Chapter 17 Appendices

Transcription

1 Chapte 17 Appendices The appendices pesented hee involve moe mathematical sophistication than is commonly pesented in feshman-level subjects. Analytic consideation of the equations esulting fom Newton s Second Law and Newton s Law of Univesal Gavitation ae pesented in these notes in a manne that should be undestood and appeciated by students who have a moe than casual inteest in the subject. The topics of the appendices ae: 17.A: Deivation of the Obit Equation Two pesentations of the esult of Equation ( ) 17.B: Popeties of Elliptical Obits The dynamical popeties of objects in elliptical obits 17.C: Analytic Geometic Popeties of Ellipses Demonstating how the esults of Appendices 17.A and 17.B ae consistent with moe familia epesentations of ellipses 17.D: A Poof that Bounded Kepleian Obits ae Ellipses Including a paameteization of the obit adius as a function of time, and a demonstation of the consistency with Newton s Law of Univesal Gavitation 17.E: Obit Radius as a Function of Time by Diect Integation An Extension of Appendix 17.D, stating fom Equation (17.3.8) (Waning! Seious but vey useful calculus techniques) 17.F: Even Moe on Keple Obits Using geomety, vecto algeba, but minimal calculus to find the obit equation, intoducing the Laplace-Runge-Lenz vecto. Appendix 17.A: Deivation of the Obit Equation Stat fom Equation (17.3.1) in the fom dθ ( 1/ ) L μ L Gm1m E + μ 1/ d. (17.A.1) What follows involves a good deal of hindsight, allowing selection of convenient substitutions in the math in ode to get a clean esult. Fist, note the many factos of the ecipocal of. Multiplying numeato and denominato by a facto of, while well- 1/15/9 17App - 1

2 intended, leaves a facto of 1/ in the numeato. So, we ll ty the substitution u 1/, du (1/ ) d, with the esult dθ L μ L μ du E u + Gm1mu 1/. (17.A.) Expeience in evaluating integals suggests that we make the absolute value of the facto multiplying u inside the squae oot equal to unity. That is, multiplying numeato and denominato by μ /L, dθ du ( μe/ L u + ( μgm m / L ) u) 1/ 1. (17.A.3) As both a check and a motivation fo the next steps, note that the left side ( dθ ) of Equation (17.A.3) is dimensionless, and so the ight side must be. This means that the facto of μgm m / L 1 in the squae oot must have the same dimensions as u, o 1 length ; so, define L / μgm1m. This is of couse the semilatus ectum as defined in Equation (17.3.1), and it s no coincidence; this is pat of the hindsight mentioned above. The diffeential equation then becomes dθ du ( μe/ L u + u/ ) 1/ ( μe/ L + 1/ u + u/ 1/ ) ( μe/ L + 1/ ( u 1/ ) ) du du du ( μe / L + 1 ( u 1) ) 1/ 1/. 1/ (17.A.4) Next, we note that the combination of tems μ E / L + 1 is dimensionless, and is in fact equal to the squae of the eccenticity ε as defined in Equation ( ); moe hindsight. The last expession in (17.A.4) is then dθ du ( ε ( u 1) ) 1/. (17.A.5) Fom hee, we ll combine a few calculus steps, going immediately to the substitution 1/15/9 17App -

3 u 1 ε cosα, du ε sinα dα, with the final integal as ε sinα dα dθ dα, (17.A.6) ( ε ε cos α) 1/ o, θ α + constant. We have a choice in selecting the constant, and if we pick θ α π, α θ + π, cosα cosθ, the esult is which is Equation ( ). 1 u 1 ε cosθ (17.A.7) Note that if we chose the constant of integation to be zeo, the esult would be 1 u 1 ε cosθ (17.A.8) + which is the same tajectoy eflected about the vetical axis in Figue 17.3, indeed the same as otating by π. The deivation of Equation (17.A.7) in the fom 1 u ( 1 ε cos θ ) (17.A.9) suggests that the equation of motion fo the educed one-body poblem might be manipulated to obtain a simple diffeential equation. That is, stat fom F μ a mm 1 ˆ d d θ G μ ˆ. dt dt Setting the components equal, using the constant of motion L μ ( dθ / dt) eaanging, and (17.A.1) d L Gmm 1 3 μ. (17.A.11) dt μ What we will do is use the same substitution u 1/ and change the independent vaiable fom t to, using the chain ule twice, since Equation (17.A.11) is a secondode equation. That is, the fist time deivative is 1/15/9 17App - 3

4 d d du d du dθ. (17.A.1) dt du dt du dθ dt Fom 1/ u we have d / du 1/ u. Combining with the angula velocity dθ / dt in tems of L and u, dθ / dt Lu / μ, Equation (17.A.1) becomes d 1 du Lu du L dt u d d θ μ θ μ, (17.A.13) a vey tidy esult, with the vaiable u appeaing linealy. Taking the second deivative with espect to, t d d d d d d dt dt dt dθ dt dt dθ μ μ dul L u du dθ L μ u. θ (17.A.14) Substituting into Equation (17.A.11), and using 1/ u, du L L 3 u u Gmm dθ μ μ 1 u. (17.A.15) Canceling the common facto of u and eaanging, du μgmm 1 u. (17.A.16) dθ Equation (17.A.16) is mathematically equivalent to the Hamonic Oscillato Equation with a constant tem. The solution consists of two pats: the angle-independent solution and a sinusoidally vaying tem of the fom u L μgm m (17.A.17) L 1 uh Acos( θ θ), (17.A.18) whee A and θ ae constants detemined by the fom of the obit. The expession in Equation (17.A.17) is the inhomogeneous solution and epesents a cicula obit. The 1/15/9 17App - 4

5 expession in Equation (17.A.18) is the homogeneous solution (as hinted by the subscipt) and must have two independent constants. We can eadily identify 1/u as the semilatus ectum, with the esult that 1 u u + u 1 + A cos ( ( θ θ )) H (17.A.19) + 1 A cos ( θ θ ) Choosing the poduct A to be the eccenticity ε and θ π (much as was done leading to Equation (17.A.7) above), Equation (17.A.7) is epoduced.. 1/15/9 17App - 5

6 Appendix 17.B: Popeties of an Elliptical Obit We now conside the special case of an elliptical obit. Figue 17.B.1 Ellipse. In Figue 17.B.1, let a denote the semimajo axis, b denote the semimino axis and x denote the distance fom the cente of the ellipse to the oigin of ou coodinate system (, θ ). We shall now expess the paametes, b and x in tems of the constants of the motion L, E, μ, and m. m1 Appendix 17.B.1: The semimajo axis: a See Equation (17.A.7) above. The majo axis A a is given by A a + (17.B.1.1) max min whee the distance of futhest appoach occus when θ, hence ( θ ) (17.B.1.) 1 ε max and the distance of neaest appoach occus when θ π, hence min ( θ π). (17.B.1.3) 1 + ε 1/15/9 17App - 6

7 Figue 17.B.: neaest and futhest appoach Thus 1 a + 1 ε 1+ ε 1 ε. (17.B.1.4) The semilatus ectum eccenticity, can be e-expessed in tems of the semimajo axis and the 1 a ε. (17.B.1.5) We can now expess the distance of neaest appoach, Equation (17.B.1.3), in tems of the semimajo axis and the eccenticity, min ( 1 ε ) a 1+ ε 1+ ε a( 1 ε ). (17.B.1.6) In a simila fashion the distance of futhest appoach is max ( 1 ε ) a 1 ε 1 ε a( 1 ε ) +. (17.B.1.7) Figue 17.B. shows the distances of neaest and futhest appoach. Using ou esults fo and ε fom Equations (17.3.1) and ( ), we have fo the semimajo axis a μ L ( 1 ( 1+ / μ ( 1 ) )) Gm1m EL Gm m Gm m E 1 and so the enegy is detemined by the semimajo axis, 1. (17.B.1.8) 1/15/9 17App - 7

8 E Gm m a 1. (17.B.1.9) The angula momentum is elated to the semilatus ectum by Equation (17.3.1). Using Equation (17.B.1.5) fo, we can expess the angula momentum (17.B.1.3) in tems of the semimajo axis and the eccenticity, Note that a esult we will etun to late. 1 μ 1 1 L μ Gmm Gmma ε. (17.B.1.1) L, (17.B.1.11) μ Gm m a ( 1 ε ) Appendix 17.B.: The location x of the cente of the ellipse: Fom Figue 17.B.1, the distance fom a focal point to the cente of the ellipse is 1 x max a. (17.B..1) Using Equation (17.B.1.7) fo max, we have that x a(1 + ε) ε εa. (17.B..) Thus, fom Equations (17.3.1), (17.B..) and (17.B.1.8), Appendix 17.B.3: The semi-mino axis: Fom Figue 17.B.1, whee ( 1 / μ ) Gm1m x εa + EL Gm1m. (17.B..3) E ( b ) b x (17.B.3.1) b 1 ε cosθ, (17.B.3.) b 1/15/9 17App - 8

9 which can be ewitten as ε cosθ. (17.B.3.3) b b b Note that fom Figue 17.B.1, x bcosθb, (17.B.3.4) so that b ε x +. (17.B.3.5) Substituting Equation (17.B..) fo x and Equation (17.B.1.5) fo Equation (17.B.3.5) yields ( 1 ε ) b a aε a into +. (17.B.3.6) The fact that b a is a well-known popety of an ellipse eflected in the geometic constuction, that the sum of the distances fom the two foci to any point on the ellipse is a constant. Thus the semi-mino axis b becomes ( b ) b x a a a ε. (17.B.3.7) ε 1 Using Equation (17.B.1.11) fo 1 ε, we have fo the semi-mino axis b al / μ Gm1m. (17.B.3.8) We can now use Equation (17.B.1.8) fo a in the above expession, yielding Gm1m 1 b al/ μgmm 1 L / μgmm 1 L (17.B.3.9) E μe Appendix 17.B.4: Speeds at neaest and futhest distances: At neaest appoach the velocity vecto is tangent to the obit, so the angula momentum is L μ v (17.B.4.1) min p and the speed at neaest appoach is 1/15/9 17App - 9

10 v L/ μ. (17.B.4.) p min Figue 17.B.3 Speeds at neaest and futhest appoach Using Equation (17.B.1.1) fo the angula momentum and Equation (17.B.1.6) fo min, Equation (17.B.4.) becomes v p min ( ) ( 1 ε) ( ε) L μgm1m 1 ε Gm1m 1 ε Gm1m 1+ ε.(17.b.4.3) μ μa 1 ε μa μa 1 A simila calculation show that the speed v a at futhest appoach, v a max ( 1 ε) L μgm1m 1 ε Gm1m 1 ε Gm1m 1 ε.(17.b.4.4) μ μa 1+ ε μa + μa 1+ ε Appendix 17.C: Analytic Geometic Popeties of Ellipses Conside Equation , and fo now take ε < 1, so that the equation is that of an ellipse. It takes some, but not a geat deal, of algeba to put this into the moe familia fom x x y a + (17.C.1) b 1 whee the ellipse has axes paallel to the x and y coodinate axes, cente at ( x, ), semimajo axis a and semimino axis b. We ewite Equation as x ε y x+ 1 ε 1 ε 1 ε. (17.C.) We next complete the squae, 1/15/9 17App - 1

11 x ε ε y ε 1 ε 1 ε 1 ε x+ + + ε x 1 ε ( 1 ε ) ( 1 ε ) ε y x + 1 ε 1 ε 1 ε y + 1. ( /1 ( ε )) / 1 ε (17.C.3) The last expession in (17.C.3) is the equation of an ellipse with semimajo axis a 1 ε, (17.C.4) semimino axis b a 1 ε 1 ε (17.C.5) and cente at ( ) ( ε /1 ε, εa, ), as found in Equation (17.B..). Appendix 17.D: A Poof (in Catesian Coodinates) that Bounded Kepleian Obits ae Ellipses 1 Intoduction: The Question: Keple s Fist Law, which states that the obits of planets ae ellipses with the sun at one of the foci, is usually pesented in fist-yea subjects in Newtonian Mechanics, but is aely poved. Thee ae many altenatives to pesenting a poof, but they often equie mathematics that fist-yea students find unfamilia usually calculus in pola coodinates, as is done in the Couse Notes, Sections 17. and Hee we set out to demonstate that an elliptical obit fo which equal aeas ae swept out in equal times (which is quantified in Keple s Second Law) does indeed satisfy Newton s invesesquae law of Univesal Gavitation. Fist we intoduce a pehaps unfamilia paameteization of the elliptical obit, showing that this paameteization is equivalent to the moe familia Catesian epesentation. We 1 This Appendix is adapted fom mateial povided by Pof. Paul Schechte of MIT. 1/15/9 17App - 11

12 then locate the foci of the ellipse, intoducing the concept of eccenticity as a consequence of the paameteization. Next we use the equal-aeas law (Keple s Second Law) to deive an expession fo time as a function of the intoduced paamete (the eccentic anomaly). With both position and time expessed as functions of this paamete, we then use basic calculus to take second time deivatives of the position and show that the acceleation is that given by Newton s Second Law and Newton s law of Univesal Gavitation. Appendix 17.D.1: Step 1: A Paametic Repesentation of the Ellipse Most students fist mathematical encounte with the ellipse is in the Catesian fom x a y +, (17.D.1.1) b 1 epesenting an ellipse with semimajo axis oigin of the Catesian coodinate system. If we define the paamete η such that a and semimino axis b, and cente at the η actan ay, (17.D.1.) bx we can infe the elations x acos η, y bsinη, (17.D.1.3) which ae seen to satisfy Equation (17.D.1.1), the Catesian equation fo an ellipse. The epesentation in (17.D.1.3) has a staightfowad geometic intepetation. The ellipse (shown as a solid cuve in Figue 17.D.1) may be thought of as the pojection of a cicle (shown as the dashed cuve), with Catesian equation x y + 1. (17.D.1.4) a a 1/15/9 17App - 1

13 Figue 17.D.1 - Popeties of an ellipse Intoducing new vaiables y ( b/ a) y and x x gives the pojected ellipse, as given in Equation (17.D.1.1). The magnitude of the y -coodinate of evey point P on the ellipse is smalle by the facto b/ a<1 than the y -coodinate of the coesponding point P on the cicumscibed cicle. In the Figue 17.D. below we show the paamete η as an ac AP, but it is also the angle made by OP and OA. Appendix 17.D.: Step : Whee is the Focus? While we e at it, whee s the othe one? Ou appoach will be to diffeentiate the position twice with espect to time to demonstate that the acceleation is given by Newton s Univesal Law of Gavitation. We know both x and y as functions of η, as in Equations (17.D.1.3), but not as functions of time. A diect consequence of the consevation of angula momentum is Keple s Second Law, which states that equal aeas ae swept out in equal times by the vecto denoting the position of the planet with espect to the sun. But the sun sits not at the cente O of the ellipse but at S, one of the foci. 1/15/9 17App - 13

14 Figue 17D. Paametes of an Ellipse The foci ae the two points lying on the majo axis such that the sum of the lengths of the two vectos dawn fom the foci to any point on the ellipse is the same. The two foci ae shown as S and S in Figue 17D.. The sum of the distances SA and SA is equal to a, twice the length of the semimajo axis. The foci ae both a distance aε fom the cente of the ellipse. The dimensionless numbe ε can ange fom zeo to unity, and is called the eccenticity, o off-centeedness of the ellipse. Conside the point B whee the semimino axis intecepts the ellipse. The segment OB has length b. The segment SB has length a ε + b. But twice the length of SB must be equal to a, allowing us to solve fo the eccenticity ε in tems of the lengths of the semimajo and semimino axes a and b, ε 1 a. (17.D..1) b 1/15/9 17App - 14

15 Appendix 17.D.3: Step 3: Paameteizing the Time In what follows, we take the sun to be at focus S. If we took the focus to be at point S, as was done in Appendix 17.B, we would need to eplace ε by ε, but the esults would be unchanged. Recalling Keple s Second Law, the aea swept out by the vecto fom the focus to the points on the ellipse is popotional to time. The total aea of the ellipse is π ab, a facto b/ a less than the aea π a of the cicumscibed cicle because all the y -coodinates of the points on the ellipse ae smalle than the coesponding y coodinates on the cicle by the same facto b/ a. As the planet moves fom point A to point P, in Figue 17.D., the position vecto sweeps out the aea bounded by the segments SA and SP, and the ac AP. The time t the planet takes to go fom A to P can be expessed as a faction of the obital peiod T, t T aea SAP. (17.D.3.1) π ab The aea swept out in time t can also be expessed as the diffeence between two aeas, ( SAP) ( OAP) ( OSP) aea aea aea. (17.D.3.) The tiangle fomed by the points O, S and P has base aε and height bsinη, the y - coodinate of point P, and hence aea ( aε)( bsin η ) /. The oddly-shaped egion bounded by the segments OA and OP, and the ac AP has an aea smalle by a facto b/ a than the aea of the coesponding secto of the cicumscibed cicle bounded by the segments OA and OP, and the ac AP. The aea of that cicula secto is just ηa /. Combining this with Equations (17.D.3.1) and (17.D.3.), we have t ηab/ εsin ηab/ η εsinη. (17.D.3.3) T πab π Appendix 17.D.4: Step 4: The Position Vecto with Respect to the Focus We will obtain the acceleation a by diffeentiating the x - and y -coodinates of the planet with espect to time t, and show that this is consistent with Newton s Law of Univesal Gavitation, GM a 3 (17.D.4.1) 1/15/9 17App - 15

16 whee M is conventional notation fo the sola mass. In Equation (17.D.4.1), note the exta powe of in the denominato equied give an oveall dependence on 1/, since we use the vecto instead of ˆ to detemine diection. This will soon be seen to be quite convenient fo the pupose of taking deivatives. The vecto fom the sun to the planet is given by ( acosη aε)ˆ i+ bsinηj ˆ. (17.D.4.) The fist tem acosη in the x -component epesents the position of the planet with espect to the cente O of the ellipse, while the second tem aε accounts fo the offset of the focus with espect to the cente of the ellipse. The length of the vecto is obtained by squaing and adding the components, ( cos ) a η ε + b sin η ( cos η εcosη ε ) a ( 1 ε ) sin η cos η sin η εcosη ε ε ( 1 cos η) a + + a + + a 1 εcosη+ ε cos η ( 1 εcosη) ( 1 εcos η). a a (17.D.4.3) Appendix 17.D.5: Step 5: Diffeentiating Twice with Respect to Time The acceleation a is given by d xˆ d y a i+ ĵ, (17.D.5.1) dt dt but we know, fom Equations (17.D.1.3), x and y as explicit functions of the eccentic anomaly η athe than the time t. We will need to use eithe implicit diffeentiation o the chain ule. Choosing the latte (they of couse yield the same esult), dx dx d dx / d dt η η dη dt dt / dη. (17.D.5.) The fist of the equations in (17.D.1.3) leads quite eadily to dx / dη asinη and Equation (17.D.3.3) gives dt / dη ( T /π)( 1 εcos ) η, with the esult 1/15/9 17App - 16

17 dx π sinη a. (17.D.5.3) dt T 1 ε cosη A simila calculation gives dy dt π cosη b. (17.D.5.4) T 1 ε cosη In finding the second deivatives with espect to time, doing some peliminay calculations will, we hope, help in the long un. Specifically, we have, stating fom Equation (17.D.5.) (the coesponding expessions fo the deivatives of y follow immediately), dx dx / dη dt dt / dη d dx dt dt dt dt d dx d dx dη dt ( dt / dη ) ( / η ) 1 d dx/ dη ( dt / dη) dη dt / dη ( d x/ dη )( dt/ dη) ( dx/ dη)( d t/ dη) 1 dt / dη / ( dt dη ) 1 d x dt dx d t. 3 dη dη dη dη (17.D.5.5) The advantage is in the faily simple fom fo the tems We also have dt T ( ε η) T d t T 1 cos, εsinη. (17.D.5.6) dη π π a dη π d x d y acos η, bsinη. (17.D.5.7) dη dη Combining, 1/15/9 17App - 17

18 dx 4π a cos 3 ( 1 cos ) ( sin )( sin ) dt T η ε η η ε η 4π a 4π x 3[ ε cos η]. 3 T T (17.D.5.8) Similaly, d y 4π b sin 3 ( 1 cos ) ( cos )( sin ) dt T η ε η η ε η 4π b 4π y 3 [ sin η]. 3 T T (17.D.5.9) Combining the esults given in (17.D.5.8) and (17.D.5.9), we have π a. (17.D.5.1) T 3 Keple s Thid Law pedicts T 4 π a / GM (note that a is NOT the magnitude of the acceleation), and so Equation (17.D.5.1) becomes GM a 3, (17.D.5.11) which is the combination of Newton s Second Law and Newton s Law of Univesal Gavitation, and is Equation (17.D.4.1). Appendix 17.D.6 Finge Benefits Most of the poofs that the obits of planets ae ellipses give the distance fom the sun as a function of the pola angle θ athe than as a function of time (see the deivations in Chapte 17). The above poof gives and t as functions of the eccentic anomaly η. Fo many puposes one needs to know position as a function of time, not just distance as a function of pola angle. We would like to complete ou desciption of planetay motion by deiving θ as a function of η. The x -component of the vecto fom the sun to the planet is given by ( cos ) Substituting a (1 ε cosη ) fom (17.D.4.3), we find cosθ a η ε. (17.D.6.1) 1/15/9 17App - 18

19 cosη ε cosθ. (17.D.6.) 1 ε cosη We then get the two expessions ( 1+ ε )( 1 cosη ) 1 cos θ, 1 εcosη ( 1 ε )( 1+ cosη ) 1+ cos θ. 1 εcosη (17.D.6.3) Then, using the tigonometic half-angle identities, which gives us ( + ε )( η ) θ 1 cosθ 1 1 cos tan, (17.D.6.4) 1 + cos θ 1 ε 1 + cos η θ 1+ ε η tan tan, 1 ε (17.D.6.5) the elation between the pola angle and the eccentic anomaly. A futhe finge benefit to using the eccentic anomaly is in calculating integals in pola coodinates, specifically the aea of an ellipse. We have found the standad esult, aea π ab, by using the known fom fo the aea in Catesian coodinates and multiplying by the atio b/ a. Howeve, in pola coodinates, the expession fo the aea, π a ( 1 ε ) π dθ dθ, (17.D.6.6) 1 aea ( + ε θ ) 1 cos is an integal is not often done in feshman-level calculus (although the technique is included in the and texts at MIT). Using the eccentic anomaly tuns out to simplify the integal geatly. Note that in this Appendix, the oigin is taken to be on the positive x -axis, and so we need to eplace ε by ε in Section To stat, e-expess the adius, the distance fom the sun to the planet, in the fist integal in tems of η, as given in Equation(17.D.4.3). Then, 1/15/9 17App - 19

20 Fom the half-angle fomulas, we have a a aea 1 cos 1 cos π π θ ( ε η) dθ ( ε η) d d dη η. (17.D.6.7) 1 εcosη 1 ε cos ( η/ ) sin ( η/ ) cos ( η/ ) + sin ( η/ ) 1 tan ( η /) 1 ε 1+ tan ( η /) ( 1 ε) + ( 1 ε) tan ( η/ ). 1+ tan ( η /) (17.D.6.8) The tem fom the change of vaiables, dθ / dη, is found by diffeentiating Equation (17.D.6.5), dθ θ dη 1+ ε η sec sec 1 ε θ 1+ ε η dθ 1+ tan dη 1+ tan 1 ε 1+ ε η 1+ ε dθ 1+ tan dη 1+ tan 1 ε 1 ε η. (17.D.6.9) Thus we see that, afte some mino algeba, 1+ ε η 1+ tan 1 ε 1 ε dθ + dη η 1 ε 1+ tan η ( ε) + ( + ε) 1 1 tan d 1, η dθ η ε 1+ tan (17.D.6.1) so that the second integal in Equation (17.D.6.7) becomes a π aea ( 1 cos ) 1 ε η ε dη πa 1 ε πab, (17.D.6.11) as detemined as descibed above. 1/15/9 17App -

21 Appendix 17.E: Obit Radius as a Function of Time by Diect Integation Intoduction In pevious pesentations, in Chapte 17 and the Chapte 17 Appendices, we have only calculated the distance of a planet fom the sun as a function of angle, epoducing the classical foms fo conic sections, but have only hinted at the possibility of finding the distance as a function of time. This latte dependence can be of geat utility, but at the cost of a geat deal of math. This will not stop us. In what follows, we ll ty to epoduce the notation of Chapte 17 as closely as possible, including that of Appendix 17.D, and intoduce new notation that should be consistent with othe souces (of couse, thee s no way we can guaantee consistency with all souces). Although the poblem may be genealized, we ll make efeence to the case of a single object moving subject to the gavitation of the sun. This object is likely to be a planet o a comet, but fo the puposes of this discussion, the motion of othe objects, specifically objects in unbound obits, and objects subject to the eath s gavitation, could be found by extension. In fact, in an example that will not be pesented hee, the expansion of the univese could be modeled by equations of the same mathematical fom. As a disclaime at the beginning, we will see that we can find explicit foms only two vey special cases: cicula motion (necessaily unifom) and the case of an object with escape velocity on a linea tajectoy. Fo all othe cases, ou solutions will give the distance as an implicit function of the time t. () t Fo a given sola mass and object mass, the tajectoy may be detemined by the total mechanical enegy and the angula momentum. Using these paametes exclusively leads to athe cumbesome expessions, and we will find it convenient to use othe paametes, specifically the semilatus ectum and eccenticity fo non-zeo angula momentum and the semimajo axis fo bound obits. We will also use common notation in evaluating needed integals, which will allow us to compae solutions in limiting cases. Appendix 17.E.1 Basics Take the mass of the sun as M sun M m and let the mass of the planet be m 1, with educed mass μ mm 1 / ( m1+ m). In many cases, it would be possible to go to the limit m m, μ m ; this pesentation will use the moe geneal fom fo the educed mass. 1 1 The semilatus ectum of the obit tajectoies in tems of the angula momentum magnitude L is fo 1/15/9 17App - 1

22 L. (17.E.1.1) μ Gm m 1 The eccenticity ε of a conic section is given by EL ε 1+. (17.E.1.) μ ( Gm m ) Fo a bound obit, the semimajo axis a > is elated to the semilatus ectum by 1 a 1 ε and the total mechanical enegy fo a bound obit is E (17.E.1.3) Gm m a 1 <. (17.E.1.4) We can then poceed to Equation (17.3.8), epoduced hee as 1 1 d 1 L G m m E + dt μ μ. (17.E.1.5) Equation (17.E.1.5) is a sepaable equation in expessed as and t, and may in geneal be d 1 L Gm m E + μ 1 dt. (17.E.1.6) μ Equation (17.E.1.6) could be fomidable, depending on the elative values of the paametes, specifically the total mechanical enegy E and the obital angula momentum L. We ll stat by looking at the special cases that make the integal simple, pimaily to demonstate the calculus techniques involved and to intoduce some useful notation. Appendix 17.E. No Angula Momentum 1/15/9 17App -

23 Setting L squae oot, in Equation (17.E.1.6) manifestly simplifies the agument of the d Gm m E + 1 dt. (17.E..1) μ Befoe jumping into the integal without a life vest, howeve, we would be wise to conside the thee cases E, E < and E > sepaately. The tajectoies of such objects would be degeneate conic sections, paabolas, ellipses o hypebolas extended so fa that they become confined to staight segments o ays. E..1 Zeo Angula Momentum, Zeo Mechanical Enegy. We should ecognize this situation as that of an object (not likely to be a planet) launched o thown with exactly the needed escape velocity. Setting E in Equation (17.E..1) yields, almost immediately, 1 1/ d dt Gm m μ 1 1 3/ t Gm m 3 μ 1 (17.E..) whee any constants of integation have been adjusted so that. Unlike most of the expessions aising fom the moe geneal situations, Equation (17.E..) can be eadily solved fo ( t ), () t 1/3 9Gm1m μ t /3. (17.E..3) It should be noted that in Equation (17.E..3), d as t. This indicates that fo dt launching an object fom a planet, o anything expelled fom the suface of the sun, the d limit is unphysical fo classical physics. Howeve, cosmologists do conside dt d this limit. We also see, as a simple check, that as t and, dt, consistent with zeo net mechanical enegy. 1/15/9 17App - 3

24 An altenative method of doing the integal in Equation (17.E..3), one that allows compaison to the othe degeneate obits, will be given below (Subsection E..4). E.. Zeo Angula Momentum, Negative Mechanical Enegy This would then be the case whee an object is launched vetically upwad, but with insufficient enegy to escape. The object (planets ae not likely to do this comets come close) would then move some distance away, and then etun. O, an object dopped fom est in an invese-squae gavitational field (but with no initial tangential velocity) would plummet. To simplify the calculations, use Equation (17.E.1.4), epoduced hee; E Gm m a 1, (17.E..4) whee a is the semimajo axis of the bound obit, a degeneate ellipse, as descibed above. We can ecognize that fo a pojectile launched fom a (nonotating) suface, a is the futhest distance the pojectile ises fom the cente of the attacting body. Using Equation (17.E..4) in Equation (17.E..1) leads to d ( a) 1/ 1/ d Gm1m dt μ Gmm a aμ 1 dt. (17.E..5) The integal aising fom the second equation in (17.E..5) may not be standad, and is not in all tables (and sometimes gives pause to compute integation pogams), but is not difficult. Clealy, it would help to make a substitution so that both numeato and denominato in the squae oot on the left ae squaes. One choice is to make the substitution η asin. (17.E..6) The facto of in the angle η / will tun out to be a slight convenience, and will be seen to match conventional usage, specifically that in Appendix 17.D. Fo states, we have fom which d a sin η cos η d η, (17.E..7) 1/15/9 17App - 4

25 ( η ) sin /. ( η ) asin / η η d asin cos d η a a asin η / (17.E..8) a This is easily integated, using standad methods and the half-angle fomula η 1 sin ( 1 cosη ), (17.E..9) which leads to dη Gm1m η sinη 3 t. (17.E..1) a μ Equations (17.E..6) and (17.E..1) then give both and t as functions of include the paamete a. We cannot solve fo ( t ), but we can solve fo of the enegy (ecall that E < ), t η, and in tems 3 1 E 1 sin E E E t 1 Gm1m μ Gm1m +.(17.E..11) Gm1m Gm1m Equation (17.E..11) is a teible mess, and not eally of geat use. Fom now on in this pesentation, such substitutions will not be made. Of fa moe use is to conside Equations (17.E..6) and (17.E..1) togethe, epoduced hee; η asin Gm1m η sin η. 3 t a μ (17.E..1) With the choice of integation constants such that ( η ), the pojectile would etun to at η π, and 1 t η, we see that 3 4π μa t( η π) T, (17.E..13) Gm m the peiod pedicted by Keple s Thid Law as given in Equation ( ). 1/15/9 17App - 5

26 Befoe moving on, anticipating late sections, it is indeed possible to pefom the needed integal in (17.E..5) by othe methods, specifically by ewiting the second expession as d d d Gmm 1 dt a a a a+ a a aμ In the last expession in (17.E..14), make the standad substitution.(17.e..14) a acos η, a 1 cos η, d asinηdη (17.E..15) (which is, of couse just Equation (17.E..6) ewitten) to obtain d ( ) a a ( 1 cosη ) a dη, (17.E..16) equivalent to Equation (17.E..1). At this point, note that the substitution in (17.E..15) is that given by Equation (17.D.4.3) with ε 1. This is not a coincidence, and we ae not making this up. E..3 Zeo Angula Momentum, Positive Mechanical Enegy We anticipate that fo this case, the tajectoy will be unbound, a degeneate hypebola. As pesented hee, we ll have to use a slightly diffeent calculus technique, and we won t be able to conside the peiod of the non-peiodic obit o the semimajo axis. Howeve, fo mathematical convenience we will intoduce a paamete with dimensions of length and call it a. If we epeat the calculation fo negative mechanical enegy, but with E Gmm /a 1, a > to epesent a positive mechanical enegy, we obtain d ( a) 1/ + 1/ d Gm1m dt μ Gmm a+ aμ 1 dt. (17.E..17) To employ simila means as used above, we would need the sum a+ to be the squae of some convenient function, and no cicula tig function would suffice. Howeve, if we use hypebolic functions, the ensuing calculus simplifies quite a bit. A complete discussion of hypebolic functions cannot be pesented hee. The featues that we will want to use ae: 1/15/9 17App - 6

27 dcoshu dsinhu sinh u, cosh u, cosh 1, sinh du du cosh u sinh u 1 (17.E..18) Accodingly, if we make the substitution 1 u ( u ) sinh cosh 1. η η η η (17.E..19) asinh, d acosh sinh d we obtain η d asinh dη a( coshη 1) dη. (17.E..) a+ Using Equation (17.E..) in (17.E..17) and integating yields Gm1m sinhη η 3 t, (17.E..1) a μ whee once again the constants of integation have been selected so that t η. An equation simila to Equation (17.E..11) is not likely to help anyone, and won t be given hee. As a simple check, note that d Gm m a + Gm m dt a μ a μ Gm m a μ 1 3 ( η ) coth / ( η ) ( η ) cosh / sinh / (17.E..) (yes, thee eally is such a thing as a hypebolic cotangent ). As η,,, coth η / 1, d / dt Gm1m / aμ E / μ, indicating that in this limit the enegy is all kinetic. We should also ecognize that if we had used, in Equations (17.E..17) t d d + a + a a (17.E..3) 1/15/9 17App - 7

28 and made the substitution diectly. + a acoshη, we would obtain Equation (17.E..) E..4 Limiting cases We should expect that in the limit E, the esults of Subsections E.. and E..3 should educe to that of Subsection E..1, and this is indeed the case. One way to see this is to conside that in both cases of nonzeo enegy, in the limit as a is vey lage, η fo finite, and in this limit η η 1 cos η, coshη η sinη η η η η sinhη η η+ η η η 6 6 aη /, (17.E..4) and combining these esults with eithe of the expessions in (17.E..1) o (17.E..1) yields, afte some algeba, the esult of (17.E..3). Equation (17.E..4) suggests that if we use the substitution aη / in the fist equation in (17.E..), we may get a esult simila to those in the expessions in (17.E..1) and Equation (17.E..1). Indeed, in doing so, with d aη dη and doing the faily simple esulting integal leads to 3 η 6 Gm1m 3 t. (17.E..5) μa The paameteizations that led to aη / and Equation (17.E..5) involve a paamete a >, but thee is no length scale in Equation (17.E..). It tuns out that the given paameteizations esult in solutions to Equation (17.E..), independent of a. This is of couse completely consistent; we can use any length scale we choose. F heaven s sake, lets daws a pictue. Okay. In Figue 17.E.1, the time axis is hoizontal and the distance axis is vetical. The scales ae abitay. The lowe cuve, ed if viewed in colo, fo E <, has been extended fo two bounces o collapses. Whethe o not this is useful depends on the situation. Note that in this idealization, the object would pass though the oigin without changing its diection of motion, but in ou notation, so the sign of does not change, but 1/15/9 17App - 8

29 the sign of the (infinite) velocity does. The uppe (geen) cuve is fo intemediate (blue) fo E. E > and the Figue 17.E. is the same, on an expanded scale nea the oigin. Pat of the point in including this figue is to show that if one wants to send a pojectile in a pescibed tajectoy, one has to be caeful; at the oigin the cuves ae indistinguishable. Figue 17.E.1 Figue 17.E. Appendix 17.E.3 Non-Zeo Angula Momentum 1/15/9 17App - 9

30 As in Section E., we ll stat by consideing the simplest mathematical case, that with E. Again, in doing so we ll gain some insight into the natue of the mathematical techniques involved, and find some advantageous notation fo the othe cases. E.3.1 Non-Zeo Angula Momentum, Zeo Mechanical Enegy Fo E, we know that the tajectoy will be a paabola. Equation (17.E.1.6) becomes Gm m d 1 1 L μ dt, (17.E.3.1) μ which can be simplified as d L μgmm d 1 Gm1m dt μ Gmm 1 dt μ (17.E.3.) whee L is the semilatus ectum (Equation (17.E.1.1) in this Appendix and μ Gm m 1 Equation (17.3.1) fom the Couse Notes). / + /, and The needed integal is found by setting / + / 1 d /d + d, (17.E.3.3) / / which is eadily integated to obtain 3/ Gm1m ( / ) + / /( + ) t.(17.e.3.4) 3 3 μ Equation (17.E.3.4) is pehaps had to ecognize as a paameteization in time of a paabola in space, but we can identify some featues that we would expect. In the limit L,, we epoduce the esult found in (17.E..). The constants of integation give / at t, and this is the minimum value of. Fo a paabola with eccenticity ε 1, the minimum distance is /1 ( + ε ) /, consistent with the above esult. min 1/15/9 17App - 3

31 Once moe anticipating a esult fom Section E.4, we can pefom the integation in the second expession in (17.E.3.) by setting η η min min, d min η dη, d min η 1+ dη, (17.E.3.5) leading to d 1 η 1+ Gmm μ 3/ min dη 1 dt, (17.E.3.6) fom which η 6 Gm m μ 3 3/ 1 min η + t. (17.E.3.7) min Note that the constants o integation ae still set so that at t, η and. Non-Zeo Angula Momentum, Non-Zeo Mechanical Enegy The two cases, E < and E >, could be consideed in one calculation, at the expense of intoducing moe sophisticated math (complex vaiables) than we need fo the cuent pesentation. Accodingly, the cases will be consideed sepaately. E.3. Non-Zeo Angula Momentum, Negative Mechanical Enegy We know that these tajectoies should be ellipses with one focus at the sun. Fom the pevious esult, we expect that using the semilatus ectum will be useful. Howeve, invoking a geat deal of hindsight, using the semimajo axis a will simplify the calculations, and epoduce moe conventional notation. So, using Equation (17.E..4), E Gmm, and Equation a ε, Equation (17.E.1.6) becomes /a (17.E.1.3), 1 ( 1 ) d ( ε ) 1/ 1 / / a a + d ( ε ) a 1 + a Gm1m dt μ Gmm 1 dt aμ. (17.E.3.8) 1/15/9 17App - 31

32 The needed integal on the left above is done by completing the squae in the agument of the squae oot in the denominato, ( 1 ε ) ( 1 ε ) ( ) a + a a a a+ + a ε a. a (17.E.3.9) This suggests, almost necessaily, the substitution leading to a 1 εcos η, d aε sinηdη, (17.E.3.1) d ( ε ) a 1 + a and hence, fom Equation (17.E.3.8), 1 a aε ( ) 1 ε cosη aεsinηdη ( 1 cos ) sinη a ε η dη (17.E.3.11) Gm1m Gm1m π η εsinη t π t t 3 3 μa 4π μa T (17.E.3.1) whee T is the peiod of the obit. It should be noted that the substitution used in Equation (17.E.3.1) and the esult of Equation (17.E.3.1) ae those in Appendix 17.D; this is not a coincidence. Note that the eccenticity does not appea on the ight in the above esult. If ε, the case fo a cicle, the integal in Equation (17.E.3.11) would be impope, and this case might be consideed sepaately. Howeve, since we have a esult involving only the semimajo axis a, with the peiod a function of a, and a simple function of the eccenticity ε, thee is no need to conside the special case sepaately. The othe limiting case, ε 1, L, is seen to epoduce the esult of Section E. above. E.3.3 Non-Zeo Angula Momentum, Positive Mechanical Enegy We know that these tajectoies will be hypebolas with one focus at the sun. As in Subsection E..3 above, intoduce the paamete a > so that the total mechanical enegy is E Gmm /a>. The paamete a 1 can no longe be intepeted as the semimajo axis of a hypebola. Equation (17.E.1.3) is no longe valid, but Equations 1/15/9 17App - 3

33 (17.E.1.1) and (17.E.1.) ae, and Equation (17.E.1.3) becomes ( a ε 1) (ecall that ε > 1). Equations (17.E.3.8) ae then d ( ε ) 1/ 1 / / a a + d ( ε ) + a a 1 Gm1m dt Gmm 1 dt aμ μ. (17.E.3.13) Completing the squae is slightly simple fo positive enegy; adding and subtacting in the agument of the squae oot in the denominato on the left yields d Gmm 1 dt a a aμ + As in Section E..3, a convenient substitution is fom which ε a. (17.E.3.14) aε cosh η a, d aεsinhηdη, (17.E.3.15) d + a a ε ( ) a ε coshη 1 aεsinhηdη aεsinhη ( cosh 1) a ε η dη (17.E.3.16) and, in tems of the eccenticity, the paamete a and the paamete η, Gm1m εsinhη η 3 t. (17.E.3.17) a μ Although the squae oot multiplying the time t does have dimensions of [ ] 1 time, its physical intepetation might be had to qualify, so we ll leave it as pesented. Once again, plots ae in ode. The plot scales have been adjusted so that each adius is at the same minimum value at t. One cuve (ed, if viewed in colo) is clealy oscillatoy, coesponding to an elliptic obit, E >. The lowe cuve below (geen) coesponds to a hypebolic obit, E < and the middle cuve, (blue) coesponds to a paabolic obit, with E. 1/15/9 17App - 33

34 It is not clea fom this figue that as t, the slope of the cuve fo the paabolic obit d appoaches dt, but this is the case. Figue 17.E.3 d Note that as the enegy inceases, at t deceases. This makes physical sense, in dt that at t the gavitational potential enegy is the same fo each, and hence tajectoies with highe enegy must be moving faste (lage L ), and will not fall in as fast. The plot on the left below shows, in an expanded scale, the behavio fo smalle values of t. The plot on the ight shows that the hypebolic tajectoy will eventually intesect the paabolic tajectoy, and should give some indication that fo the paabolic tajectoy, the slope does appoach. Figues 17.E.4A and 17.E.4B 1/15/9 17App - 34

35 E.3.3 Limiting Cases The limits we want hee ae E while maintaining L. We have to be a bit ticky fo these cases. Conside Equation (17.E.3.1); if we simply set ε 1, we meely epoduce Equation (17.E..15). What we do is sepaate tems that emain finite and those that we ve seen befoe. Such a vague statement needs explanation via an example. What follows epesents a geat deal of both hindsight and foesight. Rewite the substitution used in Equation (17.E.3.1) as ( 1 εcosη) ( 1 ε) ε( 1 cosη ) a a + a. (17.E.3.18) The fist tem on the ight will emain finite as a and ε 1, and is equal to the minimum distance (the peihelion), which in these notes we denote min. Expanding cosη and keeping the lowest nonvanishing tem yields η min + aε, η ( min ) ( min ) (17.E.3.19) aε a whee in the last step ε 1 has been used. Using this same technique in Equation (17.E.3.1), Gm1m t μ a 3/ + ( 1 ) 3/ 3/ min min 3 η η εη ε 6 a ε η+ εa a η+ εa 1/ 3/ a η + a 1/ 3/ 3 η 6 3 η. 6 3 η 6 (17.E.3.) Substituting the esult of Equation (17.E.3.19) fo η in tems of and ε 1 gives Gm1m 3/ t min min ( min ) min ( min μ which is the same as Equation (17.E.3.4). + + ),(17.E.3.1) 6 3 1/15/9 17App - 35

36 The same limiting pocedue applied to Equations (17.E.3.15) and (17.E.3.17) (we need 3 coshη 1 + η /, sinh η η+ η / 6 ) will give the same esult. A needed intemediate calculation, to the effect that even though the paamete a is not physical, the poduct a 1 ε fo a hypebola is, as may be found fom solving Equation (17.E.1.5) fo min d / dt with E Gmm /a 1. The fact that we obtain the same expessions in the limiting cases is eflected in the gaph on the ight above, following Subsection E.3.3. Appendix 17.E.4 Synopsis and a Few Othe Points of Inteest The paameteizations fo all six cases ae summaized below: E..1 & E..4 - Zeo Angula Momentum, Zeo Mechanical Enegy Tajectoies ae degeneate paabolas. 3 η η Gm1m a, t, a> abitay. (17.E.4.1) 3 6 μa E.. Zeo Angula Momentum, Negative Mechanical Enegy Tajectoies ae degeneate ellipses. Gm1m a( 1 cos η), t η sin η, a Gmm 3 1 /E >.(17.E.4.) a μ E..3 Zeo Angula Momentum, Positive Mechanical Enegy Tajectoies ae degeneate hypebolas. Gm1m + a acosh η, t sinh η η, a Gmm 3 1 / E >.(17.E.4.3) a μ E.3.1 Non-Zeo Angula Momentum, Zeo Mechanical Enegy Tajectoies ae paabolas. 3 η Gm1m 3/ η min min, t min η +. (17.E.4.4) μ 6 E.3. Non-Zeo Angula Momentum, Negative Mechanical Enegy Tajectoies ae ellipses. Gm1m π a( 1 εcos η), t t η εsin η, a is semimajo axis.(17.e.4.5) 3 μa T 1/15/9 17App - 36

37 E.3.3 Non-Zeo Angula Momentum, Positive Mechanical Enegy Tajectoies ae hypebolas. Gm1m aεcosh η a, t εsinh η η, a Gmm 3 1 / E >.(17.E.4.6) a μ You should note the similaities and diffeences in the above paameteizations. The paamete η is used fequently in such calculations (but some souces use θ, which we use fo something completely diffeent). This paamete is called many things; in othe notes we ve used eccentic anomaly; development angle and evolution angle ae also common. aμ With these paameteizations, we see that dt / dη d / dη (o Gm m min fo the ellipses). The adial component of velocity as a function of η is then 1 instead of a d d / dη Gm m 1 d Gm m d dt dt / dη aμ dη aμ dη 1 1 ln, (17.E.4.7) sometimes a convenient simplification. In fact, to some extent the above paameteizations wee developed with this in mind. In the above paameteizations, all have ( η ) an even function of η and t η an odd function. This makes easy allowance fo negative times, and any of the gaphs above may be flipped about the vetical axis to account fo ealie times. Appendix 17.F: Even Moe on Keple Obits We ve seen so fa that fo a Keple Obit, we have two constants of the motion: the angula momentum and the total enegy. Since the angula momentum is a vecto with thee components, these constitute a total of fou scala constants of the motion. The Keple Poblem has six degees of feedom (thee position, thee velocity), and so we expect to be able to find two moe scala constants of the motion. We might expect to be able to futhe identify any obit by a vecto in the plane of the obit, pependicula to the angula momentum, and this is indeed the case. Symmety suggests that this vecto would be along the majo axis, and we ll see that this is the case as well. What follows uses a good deal of vecto algeba, but minimal calculus, and leads to the obit equation in a supisingly simple fom. We ll need two esults fom vecto algeba that we haven t had to use yet. Specifically, fo vectos a, b and c, we have 1/15/9 17App - 37

38 a b c b a c c a b (17.F.1) a b c b c a c a b. These elations ae not had to deive in Catesian coodinates; the deivations will not be epoduced hee. As a check, howeve, note that the vecto on the ight side of the fist elation is pependicula to both a and b c. If a, b and c ae non-coplana, the common magnitude of the tiple poducts in the second elation is the expession fo the volume of a paallelepiped with the thee vectos foming the sides. Let s stat with the known constant angula momentum, L p μ v, (17.F.) and e-expess this quantity in a way that will allow us to use Newton s Laws. Specifically, conside the velocity in tems of pola coodinates, d dˆ v ˆ + dt dt (17.F.3) so that the angula momentum can be expessed as fom which dˆ ( ˆ dˆ L μ ) μ ˆ dt dt dˆ 1 L ˆ. dt μ (17.F.4) (17.F.5) The advantage to this opeation is that we now have an explicit scala facto of 1/, which can and should be elated to the same facto that appeas in Newton s Law of Gavitation. Howeve, in ode to use that law, we need a vecto elation involving ˆ, and so we ll coss ˆ into both sides of Equation (17.F.5), yielding ˆ ˆ ˆ d ˆ L dt μ (17.F.6) dˆ 1 d. L dt Gm1m dt 1/15/9 17App - 38

39 In the above, the fist elation in Equation (17.F.1) was used to simplify the left side, and μ d / dt Gmm / was used on the Newton s Law of Gavitation, in the fom 1 ˆ ight side. Note the cancellation of the facto of the educed mass μ. Equation (17.F.6) may now be integated to obtain v L ˆ + A Gm1m v L A ˆ, Gm m 1 (17.F.7) whee A is a constant vecto. Since ˆ is in the plane of the obit, and L is pependicula to the plane of the obit, A must lie in the plane of the obit, as indicated above. Futhe, by consideing exteme points of the obit, whee v ˆ, and hence v L ˆ, we see that at these points A is in the diection paallel to the majo axis. Since A is a constant vecto, A must always be in this diection. By consideing the vecto A at peihelion (o at any point on the obit of a cicula obit), we can see that the diection of A is that fom the peihelion point to the focus; we ll need this esult below, when we find the obit equation. The magnitude of is eadily found by calculating A A ˆ ˆ ˆ 1 Gm m + A A v L v L v L 1 1 ( Gm m ). (17.F.8) The fist dot poduct is manifestly 1. The middle tem, the coss tem, is found using the second elation in Equation (17.F.1), 1 1 ˆ ( v L) L ( ˆ v) L ( μv) L. (17.F.9) μ μ The thid tem is most easily evaluated by ecalling that v L v L vl. Combining, we see that v L, so that v L vl and 1/15/9 17App - 39

40 Thus, the constant vecto A the eccenticity. A L + Gm m μ Gm m ( Gm m ) ( Gm m ) 1 1 L 1 Gmm 1 1+ μv μ 1 LE 1 ε. + μ 1 v L (17.F.1) is diected along the majo axis and has magnitude equal to The obit equation is now found algebaically by taking the dot poduct of A and ; 1 A A cosθ ˆ ( v L Gm m 1 ). (17.F.11) The fist tem is meely the magnitude of the position vecto. The second tem, epeating the calculation of Equation (17.F.9) with instead of ˆ, is 1 L Gm m μ 1, (17.F.1) with the esult A cosθ ε cosθ. (17.F.13) Solving fo gives the obit equation in the fom 1 ε cosθ. (17.F.14) It should be noted that what we call the vecto A is a negative scala multiple of the Laplace-Runge-Lenz vecto (yes, it took thee people to come up with this). Specifically, the L-R-L vecto is in many souces given as Ou choice of the fom fo ALRL p L μgm1m μgm1m A. (17.F.15) A allows Equation (17.F.11) without intoduction of exta minus signs. A ε and the diection of A to lead to 1/15/9 17App - 4

41 1/15/9 17App - 41