On the Motion of Planets and the Determination of Orbits (an English translation of De Motu Planetarum et Orbitarum Determinatione) Leonhard Euler All footnotes are comments by the translator, Patrick Headley1. 1. At this time it is agreed that planets move in ellipses, with the sun located at one of the foci, and that the motion is arranged so that times are proportional to areas described around the sun. Two questions about the motion of the planets arise, one asking for properties of the ellipse, namely, the position of an apsis and the eccentricity, and the other for the motion of the planet itself in its orbit. Both questions are studied here, and, as far as difficulties of calculation permit, I will try to resolve them. 2. In fact I will first take the orbit of the planet to be known, and I will direct Fig. 1 my attention to determining the motion of the planet in it. Therefore let ADB be half the orbit of a planet at P whose aphelion is at A and perihelion at B, with the sun at focus S of the ellipse (see footnote2). Furthermore, let C be the center of the orbit, and let the semiaxis AC or BC equal a, and the distance from focus S 3 to centerpC, or eccentricity CS (see footnote ), equal b, so that the minor semiaxis CD is a2 − b2. We now assume the planet has arrived at P from aphelion A, and then progressed in a brief interval of time to p. From P and p lines are 1Mathematics Department, Gannon University, 109 University Square, Erie, PA 16541, [email protected] 2The figures accompanying this article can be found on page 4 of eulerarchive.maa.org/publications/journals/figures CASP07.pdf 3In the modern definition, the eccentricity would be the ratio of CS to AC. 1 drawn top S, as well as perpendiculars to the axis AB. Letting CQ = r, then 2 2 2 2 (a −b )(a −r ) br PQ = a and PS = a + a . 3. In this situation, the angle ASP represents the true anomaly (see footnote4), which will be called z. The mean anomaly, however, is proportional to the time in which the planet moves from A to P , or to the area ASP . Therefore, the area ADB will be to the area ASP as two right angles are to the mean anomaly. We now consider a circle of radius 1, with an arc equal to the true anomaly z. If we wish to find the mean anomaly, equal to the arc x, the ratio of area ADB to two equal right angles, orp to twice the area of the semicircle,p will be the same as that of AC · CD to 2, or a a2 − b2 to 2. Therefore a a2 − b2 : 2 = Area ASP : x, so that x = 2 Areap ASP. a a2−b2 4. With the true anomaly equal to angle ASP , the differential dz will be equal to P Sp. The angle P Sp is equal to twice the area P Sp divided by the square of 2 Area PSp 2a2P Sp PS; that is to say, dz = br 2 = (a2+br)2 . But, from the equation shown ear- (a+ a ) lier, dx = p2 P Sp . It remains therefore to express the area P Sp in a convenient a a2−b2 form, and it will be done from considerationp of the entire area.p For, indeed, Area (b+r) (a2−b2)(a2−r2) dr (a2−b2)(a2−r2) ASP = PQ·QS + R PQ · Qq = − R , whose 2 p 2a a −dr(a2+br) a2−b2 differential is p , which therefore is equal to P Sp. From this there- 2a a2−r2 −dr(a2+br) p fore the differential of the mean anomaly is dx = 2 2 2 , and the differential p a a −r 2 2 of the true anomaly is dz = −a dr pa −b . (a2+br) a2−r2 5. A relation between the mean anomaly and the true anomaly is contained in these two equations. To express this it is necessary to integrate both equations so that in the end an equation between z and x can be derived. Concerning the p−dr pbr dr first equation, it transforms at once to dx = 2 2 − 2 2 2 , whose integral is p p a −r a a −r a2−r2 b 2 2 x = A: a + a2 a − r , where A signifies the arc of a circle whose sine is the subsequent quantity, the whole sine (see footnote5) being 1. Therefore let this 4In the modern definitions, the anomalies are measured from the perihelion and not the aphe- lion. 5The whole sine, or sinus totus, is a value chosen as a radius, so that Euler’s sine is what would now be considered the radius multiplied by the sine. The tables that Euler used had varying choices for the whole sine. The logarithms of sine values that Euler uses in this article assume a whole 2 p a2−r2 bs sine a equal s, so that x = A:s + a . p 2 2 6. The equation of the other differential is dz = −a dr pa −b , which is com- (a2+br) a2−r2 pletely integrable, but can also be integrated in several ways using series. The chosen method will be examined before proceeding. It yields a series in which the powers of b increase in the numerator, so that for small eccentricities it suf- fices to use two or three initial terms. p 2 2 1·b2 1·1·b4 7. Thus, I first convert a − b into a series, which will be a − 2a − 2·4·a3 − 1·1·3·b6 b2 b4 b6 a 1 br b2r2 b3r3 5 etc. = a− − 3 − 5 etc. Next, 2 = − 3 + 5 − 7 + etc. Thus, 2·4·6·a 2a 8a 16a a +br ap a a a a a2−b2 br b2(a2−2r2) these two series multiplied term-by-term will give a2+br = 1− a2 − 2a4 + b3r(a2−2r2) b4(a4+4a2r2−8r4) b5r(a4+4a2r2−8r4) 2a6 − 8a8 + 8a10 − etc. If now each of the terms of this series is preceded by p−dr , the differential dz of the true anomaly will be a2−r2 obtained. All of the terms after the first will be completely integrable, so it is found p p p p p a2−r2 b a2−r2 b2r a2−r2 b3(a2+2r2) a2−r2 b4r(a2+2r2) a2−r2 that z = A: a − a2 + 2a4 − 6a6 + 8a8 − etc. p a2−r2 8. The arc or angle whose sine is a will be called V . Then x = b V + a sin V , and z will be determined from V , its sine, and sines of multiples b b2 b3 of V as follows: z = V − a sin V + 4a2 sin 2V − 12a3 (sin 3V + 3 sin V ) + b4 b5 b6 32a4 (sin 4V + 4 sin 2V ) − 80a5 (sin 5V + 5 sin 3V + 10 sin V ) + 192a6 (sin 6V + b7 6 sin 4V + 15 sin 2V ) − 448a7 (sin 7V + 7 sin 5V + 21 sin 3V + 35 sin V ) + etc. (see footnote6) The rule for this series extends easily, for the denominators consist of the numbers in this series: 1 · 1, 2 · 2, 3 · 4, 4 · 8, 5 · 16, 6 · 32, etc. 9. Thus, the true anomaly z will be most conveniently determined from the value x of the mean anomaly if, first, the angle V is determined from angle x b through the equation x = V + a sin V , and this then is substituted into the other equation producing the true anomaly z. It is difficult, however, to see how to determine V from x with this equation, since the equation is transcendental, and sine of 1010, but the trigonometric values assume a whole sine of 107. As a result, Euler’s use of the decimal point (actually a comma in the original text) seems somewhat erratic to the modern reader. For more information on the sinus totus see Gonzalez-Velasco,´ Enrique A. Journey through Mathematics. New York, Springer, 2011. 6Euler actually uses the symbol R for sin in this and subsequent formulas, in spite of using it for integration earlier in the text. 3 therefore V cannot entirely be represented algebraically using x. It is therefore necessary, in order for V to be determined accurately and with as little work as possible, that it is presented in the following way which seems to me to be most b b b convenient, namely, that V = x− a sin(x− a sin(x− a sin(x− etc. It will be very a easy to find the angle V from this, since, after log sin x is found, from this log b is subtracted, and then from this the logarithm 6:4637261 is subtracted; this number is required for logarithms of natural numbers so that the corresponding number will give the angle in minutes (see footnote7). Then this angle is subtracted from the mean anomaly x, and the logarithm of the sine of the resulting angle is found, a from which log b and 6:4637261 are subtracted, and the number corresponding to the resulting logarithm will give the number of minutes to be subtracted from x. If the value is to be estimated correctly, the resulting angle is then treated in the same manner until at last it is neither increased nor decreased any further. Then that angle will be the true value of V itself. a 10. Angle V having been found in this way, both log b and 6:4637261 are subtracted from the logarithm of its sine.