Copyght 004 4 Potental Theoy We have seen how the soluton of any classcal echancs poble s fst one of detenng the equatons of oton. These then ust be solved n ode to fnd the oton of the patcles that copse the echancal syste. In the pevous chapte, we developed the foalss of Lagange and Halton, whch enable the equatons of oton to be wtten down as ethe a set of n second ode dffeental equatons o n fst ode dffeental equatons dependng on whethe one chooses the foals of Lagange o Halton. Howeve, n the ethods developed, the Haltonan equed knowledge of the Lagangan, and the coect foulaton of the Lagangan equed knowledge of the potental though whch the syste of patcles oves. Thus, the developent of the equatons of oton has been educed to the detenaton of the potental; the est s anpulaton. In ths way the oe coplcated vecto equatons of oton can be obtaned fo the fa sple concept of the scala feld of the potental. To coplete ths developent we shall see how the potental esultng fo the souces of the foces that dve the syste can be detened. In keepng wth the celestal echancs thee we shall estct ouselves to the foces of gavtaton although uch of the foals had ts ogns n the theoy of electoagnets - specfcally electostatcs. The ost notable dffeence between gavtaton and electoagnets (othe than the obvous dffeence n the stength of the foce) s that the souces of the gavtatonal foce all have the sae sgn, but all asses behave as f they wee attactve. 5
4. The Scala Potental Feld and the Gavtatonal Feld In the last chapte we saw that any foces wth zeo cul could be deved fo a potental so that f F = 0, (4..) then F = V, (4..) whee V s the potental enegy. Foces that satsfy ths condton wee sad to be consevatve so that the total enegy of the syste was constant. Such s the case wth the gavtatonal foce. Let us defne the gavtatonal potental enegy as Ω so that the gavtatonal foce wll be F = Ω. (4..) Now by analogy wth the electoagnetc foce, let us defne the gavtatonal feld G as the gavtatonal foce pe unt ass so that G = F / = ( Ω / ) Φ. (4..4) Hee Φ s known as the gavtatonal potental, and fo the fo of equaton (4..4) we can daw a dect copason to electostatcs. G s analogous to the electc feld whle Φ s analogous to the electc potental. Now Newtonan gavty says that the gavtatonal foce between any two objects s popotonal to the poduct of the asses and nvesely popotonal to the squae of the dstance sepaatng the and acts along the lne jonng the. Thus the collectve su of the foces actng on a patcle of ass wll be GM ( ) Gρ( ) Fg () = = d, (4..5) whee we have ncluded an expesson on the ght to ndcate the total foce asng fo a contnuous ass dstbuton ρ(). Thus the gavtatonal feld esultng fo such a confguaton s GM ( ) Gρ( ) G g () = = d. (4..6) 5
The potental that wll gve se to ths foce feld s Φ() = GM = Gρ(') d '. (4..7) The evaluaton of the scala ntegal of equaton (4..7) wll povde us wth the potental (and hence the potental enegy of a unt ass) eady fo nseton n the Lagangan. In geneal, howeve, such ntegals ae dffcult to do so we wll consde a dffeent epesentaton of the potental n the hope of fndng anothe eans fo ts detenaton. 4. Posson's and Laplace's Equatons The basc appoach n ths secton wll be to tun the ntegal expesson fo the potental nto a dffeental expesson n the hope that the lage body of knowledge developed fo dffeental equatons wll enable us to fnd an expesson fo the potental. To do ths we wll have to ake a clea dstncton between the coodnate ponts that descbe the locaton at whch the potental s beng easued (the feld pont) and the coodnates that descbe the locaton of the souces of the feld (souce ponts). It s the latte coodnates that ae sued o ntegated ove n ode to obtan the total contbuton to the potental fo all ts souces. In equatons (4..5-4..7) denotes the feld ponts whle ' labels the souces of the potental. ( ) Consde the Laplacan {.e. the dvegence of the gadent [ = ]} opeatng on the ntegal defnton of the potental fo contnuous souces [the ght-ost te n equaton (4..7)]. Snce the Laplacan s opeatng on the potental, we eally ean that t s opeatng on the feld coodnates. But the feld and souce coodnates ae ndependent so that we ay ove the Laplacan opeato though the ntegal sgn n the potental's defnton. Thus, Φ() = G (') d ' ρ. (4..) Snce the Laplacan s the dvegence of the gadent we ay ake use of the Dvegence theoe Hdv = H da, (4..) V 5 S
to wte Gρ(')[ ( ' )]d = A Gρ(')[ ( ' )] da. (4..) Hee the suface A s that suface that encloses the volue. Now consde the sple functon (l/) and ts gadent so that [ ˆ da ] da = = dω = ω A A. (4..4) A The ntegand of the second ntegal s just the defnton of the dffeental sold angle so that the ntegal s just the sold angle ω subtended by the suface A as seen fo the ogn of. If the souce and feld ponts ae dffeent physcal ponts n space, then we ay constuct a volue that encloses all the souce ponts but does not nclude the feld pont. Snce the feld pont s outsde of that volue, then the sold angle of the enclosng volue as seen fo the feld pont s zeo. Howeve, should one of the souce ponts coespond to the feld pont, the feld pont wll be copletely enclosed by the suoundng volue and the sold angle of the suface as seen fo the feld pont wll be 4π steadans. Theefoe the ntegal on the ght hand sde of equaton (4..) wll ethe be fnte o zeo dependng on whethe o not the feld pont s also a souce pont. Integands that have ths popety can be wtten n tes of a functon known as the Dac delta functon whch s defned as follows: δ() 0 0 δ()d. (4..5) If we use ths notaton to descbe the Laplacan of (l/) we would wte and ou expesson fo the potental would becoe (/ ) = 4πδ(), (4..6) Φ() = Gρ(')( ' ) d = 4πG δ(' ) ρ V ' v (')d. (4..7) 54
Ths ntegal has exactly two possble esults. If the feld pont s a souce pont we get Φ() = 4πGρ(), (4..8) whch s known as Posson's equaton. If the feld pont s not a souce pont, then the ntegal s zeo and we get Φ() = 0. (4..9) Ths s known as Laplace's equaton and the soluton of ethe yelds the potental equed fo the Lagangan and the equatons of oton. Ente books have been wtten on the soluton of these equatons and a good deal of te s spent n the theoy of electostatcs developng such solutons (eg. Jackson 4 ). All of that expetse ay be boowed dectly fo the soluton of the potental poble fo echancs. In celestal echancs we ae usually nteested n the oton of soe object such as a planet, asteod, o spacecaft that does not contbute sgnfcantly to the potental feld n whch t oves. Such a patcle s usually called a test patcle. Thus, t s Laplace's equaton that s of the ost nteest. Laplace's equaton s a second ode patal dffeental equaton. The soluton of patal dffeental equatons eques "functons of ntegaton" athe than constants of ntegaton expected fo total dffeental equatons. These functons ae known as bounday condtons and the functonal natue geatly coplcates the soluton of patal dffeental equatons. The usual appoach to the poble s to fnd soe coodnate syste wheen the functonal bounday condtons ae theselves constants. Unde these condtons the patal dffeental equatons n the coodnate vaables can be wtten as the poduct of total dffeental equatons, whch ay be solved sepaately. Such coodnate systes ae sad to be coodnate systes n whch Laplace's equaton s sepaable. It can be shown that thee ae thteen othonoal coodnate faes (see Mose and Feshback l ) n whch ths can happen. Unless the bounday condtons of the poble ae such that they confo to one of these coodnate systes, so that the functonal condtons ae ndeed constant on the coodnate axes, one ust usually esot to nuecal ethods fo the soluton of Laplace's equaton. Laplace's equaton s sply the hoogeneous fo of Posson's equaton. Thus, any soluton of Posson's equaton ust begn wth the soluton of Laplace's equaton. Havng found the hoogeneous soluton, one poceeds to seach fo a patcula soluton. The su of the two then povdes the coplete soluton fo the nhoogeneous Posson's equaton. 55
In ths book we wll be lagely concened wth the oton of objects n the sola syste whee the donant souce of the gavtatonal potental s the sun (o soe planet f one s dscussng satelltes). It s geneally a good fst appoxaton to assue that the potental of the sun and planets s that of a pont ass. Ths geatly facltates the soluton of Laplace's equaton and the detenaton of the potental. Howeve, f one s nteested n the oton of satelltes about soe non-sphecal object then the stuaton s athe oe coplcated. Fo the pecson equed n the calculaton of the obts of spacecaft, one cannot usually I assue that the dvng potental s that of a pont ass and theefoe sphecally syetc. Thus, we wll spend a lttle te nvestgatng an altenatve ethod fo detenng the potental fo slghtly dstoted objects. 4. Multpole Expanson of the Potental Let us etun to the ntegal epesentaton of the gavtatonal potental ρ(')d Φ( ) = G. (4..) ' Assue that the oton of the test patcle s such that t neve coes "too nea" the souces of the potental so that ' <<. Then we ay expand the denonato of the ntegand of equaton (4..) n a Taylo sees about ' so that ' + (/ ) ' + ' j' ' j' k ' 6 j j j k j k (/ ) (/ ). (4..) o n vecto notaton [ ' ] (/ ) [' (/ )] [' ' : (/ )] = + [' ' ' (/ )] 6. (4..) In Chapte we defned the scala poduct to epesent coplete suaton ove all avalable ndces so that the esultng scala poduct of tensos wth anks and n was n. Howeve, n ode to ake clea that ultple suatons ae needed n equaton (4..), I have used ultple "dots". The defnton of ths notaton can be seen fo the explct suaton n equaton (4..) o can be defned by A B: A' B' = (A B')(B A'). (4..4) 56
Usng ths expanson to eplace the denonato of the ntegal defnton of the potental [equaton (4..)] we get Φ() = G ρ(')d = G ' V ' ρ(')dv' + ' ρ(')d ' ' ρ(')d : + +. (4..5) Ths expanson allows the sepaaton of the dependence of the feld coodnates fo the souce coodnate. Thus the ntegals ae popetes of the souce of the potental only and ay be calculated sepaately fo any othe aspect of the echancs poble. Once known, they gve the potental explctly as a functon of the feld coodnates alone and ths s what we need fo specfyng the Lagangan. We can ake ths cleae by e-wtng equaton (4..5) as { M(/ ) P (/ ) + (/ ) (/ + + } S Q Φ ( ) = Q : ). (4..6) 6 Ths expanson of the potental s known as a "ultpole" expanson fo the paaetes M, P, Q, and S whch ae known as the ultpole oents of the souce dstbuton. Fo the gavtatonal potental the unpole oent s a scala and just equal to the total ass of the souces of the potental. The vecto quantty P s called the dpole oent and Q s the tenso quadupole oent, etc. The hghe ode oents ae n tun hghe ode tensos. The epeated opeaton of the del-opeato on the quantty (l/) also poduces hghe ode tensos, whch ae sply geoety and have nothng to do wth the ass dstbuton tself. The fst two of these ae (/ ) = ˆ / (/ ) = ( ˆˆ) /. (4..7) As one consdes hghe ode tes the geoetcal tensos epesented by the ultple gadent opeatos contan a lage and lage nvese dependence on and theefoe play a successvely dnshed ole n detenng the potental. Thus we have effectvely sepaated the postonal dependence of the feld pont fo the ass dstbuton that poduces the vaous ultpole oents. 57
By way of exaple, let us consde two unequal ass ponts sepaated by a dstance l, located on the z-axs, and wth the coodnate ogn at the cente of ass (see Fgue 4.). Fo the defnton of the ultpole oents, we have M P Q S v' ρ(')d = ' ρ(')d = ' ' ρ(')d = ' ' ' ρ(')d = + ( + ) ( + ) ( + ) l kˆ = 0 ( z + z ) kˆkˆkˆ l kˆkˆ, (4..8) Fgue 4. shows the aangeent of two unequal asses fo the calculaton of the ultpole potental esultng fo the. 58
whch, when cobned wth the coodnate epesentaton of equaton (4..6), yelds a sees expanson fo the potental of the fo Φ() = G ( + ) l + ( + ) ( cos θ) + +. (4..9) Unless the feld pont coes patculaly close to the souces, ths sees wll convege quckly. We can also ake use of a pleasant popety of the gavtatonal foce, naely that thee ae no negatve "chages" n the foce law of gavtaton. Thus we ay always choose a coodnate syste such that P(') = ' ρ(')d = 0. (4..0) Ths eans that fo celestal echancs thee wll neve be a dpole oent of the potental as long as we choose the coodnate fae popely. Ths s usually done by takng advantage of any syety pesented by the object and locatng the ogn at the cente of ass. Not only does the dpole oent vansh, but fo objects exhbtng plane syety all odd oents of the ultpole expanson vansh fo the gavtatonal potental. Ths cetanly enhances the convegence of the sees expanson fo the potental and eans that the fst te that ust be ncluded afte the pont-ass potental te s the quadapole te. The ncluson of ths te eans that the eo n the potental wll be of the ode O(/ 5 ). Even though the potental epesented by a ultpole expanson conveges apdly wth nceasng dstance, the contbuton of such tes can be sgnfcant fo sall values of. Thus thee s geat nteest n detenng the agntude of these tes fo the potental feld of the eath so that the obts of satelltes ay be pedcted wth geate cetanty. We have now descbed ethods wheeby the potental can be calculated fo an abtay collecton of ass ponts to an abtay degee of accuacy. The nseton of the potental nto the Lagangan wll enable one to detene the equatons of oton and the soluton of these equatons then consttutes the soluton of any classcal echancs poble. Theefoe, let us now tun to the soluton of specfc pobles found n celestal echancs. 59
Chapte 4: Execses. The potental enegy of the nteacton between a ultpole T and a scala potental feld Φ s gven by U = T () ul () Φ, whee T () s a tenso of ank () and L () Φ descbes applcatons of the del opeato to the scala potental Φ. The sybol u stands fo the ost geneal applcaton of the scala poduct, naely the contacton (.e., the addton) of the two esultng tensos ove all ndces. a: Consde fou equal asses wth Catesan thee densonal coodnates ass # X Y Z # - 0 0 # + 0 0 # 0 #4 0 - -. Fnd the total self enegy of the syste. b: Fnd the potental enegy of the above syste wth a ffth dentcal ass located at (0,0,0).. Gven that the nteacton enegy of a dpole and quadupole ay be wtten U pq = P Q as o. Uqp = Q: Φ p show that U pq = U qp.. Use a ultpole expanson to fnd the potental feld of thee equal ass ponts located at the vetces of an equlateal tangle wth sde d. Restct you soluton to the plane of the tangle and keep only the fst two tes of the expanson. 4. Fnd the nteacton enegy of a 0 kg sphee wth the Eath-Moon syste when the thee ae located so as to fo an equlateal tangle. Assue the Eath and Moon ae sphecal. Copae the elatve potance of the fst two tes of the ultpole expanson fo the Eath-Moon potental. 60