Re-establishing Kepler s first two laws for. planets from the non-stationary Earth

Transcription

1 Re-establishing Kepler s first two laws for planets from the non-stationary Earth W Y Hsiang, H C Chang, H Yao 3 an P S Lee 3 Department of Mathematics, University of California, Berkeley, CA 947, USA. Department of Mathematics, National Taiwan University, Taipei, Taiwan 6, ROC an 3 Department of Physics, National Taiwan Normal University, Taipei, Taiwan 6, ROC yao@phy.ntnu.eu.tw PACS numbers:.4.e,.65.+g, 45.5.Pk, 95..Ce, 95..Eg Abstract The Earth itself is not stationary but keeps revolving, an its motion further satisfies the law of equal area accoring to the heliocentric octrine. That satisfaction can be use to construct the mathematical relationships between the planet Sun an Earth Sun istances. The law of equal area for planets can hence be re-establishe naturally from the moving Earth using the observe angular spee of a planet over the Sun. Furthermore, for the perioicity of a planet to the Sun, the istance from each planet to the Sun may be expresse as an angular perioic function. By coorinating with the observe ata, this perioic istance function epicts an exact elliptical path. Here, we apply relatively simple mathematical skills to illustrate the invariant forms of planetary motions an inicate the key factors use to analyze the motions in complicate planetary systems.. Introuction Mathematical moels were wiely employe to escribe natural phenomena uring Plato s era ( BC). Spherical geometry was also applie to astronomy. Geometry is a part of cosmology, an its theory represents a realization of the structure of the entire universe. Therefore, a knowlege of geometry is crucial for unerstaning astronomy []. Almagest, the cosmology literature written by Ptolemy (85 65 AD), firmly establishe the Greek trigonometry theory that was in place for more than a thousan years. He took the concept of epicycle an eferent to epict the motion of planets, which was wiely accepte by the public as absolute truth at that time. Until his pursuit of the notion of mathematical harmony an symmetry as perfection, Nicolaus Copernicus ( ) strongly suspecte human manipulation an complexity in the epicycle, eferent, an equant. The heliocentric octrine was establishe an le to the revolution of astronomy [, 3]. Johannes Kepler (57 63) was eeply enlightene by Copernicus heliocentric theory an fully supporte the octrine throughout his life. He further propose three laws for the planets, which escribe

2 the Sun as the center of the system, an establishe moern astronomy theory. His theories provie a concrete basis for Isaac Newton s (64 77) ynamics. Kepler s laws of equal areas an ellipses were publishe in New Astronomy in 69 [4]. The book s content is extremely obscure an generally presents complicate geometry rather than simple mathematical forms. Most researchers have reiscovere an obtaine Kepler s laws of planetary motion, either from Newton s laws of motion an universal gravitation [5-7], or from the principles of the conservation of energy an angular momentum [8-]. To emphasize Kepler s important influence, others erive, algebraically or graphically, the inverse-square law of gravitation from Kepler s first two laws [-3]. However, very few articles have iscusse how Kepler originally erive his laws of planetary motion [4-7]. In Copernicus an Kepler s astronomical system, the Earth is no longer stationary, which makes the etermination of planet position more complicate. Fortunately, the establishment of the rules for the motion of the Earth, which are the laws of equal areas an ellipses [4, 8], present the Earth as the starting point for epicting the positions of other planets. These rules have become the basis for iscovering the laws for other planets. This process, in fact, was reflecte in New Astronomy, where Kepler ivie its contents into two main parts or two inequalities. Each part treate one irregularity or non-circularity for the Earth an other planets, respectively. To reveal the funamental spirit of Kepler s first two laws, the present stuy uses simple, innovative approaches, such as trigonometric functions an the law of sines, to re-establish the laws of equal areas an ellipses for planets other than the Earth. It clarifies an simplifies the evelopment of planet laws, which were originally ifficult to interpret, enabling researchers to unerstan the intimate relations an analytical methos among geometry, astronomy, an physics. Moreover, this stuy allows researchers to practically realize the plentiful insights in major scientific evelopments, immerse in the joy of reiscovering scientific theories by previous great scientists, an cultivate the extensive an eep scientific prospects of these theories.. The Law of Equal Areas for Mars The perio of Mars orbiting aroun the Sun is approximately 687 ays, which inicates that Mars will return back to the same position after 687 ays; the perio of the Earth orbiting aroun the Sun is 365 ays. The perios of these two planets revolving aroun the Sun o not have a common value an are mutually prime numbers. This suggests the corresponing Earth position is ifferent for every Martian year, or when Mars returns to its original position. As inicate by figure, if S represents the Sun, M is Mars or the position of Mars on the next Martian year, an E i an E j are the corresponing positions of the Earth before an after one Martian year, respectively. Therefore, a square (SE ime j) can be forme by the Sun, Mars, an two other positions of the Earth. The SE i an SE j lines represent the istance between the Sun an the ifferent positions of the Earth, r i an r j, respectively; this is known as the Earth Sun istance. The SM line represents the istance between the Sun an Mars,, known as the Mars Sun istance.

3 M αi αj Ei μi μj Ej r i θ r j S Figure. The illustration of the Sun (S), Mars (M), an two corresponing positions of the Earth, (E i) an (E j), in an ajacent Martian year. () Mars Sun Distance We ranomly selecte the ate of E i as 5 a.m. on May 3, 95, an the corresponing ate of position E j as 4 a.m. on March 3, 95. The time ifference between E i an E j is a Martian year. The angle between S an M from the E i position is SE im = μ i. From the observation an astronomical ata from the Multiyear Interactive Computer Almanac (MICA) software [9], one foun that Mars ecliptic longitue was 7.557, an the Sun s longitue was 5.9, which inicate that μ i = =.656. Similarly, the angle between S an M at E j was SE jm = μ j = = 4.4. The corresponing Sun s longitue to the Earth is the projection point of the Sun on a celestial sphere an is marke as an ecliptic longitue while observing the Sun from the Earth. The longitue is set as while observing the Sun from the Earth on the vernal equinox an the Sun s longitue is set as 9 on the summer solstice as shown in figure. The longitues of the Sun an any other planets can be etermine from the Earth. π+θ S E θ spring equinox Figure. The Sun an planets can be irectly or inirectly observe from the Earth at any moment.

4 The E ise j = θ in Fig. is observable because the observe longitue of S as seem from the Earth at E i an E j were 5.9 an 9.473, respectively. Thus, E ise j = θ = = () Hence, θ can be confirme. The 3 observable parameters, μ i, μ j, an θ, together with the law of equal areas for the Earth, will be applie hereafter to calculate the Mars Sun istance. If E jms = α j, E ims = α i because the total internal angles are 36 for a quarilateral; thus, μ i + μ j + θ + (α i + α j) = 36, α i = 36 μ i μ j θ α j. Let β = 36 μ i μ j θ, β is an observable value, an α i = β α j. () On the other han, the quarilateral SE ime j can be consiere as the combination of SE im an SE jm, where SM is a common sie. By applying the law of sines, sin i = r i, i sin j = r j j, the Mars Sun istance can be expresse as = sin j rj. j (3) an the ratio between the sines of α i an α j may be written as follows: i j = r r i j sin i, sin j (4) If the law of equal areas for the Earth was establishe as mentione in the first half of New Astronomy [4], the following relationship will hol [8] r i r j = ω j ω i. This relation inicates that the ratios of the Earth Sun istances, which were originally ifficult to measure, can be calculate by the angular velocities ω i an ω j of the Earth at ifferent positions, which may be measure from aily observations. Therefore, r i/r j can be obtaine. Because μ i an μ j are also known, the ratio in (4) can be set as an observable value k. Then, sinα i = k sinα j. (5) Replacing (5) with (), we have sinβcosα j cosβsinα j = k sinα j.

5 Diviing by cosα j on both sies of the above equation, α j can be foun as follows: α j = sin tan. k cos (6) Hence, α j turns out to be an observable value because β an k are all observable. Combining (3) an (6), the Mars Sun istance can be irectly represente by the Earth Sun istance r j. For 5 ranomly selecte observation ates, the Mars Sun istances from (6) an (3) are liste in table. The Earth Sun istance is set as r j = r = on March 3, 95 (figure 3). Table. Five ifferent ranomly selecte ates use to calculate the Mars Sun istance. (The quote an non-quote ates have a ifference of one Martian year.) Time μ i μ j θ β k α j r j 3 May95 (3 March 95) June 95 (9 May 954) 5 August 954 ( July 956) November 956 (9 September 958) 7 January959 (4 November 96) M Ej r j r Ei i r j3 Ej3 r i3 3 S Ei r i r Ej M Ei3 M3 Figure 3. The Earth Sun istance r can be use to represent the Mars Sun istance. (r j = r = is set to be a normalize value.)

6 () Angular Spee of Mars ω The angular spee of Mars revolving aroun the Sun represents the angular change of Mars with respect to the Sun within a certain perio of time, such as within one ay. This value cannot be irectly achieve by observation; the inirect relations with observable values must be etermine. M M Ei Ei φ a b Ej c Ej S Figure 4. Two quarilaterals SE im E j an SE im E j are forme by the Sun, Earth, an Mar within two ays. The angle swept by Mars within one ay from M to M is φ = b + c a. Two quarilaterals SE im E j an SE im E j are forme by the Sun, the Earth an Mars within two ays, as shown in figure 4. The value of aily angular spee of Mars with respect to the Sun is the value of angle φ swept by Mars moving from M to M. The angles SE im, SE jm, an E ise j in quarilateral SE im E j enote by μ i, μ j, an θ in figure are also observable. The angle E jm S can further be approache from (6) like α j in figure. As for the SE jm, using the relation of interior angles, as shown in table, we obtaine M SE j = a = 8 SE jm SM E j = = Similarly, SE jm forme by the Sun, the Earth, an Mars on the secon ay gave M SE j = b = 8 SE jm SM E j = = The angle E jse j forme by the Sun to E j an the Sun to E j is similar to θ in figure. It coul be obtaine from () as follows: E jse j = c = =.988. Therefore, the angles swept by Mars with respect to the Sun within one ay was φ = b + c a = =.469. Furthermore, its value was the same as that of angular spee ω of Mars on that ay. Table lists the calculate angular velocities ω per ay of Mars on five ifferent ates from table.

7 Table : The calculate angular spee ω of Mars on five ifferent ates from the observe ata shown in table. Time ω 3 May June August November January (3) The Law of Equal Areas for Planets The law of equal areas for a planet inicates that the line joining a planet an the Sun sweeps out an equal area in the same perios of time, as shown in figure 5, i.e., A. S θ A E s E Figure 5. The law of equal areas for a planet. Thus, the area velocity is A = t t =, where ω is the angular spee of a planet aroun the Sun. Hence, to prove the law of equal areas for a planet, it is necessary to only show that the prouct of the square of the istance from a ifferent planet to the Sun an the corresponing angular spee of that planet is a constant. Namely, i ω i = j ω j. (7) This metho is the same as that employe to inspect the equivalence of j / i an ω i/ω j. Combining the Mars Sun istances an angular velocities of Mars on ifferent ates from tables an, the corresponing values of the ratios of j / i an ω i/ω j can be obtaine, as shown in table 3, where the reference ate is set as May 3, 95.

8 Table 3. The ratios of the square of the Mars Sun istance j / i an the corresponing ratios of angular velocities ω i/ω j on five ifferent ates obtaine from tables an. Time ω j / i ω i/ω j 3 May June August November January The ratios of j / i an ω i/ω j for Mars are almost ientical to the ifference of less than % from the last two columns in table 3, which inicates that Mars strictly obeys Kepler s law of equal areas from the acknowlege astronomical ata. This is an exciting an unsurprising result. 3. The Law of Ellipses for Mars From the perspective of analytical geometry, the relationships between the Cartesian coorinates (x, y) an polar coorinates (r, ) for an ellipse with one of the foci locate at ( c, ), as shown in figure 6, are x rcos c, y rsin. (8) (x, y)=(r, ) (-c, ) Figure 6. The relationships between the Cartesian coorinates (x, y) an polar coorinates (r, ) for an ellipse. The equation of an ellipse in Cartesian coorinates is or x a b y (9) b x a y a b.

9 Substituting (9) with (8) gives r( a ccos ) b ; r( a ccos ) b. Taking the positive value of r, a ccos a ( ecos ), r b b where e = c/a is the eccentricity [9]. For general situations, where along the x-axis, the equation of the ellipse in polar coorinates can then be expresse as a [ ecos( )] c c cos c sin, () r b where or c c a e / b 4 e b c c a c c c. () Therefore, () has the same form of ellipse as that of (9). From the other sie of the perspective, because the motion of Mars aroun the Sun is perioic, the istance function ( ) from a planet to the Sun, or its reciprocal /( ), can also be escribe as a perio function of. That is, it can be expresse as an infinite Fourier series of sines an cosines with ifferent multiple angles as follows []: a a cos n b sin n. n n n In the ieal case, this function can be approximate by a single perio of the trigonometric functions: a a cos b sin. () That is, this simplifie perioic () is equivalent to the equation of the ellipse in polar coorinates as shown in (). To etermine the 3 unknown a, a, an b as shown in (), three sets of ata are require to set up simultaneous linear equations with 3 unknowns. After solving these sets of equations, the equation for the ellipse an its corresponing eccentricity can be obtaine. The law of ellipses for a planet will be spontaneously reveale. The position of Mars M on May 3, 95, is now selecte as a reference point (figure 7; table ). In SE jm, the angle SE jm = μ j is observable, an SM E j = α j is calculable, which can be achieve from

10 (6) an is liste in table. Therefore, θ j = 8 μ j α j = = Similarly, θ j = = 8.85 from the observe μ j an calculate α j with respect to Mars M on June, 95, from table. Furthermore, the angle swept by the Earth from E j to E j was E jse j = by the two observable angular positions of the Earth to the Sun. Finally, the angle swept by Mars from M to M was obtaine as M SM = ψ = E jse j + θ j θ j = = 4.3, where the line connecting M to S was set to be the horizontal axis. The angles ψ swept by Mars on August 5, 954, an November, 956, shown in table coul also be etermine in a similar manner (table 4). M E i E j θ j ψ E i M S θ j E j Figure 7. The angular change (ψ) of Mars to the Sun on the two Mars opposition Table 4. The angle ψ swept by Mars moving from the reference position, which was selecte on May 3, 95, to the other three positions shown in table 3. Time ψ 3 May 95. June August November By replacing () with the Mars Sun istances an the corresponing angles ψ swept by Mars on Jun, 95, August 5, 954, an November, 956, respectively, we have a a cos b,

11 a a cos b, 3 a a cos 3 b 3. Solving the simultaneous linear equations in 3 unknowns, we obtain a =. 66, a =.4, b =.47. Thus, the perioic equation of the reciprocal of the Mars Sun istance is.66.4cos ψ +.47sin ψ. (3) To emonstrate the generality of the above perioic equation built by 4 Mars positions, we ranomly selecte 5 aitional ates an calculate the corresponing angles ψ swept by Mars, as shown in the first two columns of table 5. The Mars Sun istances ʹ were estimate from perioic (3). By comparing ʹ with, which were calculate by the Mars Sun istance (3), one coul etermine that ʹ an were almost equivalent. The valiity of perioic equation / of ψ in (3) can hence be asserte. Table 5. Five ranomly selecte ates comparing the Mars Sun istance ʹ from perioic (3) with the Mars Sun istance from (3) by the law of equal areas for the Earth. Time ψ ( -)/ (%) 7 January February March April June () or (3) has the same form as (). Therefore, the elliptical motion of Mars revolving aroun the Sun as one of the foci can be verifie, an the eccentricity of Mars can also be obtaine by () as e= c c a b.4.47 c a The result is exactly that of the well-known eccentricity of Mars,.93, thus again confirming the law of ellipses for Mars.

12 This concise metho can also be applie to the other 4 planets incluing Jupiter, Saturn, Mercury, an Venus to etermine the laws of equal areas an ellipses. Here we propose only Jupiter as an example to escribe its conformity. 4. The Laws of Planet for Jupiter () The Law of Equal Areas Referring to figures an 3 an combining (3) an (6), the Jupiter Sun istance can also be represente by the Earth Sun istance r j, where the Earth Sun istance is set to be r j = r = on August 6, 96. For the other 4 ranomly selecte observation ates, the calculate Jupiter Sun istances from (3) an (6) are liste in table 6. Table 6. Five ranomly selecte ates use to calculate the Jupiter Sun istance. (Each ate has one Jupiter orbital perio ifference.) Time μ i μ j θ β k α j r j 6 August October December February April Applying the relationships as shown in figure 4, one may obtain the angular spee ω of Jupiter at the ifferent ates shown in table 6. By verifying whether the prouct ω is a constant as shown in (7), or whether the ientity j / i = ω i/ω j hols (table 7), we can establish the law of equal areas for Jupiter. From table 7, the law of equal areas for Jupiter can be certifie.

13 Table 7. The ratios of the square of Jupiter Sun istance j / i an the corresponing ratios of angular spee ω i/ω j on five ranomly selecte ates are nearly ientical, which affirms the law of equal areas for Jupiter. Time ω j / i ω i/ω j 6 August October December February April () The Law of Ellipses After the position of Jupiter on August 6, 96 was selecte as a reference point in figure 7, the angles ψ swept by Jupiter on October 9, 964, December 8, 966, an February 6, 969 shown in table 7 coul be etermine by the same metho as that employe in the case of Mars. The results are liste in table 8. Table 8. The angle ψ swept by Jupiter moving from the reference position selecte on August 6, 96, to the other three positions shown in table 7. Time ψ 6 August October December February By replacing () with the Jupiter Sun istances an the corresponing angles ψ swept by Jupiter on October 9, 964, December 8, 966, an February 6, 969, respectively, we obtain simultaneous linear equations with 3 unknowns. By solving them, the perioic equation of the reciprocal of the Jupiter Sun istance can then be achieve as = cos ψ +.58 sin ψ. (4)

14 By comparing the Jupiter Sun istances ʹ estimate from perioic (4) with calculate by the Mars Sun istance (3) at 5 ifferent selecte ates, we can etermine that ʹ an are nearly the same as shown in table 9. Thus, the valiity of perioic equation / of ψ in (4) can be confirme, an the elliptical motion of Jupiter revolving aroun the Sun as a focus may also be asserte by combining () an (4). Table 9. Five ranomly selecte ates use to compare the Jupiter Sun istance ʹ from perioic (4) with the Jupiter Sun istance from (3) by the law of equal areas for the Earth. Time ψ ( -)/ (%) 9 April July September November January The eccentricity of Jupiter can be calculate by () as e = a a b The result is nearly the same as the well-known eccentricity of Jupiter,.48, thus again verifies the law of ellipses for Jupiter. 5. Conclusions In this stuy, we treat the Earth as a reference point to etermine the law of motions for the other planets. The fact that the Earth has regular motion, which fulfills the law of equal area, enables us to establish the mathematical relation of planet Sun istance an Earth Sun istance, as shown in (3). The laws of equal areas for the other planets can be easily an naturally constructe by combining this relation with the angular spee of the planet aroun the Sun. The perioicity of the planet aroun the Sun inicates that the planet Sun istance can be represente as the perioic function of an angle. The angular position of the observing planet an the law of equal areas are use to etermine the istance of the planet to the Sun an to buil the perioic function of each planet. The trajectory equation of planet istance may thus be obtaine. The planet orbits are prove to be ellipses, which take the Sun as a focus; thus, the law of ellipses for planets is reiscovere. We have applie relatively simple geometry, trigonometry, an basic algebra to escribe the invariant properties of planetary motion. These proceures allow researchers to comprehen the magnitue of the

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