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A comoving source's spectral luminosity $L_{{\nu }_e}$ specifies emitted power per unit emitted frequency. The cosmological redshift reduces photon energy and arrival rate by two factors of $1+z$, but also narrows the received frequency interval by one factor. Consequently its observed spectral flux density is $L_{{\nu }_e}/[4\pi a_0^2{r}^2(1+z)]$, with ${\nu }_e=(1+z){\nu }_0$. Substituting luminosity distance $d_L=(1+z)a_{0}r$ gives the displayed relation. Integrating over received frequency recovers the bolometric radiative flux $L/(4\pi d_L^2)$. A fixed-frequency comparison must evaluate the source spectrum at the shifted emitted frequency.

For an unresolved, isotropically emitting rest-frame blob with power-law spectrum $F_{\nu }^{\prime }\propto {\nu }^{-\alpha }$, the observed spectral flux density at a fixed frequency is enhanced by $D^{3+\alpha }$. The factors follow from Lorentz-invariant specific intensity and the frequency shift. The bolometric or corresponding-logarithmic-bin luminosity transforms with $D^4$. A spatially steady continuous jet has a different equal-arrival-time sampling and can have exponent $2+\alpha$ instead.

The luminosity distance is defined operationally by the bolometric relation $F=L/(4\pi d_L^2)$ for an isotropic source. It incorporates geometric spreading, reduced photon energy and dilated photon arrival times. If $D_M$ is the transverse comoving distance with the present scale factor included, then $d_L=(1+z)D_M$ in a spacetime with an FLRW metric with photon number conserved.

Let ${\nu }_e=(1+z){\nu }_o$ be the emitted frequency. A narrow emitted band carries energy $P({\nu }_e)d{\nu }_{e}d{t}_{e}d\Omega$. At the observer the energy is smaller by $1+z$, the time interval is $d{t}_o=(1+z)d{t}_e$, the frequency interval is $d{\nu }_o=d{\nu }_e/(1+z)$, and the illuminated area is $D_M^{2}d\Omega$. Hence the spectral spectral flux density per unit observed frequency isFor the specified power law, $P((1+z){\nu }_o)=(1+z{)}^{-\alpha }P({\nu }_o)$, so the power-law spectral K correction isThere is no extra $4\pi$ because $P$ is power per steradian. For total spectral luminosity $L_{\nu }=4\pi P_{\nu }$, that factor would instead appear in the denominator. The frequency-band factor is essential: bolometric flux has two redshift-dimming factors, but spectral flux has one of them compensated by the bandwidth change.

For a blob in a relativistic jet, take two emission events separated by time $\Delta t_e$ in the galaxy frame. The blob moves transversely by $u\Delta t_e\sin \theta$, while its second photon has $u\Delta t_e\cos \theta$ less distance to travel to the distant observer. Therefore the arrival interval is $\Delta t_{\mathrm{obs}}=\Delta t_e(1-\beta \cos \theta )$ andDifferentiating with respect to viewing angle gives $d{\beta }_{\mathrm{app}}/d\theta =\beta (\cos \theta -\beta )/(1-\beta \cos \theta {)}^2$. The maximum occurs at $\cos \theta =\beta$ and equals $\beta \gamma$. Hence apparent superluminal motion is possible precisely when $\beta \gamma >1$, or $\beta >1/\sqrt{2}$. This is an arrival-time effect; the material and every local signal still travel slower than light.

For $\gamma \gg 1$, $1-\beta \simeq 1/(2{\gamma }^2)$ and $1-\beta \cos \theta \simeq ({\gamma }^{-2}+{\theta }^2)/2$. Thus the Doppler factor isThe Lorentz-invariant specific intensity gives $I_{\nu }/{\nu }^3=I_{{\nu }^{\prime }}^{\prime }/{\nu }^{\prime 3}$. Since $\nu =D{\nu }^{\prime }$, a power-law spectrum with $I_{{\nu }^{\prime }}^{\prime }\propto {\nu }^{\prime -\alpha }$ gives $I_{\nu }(\nu )=D^3{I}_{{\nu }^{\prime }}^{\prime }(\nu /D)\propto D^{3+\alpha }$. This is Doppler boosting of a moving blob: the spectral flux density of an unresolved, intrinsically isotropic blob has the same boost. Comparing identical approaching and receding blobs, with no differential absorption,A spatially steady continuous jet can have a different boost exponent because the emitting material contributing at one arrival time differs; it must not silently replace the specified blob model.

Using the stipulated peak of the photon-photon pair production cross section, not the exact head-on threshold, givesThe relevant photons are near infrared. Interpret the quoted $F_2$ as target-band energy flux, or energy flux per logarithmic frequency interval, consistent with its units $\mathrm{W}{\mathrm{m}}^{-2}$; it is not a spectral flux density per hertz. For isotropic emission $L_2=4\pi d^2{F}_2$. In a stationary source of radius $R$, the order-of-magnitude photon energy density is $L_2/(4\pi R^{2}c)$, soThe luminosity variability size bound gives $R\lesssim c\Delta t$. Taking the largest allowed radius gives the smallest stationary-source optical depth in this estimate:This assumes the gamma rays and target photons occupy the same region, with an approximately isotropic target field and an order-unity geometric escape factor. A smaller region makes the gamma-ray compactness problem worse.

Here $d=3\times 10^{25}\mathrm{m}$, $R\sim 3\times 10^{11}\mathrm{m}$, and ${ϵ}_{20}=1.67\times 10^{-19}\mathrm{J}$. ConsequentlySuch a stationary co-spatial source would absorb almost all of its high-energy photons.

Let primes denote the blob rest frame and continue to ignore cosmological redshift. For fixed observed gamma-ray energy, ${ϵ}_1^{\prime }={ϵ}_1/D$. The co-moving target energy at the assumed cross-section peak is therefore ${ϵ}_2^{\prime }=D{ϵ}_{20}$. The internal pair-production target energy in a relativistic blob acquires a second Doppler shift when this same target population is observed along our line of sight:It would be incorrect to transform only the gamma ray while treating the internal target radiation as a stationary external field. The target photons responsible for collisions span directions in the blob frame; the observed energy here labels the corresponding part of the internally emitted radiation spectrum.

relativistic beaming both increases the allowed rest-frame source radius, $R^{\prime }\lesssim Dc\Delta t$, and lowers the inferred rest-frame luminosity, $L_{\log }^{\prime }\sim L_{\mathrm{iso},\log }/D^4$. The beamed gamma-ray optical depth then follows from ${ϵ}_2^{\prime }=D{ϵ}_{20}$:For an unbroken power-law spectrum $F_{\nu }\propto {\nu }^{-\alpha }$, $F_{\log }=\nu F_{\nu }\propto {\nu }^{1-\alpha }$, so ${\tau }^{\prime }\propto D^{-(4+2\alpha )}$. In particular, $\alpha =1$ gives the stated $D^{-6}$. Spectral breaks, an external target field or non-coincident emission regions change this inference. Under the requested scaling, transparency requiresThis minimum uses the most favourable nearly axial view, $D\simeq 2\gamma$. The exact axial relation gives ${\gamma }_{\min }=(D_{\min }+D_{\min }^{-1})/2\simeq 8.94$; a less favourable viewing angle requires a larger Lorentz factor. The numerical bound is conditional on the stipulated opacity scaling and internal-source geometry.

For isotropic specific intensity $J_{\nu }$, absorption cross-section ${\sigma }_{\nu }$, and threshold frequency ${\nu }_0$, the photoionization rate per target atom isFor a directed incident flux, replace $4\pi J_{\nu }$ by its spectral flux density. The factor $h\nu$ converts energy flux into photon flux. This rate alone does not imply photoionization equilibrium.

A photometric passband is the wavelength-dependent response used for a brightness measurement. An energy-weighted measurement is $I=\int {\eta }_{\lambda }{F}_{\lambda }d\lambda$, with spectral flux density $F_{\lambda }$. Photon-counting conventions can instead include a factor $\lambda /(hc)$; the calibration must use the same convention as the measurement.

A radiation spectrum whose spectral flux density follows a power law over a stated frequency range. For $F_{\nu }\propto {\nu }^{-\alpha }$, the spectral energy distribution satisfies $\nu F_{\nu }\propto {\nu }^{1-\alpha }$. Thus $\alpha =1$ gives equal energy flux in equal logarithmic frequency intervals. A spectral break invalidates extrapolation with a single index.

Spectral luminosity is energy radiated per unit time per unit frequency. Its frequency integral is the total luminosity. For an isotropic source at small distance $r$, neglecting attenuation and cosmological effects, the spectral flux density is $L_{\nu }/(4\pi r^2)$.

The system-equivalent flux density is the unpolarized spectral flux density that would produce an antenna temperature equal to $T_{\mathrm{sys}}$. With effective collecting area $A_e$, $\mathrm{SEFD}=2k_B{T}_{\mathrm{sys}}/A_e$. The factor two follows from one polarization of waves measurement of an unpolarized source. State effective area and polarization of waves conventions when comparing sensitivities.^[ref]

With constant surface brightness over a centred angular rectangle of widths ${\Theta }_x,{\Theta }_y$, and a nearly uniform primary beam over that rectangle, the Fourier transform separates into two one-dimensional integrals. The result is $V(u,v)=S_{\nu }\mathrm{sinc}(\pi {\Theta }_{x}u)\mathrm{sinc}(\pi {\Theta }_{y}v)$, where $\mathrm{sinc}x=\sin x/x$. The zero-spacing value is the integrated spectral flux density, while the first axial null is at $|u|=1/{\Theta }_x$. Long baselines therefore do not measure the total flux of a resolved rectangle.