X-Ray Notes, Part III

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Maryann Collins
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1 oll 6 X-y oe 3: Pe X-Ry oe, P III oe Deeo Coe oupu o x-y ye h look lke h: We efe ue of que lhly ffee efo h ue y ovk: Co: C ΔS S Sl o oe Ro: SR S Co o oe Ro: CR ΔS C SR Pevouly, we ee he SR fo ye hv pxel ue o o Poo R.V. If he e vlue of he phoo ou fo pxel, he he l o oe o of fo h pxel wll e: S SR S The poly ey fuo fo he Poo R.V. : e p k k k! k k We wll ow ee he poly uo of eee phoo. Suppoe he e x-y phoo v he eeo e Poo h he eeo h effey, ee pevouly. We vew he eeo y o ye whh he phoo eee wh poly p :

2 oll 6 X-y oe 3: Pe whee Q k P { k phoo e eee} P Bol k k Poo k k k! k e p p k!! k! k k e p p k!! k k e p p e k! p k e p k! Poo p { k phoo e eee k phoo e e} P{ k phoo e e} Thu, he eee phoo e lo Poo ue, u wll hve poly he SR of he eee phoo ow: SR e Coe:. I lo ey o how h he ue of phoo h e o eee lo Poo poe wh pee poly -.. The u of Poo poee lo Poo. 3. Flly, f he e phoo e Poo, he he ue of phoo h eh he eeo wll lo e Poo. Aeuo poee h epeely ffe phoo wok exly ove.

3 oll 6 X-y oe 3: Pe 3 Ce Poo Poee Coe x-y phoo h e wh he eeo eee howe of phoo: We oel he ue of lh phoo,, he ue of o vle: X whee X he ue of lh phoo fo x-y phoo ue he ue of x-y phoo Poo. Fo ow, lo vewe he of lh oveo poe. We ow eee he he of : X X X X j j flly:

4 oll 6 X-y oe 3: Pe X X Th h wo opoe: he f epee he ve he ue of phoo fo ve u leh he eo epee he vo h oe fo he he leh of he u. Th lo e we : we ow we ew expeo fo he SR: SR Th l p of h expeo e houh of he SR eo e. If he vey le, he h poe wll eul eelly o lo of SR. Fo ol e Poo poee, e..: Z W whee Z Poo he: Z W Z Z W follow h: W W he SR e we :

5 oll 6 X-y oe 3: Pe 5 SR Fo ol e poee, h uo oue: SR 3... If y of he e e ll o eve loe o oe he he SR wll e eue. Thu, he e of eeo ye, po h he pou of ll of he kep le. Fo exple, we w >>, >>, 3 >>, xple Coe ll ee h poue 5 lh phoox-y phoo eee h ke ouhly lh phoo o ove lve hle ple o oevle he evelope fl h,. The SR euo fo wll he e: The e, whh le, oe oue o y lo of SR. The e he oue of SR euo. Ovell Sye Repoe SR xple Le oe he follow ye:

6 oll 6 X-y oe 3: Pe 6. The oue fuo e: exp whee h u of phoo.. The o fuo : exp whee exp epee he leo h we e y o ee. The o he o fuo C. 3. We ll ee he eoe epoe fuo he eo of lh pue y he peue of he fe op oupl: exp h.. Le,. 5. z: he oje fo fo he oue fo fo The e 5 lh phooeo,. e pue y he opl oupl, he opol 3 phoophoo he lh plfe,.5 fo he fe loe, opl oupl effee of he CCD. The fuo he ue : **exp exp ** exp ** ** h I Th ovoluo o ely olve he Foue o: exp, 8 exp exp, exp ρ δ ρ ρ δ ρ ρ v u v u I k o he e o: exp 8 I Fo h expeo, le he leo o fe he ye epoe wll e:

7 oll 6 X-y oe 3: Pe C whh eue y ou % fo he pu fuo. The ue of x-y phoo h fll wh pxel he e of he opl oupl : 8 The fl SR wh ou 3 : 8 CR whee he SR euo fo : wh he opol lh plfe, he SR euo fo : whh eoe oo pe of e ye e efe, fo exple efoe he low opoe lu he hu eye.

8 oll 6 X-y oe 3: Pe 8 Copo See X-y Coe he follow oje wh x-y opque oe: he oupu e h look lke h: o whh we efe o C ΔS S o o oe o CR ΔS. ow, oe h he ee phoo hee oe fo of he ee phoo wll e fow wll eee ol phoo he fl e. The uo of he ee phoo wll look oeh lke he oje ovolve wh he fow e uo. The fl e wll e he u of he e phoo he ee phoo. By e S he ee phoo wll eue oh he o he o o oe o. How y phoo e ee? Deve fo ovk, Pole 3. Le look oje of leh l hv euo oeffe. Le e pe

9 oll 6 X-y oe 3: Pe 9 he ue phoo e upo he oje h he ue of phoo h hve o ee eph x x. The ue of ee phoo evl x wll e: x x he ol ue of ee phoo wll e: x l x x l l x x exp x x exp l oe h -exp-l he ol ue of phoo h e wh he oje. Ave oe X-Ry I Thee e wo k of ve oe h we wll oe. The f zeo-e ve oe, fo exple, eleo oe ze e. I h e, he e vlue oe he, u he ve oe. I o e, he ve oe wll e epee of he Poo vo he eeve phoo hu, he ve wll :

10 oll 6 X-y oe 3: Pe whee he ve of he ve oe he Poo ve. Thu, he SR : S SR The ohe k of ve oe h we wll oe e. Se e lo Poo ue, zeo e, we ve ue efoe, oe ffe he o. Coe he e whee we hve e phoo ee phoo l ffeee of Δ. The ol o w: ou eue o : C C Δ Δ We lo kow h he ve of he kou l wll e he u of he ve of he o oue Poo poee: hu, he eue o o oe o wll e: CR Δ C C The e CR euo fo o Ψ, whee e phoo o e phoo. Ψ he o of

11 oll 6 X-y oe 3: Pe SR Reuo To e e of how y phoo e ee ke he eeo, we look exple wh oop oje wh euo oeffe e opoe : F, he ue of ee phoo eee eh eel hke : z z z e z z We ke vey of upo:. Ioe olque. Aue pllel y eoey fo e ey 3. Aue eey epee. ele ulple e 5. Aue oop e Thu, fo he ue of ee phoo, oe fo, Fz, wll e pue: Ω z F z e Lz The ue of ee phoo he eeo wll he e:

12 oll 6 X-y oe 3: Pe G z z e z e e z z z z F L L z z L L L Ω Ω whee G eoe, oje epee fo whh h u of leh. Fo h we oje, G.5L h eul fo Ωz fo lo, l oje G. Theefoe: G Ψ If we ke ypl vlue fo euo oeffe fo we kev,.6, L we wll le G.L, he: Ψ.8 eul euo of SR of:.5 Ψ 5% euo SR. Se Reuo G The o oo wy of eu e houh he ue of e euo : whee he e ou of oe hh el lke P, W h wll lok y phoo h ke. The wok pplly y u ow o he epe le fo ee phoo, Ω:

13 oll 6 X-y oe 3: Pe 3 We efe e euo fo: Ω' z z R Ω z z whee Ω z he epe le of he e euo. I o o eu he e, h lo eul euo of e phoo. We efe effey of he y oe e phoo loke y eue y he flle el: equo, h wll e: The CR wll ow e: exp f h CR C R C RΨ whee R Ψ he ew SR euo fo.

14 oll 6 X-y oe 3: Pe The Relohp of Poo Poe o he xpoel R.V. Le T e expoel R.V. h ee he e ewee eve Poo poe. The evo follow. Rell h he poly h eve ou evl Δ wll e p λδ. Alo, oe h he poly h o eve ou evl Δ wll e q - λδ. ow, uppoe he we w o kow wh he poly h o eve oue ewee. Th he e y h we hve Δ vevl whh o eve ou. If hee evl e epee h y h he phoo o e wh eh ohe o e o oe oup o wheve, he poly h o eve oue ewee wll e q : Δ { o eve ou, } Δ P λ we eee h fuo Δ : P λδ λ λ { o eve ou, } λδ e The poly ey fuo of T, f, ee he poly h eve ou e poly uo fuo of T el of f ee he poly h eve ou y e wll e equl o: l Δ λ e, fo F P{ o eve ou, }, fo < he poly ey fuo he evve of h fuo: λ λe, fo f, fo < The expoel R.V. ouou R.V. of he e ewee eve ee : whh h e ve of: T ~ xpoelλ T λ T λ

15 oll 6 X-y oe 3: Pe 5 eoyle Popey The expoel R.V. eoyle, e h uo ey of eve e he fuue o ffee y p eve, h, y po e, he e ul he ex eve expoel R.V. wh pee λ. Th he e y h ju eue we hve ee eve lo e, we e o oe lkely o hve eve oo. Ju lke he le flly. Speflly, { T > T > } P{ T > } P whh y ve h eve h oue y e, he poly h eve wll o ou y e wll e e he poly h o eve ou,. Poof: P { T > T > } F F exp λ exp λ exp λ F P { T > }

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