m 1 r = r 1 - r 2 m 2 r 2 m1 r1

Transcription

1 Topic 4: Two-Body Cental Foce Motion Reading Assignment: Hand & Finch Chap. 4 This will be the last topic coveed on the midtem exam. I will pass out homewok this week but not next week.. Eliminating the cente of mass and the equivalent -body poblem m = - m We will be consideing the motion of two paticles acting unde the influence of a cental foce. We can elimanate the motionof the cente of mass, and educe this to an equivalent -body poblem. Conside the cental foce: F = F ; _p = _p ; whee F _. The cente of mass coodinate is: R = m + m ; M = m + m M d R dt = _p + _p dr =0and = const (no extenal foces) M dt L = R P = const whee (P = p + p ) (no extenal toques) ) The motion of the cente of mass is unifom and ignoable. m m

2 So, we will now measue and elative tothe cente of mass: = T = m _ + m _ and _ R =0in this efeence fame. Consevation of linea momentum becomes m = m Since and ae elated, we have a single degee of feedom. Let's educe down to a single position coodinate, and wite an expession fo the kinetic enegy: = = + m m m = m +m and so the kinetic enegy is = T = m m = m m + m m + m m m + m = m m _ = m + m μ + m m m + m whee we define the educed mass, μ as μ = m m m +m. We have educed the poblem to finding the motion of a single paticle of mass μ in a cental field descibed by U (), so to an equivalent dof poblem. _. Some Chaacteistics of Cental Foce Motion We can deive some inteesting chaacteistics of cental foce motion (fo any U() - not just invese squae) by consideing consevation of angula momentum: l = p _l = _ p + _p = 0 + F = 0 since fo a cental foce F k. So l = const as we have peviously deived. Let = ^e whee ^e is a unit vecto in the -diection. Then, l = μ^e _ =μ^e (_^e +

3 φ e^ ^ e dt so and we have a elationship between ^e ; d^e dt ; l The ate of sweeping out aea: dffi = j d^e dt jdt j^e j j d^e dt j = dffi dt = _ ffi jlj = m _ ffi = const +d dφ dφ da = (dffi)+ (dffi) d da lim dt!o dt = _ ffi = const So the aeal velocity (ate of sweeping out aea) is constant. This is Keple's second law, which he deived empiically by obseving the motion of the planets. Note though, that this is valid fo any cental foce, not just..3 The Lagangian Appoach to the Geneal Cental Foce Poblem Conside a consevative foce: ; V = V () L = μ _ + _ ffi V() 3

4 The Eule-Lagange equations ae fist the familia consevation of l: and We also have enegy l = μ _ ffi = const μ l μ = 0, and T is homogeneous and quadatic in _ H = E = T + V = const l E = μ _ + + V () =const μ A useful way to look at this is as a "-D" equation fo motion of a paticle of mass μ in an effective potential, since the equation of motion involves only and its deivatives; V eff = E = T + V eff l μ + V () l Hee is a "centifugal" tem aising fom the kinetic enegy - ie. a fictitious foce due μ to the motion in a non-inetial fame. We can deduce quite a bit about the chaacte of the motion by examining the fom of the effective potential. Take fo example the invese squae foce, V = k ; V eff = l μ k Sketch the effective potential as a function of : E > 0 Unbound motion E > 0 Bound obit with tuning points min ; max, the apsidal distances E 3 is constant, coesponding to cicula motion. _ =0: E=V eff = const f + = 0 and = l! balance of centifugal foce μ3 This type of analysis is a common appoach that can be applied to othe-shaped potentials. Lets look at a couple of examples: Now we will descibe the obit quantitatively. We could (as one usually does) solve the equations of motion to get (t);ffi(t). This constitutes a complete solution. Instead, we can make a simple tansfomation, u =. We can eliminate time, and get (ffi) - the equation fo the obit. This is moe elegant. _ = _ ffi d dffi = l d μ dffi 4

5 let u = ; du = d fom the equation of motion fo : _ = l du μ dffi = d _ffi d = dffi d l du dt dffi dt dffi μ dffi 00 = l μ u u 00 d dffi l μ l μ μ u u 00 l μ u3 5

6 which we can simplify to the following diff. eq: u 00 + u = We havedeived a diffeential equation fo the obit if the foce law is given. Fo V _ o we get a linea equation. 3 Note that this equation has only d in it ie. is invaiant toffi! ffi, and hence implies dffi that the obit is symmetic about the tuning points (choose the tuning point at ffi = 0), and theefoe the solution is independent of eflection about the apsidal vectos. We will use this diffeential equation late. Now we will look at the invese squae law, but use a diffeent appoach to get (ffi)..4 The Invese Squae Foce, V=-k/ We will use a cleve tick to get the equation fo the obit. Conside d dt let F = f () ^e whee f() = k= (l p) =(dl p)+(l _p)=l F dt so l = p = μ(^e d^e dt ) d dt (l p) =l F=μ f ()(^e d^e dt ) ^e = μ f () d^e since ^e? d^e =dt dt Now define avecto A (called the Runge-Lenz vecto) A p l μk^e (Runge-Lenz vecto) 6

7 this is a constant vecto fo a foce, since d l p μ f () ^e =0if f()= k= dt This vecto lies in the plane of the obit, since A l = 0 so it is a fixed vecto in the obital plane. Take alookatthelength of A: A = μ k μk (p l) +(p l) (p l) μk (p l) = μk l (p ) = μk l = μl V () (p l) (p l) = l [p (l p)] = l p l (p l) p Λ = l p =μl p μ =μl T 7

8 so, and A = p l + μ k μk l = μl (T + V )+μ k E = A μl μk l and the constancy of A implies (o is elated to) enegy consevation. Now A has thee components, but only one can be independent, because that it is both fixed in the obital plane, and the enegy identity detemines its magnitude. We can get the equation fo the obit if we conside A = (p l) μk ^e = l μk = A cos ffi (ffi is the angle between and A). This is the pola equation fo a conic section. This is a geneal poof of eple's fist law with the eccenticity " = A=μk [Keple's fist law states that the planets move in elliptical obits with the Sun at one focus]..4. Bound Obits (E<0) Elliptical motion applicable to planets, satellite, etc. Fo cental foce motion we deived whee k = Gm m The equation fo an ellipse is l μk =+ A cos ffi μk a( e ) = ff =+"cos ffi a = semi majo axis, b = semi mino axis, and b = a( " ) =. Hee min and max ae measued fom the focus to the obit. min;max ae elated to the axes of the ellipse by min = a( ") = ff +" max = a (+")= ff " So we can elate the paametes of the ellipse to the dynamical constants: " = A μk ff = l μk 8

9 We can also find some useful expessions fo l; E: l μ = a " Gm m μ whee M = m + m. Also, l μ = GMa " Λ = (specific angula momentum) so, Finally, E = A μl μk l = μk " μk l = μk ( " ) l = μk ( " ) μka ( " ) = k a = Gm m a E μ = GM a (specific enegy) l μ = _ da ffi = dt = ßab P hee da is the aea swept out (dffi), and P is the obital peiod. Since b = a ( " ) =, da dt = na " = whee we define n = ß P. So, da dt =(GMa)= " = 9

10 and and we have shown that GMa = n a 4 ; GM = n a 3 P _ a 3=! Keple's thid law The squae of the peiod is popotional to the cube of the majo axis of the elliptical obit. Note this is fo constant m + m ; so Keple was coect in the limit that m planet + m sun ß m sun. Also note that this is independent of the eccenticity of the obit. Fo the Eath obiting the Sun, P =y; a =:5 0 8 km =) M = 0 30 kg. N.B. The solution fo invese squae foce is a closed obit. This is not tue in geneal. The only othe case is V _, the SHO (x and y motions have the same peiod)..4. Time Dependence We have not specified (t), but only deived the equation fo the obit and peiod. To do this we will intoduce a simple geometic constuct, the eccentic anomally, Ψ(the tue anomally is geneally denoted by Φ]. Cicumscibe the ellipse of the obit with a cicle of adius a and poject onto this cicle: a cos Ψ = cos ffi + a" since we know fom the equation fo an ellipse that = a ( " ) +"cos ffi ; a cos Ψ = a ( " ) cos ffi + a" +"cos ffi a cos ffi + a" = +"cos ffi cos Ψ = cos ffi + " +"cos ffi 0

11 and the invese tansfomation: also we can deive the elations cos ffi = cos ffi " " cos Ψ sin Ψ = ( " ) = sin ffi +"cos ffi combine these: so sin Ψ sin ffi = sin ffi = ( " ) = sin Ψ " cos Ψ ( " ) sin ffi sin Ψ ( + " cos ffi)( "cos Ψ) ( + " cos ffi)( "cos Ψ) = " = a ( " ) = a ( " cos Ψ) +"cos ffi Take the time deivative ofsinffi cos ffi ffi _ =» " = cosψ( "cos Ψ) " sin Ψ ( " cos Ψ) _Ψ _ffi = " = cos Ψ " Ψ cos _ ffi ( " cos Ψ) = " = _Ψ " cos Ψ and ffi _ = " = _Ψa ( " cos Ψ) The aea swept out by the adius vecto is so combining with the above, and integating both sides, da dt = _ ffi = na " = " = _Ψa ( " cos Ψ) = na " = Z n = _Ψ( "cos Ψ) Z ( " cos Ψ) dψ = ndt n (t t o )=(Ψ "sin Ψ) This is a complete solution, since this gives us paametic solutions fo (t);ffi(t) since we know (Ψ) ;ffi(ψ) :

12 .5 Unbound Obits Scatteing E > 0 and attactive foce: k>0 F= k ^e ; V = k The equation fo the obit becomes: l =+ A cos ffi μk μk emembe A μk is the eccenticity. We have E > 0=) A μl > μk l A>μk=)"> eccenticity geate than unity descibes the equation fo a hypebola:!; cos ffi m = μk A μ k A = μl E A =) sin ffi m = p μe l A which gives an equation fo ffi m. μ k cot ffi m = E l.6 Repulsive Potential ff paticle scatteing Conside the case of a heavy nucleus, whee m ff is the mass of the alpha-paticle (a helium nucleus = potons). The nucleus is heavy (ie gold), so that m ff << m nucleus, and μ! m ff. In this case, k = Ze, whee Z is the atomic numbe of the heave nucleus. The foce is of couse Coulomb epulsion: F = Ze ^e

13 V = Ze A lies along the symmety axis of the scatte. We have l =+ A cos ffi μk μk The diffeence between this and the attactive case is k<0. Define the scatteing angle, to be the angle between the incoming and outgoing diections. Define the impact paamete, b = ß ffi m ß tan = tan ffi m = cot ffi m = mff E jkj l The angula momentum of the incoming paticle with espect to the fixed scatteing cente is l = m ff v b and of couse so E = m ffv l = p Em ff b putting this into the expession fo, we get mff jkj tan = E b p = Em ff so we have a one-to-one coespondence between the impact paamete, b, and the scatteing angle. 3 k Eb

14 In the case of ff-paticle scatteing, b cannot be detemined expeimentally. In the expeiment pefomed by Ruthefod (and in geneal with these types of expeiments), a beam of paticles was incident on a thin foil, and the faction of paticles scatteed though vaious angles, d obseved. This leads us to define an expeimentally-motivated quantity called the coss section. Definition of coss section: N = # of incident paticles/unit aea stiking a thin foil containing n scatteing centes/unit aea. So we deduce that dn N = ndff dff has the units of aea, dn =# of paticles scatteed though and angle between and + d. take the deivative fom the pictue, dff =ßb db tan = k Eb = cos db = Ze = Ze mv b mv b d = Ze mv b db mb v cos d Ze b = Ze mv cos sin 4

15 so using dω =ßsin d dff = ßmv cos ( )(Ze ) (Ze ) 3 (mv ) cos3 ( ) 3 sin 3 ( ) d sin = ß (Ze ) (mv ) cos sin d 4 = 4 ß (Ze ) sin (mv ) sin d 4 dff Ze dω = csc 4 mv This is the Ruthefod scatteing law Ruthefod found ageement with expeiment (and deived q q = Ze ) so long as the peiapse distance is 0 cm. Note that quantum mechanically, the concept of a coss section is still valid, but a definite tajectoy is not quantum mechanical Ruthefod scatteing tuns out to have the same answe fotuitously. If we ty to compute the total coss section Z ff tot = ß sin d dff dω dff ο dω 4 and ß sin d =dω ο so it diveges at small due to the infinite ange of the Coulomb foce. Really sceening fom the electons makes this poblem go away. Astonomical Illustations Petubations to the Keple obit. Geneal Relativitistic Coections to the Planetay Obits One of the classic tests of geneal elativity is the petubation it makes to the motion of the planets, which is measuable. GR makes a small change to the adial foce. To calculate the petubation to the obit, etun to the geneal obit equation we deived ealie: fom befoe we found that _ = l μ u0 so u 00 + u + μ l dv du =0 u0 + u + μ lv =const l T = μ _ + μ T = l μ u0 + l μ u 5

16 so and u0 + u + μ l V = μ l E» μ u 0 = (E V ) u l Integate between peihelia (note now the obit is not necessaily closed, so peihelion can happen at diffeent ffi) I du ffi = Φ μ (E V ) u Ψ = l if V = Gm m then ffi =ß=)wehave a closed obit fo a pue / potential. let V! V o + ffiv, ffi! ß + ffiffi so note, we I ffi + ffiffi = I ffiffi I du f j ψ j V o ffiv + highe ode l fμ(e V ) l u g = lu! they diffe by so: μ ffiffi I ffiv μ u fμ(e V o ) l u g = Now we do a tick convet this to an integal ove ffi aound the unpetubed obit: (note, the denominato of the integand is efeenced to the unpetubed obit)! du du dffi = Φ μ (E Vo ) u Ψ = l ffiffi I» I ffiv μ ffiv u l ( du) = μ l dffi Let the putubation be of the fom ffiv () =fl= 3 : Using the equation fo an elliptical obit = μk ( + " cos ffi) l ffiffi Z ß 0 = μk l 3 ( + " cos ffi) dffi l 3 = 6ßμ kfl Geneal elativity gives a petubation of this fom to Newtonian gavity, with fl = l GMsun μ c 6 l 4

17 so ffiffi = 6ß μ l μl 4 GMsun c = 6ß μ (GM sun ) GMμ l c whee [M ß M sun ] fom befoe, we deived the specific angula momentum; Using this above we get l μ = GMa " Λ = ffiffi = 6ßGM sun a ( ") c pe obit Note, the putubation is the biggest fo the lagest " and the smallest a. Fo mecuy, the contibution to the pecession of its peihelion due to GR is /centuy. Thee is a /centuy pecession due to othe Newtonian petubations which can be calculated. The GR tem agees with measuement to 0.4%, which is consideed an impotant confimation of the theoy. It was suggested by Dicke and othes that othe effects may contibute, making ageement with GR coincidental. One of these effects is the possibility of a non-spheical sun. This has, howeve, been shown not to be an impotant effect. 7