Sterrekunde

Volgens my berekeninge volg Jupiter se mane nie Kepler se 3de wet nie - waarom is dit?

Volgens my berekeninge volg Jupiter se mane nie Kepler se 3de wet nie - waarom is dit?


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Ek word gevra om data oor die wenteleienskappe van die vier Galilese satelliete van Jupiter te versamel en aan te toon dat hulle dieselfde skaal as in Kepler se 3de wet gehoorsaam.

My benadering vir maan Io:

Aanlyn het ek gevind dat die afstand van Io na Jupiter 422.000 km is ~ = 0.00282089577 AU wat

$ 2,82 times10 ^ {- 3} AU $

Io se baan om Jupiter is 1,77 Aardae; 1,77 / 365 ~ = $ 4,85 times10 ^ {- 3} , mathrm {Aarde jaar} $

Volgens Kepler se 3de wet is $ P ^ 2 ( mathrm {Aarde jaar}) = a (AU) ^ 3 $

Dus

$ (4.85 times10 ^ {- 3}) ^ 2 = (2.82 times10 ^ {- 3}) ^ 3 $ wat uiteraard nie waar is nie, selfs net deur na die magte van 10 te kyk nadat die eksponent aan elke kant versprei is.


'N Mens moet die massa van die voorwerp wat om die draai is, in ag neem. Kepler se derde wet geld vir alle planete wat om die son wentel, en vir alle mane wat om Jupiter wentel, maar nie oor verskillende putte van die swaartekrag nie. Dit is eers in Newton begryp en moes 'n interessante probleem (waaraan ek nog nooit gedink het nie) al in Kepler se tyd gehad het sedert die wenteltydperke en relatiewe afstande van die vier Galilese mane toe bekend was. En natuurlik die baan en relatiewe afstand van die aarde se eie maan. Miskien was dit selfs 'n belangrike inspirasie vir Newton se denke oor swaartekrag?


Kepler se derde wet is dat $ R ^ 3 / P ^ 2 $ 'n konstante is. Dit is egter nie 'n universele konstante nie; dit hang af van die massa van die liggaam wat wentel. $$ frac {R ^ 3} {P ^ 2} simeq frac {GM} {4 pi ^ 2}, $$ waar $ M $ die massa van die liggaam is (as ons aanneem dat $ M gg $ die massa van die mane).

Afhangend van u verfyndheid, kan u probeer om $ R ^ 3 $ teen $ P ^ 2 $, of beter, $ log R $ teen $ log P $ te beplan, om ooreenstemming met Kepler se derde wet te toon.


Dit is waarna hulle soek. Ek het dit met syfers probeer en dit het gewerk. Dankie vir u insette, u plaas my op die regte pad.

M1 + M2 = A3 / P2

Konsekwente eenhede moet gebruik word om hierdie vergelyking te laat werk. As die data nie in 'n konsekwente eenheidstelsel gegee word nie, moet dit omgeskakel word.

Die massas moet gemeet word in sonmassas, waar een sonmassa 1,99 X 1033 gram of 1,99 X 1030 kilogram is.

Die semi-hoofas moet gemeet word in Astronomiese eenhede, waar 1 AU 149,600,000 kilometer, oftewel 93,000,000 myl, is.

Die wentelperiode moet in jare gemeet word, waar 1 jaar 365,25 dae is.


Volgens my berekeninge volg Jupiter se mane nie Kepler se 3de wet nie - waarom is dit? - Sterrekunde

Aan die einde van hierdie afdeling is u in staat om:

  • Stel Kepler se wette van planetêre beweging.
  • Lei die derde Kepler-wet af vir sirkelbane.
  • Bespreek die Ptolemeïese model van die heelal.

Voorbeelde van gravitasiebane is volop. Honderde kunsmatige satelliete wentel om die aarde saam met duisende stukke puin. Die baan van die Maan oor die Aarde het mense van oudsher geïntrigeer. Die wentelbane van planete, asteroïdes, meteore en komete oor die son is nie minder interessant nie. As ons verder kyk, sien ons byna onvoorstelbare getalle sterre, sterrestelsels en ander hemelse voorwerpe wat om mekaar wentel en deur swaartekrag in wisselwerking tree.

Al hierdie bewegings word deur swaartekrag beheer, en dit is moontlik om dit in verskillende mate van presisie te beskryf. Presiese beskrywings van komplekse stelsels moet op groot rekenaars gemaak word. Ons kan egter 'n belangrike klas wentelbane beskryf sonder die gebruik van rekenaars, en dit sal insiggewend wees om dit te bestudeer. Hierdie wentelbane het die volgende kenmerke:

  1. 'N Klein massa m wentel 'n veel groter massa M. Dit stel ons in staat om die mosie te beskou asof M stilstaan ​​— in werklikheid asof uit 'n traagheidsverwysingsraamwerk aangebring M—Sonder noemenswaardige fout. Massa m is die satelliet van M, as die wentelbaan gravitasiegebonde is.
  2. Die stelsel is geïsoleer van ander massas. Dit stel ons in staat om klein effekte as gevolg van massas van buite te verwaarloos.

Die voorwaardes word bevredig, tot goeie benadering, deur die Aarde se satelliete (insluitend die Maan), deur voorwerpe wat om die Son wentel, en deur die satelliete van ander planete. Histories is planete eers bestudeer, en daar is 'n klassieke stel van drie wette, genaamd Kepler se wette van planetêre beweging, wat die wentelbane beskryf van alle liggame wat aan die twee vorige toestande voldoen (nie net planete in ons sonnestelsel nie). Hierdie beskrywende wette is vernoem na die Duitse sterrekundige Johannes Kepler (1571–1630), wat dit bedink het na deeglike studie (meer as twintig jaar) van 'n groot hoeveelheid noukeurig opgetekende waarnemings van planeetbeweging gedoen deur Tycho Brahe (1546–1601). Sulke noukeurige versameling en gedetailleerde opname van metodes en data is kenmerke van goeie wetenskap. Data vorm die bewyse waaruit nuwe interpretasies en betekenisse gekonstrueer kan word.


Kepler se 3de wet en die Galilese mane

Hierdie webquest is ontwerp om 'n insiggewende, onderhoudende aktiwiteit te wees om die begrip van Kepler se 3de wet . Dit behoort nie meer as 2 ure en kan tuis of in die loop van 'n paar klasbyeenkomste voltooi word (veral as die studente hul eie data versamel - sien Bronne). Dit sal goed inpas by 'n laat sterre- of fisikaklas op hoërskool of vroeë kollege, mits die studente het die nodige wiskunde-agtergrond (algebra en meetkunde, minstens en verkieslik 'n mate van trigonometrie). Ideaal gesproke sal hierdie aktiwiteit na 'n lesing oor Kepler se drie wette en / of swaartekrag gedoen word, maar 'n voorafgaande vertroudheid met hierdie wette is nie nodig nie. Dit is ten einde die aantal studente / klasse waarop hierdie webvraag toegepas kan word, te maksimeer.

Oor die algemeen wil sy aktiwiteit die hoeveelheid wiskunde wat hierdie berekeninge benodig, verminder. Enigeen van u met 'n graad Fisika of Sterrekunde, of selfs diegene wat dit op universiteitsvlak geneem het, weet hoeveel wiskunde regtig behels om sekere van hierdie vergelykings te gebruik en te gebruik. 'N Vinnige besoek aan die wiki-bladsy van die wette van planetêre beweging wys hoe ingewikkeld swaartekrag regtig kan word.

As die studente 'n uitdaging wil hê, is daar 'n klein ekstra kredietvraag aan die einde van die inleiding. Dit is nie heeltemal afgelei van die eerste beginsels nie, maar dit vereis 'n sekere mate van gemak met eenheidsomskakeling en algebra.

Daar is hulpbronne (sybalk) tot u beskikking wat 'n opfris oor hierdie (en verwante) onderwerpe, idees vir bespreking en 'n skakel na die webwerf bevat, waar u die program wat ek gebruik het om die data vir hierdie aktiwiteit te versamel, werklik kan bekom.


Jupiter's Satellites

Jupiter en sy satelliete word soms 'n 'miniatuur-sonnestelsel' genoem, maar die waarheid is ingewikkelder. Drie van Jupiter se vier satelliete is opgesluit resonant wentelbane. Dit het interessante gevolge vir ons weeklikse waarnemings en vir die geskiedenis en die lot van die Joviese stelsel.

Jupiter het vier helder satelliete wat maklik met 'n teleskoop gesien kan word. Hierdie liggame beweeg rondom die planeet in byna sirkelvormige wentelbane in volgorde van toenemende afstand vanaf Jupiter, word hulle genoem Io, Europa, Ganymedes, en Callisto. Kepler se derde wet impliseer dat satelliete met kleiner wentelbane vinniger beweeg. Dus, Ganymedes, Europa en Io, wat almal nader aan Jupiter is as Callisto, moet almal vinniger beweeg as Callisto.

As ons hierdie voorspelling probeer toets deur elke Maandagaand waarnemings te maak, sal ons die satelliete vind op die posisies soos in Fig. 1. As u na hierdie figuur kyk, kan u iets vreemds opmerk. Dit lyk asof die drie binnesatelliete, Io, Europa en Ganymede skaars beweeg van een week na die volgende, terwyl Callisto oral spring. Dit is nie wat Kepler se derde wet ons laat verwag het! Wat gaan aan?

Die antwoord behels twee feite. Die een is pure toeval, die ander is 'n diep waarheid oor Jupiter se satelliete.

Eerstens die pure toeval: dit neem Ganymedes 7,155 dae om om Jupiter te wentel. Dit is effens meer as een week, dus as ons Jupiter se satelliete elke week op dieselfde dag waarneem, sien ons Ganymedes in amper presies dieselfde plek elke keer. Terwyl ons nie kyk nie, gaan Ganymede Jupiter om en keer amper terug na waar dit die vorige week was. Dit is 'n kwessie van geluk, dit gebeur net so dat die week van 7 dae wat ons op Aarde gebruik, amper presies gelyk is aan Ganymedes se wentelperiode.

Tweedens, die diep waarheid: die wentelperiode van Europa (3.578 dae) is die helfte van Ganymedes en Io se wentelperiode (1.789 dae) is die helfte van Europa's! In die tyd wat dit Ganymedes neem om een ​​wentelbaan te maak, maak Europa twee bane en Io vier bane. As ons dus een keer per week waarneem, sien ons dit al drie van hierdie satelliete byna presies waar hulle die vorige week was. Die verhouding tussen Ganymedes, Europa en Io se wenteltydperke is nie toevallig is die kans dat sulke hemelse uurwerk toevallig plaasvind baie klein.

As die tyd tussen ons waarnemings presies ooreenstem met die wenteltydperk van Ganymedes, sou ons die drie binnesatelliete elke week op dieselfde plekke sien. Maar die tyd tussen ons waarnemings is 0,155 dae (of 3 uur, 43 minute) korter as die wenteltydperk van Ganymedes, dus Ganymedes voltooi sy reis rondom Jupiter nie heeltemal nie. Die resultaat is net soos om een ​​keer in die 59 minute 'n horlosie te fotografeer. 'N Reeks sulke foto's wys hoe die minute hand stadig beweeg agtertoe omdat dit sy reis om die draaiknop tussen foto's nie heeltemal voltooi nie. Op dieselfde manier wys ons weeklikse waarnemings dat Io, Europa en Ganymede almal stadig agteruit beweeg.

Fig. 1 toon die buitenste satelliet, Callisto, wat oral voorkom. Dit gebeur omdat Callisto se orbitale periode van 16.689 dae ietwat is meer as twee weke, vind waarnemings met weeklikse tussenposes Callisto aan min of meer teenoorgestelde kante van Jupiter.

Fig. 1. Jupiter se satelliete om 20:30 (HST) op laboratoriumaande gedurende die herfs van 2010. Die satelliete word geïdentifiseer deur hul voorletters.

RESONANSIES

Die verhouding tussen die wenteltydperke van Io, Europa en Ganymedes is 'n voorbeeld van a resonansie. Meer algemeen sê ons dat twee wentelbane resonant is as die verhouding van hul periodes 'n verhouding van heelgetalle is. Pluto se wenteltyd is byvoorbeeld 247,7 jaar, terwyl Neptunus se wentelperiode 164,8 jaar is. Die verhouding 247,7: 164,8 is gelyk aan 3: 2, en Pluto voltooi dus twee bane om die son, terwyl Neptunus presies drie bane neem. Hierdie resonansie verklaar hoe Pluto en Neptunus wentelbane kan kruis sonder om te bots: Pluto kom slegs binne die baan van Neptunus as Neptunus aan die ander kant van die sonnestelsel is. Dit is ook moontlik om resonansies tussen wentelbeweging en rotasie te hê, byvoorbeeld, die wentelperiode van die maan en rotasietydperk is albei 27,3 dae, dus is die verhouding presies 1: 1.

In die geval van Jupiter se satelliete is dit waarskynlik dat Io, Europa en Ganymedes hul resonansie ontwikkel het as gevolg van aantrekkingskrag vir swaartekrag. Een moontlike scenario begin met Io, Europa en Ganymedes wat almal nader aan Jupiter wentel as vandag. As gevolg van die getye wat Io op Jupiter geskep het, het die baan van Io stadig na buite gedryf, en sodoende sou dit uiteindelik 'n 2: 1-resonansie met Europa nader. Sodra dit gebeur het, sou die wentelbane van die twee satelliete deur swaartekrag "gesluit" word, en albei saam na buite sou dryf. Uiteindelik, namate Europa se baan groter geword het, sou dit 'n 2: 1-resonansie met Ganymedé bereik het, en die wentelbane van al drie satelliete sou in hul huidige verhouding vassteek.

Resonansies speel 'n belangrike rol in die sonnestelsel. Sommige van die gapings in Saturnus se ringe kom byvoorbeeld voor as gevolg van resonansies tussen deeltjies in die ringe en Saturnus se satelliete. Daar is ook gapings in die asteroïedegordel as gevolg van resonansies tussen asteroïdes en Jupiter.

WAARNEMING VAN DIE SATELLIETE

Om te sien dat Io en Europa regtig onderskeidelik vier en twee bane voltooi, moet ons Jupiter tussen ons weeklikse laboratoriumvergaderings in die loop van die tyd wat Ganymedé een baan voltooi, voltooi. Stellarium egter [www.stellarium.org] en soortgelyke planetariumprogramme kan gebruik word om Jupiter se satelliete te vertoon en hul beweging te bespoedig. Ons kan dit op 'n bewolkte nag probeer.

Ons sal voortgaan om Jupiter se satelliete waar te neem en te skets wanneer dit gerieflik is. U kan u sketse vergelyk met die voorspellings hierbo om te bevestig dat die satelliete op hul verwagte posisies verskyn.

Gebeurtenisse op laboratoriumnagte

Terwyl Jupiter se satelliete wentel, gaan hulle gereeld voor of agter die planeet deur, en ook deur Jupiter se skaduwee of gooi hulle hul eie skaduwees op Jupiter se skyf. 'N Deurgang voor die planeet word 'n genoem transito, terwyl 'n gang agter 'n okkultasie. 'N verduistering kom voor wanneer 'n satelliet deur Jupiter se skaduwee gaan, terwyl a skadu-transito kom voor as satelliet se skaduwee op Jupiter val.

Die onderstaande tabel bevat 'n lys van verskillende gebeure rakende Jupiter-satelliete wat ons tydens laboratoriumaande kan waarneem. Alle datums en tye word in HST.


Vraag Waarom het ons 'n reeks fisika-wette? Is daar iets wat dit nie volg nie?

* opgestel tydens die oerknal *. Watter weergawe van die BB het vandag fisiese wette geskep wat in die wetenskap gedokumenteer is? Voorbeeld van Newton se bewegingswette, swaartekrag, c-konstante, alfa-konstante, ens. In 1948 was daar 'n heel ander weergawe van die BB met behulp van kwantummeganika en algemene relatiwiteit wat die oorsprong van die heelal en alle elemente in die Periodieke Tabel verduidelik het. maar ek weet nie dat hierdie model die oorsprong van alle bekende fisiese wette wat vandag in die wetenskap waargeneem en gemeet word, verklaar het nie.

'Negentien jaar na die ontdekking van Edwin Hubble dat dit lyk asof die sterrestelsels met geweldig hoë snelhede van mekaar af weghardloop, is die prentjie wat aangebied word deur die uitbreidende heelal-teorie - wat veronderstel dat alle materie in sy oorspronklike toestand saamgedruk is in een soliede massa van hoë digtheid en temperatuur — gee ons die regte omstandighede om al die bekende elemente in die periodieke stelsel op te bou. Volgens berekeninge moes die vorming van elemente vyf minute na die maksimum kompressie van die heelal begin het. Dit is ongeveer tien minute later ten volle bereik. ” —Wetenskaplike Amerikaner, Julie 1948

Die BB-wiskunde wat gebruik word, verskil tans baie: kwantumveldteorie, leptogenese, inflasie, multiverse, ens.


Volgens my berekeninge volg Jupiter se mane nie Kepler se 3de wet nie - waarom is dit? - Sterrekunde

Simulasies van fisiese stelsels is wyd beskikbaar, sonder enige koste, en is gereed om in ons klaskamers gebruik te word. , 2 Sulke simulasies bied 'n toeganklike hulpmiddel wat gebruik kan word vir 'n verskeidenheid interaktiewe leeraktiwiteite. Die Jovian Moons Applet 2 stel die gebruiker in staat om die posisie van Jupiter se vier Galilese mane op te spoor met 'n verskeidenheid kykopsies. Vir hierdie aktiwiteit word data verkry uit die wentelperiode en wentelradiakaarte. In vroeëre eksperimente is teleskope gebruik om die wentelbeweging van die Galilese mane vas te lê, alhoewel waarneming van astronomiese gebeure en die meting van hoeveelhede moeilik kan wees om te bereik as gevolg van 'n kombinasie van koste, opleiding en waarneming van toestande. Die applet laat toe om 'n geskikte stel data te genereer en data-analise wat Kepler se derde wet van planetêre beweging verifieer, wat lei tot 'n berekende waarde vir die massa van Jupiter.


Ek hou van Kieran Hunt se antwoord, maar ek gaan 'n ander antwoord gee, al stem ek saam met wat hy gesê het.

In 'n baie regte sin, gehoorsaam ons sonnestelsel nie Kepler se wette nie, want daar is baie liggame. Die planete, en nog meer, die mane in ons sonnestelsel volg nie presies Kepler se 3 wette nie, maar hulle volg dit meestal redelik naby. Ons maan het 'n baie vreemde, wankelende wentelbaan omdat dit deur die son en die aarde aangetrek word. Maar die planete in ons sonnestelsel volg Kepler se wette goed genoeg sodat Kepler sy wette getoets en geverifieer het.

In 'n binêre sterstelsel is die eindresultaat waarskynlik redelik soortgelyk. Stel jou voor dat Jupiter 'n ster was - verder uit. Dit sou afhang van hoe groot en hoe naby, maar as dit ver genoeg was, kon die aarde steeds om die son wentel, terwyl Jupiter en die son om mekaar wentel. Daar is twee hoofsoorte binêre stelsels. Een, waar die sterre naby is en die planete om die massamiddelpunt van die 2 sterre wentel. Die ander, waar die sterre ver genoeg van mekaar af is, waar die planete afsonderlik om een ​​van die sterre kan wentel. Sien foto hieronder:

Is dit altyd die geval? en, kan ek sweer dat die outeur daarvan reg is? Wel, nee, maar ek wil daarop let dat, selfs in 'n binêre sterstelsel, planete meestal Kepler se wette van twee liggaamsbane sal volg in een van die twee voorbeelde in die foto hierbo.

Die probleem met 3-liggaam wiskunde in 'n sonnestelsel wat al 'n miljard jaar bestaan, is dat die chaos en die onvoorspelbaarheid nie so lank in die stelsel bly nie. Die kleinste van die drie lyke sal waarskynlik uitgegooi word, of dit sal te kort in een van die ander twee lyke val, of dit vind 'n wentelende resonansie. 'N Langdurige 3-liggaamstelsel sal waarskynlik 'n son-aarde-maan-stabiliteit hê, of 'n stabiliteit van die Son-Neptunus-Pluto of miskien 'n Jupiter-, Sun-, L4- of L5-baan. Die onstabiliteit van die drie-liggaamsprobleem wat al eeue lank 'n raaisel vir wiskundiges is, duur nie baie lank in sonnestelsels nie.

Wysig, ek wil wel byvoeg, dat L4 / L5 om die kleiner ster in 'n binêre stelsel wentel, waarskynlik 'n bykomende scenario is wat ons sal sien afhangende van die grootteverhouding van die twee sterre, maar 'n stabiele L4- of L5-baan is vriendelik soortgelyk aan 'n Kepler-baan.

Daar is 'n paar wiskundige wette vir drie liggaamsprobleme, dink ek nie, en dit is baie kompleks. Lagrange-Euler is die een vir L4-L5. Bietjie oor my betaalgraad.

'N Ander manier om dit te verklaar, is 'n hoof in die chaostorie wat eilande van stabiliteit of chaotiese aantrekkers genoem word. Ek sal hierdie skakel laat verklaar, want ek dink dit doen dit redelik goed.

Terwyl u agterkom dat geen werklike N-liggaamstelselsbane stabiel is nie (herhaal hulself presies), kom u agter dat dit in patrone beland. Terwyl die stelsel van Jupiter se innerlike mane: Io, Europa en Ganymede byvoorbeeld nooit heeltemal dieselfde pad herhaal nie, slaag hulle wel daarin om met mekaar te "resoneer" en in 'n ritme te vestig. Vandaar die naam "orbitale resonansie".

Dus, selfs in 'n binêre stelsel, sou u meestal die algemene voorspelbaarheid van Kepler se wette sien.

Hier is 'n redelike goeie beeld van die onvoorspelbaarheid van die 3-liggaamsprobleem, maar u sal waarskynlik nie so 'n baan so sien nie.

Klein punt om by te voeg, maar dit is wiskundig moontlik om alternatiewe oplossings vir die 3 liggaamsprobleem te skep (byvoorbeeld die figuur 8). Ek dink nie dit gaan in die heelal baie gereeld gebeur nie. Die figuur 8 of die bal van die gare volg natuurlik nie Kepler se wette nie.


Ontsnapmetode vir die massa van die Jupiter:

Tweedens gebruik ons ​​die Escape-snelheidsmetode om die massa van die Jupiter te bereken. Terwyl ons hierdie vergelyking gebruik, kan ons die massa van Jupiter maklik uitvind. Verder is die waardes van ontsnap snelheid, gravitasiekonstante en radius van die planeet verpligtend om die massa deur hierdie vergelyking te skat. Verwys na die onderstaande vergelyking van ontsnap snelheid.

V² = 2 G M / R

Terwyl u die bostaande formule herrangskik om die massa te vind, dus:

M = RV² / 2G

G = Gravitasiekonstante = 6,67408 × 10 ^ -11 m3 kg-1 s-2

M = massa van die Jupiter in kg

R = Radius van die planeet = 69.911km

V = ontsnappingssnelheid (60 km / s op die Jupiter-oppervlak)

M = (6,991X10 ^ 7X (60X1000) ^ 2) / (2X6,67408 X 10 ^ -11)

M = 1,9 X 10 ^ 27 kg.

Ten slotte, met behulp van die ontsnappingssnelheidsmetode, is die massa van Jupiter 1,9 X 10 ^ 27 kg.


Volgens my berekeninge volg Jupiter se mane nie Kepler se 3de wet nie - waarom is dit? - Sterrekunde

Die aanhangsels agter in u handboek bevat baie nuttige inligting, waaronder oorsigte van eenvoudige probleemoplossing. Lees dit deur middel van onderwerpe soos eenhede of wetenskaplike notasie.

Hierdie oefenberekeninge moet voorheen voltooi word middag op Donderdag 8 Februarie 2018. Meer oefenoefeninge sal binnekort daarna geplaas word. Wees seker dat u streef na organisasie, akkuraatheid, netheid en duidelikheid in u werk.

1. Kepler se derde wet sê dat die waarde van 'n 3 / P2 dieselfde is vir alle voorwerpe wat om die son wentel. (a) Wat is die waarde van 'n 3 / P2 vir die aarde? (b) Gebruik die Saturnus-gegewens op bladsy A-10 (Tabel E-2) aan die agterkant van u handboek om 'n 3 / P2 vir daardie planeet te bereken en toon aan dat u antwoord dieselfde getal is as die antwoord op deel (a ).

2. Die handtekening hieronder is die van Clyde Tombaugh, wat Pluto in 1930 ontdek het. As Pluto gemiddeld 39,48 AE van die son af is, hoeveel jaar het Pluto dan nodig om een ​​keer om die Son te wentel, volgens Kepler se derde wet? Hoe vergelyk u antwoord met die tydperk vir Pluto op bladsy A-10 agter in u handboek?

3. Daar is rekords van Comet Halley (hieronder getoon) wat duisende jare teruggaan. As hierdie voorwerp een keer elke 76 jaar om die Son wentel, wat is die gemiddelde afstand tussen die Son en die komeet?

4. Kepler se wette geld ook vir ander voorwerpe as die planete wat om die son wentel en vir ander eenhede as AU en jare. Beskou die drie Galilese mane van Jupiter hieronder. Verifieer die derde wet van Kepler deur 'n 3 / P2 vir elke voorwerp te bereken met behulp van die gegewe data. Kies enige kombinasie van eenhede, maar gebruik dieselfde vir alle berekeninge. Let daarop dat u waardes van 'n 3 / P2 nie dieselfde sal wees as die van die aarde nie!


Kepler en Kircher oor die Harmonie van die Sfere

Op 29-30 Oktober 2007 het die Giorgio Cini-stigting 'n konferensie aangebied oor 'Vorme en strominge van Westerse Esoterisme'. Dit het plaasgevind in die pragtige kwartiere van die Stigting op die eiland San Giorgio Maggiore in Venesië. Ek wou hê dat my bydrae 'n musikale komponent sou hê, maar my eerste idees (vir iets oor esoterisme in die 17de-eeuse klavesimbelmusiek) moes weggegooi word omdat die Stigting nie 'n klavesimbel kon bied nie. Ek het dus navorsing gedoen wat ek vir my boek gedoen het Athanasius Kircher se teater van die wêreld, en dit by die klavier geïllustreer, insluitend die speel van Kircher se een volledige instrumentale komposisie, waarvan die partituur in daardie boek voorkom. Die vraestel is vertaal en in Italiaans gepubliseer. Dit is die oorspronklike Engelse weergawe.

The Harmony of the Spheres, 'n transdissiplinêre idee wat kosmologie, sterrekunde, wiskunde en musiekteorie verenig, was 'n belangrike middel in die Pythagorese stroom in die intellektuele geskiedenis van die Weste. Hierdie artikel fokus op twee figure wat grootliks daartoe bygedra het in die vroeë fase van die wetenskaplike rewolusie. Deur te leer en te geneig, het albei onder Neoplatoniese en Hermetiese invloede gekom, albei was aanhangers van die stroom Christelike esoterisme wat 'n dieper begrip van die geskape wêreld gesoek het. Maar soos ons sal sien, was hul houding teenoor hemelse harmonieë in skrille kontras met mekaar.

Elke boek oor Johannes Kepler (1571–1630), en die meeste boeke oor die geskiedenis van sterrekunde, maak melding van die teorie van hemelse harmonie wat Kepler ontwikkel het in Harmonieke Mundi (1619). 1 Hulle gee gereeld, as 'n nuuskierigheid, sy notasie van die planetêre "liedjies" weer:

Die belangrikheid van hierdie werk vir die geskiedenis van die wetenskap is onbetwis. Dit voltooi die later genoemde drie Kepleriaanse wette van planetêre beweging: 1. Elke planeet beweeg in 'n ellips met die son op een fokus. 2. Die radiusvektor van elke planeet beweeg oor gelyke gebiede in gelyke tydsintervalle. 3. Die vierkant van die rewolusietydperk van 'n planeet rondom die son is eweredig aan die kubus van die gemiddelde afstand van die planeet tot die son. 2 Kepler was self nie in staat om 'n fisiese verklaring vir hierdie wette te gee nie, maar hulle het die basis gevorm vir Isaac Newton (1643–1727) om sy teorie van universele gravitasie te ontwikkel, wat die geldigheid daarvan bevestig en die onsterflikheid van hul ontdekker verseker.

Dit was die gegewens van die nuwe sterrekunde, soos vervat in die waarnemings en tabelle van sy meester Tycho Brahe (1546-1601), wat hierdie gevolgtrekkings op Kepler afgedwing het na baie jare se intense navorsing en meditasie. Hulle het twee radikale nuwighede in die rangskikking van die sonnestelsel nodig gehad: eerstens die aanvaarding van die Copernicaanse of heliosentriese stelsel en tweedens die elliptiese wentelbane met hul veranderlike snelhede van planeetbeweging, wat die episiklusse en ekwivalente wat die Ptolemeïese of geosentriese stelsel deurmekaar gemaak het, afgeskaf het. . Alhoewel daar 'n paar presedente vir heliosentrisme in die antieke wêreld was, was die tweede gevolgtrekking teen die hele astronomiese tradisie, en veral teen die beginsel wat deur Aristoteles verklaar en deur Ptolemeus aanvaar is: dat alles in die hemel in perfekte sirkels beweeg. Selfs Copernicus het dit nie oortree nie.

Kepler se wette "het die voorkoms" meer suksesvol gered as enige vorige teorie, maar dit was nie genoeg vir hom nie, gedrewe omdat hy deur 'n lewenslange passie was om die goddelike rede agter die verskynings te ontdek. Nadat hy die Copernicaanse rangskikking van die planete rondom die son reeds geregverdig het deur middel van 'n geometriese argument wat die vyf platoniese vaste stowwe betref, het hy 3 die onreëlmatigheid van hul wentelbane aangespreek. Waarom moes God hierdie elliptiese eerder as sirkelvormig gemaak het, en so uiteenlopend in hul grade van elliptika?

By die soek na die antwoorde op hierdie vrae was Kepler se basiese aanname die Pythagorese: dat die sleutel tot die kosmos in aantal lê. 'N Sekondêre idee, ewe Pythagoreas van oorsprong, was dat harmonie getal betekenis gee, kwantiteit met kwaliteit. Dit bevoordeel sekere getalle bo ander, naamlik dié wat, vertaal in musikale terme, die intervalle lewer wat ons as konsonant, aangenaam en musikaal nuttig beskou. Harmonieke Mundi is 'n triomf van vernuf om hierdie beginsels in die gegewens van die nuwe sterrekunde in te lees, en sodoende laasgenoemde te regverdig.

Wetenskaphistorici is deeglik bewus van hoe Kepler se argument werk, en van die verband tussen die planetêre liedere en die Eerste Wet, maar vir sommige lesers kan dit nuttig wees om dit hier te verduidelik. In hierdie oordrewe diagram van 'n planeet se elliptiese wentelbaan word gesien dat die beweging daarvan versnel soos dit perihelium (die naaste aan die son) nader, en dat dit vertraag as dit weg beweeg na die aphelie (die verste van die son af).

Na aanleiding van Kepler se tweede wet, hang die versnelling daarvan af van die mate van elliptika van sy baan. Mercury se baan is byvoorbeeld baie elliptieser as die van Venus, wat amper sirkelvormig is. Daarom is die verskil tussen Mercurius se ekstreme posisies veel groter as dié van Venus, en die musikale interval wat daardie verskil baie wyer uitdruk.

Die natuurfilosowe van die oudheid het geglo dat die planete nie in hul wentelbane stil is nie. As ons die vraag of hulle deur die lug beweeg of deur 'n fyner medium soos eter, ter syde stel, lyk dit logies dat hierdie groot liggame 'n geluid moet maak, net soos bewegende liggame op aarde doen en die vele teorieë van die Harmonie van die Sfere bly as pogings om te spesifiseer wat die klank kan wees, vertaal in die taal van musiek.

Daar is twee hoofskole oor hoe hierdie vertaling gemaak moet word. Die eerste veronderstel dat die relatiewe afstande van die planete van die aarde harmonies verband hou, asof dit verskillende punte op 'n tou is. Hierdie teorie is afgelei van Pythagoras se skool, waarin die afstand van die aarde vanaf die maansfeer gereken is op 126 000 stade. As ons hierdie afstand as ekwivalent aan 'n heel toon neem, is die afstande na die ander planetêre sfere eweredig soos die intervalle van 'n diatoniese skaal. 4 Die tweede skool is van mening dat dit die bewegings van die planete is wat harmonies verband hou, en hulle verskillende rewolusiesnelhede stem ooreen met die toonhoogteverskille. Dit veronderstel almal 'n stilstaande en stille aarde, alhoewel dit nie seker was of die omwentelinge ten opsigte van die aarde bereken moes word nie, in welke geval Saturnus, die verste om te reis, die vinnigste sou beweeg, of relatief tot die diereriem, in welke geval Saturnus die stadigste planeet wees, wat 30 jaar neem om een ​​stroombaan te maak, en die maan met sy siklus van 28 dae die vinnigste. 5

Daar is ander skemas, veral dié van die Arabiese sterrekundiges en die verskillende tolke van die "skaal" van Plato's Timaeus, maar hulle hoef ons nie hier te bekommer nie. Wat die gevolg is van elke skema vóór Kepler, is dat die planetêre toon afkomstig is van 'n bestaande skaal of intervalvolgorde wat onmoontlik op enige wetenskaplike, kwantitatiewe manier geldig is, omdat die bekende verhoudings van afstande of bewegings baie verskil van die verhoudings. van die toon wat gebruik word om dit voor te stel. Dit is hier waar Kepler se benadering van al sy voorgangers verskil het: sy werk van 1619 was die eerste keer dat 'n teorie van hemelse harmonie direk uit astronomiese waarneming afgelei is.

Tot dusver het hierdie teorieë byna eenparig 'n enkele, onveranderlike toon aan elke planeet toegeken, soos 'n mens sou verwag uit 'n perfekte sirkelbaan. 6 Met 'n geïnspireerde sprong van die verbeelding het Kepler egter gesien dat die planetêre toon nou moet wissel, hul toonhoogte styg en daal in verhouding tot hul versnelling en vertraging. Hy het die presiese hoeveelheid bereken deur die daaglikse beweging van 'n planeet by perihelium met sy daaglikse beweging by aphelie te vergelyk, uitgedruk as grade van 'n sirkel. Dit het 'n eenvoudige proporsie gegee, wat soos alle verhoudings in musiekintervalle vertaal kon word deur die twee terme as verskillende snaarlengtes te beskou.

Saturnus se hoekbeweging by aphelion, na aanleiding van Kepler se data, is byvoorbeeld 106 minute boog. By perihelion is dit 135 minute. Die verhouding van die twee hoeveelhede, 106: 135, is ongeveer 4: 5. Twee stringe van relatiewe lengtes 4 en 5 klank 'n groot derde uitmekaar. Daarom is Saturnus se "liedjie" vervat binne die limiet van 'n groot derde (sien Figuur 1). 7

Die ooreenstemmende syfers vir Jupiter is: beweging by aphelion 270 minute beweging by perihelion 330 minute. Die verhouding 270: 330 is ongeveer 5: 6, dus is die musikale interval 'n klein derde.

In die geval van Venus, wat 'n byna sirkelvormige wentelbaan het, is die toonhoogteverskil 24:25, 'n interval kleiner as 'n halftoon wat Kepler as eenstemmig noem. In die geval van Mercury se baan dek die musikale voorstelling daarvan 'n oktaaf ​​plus 'n klein derde (alhoewel dit verkeerd is om aan te neem, soos die notasie suggereer, dat die opwaartse en afwaartse gang daarvan anders is).

Kepler kon nou sy behoefte bevredig om goddelike rede te vind in die planetêre bewegings: dit was God se begeerte dat die kosmos 'n verskeidenheid toon en harmonieë moes voortbring. Met ietwat geforseerde argumente het hy beide die hoof- en die klein modus gevind, maar helaas was die musiek van al die planete wat tegelyk gesing het, volgens die sewentiende-eeuse standaard verskriklik onoordeelkundige. Aangesien die ses planete amper nie op die note van 'n perfekte drieklank saamval nie, het Kepler al die gevalle waarin vyf of selfs net vier daarvan doen, getabelleer en baie bladsye gevul in 'n desperate poging om die gegewens aan te pas by die tradisionele harmonie. In fact, his planetary music, when transposed within our range of auditory perception, sounds much more like twentieth-century electronic music, as one can hear from the recording made in 1979 by two professors at Yale University, John Rodgers and Willie Ruff. 8

None of the believers in the Harmony of the Spheres contended that we can hear it on earth. Tycho Brahe himself, not contesting the existence of the heavenly music, had used our deafness to it as sure evidence that the heavens cannot be filled with air. 9 Kepler could not leave it at that. Having taken such pains to establish the existence of an entirely new kind of planetary music, he had to integrate it with his search for meaning and purpose in the cosmic ordering: someone, besides God, had to benefit from it. In the final chapter of his book, he refers to Tycho’s surmise that the planets might be inhabited, and suggests that the intellect best able to appreciate the planetary harmonies might reside in the place from which they are measured, namely the Sun. “What use is this furnishing, if the globe is empty? Do not the very senses themselves cry out that fiery bodies inhabit it, which have the capacity for simple minds, and that in truth the Sun is, if not the king, at least the palace of the ‘intellectual fire’?” 10

By modern criteria, Kepler seems to have had a split personality, half scientist, half mystic. His obsession with cosmic harmony puts him in the same category as Robert Fludd, author of Utriusque cosmi historia (1617) and other encyclopedic works of Christian Hermetism yet in the Appendix to Harmonices Mundi, Kepler attacks Fludd’s system on the grounds that “what he endeavors to teach us as harmonies are mere symbolism…rather than philosophical or mathematical.” 11 The immense value of Kepler’s discoveries, to his own way of thinking, was anything but a split: it lay in the fact that his Neoplatonic intuitions were backed up by hard, scientific data.

To his sorrow, they were received in profound silence by the scientific world, in which the Harmony of the Spheres was as irrelevant as the quest for the unicorn. The heliocentrists, Copernicus and Galileo, had ignored the time-honored myth, and it played no part in the rapid triumph of their cosmology. It would take Newton to sift Harmonices Mundi and extract the scientific wheat from the speculative chaff. However, after Kepler’s death his work found one careful reader: Athanasius Kircher (1602-1680), whose combination of a scientific mentality with Christian piety and a Hermetic-Neoplatonic philosophy resembled Kepler’s own.

It is instructive to see these esoteric inclinations occurring across the sectarian divide that separated the heterodox Lutheran 12 Kepler from the Jesuit Kircher, and to compare the consequences of it for our subject. Take first the Copernican question. In his standard history of the Copernican Revolution, Thomas S. Kuhn writes that “Protestant leaders like Luther, Calvin and Melanchthon led in citing Scripture against Copernicus and urging the repression of Copernicans. […] For sixty years after Copernicus’ death there was little Catholic counterpart for the Protestant opposition to Copernicanism.” 13 His system was known in the Catholic universities, and his calculations aided in the preparation of the new Gregorian Calendar of 1582. For a while, the Church held no official position on the subject, and free debate prevailed among those able to comprehend the mathematical arguments pro en con. In 1584 Giordano Bruno published his cosmological ideas, including a defence of Copernicus, in his Cena de le Ceneri, and lived, for the time being, unmolested.

Meanwhile, the Lutheran astronomer Tycho had become increasingly dissatisfied with the Aristotelian model of the heavens. His observation of comets had persuaded him that the heavens did not consist of solid, crystalline spheres, but that comets, planets, and the earth all floated in a rarefied ether. This conclusion freed him from dependence on either the Aristotelian-Ptolemaic system or on the Copernican, while his aristocratic and independent nature induced him to invent his own solution. By 1587 he was writing to his correspondent Christoph Rothmann about “a certain theory concerning the arrangement of the heavenly revolutions other than the Ptolemaic or Copernican, far more agreeable than these, and recently ascertained by me, informed by experience itself.” 14

While Tycho’s system was indeed based on his observations, and these of a precision hitherto unequalled, he too subscribed to Neoplatonic notions of a living and harmonious cosmos. He wrote: “As that divine philosophy of the Platonists seems to have appropriately realised, heaven is animated, and the heavenly bodies are themselves animated, endowed with the living spirit of a particular heaven.” 15 He rejected Copernicus’ system, but mainly on aesthetic grounds because he found it ill-proportioned when compared with the ratios, symmetries, and harmonies found in the microcosm. Referring to the heliocentric hypothesis, he says that “That ungeometric, and asymmetric, and disordered way of philosophising would produce something very foreign to divine wisdom and providence.” 16 His own solution, known as the Tychonian system, has the planets revolving around the sun, while the sun, together with the moon and the fixed stars, revolves around an unmoving earth.

It was this cosmology that was eventually adopted by the Society of Jesus, and thus of necessity by Kircher. Originally, the Jesuits had no official position on the matter, except that the Society’s rules required that “In matters of any importance professors of philosophy should not deviate from the views of Aristotle, unless his view happens to be contrary to a teaching that is accepted everywhere in the schools or especially if his opinion is contrary to the orthodox faith.” 17 Nonetheless, by the early years of the seventeenth century the Society had become one of the Copernican system’s main promoters, albeit unintentionally, because of the excellent astronomical teaching of their colleges in which all systems were studied from a mathematical point of view, even if only to refute them. 18 Jesuit scientists shared in the excitement about the discoveries that Galileo was making through his telescope, such as the four satellites of Jupiter and the phases of Venus, and when in 1611 Cardinal Bellarmine (himself a Jesuit) asked them to evaluate the discoveries, they confirmed them, despite their deviation from Aristotelian orthodoxy. 19

Kepler had long been convinced by Copernicus, and in his Astronomia Nova (1609) could shrug off the objections of his fellow Protestants in the following bold words:

This was exactly the kind of attitude that led, under the Catholic hegemony, to the prohibition placed upon Galileo in 1616, not to “hold or teach” the Copernican system. As is generally acknowledged by scholars today, it was not because the geocentric system was official dogma, but because Galileo, as a layman, had presumed to interpret the Bible and the Church Fathers as suited his scientific program. Rivka Feldhay, in her useful summary of the “Trials” of Galileo, writes that from the point of view of the church authorities, “an attempt to prove the motion of the earth might result in an encroachment on the domain of scholastic philosophers and theologians, who, in fact, had been unchallenged by the traditional form of astronomy. It could also be perceived as a threat to the monopoly of priests in the interpretation of the Scriptures which the decrees of the Council of Trent for the first time had anchored in canon law.” 21

The prohibition had the immediate effect of placing Copernicanism itself under a ban in Catholic lands. The General of the Jesuits, Claudio Aquaviva (1543–1615) had already been tightening the screws on the Order’s members to enforce Aristotelian and Thomist orthodoxy. 22 After the prohibition of 1616, the Jesuit scientists had to find some non-Copernican system within which to work, and the Tychonian, which had room for recent discoveries but did not require a re-interpretation of the Scriptures, was the best they could find.

This was not a happy situation for the scientists, and its consequences are starkly summed up in the words of Robert Blackwell: “Jesuit science thus died on the vine, just as the first blossoms appeared.” 23 Blackwell writes of the typical predicament of Orazio Grassi (1583–1654), who held the Chair of Mathematics at the Collegio Romano (the Jesuit college in Rome), and who had had a long controversy with Galileo:

This, then, was the system that Athanasius Kircher was obliged to adopt in his published works, whatever he thought in private: 25 a constriction that would naturally affect any theory he might have on the Harmony of the Spheres. In his early work on optics, Ars Magna Lucis et Umbrae (1646), Kircher outlines philosophical principles hardly different from Kepler’s. Celestial bodies (he writes) are placed by the Creator to complement discord with concord, consonance with dissonance, and sometimes to give absolute harmony. (This is exactly what Kepler found in combining the planetary songs.) As we see, the sun encourages growth and procreation, then in the autumn when it retires, things decay. But God has put the moon there to perform twelve circuits to each one of the sun, and to supplement the want of sunlight. The combination of influences is responsible for all the generation in our world. 26 “For the same reason, the rest of the planets have various courses, aspects, and anomalous movements relative to the earth and the sun, so that by their approach and departure from the sun, moon, and earth, and by the various mixtures of light and qualities, they cause various effects here below.” 27

Towards the end of Ars Magna Lucis, Kircher draws up a chart, based on data from Tycho Brahe’s observations and conjectures about the distances of the planets from the sun and from the earth, and his estimates of the diameters of the planets and the sun. 28 This was bound to give different figures from Kepler’s elliptical orbits and heliocentric system, but the most notable thing about the chart is its emphasis on proportion. Kircher tabulates the proportions of the earth’s radius to the radii of the sun, moon, and planets the proportions of the earth’s volume to the volumes of the same and the proportions of the sun’s diameter to the radii 29 of the planets.

In the sciences of the classic Quadrivium (Arithmetic, Geometry, Music, and Astronomy), proportion is studied in the context of musical intervals, and consequently, proportional tables immediately put one in mind of intervallic studies. What leaps out of this chart is that the great majority of the proportions give non-harmonic intervals, not used in the musical system. 30 There is no possibility of deriving a theory of the Harmony of the Spheres from them, and Kircher perhaps intended to show the absurdity of any such attempt.

Kepler’s harmonies receive specific attention in Kircher’s encyclopedic work on music, Musica Universalis (1650), whose tenth and last book, “Decachordon Naturae,” promises to demonstrate “that the nature of things in all respects observes musical and harmonic proportions, and that even the nature of the universe is nothing other than the most perfect music.” 31 Introducing the theme of the Harmony of the Spheres, Kircher writes that many have tried to specify the celestial harmonies, but that all their efforts are flawed. 32 Yet according to Pythagoras, Seneca, Saint Augustine, Cicero, Plato, Philo, Boethius, and many others, the world must be harmonious or (to draw on Kircher’s favored metaphor), if the universe is the Temple of God and the Church of the Blessed, then it cannot lack for singers and organs. 33

Modern astronomy, Kircher continues, has exploded the ancient belief that the celestial bodies make audible harmony, since the heavens have no solidity, nor is the order of the spheres the same as the ancients thought. Having thus dismissed the ancients, he turns to Kepler, who replaced Ptolemy’s theories with a new structure of the heavens, yet wrapped it in almost unintelligible, mystical terms. Kircher summarizes Kepler’s theory of the Platonic solids as dictating the planetary orbits, with a diagram, and concludes “I truly do not see how the intended harmony of the heavens can be proven from these [speculations] by Philosophers and Mathematicians, since one could rather say that the heavens are forced into his violently distorted five solid bodies, than that the bodies are applied to the heavens.” 34

It was the inaccuracies in Kepler’s scheme that displeased Kircher, as indeed it had displeased Kepler, who, finding that the orbits did not fit perfectly between the five solids, was set on the path that led to the solutions of Harmonices Mundi. Turning to the latter, Kircher reproduces Kepler’s astronomical data and the “songs” derived from them, but refuses to grant that the proportions between perihelion and aphelion motion deserve to be called harmonic. They are simply not accurate enough. Saturn’s proportion of 135:106 is nie a major third, says Kircher that would require the latter figure to be 108. For Jupiter’s interval to be a minor third, its proportion should be not 270:330 but 270:324. In short, there are no perfect consonances in Kepler’s data.

Kircher passes from Kepler’s theories to those of the Bohemian astronomer Anton Maria Schyrleus de Reita, which need not concern us here. 35 He then tells his readers what the heavenly harmony really consists of. (Because of Kircher’s verbose writing, I give a précis 36 rather than a complete translation.) The heavenly harmony (he says) cannot be shown in numbers of motions or the sensible collision of heavenly bodies, but only in their admirable disposition, and their ineffable proportion one to another, so that to take one away would cause the whole to perish. It is also in the exact quantity and magnitude of each body for achieving the desired effect. Thus the sun, moon, and earth have the requisite distances and magnitudes for perfect mutual influence, aid, and preservation. (Kircher gives no figures for any of these.) An example is the temperature on the earth, ideal for human life which would be impossible if the sun were closer or further away.

The distances between the sun, earth, and planets are such as to balance the sun’s heat with the moon’s coldness. For example, in summer the sun is strong, the moon weak, causing a variety and mixture of consonance and dissonance. The influence of the sun and moon is like a perfect octave. However, God has added Venus to give support with virtues such as vary the lunar influences meanwhile, Mercury modifies that which is noxious in the sun. The changing distances from the earth bring about different effects.

Moreover, God has placed two dissonant bodies, Mars and Saturn, from whose pestiferous evaporations all the earth’s ills come. Yet between them is the benign star of Jupiter. The malefic planets act like caustic medicines which attract sick matter and liberate it, so that there is no ill in nature that does not turn to good.

In musical terms, Mars and Saturn are dissonances, tied in perfect syncopation to Jupiter, while Mercury sounds a dissonance between the concords of Venus and the moon. The seven planets together give a perfect “tetraphony” or four-part harmony that Kircher now illustrates with a short musical example:

This trivial phrase may compare poorly with Kepler’s spectacle of ever-changing harmonies, but perhaps it was deliberately poor, just as the tables of proportions in Ars Magna Lucis were conspicuously un-harmonic: they showed, as Kircher undoubtedly believed, that the heavenly harmonies could not possibly be reproduced in earthly music.

The solar system of Kircher’s day had become much more complex than the seven traditional planets. Although Uranus, Neptune, and Pluto still lay undiscovered, the primitive telescope had revealed four moons around Jupiter, and twin bulges or adjacent satellites (actually, the rings) of Saturn. Wanting to find a rationale for these phenomena, Kircher hit on the idea that the heavenly bodies were grouped in “choirs.” The outermost one was the Choir of Saturn, in which the planet was given two “moons” to supplement the light of the distant sun. Next came the Choir of Jupiter, the only instance in which Kircher offers a harmony based on astronomically determined numbers. According to Reita’s figures, Jupiter’s moons were distant by 3, 4, 6, and 10 diameters of their planet. “Whatever is requisite for music certainly lies concealed in these numbers: for the distances of each body correspond precisely to a harmonic quantity: 3:4:6:10.” 37 But the real purpose of the “Jovian Choir” was to cast an ever-changing variety of light and shade and thus to moderate the influences that Jupiter sends down to our world. Then there is the “The Solar or Apolline Choir,” which “contains in itself Venus, Mercury, the moon, the earth, and is parallel, as it were, to the Jovian Choir of which enough has been said at the beginning, so we will not repeat it here.” 38 The one planet left out of any choir is Mars, whose eccentric orbit carries it now close to Jupiter, now to the sun, bringing to each its “syncopations” and baleful influences.

To deter those who might suspect other purposes in such a complicated arrangement, Kircher draws a “corollary” that seems directly aimed at Kepler’s bold speculations about other inhabited spheres:

Kircher’s vision of a harmonious cosmos was second to none in its elaboration and imaginative power, of which I have given only a slight sampling here but whereas Kepler had presented his planetary songs as factual, Kircher’s choirs were mere figures of speech, his “Decachord of Nature” a metaphor for the Hermetic principle of correspondences that he believed to underlie all of creation.

In conclusion, I will mention some of the later developments of Kepler’s and Kircher’s ideas. Kepler’s faith in an astronomical rationale for the Harmony of the Spheres lay latent for nearly three centuries, until with the dawn of the twentieth century a few isolated researchers began reconsidering it. The first of these was Emile Abel Chizat (1855–after 1917), a French composer and impresario. 40 His approach consisted in a revision of the first type of planetary music, as described above, which compares the planetary distances to intervals on a hypothetical string. Unlike the Greek and medieval theorists, whose musical system was limited to two or three octaves, Chizat found that it took over seven octaves to notate the intervals of the planets from Mercury to Neptune, including the asteroids Hungaria, Vesta, Ceres, Psyche, and Ismene, and to discover that they fell into place in a gigantic major chord.

I will only mention briefly the theories of some other twentieth-century researchers: W. Kaiser, who found harmonies not in the distances between the planets, as Chizat did, 41 but in their mean distances from the sun Alexandre Dénéréaz, who constructed a scale based on taking the Golden Section of the planetary distances 42 Rodney Collin, who used as his data the conjunctions of the planets 43 Thomas Michael Schmidt, who derived significant (musical) harmonies by comparing the time-periods of the planets’ rotation around the sun. 44 More relevant to this study are those who addressed themselves specifically to Kepler’s harmonies.

In 1909 Ludwig Günther revisited Harmonices Mundi, corrected Kepler’s values according to modern astronomy, and applied their principle to Uranus and the asteroids Ceres, Vesta, Pallas, and Juno. 45 This exercise was completed by Francis Warrain in his book on Kepler, published in 1942, which included the perihelion and aphelion values for Neptune (discovered 1843) and Pluto (1930). 46 Finally, Warrain’s data were analyzed by Rudolf Haase following the methods of Hans Kayser, the re-founder of the science of Harmonics in modern times. 47 Haase took the aphelion value of Saturn as the fundamental of a theoretical harmonic series, and related all the other values to it in terms of the tones to which they corresponded, irrespective of octave displacements. He found that the great majority of them fell on the tones C, D, E, and G, thus validating the belief that the planetary orbits accord with the laws we know as harmonic. Haase’s approach to the data, and the conclusions he draws from it, are quite different from Kepler’s, expressed as they are in secular and scientific terms and free from the anachronistic influences of musical practice, but they show the continuing vigor of Kepler’s example.

These scattered instances pale in comparison with the recent publishing campaign of John Martineau (born 1967). His books, illustrated with finely-drawn geometrical diagrams, present a mass of evidence that the solar system is in fact designed in accordance with the principles sensed by Pythagoras, the Platonists, and especially Kepler. 48 For instance, in The Harmony of the Spheres Martineau shows that Kepler was right in principle, both in his interpretation of the planetary orbits as governed by simple geometrical figures and in his conviction that simple musical proportions control their orbits only these principles need to be tested against contemporary astronomical data, whereupon they prove far more fruitful and accurate than they ever were in the past. A Book of Coincidence collects an astounding number of instances of the geometrical and harmonic placement and interrelation of the planets, any one of which might be dismissed as coincidence, but which, taken as a whole, confirm that, in Plato’s words, “God always geometrizes.” 49

Kircher would have been delighted by these discoveries. While renouncing the attempt to transcribe the heavenly music in earthly terms, he readily embraced it as a metaphor for the intelligent design of creation. Whereas Kepler’s God had taken delight in assembling a cosmos out of geometric solids and making music out of its motions, Kircher’s God was more a scientist than an artist or musician, calibrating the planetary motions and distances in exactly the right proportions to facilitate life on earth. Concord and discord were merely the musical equivalent of benefic and malefic planetary influences harmony, of the indescribable complexity and ultimate benevolence of God’s design. These principles, as Kircher believed, could survive any revision of the figures, and even stand aloof from the debate over the Copernican system, of which he himself was a dutiful opponent.

Such an attitude to the Harmony of the Spheres, even if excluded from scientific discourse, served as a fruitful metaphor for three centuries of poets. 50 And this was not the end of it. In the 1990s, Kircher’s notion of the finely-calibrated earth resurfaced among a few influential biologists, already leaning towards the “Anthropic Principle” (that the only universe we can know is one that happens to contain humans), and to “Gaia Theory” (that the earth is best studied as if it were itself a living organism). 51 They observed that the presence and variety of the biosphere depends on a delicate equilibrium of earth’s characteristics, such as its distance from the sun, gravity, atmosphere, oxygen, water, ocean salinity, axial inclination, presence of the moon, etc. If any of these were even slightly different, life could not have evolved as it has done: a situation playfully christened “The Goldilocks Effect.” 52 For Kircher, this could only be the work of a concerned, personal God, and its sole purpose was to serve man, whose purpose in turn was to serve and love God. Today’s scientists prefer non-theistic explanations, but the phenomenon of earth’s fine-tuning remains as a challenge to cosmologists, who may find themselves unwittingly continuing where Kepler and Kircher left off.

1 Johannes Kepler, Harmonices Mundi Libri V, Linz: J. Planck, 1619. I refer to the definitive English edition: The Harmony of the World, translated with an Introduction and Notes by E.J. Aiton, A.M. Duncan, and J.V. Field, Philadephia: American Philosophical Society, 1997 (Memoirs of the American Philosophical Society, vol. 209). For clarifications of Kepler’s often obscure text, I am indebted to Bruce Stephenson, The Music of the Heavens: Kepler’s Harmonic Astronomy, Princeton: Princeton University Press, 1994.

2 Definitions from Van Nostrand’s Scientific Encyclopedia, 3rd ed., Princeton: D. Van Nostrand Co., 1958, p. 930, s.v. “Keplerian Laws of Planetary Motion.” The first two laws were enunciated in Kepler’s Astronomia nova, Prague, 1609.

3 In Kepler’s Mysterium Cosmographicum, Tübingen, 1596.

4 Examples of this approach include the systems of Pliny, Martianus Capella, Censorinus, Theon of Smyrna, and Achilles Tatios.

5 This is the approach of Boethius, Nicomachus of Gerasa, and probably Cicero (in The Dream of Scipio).

6 A rare exception is Giorgio Anselmi Parmensis (before 1386-between 1440 and 1443), De Musica, red. Giuseppe Massera, Florence: Olschki, 1961, who anticipated Kepler in describing the planetary music as polyphonic and continually changing.

7 The planetary songs should be imagined as glissandi moving up and down between the given limits, not as scales with distinct tones, as Kepler’s notation suggests.

8 See John Rodgers and Willie Ruff, “Kepler's Harmony of the World: A Realization for the Ear,” American Scientist, 67 (1979). The recording was released on a long-playing record, and has been reissued as a compact disc. It includes the harmonies of the outer planets.

9 Tycho Brahe, letter to Johannes Rothmann, August 17, 1588, cited in Adam Mosley, Bearing the Heavens: Tycho Brahe and the Astronomical Community of the Late Sixteenth Century, Cambridge: Cambridge University Press, 2007, p. 89.

10 The Harmony of the World, p, 496.

11 The Harmony of the World, bl. 505. The Fludd-Kepler debates are well known from their treatment in Wolfgang Pauli, “The Influence of Archetypal Ideas on the Scientific Theories of Kepler,” in C.G. Jung and W. Pauli, The Interpretation of Nature and the Psyche, New York: Pantheon Books for the Bollingen Foundation, 1955, pp. 149-240, and Frances A. Yates, Giordano Bruno and the Hermetic Tradition, London: Routledge & Kegan Paul, 1964, pp. 440-444.

12 Although a Lutheran by faith, Kepler’s personal beliefs kept him from being a regular communicating member of his church. Max Caspar writes: “…he had arrived at a conception of the doctrines concernings ubiquity [of the body of Christ] and the Eucharist, which deviated from the Augsburg Confession in which he had been reared regarding ubiquity, he leaned toward the Catholic doctrine, but regarding the sacrament, toward the Calvinist.” Max Caspar, Kepler, trans. C. Doris Hellman, London: Abelard-Schuman, 1959, pp. 82-83.

13 Thomas A. Kuhn, The Copernican Revolution: Planetary Astronomy in the Development of Western Thought, New York: Vintage Books, 1959, p. 196.

14 Letter in Tychonis Brahe Dani Opera Omnia, red. J. Dreyer et al., Copenhagen: Nielsen & Lydiche, 1913-1929, VI, 88.15-25, cited in Mosley, Bearing the Heavens, bl. 79.

15 Tychonis Brahe Opera Omnia, VI, 221.45-49, cited in Bearing the Heavens, bl. 144.

16 Tychonis Brahe Opera Omnia, VI, 222.27-31, cited in Bearing the Heavens, bl. 145.

17 Decree 41 of the Fifth General Congregation of the Society of Jesus (1593-94), as cited in Richard J. Blackwell, Behind the Scenes at Galileo’s Trial, Notre Dame: University of Notre Dame Press, 2006, pp. 208-209.

18 See John Gascoigne, “The Role of the Universities,” in Reappraisals of the Scientific Revolution, red. David C. Lindberg and Robert S. Westman, Cambridge: Cambridge University Press, 1990, pp. 207-260 here cited, p. 214.

19 See Rivka Feldhay Galileo and the Church. Political Inquisition or Critical Dialogue? Cambridge: Cambridge University Press, 1995, p. 249.

20 Kepler, Astronomia Nova, in Gesammelte Werke, Munich: C.H. Beck, 1937, III, 34, cited in Richard J. Blackwell, Galileo, Bellarmine, and the Bible, Notre Dame: University of Notre Dame Press, 1991, p. 56.

21 Galileo and the Church, bl. 36.

22 See Galileo, Bellarmine, and the Bible, pp. 138-139.

23 Galileo, Bellarmine, and the Bible, bl. 142.

24 Galileo, Bellarmine, and the Bible, bl. 156.

25 On Kircher’s leanings toward Copernicanism, see Galileo and the Church, bl. 203 Galileo, Bellarmine, and the Bible, pp. 158, 163-164. On Kircher’s astronomy in general, see Davide Arecco, Il sogno di Minerva: La scienza fantastica di Athanasius Kircher (1602–1680), Padova: CLEUP Editrice, 2002, pp. 93-100 Giuseppe Monaco, “Tra Tolomeo e Copernico,” in Athanasius Kircher: Il Museo del Mondo, red. Eugenio Lo Sardo, Rome: Edizioni de Luca, 2001, pp. 142-158.

26 Summarized from Ars Magna Lucis et Umbrae, Rome, 1646, pp. 47-48.

27 “Eandem ob causam reliqui Planetae varios ad terram, Solemque habitus, repectusque, variamque motum anomalian sortiti sunt ut accessu, recessuque ad Solem, Lunam et terram ex varia liminis, qualitatumque mistura, varios quoque in inferioribus effectus causentur.” Ars Magna Lucis, bl. 48.

29 Sic, though a comparison of diameters or of radii is intended, the proportions being the same in both cases.

30 For example: the proportions of radii are 17:5, 8:3, 11:6, 5:26, 11:6, 5:12, 11:31, and 3:13.

31 “Naturam rerum in omnibus ad Musicas & harmonicas proportiones respexisse, atque adeò Naturam universi nil aliud nisi Musicam perfectissimam esse ostenditur.” A. Kircher, Musurgia Universalis, Rome, 1650, II, p. 364.

32 Musurgia Universalis, II, p. 373.

33 Musurgia Universalis, II, p. 376. See the well-known engraving of the “Organ of the World’s Creation” (Musurgia Universalis, II, opposite p. 366) in which the creations of the six days are symbolized as registers of an organ. A reproduction is in Athanasius Kircher: Il Museo del Mondo, bl. 266.

34 “Verùm quomodo ex his à Philosophis & Mathematicis intenta coelorum harmonia demonstrari possit non video, cum ipse in hoc potius coelos ad sua 5 corpora solida violenter detorta attraxisse, quam corpora coelis applicasse dici possit.” Musurgia Universalis, II, p. 377.

35 Reita or Rheita was the author of Oculus Enoch et Eliae, sive, Radius sidereomysticus, Antwerp, 1645, which proposed an algebraic solution to the (geocentric) planetary distances. Kircher explains it with apparent approval.

36 Musurgia Universalis, II, pp. 381-382.

37 “Certè sub hisce numeris quicquid in musica desiderari potest abditum est, cùm & distantiae vniuscuiusque corporis quantitate harmonicè prorsus correspondeant.” Musurgia Universalis, II, p. 386.

38 “Chorus Solaris siue Apollineus sub se continet Venerem, Mercurium, Lunam, Terram, estque Iouiali choro quasi parallelus de cuius harmonia cùm in principio sat dictum sit, hic eadem repetere noluimus.” Musurgia Universalis, II, p. 388. To make the earth merely one of four “choristers” to the sun steers perilously close to heliocentricity.

39 “Sequitur etiam, ibi homines ob excessiuam luminis intensionem, & ob temperamentum loci humanae naturae incongruum habitare minimè possit, qui verò ibi diuersae naturae creaturas conditas esse volunt cum de ijs nihil nobis constet, sed nec constare possit, imo in Fide periculosum videatur, quis non videt id non nisi id temere & absque vllo fundamento à nouitatum sectatoribus confictum excogitatumque?” Musurgia Universalis, II, p, 387.

40 See Azbel [Chizat’s pseudonym], Harmonie des mondes, Paris: Hughes Robert, 1903. English translation in Godwin, Harmony of the Spheres (see note 47 below), pp. 400-401.

41 Kaiser’s theories are discussed in Hans Kayser, Lehrbuch der Harmonik, Zurich: Occident Verlag, 1950, pp. 214-216.

42 Alexandre Dénéréaz, La Gamme, ce problème cosmique, Zurich, Hug, n.d.

43 Rodney Collin, The Theory of Celestial Influence, London: Watkins, 1980, pp. 78-87.

44 Thomas Michael Schmidt, Musik und Kosmos als Schöpfungswunder, Frankfurt, Verlag Thomas Schmidt, 1974, pp. 174-185.

45 Ludwig Günther, Die Mechanik des Weltalls, Leipzig, 1909, pp. 142-143.

46 Francis Warrain, Essai sur l’Harmonices Mundi ou la Musique du Monde de Johannes Kepler, 2 vols., Paris, 1942.

47 Rudolf Haase, Aufsätze zur harmonikale Naturphilosophie, Graz: Akademische Druck- und Verlangsanstalt, 1974. The relevant articles are translated in Cosmic Music: Musical Keys to the Interpretation of Reality, red. Joscelyn Godwin, Rochester, Vt.: Inner Traditions International, 1989. For further documentation and discussion of the present subject, with English translations of Kepler’s and Kircher’s texts, see also my books Music, Mysticism and Magic: A Sourcebook, London: Routledge, 1985 Harmonies of Heaven and Earth: The Spiritual Dimension of Music from Antiquity to the Avant-Garde, London Thames & Hudson, 1987 The Harmony of the Spheres, A Sourcebook of the Pythagorean Tradition in Music, Rochester, Vt.: Inner Traditions International, 1993 L’ésotérisme musical en France, 1750–1950, Paris: Albin Michel, 1991 (translated as Music and the Occult: French Musical Philosophies 1750–1950, Rochester, NY: University of Rochester Press, 1995) The Mystery of the Seven Vowels in Theory and Practice, Grand Rapids: Phanes Press, 1991 (Italian translation by Francesca Maltagliati: L’α e l’ω: Il mistero delle sette vocali del nome di Dio, Casaletto Lodigiano: Mamma Editori, 1998) Athanasius Kircher’s Theatre of the World, London: Thames & Hudson, forthcoming (2008).

48 John Martineau, A Book of Coincidence. New Perspectives on an Old Chestnut, Presteigne: Wooden Books, 1995 A Little Book of Coincidence, Presteigne: Wooden Books, 2001 Ofmil C. Haynes [pseudonym?], The Harmony of the Spheres, Presteigne: Wooden Books, 1997.

49 Plato’s dictum is reported by Plutarch, Convivialium disputationum, 8,2. Among recent attempts to reconcile ancient cosmological traditions with the findings of modern science, with an emphasis on harmony, Italian readers will appreciate the work of the erudite musician Roberto Caravella, Sphaerae: trattato sull’iperrealtà, Casaletto Lodigiano: Mamma Editori, 2001.

50 For insights into this historical process, see Fernand Hallyn, La Structure poétique du monde: Copernic, Kepler, Paris: Editions du Seuil, 1987 English translation: The Poetic Structure of the World: Copernicus and Kepler, New York: Zone Books, 1990, especially pp. 250-251 which treat Kircher.

51 A.J. Watson, “Co-evolution of the Earth's Environment and Life Goldilocks, Gaia and the Anthropic Principle,” in James Hutton - present and future, red. G.Y. Craig and J.H. Hull, London: Geological Society, 1999 (Special Publications, no. 150), pp. 75-88.

52 Referring to the fairytale Goldilocks and the Three Bears, in which Goldilocks finds the Bears’ porridge to her satisfaction when it is not too hot, not too cold, but “just right.”


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