Derivations

Question 1

background subtraction errors

Answer 1

N(s+b) = N(s) + N(b) = total counts from source plus background

N(s) = N(s+b) - N(b)

= [N(s) + N(b)] -N(b)

σ^2(S) = N(s+b) + N(b)

= N(s) + 2N(b)

Question 2

N(s) =

Answer 2

rate x Δt

where Δt = integration

Question 3

A(jI) =

Answer 3

einstein coefficient

Question 4

I(jI) =

Answer 4

photon emission rate per unit volume

Question 5

line broadening derivation

Answer 5

[λ-λ(0)]/λ(0) = v/c

φ(th) ∝ exp(-mv^2/2kT)

substitute for v

= φ(0) exp(-([λ-λ(0)]/Δλ(th)^2)

[Δλ(th)/λ(0)]^2 = 2kT/mc^2 = (v(th)/c)^2

Question 6

v(TOT)^2 = v(th)^2 +v(turb)^2

Answer 6

φ(TOT) = φ(th) * φ(turb)

[Δλ(turb)/λ(0)]^2 = [v(turb)/c]^2

convolution theorem

fourier of φ(TOT) ∝ e^-π^2(Δλ(th)^2+Δλ(turb)^2)s^2

take the inverse Fourier transform

Δλ(TOT)^2 = Δλ(th)^2 + Δλ(turb)^2

Δλ(TOT)^2 = λ(0)^2 v(TOT)^2/c^2

v(TOT)^2 = v(th)^2 +v(turb)^2