Formulas Flashcards
X(ij) =
P(ij) + T(ij) + E(ij) + C(ij)
P(ij) = contribution from pixel charge due to photons from the astronomical source
T(ij) = contribution from pixel charge due to thermal effects
E(ij) = contribution from readout process
C(ij) = contribution from pixel charge produced by cosmic ray hits
where X(ij) is in DN
B(ij) =
T(ij) + E(ij)
N(ij) =
[X(ij) -B(ij)]/F(ij)
X(ij) - ‘raw’ image
N(ij) - DN per pixel due to astronomical sources
B(ij) - background
F(ij) - flat field
N =
n/g ± σ(0)
N is the DN read out
g is the gain
n is the number of photo-electrons
σ(0) is the readout noise
σ(n) =
√Ng in e-
√N/g in DN
σ^2(N) =
σ(0)^2 + N/g
n(ij) =
N(ij) g
where n(ij) is the number of photo-electrons produced by a pixel
n(ph) =
n(ij)/Q
where Q is the quantum efficiency
λ(mean) =
[ ∫ P(λ)s(λ)dλ]/[ ∫P(λ)s(λ)dλ]
s(λ) is the normalised source spectrum
P(λ) is the probability of detecting a photon of wavelength λ
J =
[n(ph) h v(mean)]/[A(pix) Δt]
=
F (D/θf)^2
where θ = y/f
y is the pixel size
A is the pixel area
F is the flux
f the focal length
θ the angle subtended by the object being imaged
Δt is the exposure time
dynamic range =
full well/total noise
total noise^2 =
sum of all noise sources squared
i.e.
σ^2(dc) + σ^2(r) + …
max CTE =
CTE^(charge transfers)
charge transfers =
for a 2048 x 2048 pixel format
2047 + 2047 = 4094
max fraction of CTE lost =
1 - max CTE
O(x(2)) =
of a single source
I(x(1)) PSF(x(2)-x(1))
I =
F^-1[F(0)/F(PSF)]
O(λ(2)) =
single line
I(λ(1)) ILP(λ(2)-λ(1))
O(λ(2)) =
a spectrum
I*ILP = (∞ ∫ 0) I(λ(1)) ILP(λ(2)-λ(1))dλ(1)
G(u) ∝
(∞ ∫ -∞) F(v(LOS)) S(u-v(LOS))dv(LOS) = F*S
CCF(v(LOS)) =
(∞ ∫ -∞) G(u) S(u-v(LOS))du
CCF(τ) =
(∞ ∫ -∞) f(t) g(t-τ)dt
τ is a variable time lag/shift
ACF(τ) =
(∞ ∫ -∞) f(t) f(t-τ)dt
τ = nT
max ACF
τ = (2n+1)T/2
get a min ACF
PSD(v) ∝
|F[f(t)]|^2
PSD(v)^2 =
C(v)^2 + S(v)^2
Wavelet transform W(a,b)
see formula sheet
changing a -> 1/freq change
changing b -> time shift
Resolving power =
R = λ/Δλ = Nn
N = total number of lines on the grating
n = order number of spectrum
Angular dispersion
dθ/dλ = n/acosθ
a = spacing between lines of grating
θ = angle at which spectral feature is formed
Grating response width
W = λ/Na
Interpolation
N(i’) = N(i(1)’) + m(1-Δi’)
m = [N(i(2)’) - N(i(1)’)]/[i(2)’-i(1)’]
Δi’ = i(2)’-i(1)’
sky background
N(b)(λ) = mλ + c
total real counts in the source
(λ2 Σ λ=λ1) [N(λ) - N(b)(λ)]
Significance =
SNR in σ = Ns/√[Ns+2Nb] ~ √Ns
line profile =
φ(λ) = [Bc - Bλ]/Bc
Bc = continuum intensity
Bλ = line intensity
equivalent width =
W =(∞ ∫ 0) [Bc - Bλ]/Bc dλ
Optical depth
Bλ/Bc = e^(-τ(λ))
-> W = (∞ ∫ 0) 1-e^(-τ(λ)) dλ
Weak spectral line
τ(λ) «_space;1
-> W = (∞ ∫ 0) τλ dλ
a narrow slab of absorbing gas has optical depth
τ(λ) = nσ(λ) Δs = σ(λ)N
Δs = thickness
column density N = nΔs
σ(λ) = absorption cross-section
Boltzmann factor
p(i) = p(0)g(i) exp[-Ei/kT]
E(i) is the energy of this state
g(i) is the degeneracy
p(0) is a constant found by normalising the probability over all possible states of energy E(i)
the average number of transitions per second per unit volume
n(ij) = n(i) n(e) C(ij)
C(ij) is the collisional rate coefficient and depends on the electron speed and the atomic properties
n(e) is the number of electrons per unit volume
n(i) is the number of atoms in state i per unit volume
in the case where upwards transitions are collision-dominated and downward transitions are due to spontaneous radiation
coronal approximation
n(i) n(e) C(ij) = n(j) A(ji) = I(ij)
detailed balance
n(i) n(e) C(ij) = n(j) A(ji) + n(j) n(e) C(ji)
emission lines from a hot gas are broadened due to random particle motions
[λ - λ(0)]/λ(0) = v/c
turbulent broadening
(Δλ(turb)/λ(0))^2 = (v(turb)/c)^2
