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Angular resolution describes the smallest angular separation or structure an observation can distinguish. A physical scale $r$ at distance $D$ subtends approximately $r/D$ radians when $r\ll D$. Resolving a black-hole sphere of influence therefore involves distance as well as its physical radius, and useful kinematics also require adequate surface brightness.

Diffuse X-ray background is X-ray emission spread over the instrument's observed sky rather than assigned to a single resolved source. Diffuse appearance at finite angular resolution does not prove emission by a continuous medium: many unresolved point-like astronomical sources can also give a smooth signal. Instrumental background is a separate contribution, even though both enter measured counts.

Grazing-incidence X-ray optics redirect X-rays by shallow specular reflections along mirror surfaces. Riccardo Giacconi's lecture describes coaxial paraboloidal and hyperboloidal mirrors using two reflections, and the later Einstein observatory's imaging capability. Focusing separates collecting area from detector area: the collected source signal can remain large while the detector background is drawn from a smaller region.^[ref]

An ideal lossless geometrical optics calculation explains the shallow angle. Take vacuum refractive index one and a material with real refractive index $n=1-\delta$, where $0<\delta \ll 1$. Let $\theta$ be the grazing angle measured from the surface. Snell's law written for angles from the normal requires $\cos \theta =n\sin {\theta }_t$. A propagating transmitted ray exists only if $\cos \theta \le n$. Thus the ideal threshold isThe last step uses the Taylor series $\cos \theta =1-{\theta }^2/2+O({\theta }^4)$. For a chosen $\delta =2\times 10^{-5}$, ${\theta }_c\simeq 0.006325$ radians, or $0.3624^{\circ }$. Absorption makes the refractive index complex and softens this ideal threshold; surface roughness and alignment errors also reduce throughput or worsen the point spread function. These are reasons to calibrate X-ray effective area and angular resolution, rather than assume that reflection is perfect.

The benefit can be quantified without specifying a historical telescope. Suppose a source retains fraction $f$ of its counts inside an extraction region, while a uniform detector background contributes $qat$ counts over detector area $a$. In the background-dominated limit of the same independent on-off model,Reducing $a$ by a factor of 100 at fixed $f$ and X-ray effective area improves this ideal ratio tenfold. A collimator restricts admitted directions; focusing optics bring selected rays together. An image also separates nearby sources and identifies clean background regions, connecting optical design directly to a better constrained inverse problem.

An interferometric baseline is the displacement $\mathbf{b}={\mathbf{r}}_2-{\mathbf{r}}_1$ between two antennas. For a unit line of sight ${\mathbf{s}}_0$ toward the phase centre, its projection onto the plane of the sky isHere $c$ is the speed of light. The signed delay depends on which receiver is delayed; the geometric path difference's magnitude does not. If ${\mathbf{e}}_{\theta }$ is a unit tangent in the direction being imaged, the dimensionless coordinate $u=\mathbf{b}\cdot {\mathbf{e}}_{\theta }/\lambda$ is a spatial frequency, measured in cycles per radian of small sky offset. A single pair gives fringes of angular period $1/|u|$ along that tangent. This period specifies one measured oscillation; it does not specify a whole image's angular resolution or eliminate ambiguous source arrangements.

The left panel separates the component producing the reference delay from the component measuring angular structure. The right panel shows a possible Earth-rotation aperture synthesis track, with its opposite complex conjugate track. Those opposite samples provide the symmetry required by a real brightness distribution; they are not independent extra observations.

Many short frames freeze atmospheric image fluctuations. Selecting the sharpest frames, registering them and combining them can improve angular resolution. Selection discards photons and can create a read-noise penalty, so adequate reference-source photon counts are essential. It does not guarantee that every high-order wavefront error disappears.

Independent visibility samples are combined with weights proportional to inverse noise variance. This maximizes the point-source sensitivity for correctly calibrated samples. Dense regions of the sampled Fourier plane receive more total weight; tapering the long baselines can further emphasize extended structure at the cost of angular resolution. A source-template filter instead also weights by predicted source response.

The observed population is the parent population filtered by an astronomical selection function and measurement errors. Malmquist bias arises because a flux-limited galaxy selection admits luminous objects from larger volumes: in Euclidean space $d_{\max }\propto L^{1/2}$ and $V_{\max }\propto L^{3/2}$. Thus observed luminosity counts are weighted by $V_{\max }$. Even a nominal standard candle with intrinsic scatter is preferentially observed on its bright side near the limit; ignoring this can underestimate distances. A volume-limited sample or a justified $1/V_{\max }$ weighting helps, subject to completeness, geometry and evolution assumptions. Separately, Eddington bias occurs when noise scatters a steeply rising population across a detection threshold: more faint objects enter than bright objects leave.

Lutz-Kelker bias concerns noisy stellar parallaxes and population/selection weighting. Locally uniform spatial density gives $p(r)\propto r^2$ and, by changing variables, $p(\varpi )\propto {\varpi }^{-4}$ over the model's valid finite range. More distant stars are available to scatter into a parallax-limited sample than nearby stars to scatter out. Inverting an uncertain parallax also introduces a distinct nonlinear-estimation problem. There is no universal magnitude correction for every star: spatial density, luminosity, extinction and how the sample was selected all matter. The classical treatment motivates this issue; it should not replace an explicit generative model.^[ref]

Bayes' theorem combines the measurement likelihood function with a justified prior distribution. For example, with a Gaussian parallax error and consistent distance/parallax units,A proper distance prior for parallax inference might have $p(r)\propto r^2{e}^{-r/L}$ with a justified scale $L$; extending $r^2$ to infinity gives an improper posterior here. Negative measured parallaxes remain valid noisy data. For selection conditioning in Bayesian inference, distinguish individual inference from population inference: if selection is determined entirely by the exact data already conditioned on, it adds no further information about that individual's parameters. For population parameters $\Lambda$, the selected-data likelihood must instead include the detection normalization $\alpha (\Lambda )=\int S(d)p(d\mid \Lambda )dd$. Applying a cut twice is also an error.

Selection can depend on surface brightness, angular resolution, dust, sky coverage, colour and observing cadence as well as total flux. Crowding and unresolved binaries bias distance calibrators. Transit surveys favour large planets and short periods, with an additional geometric transit probability approximately $R_{\star }/a$ for small planets on circular orbits; injection/recovery tests quantify the remaining transit survey completeness. Cluster catalogues select noisy mass proxies from a steep abundance distribution. Planck's 2013 SZ catalogue explicitly characterized detection completeness and reliability rather than identifying every detection with an unbiased mass census.^[ref] Calibrators, artificial-source tests and a forward model of detection, measurement and follow-up are the practical checks on any correction.

Spatial geometry. An embedding of the three-sphere isTaking differentials and collecting terms gives the induced metric tensorPut $\bar{r}=A\sin \chi$. Then $d\bar{r}=A\cos \chi d\chi$, soThe areal radius $\bar{r}$ is not a global radial coordinate on the entire three-sphere: its maximum occurs at the equator and two hemispheres share each smaller value. Use separate charts there; the coordinate singularity at $\bar{r}=A$ is not a physical singularity.

For Euclidean space and hyperbolic space of curvature $-1/A^2$, respectively,For the latter, $\bar{r}=A\sinh \chi$ gives $d{\ell }_{-}^2=A^2[d{\chi }^2+{\sinh }^2\chi d{\Omega }^2]$. In all three cases the induced two-metric at fixed $\bar{r}=\bar{R}$ is ${\bar{R}}^{2}d{\Omega }^2$. Its surface area is thereforeThis equality compares the same areal radius, not the same proper radial distance. For a proper radial distance $\ell$, the corresponding areas are $4\pi A^2{\sin }^2(\ell /A)$, $4\pi {\ell }^2$, and $4\pi A^2{\sinh }^2(\ell /A)$.

Homogeneity, isotropy and redshift. The FRW metric has constant-time spatial slices of constant curvature $k/a^2(t)$, with $k=+1,0,-1$ selecting the three geometries above. Their isometries act transitively on points and isotropically about every point, implementing the cosmological principle. Multiplication by one common time-dependent scale factor preserves these properties. With the source's $k=\pm 1$ normalization, $r$ is dimensionless and $a$ has units of length; using a dimensionless scale factor instead rescales the spatial curvature parameter. The origin of the spherical coordinates is arbitrary; it does not identify a physical centre. Cosmic time is the proper time of the comoving observers, and the absence of time-space cross terms makes their worldlines orthogonal to the homogeneous slices.

For a radial null geodesic between comoving endpoints,The next wave crest travels between the same endpoints. Subtracting the two path integrals, for wave periods short compared with the expansion time, gives $\delta t_0/a_0=\delta t_E/a_E$. Thus the cosmological redshift isBoth observers' clocks are cosmic proper-time clocks. Peculiar velocities and local gravitational-potential shifts have been left out of this homogeneous calculation.

Angular size and brightness. For a galaxy small compared with its distance, the angular diameter distance is $D_A=a_{E}R$ and its angular diameter isHere the emitted proper diameter is measured transverse to the line of sight; radial proper distance is a different integral when $k\ne 0$. Ignoring absorption, lensing caustics and intrinsic evolution, photon-number conservation spreads an isotropic-equivalent luminosity across the sphere $4\pi a_0^2{R}^2$. Each photon loses energy by $1+z$, and its arrival interval is stretched by the same factor. The received bolometric flux isThe disc's proper projected area is ${\mathcal{A}}_d=\pi d^2/4$, and its observed solid angle is ${\Omega }_d\simeq {\mathcal{A}}_d/D_A^2$. Therefore its surface brightness isThis is cosmological surface-brightness dimming. The normalization uses $L$ as isotropic-equivalent luminosity along the viewing direction; another disc emission law changes that normalization but not the redshift power. This is a bolometric result. In a specified observed frequency band the emitted spectrum is sampled at a shifted frequency and requires a spectral correction.

The distinction in telescope aperture and resolved surface brightness matters here. A larger telescope collects more photons and improves angular resolution, but cannot restore the incoming radiance lost by cosmological dimming. For aperture diameter $D$ and a diffraction resolution element of solid angle ${\Omega }_{\mathrm{res}}\sim (\lambda /D{)}^2$, the collected power from uniform resolved brightness scales asindependent of $D$. This is consistent with conservation of etendue. It explains why increasing aperture while keeping every gain in resolution does not increase counts per native resolution element from a smooth extended source. It does not make larger telescopes useless: at a fixed angular bin the counts grow as $D^2$, so a photon-noise-limited or uniform-sky-noise-limited signal-to-noise ratio grows roughly as $D\sqrt{t_{\exp }}$. They also resolve smaller structures and improve point-source sensitivity. Aperture, exposure and angular binning help measure a dim galaxy, without changing its intrinsic $(1+z{)}^{-4}$ brightness penalty.

Many lower-luminosity elliptical galaxies have dense central stellar cusps, giving far more photons and kinematic tracers near a central black hole than the shallow depleted cores common in massive elliptical galaxies, as measured in Central galaxy profiles. Thus central surface brightness can make the smaller black hole easier to measure, despite its smaller mass, whereas sparse central tracers and uncertain stellar orbital anisotropy weaken measurements in a large core. This is not a universal mass ordering: the black-hole sphere of influence $r_{\inf }\simeq G{M}_{\bullet }/{\sigma }^2$ must still be sufficiently resolved in angle, so distance, velocity dispersion and angular resolution also matter.

Use Fraunhofer diffraction, appropriate to observation in the far field or in the focal plane of a lens, and uniform illumination across the slit. At observation angle $\theta$, points separated by transverse coordinate $u$ acquire a phase difference $ku\sin \theta$, where $k=2\pi /\lambda$ is the wavenumber. The superposition principle for the coherent wave amplitudes givesThe intensity is proportional to the squared modulus of the wave amplitude, soThe limit at $\beta =0$ is one. The noncentral zeros satisfy $w\sin \theta =m\lambda$ for nonzero integers $m$ for which the angle is physically allowed. In the Rayleigh criterion, two mutually incoherent point images are just resolved when the central maximum of one diffraction pattern lies at the first minimum of the other. For single-slit diffraction this givesThe exact statement here takes one image on the slit normal; for nearby paraxial images the angular resolution is $\lambda /w$. The first propagating minimum exists only for $w\ge \lambda$. The numerical factor $1.22$ belongs to a circular aperture and does not apply to this slit.

Two nearby directions separated by ${\theta }_R$ focus at transverse separation $s\simeq f_m{\theta }_R$. To avoid a receiver covering both geometric image centres at once, its diameter $d$ should not exceed this separation along the orbital direction:This is the usual receiver-size estimate from angular resolution. The Airy disk first-minimum diameter is $2f_m{\theta }_R=53.8\mathrm{mm}$, but that is a spot-size definition, not the same as the receiver's allowed angular acceptance. At the Rayleigh criterion, the images already overlap; integrating over a finite receiver further reduces contrast. Consequently $26.9\mathrm{mm}$ is an order-of-resolution upper scale, not a sharp universal boundary for a specified signal discrimination probability. A detailed limit would require the receiver response and a resolving criterion.

For viewing near the nadir, the required angular resolution isUsing the Rayleigh criterion with the first minimum of the single-slit diffraction pattern gives $\delta \theta \simeq \lambda /w$. Thus the minimum collecting width in that model isThe numerical coefficient depends on the aperture shape and the chosen resolution criterion. If the mirror has an unobstructed circular aperture, the Airy disk version of the Rayleigh criterion is $\delta \theta =1.22\lambda /D$, giving $D_{\min }=0.12298\mathrm{m}\simeq 12.3\mathrm{cm}$. The question does not specify mirror shape; $10.1\mathrm{cm}$ follows the slit-width model of part (i).

Radio astronomy studies celestial sources through their radio waves. The long wavelengths make angular resolution expensive: a filled telescope of diameter $D$ has a characteristic diffraction scale of order $\lambda /D$. Aperture synthesis instead combines measurements from separated antennas, using their separations to measure fine structure while their individual collecting areas determine how much radiation they receive. A kilometre-wide arrangement of small antennas does not collect the power intercepted by a filled kilometre-wide dish.

The shortest useful projected baseline sets the lowest sampled spatial frequency, of order $b_{\min }/\lambda$, and hence an angular scale $\lambda /b_{\min }$. Operational maximum recoverable source sizes also depend on source shape, coverage, weighting and primary beam. The longest baseline sets a different scale, the synthesized angular resolution. The number of antennas alone does not fix either limiting baseline or supply zero-spacing total power.

For a diffraction-limited telescope, collecting area scales as $D^2$ while a native angular resolution element scales as $(\lambda /D{)}^2$. Counts from a smooth resolved source per native resolution element therefore do not increase with aperture, at fixed radiance, exposure and throughput. Counts in a fixed angular bin instead grow as $D^2$, and the photon-noise-limited signal-to-noise ratio grows as $D\sqrt{t}$. Conservation of etendue and cosmological surface-brightness dimming do not forbid this latter gain.

Choose a signed angular coordinate along the source-lens line. The point-mass gravitational lens becomesFor $\beta \ne 0$, multiplication by $\theta$ gives the quadratic equation ${\theta }^2-\beta \theta -{\theta }_E^2=0$. Its positive discriminant yields the two displayed roots. Their product is ${\theta }_{+}{\theta }_{-}=-{\theta }_E^2$, so they lie on opposite sides of the lens. With $\beta >0$, ${\theta }_{+}>{\theta }_E$ and $-{\theta }_E<{\theta }_{-}<0$: the outer image lies on the source side. For $\beta <0$, the outer image is instead the minus root. These are image directions seen by the observer, not two emitting objects.^[ref]

For a dimensionless example, set $u=\beta /{\theta }_E=0.6$ and $x=\theta /{\theta }_E$. Solving $u=x-1/x$ gives $x_{+}\simeq 1.344$ and $x_{-}\simeq -0.744$, with separation $\sqrt{u^2+4}\simeq 2.088$. The lensing magnification follows from image parity in gravitational lensing: ${\mu }_{+}\simeq 1.442$, ${\mu }_{-}\simeq -0.442$, and total amplification $\simeq 1.884$. Changing $M$ or the distances rescales the angles through the Einstein angular radius, while these dimensionless values remain the same.

For $u\gg 1$, the quadratic equation gives $x_{+}\simeq u+1/u$ and $x_{-}\simeq -1/u$. Using $\mu =1/(1-x^{-4})$, the outer image tends to unit lensing magnification and the inner image has $|{\mu }_{-}|\simeq u^{-4}$. Two mathematical images can therefore look like one at finite sensitivity or angular resolution. Furthermore, the inner impact parameter eventually becomes too small for the stated weak-field approximation or passes through the opaque lens: the ideal formula must then be replaced by an appropriate physical model.

very-long-baseline interferometry correlates accurately time-referenced radio signals recorded at widely separated collectors, giving angular resolution of order wavelength divided by projected baseline. Geometric delays, frequency standards, atmospheric propagation and source structure enter the calibration. Compact maser positions and motions are one application.