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 Departmet Rowa Uiversity Glassboro, NJ 0808 Osler@rowa.edu Preface The followig summary of E 105 was costructed by abbreviatig the collectio of Notes. Thus, there is cosiderable repetitio i these two items. We hope that the reader ca profit by readig this syopsis before tacklig Euler s paper itself. I. Plaetary Motio as viewed from the earth vs the su ` Euler discusses the fact that plaets observed from the earth exhibit a very irregular motio. I geeral, they 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 jourey. However, if we observe the plaet from the stad poit of a observer o the su, this retrograde motio will ot occur, ad oly a west to east path of the plaet is see. II. The aphelio ad the perihelio From the su, (poit O i figure 1) 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 say 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 plaetary motio The plaet s speed is slowest at the aphelio ad fastest at the perihelio. The plaet obeys 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 The more elliptic the orbit is, the greater is this variatio i speed. If the orbit were a circle, the speed would always be costat ad equal agles would be swept out i equal times. We imagie a fictitious compaio plaet (poit X i Figure 1) 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 says: After these two plaets have passed by the aphelio, the false plaet will appear to go faster tha the true ad the real plaet will imperceptibly 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 they rejoi agai at the Aphelio. Figure 1: The plaet P as observed from the su at O,
3 V. The mea aomaly x, the true aomaly z ad the equatio of the ceter Astroomers call the agle x made by the fictitious plaet X the mea aomaly. The agle made by the true plaet P is z ad is called the true aomaly. The differece of these two agles is x z ad is called by astroomer s the equatio of the ceter. x z is zero at the aphelio ad gradually 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 try 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 eccetricity of the ellipse. Euler otes that Ad iversely, we will have to determie the eccetricity by the biggest equatio. This meas that we will observe the maximum of x z, ad from this value, determie the eccetricity of the orbit. This is the mai purpose of this paper, to determie the eccetricity of a plaet by observatios that have determied the maximum of x z. The followig sequece of five figures shows the progressio of the plaet P ad the fictitious plaet X as they move from the aphelio to the perihelio. Figure a: x z = 0 at aphelio
4 Figure b: x z growig Figure c x z is a maximum Figure d: x z shrikig
5 Figure e: x z = 0 at the perihelio VII. The focus ad the eccetricity of the ellipse Euler otes that this eccetricity equals the distace betwee the two foci of the a ellipse divided by the legth of the major axis. I Figure 3 we see that this is. a Figure 3: Features of the ellipse used i E105
6 Whe 0 1 the orbit is a ellipse, whe 1 it is a parabola, ad whe 1 the orbit is a hyperbola. 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 VIII. The eccetric aomaly y, equatios of the ellipse ad Kepler s equatio a 1. Euler itroduces the eccetric aomaly y which is show i Figure. This agle y has the property that the equatio of the ellipse traced by the plaet at P ca be writte parametrically as u a acos y ad v bsi y. Euler gives without derivatio the followig equatios (A6) r a( 1 cos y). (A7) (A8) (A9) b a 1. cos y cos z. 1 cos y 1 si y si z 1 cos y 1 si y (A10) ta z cos y Equatios (A6) through (A10) are derived below i Appedix I. He also lists Kepler s equatio (A11) x y si y, 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 closely 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( 1 cos y), we see that we eed y 90, ad from (A11) x y si y, we have x y /. From (A8) with y 90 we get cos z ad (9.1) z arccos 90 arcsi. X. The true aomaly z i terms of the eccetric aomaly y ad the eccetricity
7 Euler has previously obtaied the relatio (10.1) z y si y 1 si y 4 1 4 si 4y 4 8 4si y 1 6 si 6y 6 3 6si 4y 1 3 si 3y 3 4 3si y 1 5 si 5y 5 16 5si 3y 15si y etc 10si y XI. Usig calculus fid whe the maximum of x z occurs i terms of ad y. See (11.4). With y, fid λ i (11.5) ad (11.6). Euler sets the problem: From the eccetricity ad the eccetric aomaly y, fid the maximum of the equatio? Startig with Kepler s equatio simple calculus to fid x y si y ad (A8) cos z 1 cos y, Euler uses cos y (11.3a) ad 1 cos y 1 1 1 (11.4) cos y. Note that this last result gives the exact value of y for which that for small eccetricity, cos y, ad so y. 4 Now we let be that small chage i the agle by writig si si y / ) cos y ad (11.4) we get (11.5) si 1 1 Thus kowig the eccetricity, we ca calculate x z is a maximum. Note y ad thus usig from (11.5)) ad the eccetric aomaly from y. Fially, the true aomaly z ca be foud from (10.1) or by ivertig ay of (A8), (A9) or (A10). XII. Determie more formulas for λ i terms of the eccetricity.
8 Euler fids 1 1*3 3 1*3*7 5 1*3* 7 *11 7 (1.3) si 4 4*8 4*8*1 4*8*1 *16 ad 1 49 4 133 6 (1.4) cos 1. 3 048 65536 XIII. Fid the mea aomaly y directly i terms of at the maximum of the equatio. Get (13.1) through (13.3). Euler otes that havig foud from the previous sectio, we ca ow fid y 90, the we ca fid from Kepler s Equatio (A11) x y si y, ad z from (10.1) or ay of (A8), (A9) or (A10). However, he would ow like to fid x ad z directly from. So Euler begis with the problem: Beig give the eccetricity, fid the mea aomaly, to which correspods the maximum of the equatio. Without showig all the details of series maipulatios Euler arrives at 5 5 3 1383 5 (13.1) x 90 etc. 4 384 40960. Aother fidig is (13.), with (13.3) si 1 1. XIV/ Fid the true aomaly z directly i terms of the eccetricity. See (14.1) to (14.3). Euler ow tries to fid the true aomaly z from the eccetricity. He defies the ew variable through the equatio (14.1) z 90. After several series maipulatios which are ot explaied i detail he arrives at (14.) 3 1 3 3409 5 etc. 4 18 40960. Euler also has (14.3) si 1 1 3 (1 ).
9 See Figure 4 which illustrates these variables. Figure 4 XV. Fid the maximum of the equatio x z directly i terms of the eccetricity. See (15.1) through (15.3). Euler ow raises the questio Beig give the eccetricity of the plaet s orbit, fid the greatest equatio. From (13.) ad (14.1) we get (15.1) x z cos, which ca be expressed as 11 3 599 5 x z. (15.) 48 510 But whe the distace from the plaet to the su is equal to half the major axis, the equatio is 1 3 3 5 (15.3) x z arcsi 6 40 XVI. Give the maximum of the equatio x z, determie the eccetricity. This ca oly be doe by umerical guessig.
10 From (A6) r a( 1 cos y) ad (11.3a) 1 cos y 1 we have the distace from the su to the plaet at the maximum value of x z is (16.1) r a 1. (Note that it is less tha a.) If the value of x z is called m ad is give, it becomes very difficult to determie the eccetricity from this. Euler states that we must use the equatio ad try to determie by substitutig umbers for ad usig trial ad error to approximate the result by calculatig values above ad below m. I this way we ca get bouds o a solutio. XVII. Fid the eccetricity as a series i powers of the maximum m = x z. Euler ow cosiders fidig series for the eccetricity i powers of the greatest equatio m x z. These will be valuable whe is small. So he starts with (15.) m ad iverts to get 11 48 599 510 3 5 1 11 3 587 5 (17.1) m m m. 16 768 15 Euler remids the reader that the value obtaied from this equatio must have 4.6855749 added to logarithm of the result to covert agles i secods to radias. (See sectio XI.) The mea aomaly x ca the be calculated from 5 5 3 1 5 (17.) x 90 m m m. 9 9 8 3 5 Euler remarks that whe is small oly the first term 5 m 8 eed be added to 90. XVIII. A sample calculatio for the plaet Mercury. Fid x z whe the mea aomaly y is 90 degrees. I this sectio Euler does a umerical example of the use of the above results. He 797 chooses the plaet Mercury which has a eccetricity of 0.0589. 3871 Now log 0.686364849 9.31363515 10. He makes the approximatio by assumig that the maximum of the equatio occurs where the eccetric aomaly y is
11 90 degrees. ( 0.) I this case, from (13.) we get the mea aomaly x 90. Euler writes the result as where it appears that the symbol 3 meas 90 degrees. Usig (9.1) Euler calculates ad fids that. Thus x z = 3.40.4, which is early two miutes less tha the (kow) maximum of the equatio. XIX. Calculate the maximum of x - z for the plaet Mercury 797 Agai we start with Mercury with the eccetricity 0.0589. To fid 3871 1 1 the maximum of the equatio Euler begis usig (11.5) si ad usig logarithms he fids log(si ) 8. 718609. Thus ad the eccetric aomaly is y 90 59' 55' '. From x 90 cos (Kepler s equatio) Euler fids x 104 46' 44' '. To fid the true aomaly z Euler uses (14.3) 1 1 si ad 3 obtais 8 55' 5. Next Euler adds = 14.46.44 to obtai the maximum of the equatio x z = 3.4.36, which does ot differ a secod from the result foud i the tables. Euler eds by fidig the distace Mercury is from the su whe the maximum of the equatio occurs. He obtais this from (16.1) r a 1 with 38710 = a. XX. Euler explais his table The eccetricities are give every 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.
1 XXI. Euler explais how to use liear iterpolatio to obtai the maximum of the equatio from a give eccetricity. Euler uses simple liear iterpolatio for the plaets Earth ad Mars. XXII. Euler explais how to use liear iterpolatio to obtai the eccetricity whe the maximum of the equatio is give give. Euler uses the plaet Mercury for a sim-le sample calculatio. XXIII. Fid the maximum of the term cos λ ad metio the value of the eccetricity 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 1. Euler the uses simple calculus to fid whe cos is a maximum ad discovers that it occurs whe the eccetricity is = 0.9375645, ad the actual maximum value is = 48.18.10.40. This last result is etered as the fial lie i Euler s table.
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