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1 1 2 flux( ) flux 2.1 flux Flux( flux ) Flux [erg/sec/cm 2 ] erg/sec/cm 2 /Å erg/sec/cm 2 /Hz 1

2 Flux -2.5 Vega Vega ( Vega ) AB cgs F ν [erg/cm 2 /s/hz] m(ab) = 2.5 logf ν 48.6 V-band 2.2 Flux Suprime-Cam SDSS Johnson, Kron-Cousins, SDSS, HST 2MASS, MKO, CIT, UKIRT flux flux flux flux 2.3 CCD data = ((object + sky) t obs flat + dark t dark ) gain object = (data bias) gain dark t dark flat t obs data t dark, t obs bias, sky: sky + bias (bias), dark, flat: gain, (flat): BIAS: 0 CCD DARK FLAT 2

3 CCD data = ((object + sky) t obs flat + dark t dark ) gain object = (data bias) gain dark t dark

3 STANDARD STAR object = (data bias) gain dark t dark flat t obs 1. (bias) 2. (dark) 3. (flat) Flux ( ) sky Flux 2.4 bias,dark,flat flat Flux F obs = F 0 exp( kx) 3

(dark) 3. (flat) 4. 5. Flux ( ) sky Flux 2.

4 m obs = m 0 + k x + m x ( ) z 1/cos(z) = sec(z) k m R R R flux R flux R (V-R) R = R + b(v R) b Suprime-Cam Suprime-Cam i SDSS i

5 m/s Flux flux flux X 1, X 2,..., X n Y = f(x 1, X 2,..., X n ) Y X Y y 1, (x 1 ) 1, (x 2 ) 1,..., (x k ) 1 X n X n data = ((object + sky) t obs flat + dark t dark ) gain + bias object, sky, dark, bias 5

6 P(k)(x) = kx e x x! 1 k FITS data = ((object + sky) t obs flat + dark t dark ) gain dark t t dark t t P(dark) t) flat ((object + sky) t obs flat + dark t dark ) + bias N(m, σ 2 ) = 1 e (x m)2 2σ 2 2πσ ax N(am, (aσ) 2 ) X + Y N(m X + m Y, σ 2 X + σ2 X ) X,Y Cov(X,Y) E(aX + by ) = ae(x) + be(y ) V (ax) = a 2 V (X) V (X + Y ) = V (X) + V (Y ) + 2Cov(X, Y ) X Y Cov(X, Y ) = 0 V (X + Y ) = V (X) + V (Y ) Cov(X + Y, Z) = Cov(X, Z) + Cov(Y, Z) 1 LaTeX 6

sky) t obs flat + dark t dark ) + bias N(m, σ 2 ) = 1 e (x m)2 2σ 2 2πσ ax N(am, (aσ) 2 ) X + Y N(m X + m Y, σ 2

7 Cov(X, Y ) = Cov(Y, X) Cov(X, X) = V (X) 2 data = ((object + sky) t obs flat + dark t dark ) gain 3 ((object + sky) t obs flat + dark t dark ) gain ((object + sky) t obs flat + dark t dark ) gain 2 + bias + bias + V (bias) σ p(x) E(X) V (X) = σ = V (X) (X E(X)) 2 p(x)dx (Signal) (Noise) S/N S/N 3 28 S/N=5 S/N=5 28 S/N m S/N S/N object t obs flat, sky t obs flat, dark t dark, bias bias (readout noise) r S/N S object t obs flat/gain N object t obs flat, sky t obs flat, dark t dark, bias ((object + sky) t obs flat + dark t dark ) gain 2 + V (bias) 2 Cov(X, Y ) E((X E(X))(Y E(Y ))) 3 7

2 (Signal) (Noise) S/N S/N 3 28 S/N=5 S/N=5 28 S/N m 1.

8 S/N S/N = object t obs flat/gain ((object + sky) tobs flat + dark t dark )/gain 2 + r 2 S/N t = t obs = t dark S/N object flat gain r t S/N S/N object flat (object + sky) flat + dark t S/N ( flux mean median clipped mean clipped mean CCD N(0,1) N x( a=100) H h=h/(n+h) f(x) = (1 h) 2π e x2 2 + hδ(x a) E(X)=ah (N-H+1)/2 erf(x) = 1 2(1 h) erf(x) = x e z2 2 dz h H x

2 1 ( flux mean median clipped mean 3.2.1 clipped mean CCD N(0,1) N x( a=100) H h=h/(n+h) f(x) = (1 h) 2π e x2 2

9 N E X E = 1 N N i=1 X i X i 1/ N V (E) = 1 N 2 N V (X i ) = 1 N i=1 f(x) y g(y) y (N-1)/2 y (N-1)/2 y k=(n-1)/2 ( y ) k ( ) k g(y) = N C k N k C k f(y) f(x)dx f(x)dx y N π/2n π 2N 1.25 N 25% N N=3,5,7 16%,20%,21% clipped mean (clip) clip 1/ N(1 h) clipped mean S/N S/N X i N(m, σ 2 i ) X = w i X i wi = 1 σ i X E(X) E(X) = w i E(X i ) = w i m = m V (X) = w 2 i V (X i ) = w 2 i σ 2 i w i 9

25 N 25% N N=3,5,7 16%,20%,21% clipped mean (clip) clip 1/ N(1 h) clipped mean 3.2.2 S/N S/N X i

10 L = w 2 i σ 2 i λ( w i 1) L w i = 2w i σ 2 i λ = 0 w i = λ 2σ 2 i wi = 1 λ 1 2σ 2 i = 1 λ = 1 1 2σ 2 i w i = 1 σ 2 i 1 σ 2 i S/N flux Readout i Y i Y i p(object i ) + p(sky i ) object, sky flux a i m=object+sky Y i p(a i m) N(a i m, a i m) X i = Y i a i N(m, m a i ) w i = a i m sum ai m = a i ai X = a i X i Yi = ai ai S/N flux Readout S/N flux 0.5, 1,

object, sky flux a i m=object+sky Y i p(a i m) N(a i m, a i m) X i = Y i a i N(m, m a i )

11 3.2.3 S/N -a (0 < a < 1) v (x ) = (1 a)v(x) + av(x + 1) x+0.5, y ADU 43ADU S/N σ v (x ) = (1 a)v(x) + av(x + 1) V ((1 a)v(x) + av(x + 1)) = (1 a) 2 σ 2 + a 2 σ 2 a= σ 2 σ 2 S/N [0,N-1] N V (v(0) v(n 1)) = V (v(0)) +...V (v(n 1)) = Nσ 2 v (0) = (1 a)v(0) + av(1) v (1) = (1 a)v(1) + av(2)... v (N 1) = (1 a)v(n 1) + av(n) Cov(v (0), v (1)) = Cov((1 a)v(0) + av(1), (1 a)v(1) + av(2)) 4 11

2 a=0.5 0.5σ 2 σ 2 S/N [0,N-1] N V (v(0) +... + v(n 1)) = V (v(0)) +.

12 Cov(v (0), v (1)) = (1 a) 2 Cov(v(0), v(1)) + a 2 Cov(v(1), v(2)) + a(1 a)cov(v(1), v(1)) + a(1 a)cov(v(0), v(2)) v(x) Cov(v(1), v(1)) = V (v(1)) = σ 2 Cov(v (0), v (1)) = a(1 a)σ 2 V (v (0) + v (1)) = V (v (0)) + V (v (1)) + 2Cov(v (0), v (1)) = 2(1 a(1 a))σ 2 v (0) +...v(n 1) = (1 a)v(0) + v(1) v(n 1) + av(n) ((1 a) 2 + (N 1) + a 2 )σ 2 N S/N RMS RMS 12

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