ࡱ> 02-./5@ Hbjbj22 "XXX@x x x x  " ^``````$RS     ^ ^b:: @mx  :^0:  : :$  x  x Diametrul aparent al Soarelui si al Lunii MOTTO: "Universul ... este scris intr-o limba matematica si caracterele sunt triunghiuri, cercuri si alte figuri geometrice, mijloace fara de care ar fi cu neputinta sa intelegem ceva." - Galileo Galilei a. Sursa de lumina. Imaginea sursei Privirea omului este atrasa, instinctiv, de lumina, deci de sursele de lumina; totusi, vizarea directa a acestora poat produce, de multe ori, un efect nedorit asupra vederii noastre, ajungandu-se uneori chiar la pierderea totala sau partiala a vederii. Se stie ca nu putem privi Soarele in plina zi decat daca lumina sa este filtrata printr-un strat de nori; un bec privit direct, sau chiar o lumanare, ne poate produce o senzatie neplacuta.  INCLUDEPICTURE "http://director2.lefo.ro/curriculum/astronomie/Eclipsa1/figura1-1.jpg" \* MERGEFORMATINET Figura 1.1 Lasand pentru mai tarziu problema generala a protectiei vederii, sa ne indreptam acum atentia asupra posibilitatii de a privi indirect o sursa puternica, printr-un procedeu foarte simplu de realizare a unei imagini (mai putin luminoase) a sursei; procedeul este ilustrat de figura 1.1. Pentru a-l pune in aplicare, avem nevoie de un paravan mobil (un simplu carton) in care s-a creat - cu un ac - un mic orificiu; seara (sau ziua, cu storurile trase), aprindem o lumanare in camera. Apropiind paravanul mobil de un perete (sau de ecran - un paravan fix), vom vedea pe acesta din urma, clar, imaginea sursei de lumina (flacara lumanarii). Experienta poate fi realizata folosind orice alta sursa, cu conditia ca ecranul sa fie umbrit de paravanul mobil, adica sa fie ferit de lumina "directa" a sursei. Marimea imaginii depinde de pozitia paravanului gaurit, mai precis de distantele sursa-paravan si paravan-ecran, dar despre acest aspect putem spune mai multe dupa ce formulam o explicatie a experientei efectuate. Explicatia formarii imaginilor prin procedeul din figura 1.1 face apel la cateva concepte matematice elementare, constituind o prima modelare matematica a naturii inconjuratoare. Mai intai, vom considera ca orificiul din paravanul mobil este atat de mic, incat poate fi asimilat cu un punct. Apoi, consideram ca sursa de lumina este un domeniu geometric, format dintr-o multime infinita de puncte luminoase; aceste puncte pot fi numite surse punctuale1 de lumina, considerand ca au dimensiuni infinit mici. Asemanarea imaginii cu obiectul-sursa sugereaza faptul ca ea se formeaza prin proiectarea sursei pe ecran cu ajutorul unor drepte care trec prin orificiul punctual al paravanului mobil; de aici rezulta, evident, ca: intr-un mediu omogen, lumina se propaga, de la orice sursa punctuala spre orice punct din spatiu, in linie dreapta; traiectoria luminii, intre doua puncte date, se mai numeste "raza de lumina". Desi nu are o realitate "fizica", notiunea de sursa punctuala de lumina este o aproximatie foarte utila in multe situatii din fizica si astronomie, permitand introducerea si utilizarea eficienta a unui aparat matematic. Spre deosebire de sursele punctuale de lumina, notiuni matematice despre care, pentru a fi "conectate" cu realitatea fizica, se spune ca "au dimensiuni infinit mici", sursele reale "au dimensiuni finite". Revenind la problema marimii imaginii de pe ecran, este evident ca, daca se aseaza ecranul paralel cu axa sursei, triunghiurile cu varful in orificiul paravanului si avand ca baze sursa, respectiv imaginea ei, sunt triunghiuri asemenea. Cititorul poate deduce singur relatiile dintre marimile implicate in aceasta asemanare, intre care marimea imaginii, a sursei, distanta sursa - paravan si distanta paravan - ecran. Desigur, cititorul care are cunostinte solide de geometrie elementara poate studia si cazurile, mai complicate, in care ecranul nu este paralel cu axa sursei. b. Diametrul unghiular al Soarelui Luand Soarele ca sursa, putem forma imaginea sa utilizand un ecran mobil, care sa poata fi orientat perpendicular pe directia spre Soare (fig. 1.2). Orientarea celor doua cartoane (paravan si ecran) este relativ simpla, daca urmarim ca ele sa fie aproximativ paralele, iar umbra paravanului sa acopere ecranul.  INCLUDEPICTURE "http://director2.lefo.ro/curriculum/astronomie/Eclipsa1/Figura1-2.jpg" \* MERGEFORMATINET Figura 1.2 Daca orientarea este buna, vom obtine pe ecran un mic cerc, slab luminat, care este imaginea Soarelui; marimea imaginii depinde, evident, de distanta dintre paravan si ecran. Ne putem convinge, daca mai este nevoie, variind aceasta distanta, in limita permisa de lungimea bratelor; la o distanta de aproximativ 1 m intre ecran si paravan, diametrul imaginii Soarelui este de aproximativ 1 cm. Desigur, imaginea obtinuta in acest fel nu este nici pe departe satisfacatoare, daca vrem sa studiem suprafata Soarelui; totusi, chiar atat de modesta cum este, ea devine utila in cazul unei eclipse de Soare. intr-adevar, in aceasta situatie, procedeul rudimentar din figura 1.2 permite o urmarire a desfasurarii eclipsei partiale, lipsita de pericol pentru vederea noastra. Dar obtinerea imaginii Soarelui prin acest procedeu ofera posibilitatea efectuarii unei masuratori astronomice efective: este vorba de determinarea (masurarea indirecta) diametrului unghiular al Soarelui. Diametrul unghiular al Soarelui este unghiul maxim format de razele vizuale2 tangente la suprafata Soarelui.  INCLUDEPICTURE "http://director2.lefo.ro/curriculum/astronomie/Eclipsa1/Figura1-3.jpg" \* MERGEFORMATINET Figura 1.3 Avand in vedere drumul razelor de lumina prin orificiul paravanului, este evident (fig. 1.3) ca cele doua unghiuri cu varful in orificiul paravanului sunt egale, fiind opuse la varf. Ori, unul din cele doua unghiuri este chiar diametrul unghiular al Soarelui! Sa consideram triunghiul accesibil, cu varful in orificiul paravanului; el are ca baza segmentul d (diametrul imaginii Soarelui) si ca inaltime un segment de lungime D (distanta dintre paravane). Daca directia spre centrul Soarelui este perpendiculara pe cele doua paravane, este evident ca triunghiul considerat este isoscel (bisectoarea unghiului din varf este si inaltime); in acest caz, triunghiul este complet determinat de segmentele d si D. in consecinta, masurarea celor doua segmente determina toate elementele triunghiului, deci si unghiul u. Dar, unghiul u fiind foarte mic, incercarea de a-l masura direct - pe o figura realizata la scara, pe hartie - cu un raportor, nu poate duce decat la rezultate eronate in mod grosolan. c. Relatii exacte si aproximatii Este mai indicat sa se determine masura unghiului u prin calcul; acest lucru trebuie sa fie posibil, deoarece triunghiul care-l cuprinde este bine determinat. Tocmai pentru a rezolva astfel de cazuri a fost creata ramura matematicii numita trigonometrie; ea stabileste, printre altele, relatiile dintre lungimile laturilor unui triunghi si masurile unghiurilor sale. Pentru a se face mai functionale aceste relatii, au fost create asa-numitele functii trigonometrice care, dupa cum vom arata in alt paragraf (1.1.3 c), ar fi fost mai potrivit sa se numeasca functii goniometrice, deoarece sunt asociate fiecarui unghi. Utilizand una dintre aceste functii (tangenta) si functia inversa asociata ei (arctangenta), deducem imediat, din triunghiul considerat:  INCLUDEPICTURE "http://director2.lefo.ro/curriculum/astronomie/Eclipsa1/Formul1-1.jpg" \* MERGEFORMATINET (1.1) Desi relatia (1.1) ne ofera solutia exacta3 a problemei determinarii diametrului unghiular, este momentul sa luam in considerare si o alta varianta de calcul, mai simpla - aproximativa, e adevarat! - care poate fi aplicata datorita unei particularitati a situatiei noastre. De altfel, in modelarea matematica a fenomenelor naturale, aproximarile de diferite naturi sunt frecvente; chiar asimilarea unor obiecte reale cu unele obiecte matematice comporta, din start, un grad de aproximatie. Acest aspect a fost subliniat de noi in descrierea formarii imaginilor simple ale surselor luminoase. Vom aborda acum un alt gen de aproximatii, care apare mai tarziu, pe parcursul tratarii matematice a modelelor create pentru fenomenele naturale. Aceste aproximatii, desi nu sunt intotdeauna necesare din punct de vedere matematic, sunt sugerate de contextul concret al fenomenelor studiate si pot simplifica - de multe ori radical - aparatul matematic utilizat. in cazul de fata, diametrele unghiulare ale astrilor sunt (unghiuri) deosebit de mici; chiar Soarele si Luna au diametre unghiulare numai de ordinul unei jumatati de grad (30'). in astfel de situatii, putem stabili relatii mai simple intre unghiuri si lungimi, pe baza unei aproximatii la indemana, fara a face apel la functiile trigonometrice.  INCLUDEPICTURE "http://director2.lefo.ro/curriculum/astronomie/Eclipsa1/figura1-4.jpg" \* MERGEFORMATINET Figura 1.4 Pentru aceasta, sa consideram un cerc de raza D (fig. 1.4), in care sa inscriem diferite poligoane regulate; evident, exprimarea laturii in functie de raza trebuie sa se faca prin intermediul functiilor trigonometrice ale unghiului la centru corespunzator laturii respective. Sa ne gandim insa, acum, la un poligon regulat astfel construit incat fiecare latura sa se "vada" din centru sub un unghi de ... 1"! Acest poligon are 360( ) 60(') ( 60(") = 1 296 000 laturi; evident, orice incercare de desenare a lui nu va putea decat sa reproduca cercul initial, cu care, practic, poligonul nostru coincide! Dar lungimea cercului este lc = 2(((D; rezulta ca la 1" unghi la centru corespunde, pe cerc, o lungime de  INCLUDEPICTURE "http://director2.lefo.ro/curriculum/astronomie/Eclipsa1/Formul1-2.jpg" \* MERGEFORMATINET  Cata vreme unghiurile sunt foarte mici, lungimile coardelor cercului pot fi aproximate prin lungimile arcelor corespunzatoare, care sunt proportionale cu unghiurile; prin urmare, regula de trei, simpla, arata ca la un unghi (la centru) de u" va corespunde o lungime  INCLUDEPICTURE "http://director2.lefo.ro/curriculum/astronomie/Eclipsa1/Formul1-3.jpg" \* MERGEFORMATINET (1.2) Relatia (1.2) este aplicabila, daca u este mic, in orice situatie in care lungimea d este privita normal4 de la distanta D. In consecinta, pentru calcularea diametrelor unghiulare ale Soarelui si Lunii, prin masurarea segmentelor d si D (fig. 1.3), putem folosi relatia aproximativa:  INCLUDEPICTURE "http://director2.lefo.ro/curriculum/astronomie/Eclipsa1/formul1-4.jpg" \* MERGEFORMATINET (1.3) Relatiile (1.2), respectiv (1.3), aplicabile doar pentru unghiuri mici, au avantajul ca nu necesita utilizarea functiilor trigonometrice ale unghiurilor respective; acest avantaj, in conditiile de azi - caracterizate prin utilizarea curenta a calculatoarelor electronice - nu mai este esential. Totusi, astfel de relatii - aproximative, dar suficient de precise - au si un alt avantaj: acela de a face mai evidente legaturile dintre diferitele marimi care intervin in ele, legaturi care par uneori "nebuloase" datorita aparatului matematic. d. Erori absolute si erori relative Cititorul care doreste sa efectueze practic acesta determinare va trebui sa masoare cu cea mai mare acuratete lungimile d si D; ori, in conditiile sugerate de figura 1.2, acest lucru este greu de realizat, chiar daca masuratorile sunt efectuate de o echipa. Este bine ca ansamblul celor doua paravane sa fie in prealabil rigidizat printr-o structura din lemn sau din metal; sau, si mai bine, se poate construi un tub solar dintr-un cilindru de cativa centimetri diametru, din carton sau material plastic. Unul din capetele tubului se inchide cu un carton perforat in centru, iar pe celalalt capat se lipeste, cu banda adeziva, o bucata de hartie de calc, bine intinsa si asezata perpendicular pe axul tubului. In acest caz, D este chiar lungimea tubului solar, iar singura marime care ramane de masurat este d; orientarea tubului pe directia spre Soare se face urmarind ca umbra lasata de tub pe sol sa fie minima. O echipa de doi observatori va putea asigura, pentru cateva secunde, atat fixitatea necesata a tubului solar cat si masurarea cat mai atenta a lui d. Desi "solar", tubul confectionat de noi poate fi utilizat si pentru determinarea diametrului unghiular al Lunii. Recomandam cititorului sa faca si aceasta determinare! O alta chestiune este aceea de a alege cea mai potrivita marime pentru "tubul solar" sau pentru structura rigida echivalenta. Prima tentatie este de a alege o lungime minima a acestuia, pentru a reduce la minim dificultatile de constructie si de manevrare; dar, daca aceasta lungime este mica, se micsoreaza si diametrul imaginii Soarelui! Ori, masurand acest diametru cu o rigla, eroarea ("absoluta") probabila este cam 0,5 mm; daca aceasta eroare se face asupra unui diametru de 1 cm, ea reprezinta 5 % din marimea masurata, iar daca se face asupra unui diametru de 2 mm, ea reprezinta 40 % ! Eroarea absoluta este un parametru caracteristic pentru instrumentul de masurare utilizat, in timp ce eroarea "relativa" (5 %, respectiv 40 % in cele doua cazuri mentionate) depinde de circumstantele in care se realizeaza masuratoarea. Cu un instrument dat, vom face mereu - aproximativ - aceeasi eroare absoluta; totusi, eroarea relativa poate fi redusa substantial, daca ne incadram in circumstante adecvate acestui scop. Astfel, in cazul nostru, pentru a reduce cat mai mult eroarea relativa de masurare, este necesar sa avem o imagine cat mai mare a Soarelui, deci sa luam o lungime cat mai mare pentru tubul solar. Totusi, nu trebuie sa cadem in cealalta extrema: daca aceasta lungime este prea mare, imaginea Soarelui - mai mare, e adevarat - va fi prea putin luminoasa si vom avea probleme serioase incercand sa-i vedem marginile! e. Diametrul liniar al astrilor; alte aplicatii Diametrul unghiular al unui astru depinde, in mod esential, de distanta dintre observator si astrul respectiv; acest diametru nu ofera, deci, o informatie substantiala despre marimea astrului observat.  INCLUDEPICTURE "http://director2.lefo.ro/curriculum/astronomie/Eclipsa1/figura1-5.jpg" \* MERGEFORMATINET Figura 1.5 Totusi, daca distanta pana la astru a putut fi determinata printr-un procedeu oarecare, cunoasterea diametrului unghiular isi dovedeste din plin utilitatea. Fie, in figura 1.5, u"/2 raza unghiulara a astrului, d distanta astru-observator, iar r raza (liniara) a astrului. Relatia (1.2), aplicata in acest caz, capata forma  INCLUDEPICTURE "http://director2.lefo.ro/curriculum/astronomie/Eclipsa1/formul1-5.jpg" \* MERGEFORMATINET (1.4) de unde rezulta imediat raza astrului observat:  INCLUDEPICTURE "http://director2.lefo.ro/curriculum/astronomie/Eclipsa1/formul1-6.jpg" \* MERGEFORMATINET (1.5) Analog, tot din (1.4) se poate exprima si distanta astru-observator:  INCLUDEPICTURE "http://director2.lefo.ro/curriculum/astronomie/Eclipsa1/formul1-7.jpg" \* MERGEFORMATINET (1.6) Cititorul poate deduce singur si relatiile exacte corespunzatoare. Aflarea diametrului liniar al astrilor - pe baza relatiei (1.4) - evidentiaza, de pe acum, importanta deosebita pe care o are determinarea distantelor cosmice. Din pacate, aceasta problema este foarte greu de rezolvat si, de aceea, formarea unei imagini sigure despre Univers a durat deosebit de mult, la scara istoriei. 2. Activitati practice: (1) Construiti un tub solar, asa cum este indicat in paragraful e. (2) Transmiteti pe retea dimensiunile exacte ale tubului construit de voi. Calculati si transmiteti eroarea relativa a instrumentului vostru. (3) Determinati diametrul unghiular al Soarelui si al Lunii si transmiteti rezultatul. (4) Continuati determinarile pe cit posibil zilnice pentru diametrul unghiular al Soarelui si al Lunii, urmind ca in perioada de partajare a rezultatelor sa transmiteti toate datele obtinute de voi sub forma: Data |Diametru Soare | Diametru Luna (5) Utilizand Anuarul astronomic editat in fiecare an de Institutul Astronomic sau orice alta sursa analizati si explicati datele legate de diametrul aparent al Soarelui. 3. 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