Solar Orbital and Solar Activity

Transcription

1 Solar Orbtal and Solar Actvty Aurel Rusu Duma Translated from Romanan by Vorca Zamarcaru 1 Introducton In the artcle Planetary Orbtals we have seen that each astronomc body (AB) wthn the structure of a planetary system (PS) has an assocated orbtal, noton we remnd the reader: The abstract object made-up from the reunon of all the nvarant attrbutes whch s characterstc for the moton of an AB wthn a PS, shall be called orbtal. An orbtal s composed of the followng nvarant attrbutes: Spatal zone where the orbtal moton s ranged. Orbtal perod or ts reverse orbtal frequency. As the Sun also s an AB of PS, even ts central element, the Sun also wll have an orbtal,.e. a spatal area (a volume) whch encloses the moton (trajectory) of the Sun around the mass center (MC) of PS. Also, ths moton wll be assgned a frequency dstrbuton (a spectrum), as we wll see later. Comment 1.1: As establshed n Introducton to objectual phlosophy, the mass center of a planetary system s the nternal translaton reference T of the system, the null rotaton (revoluton) pont and applcaton pont of the resultant of all nteracton forces wth the planetary system external systems. For a closer star, or for the galactc center, the whole mass of our planetary system s concentrated n ths pont. Table 1.1 AB m [kg] a [UA] q T [days] - Sun Mercury Venus Earth Mars Jupter Saturn Uranus Neptune Pluto * In Table 1.1 the man attrbutes of AB of our planetary system structure are gven, where m s the mass of AB, a s the major sem-axs of the orbt, s the numercal eccentrcty of the orbt, T s the orbtal perod and q wll be defned below. If we note the sun mass as m S and planets masses as m, ( s the ndex n Table 1.1 startng wth 1 for Mercury 1 ) then q m S m. 1 The soft for mathematcal modelng was Mathcad, and the conventon for the ndces of a vector was ORIGIN=1.

2 2 The moton of PS elements n relaton to the MC of the planetary system Gven a PS represented n Fg. 2.1, consstng of small-szed n AB (planets), wth masses m 1, m 2,...m n orbtng around a large-szed AB (Sun) wth mass m S. To an external 2D (twodmensonal) reference system wth O orgn and XY 2 axes, the PS elements have the vectors of spatal poston r 1, r 2, rn and r S. r 1 r2 r S r r MC n Y m 1 m S m 2 m n O X Fg. 2.1 In these condtons, the mass center (MC) of the PS has the poston vector r MC gven by the equaton 3 : mr 11mr 2 2mr n nmsrs rmc m1m2 mn ms (2.1) If we move (shft) the O orgn of the reference system n the MC, ths s equvalent to wrtng n the equaton 2.1 rmc 0. Denotng m1 m2 mn ms m T, where m T s the total mass of the PS, the equaton 2.1 becomes: ms m1 m2 m rs r1 r2 mt mt mt mt r n (2.2) hence the poston vector of the Sun to the MC: m1 m2 mn rs r1 r2 ms ms m n S (2.3) ms By denotng q [1, n], the equaton 2.3 becomes: m r n S (2.4) 1 q r We know that planets move n ellptcal trajectores wth MC n the focus pont and r as the poston vector, hence the Sun shall also move throughout the ellptcal trajectores 4, wth 2 We consder the smplfed case where the motons of the AB formng the PS are coplanar, ncluded n the XY plane. 3 Rchard Ftzpatrck - An Introducton to Celestal Mechancs, Cambrdge Unversty Press Sun trajectory s ellptcal n a couple consstng of the Sun and a certan planet, but on the whole, by the composton of all ellptcal motons, solar trajectory s more complex, as we shall see.

3 MC focus pont also, but wth r S as poston vector. Mathematcal modelng of these motons (trajectores) shall be done n smplfyng condtons, namely: 1. Moton n ellptcal trajectory n the XY plane s the result of the projecton on the XY plane of a crcular moton wth radus a and constant angular velocty ( t), moton set n a plane that ncludes the X axs and s nclned by the angle n relaton to XY plane. In these crcumstances we have a major sem-axs of the orbt collnear wth the X axs, b a cos( ) collnear wth the Y axs (mnor sem-axs of the orbt), sn( ) and 2 () t t. T 2. All ellptcal trajectores are ncluded n the XY plane and have the apsdes axs (perhelon-aphelon axs) collnear wth the X axs. 3. Startng pont of trajectores calculaton (t = 0) s the perhelon. In these crcumstances (, j beng the unt vectors of X and respectvely Y axs), the poston vector of the Sun for a m S -m couple wll be: rt () x() t y() tj (2.5) where: a x() t (cos( t) ) (2.6) q a y( t) sn( t) cos(arcsn( )) (2.7) q Global (resultant) poston vector of the Sun wll be the vector sum of all ndvdual poston vectors 5,.e.: 9 r() t r() t (2.8) or fnally: 1 a 2 xt ( ) cos( ) t 1 q T (2.9) a 2 yt ( ) sn( ) cos(arcsn( )) Wth the data n Table 1.1, x(t) and y(t) resultng n km. 9 8 t 1 q T (2.10) 3 The ndvdual contrbutons of planets to Sun moton The most mportant couple n our planetary system s Sun-Jupter, both n the ntensty of the nteracton 6 and n the rato of masses (q 5 ), whch determnes the largest contrbuton to the poston vector of the Sun n ts orbtal. Wth the data n Table 1.1 and accordng to relatons 2.9 and 2.10 (n whch =5 both ntally and fnally), assumng that we neglect the other planets, for the Sun the trajectory n Fg. 3.1 results. We note that on the X axs the 5 poston of the Sun n relaton to MC s between km (at aphelon of Jupter) and km at Jovan perhelon. If we consder that the Sun radus s km, then the MC of Sun-Jupter couple s outsde the Sun, at a dstance between 5 The ndvdual contrbutons of each planet n the PS. 6 See Fgure 4.1 and Table 4.1 of Planetary Orbtals, reproduced for convenence n Annex 1.

4 km at Jovan perhelon. km at Jovan aphelon and Fg The trajectory of the Sun n relaton to MC of Sun-Jupter couple Fg. 3.2 Varaton of solar poston vector module for the Sun-Jupter couple 2 2 In fg. 3.2 the varaton of poston vector module of the Sun rt () xt () yt () s dsplayed durng a complete revoluton perod of Jupter. The tme axs s graded n days. Usng the same procedure we wll fnd n Table 3.1 for all couples Sun-planet r mn and r max postons between whch the poston of the Sun vares accordng to each couple. In the fourth column of Table 3.1 by addng we get r MM [km], radus of spatal range wthn whch the moton of the Sun s framed,.e., the frst attrbute of solar orbtal: r (3.1) MM 6 rmm (3.2) Ths radus s reached when all the planets are algned to aphelon. It s mportant to note that due to the moton of the Sun around the MC of our planetary system, the dstance between the surface of the Earth and of the Sun vares over tme (n addton to the annual 6 5 varaton determned by Earth orbt eccentrcty) by up to km.e. 0.59% of the UA. Ths may nfluence n the same proporton the solar constant (the amount of energy receved from the Sun to the Earth's crust). Comment 3.1 Varaton of the dstance between Earth and Sun due to Sun s moton on ts orbtal s extremely slow due to the fact that gant planets have the most mportant contrbuton to ths varaton (notceable contrbuton), and they have very long perods of revoluton (n human tme scale).

5 Table 3.1 Planet r mn [km] r max [km] 1 Mercury Venus Earth Mars Jupter * * Saturn * * Uranus * * Neptune * * Pluto * * Analyss of Sun s motons n ts orbtal To start wth we shall analyze the solar trajectory determned by tellurc planets (Mercury, Venus, Earth + Moon and Mars) on a temporal nterval of three complete Martan perods (2061 days). That means that we wll have n the relaton 2.8: 4 rt () rt () (4.1) 1 In Fg. 4.1 we see the trajectory of Sun s center caused by revoluton motons of the tellurc planets wth axes graded n km, revealng very steep varatons of poston, and n Fg the value of poston vector module rt () xt () yt () n km, correspondng to the trajectory n Fg. 3.1, the tme axs beng graded n days. On the bass of the trajectory n Fg. 4.1 and of the value of the poston vector n Fg. 4.2, we wll calculate the velocty and acceleraton correspondng to these motons. Comment 4.1: When calculatng velocty and acceleraton two methods wth the same fnal results can be used: d d 1. Classcal methods of dfferental calculus,.e. vt () rt () and at () vt () dt dt 2. Objectual method of calculaton wth fnte dfferences (see chapter 2 of Introducton to rt () rt ( t) vt () vt ( t) Objectual Phlosophy),.e. vt () and a() t, where t (one day) s t t temporal ncrement n the relatons 2.9 and 2.10 durng the mathematcal modelng. In ths case, the concept of dervatve s replaced wth the densty concept of the dervatve dstrbuton. Usng the same software for mathematcal modelng - Mathcad - applyng the classcal method, the calculaton takes tens tmes longer than the calculaton wth fnte dfferences. The graphs presented reveal an nterestng observaton, namely when the Sun poston n relaton to the MC passes through ts mnmum values, hgher pronounced acceleraton values appear. But we know that acceleraton means energy nput (n ths case knetc), energy that the Sun wth all ts nternal structure receves from the planets around t (through gravtatonal nteractons). In the case of tellurc planets ths energy s very low compared to the energy receved from the gant planets (as we shall see below), but the scenaro s repeated n these planets also,.e. the maxmum energy ntake occurs also at tmes of the solar poston vector mnmum.

6 Fg. 4.1 Sun's trajectory determned by the tellurc planets x and y n km Fg. 4.2 Varaton of the poston vector module accordng to trajectory n Fg. 4.1 Fg. 4.3 Sun velocty n km /day correspondng to Fg. 4.2 Fg Acceleraton n km/day 2 correspondng to velocty varaton n Fg. 4.3 It s now tme to consder solar trajectory determned by the gant planets (Jupter, Saturn, Uranus and Neptune) over a temporal nterval of days (three complete cycles of the planet Neptune). We wll have n the relaton 2.8:

7 8 rt () rt () (4.2) 5 Fg. 4.5 shows how complex the sun's trajectory s due to gant planets revolutons and Fg. 4.6 the varaton of poston vector module n the same stuaton. Fg. 4.5 Solar trajectory due to gant planets Fg. 4.6 Varaton of the solar poston vector due to gant planets [km] Furthermore, n Fgures 4.7 and 4.8 we have solar velocty and acceleraton caused by gant planets motons, tme axs stll beng n days. Fg. 4.7 Sun velocty derved from Fg. 4.6 [km / day]

8 Fg. 4.8 Sun acceleraton [km/day 2 ] The graphs presented clearly reveal that under the nfluence of gant planets also abnormally hgh peaks of solar acceleraton appear (superposed over a quas-snusodal varaton) much hgher acceleratons than n the case of tellurc planets. Also, these peaks concde wth the mnmum values of solar poston vector n relaton to MC. To confrm the statements above n Fg. 4.9 and 4.10 the varatons n Fg.. 4.6, 4.7 and 4.8 are shown n detal for the frst 5000 days (13.7 years) n the nterval of days (494 years). Fg. 4.9 Detal n Fg. 4.6 and 4.7 for the frst 5,000 days Fg Detal n Fg. 4.8 for the frst 5,000 days Let us now analyze the above data n terms of those set n chap. 7 of Introducton to Objectual Phlosophy. There we saw that force ultmately means an energy flux transmtted to the materal system (MS) over whch the force s appled. The transmtted energy flux s dstrbuted on the nternal structure of MS (nternal state varaton), then the external state varaton appear (acceleraton, see Energetc Acton under the same chapter 7). Sun s velocty of 1 km/day means m/s, and acceleraton of 1 km/day 2 means 1.389*10-7 m/s 2, so when the Sun has an acceleraton of 2.77 km/day 2, t means a=3.848*10-7 m/s 2. However, gven that m S =1.989*10 30 kg, then F ms a,.e. F=7.654*10 23 N,.e. an energy flux of 7.654*10 23 J/s. It s natural that ths knetc energy ntake globally dstrbuted throughout the nternal structure of the Sun modfy the nternal solar knetc processes (nternal acton of energy flux transmtted), processes whose externally vsble effects appear

9 to us as "solar actvty",.e. spots, eruptons, varatons of the magnetc feld and solar wnd, etc.. We also see n Fg. 4.8 that at certan tmes, the acceleraton peak s hgher than that of day 3000 (e.g. on day t reaches 10.8 km/day 2 ). 5 Spectral analyss of solar orbtal To determne the second attrbute of solar orbtal - orbtal frequency - we wll analyze n the (spectral) frequency doman the Sun s moton around MC. The most sgnfcant parameter of those analyzed so far s the acceleraton 7 as t gves us drect nformaton on the energy receved by the Sun durng ts evoluton on the orbtal. Therefore, we wll analyze n the frequency doman the values array of solar acceleraton smlar to that n Fg. 4.8, but wth the contrbuton of tellurc planets also on an nterval of 2 17 = days (359 years). The algorthm for spectral analyss s the Fast Fourer Transform (FFT (x)) algorthm applyng to an x vector wth the number of elements of the form 2 n. In our case, the x vector s a(t) n Fg We note that n regard to Fg. 4.8 a knd of nose appeared that actually represents the contrbuton of tellurc planets, where we can apprecate the sze of ths contrbuton compared to the contrbuton of gant planets. Also, out of the length of days of Fg. 4.8, here only days are rendered for the reasons mentoned above concernng the requrements of FFT analyss. Fg. 5.1 Solar acceleraton dependng on the moton of all the planets Fg. 5.2 shows how complex the spectral dstrbuton of solar acceleraton s, where j s the ndex of spectral component value resultng after the FFT analyss, y j beng the spectral component module and f j ts assocated frequency n Hz. Out of the spectral components amount only few (the most obvous) were selected, and ther values accompaned by bref comments can be seen n Table 5.1. Examnng the fgures presented, especally the table, we see that the solar orbtal spectrum s smlar to the spectrum of ampltude modulated sgnal, n whch, besdes the man frequences, sum and dfference frequences appear. Untl now, due to the complexty of the spectrum only a few components were analyzed, enough to start wth for the purpose of ths artcle, namely that to show very close lnk between the moton of the Sun n ts orbtal and solar actvty caused by ths moton. 7 Frequences of the spectral components resultng from the analyss are the same even f we analyze the velocty or poston of the Sun, only the component ampltudes and shape of the spectrum beng dfferent.

10 k Table 5.1 Frequency [Hz] Fg. 5.2 Spectrum a(t) n Fg. 5.1 n unfltered verson Freq.*10 12 [Hz] Amplt. comp. rel. val. 8 T [days] T [years] Comments 1,923* f Ne 3.772* f Ur 1.076* f Sa * Frst major component f Ju -f Sa = * f Ju -f Ur = * The most mportant component f Ju -f Ne = * Very close to f Ju * f Ju * * * * f Ma * Contrbuton of Earth + Moon (synodc perod of Jupter from Earth) * f Te * Contrbuton of Venus * f Ve * Contrbuton of Mercury * f Me Table 5.1 and Fg. 5.2 show the contrbutons of the frst three planets (the contrbuton of Mars s neglgble) to the solar orbtal spectrum. Note the major contrbuton of Venus, that naturally results f we revew Fg. 4.1 and Table 4.1 of the Planetary Orbtals (for convenence we have reproduced these tems n Annex 1 to ths artcle). From ther 8 Spectral component ampltude vares dependng on the method of analyss, analyss nterval, or on applyng or not a flter to the data, etc. Rapport of relatve values s mantaned, as well as major component frequences.

11 examnaton t becomes very clear that the second couple n ntensty of the gravtatonal nteracton after couple Sun-Jupter s Sun-Venus, hence the contrbuton of Venus to the solar orbtal spectrum must also be mportant. The frst f 1 spectral component wth perods of 19.9 years s very close to the dfference between the orbtal frequences of Jupter and Saturn. The second f 2 component wth the perod of 13.8 years s very close to the dfference between the orbtal frequences of Jupter and Uranus. Note that the most ntense f 3 spectral component wth a perod of 12.8 years corresponds to the average frequency of quas-snusodal varaton of Sun's acceleraton, frequency very close to the dfference between the orbtal frequences of Jupter and Neptune. For other components the relatonshp wth orbtal frequences has not yet beng found. 6 Possbltes for predcton of solar actvty It s tme to draw the reader's attenton that so far we have only made a theoretcal (pure mathematcal) analyss of the moton of planets wth gven masses and perods of revoluton on ellptcal trajectores, but all these motons were not real planetary motons n real tme but n smplfed condtons 1, 2 and 3 of par. 1. In realty, the planets move n ellptcal trajectores n relaton to MC of the PS, but the orbts are not coplanar and apsdes axes do not concde ether. However, at present the planets trajectores are well known (wthout ths nformaton space mssons would not be possble), so the relatons 2.9 and 2.10 can be adjusted so that they be vald n real tme. Annex 2 shows the apsdes axes rotaton smulaton results to demonstrate that ths rotaton, n contrast wth the smplfyng condtons n par. 1, does not alter ntal results n terms of qualty. Only the temporal dstrbuton of the acceleraton peak ampltude changes but not ts spectral components. Nether the relatvely low nclnaton of planetary orbts compared to the stuaton when they were coplanar can change the frequency dstrbuton of solar acceleraton. If we accurately determne the poston of each planet at the present tme, we can determne wth the same accuracy the poston of the Sun n relaton to MC of PS at that tme and ts future evoluton. The purpose of ths knowledge s that we can very accurately determne abnormal acceleraton momenta, and as a result, we can predct abnormal events n solar actvty. As future evolutons of Sun acceleraton may be predcted on reasonable terms from astronomcal vewpont, prevous trajectores can also be analyzed and past postons of the Sun wth mportant events n the hstory of solar actvty can be correlated, n order to understand what results a certan acceleraton n the past compared to the effects observed and recorded by observers can lead to. In ths way one can make a calbraton of the evoluton of solar actvty accordng to ts orbtal moton. Unfortunately, at present we do not know how long t takes for the dstrbuton of knetc energy mpulse on Sun s nternal structure (how long t takes to change the nternal state), that n order to predct what external effects wll follow and when they wll occur, dependng on the sze of mpulse whose magntude and tme of occurrence are predctable by solar orbtal method. 7 Conclusons 1. The domnant couple of AB n our planetary system s Sun-Jupter, whose MC s outsde the Sun at a dstance rangng from km to Jovan aphelon and km at Jovan perhelon. Followng the moton of Jupter n ts ellptcal trajectory, the Sun wll also run an ellptcal moton between these two lmts (see Fg. 3.1). We say that ths s the contrbuton of Jupter to the solar moton n the solar orbtal. Lkewse, all the planets n PS wll each have a contrbuton to the moton of the Sun whose fnal poston vector s gven by the relaton 2.8.

12 6 2. Maxmum lmt of solar poston vector rmm km s the radus of the crcular doman n whch all the possble postons of the Sun are ranged, ths doman beng the frst attrbute of the solar orbtal abstract object. 3. Sun s moton n ts orbtal s characterzed by several temporal dstrbutons: - Temporal dstrbuton of x(t) and y(t) coordnates of solar center n relaton to the MC of PS; - Temporal dstrbuton of solar poston vector module; - Temporal dstrbuton of Sun s velocty on ts orbtal; - Temporal dstrbuton of solar acceleraton; 4. Each temporal dstrbuton may be assocated a frequency dstrbuton. We chose as sgnfcant the frequency dstrbuton of solar acceleraton as the acceleraton s on the one hand energy ntake (f postve), and on the other hand as mpulses wth abnormally large values appear, whch may affect the nternal state of the Sun. 5. Solar acceleraton spectral analyss shows the presence of some spectral components wth frequences (perods) derved from the orbtal frequences of the planets n PS, or dfferences between the Jupter frequency and other gant planets frequences. These frequences are the components of the second attrbute of solar orbtal abstract object; 6. Spectral components of solar orbtal are frequences (perods) of some perodc processes of Sun s moton overall, moton that could nfluence nternal solar processes and therefore can generate known perodcty n solar actvty. 7. Solar acceleraton abnormal peaks occurrng at certan tmes, can lead to abnormal phenomena of solar actvty. If the mathematcal modelng of solar moton s consstent wth the moton of the planets n real tme the momenta of acceleraton peak occurrence are predctable, therefore ther effects on solar actvty also. 8. Currently, accordng to the data collected by observers snce 1755, 23 complete cycles 9 of sunspots (surnamed solar cycles) have been revealed. The duraton of these cycles vares between 9 and 12.6 years, wth an average of 10.6 years per cycle. Spectral analyss of Sun s moton n ts orbtal reveals much more possble cycles, ncludng the one found so far (average of f 3 and f 7 components, frst and thrd component n ntensty s 10.3 years). Unfortunately, modern methods and means of solar parameters measurement (especally from satelltes) exst only snce few decades ago, an nterval nsuffcent for clear promnence of longer cycles, such as the one of 19.9 years for example, or of cyclcal varatons of other physcal measures characterstc for solar actvty and whch could not be determned n the past. 9. Spatal-temporal modulaton of the solar poston n relaton to MC of PS causes a correspondng modulaton of the solar gravtatonal feld as the gravtatonal nteracton s dependent on the dstance between AB couples. For gant planets whch are far away from the Sun ths feld varaton s not so mportant, but for those nearby, especally for Mercury, can be one of the causes of the perhelon advance. 9 Wkpeda Lst of solar cycles

13 Annex 1 - Dstrbuton of gravtatonal nteractons ntensty on AB couples n our planetary system Table X.1 n AB Couple n AB Couple 1 Sun - Mercury 24 Venus Pluto 2 Sun Venus 25 Earth - Mars 3 Sun Earth 26 Earth - Jupter 4 Sun - Mars 27 Earth - Saturn 5 Sun Jupter 28 Earth - Uranus 6 Sun Saturn 29 Earth - Neptune 7 Sun Uranus 30 Earth - Pluto 8 Sun - Neptune 31 Mars - Jupter 9 Sun Pluto 32 Mars - Saturn 10 Mercury - Venus 33 Mars - Uranus 11 Mercury - Earth 34 Mars - Neptune 12 Mercury - Mars 35 Mars - Pluto 13 Mercury - Jupter 36 Jupter - Saturn 14 Mercury - Saturn 37 Jupter - Uranus 15 Mercury - Uranus 38 Jupter - Neptune 16 Mercury - Neptune 39 Jupter - Pluto 17 Mercury - Pluto 40 Saturn - Uranus 18 Venus - Earth 41 Saturn - Neptune 19 Venus - Mars 42 Saturn - Pluto 20 Venus - Jupter 43 Uranus - Neptune 21 Venus - Saturn 44 Uranus - Pluto 22 Venus - Uranus 45 Neptune - Pluto 23 Venus - Neptune 1,0E+24 1,0E+23 1,0E+22 1,0E+21 1,0E+20 1,0E+19 1,0E+18 1,0E+17 1,0E+16 1,0E+15 1,0E+14 1,0E+13 1,0E+12 1,0E+11 1,0E+10 1,0E+09 1,0E Fmn Fmax Fg. X.1 Dstrbuton of gravtatonal nteractons ntensty on couples n Table X.1

14 Annex 2 Sun moton n case of planetary apsdes axes rotaton Wth regard to the condtons n par. 1 on the calculaton of planetary trajectores, n whch we determned that ellptcal trajectores have collnear apsdes axes, n ths annex we shall analyze the moton of the planets closest to the actual condtons,.e. wth apsdes axes nclned wth angles shown n the table of planetary orbtals elements n astronomy books 10. The rotaton of apsdes axes s equvalent n vewpont of relatons 1.6 and 1.7 wth the rotaton of coordnate axes wth angle, after whch, compared to the old x and y coordnates, the new x and y coordnates are: x' xcos ysn y' xsn ycos (X.2.1) whch entered nto relatons 2.9 and 2.10, the new relatons of poston calculaton (Pluto excluded as ts contrbuton s mnor, smlar to that of Mars) become: 8 a xt ( ) cos t cossn t cosarcsnsn( ) q T T 8 a yt ( ) cos t snsn t cosarcsncos( ) q T T whose graphcal representaton, also for days (to be compatble wth the FFT analyss) s gven by Fg. X.2.1 for poston, X.2.2 for poston vector module, X.2.3 for velocty and X.2.4 for acceleraton. Fg. X Solar trajectory due to all the planets (excludng Pluto) 10 Imke de Pater, Jack J. Lssauer Planetary Scence, Cambrdge Unversty Press, 2001 (p. 6) *** - Efemerde astronomce pentru anul 2013, Bumbeşt Ju, 2012 (p. 150)

15 Fg. X.2.2 Fg. X.2.3 Fg. X.2.4 Fg. X.2.5 Solar acceleraton spectrum for planetary apsdes axes rotaton

16 Table X.2.1 Major spectral components n case of apsdes axes rotaton k Frequency [Hz] Freq.*10 12 [Hz] Amplt. comp. rel.val. T [days] T [years] Comments * (51) Frst major component f Ju -f Sa = * (31.8) f Ju -f Ur = * (68.4) The most mportant component f Ju -f Ne = * (18.7) Very close to f Ju * (28.3) * (22.7) * (44.8) * (15) Contrbuton of Earth+Moon * (30) 237 Contrbuton of Venus * (6.5) 89.8 Contrbuton of Mercury In Fg. X.2.5, and Table X.2.1 one can see that the varaton n drecton of apsdes axes does not affect n terms of qualty the major spectral components and n terms of quantty the dfferences are qute small. For comparson, n the column showng the component ampltude the value n the case of apsdes axes algnment s ncluded n brackets, thereby t can be seen low nfluence of rotaton of these axes. The most mportant nfluence s exerted by apsdes axes rotaton on the temporal dstrbuton of solar acceleraton peaks,.e. of the moments when mportant and abnormal events of solar actvty may occur.