Risoluzione angolare

La risoluzione angolare è il minimo angolo che un sistema ottico (come una lente, un microscopio o un telescopio) è in grado di distinguere, senza che il fenomeno della diffrazione confonda l'immagine. Il potere risolvente angolare è la capacità del sistema ottico di distinguere due immagini in base al diverso angolo con cui vengono proiettate.

Definizione matematica

Si definiscono risolti due punti immagine se il massimo centrale della figura di diffrazione di uno cade (almeno) sul primo minimo della figura di diffrazione dell'altro.

Generalmente si tratta con sistemi ottici circolari (ad esempio sistemi composti da lenti e specchi circolari); in questo caso l'angolo $\varphi _{0}$ per cui si verifica questa condizione vale:

\varphi _{0}\cong 1,22{\frac {\lambda }{D}}

dove $\lambda$ è la lunghezza d'onda della radiazione e $D$ la sezione dell'apertura attraverso la quale passa la luce.

Il potere risolvente angolare è l'inverso di questo angolo:

{\frac {1}{\varphi _{0}}}\cong {\frac {D}{1,22\lambda }}

Potere risolvente lineare

Il potere risolvente lineare di un sistema ottico è la capacità di distinguere due punti oggetto in base alla loro distanza lineare. Di conseguenza, la risoluzione lineare è la minima distanza tra due oggetti affinché il sistema ottico li possa distinguere.

Chiamando $\varepsilon$ questa distanza, e $s$ la distanza tra il piano su cui giacciono gli oggetti e la lente (o specchio), possiamo esprimere la risoluzione angolare:

\varepsilon =s\tan {\varphi _{0}}\cong s\tan {\left(1,22{\frac {\lambda }{D}}\right)}

Per angoli molto piccoli (come nel caso dei telescopi), per i quali si può approssimare $\tan {\theta }\cong \theta$ , questa si può semplificare:

\varepsilon \cong 1,22{\frac {s\lambda }{D}}

Alcuni esempi:

Caso pratico	Diametro telescopio (m)	Distanza oggetto (m)	Minimo dettaglio osservabile (m)
Hubble/Superficie terrestre	2,4	600×10³	${\frac {1,22\times 600\times 10{^{3}}\times 555\times 10^{-9}}{2,4}}=0,17$
Hubble/Luna	2,4	380×10⁶	107

Diametro D necessario per avere risoluzione $\varepsilon$ su corpo distante s:

$D={\frac {1,22\times s\times \lambda }{\varepsilon }}$

Caso pratico	Diametro telescopio (m)	Distanza oggetto (m)	Minimo dettaglio osservabile (m)
Telescopio orbitale verso superficie terrestre	$D={\frac {1,22\times 600\times 10^{3}\times 555\times 10^{-9}}{10^{-3}}}=406$	$600\times 10^{3}$	$10^{-3}$
Telescopio terrestre verso Luna	$D={\frac {1,22\times 380\times 10^{6}\times 555\times 10^{-9}}{1}}=257$	$380\times 10^{6}$	$1$

Bibliografia

Claudio Oleari e Andrea Peri, Schede di OTTICA, 2006.

Voci correlate

Collegamenti esterni

(EN) resolution, su Enciclopedia Britannica, Encyclopædia Britannica, Inc.

Portale Astronomia

Portale Fisica

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