PHYS60 Fa 08 Notes - 3 Centa foce motion The motion of two patices inteacting by a foce that has diection aong the ine joining the patices and stength that depends ony on the sepaation of the two patices is equivaent to a centa foce motion In the cente-of-mass efeence fame, the system is competey detemined by the eative position of the two patices Futhemoe, because the Lagangian is spheicay symmetic about the cente-of-mass, the angua momentum about the cente-of-mass is conseved Hence, the eative motion ies in a pane that has its noma in the diection of the angua momentum vecto The system can then be descibed by two geneaized co-odinates Let the two patices have masses, m and m, and position vectos and in the cente-of-mass fame The Lagangian is L m m U (3) Howeve, because the cente-of-mass is at the co-odinate oigin, and ae not independent, but ae eated by m m 0 (3) In tems of the patice sepaation,, (33) we have m, (34) m m and m (35) m m The Lagangian is then m m L m m U mm mm (36) mm U m m The quantity mm, (37) m m is caed the educed mass of the system We have educed the motion of two patices to an equivaent pobem of a singe patice of mass μ acted on by a foce diected aong the ine fom the patice to a fixed point
The geneaized momentum conjugate to the position vecto has components L pi, (38) i and hence p (39) The angua momentum Lp, (30) is othogona to both and p Hence, the motion is in the pane containing the oigin and has noma paae to L The motion has two degee of feedom and can be descibed by using poa coodinates (, ) In these co-odinates, the Lagangian is L U (3) Since this does not depend expicity on, ie is ignoabe, thee is a fist intega of motion L p, (3) whee is a constant In othe wods, the angua momentum about an axis though the oigin noma to the pane of motion is conseved Since the ate at which the position vecto sweeps out aea is da d, (33) dt dt consevation of angua momentum eads to Kepe s second aw of obita motion (a esut that is independent of the paticua dependence of the foce on ) d O Aso, because the Lagangian does not expicity depend on time, the Hamitonian is conseved Futhemoe, the two conditions fo the Hamitonian to be equa to the enegy ae satisfied Hence
3 U E U Re-aanging to give an expession fo, we find EU (34) (35) In genea, this is not an easy equation to sove fo (t) Often we ae inteested in the tajectoy taken by the patice The Lagange equation of motion is U (36) Using equation (3), we can eiminate the angua veocity to get U (37) 3 Aso fom equation (3), we have d d, (38) d d and d d d d d d dtd dtd dd (39) Fom the fom of the ast tem in equation (39), we see that it is now usefu to make the substitution u (30) We now have so that equation (37) becomes d u u, (3) d du U u d u This equation is easiy soved fo a gavitationa o Couomb foce (3) Panetay motion To investigate the motion of a panet aound a sta (both teated as point masses), U ku, (3) whee fo the gavitationa foce
4 k Gmm (3) Equation (3) fo this paticua case is du k u, (33) d which has soution k u Acos 0 (34) Hee A and 0 ae constants We see that the soution is fomay peiodic, but not necessaiy cosed (eg conside foce-fee motion fo which k = 0) We can take 0 to be zeo by measuing fom a position in which u takes a stationay vaue Equation (34) fo the obit can be witten as Gmm Acos, (35) o in an atenative fom as cos, (36) whee, (37) Gmm and is a constant It can be shown that equation (36) descibes a conic section with one focus at the oigin is caed the eccenticity of the obit, and is caed the atus ectum, which is a ength The enegy of the obit is d Gmm d du u Gmm u E d (38) Hence the enegy is negative if, and is minimum fo 0 Negative enegy means that the obit is bound, ie the patice neve eaches infinity In this case, the obits ae eipses The minimum enegy occus fo a cicua obit If, the obit is unbound and hypeboic If, the obit is a paaboa To show that equation (36) is that of an eipse fo, conside the foowing constuction
5 O Latus ectum O P min The minimum vaue of occus when = 0, and the maximum vaue occus when = π Daw a ine joining these two points (this wi be the majo axis of the eipse) Mak off a distance min fom each end of this ine (these ae the positions of the two foci of the eipse, O and O) Fom the tiange OPO If the sum of the engths OP and PO is independent of then the obit is an eipse The distance between O and O is max min Fom the cosine ue the distance between O and P is cos max min max min cos cos cos cos cos Hence the sum of the engths OP and PO is cos max min max min cos cos cos cos cos This is independent of and so the obit is an eipse This poves Kepe s fist Law Once we know that the obit is an eipse, we eate the paametes α and to the engths of the semi-majo and semi-mino axes Ceay
6 amin max Aso by consideing the ight tiange AOB in the figue beow (39) B b a O A O min we see that and so b a b a min, (30) max min max min max min, a (3) which contains the usua definition of eccenticity The enegy of the obit can be witten in tems of the engths of the axes by using equations (37), (38), and (39) We find that Gmm E (3) a a Gmm Note that the enegy depends on the ength of the semi-majo axis but not the semi-mino axis The angua momentum is eated to the axis engths by Gm m Gmm a Gmm b (33) a Using this ast expession, we can wite the semi-mino axis in tems of the constants of the motion b (34) E
7 Since the aea of the eipse is ab, Kepe s second aw enabes us to find the obita peiod Gmm 3 P ab A E E Gmm E (35) By squaing and using equation (3), we find the moe famiia fom of Kepe s thid aw 4 3 P a (36) Gm m Again, we note that this expession invoves ony the semi-majo axis The point on its obit at which a panet has its cosest appoach to the Sun is caed the peiheion The point at which the panet is futhest fom the Sun is the apheion Simiay, fo sateites obiting the Eath, the tems peigee and apogee ae used In tems of the semi-majo axis and the eccenticity, and Hence min max a, a (37) (38) a, (39) max min max min a, (30) and max min (3) max min Finay, note that the distance of a focus fom the cente of the eipse is max min AO a (3) 3 The effective potentia The tota enegy, (33) E U is the same as that of a patice of mass μ expeiencing one dimensiona motion with potentia enegy Ueff U (34)
8 Hence, Ueff() is an effective potentia enegy The foce coesponding to the effective potentia enegy is dueff du Feff 3 d d (35) 3 is commony (but incoecty) caed the centifuga foce This The tem foce and the coesponding centifuga potentia enegy aise fom consevation of the angua momentum of the motion Pat of the usefuness of the effective potentia concept is that it aows gaphica epesentation of the ocations of equiibium points and tuning points Since the kinetic enegy cannot be negative, we have EUeff 0, (36) with equaity ony when Ueff E Fo bound obits, the tuning points (whee 0 ) ae caed apsides Note that the motion is not necessaiy peiodic, and hence the apsides do not have to occu at the same angua position The effective potentia enegy is aso usefu fo studying the effects of petubations to the system EXAMPLE Suppose the inteaction potentia is U k Gaph the effective potentia enegy Identify the equiibium position, and epesentative tuning points Find the fequency of osciation fo a sma petubation fom equiibium (that does not change the angua momentum) Soution The effective potentia enegy is Ueff k Ceay, this invoves a numbe of paametes, whose vaues ae unspecified To simpify the anaysis, it is usefu to intoduce a ength scae detemined fom the vaue of at which the two contibutions to the potentia ae equa This occus at Making the substitution sr, R k 4 Ueff s U0 s s,
9 whee A pot of Ueff U0 against s ooks ike 0 9 U k 0 8 7 6 U eff /U 0 5 4 3 Ceay, the effective potentia enegy has a minimum at s =, which coesponds to a point of stabe equiibium Aso, we see that fo E > U0, the patice is tapped between two tuning points These tuning points ae found by soving E U0 s s fo s This equation can be e-aanged to 4 E s s 0, U0 which has soutions The equation of motion is which in tems of s is 0 0 3 4 5 s s E E U0 U0 k 0, 3
0 s s 0 3 k s The equiibium is at s = Expanding about the equiibium by setting s = +, and ineaizing the esuting equation we get k 4 0 Hence the fequency of osciation is k 4 Stabiity of cicua obits Fo a genea potentia, the equation of adia motion is du (33) 3 d Fo a given angua momentum, cicua obits exist ony if thee ae ea soutions of du d (33) 3 Suppose thee is a soution at = 0 We can test fo stabiity by consideing a sma petubation such that = 0 + The equation of motion is du d U 3 0 0 d d 0 3 d U 4 0 0 d (333) Dopping tems of second ode and highe, and assuming a hamonic time behavio, the fequency of sma osciations,, satisfies 3 d U 4 0 (334) 0 d Fo stabiity has to be ea The cicua obit is stabe if 3 du 4 0 0 (335) 0 d Using equation (33) to find 0, the stabiity condition becomes d 3 du 0, d d 0 and the fequency of sma osciations,, satisfies (336)
d du 3 3 0 d d 0 (337) If we appy these esuts to the potentia U k, we find that d 3 du 3 4 k0, (338) d d 0 and k 4 (339) This in ageement with the esut found above 5 Apsida pecession Two point masses inteacting by Newtonian gavity obit the system CM peiodicay and so the apsides occu at the same angua positions Howeve, thee ae a numbe of easons why panetay obits ae not exacty pana eipses The soa mass distibution has a sma deviation fom isotopy due to soa otation Each panet aso fees the gavitationa attaction of the othes Newtonian gavity is not an exact theoy and thee ae sma deviations fom the invese squae aw due to effects of genea eativity The easiest of these effects to take into account is the sma coection due to genea eativity This adds a tem popotiona to -4 to the foce The foce emains a centa foce and angua momentum consevation hods The equation fo the obit when the sma genea eativistic coection is incuded is du GmM 3GM u u d c (34) Hee m is the mass of the panet, which is assumed sma compaed to M, the mass of the Sun Because the genea eativistic coection is sma, this equation can be soved by a petubation method invoving expansion in a dimensioness sma paamete To find a convenient sma paamete, conside a cicua obit in the absence of the GR tem The soution to equation (34) is then Gm M u (34) In this case, the atio of the GR tem to the Newtonian tem is GmM 3 (343) c This atio povides a convenient dimensioness paamete to use in the expansion The equation fo the obit is
We ook fo a soution of fom du u u (344) d Substituting this into equation (344), we have u u u (345) 0 u u du 0 du 0 u 0 u d d (346) Dopping tems of second ode and highe, equation (346) becomes du du u d d Equating tems of the same ode in λ, we have 0 0 u 0 u du0 u 0, (347) d (348) and du u0 u (349) d We have aeady found the soution fo u0: u0 cos (340) Using this in equation (349), we have du d u cos cos (34) cos cos This equation is simia to that fo simpe hamonic motion with a peiodic focing tem Because the cos tem on the ight hand side has the same fequency as the soution of the homogeneous equation it acts to incease the ampitude of the hamonic motion The paticua intega is u, p sin cos (34) 6 To ode λ, the soution fo u() is u cosacos sin cos (343) 6 Once again, we have appied the condition that 0 is an apside The constant A is detemined by the vaue of u at 0 as befoe, so that We can take this to be
3 u 3 6 (344) The apsides occu when u is stationay, ie, when cos cos cos sin du sin sin sin sin cos 0 d 3 3 (345) When 0, the soutions ae n fo intege n Hence when 0, we ook fo soutions of fom n n Substituting this into equation (345), and keeping ony tems that ae fist ode in, we get n n (346) Hence the position of the apsides ae given by n (347) Since the apsides atenate between peiheion and apheion, the peiheion shifts by an ange GmM 6 GM 6 c a c 6 Anomaous Peiheion pecession of Mecuy The buk of the pecession of the peiheion of Mecuy is due to the pecession of the equinoxes that aises fom tida foces of Moon and Sun on non-spheica Eath Eath s otation axis changes diection with a peiod of 5,800 yeas The diffeence between the obseved amount and that due to the pecession of the equinoxes is caed the anomaous pecession The contibutions to the anomaous pecession ae given in the tabe Amount (acsec/centuy) Cause 533035 Gavitationa tugs of the othe panets 0086 Obateness of the Sun 49779 Genea eativity 5753 Tota pedicted 5740 ± 065 Obseved Einstein, Abet (96) "The Foundation of the Genea Theoy of Reativity" Annaen de Physik 49: 769 8
4 EXAMPLE PROBLEMS A patice moves in a cicua obit in a foce fied given by F () k/ Show that, if k suddeny deceases to haf its oigina vaue, the patice s obit becomes paaboic Soution: Let the adius of the cicua obit be 0 The initia enegy is E T U k k k Assuming that the kinetic enegy does not change when k is educed to k/, the fina enegy is E T U k k 0 Hence, the obit becomes paaboic Two gavitating masses m and m (M = m + m) ae sepaated by a distance 0 and eeased fom est Show that when the sepaation is, the speeds ae v m, v m Soution: Since the patices ae eeased fom est, the angua momentum is zeo and the masses move aong a staight ine In the CM fame the enegy is E μ Gm m Gm m Hee µ is the educed mass Hence μ Gm m Gm m Gm m The speeds of the patices ae eated to the magnitude of the eative veocity by Hence v m M, v m M v m M Gm m μ m G M, v m M Gm m μ m G M
5 3 A communications sateite is in a cicua obit aound Eath at adius R and veocity v A ocket accidentay fies quite suddeny, giving the ocket an outwad adia veocity v in addition to its oigina veocity (a) Cacuate the atio of the new enegy and angua momentum to the od (b) Descibe the subsequent motion of the sateite and pot T(), V(), U(), and E() afte the ocket fies Soution: Let the masses of the sateite and Eath be m and M Since the Eath is much moe massive than the sateite, we ignoe the sma diffeence between m and the educed mass (a) Since the impuse is diected adiay, the angua momentum = mrv is unchanged The initia enegy is E GMm R The impuse inceases the kinetic enegy by an amount equa to the initia kinetic enegy Hence the enegy afte the impuse is E E T GMm R GMm R 0 Hence the atios ae E/E0 = 0, and /0 = (b) The sateite foows a paaboic obit afte the impuse The enegy of the sateite afte the impuse is E m R m v GMm 0 Hence the kinetic, gavitationa potentia and effective potentia enegies ae T GMm GMm R R, V GMm GMm R R, and U R m v GMm Since the initia obit is cicua v GM R, and so U R GMm R GMm GMm R R R These enegies ae potted beow against /R
6 0 08 06 T V U E Enegy in units of GMm/R 04 0-00 -0-04 0 3 4 5 6 7 8 9 0 /R -06-08 -0 4 Conside a comet moving in a paaboic obit in the pane of the Eath s obit If the distance of cosest appoach of the comet to the Sun is βe, whee E is the adius of Eath s (assumed) cicua obit, show that the time the comet spends within the obit of the Eath is given by β β 3π yea If the comet appoaches the Sun to the distance of the peiheion of Mecuy, how many days is it within Eath s obit Soution: Since the comet s obit is a paaboa, it has eccenticity equa to The equation of the obit is β cosθ Hence, the comet is inside the Eath s obit if θ θ cos β The aea swept out by the position vecto of the comet whie it is inside the Eath s obit is A dθ 4β cosθ dθ 4β cosθ dθ β secθ dθ On making the substitution xtanθ, this is Aβ whee x tanθ x dx β x x 3,
7 Since we find Hence cos θ cos θ β, tanθ β β A 3 β β β β β We can find the time deivative of A by consideing the enegy and angua momentum of the obit, E m 0 θ GM Hence, at peiheion 0 β GM θ, β and so the angua momentum pe unit mass if the obit is θ β GM β GMβ The ate at which the aea is swept out is / Hence the time inside Eath s obit is τ 4 β β β β β 4 ββ 3 GMβ 3 GM The yea is the obita peiod of the Eath A simia anaysis shows that yea π GM Hence τ yea 4 β β β β 3 π 3π If the comet s cosest appoach to the Sun is at same distance as the peiheion of Mecuy, then β = 0387, and so τ = 0084 yeas =76 days 5 A patice of unit mass moves fom infinity aong a staight ine that, if continued, woud aow it to pass a distance b fom a point P If the patice is attacted towads P with a foce vaying as k, and if the angua momentum about P is k b, show that the tajectoy is given by
8 bcothθ Soution: Since the foce is attactive, the potentia enegy is U The equation fo the tajectoy is d u dθ u k u b u, whee is the angua momentum of the patice about P Mutipying by du dθ, we get du d u dθ dθ d dθ du dθ du dθ b u u, which on integating gives du dθ u u b 4 C, whee C is a constant of integation We can use consevation of enegy and consevation of angua momentum to find C Let the speed at infinity be v0 The angua momentum is Hence b v k b v k b Consevation of enegy gives v k 4b k 4 Hence, the distance of cosest appoach to P, p, is given by k 4b k b k 4 Soving fo p, we get b At cosest appoach du dθ 0 Hence C b b 4b 4b The equation fo the obit is then du dθ b b u b u b b u Hence du dθ b u b Make the substitution bu tanh φ to get dφ b cosh φ dθ tanh φ, b
9 which gives so that dφ dθ, bu tanh θφ, whee ϕ0 is a constant of integation If we take θ = 0 to coespond to when the patice is at infinity, then ϕ0 = 0, and so bcoth θ The tajectoy is shown beow The obit asymptoticay becomes cicua with adius b y 0 05 00-0 x 3-05 -0 n 6 A patice moves unde the infuence of a centa foce given by F k If the patice s obit is cicua and passes though the foce cente, show that n = 5 Soution: Let the obit have adius a The situation is as shown beow:
0 O C a Hee C is the cente of the cice and O is the foce cente Since the tiange is a ight tiange, the equation fo the obit is acos The enegy of the obit is k E m, n n and the angua momentum is m Using the equation fo the obit, k E m a a m acos n acos 4 sin 4 cos 4 n k 8amcos n a cos 4 n n The enegy is a constant of the motion, and hence must be independent of θ This is possibe ony if n = 5 and E = 0 The atte equiement eads to km 8a 7 A patice moves in an eiptica obit in an invese-squae aw centa foce fied If the atio of the maximum angua veocity to the minimum angua veocity of the patice in its obit is n, then show that the eccenticity of the obit is n n
Soution: Let the angua veocity be The angua momentum of the obit is conseved Hence, the maxima and minima of the angua veocity occu at peiapsis and apoapsis, espectivey, and ae eated by max min min max Fom this, we get max n min The eccenticity is max max min min n max min max n 8 Discuss the motion of a patice moving in an attactive centa-foce fied descibed by k 3 F Sketch some of the obits fo diffeent vaues of the tota enegy Can cicua obits be stabe in such a foce fied? Soution: Assume that the patice has unit mass The enegy of the obit is k E Using consevation of angua momentum, to eiminate and epace the time deivative by a deivative with espect to, we get min d k du E k u, d d whee u We see that if k 0, then the obit is unbound By diffeentiating the enegy expession with espect to, we get which has soutions du k u d 0, u k u u A k u k 0cos 0, if, 0 0, if, 0cosh 0, if In these expessions, u0, A and 0 ae constants, and
k The tota enegy in each case is ku0, if k, E ka, if k, ku0, if k The pot beow shows thee obits fo cases with k The obit with the owest enegy is shown by the back ine and the obit with the highest enegy is shown by the bue ine The pot beow shows an obit fo a case with k The obit is a spia
3 The pots beow shows thee obits fo cases with k These obits ae bound Cicua obits ae ony possibe if k In this case the effective potentia is identicay zeo Hence, thee is no estoing foce to stabiize the obit We concude that the cicua obits ae unstabe A adia petubation wi give a spia tajectoy as in the second of the thee pots above 9 An Eath sateite has a peigee of 300 km and an apogee of 3,500 km above Eath s suface How fa above the Eath is the sateite when (a) it has otated 90 aound Eath fom peigee and (b) when it has moved hafway fom peigee to apogee? Soution: The adius of the Eath is 637 km Hence the peigee and apogee distances ae min = 667 and max = 987 km espectivey The semi-majo axis is the aveage of these two vaues Hence a = 87 km The eccenticity of the obit is max min 0934 max min Peigee (a) (b) Apogee (a) Using the equation fo the obit a cos, we find that when 90, a 796 km Hence the sateite is 590 km above the Eath suface (b) When the sateite is hafway fom peigee to apogee, its distance fom the cente of the Eath is a Hence the sateite is 900 km above the Eath suface