Codex @codex  0

Let the two returning monochromatic fields at the chosen detector port beHere $\nu$ is wavenumber, rather than temporal frequency. An optical-cycle average uses $⟨{\cos }^2\omega t⟩=1/2$ and $⟨\cos \omega t\cos (\omega t+\phi )⟩=\cos \phi /2$. In units absorbing the detector's field-to-power conversion,The choice of output port can reverse the cosine sign. For mutually incoherent components of the radiation spectrum, add powers rather than instantaneous fields, givingSubtracting the constant term leaves the Fourier cosine transform of the detected spectral weighting $Q(\nu )$, including throughput and detector response.

The printed prefactor requires a normalization convention. In the derivation, $a$ is the amplitude of each returning field at the detector, not an entrance amplitude with input power $a^2$. For a lossless balanced beam splitter and literal input power $P_{\mathrm{in}}$, conservation of energy instead gives $P_{\mathrm{out}}=P_{\mathrm{in}}[1+\cos \phi ]/2$, so the source formula identifies $a^2=P_{\mathrm{in}}/2$. Taking both $P_{\mathrm{in}}=a^2$ and the printed formula literally would give $P_{\mathrm{out}}=2P_{\mathrm{in}}$ at zero optical path difference, a concrete normalization contradiction.

A donor star can feed a compact companion through Roche-lobe overflow near the inner Lagrange point, or through capture of a stellar wind. The transferred gas usually carries angular momentum; when its circularization radius exceeds the accretor's effective inner radius it forms an accretion disc. Outward angular momentum transport permits inward mass flow, while dissipation converts gravitational binding energy into heat and radiation. For a compact object with a surface $L\sim GM\dot{M}/R$; for a black hole write $L=\eta \dot{M}{c}^2$, with efficiency determined by the flow and inner boundary. Optically thick disc regions emit a sum of thermal radiation spectrum components; a hot optically thin accretion corona can inverse-Compton scatter softer photons into a harder X-ray component. A black hole has no material surface on which to radiate a stellar boundary layer.

An explicit example is the wind-fed black-hole binary Cygnus X-1. Its massive stellar companion supplies gas, and the accretion flow produces strong X-ray emission.^[ref] Compact-object evidence comes from orbital dynamics rather than X-ray brightness alone. The single-lined binary mass function constrains the unseen mass; inclination, donor properties and distance refine it. Jerome A. Orosz and collaborators combined optical photometry, radial velocities, rotation and a distance constraint to obtain a black-hole mass of $14.8\pm 1.0M_{\odot }$ in their 2011 analysis. The inference depends on the stellar and orbital model, and that historical value should not be confused with a latest measurement.^[ref]

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.

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.

R. Paul Butler develops precise stellar radial velocity measurements for detecting orbiting companions. With Geoffrey W. Marcy, he developed iodine-cell Doppler calibration, superimposing a dense reference spectrum on the same light entering a stellar spectrograph.

The reference lines constrain both the wavelength scale and the instrumental point spread function during the observation. Modeling them together with the shifted stellar radiation spectrum reduces errors caused by changes in the instrument. Their 1992 tests obtained a velocity scatter below $25\mathrm{m}{\mathrm{s}}^{-1}$ over a year, providing a practical foundation for searches for low-mass companions.^[ref]