Notes on Euler s paper. E Memoire sur la plus grande equation des planetes. (Memoir on the Maximum value of an Equation of the Planets)

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1 Notes o Euler s paper E05 -- Memoire sur la plus grade equatio des plaetes (Memoir o the Maximum value of a Equatio of the Plaets) Compiled b Thomas J Osler ad Jase Adrew Scaramazza Mathematics Departmet Rowa Uiversit Glassboro, NJ 0808 Osler@rowaedu I Plaetar Motio as viewed from the earth vs the su ` Euler discusses the fact that plaets observed from the earth exhibit a ver irregular motio I geeral, the move from west to east alog the ecliptic At times however, the motio slows to a stop ad the plaet eve appears to reverse directio ad move from east to west We call this retrograde motio After some time the plaet stops agai ad resumes its west to east joure However, if we observe the plaet from the stad poit of a observer o the su, this retrograde motio will ot occur, ad ol a west to east path of the plaet is see II The aphelio ad the perihelio From the su, (poit O i figure ) the plaet (poit P ) is see to move o a elliptical orbit with the su at oe focus Whe the plaet is farthest from the su, we sa it is at the aphelio (poit A ), ad at the perihelio whe it is closest The time for the plaet to move from aphelio to perihelio ad back is called the period III Speed of plaetar motio The plaet s speed is slowest at the aphelio ad fastest at the perihelio The plaet obes Kepler s secod law The radial lie from the su to the plaet sweeps out equal areas i equal times IV The fictitious plaet which moves with costat speed

2 The more elliptic the orbit is, the greater is this variatio i speed If the orbit were a circle, the speed would alwas be costat ad equal agles would be swept out i equal times We imagie a fictitious compaio plaet (poit X i Figure ) that circles the su with the same period as our plaet, but with uiform motio Further, we assume that both the real ad the fictitious plaet reach the aphelio ad perihelio poits at the same time As Euler sas: After these two plaets have passed b the aphelio, the false plaet will appear to go faster tha the true ad the real plaet will imperceptibl icrease its speed util it will have caught the false oe at the perihelio The it will pass its parter i speed, ad will leave it behid util the rejoi agai at the Aphelio Figure : The plaet P as observed from the su at O, V The mea aomal x, the true aomal z ad the equatio of the ceter

3 Astroomers call the agle x made b the fictitious plaet X the mea aomal The agle made b the true plaet P is z ad is called the true aomal The differece of these two agles is x z ad is called b

3 3 Astroomers call the agle x made b the fictitious plaet X the mea aomal The agle made b the true plaet P is z ad is called the true aomal The differece of these two agles is x z ad is called b astroomer s the equatio of the ceter x z is zero at the aphelio ad graduall icreases util it reaches a maximum ear b, the it decreases to zero agai at the perihelio VI The maximum of the equatio of the ceter x z We will tr to fid the maximum of x z ad the value of the agle x at which this occurs This maximum value must be a fuctio of the eccetricit of the ellipse Euler otes that Ad iversel, we will have to determie the eccetricit b the biggest equatio This meas that we will observe the maximum of x z, ad from this value, determie the eccetricit of the orbit VII The focus ad the eccetricit o the ellipse Euler otes that this eccetricit equals the distace betwee the two foci of the a ellipse divided b the legth of the major axis I Figure we see that this is a Figure : Features of the ellipse used i E05

4 4 Whe 0 the orbit is a ellipse, whe it is a parabola, ad whe the orbit is a hperbola The distace from the su to the aphelio is a a ad the distace from the su to the perihelio is a a The legth of the semi-mior axis is See Appedix I for derivatios of these features of the ellipse VIII The eccetric aomal, equatios of the ellipse ad Kepler s equatio a Euler itroduces the eccetric aomal which is show i Figure This agle has the propert that the equatio of the ellipse traced b the plaet at P ca be writte parametricall as u a acos ad v bsi Euler gives without derivatio the followig equatios (A6) r a( cos) (A7) (A8) (A9) b a cos cos z cos si si z cos si (A0) ta z cos Equatios (A6) through (A0) are derived below i Appedix I He also lists Kepler s equatio (A) x si, which is derived below i Appedix II IX Begi to fid the equatio of the ceter whe r = a Fiish i sectio XV Euler ow wishes to examie closel the equatio of the ceter x z I particular, he wishes to fid the values of the agles x ad z whe r a From (A6) r a( cos), we see that we eed 90, ad from (A) x si, we have x / Here of course, the eccetricit is a agle measured i radias To covert a agle i secods of arc to radias we must multipl b 80 Sice Euler does arithmetic with commo logarithms we ote 3600 that log Euler uses the value which is

5 log (Probabl Euler s log tables do ot have egative values) From (A8) with 90 we get cos z ad (9) z arccos 90 arcsi X The true aomal z i terms of the eccetric aomal ad the eccetricit Euler has previousl obtaied the relatio 3 z si si si 3 3si (0) si 4 4si si 5 5si si 6 6si 4 5si etc 6 3 See Appedix III for a Mathematica derivatio of this result 0si XI Usig calculus fid whe the maximum of x z occurs i terms of ad See (4) With, fid λ i (5) ad (6) Euler sets the problem: From the eccetricit ad the eccetric aomal, fid the maximum of the equatio? From Kepler s equatio x si we have () dx ( cos) d ad differetiatig (A8) cos cos z we have cos ( cos )( si ) ( cos )( si z dz ( cos ) si ) d ( )si () si zdz d ( cos ) si But b (A9) si z, so () becomes cos

6 6 (3) dz d cos Sice at the maximum of x z, we have dx dz, the from () ad (3) we have cos (3a) ad Thus we have cos cos 4 4 (4) cos Note that this last result gives the exact value of for which that for small eccetricit, cos, ad so 4 Now we let be that small chage i the agle b writig si si / ) cos ad (4) we get (5) (6) si si si si si Thus kowig the eccetricit, we ca calculate x z is a maximum Note ad thus usig from (5) or (6) ad the eccetric aomal from Fiall, the true aomal z ca be foud from (0) or b ivertig a of (A8), (A9) or (A0) XII Determie formulas for λ i terms of the eccetricit Let (5) that si m, ad we ow have cosm It follows at oce from

7 7 cos m si, () si m ad from si cos that () cos cosm cosm si m cosm For small values of the biomial theorem gives us (a) *3 4 *3*7 6 *3*7* 8 4 4*8 4*8* 4*8**6 4 It ow follows from (5) that *3 3 *3*7 5 *3*7 * 7 (3) si 4 4*8 4*8* 4*8**6 Euler also fids (4) cos, without showig the details of series maipulatios XIII Fid the mea aomal directl i terms of at the maximum of the equatio Get (3) through (33) Euler otes that havig foud from the previous sectio, we ca ow fid 90, the we ca fid from Kepler s Equatio (A) x si, ad z from (0) or a of (A8), (A9) or (A0) However, he would ow like to fid x ad z directl from So Euler begis with the problem: Beig give the eccetricit, fid the mea aomal, to which correspods the maximum of the equatio Without showig all the details of series maipulatios Euler arrives at (3) x 90 etc From Kepler s equatio x si, ad we have (3), with ()

8 8 (33) si 4 XIV/ Fid the true aomal z directl i terms of the eccetricit See (4) to (43) Euler ow tries to fid the true aomal z from the eccetricit He defies the ew variable through the equatio (4) z 90 After several series maipulatios which are ot explaied i detail he arrives at (4) etc See Figure 3 which illustrates these variables From (A8) cos z cos cos ad (4) we get si si si Replacig si with () ad simplifig we get (43) si 4 3 ( )

9 9 XV Fid the maximum of the equatio x z directl i terms of the eccetricit See (5) through (53) Euler ow raises the questio Beig give the eccetricit of the plaet s orbit, fid the greatest equatio From (3) ad (4) we get (5) x z cos, which ca be expressed as x z (5) But whe the distace from the plaet to the su is equal to half the major axis, the equatio is (53) x z arcsi 6 40 Thus the maximum of x z surpasses this b a quatit = XVI Give the maximum of the equatio x z, determie the eccetricit This ca ol be doe b umerical guessig

10 0 4 From (A6) r a( cos) ad (3a) cos we have the distace from the su to the plaet at the maximum value of x z is 4 (6) r a (Note that it is less tha a) If the value of x z is called m ad is give, it becomes ver difficult to determie the eccetricit from this Euler states that we must use the equatio ad tr to determie b substitutig umbers for ad usig trial ad error to approximate the result b calculatig values above ad below m I this wa we ca get bouds o a solutio XVII Fid the eccetricit as a series i powers of the maximum m = x z Euler ow cosiders fidig series for the eccetricit i powers of the greatest equatio m x z These will be valuable whe is small So he starts with (5) m ad iverts to get (7) m m m Euler remids the reader that the value obtaied from this equatio must have added to logarithm of the result to covert agles i secods to radias (See sectio XI) The mea aomal x ca the be calculated from (7) x 90 m m m Euler remarks that whe is small ol the first term 5 8 m eed be added to 90 (Euler probabl used usig (3) ad (5) to obtai (7)) XVIII A sample calculatio for the plaet Mercur Fid x z whe the mea aomal is 90 degrees I this sectio Euler does a umerical example of the use of the above results He 797 chooses the plaet Mercur which has a eccetricit of

11 Now log He makes the approximatio b assumig that the maximum of the equatio occurs where the eccetric aomal is 90 degrees ( 0 ) I this case, from (3) we get the mea aomal x 90 Euler writes the result as where it appears that the smbol 3 meas 90 degrees Usig (9) Euler calculates ad fids that Thus x z = , which is earl two miutes less tha the (kow) maximum of the equatio XIX Calculate the maximum of x - z for the plaet Mercur 797 Agai we start with Mercur with the eccetricit To fid the maximum of the equatio Euler begis usig (5) si ad usig logarithms he fids log(si ) Thus ad the eccetric aomal is 90 59' 55' ' From x 90 cos (Kepler s equatio) Euler fids x 04 46' 44' ' To fid the true aomal z Euler uses (43) 4 si ad 3 obtais 8 55' 5 Next Euler adds = to obtai the maximum of the equatio x z = , which does ot differ a secod from the result foud i the tables Euler eds b fidig the distace Mercur is from the su whe the maximum of the equatio occurs He obtais this from (6) 4 r a with 3870 = a XX Euler explais his table The eccetricities are give ever hudredth i the first colum, ad the correspodig agle of the maximum of x z is give i the secod colum The last colum also provides the logarithm of the distace from the plaet to the su, where its equatio is the greatest

12 XXI Euler explais how to use liear iterpolatio to obtai the maximum of the equatio from a give eccetricit Euler uses simple liear iterpolatio for the plaets Earth ad Mars XXII Euler explais how to use liear iterpolatio to obtai the eccetricit whe the maximum of the equatio is give give Euler uses the plaet Mercur for a simple sample calculatio XXIII Fid the maximum of the term cos λ ad metio the value of the eccetricit whe x z = 90 degrees Euler otes that i our equatio x z cos, both λ ad μ icrease as icreases, but this is ot true for the term cos I fact, this term is zero whe 0 ad whe Euler the uses simple calculus to fid whe cos is a maximum ad discovers that it occurs whe the eccetricit is = , ad the actual maximum value is = This last result is etered as the fial lie i Euler s table Euler s Table Euler s table is meat to be used b astroomers who seek the eccetricit of a plaet from observatios of the maximum of x z (equatio of the ceter) For this purpose ol the first two colums of this seve colum table are eeded The

13 3 astroomer would fid the maximum of x z i colum two ad read off the eccetricit from colum oe Appedix I: Features of the ellipse I Figure we see the ellipse with focus at the origi O of the uv-plae We imagie that the su is at poit O ad the plaet is at poit P Followig Euler, we will use the variables: = eccetricit of the ellipse z = true aomal, (usuall the polar agle ) = eccetric aomal, (ofte E is used) a = semi-major axis b = semi-mior axis r = OP (usual polar radius) t = time for plaet to move from A to P T = period of the plaet = mea agular velocit T x t = mea aomal Defiitio: Let O be a fixed poit (focus) ad u d be a fixed lie (directrix) The locus of all poits P such that the ratio of the distace from the focus to the distace from the directrix is a costat (eccetricit) is called a coic sectio Thus we have OP (a costat) DP If 0 the curve is a ellipse If the curve is a parabola If the curve is a hperbola Sice DP d r cos z we have OP r, DP d r cosz ad solvig for r we get (A) r d cosz At the perihelio we have from the defiitio r d r, ad solvig for r we get

14 4 (A) d r perihilio At the aphelio we have r d r so we get (A3) d r aphelio Sice the major axis of our ellipse is a r Thus it follows that d d d perihilio raphelio, d a, ad so our equatio for the ellipse (A) becomes (A4) r a cos z Returig to (A) we ow have r d a a a perihilio This last relatio tells us that the distace from the focus to the ceter of the ellipse is a as show o the figure From Figure we ca ow calculate the distace OU (A5) OU r cos z a acos We ca rewrite (A4) as r r cos z a, ad usig (A5) to elimiate r cos z we get r ( a a cos ) a which simplifies to (A6) r a( cos)

15 5 From this we ifer that whe /, the r a It follows the from Figure that we have a right triagle OCb with legs a, ad b ad hpoteuse a It follows that the semi mior axis is give b (A7) b a Notice also that OU a a cos cos z, OP r ad replacig r b (6) we get (A8) Also cos cos z cos PU bsi si z, OP r ad usig (A6) ad (A7) this becomes (A9) si z si cos From (A8) ad (A9) we get si (A0) ta z cos Equatios (A6) through (A0) appear i the last few lies of sectio VIII of E05 Appedix II: Derivatio of Kepler s equatio Kepler s equatio is (A) x si where x t is the mea aomal This equatio is the mathematical statemet of Kepler s secod law of plaetar motio: The plaet sweeps out equal areas i equal times Kepler s equatio (A) appears ear the ed of sectio VIII (To see Kepler s origial derivatio, which is much like ours see to [])

16 6 Referrig to Figure we see that this traslates to (A) Now Area OAP Area of full ellipse t T Area OAP = a b Area OAQ Area OAP = a b (Area Triagle OCQ + Area Sector CAQ) b Area OAP = ( ( a) asi + a ) a (A3) Area OAP = ab si Usig (A3) to simplif (A) we get ab si ab si t T t T t x This is Kepler s equatio Appedix III: Mathematica verificatio of Euler s equatio relatig the true aomal z to the eccetric aomal i powers of the eccetricit Euler has previousl obtaied the relatio 3 z si si si 3 3si si 4 4si si 5 5si 3 0si si 6 6si 4 5si etc 6 3 We used Mathematica to verif this result Startig with (A8) we ca write cos z cos cos Euler s relatio is the Talor s series i powers of of this expressio The Mathematica code is z[_,_]:=arccos[(+cos[])/(+*cos[])]

17 7 Series[z[,],{,0,5}] ArcCos[Cos[]]- Si +/ Cos[] Si -/6 ((+Cos[ ]) Si ) 3 +/6 (5 Cos[]+Cos[3 ]) Si 4 -/40 ((8+6 Cos[ ]+Cos[4 ]) Si ) 5 +O[] 6 This simplifies to z 6 si( ) 5cos cos3 cos si si ( cos)si 6cos cos4 3 si 5 O 6 Subtractig the above from Kepler s equatio x si we get 3 x z si( ) cos si ( cos)si cos cos3 si 8 6cos cos4 si O 6 40 Differetiatig to fid the maximum we get 6 d( x d 6 ( z) 5si cos( ) 3si 3 ( si si si () cos ) cos cos ) 4 40 ( 6 ( si si )si 4si 4 si ( cos)cos () cos ) 5 3 O 6 If we set / we get d( x z) d 8 / so / is ot at the maximum of x z 4 O 6, but is close to the maximum Refereces [] Kepler, Johaes, Epitome of Coperica Astroom ad Harmoies of the World, (Traslated b Charles Gle Wallis), Prometheus Books, New York, 995, p 5

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Synopsis of Euler s paper. E Memoire sur la plus grande equation des planetes. (Memoir on the Maximum value of an Equation of the Planets)

Synopsis of Euler s paper. E Memoire sur la plus grande equation des planetes. (Memoir on the Maximum value of an Equation of the Planets) 1 Syopsis of Euler s paper E105 -- Memoire sur la plus grade equatio des plaetes (Memoir o the Maximum value of a Equatio of the Plaets) Compiled by Thomas J Osler ad Jase Adrew Scaramazza Mathematics