5 Solving Kepler eqution Conider the Kepler eqution ωt = ψ e in ψ We wih to find Fourier expnion of e in ψ o tht the olution cn be written in the form ψωt = ωt + A n innωt, n= where A n re the Fourier coefficient in the expnion e in ψ = A n innωt n= We multiply the bove eqution by inmωt nd integrte from to π/ω, noting tht { π/ω fo n, innωt inmωtdt = π/ω fo = n Thu, ll term in the ummtion vnih, except for the ce of m = n We then find tht the coefficient re given by A n = /ω π ω e in ψ innωtdt Mke the chnge of vrible to ν = ωt, nd we hve A n = π Integrting by prt, A n = [ n π π e inψν conν + n e inψν innνdν ] e coψνψ ν conνdν The firt term vnihe, becue inψ = inψπ = For the econd term, we oberve tht e coψνψ νdν = de inψν = dψν ν = dψν dν Therefore, the integrl become A n = [ π n conνdψν n ] conνdν The econd integrl vnihe, ince innπ = in = For the firt integrl, we mke the ubtitution ν = ψ e in ψ, giving A n = n π conψ ne in ψdψ
Oberve tht the integrl definition of the Beel function of the firt kind i Compring the two, we ee tht J n x = π conτ x inτdτ A n = n J nne Therefore, the olution to Kepler eqution cn be written ψωt = ωt + n= n J nne innωt The power erie expnion of J n x round x = for integer n i given by J n x = m= m x m+n m!m + n! Thu, we cn expnd ψ in power of e by 3 ψ = ωt + inωt e + inωt e + 8 in3ωt 8 inωt e 3 + 5 Yukw potentil: combintion of power-lw nd exponentil Conider the Yukw potentil V r = k r e r/ Strting from the Lgrngin, we hve L = T V = mṙ + r θ V r We hve tht θ i cyclic vrible, o p θ = l i contnt of motion Thu, we hve for the firt eqution of motion L θ = mr θ = l The econd eqution i given by d L dt ṙ L r = m r mr θ + V r = Uing the firt eqution, we cn rewrite the econd eqution m r = V r + l mr 3 = d V + l dr = d dr V,
where V i the effective one-dimenionl potentil Multiply both ide by ṙ nd integrting with repect to t give mṙ = V + E, where i the reulting integrtion contnt, nd the totl energy, ince we ee tht thi eqution i equivlent to E = mṙ + V We cn ue the effective one-dimenionl potentil V r = k r e r/ + l mr, to clify the nture of orbit We firt wnt to know when the behvior of one term dominte, which require olving the eqution k r e r/ = l mr, or equivlently, k er/ = m l r Thi i trncendentl eqution nd cnnot be olved nlyticlly However, there re three poible ce The two eqution either hve interection, interection, or interection In the ce of or interection, the Yukw potentil will dominte the behvior for ll r except poible t point In thi ce, we hve n effective potentil like tht een below 4 8 6 4 5 5 In thi ce, ll poible orbit re unbounded In the ce of interection, the Yukw potentil till dominte r nd r, but there exit rnge of r < r < r where the /r behvior dominte A poible grph of the effective potentil i hown below The two imge re from the me grph, but with different window, in order to how both the locl minimum nd the locl mximum 3
5-5 5 5 8 6 4 - -5 - - -4-6 3 4 5 In thi ce, there i locl mximum for which ll orbit re unbounded if E > V mx However, there i lo well tht llow for bounded orbit when E, with circulr orbit occurring t E = V min In ddition, if we hve < E < V mx, then one cn hve either bounded or unbounded orbit, depending on the initil vlue of r b In term of the invere rdiu u = /r, we hve the following eqution for orbit eq 334, d u dθ + u = m d l du V /u We hve V /u = kue /u, the differentil eqution i then, in the ce of the Yukw potentil, d u dθ + u = mk l e /u + u Conider the ce of ner-circulr orbit, where u = u + δθ, for contnt u nd δθ mll Firt, et fu = mk l e /u + u note tht by Tylor expnion, fu fu + u fu u=u δθ e /u + u + e /u u 3 δθ Therefore, to firt order, the differentil eqution i δ + u + δ = mk l e /u + u + δ u 3 To find u = /r, we know tht circulr orbit occur when d dr V r =, which men which give the condition d dr V r = l mr 3 + k r e r/ + k r e r/ =, e /u = l u 3 km u + u / 4
Regrouping term, we hve nd eqution of the form δ + ω δ =, which h inuoidl olution with frequency ω = mk l u 3 e /u Plugging the expreion for u ω = mk l u 3 e /u = u + u Auming u or r, we cn then find ω = u + u u Suppoe the pide occur t θ = Then the next pide will occur t θ = π ω π u π + π = π + π u r In purely circulr orbit, the pide hould occur every π rdin Ner circulr orbit, however, we hve tht the pide i dvncing by pproximtely πr / 53 Differentil cro-ection for repulive invere cube lw force Suppoe we hve repulive centrl force of the form f = kr 3 The correponding potentil i V r = kr We firt find the reltionhip between the cttering ngle Θ nd the impct prmeter uing Θ = π um du, V u E u where V u = ku nd u m = /, with the ditnce of cloet pproch Mking the chnge of vrible w = u + k E, thi integrl eily evlute to + k Θ = π in E + k E 5
To find, we firt note by conervtion of ngulomentum tht l = mv = mv m, where v m i the peed t Then, by conervtion of energy, E = mv = mv m + k rm = m v + k rm Solving for yield = + k mv = + k E Thu, ince in = π/, we hve Θ = π Subtituting in x = Θ/π nd olving for, we get Differentiting, = + k E k x E x x Hence, σθ dθ = d = k x x x x x E x x dx = k x 4x + x + x + x x E x x dx = k x E x x dx in Θ d dθ dθ = inπx d dx dx = k E x dx x x inπx 54 Truncted repulive Coulomb potentil Conider the truncted repulive Coulomb potentil { k/r, r >, V r = k/, r The cttering ngle i dependent on the impct prmeter ccording to Θ = π dr r r V r E 6
We will firt find the minimum eprtion By conervtion of ngulomentum, l = mv = mv m, where v m i the peed of the prticle t r = By conervtion of energy, then E = mv = mv m + V = m v + V, which give the reltion For >, thi become = rm + V E which h olution = rm + k E = k E =, = k k E + + E Note tht ince the dicriminnt i lwy lrger thn k/e, tking the negtive root in the qudrtic formul would led to < Next, for <, we hve Finlly, for =, = rm + k E = = k E = + k E = = k E Returning to the ce of >, we ee tht = rm k E = k E Since E > k/, then k/e < < o tht k/e > k/e Thi men, > k = E Hence, if we hve, then nd the prticle will ty within the region of the uul repulive Coulomb potentil Hence, if, we hve the reltion Thu, we hve cot Θ = E k E Θ = cot = tn E/ k k/ = cot E k k E 7
For the ce of or, we return to the originl formul, Θ = π = π r r dr V r E dr r r k re dr r r k E For the firt integrl, For the econd integrl, dr r r k re du = k E u u = co k E + u k E + 4 = co k E + + co k E k E + 4 k E + 4 dr r r k E = rm dx = x x = r x = cot t rm = π r m in < dr r r = π cot Combining two integrl, we get for <, Θ = cot co k E + + co k E k E + 4 k E + 4 Conider the pecil ce of E = k/ Then we hve = k E =, 8
So the expreion for i implified to cot Θ = nd for < <, we hve k E = = Now we cn orgnize everything to get the finl nwer cot,, Θ = in + co + co, < < A plot of thi ce cn be een below +8 +8 θ 7 6 5 4 3 3 4 5 55 Attrctive potentil The potentil in thi problem i: V r = {, r >, V, r It i obviou tht when >, there i no ctteringθ = Nontrivil reult rie when < Since thi potentil i roughly tep function, the peed of incident prticle will be tep function with jump t r = Set the peed to be v when r > nd v when r < Energy conervtion give mv = E r > mv V = E r < 9
We cn red the rtio between two velocity v E + v = V E E+V Here we define n n = E Since the potentil only chnge long rdil direction, only rdil force cting on the prticle when it pe through r = Therefore, velocity tngent to the circle t r = of the prticle remin unchnged when ping through r = Thi led to the reltion tht = n v in α = v in β in α = n in β α α β β α Θ O From the plot bove, one cn red in α = in β = = n, where the lt eqution come from ngulomentum conervtion mv = m v = n The pth of incident prticle i ymmetric repect to the center of the potentil, o we cn find from the plot tht Θ = β α Note tht Θ i negtive ince it i clockwie repect to Θ = Alo thi i the typicl feture of cttering from ttrctive potentil Plugging expreion for α nd β in, Θ = β α co Θ = coβ α co Θ = co β co α + in β in α co Θ = n + n
Rewriting the bove eqution, one cn find Θ = n in Θ + n n co Θ Then deferentil cro-ection σθ cn be obtined σθ = in Θ d dθ = n n co Θ n co Θ 4 co Θ + n n co Θ To get the totl cro-ection, we need to integrte over the ngle Θ Θmx σ T = π σθ inθdθ = π where Θ mx = co n Thi reult gree with the fct tht no cttering pper when >, o totl cro-ection i the re of the circle with rdiu, which men σ = π