SINGLE EVENTS, TIME SERIES ANALYSIS, AND PLANETARY MOTION

Transcription

1 SINGLE EVENTS, TIME SERIES ANALYSIS, AND PLANETARY MOTION John N. Harrs INTRODUCTION The advent of modern computng devces and ther applcaton to tme-seres analyses permts the nvestgaton of mathematcal and astronomcal relatonshps on an unprecedented scale. Snce nether numercal complexty nor calculaton ntensty pose nsuperable dffcultes, t becomes feasble to treat sngle events sequentally and apply detaled tme-seres analyses to the results. The followng dscusson s prmarly concerned wth the real-tme helocentrc motons of the four major superor planets (Jupter, Saturn, Uranus and Neptune) and the four terrestral planets (Mercury, Venus, Earth and Mars), plus ther varous nteractons. Shown n graphcal form n the second secton, the fnal outputs were based ntally on the sngle-event formulas provded by Bretagnon and Smon (1986) adapted to produce tme-seres data utlzng spreadsheet technques. A. THE MAJOR SUPERIOR PLANETS The methodology and formulas appled to planetary moton n ths context are provded by Perre Bretagnon and Jean-Lous Smon n Planetary Programs and Tables from to (Wllman-Bell, Rchmond, 1986). The astronomcal programs n the latter concern the determnaton of the postons of the planets as vewed from Earth (.e., geocentrc coordnates wth correctons for aberraton, nutaton, and precesson, etc). The frst stage of the computaton, however, concerns the determnaton of helocentrc coordnates whch for Jupter, Saturn, Uranus and Neptune are obtaned from the followng power seres formulas: HELIOCENTRIC LONGITUDE (L) HELIOCENTRIC LATITUDE (B) HELIOCENTRIC RADIUS VECTOR (R) The parameter V s measured n unts of 2000 julan days from the begnnng of successve fve-year ntervals; unts are radans for L and B and astronomcal unts (AU) for R. The motons and postons of Jupter, Saturn, Uranus and Neptune are obtaned from power seres data provded for fve-year ntervals, e.g., for the perod 1990 to 1995 commencng wth Julan Day the power seres data are as follows (Bretagnon and Smon 1986:124,140):

2 -2- JUPITER L) B) R) SATURN L) B) R) URANUS L) B) R) NEPTUNE L) B) R) The frst lne gves the startng year of the fve year tme-span followed by the ntal julan day h (January 1) at 0 ET. The second lne gves the seven coeffcents of the polynomal for the helocentrc longtude L, the thrd the coeffcents for the helocentrc lattude B, and the forth the coeffcents for the helocentrc radus vector R. TIME Ephemers Tme (ET) wth the varable V(t) obtaned from the followng relaton: where T 0 s the begnnng julan date for the fve year tme-span and T the requred nstant (or successve nstants) for the superor planet(s) n queston. V(t) ranges from 0 to REAL-TIME PLANETARY ORBITS Plan-vew plots of planetary orbts requre the computaton of the helocentrc longtude (L) and the helocentrc radus vector (R) for successve values of V wthn a gven tme-span. However, none of the major superor planets have sdereal perods that are shorter than fve years thus the computaton of each orbt entals the use of successve fve-year data sets. For one complete orbt of Jupter, a mnmum of two sets of data s requred; for Saturn fve, Uranus seventeen, and for Neptune thrty-three. For the nterval BP, one hundred consecutve sets of power seres data are therefore requred for each planet.

3 -3- B. THE FOUR TERRESTRIAL PLANETS In contrast to the relatvely smple power-seres methodology for the major superor planets, formulas for the terrestral planets are both cumbersome and dffcult to mplement n tmes-seres format wthout the heavy use of computng devces. Here the formulas vary from planet-to-planet and all requre tables and lengthy trgonometrc summatons. For example, n the case of Mercury the formulas and tables for the helocentrc radus vector (R), the helocentrc lattude (B) and helocentrc longtude (L) are as follows: MERCURY: HELIOCENTRIC RADIUS VECTOR ( R ) TABLE 1: = 1 to 14 r a v

4 MERCURY: HELIOCENTRIC LATITUDE ( B ) -4- TABLE 2: = 1 to 18 b a v MERCURY: HELIOCENTRIC LONGITUDE ( L ) L = U+10 {( U-1408U +114U +233U 88U ) x sn( u U U U U )}

5 -5- TABLE 3: = 1 to 25 l a v HELIOCENTRIC RADIUS VECTORS: VENUS, SUN (EARTH) AND MARS In so much as the present paper s an ntroducton rather than a detaled descrpton the correspondng formulas and tables for the longtudes and lattudes of the other terrestral planets wll not be presented here n toto. For general nformaton, however, a lmted treatment of the remanng helocentrc radus vectors for ths tro of planets s shown below; for further detals refer to the descrptons and explanatons provded by Bretagnon and Smon (1986).

6 -6- VENUS: HELIOCENTRIC RADIUS VECTOR ( R ) ) R = {( U+7096U 3360U +890U -210U x cos( u u U U U U )} {( U+131U ) x cos( u+0.48u +0.20U )} TABLE 4: = 1 to 5 r a v SUN (EARTH): HELIOCENTRIC RADIUS VECTOR [Table 5: = 1 to 50 omtted ] MARS: HELIOCENTRIC RADIUS VECTOR ( R ) [Table 6: = 1 to 29 omtted ] R = {( U-1230U -378U + 187U -153U -73U ) cos( u u U U U U )} {( U-53U -46U +14U -12U +99U )x cos( U U U U U U )} TIME Ephemers Tme (ET) wth the varable (U)t obtaned from the followng relaton:

7 -7- C. HELIOCENTRIC RADIUS VECTORS Although relatons [9] and [4] requre addtonal correctons for hstorcal research, for present purposes t s more useful to reman wth julan dates throughout snce the latter lend themselves readly to loopng and ncrementaton n a varety of complex applcatons. Moreover, although t stll remans feasble to calculate the planetary postons by applyng related formulas for the helocentrc dstances, longtudes and lattudes n standard manner, t s the helocentrc dstances that are by far the most useful. The exact sequental value for the radus vector of a planet movng n an ellptcal orbt carres wth t both correspondng orbtal veloctes and orbtal "perods" for the exact poston and tme n queston. In other words, the varable radus vector that moves between the lmts establshed by the ponts of perhelon and aphelon provdes two further related tme-seres functons. The frst descrbes the manner n whch the radus vector changes, the second the orbtal velocty tself, and the thrd though not mmedately apparent the correspondng "range" of the perod of revoluton. To put the latter n a clearer lght, the mean synodc tme (Ts) between a par of co-orbtal planets essentally the tme a faster movng nner planet (mean orbtal perod T1) takes to lap a slower outer planet (mean orbtal perod T2 ) may be obtaned from the general synodc formula: In practce, however, adjacent pars of planets are rarely both precsely at the partcular ponts n ther orbts that correspond to ther respectve mean value radus vectors. Thus the mean synodc perod remans bascally a theoretcal parameter. From a more practcal vewpont, however, for every value of the radus vector between perhelon and aphelon there are correspondng "perods" of revoluton, and as a consequence, real-tme synodc functons may be determned drectly from the resultng radus 2 3 vectors by the applcaton of the Harmonc Law ( T = R ). For the superor planets ths poses no great problem snce true radus vectors may be obtaned from power seres data and assocated tables n a relatvely straghtforward manner. For the terrestral planets the same basc approach holds, except that the more complex formulas and tables are nvolved. Both methods, however, lend themselves readly to loopng and ncrementaton and all provde the means for nvestgatng nteractve relatonshps. Examples of the latter nclude vsualzaton of the well known 2 : 1 Earth-Mars and 2 : 5 Jupter-Saturn resonances, relatonshp between dfferences n nverse orbtal veloctes of the latter par and the orbtal velocty of Mars. The latter and further complextes assocated wth Venus-Earth-Mars resonances are examned brefly n Part Two. SOURCE Part C and Relaton 10 excepted, the above formulas, tables, power seres data and general methodology are from Bretagnon and Smon(1986): TABLES FOR THE MOTION OF THE SUN AND THE FIVE PLANETS FROM TO TABLES FOR THE MOTION OF URANUS AND NEPTUNE FROM TO Perre Bretagnon and Jean-lous Smon. Servce des Calculs et de Mécanque Céleste du Bureau des Longtudes 77, avenue Denfert-Rochereau, Pars, France. Publshed by Wllmann-Bell, Inc., Rchmond, John N. Harrs. Last updated February14, sprasolars.ca