ν_{g} = \frac{eB_{⊥}}{2πγmc} = \frac{ν_{∘}}{γ}

ν′_{g} ∼ \frac{ν_{g}}{sin^{2} θ}

Δt ∼ \frac{1}{2πν_{g}γ^{3}}

ν_{c} = \frac{3}{2}γ^{2}ν_{∘} = 16.08 × 10^{6}B_{⊥}E^{2} MHz

\frac{dE}{dt} = −119.7B_{⊥}^{2}E^{2} GeV/yr

I(ν) = ∫_{0}^{∞}N(E)P (ν)dE

I(ν) = \frac{\sqrt{3}e^{3}L}{8πmc^{2}} (\frac{3e}{4πm^{3}c^{5}} )^{(p−1)/2} N_{∘}B_{⊥}^{(p+1)/2}ν^{(1−p)/2}a(p)

Π = \frac{p + 1}{p + 7/3}

R_{s} = 1.17 (\frac{E_{∘}}{ρ_{1}} )^{1/5} t^{2/5}

v_{s} = 0.4R_{s}/t

t = 3.5 × 10^{4} (\frac{E_{∘}}{10^{51} erg}) n_{1}^{−9/17} yr

v = 230n_{1}^{2/17} (\frac{E_{∘}}{10^{51} erg})^{1/17} km s^{−1}

I(ν) = I(ν_{∘}) (\frac{ν}{ν_{∘}} )^{α} e^{−τ_{ν∘}( \frac{ν}{ν∘} )^{−2.1}}

τ_{ν} = 1.643 × 10^{5}a (T_{e}, ν) (ν^{−2.1}) (EM T_{e}^{−1.35})

EM = ∫_{L} n_{e}^{2}dl

I = I_{bg}e^{−τ} + ∫_{0}^{τ} \frac{j}{κ}e^{−t}dt + I_{fg}

I = I_{bg}e^{−τ} + \frac{2kT_{e}ν^{2}}{c^{2}} (1 − e^{−τ})

S = ∫_{source} I_{ν}dΩ ≈ \frac{2kν^{2}}{c^{2}} ∫ T_{source}dΩ

S_{ν} = S(1 − e^{−τ})

Δϕ = r_{e}λaΔN_{e}

θ_{s} = \frac{1}{π} r_{e}λ^{2}ΔN_{e}\sqrt{L/a}

ρ(d) = e^{−d^{2}/2a^{2}} ⟨V_{m}⟩ = V e^{−Δϕ^{2}d^{2}/2a^{2}}

θ_{s} = \frac{\sqrt{2 ln 2}}{π}r_{e}λ^{2}ΔN_{e}\sqrt{L/a}

\sqrt{1 − l^{2} − m^{2}}

V (u, v, w) = ∫∫ I(l, m)P (l, m)e^{−2πι(ul+vm+w(\sqrt{1−l^{2}−m^{2}}))} \frac{dldm}{\sqrt{1 − l^{2} − m^{2}}}

\sqrt{θ_{r}}

\sqrt{1 − l^{2} − m^{2}}

V (u, v, w) = ∫∫∫ F (l, m, n)P (l, m)e^{−2πι(ul+vm+wn))}dldmdn

F (l, m, n) = \frac{I(l, m)δ(\sqrt{1 − l^{2} − m^{2}} − n)}{\sqrt{1 − l^{2} − m^{2}}}

[ \begin{matrix}X \\ Y \\ Z \end{matrix} ] = [ \begin{matrix}− sin L & cos L & 0 \\ 0 & 0 & 0 \\ sin L & cos L & 0 \end{matrix} ] [ \begin{matrix}x \\ y \\ z \end{matrix} ]

[ \begin{matrix}u \\ v \\ w \end{matrix} ] = [ \begin{matrix}sin H & cos H & 0 \\ − sin δ cos H & sin δ sin H & cos δ \\ cos δ cos H & − cos δ sin H & sin δ \end{matrix} ] [ \begin{matrix}X \\ Y \\ Z \end{matrix} ] s b s b b e(b⃗_{1}, t) = ∫_{P} E(s⃗, b⃗_{1}, t)ds⃗ b b s s b

E(s⃗_{1}, b⃗_{1}, t) = \frac{ξ(s⃗_{1}, t − τ_{1})}{R_{1}}e^{−2πιν(t−τ_{1})}

Γ(s⃗_{1}, s⃗_{2}, τ) = lim_{T →∞} \frac{1}{2T} ∫_{−T}^{T} ξ(s⃗_{1}, t)ξ^{⋆}(s⃗_{2}, t − τ)dt

= ⟨ξ(s⃗_{1}, t)ξ^{⋆}(s⃗_{2}, t − τ)⟩ s s s s s s Γ(s⃗, τ) = ⟨ξ(s⃗, t)ξ^{⋆}(s⃗, t − τ)⟩ s b b

γ(b⃗_{1}, b⃗_{2}, τ) = lim_{T →∞} \frac{1}{2T} ∫_{−T}^{T} e(b⃗_{1}, t)e^{⋆}(b⃗_{2}, t − τ)dt

= ⟨e(b⃗_{1}, t)e^{⋆}(b⃗_{2}, t − τ)⟩ b

γ(b⃗_{1}, b⃗_{2}, 0) = ∫ ⟨E(s⃗, b⃗_{1}, t)E^{⋆}(s⃗, b⃗_{2}, t)⟩ds⃗

= ∫ ⟨ξ(s⃗, t − τ_{1})ξ^{⋆}(s⃗, t − τ_{2})⟩e^{−2πιν(τ_{2}−τ_{1})} \frac{ds⃗}{R_{1}R_{2}}

= ∫ ⟨ξ(s⃗, t)ξ^{⋆}(s⃗, t − τ_{t})⟩e^{−2πιντ_{t}} \frac{ds⃗}{R_{1}R_{2}}

b b b b s b s b b b γ(b⃗_{1}, b⃗_{2}, 0) = γ(b⃗_{1} − b⃗_{2}, 0) = γ(b⃗_{12}, 0) b s

τ_{t} = \frac{b⃗_{12} · s⃗}{c} = \frac{u_{12}l + v_{12}m + w_{12}n}{c}

b s b s s s b b b b b b s

\sqrt{1 − l^{2} − m^{2}}

s s s b s b s s s s s s s s b b

γ(u, v, w) = ∫∫ I(l, m)e^{−2πι(ul+vm+w(\sqrt{1−l^{2}−m^{2}}−1))} \frac{dldm}{\sqrt{1 − l^{2} − m^{2}}}

γ(u, v) = ∫∫ I(l, m)e^{−2πι(ul+vm)}dldm S(u, v) = δ(u − u(t), v − v(t))

s_{o}

s_{o} S_{1} = e_{1} + ϵ_{1} S_{2} = e_{2} + ϵ_{2} = e_{1}e^{2π \frac{b_{12}·s⃗o}{c}} + ϵ_{2}

V (u, v, w) = ⟨S_{1} · S_{2}^{⋆}⟩

= ⟨e_{1} · e_{2}^{⋆}⟩ + ⟨e_{1} · ϵ_{2}^{⋆}⟩ + ⟨e_{2} · ϵ_{1}^{⋆}⟩ + ⟨ϵ_{1} · ϵ_{2}^{⋆}⟩ V (u, v, w) = Ie^{2πι^{\frac{b_{12}·s⃗o}{c}}} = Ie^{2πιw} = Ie^{2πιντ_{g}} s_{o} b

V (u, v, w) = I(l, m) [cos(2π \frac{D}{λ} sin(θ(t))) + ι sin(2π \frac{D}{λ} sin(θ(t)))]  (2.17)

\frac{Δϕ}{2π} = [τ_{g}ν_{RF}^{∘} + τ_{F ix}ν_{IF}^{∘}] + [(τ_{g} + τ_{F ix} − τ_{Int}) ν_{i}]

≡ \frac{Δϕ^{∘}}{2π} + \frac{Δϕ_{i}}{2π}

V (u, v, w) = \frac{sin(πΔντ)}{πΔντ}e^{2πν_{o}τ}  (2.19)

|V_{12}| = \sqrt{\frac{T_{1}^{a}T_{2}^{a}}{(T_{1}^{sys} + T_{1}^{a})(T_{2}^{sys} + T_{2}^{a})}}

τ_{F ST C}(t) = \frac{1}{2π} \frac{dϕ}{dν}|_{t}

w = [X_{ij} cos(H) − Y_{ij} sin(H)] cos(δ) + Z_{ij} sin(δ) ϕ = 2π \frac{ν_{RF}^{o}}{c} [ΔX cos(δ) cos(H) − ΔY cos(δ) sin(H) + ΔZ sin(δ)] + 2πτ_{F ix}ν_{IF}

ϕ = C + A sin(\frac{2π}{24.0}H + 2πψ)

ΔX = \frac{A sin(ψ)c}{cos(δ)2πν_{RF}^{o}}  (2.25)

ΔY = \frac{A cos(ψ)c}{cos(δ)2πν_{RF}^{o}}  (2.26)

τ_{g}(ν_{RF}^{o} + ν_{i}) + τ_{F ix}(ν_{IF}^{o} + ν_{i}) = τ_{corr}(ν_{BB}^{o} + ν_{i})

τ_{corr} = (τ_{Int} + τ_{F ST C}) = (τ_{g} + τ_{F ix}) + \frac{τ_{g}ν_{RF}^{o} + τ_{F ix}ν_{IF}^{o}}{ν_{i}}

\frac{ϕ}{2π} = (τ_{g} + τ_{F ix} − τ_{Int} − τ_{F ST C})ν_{i} + τ_{g}(ν_{LO1} + ν_{LO2}) + τ_{F ix}ν_{LO2}  (2.29)

\frac{1}{2π} \frac{∂ϕ}{∂ν} = τ_{F ix}

ρ_{ij}^{Obs} = g_{i}g_{j}^{⋆}ρ_{ij}^{∘} + ϵ_{ij} |ρ_{ij} ·ρ_{il}|

V (u, v, w) = ∫∫ I(l, m)P (l, m)e^{−2πι(ul+vm+w(\sqrt{1−l^{2}−m^{2}}−1))} \frac{dldm}{\sqrt{1 − l^{2} − m^{2}}}

\sqrt{1 − l^{2} − m^{2}}

F (l, m, n) = ∫∫∫ V (u, v, w)e^{2πι(ul+vm+wn)}dudvdw

F (l, m, n) = ∫∫ \{ \frac{I(l′, m′)}{\sqrt{1 − l^{′^{2}} − m^{′^{2}}}}

∫∫∫ e^{−2πι(u(l′−l)+v(m′−m)+w(\sqrt{1−l^{′^{2}}−m^{′^{2}}}−n−1))}dudvdw

} dl′dm′  (4.3)

δ(l′ − l) = ∫ e^{−2πιu(l′−l)}du

= \{ \begin{matrix}1 : & l = l′ \\ 0 : & otherwise \end{matrix}

F (l, m, n) = ∫∫ \frac{I(l′, m′)}{\sqrt{1 − l^{′^{2}} − m^{′^{2}}}} δ(l′ − l)δ(m′ − m)δ(\sqrt{1 − l^{′^{2}} − m^{′^{2}}} − n)dl′dm′

= \frac{I(l, m)δ(\sqrt{1 − l^{2} − m^{2}} − n)}{\sqrt{1 − l^{2} − m^{2}}}

\sqrt{1 − l^{2} − m^{2}}

N_{n} =B_{max}θ^{2}/λ
=λB_{max}/D^{2} (for θ = Full Primary beam)

N_{poly} = 2θ_{f}^{2}B_{max}/λ
= 2B_{max}λ/D^{2} (for θ_{f} = Full primary beam)

(sin(πlΔν/ν_{o}θ)) / ((πlΔν/ν_{o}θ))

(u, v) = (ν_{o} \frac{u_{ν}}{ν} , ν_{o} \frac{v_{ν}}{ν} )

V (u, v) = V (u_{ν} \frac{ν_{o}}{ν} , v_{ν} \frac{ν_{o}}{ν} ) FT (\frac{ν}{ν_{o}} )^{2} I (l\frac{ν}{ν_{0}} , m\frac{ν}{ν_{0}} )

\sqrt{l^{2} + m^{2}}

I^{d}(l, m) = [ \frac{∫_{0}^{∞}|H_{RF} (ν)|^{2} (\frac{ν}{ν_{o}} )^{2} I (\frac{lν}{ν_{0}} , \frac{mν}{ν_{0}} ) dν}{∫_{0}^{∞}|H_{RF} (ν)|^{2}dν} ] ∗ DB_{o}(l, m)

\sqrt{8ln2}

\sqrt{l^{2} + m^{2}}

R_{b} = 1.064\frac{θ_{b}ν_{o}}{rΔν} erf (0.833\frac{rΔν}{θ_{b}ν_{o}} )

\sqrt{u^{2} + v^{2}}

E_{i}^{p} = g_{i}^{p}E_{i,∘}^{p} + α_{i}^{q}E_{i,∘}^{q}

ρ_{ij}^{pp} = g_{i}^{p}g_{j}^{p}^{⋆}ρ_{ij,∘}^{pp} + α_{i}^{q}α_{j}^{q}^{⋆}ρ_{ij,∘}^{qq} + ϵ_{ij}

X_{ij}^{pp} = g_{i}^{p}g_{j}^{p}^{⋆} + α_{i}^{q}α_{j}^{q}^{⋆} + ϵ_{ij} |X_{ij}^{pp} − g_{i}^{p}g_{j}^{p}^{⋆}|

\sqrt{ϵ_{ij}}

S = ∑_{{i,j \atop i≠j}} |X_{ij}^{pp} − g_{i}^{p}g_{j}^{p}^{⋆}|^{2} w_{ij}^{pp}

(∂S) / (∂g_{i}^{p}^{⋆})

g_{i}^{p} = \frac{∑_{{j \atop j≠i}} X_{ij}^{pp}g_{j}^{p}w_{ij}^{pp}}{∑_{{j \atop j≠i}} |g_{j}^{p}|^{2} w_{ij}^{pp}}

g_{i}^{p}^{,n} = g_{i}^{p}^{,n−1} + λ [g_{i}^{p}^{,n−1} − \frac{∑_{{j \atop j≠i}} X_{ij}^{pp}g_{j}^{p}^{,n−1}w_{ij}^{pp}}{∑_{{j \atop j≠i}} |g_{j}^{p}^{,n−1}|^{2} w_{ij}^{pp}} ]

|S_{n} − S_{n−1}|

S = ∑_{{i,j \atop i≠j}} |X_{ij}^{pp} − (g_{i}^{p}g_{j}^{p^{⋆}} + α_{i}^{q}α_{j}^{q^{⋆}})|^{2} w_{ij}^{pp}

(∂S) / (∂g_{j}^{p}^{⋆})

(∂S) / (∂α_{j}^{q}^{⋆})

g_{i}^{p} = \frac{∑_{{j \atop j≠i}} X_{ij}^{pp}g_{j}^{p}w_{ij}^{pp} − α_{i}^{q} ∑_{{j \atop j≠i}} α_{j}^{q}^{⋆}g_{j}^{p}w_{ij}^{pp}}{∑_{{j \atop j≠i}} |g_{j}^{p}|^{2} w_{ij}^{pp}}

α_{i}^{q} = \frac{∑_{{j \atop j≠i}} X_{ij}^{pp}α_{j}^{q}w_{ij}^{pp} − g_{i}^{p} ∑_{{j \atop j≠i}} g_{j}^{p}^{⋆}α_{j}^{q}w_{ij}^{pp}}{∑_{{j \atop j≠i}} |α_{j}^{q}|^{2} w_{ij}^{pp}}

|1 − \frac{X_{ij}}{g_{i}g_{j}^{⋆}} | |1 − \frac{X_{ij}}{(g_{i}g_{j}^{⋆}+α_{i}α_{j}^{⋆})} | |α_{i}^{q}/g_{i}^{p}|

(\sqrt{Q^{2}+U^{2}+V^{2}}) / I

E_{i}^{R} =g_{i}^{R} (E_{i,∘}^{X}e^{ιδ} + E_{i,∘}^{Y}e^{−ιδ}) +

α_{i}^{L} (E_{i,∘}^{X}e^{−ιδ} + E_{i,∘}^{Y}e^{ιδ})

X_{ij}^{RX} =(g_{i}^{R}e^{ιδ} + α_{i}^{L}e^{−ιδ})g_{j}^{X^{⋆}}+

(g_{i}^{R}e^{−ιδ} + α_{i}^{L}e^{ιδ})α_{j}^{Y^{⋆}}

≡g_{i}^{X′}g_{j}^{X^{⋆}} + α_{i}^{Y′}α_{j}^{Y^{⋆}}

( \begin{matrix}cos θ/2 \\ sin θ/2 e^{ιϕ} \end{matrix} )

( \begin{matrix}g_{1} \\ α_{1} \end{matrix} )

( \begin{matrix}g_{2} \\ α_{2} \end{matrix} )

( \begin{matrix}g_{3} \\ α_{3} \end{matrix} )

( \begin{matrix}g_{i} \\ α_{i} \end{matrix} )

\sqrt{2}

( \begin{matrix}1 \\ ι \end{matrix} )

( \begin{matrix}1 \\ 0 \end{matrix} )

( \begin{matrix}cos γ \\ sin γ \end{matrix} )

( \begin{matrix}g′ \\ α′ \end{matrix} ) = ( \begin{matrix}cos ϕ & e^{ιγ} sin ϕ \\ −e^{−ιγ} sin ϕ & cos ϕ \end{matrix} ) ( \begin{matrix}e^{ιζ_{1}} & 0 \\ 0 & e^{ιζ_{2}} \end{matrix} ) ( \begin{matrix}g \\ α \end{matrix} )

|g_{i}|

ρ_{ij}^{Obs} = ρ^{Obs}(u_{ij}, v_{ij}, w_{ij}) = ∫∫_{−∞}^{+∞} I(l, m)\sqrt{\frac{η_{i}η_{j}}{T_{i}^{s}T_{j}^{s}}} e^{2πι(u_{ij}l+v_{ij}m+w_{ij} \sqrt{1−l^{2}−m^{2}}+ϕ_{i}−ϕ_{j})}

\frac{dl dm}{\sqrt{(1 − l^{2} − m^{2})}} + ϵ_{ij}

\sqrt{η_{i}/T_{i}^{s}}

\sqrt{η_{i}/T_{i}^{s}}

ρ_{ij}^{Obs} = g_{i}g_{j}^{⋆}ρ_{ij}^{∘} + ϵ_{ij}

ρ_{ij}^{∘} = ∫_{−∞}^{+∞} ∫_{−∞}^{+∞} I(l, m) e^{2πι(u_{ij}l+v_{ij}m+w_{ij} \sqrt{1−l^{2}−m^{2}})} \frac{dl dm}{\sqrt{(1 − l^{2} − m^{2})}}

|g_{i}|

S = ∑_{{i,j \atop i≠j}} |ρ_{ij}^{Obs} − g_{i} g_{j}^{⋆} ρ_{ij}^{∘}|^{2} w_{ij}

S = ∑_{{i,j \atop i≠j}} |X_{ij} − g_{i} g_{j}^{⋆}|^{2} w_{ij}

S = ∑_{{i,j \atop i≠j}} [ |X_{ij}|^{2} − g_{i}^{⋆}g_{j}X_{ij} − g_{i}g_{j}^{⋆}X_{ij}^{⋆} + g_{i}g_{i}^{⋆}g_{j}g_{j}^{⋆}] w_{ij}

\frac{∂S}{∂g_{i}^{⋆}} = ∑_{{j \atop j≠i}} [−g_{j}X_{ij}w_{ij} + g_{i}g_{j}g_{j}^{⋆}w_{ij}] = 0

g_{i} = \frac{∑_{{j \atop j≠i}} X_{ij}g_{j}w_{ij}}{∑_{{j \atop j≠i}} |g_{j}|^{2} w_{ij}}

g_{i}^{n} = g_{i}^{n−1} + α [g_{i}^{n−1} − \frac{∑_{{j \atop j≠i}} X_{ij}g_{j}^{n−1}w_{ij}}{∑_{{j \atop j≠i}} |g_{j}^{n−1}|^{2} w_{ij}} ]

|S_{n} − S_{n−1}| < δ |g_{j}| X_{ij}| |g_{j}| X_{ij}|

ρ_{ij}^{Obs} = Ig_{i}g_{j}^{⋆} ≈ I\sqrt{\frac{η_{i}η_{j}}{T_{i}^{s}T_{j}^{s}}}

|g_{i}| = \sqrt{\frac{η_{i}}{T_{i}^{s}}}

|g|

T_{i}^{s} = T_{i}^{n}(\frac{g_{i}^{ON}^{2}}{g_{i}^{OF F}^{2} − g_{i}^{ON}^{2}} )

\sqrt{T^{a}/(T^{n} + T^{s})}

∑_{{i,j \atop i≠j}} |X_{ij} − g_{i}g_{j}^{⋆}|^{2} = ∑_{{i,j \atop i≠j}} [X_{ij} − g_{i}g_{j}^{⋆}] [X_{ij}^{⋆} − g_{i}^{⋆}g_{j}]

= ∑_{{i,j \atop i≠j}} [(X_{ij}^{R} + ιX_{ij}^{I}) − (g_{i}^{p} + ιg_{i}^{I}) (g_{j}^{p} − ιg_{j}^{I})]

[(X_{ij}^{R} − ιX_{ij}^{I}) − (g_{i}^{p} − ιg_{i}^{I}) (g_{j}^{p} + ιg_{j}^{I})]

= ∑_{{i,j \atop i≠j}} [(X_{ij}^{R} − g_{i}^{p}g_{j}^{p} − g_{i}^{I}g_{j}^{I}) + ι (X_{ij}^{I} + g_{i}^{p}g_{j}^{I} − g_{i}^{I}g_{j}^{p})]

[(X_{ij}^{R} − g_{i}^{p}g_{j}^{p} − g_{i}^{I}g_{j}^{I}) − ι (X_{ij}^{I} + g_{i}^{p}g_{j}^{I} − g_{i}^{I}g_{j}^{p})]

= ∑_{{i,j \atop i≠j}}S_{0}S_{0}^{⋆}

S_{0} = [X_{ij}^{R} − g_{i}^{p}g_{j}^{p} − g_{i}^{I}g_{j}^{I}] + ι [X_{ij}^{I} + g_{i}^{p}g_{j}^{I} − g_{i}^{I}g_{j}^{p}]

\frac{∂S}{∂g_{i}^{p}} = ∑_{{j \atop j≠i}} \{[−g_{j}^{p} + ιg_{j}^{I}] S_{0}^{⋆} − S_{0} [g_{j}^{p} + ιg_{j}^{I}]\} w_{ij}

= − ∑_{{j \atop j≠i}} [S_{0}g_{j} + g_{j}^{⋆}S_{0}^{⋆}] w_{ij}

= − 2 ∑_{{j \atop j≠i}} Re (S_{0}g_{j}w_{ij})

= − 2 ∑_{{j \atop j≠i}} [(X_{ij}^{R} − g_{i}^{p}g_{j}^{p} − g_{i}^{I}g_{j}^{I}) g_{j}^{p} + (X_{ij}^{I} + g_{i}^{p}g_{j}^{I} − g_{i}^{I}g_{j}^{p}) g_{j}^{I}] w_{ij}

= − 2 ∑_{{j \atop j≠i}} [X_{ij}^{R}g_{j}^{p} − X_{ij}^{I}g_{j}^{I} − g_{i}^{p}g_{j}^{I}^{2} − g_{i}^{p}g_{j}^{p}^{2}] w_{ij}

\frac{∂S}{∂g_{i}^{p}} = −2 ∑_{{j \atop j≠i}} [Re(X_{ij}g_{j}^{⋆}) − |g_{j}|^{2} g_{i}^{p}] w_{ij}

g_{i}^{p} = \frac{∑_{{j \atop j≠i}} Re(X_{ij}g_{j}^{⋆}w_{ij})}{∑_{{j \atop j≠i}} |g_{j}|^{2} w_{ij}}

\frac{∂S}{∂g_{i}^{I}} = −2 ∑_{{j \atop j≠i}} [Im(X_{ij}g_{j}^{⋆}) − |g_{j}|^{2} g_{i}^{I}] w_{ij}

g_{i}^{I} = \frac{∑_{{j \atop j≠i}} Im(X_{ij}g_{j}^{⋆}w_{ij})}{∑_{{j \atop j≠i}} |g_{j}|^{2} w_{ij}}

g_{i} = \frac{∑_{{j \atop j≠i}} X_{ij}g_{j}^{⋆}w_{ij}}{∑_{{j \atop j≠i}} |g_{j}|^{2} w_{ij}}

T L F