05, we plot equation 6 for different values of energy. ( h =2 , , . , . , . , . , . u axis perpendicu-larly representing dV eff du to be undefined. This states that we have not reached the thresholdenergy value. For energy values less than 0 . u axisrepresenting that we have gone further below the threshold energy for coupling constant α = 0 .
05 9 y u alpha = 0.05 "h=2" u 1:2"h=1" u 1:2"h=0.5" u 1:2"h=0.25" u 1:2"h=0.1" u 1:2"h=0.05" u 1:2"h=0.005" u 1:2 Figure 7: Energy diagram for constant coupling parameter α = 0 .
05 and varying energy h Again for a constant h = 1 . α =0 . , . , . , . , -2-1.5-1-0.5 0 0.5 1 1.5 2-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 y uh = 1.5 " alpha =0.0005." u 1:2" alpha =0.005." u 1:2" alpha =0.05." u 1:2" alpha =0.5." u 1:2" alpha =1.0." u 1:2 Figure 8: Energy diagram with constant energy h = 1 . α Now we calculate the minimum values of energy and the u-nullclines graphically. Fora constant value of α we start with a bound orbit in the negative u axis and keep ondecreasing energy untill the circle reduces to a point. Then if the value of energy is furtherdecreased; suddenly we observe that the point in the negative u axis disappears. Thiscorresponds to the minimum value of energy for a closed orbit in the u − θ diagram. Inthe figure below we have plotted the minimum value of energy ( h ) upto which the point in10he negative u axis was found. The position of the point gives the value of the u -nullclineand we have measured that value for different α . -2.5-2-1.5-1-0.5 0 0.5 1 1.5 2 2.5-1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 y uPlot for Minimum Energy " alpha =1.0" u 1:2" alpha =0.5" u 1:2" alpha =0.05" u 1:2" alpha =0.005" u 1:2" alpha =0.0005" u 1:2 Figure 9: Threshold energy values for different coupling parameters α From the above graph we have made a table for coupling constant and minimum thresh-old energy for a bound orbit (that we have obtained numerically by varying the parameter h ) and the u-nullcline. Value of α Minimum value of h (+ ve ) for which closedcurve occurs in − ve axis Value of u dV eff du = 0 we get: u − α (cid:18) u (cid:19) e − u = 0and solving this equation analytically as well as graphically we get the table below. Thegraphical solution is given for a comparison.11 d V ( e ff ) / du uu nullcline x -0.0005*exp(-1/x)*(1+1/x)x -0.005*exp(-1/x)*(1+1/x)x -0.05*exp(-1/x)*(1+1/x)x -0.5*exp(-1/x)*(1+1/x)x -1.0*exp(-1/x)*(1+1/x) Figure 10: Graphical solution of u-nullclines
Value of α Value of u d V eff du >
0, hence V eff has minima at those points given in the abovetable for different α .Hence the paths shown in the above figures are stable. Thus from the above study we have found a set of values of coupling constant for a fixedenergy value above a certain threshold limit. So, it is evident that a proper tuning of thecoupling constant and energy may change the motion of a particle from closed to aperiodicor vice-versa.A possible application for this phenomena is the removal of outermost electrons of a heavyatom. The outermost electrons experience a screened coulomb or yukawa type of interactiondue to the presence of other inner electrons. Thus tuning the strength of external magnetic12eld, one can tune the angular momentum ( J )of the outermost electron which in turnaffects the coupling constant ( α = mαJ a ). For a fixed energy, a proper choice of magneticfield can cause easy removal of outermost electrons from their shells. Very heavy elementsor atoms in very excited states, can be exposed to external electric field that will alter theenergy of the outermost electron, keeping the angular momentum conserved, so that itsorbit becomes aperiodic. One can also think of electrons just below the conduction bandin a metal undergoing some periodic orbits, can be taken up to the conduction band bymodulating the coupling constant as mentioned above.In case of strongly-coupled complex plasma the caging effect on the dust particles can giverise to some of these closed orbits. This in turn may lead to some oscillating collectivebehaviors in complex plasmas.Another important application of yukawa potential is in astronomy to explain anomaly inthe period of orbits of planets[8], mean motion of planets in solar system[13] and in longtime run of satellites[9]. For astrophysical measurements based on radar signals near thesun, get also affected by the yukawa correction term in gravitational potential[16]. For allthe above cases the periodicity and closure issues discussed above are quite relevant. Acknowledgements:
RM and SS are thankful to Mrityunjay Kundu , Sayantani Bhattacharyya (presentlyat IIT Kanpur) and Abhijit Sen for their valuable suggestions and discussions. The au-thors also thank an anonymous referee of Indian Journal of Physics for several insightfullsuggestion. References:
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