j i i,i j The second term on the right vanishes by Newton s third law. If we define N and Figure 1: Definition of relative coordinates r j.

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1 Cental Foces We owe the development of Newtonian mechanics to the poblem of celestial mechanics and Newton s claim to Edmund Halley that the motion of a body to a foce that vaies invesely with the squae of the distance is in an ellipse. Although we leave aside just how Keple detemined that Tycho Bahe s data ageed with elliptical obits with the Sun at one focus, we will show that the gavitational two-body poblem yields obits that ae conic sections. Monday, 0 Octobe 0, evised 7 Decembe 0 Befoe tuning to cental foces, let us conside the motion of a set of N paticles, subject both to foces between paticles and to extenally applied foces. The equation of motion fo the ith paticle is then d(m i vi ) (no sum ove i) fiext + fi j dt j i whee fi j is the foce on the ith paticle due to the jth paticle. By Newton s thid law, fi j f ji. If we now sum ove all N paticles, then N d N ( m i vi ) fiext + fi j dt i i i,i j () The second tem on the ight vanishes by Newton s thid law. If we define N P m i vi i then Eq. N Fext fiext and () () i takes dp Fext (3) dt which states that the sum of the extenal foces causes the total momentum to change. Now conside the kinetic enegy of the paticles, given by N T m j x j x j j the y m x fom R x m whee x j v j is measued with espect to an inetial x fame. Expess the position of the jth paticle as Figue : Definition of elative coodinates xj R + j j.

2 . THE TWO-BODY PROBLEM whee R is a vecto fom the oigin to some (as yet abitay) point, and j the vecto fom this point to paticle j, as illustated in Fig.. Then, the kinetic enegy of the paticles is T m j x j x j m j (R + j ) (R + j ) j j Expanding the dot poduct gives T d M R + m j j + R (m j R + m j j + m j R j ) m j j j dt j j whee M j m j. Now we make a judicious choice fo the vecto R. Let R M m j x j M m j (R + j ) R + m j j j j j (4) so that this final sum must vanish. Equation (4) defines the cente of mass of the paticles; and if we choose R to be the position of the cente of mass, then the final tem in the kinetic enegy expession vanishes. Theefoe, T M R + m j j j (5) which is to say that the kinetic enegy may be expessed as that of a single mass point having the entie mass of the system, located at the cente of mass, and moving with the cente of mass, plus the kinetic enegy of paticles moving with espect to the cente of mass. We shall see shotly that the summation in Eq. (5) takes a simple fom in the case of a (otating) igid body, but the expession given hee is pefectly geneal.. The Two-Body Poblem We conside two bodies that ae attacted to (o epelled fom) one anothe by a foce aligned with the line that sepaates them and which depends only on the distance between them. Such a foce is called cental and we may use consevation laws to educe the twobody poblem to an equivalent one-body poblem. In the cente-of-mass fame, with the oigin at the cente of mass, m + m 0 Ô m m (6) Solving fo the motion of one body automatically yields the motion of the othe. Because the foce on m i is along the position vecto i, it exets no toque about the oigin, so the angula momentum of each mass about the oigin is conseved. This vecto is pependicula the plane defined by i and i and so the motion is confined to that plane, which we may take as the x y plane with no loss of geneality. In pola coodinates, the Lagangian is L µ ( + θ ) U() (7)

3 . INTEGRATING THE EQUATION OF MOTION whee,, and µ m m m m m + m M (8) is the educed mass and M m + m is the total mass. When one mass is much geate than the othe, the educed mass is just slightly less than the small mass. Similaly, the angula momentum of the two bodies about thei cente of mass is µ θ (9) which is one fist-integal of the motion. Since the system is subject to no extenal foces, enegy is conseved and equal to E µ ( + ) + U() µ (0) which is the second fist-integal. Execise Stating fom Eq. (4), deive Eqs Execise Show that Eq. (9) implies Keple s second law, which holds that the adius vecto to a planet sweeps out equal aeas in equal times. Johannes Keple (57 630) discoveed this law using obsevations of the obit of Mas made by Tycho Bahe. Note that it elies on no othe popety of F() than it be cental.. Integating the Equation of Motion The consevation of enegy equation, Eq. (0), may be ewitten E µ + Ueff () L m () whee the effective potential is Ueff U() + µ Veff () - GM m which makes tanspaent that we have a fist-ode diffeential equation to solve. As illustated in Fig., the effective potential contains a centifugal baie tem (unless 0) that tends to ovewhelm the attactive pat of U() as 0. Figue : Effective potential U eff fo a gavitational attaction between two spheical bodies. In a bound obit the system oscillates in the asymmetic potential well shown in the olive-coloed cuve. 3

4 3. INVERSE-SQUARE FORCES Isolating the tem and integating, we have E Ueff Ô µ/ d [E Ueff ()] dt µ (3) so that we may obtain t() by integating t t0 d µ 0 E U( ) L /µ (4) Thus, the poblem of finding t() has been educed to quadatue. In pinciple, we can then invet to find (t), and use the angula momentum equation to obtain a fist-ode diffeential equation fo θ(t). Altenatively, we can etun to Eq. (3) and use the chain ule to yield a fist-ode equation fo θ(), d θ 3. µ [E Ueff ()] L/µ Ô L d (5) µ(e Ueff ) Invese-Squae Foces Pehaps the most impotant example of cental foce motion is planetay motion unde the gavitational attaction with the Sun. Of couse, the sola system consists of many moe bodies than just one sta and one planet, and all massive bodies attact one anothe. Howeve, to a vey lage degee the Sun is the big kid on the block and we can make a lot of pogess by neglecting the inteactions of the planets with one ancicle othe and with the othe bits of flotsam (comets, asteoids, and such) left ove fom the cloud of Figue 3: Radial potential fo a planet. gas fom which ou sola system emeged about 4.6 billion yeas ago. In eality, the lage planets cause noticeable petubations to the obits we will obtain by teating each planet with the Sun as an isolated two-body system with an attactive potential of the fom p a E E U() k Gm m G µm (6) Fo a planet in a bound obit, E < 0 as we will show. Fo a given angula momentum L, the minimum enegy obit is cicula, which takes place at a single value of adius, as 4

5 3. INVERSE-SQUARE FORCES suggested in Fig. 3. Fo the same angula momentum but geate enegy, the planet oscillates between a minimum adius p at peihelion and a maximum adius a at aphelion. To solve fo the shape of the obit, thee is a sneaky substitution that simplifies the equations significantly fo invese-squae-law foces (but is not so useful in geneal). It is to eplace with the invese adius u /. Retuning to Eq. (5) witten fo the gavitational attaction between m and m, we have k d µ(e µ + ) (7) Then d d(/u) du du u Squaing Eq. (7) and substituting Eq. (8) gives ( µ du ) (u ) (E u + ku) µ (8) (9) which is stating to look a lot like the SHO equation. Diffeentiating with espect to θ poduces µ GM µ u u uu ku Ô u + u This is the simple hamonic oscillato equation with an inhomogeneous tem. The solution to the homogeneous pat is just a cosine (unless we want to shift the oigin of angle), so the full solution is GM µ u(θ) ( + є cos θ) b a q a aphelion e peihelion Figue 4: Elliptical obit. 5

6 3. INVERSE-SQUARE FORCES whee with malice of foethought I have chosen to wite the sinusoidal amplitude as є GM µ /. Solving fo, we get A + є cos θ whee A GM µ (0) This is the equation of a conic section; є is called the eccenticity of the obit. When є 0, the adius is constant and the obit is a cicle. Fo 0 < є <, the obit is an ellipse, with the cente of mass at one focus. As illustated in Fig. 4, with the choice we made fo the oigin of θ, peihelion is at θ 0, at which point the planet is at p a( є) fom the Sun, whee a is the semimajo axis of the ellipse. Half a yea late, the planet is at aphelion, whee a a( + є). We can use these expessions to ewite the equation of the obit in tems of p o a: p ( + є) a( є ) () + є cos θ + є cos θ Retuning to Eq. (9) and using Eq. () to expess u /, we find that a GM µ E () which shows that the semimajo axis of the obit depends only on the enegy, not on the angula momentum. The angula momentum, howeve, entes into the detemination of the eccenticity, which we can develop using a( є ) A and Eq. (0). Solving fo є yields / E є [ + ( )] µ GM µ GM µ a( є ) (3) (4) Thus we can expess the paametes of the obital ellipse in tems of the physical paametes E and. Table shows the obital paametes fo the planets of ou sola system. Note that most obits ae quite close to cicula. Mecuy has the most eccentic obit (since Pluto was demoted fom planethood; the eccenticity of Pluto s obit is 0.48), but it is still athe close to a cicle, as shown in Fig. 5. Execise 3 Cente of Mass inside the Sun? Figue 5: Mecuy s obit (ed) compaed to a cicle with the same semimajo axis (blue). The ellipse has been offset by aє to give it the same cente as the cicle. The eal location of the Sun in each case is shown by the cental dot of coesponding colo. Thus, the pimay effect of eccenticity is to offset the cente of the obit. Locate the cente of mass of the Sun-Jupite system. Is it 6

7 4. PERIOD body semimajo axis a (08 km) a (A.U.) є peiod (y) mass Sun Mecuy Venus Eath Mas Jupite Satun Uanus Neptune Table : Obital paametes. The mass is given in multiples of the sola mass, kg. 4. Peiod The aea of an ellipse is πab, which we can use, along with Keple s second law, to figue out the peiod of an elliptical obit. The aea swept out pe unit time is da θ ( ) dt dt µ To compute the aea we need the semimino axis, b, which we may get eithe by finding the value of θ that maximizes y o by using the fact that an ellipse is the locus of points such that the sum of the distances to the two foci is a constant. In eithe case, we find A b a є є so the peiod is µ πaµ τ a є π πab GM µ a( є ) µ π 3/ a3/ a GM µ GM which is Keple s thid law. It is also often witten τ a Keple s Laws Pehaps it is wothwhile summaizing the thee laws Keple found using Bahe s data: I. Planets follow elliptical obits with the Sun at (eally nea) one focus. II. The adial vecto sweeps out equal aeas in equal times (a consequence of angula momentum consevation). I had a had time emembeing whethe it was τ and a 3 o the othe way aound until I figued out a helpful aide-me moie. Newton s equations ae second ode in time. Theefoe, time appeas quadatically in Keple III. 7

8 5. UNBOUNDED ORBITS III. The squae of the peiod is popotional to the cube of the semimajo axis. Of couse, this last one equies that we neglect the mass of the planet compaed to the Sun, which is a fine appoximation fo ou sola system. 5. Unbounded Obits Two othe kinds of obits ae possible. When є, the obit is paabolic. Using the fist expession fom Eq. () and є, we find p + cos θ (5) which diveges as θ π. When є >, the obit is hypebolic. These obits ae illustated in Fig. 6. Paabolic obits have E 0, so they ae asymptotically unbounded. Hypebolic obits have E > 0. Comets typically have highly eccentic obits, eithe vey thin ellipses o hypebolæ. Fo example, Halley s comet obits with a peiod of 75.3 yeas and an eccenticity of Halley s was the fist peiodic comet to be ecognized. Simila comets had been obseved in 456, 53, 607, and 68; Edmond Halley (656 74) figued out in 705 that they must be the same and pedicted a etun in 758. Although he died at age 85, having become Astonome Royal in 70, he did not live to see his pediction fulfilled. Halley s was obseved on Chistmas day in 758, but didn t each peihelion until Mach, 759. Its etun Figue 6: Paabolic obit (blue) and a hypehad been delayed by Jupite and Satun. bolic obit with є. having the same peihehalley s comet last visited in 986 (you pob- lion. The staight lines ae the asymptotes of the ably missed it then) and is set to etun in 06. hypebola (not pat of the obit). Its appeaance is always an occasion and has been noted as fa back as 40 BCE. Mak Twain was bon on 30 Novembe 835, two weeks afte the closest appoach to Eath of Halley s comet. He had a singula attachment to the comet. In 909, he said I came in with Halley s comet in 835. It is coming again next yea, and I expect to go out with it. It will be the geatest disappointment of my life if I don t go out with Halley s comet. The Almighty has said, no doubt: Now hee ae these two unaccountable feaks; they came in togethe, they must go out togethe. Twain died of a heat attack on Apil 90, one day afte the comet s closest appoach to Eath. 8

9 7. 6. SUMMARY Pecession A emakable popety of elliptical obits fo an invese-squae-law foce is that the oientation of the semimajo axis is fixed in time. Betand s theoem shows that thee ae only two foce laws that poduce stable, epeating obits: F n fo n and n. See the appendix in Goldstein s book on mechanics fo a poof. A caeful test of the invese squae law using astonomical measuements, howeve, is quite challenging. Not only must one account fo the petubations caused by the othe planets, but also fo the shape of the Sun. We have assumed that we may model the Sun and the planet as point masses inteacting accoding to V () k/. It took Newton some time to show that outside a spheically symmetic mass distibution, the gavitational foce is the same as if all the mass wee concentated in a point at the cente povided that gavity obeys an invese squae law. You have poved this (o seen it poved) fo Coulomb s law. You can show it is tue by consideing a unifom thin shell of mass. But what if the Sun ween t spheically symmetic? In fact, since it is otating, isn t it likely to be a bit lage aound the equato than at the poles? This kind of distotion intoduces anothe tem in the gavitational potential of the Sun, which becomes k Q V () 3 (6) whee Q is the coefficient of a quadupola gavitational tem. Such a tem causes elliptical obits to pecess. Geneal elativity causes anothe depatue fom invese squae attaction, which also causes obits to pecess. This effect is geatest fo Mecuy, as it depends invesely on adius (o equivalently, on υ /c, which is geatest fo Mecuy). It amounts to 43 seconds of ac pe centuy. Fo compaison, the pecession of the equinox with espect to the fixed stas is ±0.050 pe centuy, and the pecession caused by the influence of the othe planets on Mecuy s motion is about 574 pe centuy. Einstein s computation of the pecession ate fo Mecuy was the fist tiumph of geneal elativity. 7. Summay We educe the two-body poblem to a one-body poblem by noting that (in the absence of extenal foces) the cente of mass does not acceleate and so we may simplify notation by woking in the cente-of-mass fame. The Lagangian of the two-paticle system educes (in the cente-of-mass fame) to µ L ( + θ ) U() whee the educed mass is µ m m /M and M m + m. The coodinate is the distance between the paticles (not the distance to the cente of mass). If the two masses ae vey diffeent, the educed mass is oughly the smalle of the two. This Lagangian is cyclic in θ, fom which we deduce that the angula momentum pθ L θ µ θ constant 9

10 Since enegy is conseved in this system, the adial equation may be expessed E µ (ṙ + l µ ) + U() µ ṙ + U eff () 7. SUMMARY which is a fist-ode diffeential equation fo (t). Dividing by the expession fo the angula momentum poduces a fist-ode diffeential equation fo θ(), which yields (on invesion) the shape of the obit. Fo the impotant special case of an invese-squae-law foce (both gavitation and Coulomb attaction o epulsion, including Ruthefod scatteing), the obits ae conic sections satisfying A + є cos θ a( є ) + є cos θ whee a is the semi-majo axis and є is the eccenticity of the obit. Fo gavitational obits, the geometic paametes ae elated to the dynamical paametes via a GMµ E є E + µ ( l GMµ ) l GMµ a( є ) Invese-squae laws and Hooke s-law sping foces ae unusual in poducing closed obits egadless of initial conditions. Closed obits ae possible with othe attactive cental potentials, but only fo caefully chosen initial conditions. 0