Codex @codex  0

Let the uniform thin galactic disk have outer radius $R_d$ and surface mass density $\Sigma =M/(\pi R_d^2)$, where $M$ is the mass calculated in a. If the redistribution retains the stated outer radius, set $R_d=16\mathrm{kpc}$; otherwise $M$ alone does not determine both $R_d$ and $\Sigma$.

A ring of radius $s$ and width $ds$ has mass $dM=2\pi \Sigma sds$. By axial symmetry its transverse gravitational force components cancel. At signed height $z$, its vertical force on a star of mass $m$ isIntegrating the rings gives, for $z\ne 0$,ThusFor a positive distance above the plane, this is $\mathbf{F}=-2\pi Gm\Sigma [1-z/\sqrt{z^2+R_d^2}]\hat{\mathbf{z}}$, directed toward the disk. With $R_d=16\mathrm{kpc}$, the inferred $\Sigma \simeq 178.3M_{\odot }\mathrm{p}{\mathrm{c}}^{-2}$ sets its normalization.

The far-field expansion gives $F_z\simeq -GMm\mathrm{sgn}(z)/z^2$, as required for a point-like total mass. Close to either face, $F_z\to \mp 2\pi Gm\Sigma$, the infinite-sheet limit. The ideal zero-thickness model has a jump in gravitational field at $z=0$; the formula is understood through its one-sided limits. Its symmetric mean is zero on the plane, and a disk of finite thickness resolves the discontinuity.

A stable sinusoidal radial-velocity semi-amplitude can be stellar reflex motion caused by a companion on a nearly circular orbit. The amplitude is compatible with a giant planet; a low-mass star or brown dwarf viewed nearly face-on can mimic the same projected motion, but requires an increasingly unlikely orientation. Starspots and magnetically suppressed convection can produce a rotational stellar activity signal on a ten-day timescale; contemporaneous photometry, activity indicators and stellar spectral-line bisector and other profile changes distinguish these from a clean common Doppler shift. Solar-like stellar oscillations ordinarily have much shorter periods, so a coherent ten-day oscillation of an otherwise ordinary solar-type dwarf is less plausible. Instrumental drift, sampling aliases and blended spectra also need checking. A sinusoid alone does not establish a planetary origin, and the data do not assign numerical probabilities to these alternatives.

For the minimum companion mass from radial velocity, take a circular two-body orbit and initially $M_{\star }=M_{\odot }$. The exact relation and its low-mass limit areWith $\sin i=1$,Here $a=[G{M}_{\star }{P}^2/(4{\pi }^2){]}^{1/3}$ is the relative orbital semi-major axis. Nonzero orbital eccentricity multiplies the inferred $M_p\sin i$ by $\sqrt{1-e^2}$ at fixed measured $K,P$, but the sinusoidal model motivates $e=0$ here.

For the radial-velocity sensitivity limit at fixed planet mass, use $K=M_p\sin i\sqrt{G/(M_{\star }a)}$ and interpret the stated detection scale as a semi-amplitude threshold $K_{\min }$. The largest distance occurs for $\sin i=1$:For an inclination $i$, this boundary is smaller by ${\sin }^{2}i$. The result assumes adequate time coverage and neglects the stipulated stellar noise. If the detection scale instead denotes a full peak-to-peak excursion, set $K_{\min }$ to half that scale; $a_{\max }$ then increases fourfold.

The stellar-mass scaling of radial-velocity planet estimates changes these inferences because Kepler's third law and the stellar reflex response depend on the stellar mass. At fixed $K,P$, $M_p\sin i\propto M_{\star }^{2/3}$ and $a\propto M_{\star }^{1/3}$, whereas the fixed-mass detectability boundary scales as $M_{\star }^{-1}$. ThereforeThe observed period and amplitude are unchanged. The stellar radius does not enter these leading dynamical estimates, but is crucial for the exoplanet transit inference.

For the transiting companion, assume an opaque spherical planet, negligible transit dilution and negligible limb darkening. Its radius ratio is $k=R_p/R_{\star }=\sqrt{\delta }=0.03873$, henceUsing the revised stellar mass and a circular orbit gives $a=0.01850\mathrm{au}$ and $A=a/R_{\star }=7.957$. Interpret the duration as the total first-to-fourth-contact duration $T_{14}$. The impact parameter from total transit duration follows from the total duration of a circular planetary transit:This is noncentral but not grazing: $b<1-k$. The small-planet straight-chord approximation gives $b\simeq 0.845$; retaining finite $R_p$ and orbital curvature gives the stated value. A duration measured between other contacts needs the corresponding contact formula. Unknown orbital eccentricity and argument of periapsis change the transverse speed and prevent a unique unconstrained $b$; the numerical estimate is conditional on the stated circular model.^[ref]

Because the transit constrains $\sin i\simeq 0.994$, the RV nondetection now bounds the actual planetary mass, rather than only its projected mass. The bulk-density upper limit from a transit and an RV nondetection isThese bounds assume the instrumental threshold can be applied to a fitted signal at the transit period; no statistical confidence level is specified. Such a large radius at this low mass is inconsistent with an ordinary compact rock-and-iron body. A low-density volatile-rich planet, possibly with an extended H/He envelope, is indicated under the planetary interpretation. The observations do not uniquely determine interior composition, atmospheric abundance or formation history; blended eclipses and activity still require validation. There is no luminosity or effective temperature supplied for the host, so a numerical planetary equilibrium temperature cannot be inferred from these data alone.

The numerical conversions above use externally documented $(GM{)}_{\odot }=1.3271244\times 10^{20}{\mathrm{m}}^3{\mathrm{s}}^{-2}$, $R_{\odot }=6.957\times 10^8\mathrm{m}$, $M_{\oplus }=5.9722\times 10^{24}\mathrm{kg}$, $R_{\oplus }=6.371\times 10^6\mathrm{m}$, $M_J=1.89813\times 10^{27}\mathrm{kg}$, $G=6.67430\times 10^{-11}{\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$ and $1\mathrm{au}=149597870700\mathrm{m}$. They are conversion constants, not additional measured system parameters.^[ref][ref]

In transit spectroscopy, starlight passes tangentially through the planet's terminator atmosphere during primary transit. Wavelength-dependent opacity changes the effective occulting radius. For a thin atmosphere with an absorption feature spanning $N_H$ atmospheric scale heights,Here $\mu$ is the mean particle mass in units of $m_H$, not a chemical potential. Large planets with warm low-gravity, low-molecular-weight atmospheres against small stars have larger signals. A spherical isothermal atmosphere has tangential column enhancement of order $\sqrt{2\pi R_p/H}$ over a vertical column, explaining sensitivity to tenuous high-altitude gas. Molecular bands, atomic lines, scattering slopes and clouds constrain the terminator, but cloud opacity, reference pressure and abundance-temperature degeneracies limit unique retrievals.^[ref]

Occultation spectroscopy instead subtracts the star-only spectrum during secondary eclipse from the star-plus-planet spectrum immediately outside eclipse. It measures the dayside planet/star flux ratio. In the blackbody approximation the thermal eclipse depth isLonger infrared wavelengths generally improve contrast for a cooler planet; molecular absorption or emission depends on the vertical temperature gradient. At optical wavelengths reflected light can dominate, with full-phase contrast $A_g(\lambda )(R_p/a{)}^2$, where $A_g$ is geometric albedo. Eclipse spectra therefore probe dayside emission or reflection, rather than the transit's slant absorption. Clouds and temperature inversions affect interpretation.^[ref]

For either measurement, the photon-limited signal-to-noise ratio scales as the spectral feature amplitude times $\sqrt{N_{\gamma }}$, with a penalty for subtracting finite in- and out-of-event samples. Finer spectral bins reduce photons per bin; bright hosts, repeated events and stable instruments improve sensitivity. Detector systematics, telluric absorption for ground observations, and variable starspots can limit the practical gain. Space observations remove telluric contamination, not stellar variability or instrumental drift. Neither technique directly measures a unique bulk composition or samples all longitudes of the planet.

Adopt circular orbits, negligible planetary masses, spherical bodies, uniform stellar brightness and no transit dilution. Interpret the measured duration as the total duration of a circular planetary transit, from first to fourth contact, $T_{14}$. These conventions matter especially for the off-centre crossing: finite planet size cannot automatically be ignored there. Unknown orbital eccentricity or limb darkening would require further information to infer unique stellar radii.

First convert the apparent magnitude difference into a fractional exoplanet transit depth, rather than treating a milli-magnitude as a part in a thousand:Both crossings are non-grazing because their transit impact parameters satisfy $b<1-k$, so the full projected planetary disc can block the star. Equal depths determine equal radius ratios, not equal absolute planet radii.

For orbital inclination $i$ measured from the sky plane, $b=(a/R_{\star })\cos i$. Choosing mid-transit as orbital phase zero, the projected separation obeysAt first and fourth contact, $z/R_{\star }=1+k$. The full crossing duration therefore isFor the numerical scale use solar radius $R_{\odot }=6.957\times 10^8\mathrm{m}$ and nominal solar mass parameter $(GM{)}_{\odot }=1.3271244\times 10^{20}{\mathrm{m}}^3{\mathrm{s}}^{-2}$, with one day equal to $86400\mathrm{s}$. These are stated conversion choices, not additional measurements of the two systems.^[ref]

Kepler's third law gives $a=[G{M}_{\star }{P}^2/(4{\pi }^2){]}^{1/3}$. For a solar-mass, solar-radius host, this gives $a/R_{\odot }=12.3051$ at five days and $6.68023$ at two days. Substitution in the contact formula yieldsThe transit duration of a small planet instead takes both $k\to 0$ and $R_{\star }/a\ll 1$, giving $T\simeq P{R}_{\star }\sqrt{1-b^2}/(\pi a)$. It predicts $3.10\mathrm{h}$ and $1.37\mathrm{h}$, respectively. The difference is appreciable for B because the fractional change in its crossing chord is larger. Durations defined by mid-ingress to mid-egress or by second to third contact should not be silently equated with $T_{14}$.

The original diagram uses a straight projected track to display the contact geometry; the duration formula above retains orbital curvature.

To infer the actual stellar radii, invert the contact formula before applying the supplied mass-radius scaling. Let $A=a/R_{\star }$ and $s=\sin (\pi T_{14}/P)$; thenThe measured durations imply $A_A=8.34661$ and $A_B=7.55743$. Normalize the stipulated constant mass-radius ratio to the Sun, so $M_{\star }/M_{\odot }=R_{\star }/R_{\odot }\equiv r_{\star }$. Kepler's third law then givesThus the stellar radius from transit duration and a linear mass-radius relation isThe same assumed scaling gives $M_A=1.79M_{\odot }$ and $M_B=0.831M_{\odot }$. A's longer crossing requires a lower mean stellar mass density, while B's shorter full-contact crossing requires a higher one. They are not both solar replicas under this model. Their planets have $R_{p,A}=0.1213R_{\odot }$ and $R_{p,B}=0.05633R_{\odot }$.

For a consistently applied point-planet approximation, the duration scales as $T\propto P^{1/3}{r}_{\star }^{2/3}\sqrt{1-b^2}$ and the inferred radii would instead be $R_A\simeq 1.98R_{\odot }$, $R_B\simeq 1.07R_{\odot }$. The latter is close to solar. Consequently the assertion of a substantial non-solar radius for B depends on which duration convention and finite-size approximation are intended. The contact-based values above use the measured depth rather than discarding it. If eccentric orbits are allowed, the transit speed introduces an additional factor depending on eccentricity and orientation, so the two durations alone cannot establish those radii uniquely.

For an outer coplanar planet around B, the common orbital inclination gives $b_2/b_B=a_2/a_B=(P_2/P_B{)}^{2/3}$. A point-planet centre must cross the stellar disc, requiring $b_2<1$. The usual coplanar transit limit is thereforeThis is a limiting value approached from below; a tangent path has no duration. For an outer planet of finite, unspecified radius ratio $k_2$, any overlap instead requires $b_2<1+k_2$, giving $P_2<P_B[(1+k_2)/b_B{]}^{3/2}$. If it had the same ratio $k_2=k$, that geometric limit would be $3.08$ days; requiring full overlap would give $2.52$ days. Because neither its radius nor a photometric detection threshold is supplied, no unique finite-size practical maximum period can be inferred beyond this conditional expression. The stellar radius cancels from the point-planet period-ratio bound.

For B's inner planet, a reflected-light planetary phase curve haswhere $A_g$ is geometric albedo and $\alpha$ is phase angle. Near exoplanet secondary eclipse the visible hemisphere is almost fully illuminated. Using the standard full-phase approximation and the contact-based $A_B$,The exact lost fraction of combined stellar and planetary light is $F_p/(F_{\star }+F_p)$, which changes the result negligibly at this contrast. The corresponding magnitude depth is about $0.0437$ milli-magnitudes. The circular orbit has the same impact distance at eclipse as at transit, and $b_B<1-k$, so the star can hide the entire planetary disc.

The inclination inferred for B is $83.92^{\circ }$, putting closest approach to full phase at $\alpha \simeq 6.08^{\circ }$. A fully precise prediction would therefore also require the reflected-light phase function, which the question does not specify; the boxed value uses $\Phi \simeq 1$. The consistent point-planet duration approximation yields about $56.7\mathrm{ppm}$ instead, through its different inferred $a/R_B$. It is important to keep the geometry used for the radius inference and the reflected-light calculation consistent.

Finally, the stellar radial-velocity semi-amplitude for a low-mass companion on a circular orbit isThe two stars' equal apparent magnitudes make a comparison at similar photon statistics plausible, though spectral-line widths, stellar variability and observing cadence can still differ. With the contact-based stellar estimates,If the planets have comparable mean densities, A is easier: their equal depth then gives $M_{p,A}/M_{p,B}\simeq (R_A/R_B{)}^3\simeq 9.99$, so $K_A/K_B\simeq 4.44$. The larger planetary mass outweighs A's longer orbital period and larger host mass. For equal planetary masses, B is easier by a factor $2.25$ in amplitude. The supplied transit data give planet radii, not masses, so the ranking is conditional on a planetary mass-radius assumption; a unique unconditional ranking would invent information.

The principal feature is the abundance of planets with radii of roughly one to a few Earth radii compared with large gas giants within the stated short-period sample. The smallest few plotted bins are of broadly similar height, followed by a substantial decline at larger sizes. Thus the figure supports a high occurrence of small planets relative to giants, rather than an indefinitely steep rise toward arbitrarily small sizes. The horizontal scale is logarithmic: one should compare counts in the displayed bins with their widths in $\ln R_p$, not mistake a bin count for a density per unit linear radius.

The vertical axis is an average number of planets per target star. This exoplanet occurrence rate need not equal the fraction of stars hosting a planet, because one star can host several. The restriction to periods below $85$ days also matters: the figure is not a census of distant gas giants or of every planet in a system. Neither the detailed reduction procedure nor the raw detection counts are specified in the caption, so they cannot be reconstructed uniquely from the histogram.

Selection effects actually make the small-planet population harder to see. A full, undiluted exoplanet transit depth is approximately $(R_p/R_{\star }{)}^2$. For independent photometric errors of fractional size ${\sigma }_f$ per sample, an idealized detection signal-to-noise ratio iswhere $N_{\mathrm{in}}$ counts in-transit samples. At fixed star and orbital period, halving planet radius reduces this amplitude by four. Fainter hosts, correlated stellar noise, gaps and transit vetting reduce transit survey completeness further. Geometric transit probability also falls with orbital distance, $p_{\mathrm{tr}}\simeq R_{\star }/a\propto P^{-2/3}$ for fixed stellar mass. For a fixed observing baseline and cadence, the small-planet circular duration law gives $N_{\mathrm{in}}\propto P^{-2/3}$ and approximately $\mathrm{SNR}\propto R_p^2{P}^{-1/3}$.

Recovering an intrinsic exoplanet occurrence rate therefore requires correcting for geometric alignments, incomplete recovery and false positives. An illustrative estimator for a radius-period bin iswhere $C_j$ is the recovery probability over the relevant target sample and $w_j$ accounts for candidate reliability. Real analyses must average these probabilities over heterogeneous targets and measurement uncertainty; the formula indicates the required weights, not a reconstruction of the unpublished inputs to this figure. A primary Kepler space telescope analysis explicitly models both false positives and incomplete detection to infer occurrence.^[ref]

Thus the apparent preference for small planets cannot be explained by their being photometrically easier to detect. Small transits are harder. A plateau in occurrence below about twice Earth's size was independently reported using synthetic-transit injection and recovery, though that study used a different $5$–$50$-day selection and is not identified here as the source of the exam's plot.^[ref]

The natural physical framework is core accretion in a protoplanetary disk. Most initial disk mass is gas, principally hydrogen and helium; a much smaller fraction is solid material. Protoplanetary dust coagulation first assembles aggregates through low-speed sticking. Settling and differential drag help concentrate solids, but bouncing, the fragmentation barrier to dust growth and radial drift of solids in a gas disk prevent a simple assumption that every collision grows a body smoothly to planetary size. Concentration in dust traps or through streaming instability, followed by gravitational collapse of a dense solid population, provides routes to planetesimals. Subsequent collisions must dissipate sufficient relative energy to bind material rather than disrupt it.

Once sufficiently large planetesimals exist, gravitational focusing enhances their accretion cross-sections:When background random velocities remain small, larger bodies grow disproportionately fast in runaway planetesimal growth. As they become massive enough to stir the surrounding population, oligarchic growth develops: several planetary embryos occupy neighboring feeding zones, while much of the remaining solid material resides in smaller planetesimals. Heating the planetesimal random motions weakens gravitational focusing, and local depletion drives the embryos toward an isolation mass. Collisions among embryos can continue assembly after gas dispersal.

This solid-building pathway naturally produces many modest cores. Producing a large gas giant requires an additional, time-sensitive step: a sufficiently massive core must acquire and cool a gaseous envelope while disk gas remains available. Gas captured by a low-mass core releases gravitational potential energy; the envelope's ability to radiate that energy governs its contraction and further accretion. A characteristic cooling estimate iswhere $E_{\mathrm{tot}}$ includes the envelope’s gravitational and internal energies. This states an energy budget rather than a calibrated accretion timescale. Increasing core mass generally facilitates faster gas growth in the appropriate envelope regime. The critical core mass for gas accretion depends on opacity, solid accretion heating and disk conditions; no one exact threshold follows from the histogram. If envelope cooling becomes rapid enough and gas supply persists, runaway gas accretion can produce a giant. If core growth or cooling is too slow, protoplanetary disk dispersal removes the reservoir first, leaving a rocky core or a smaller planet with a limited envelope.

The snow line changes the supply of condensed solids: where water ice can survive, additional solid material can help cores grow. Its location depends on the evolving disk temperature and pressure, so there is no universal orbital radius at which giants must form. Short-period planets need not have assembled entirely at their present locations. Planetary migration can carry cores and giants inward; scattering and collisions can further reshape their orbits. Pebble accretion is another way to supply solids rapidly to embryos, but its efficiency depends on particle coupling and the available pebble flow. These processes alter the distribution without requiring every core to become a giant.

Radius also is not a linear proxy for mass. A relatively small hydrogen-helium envelope can increase a planet's radius substantially, so some planets with a few Earth radii need not be purely rocky. Contraction as planets cool and irradiation-driven atmospheric escape can subsequently reduce their radii. Hence the final histogram combines solid growth, gas capture, migration and later thermal evolution. In particular, the smallest bins need not all share one composition or one formation location.

Direct fragmentation through gravitational instability of an astrophysical disk is an alternative route to massive gaseous objects. It requires a sufficiently self-gravitating disk, described locally by Toomre's stability criterion, together with sufficiently rapid cooling for bound fragments to survive. It does not naturally explain the dominant population of small close-in cores by itself, although migration or further evolution can change the locations of fragments. The observed abundance of small planets is therefore consistent with solid cores commonly forming while only a minority acquire a massive gas envelope before disk dispersal. It does not uniquely prove a particular prescription for each uncertain growth stage.

The physical explanation is widespread core formation combined with restricted access to runaway gas accretion. The strongest observational conclusion concerns small versus giant planets in the stated period range, after accounting for selection; the figure alone does not determine a universal core threshold, a unique mass distribution, or the population beyond the period cut.

A protoplanetary disk is a flattened, rotating reservoir of gas and solids around a young star. Its radial gravity is usually dominated by the host, making its rotation close to Keplerian rotation, ${\Omega }_K=(G{M}_{\star }/r^3{)}^{1/2}$. Gas pressure provides vertical support and a small radial correction to the orbital speed; it does not make the disk a nonrotating spherical atmosphere.

At height $|z|\ll r$, vertical hydrostatic equilibrium in the stellar gravitational field givesFor a locally vertically isothermal ideal gas, $P_g={\rho }_g{c}_s^2$ with $c_s^2=k_{B}T/(\mu m_H)$, where $\mu$ is mean molecular weight. Integration givesHere ${\Sigma }_g$ is the vertically integrated gas surface density of a disk, and $H$ is the isothermal disk scale height. The disk aspect ratio $H/r=c_s/v_K$ is small in a geometrically thin disk. For $T\propto r^{-q}$ it scales as $r^{(1-q)/2}$, so $q<1$ produces disk flaring. This illustrative local vertical model does not determine a unique radial surface-density law or temperature normalization from the information given.

The gas is predominantly molecular hydrogen and helium. A much smaller component consists of heavier elements, partitioned between vapor, solid refractory grains and condensed volatile ices. A gas-to-solid mass ratio of order $100$ is a common initial fiducial model, not a universal measurement applying at every disk radius or evolutionary stage. Solids can grow and settle while gas accretes or disperses, changing the local ratio substantially.^[ref]

Thermal structure organizes composition. In the hottest inner region, grains can sublimate and much of their material remains in the gas. Refractory silicate minerals and metallic grains survive at lower temperatures; farther out, water and then other sufficiently volatile species can condense when their local condensation conditions are met. The water snow line can raise the solid inventory available to planet formation. Its location depends on stellar irradiation, accretion heating, opacity and gas pressure, and evolves with time; it is not fixed merely by naming a star or adopting a solar-system distance.

Stellar radiation heats the exposed surface of a passive disk. Beneath it, shielding can leave a cooler, molecule-rich interior with a higher mass density. In the inner actively accreting disk, dissipation can instead make the optically thick midplane the hottest region. These are distinct heating regimes; a vertically isothermal profile is an analytic approximation rather than a claim that every real layer has the same temperature. Dust opacity controls how radiation is absorbed and re-emitted. Infrared and millimetre emission probe solid material, while spectral lines probe gas; neither observable alone supplies a model-free inventory of the whole disk.

Dust settling in an astrophysical disk makes larger, less tightly coupled particles more concentrated toward the midplane, while turbulence and mixing oppose that settling. For an aerodynamic stopping time $t_s$ short compared with ${\Omega }_K^{-1}$, terminal vertical balance gives $v_z\simeq -{\Omega }_K^2{t}_{s}z$. Small grains remain better coupled to the gas. Radial drift also changes the distribution of solids and their access to snow lines and embryos.

The disk evolves through accretion and angular momentum transport, potentially involving turbulent stresses, magnetorotational instability or magnetically driven winds where conditions permit. Protoplanetary disk dispersal eventually removes gas through accretion and mass loss, including photoevaporation. In standard evolutionary pictures this occurs on a scale of a few million years, with substantial system-to-system variation. That finite reservoir lifetime is central to whether an embryo acquires a massive gaseous envelope.^[ref]

The essential structure is a nearly Keplerian gas disk with vertical pressure support, radial and vertical thermal gradients, and an evolving solid population concentrated by settling, growth and drift. Its composition is set jointly by the original elemental inventory, local condensation conditions and transport.

Introduce the constant $\nu >0$ by $\psi (r,t)=\nu {\rho }_g(r,t)$, and set $q=\gamma E_{\mathrm{SN}}$, the injected energy per unit mass of newly formed stars. Before a local wind switches on, no return or matter exchange impliesIn this independent-volume supernova quenching model, the accumulated injection per volume isFor any specified positive binding energy per unit remaining gas mass $B(r)$, the local threshold isGas conversion must be included in this comparison; replacing the integrated star formation by its initial rate times $t$ is only a short-time approximation.

For a quenching front in a fixed galactic potential, take $B(r)=-{\Phi }_{\mathrm{grav}}(r)=2k[1+\ln (r_0/r)]$. This treats the gravitating profile as fixed: newly formed stars initially replace gas at the same radius, and later changes of the potential caused by expelled gas are neglected. Gas first becomes unbound at the outer edge, atThereafter the inner edge of the quenched exterior region isThere is no quenched radius inside the galaxy before this onset. The remaining star-forming region is $0<r<r_{\mathrm{quench}}$; the front moves inward. The ideal logarithmically divergent central potential delays complete removal to infinite time in this fixed-profile model. It should not be interpreted as a physical unresolved central core.

The local binding-energy allocation in galactic feedback also admits a different convention, because it calls $\varphi$ an energy of a volume element without specifying how the nonlocal self-gravitating energy is assigned to that element. If the intended prescription is instead the shell-assembly contribution from part (a),then $B_{\mathrm{shell}}=2k$ at every radius. The same time $t_{\mathrm{start}}$ applies everywhere: all shells quench simultaneously, and there is no continuously propagating radial front. This is a shell-energy toy model, not the escape potential including exterior shells.

Both results follow from explicit definitions. A convention assigning $\frac{1}{2}{\rho }_g{\Phi }_{\mathrm{grav}}$ locally replaces $B$ by $B/2$ in $Q(t)=B(r)$; it is again an energy allocation rather than test-parcel escape. Updating the potential after gas loss or choosing a fixed initial-gas binding budget instead of the remaining-gas budget would require an additional closure. The stated independent-volume assumptions do not uniquely choose between these. The displayed escape-potential result and shell prescription must therefore not be presented as interchangeable universal laws.

To test radial stability of an axisymmetric circular orbit, let $\Phi (r)$ be the time-independent gravitational potential per unit stellar mass in the midplane. Axial symmetry conserves the specific angular momentum $j=r^2\dot{\phi }$. The radial equation isA circular orbit at $r_0$ obeys $j_0^2=r_0^3{\Phi }^{\prime }(r_0)$. Choose the guiding circular orbit with the same conserved angular momentum as the perturbed star. Setting $r=r_0+x$ and expanding at fixed $j=j_0$ giveswhere the radial epicyclic frequency isHere ${\Omega }^2(r)={\Phi }^{\prime }(r)/r$ and $j_c=r^2\Omega$ refer to the family of circular orbits. They are differentiated as functions of radius only after the perturbation equation has been expanded at conserved stellar $j$.

For a radially stable circular orbit, ${\kappa }^2>0$, so to linear orderThus the orbit performs small simple harmonic radial oscillations about its guiding radius.

Symmetry alone is insufficient to guarantee stability: ${\kappa }^2<0$ gives exponentially growing displacement, and ${\kappa }^2=0$ is marginal. The intended statement assumes a stable unperturbed orbit.

For a Kepler potential $\kappa =\Omega$; for a flat circular-speed curve $\kappa =\sqrt{2}\Omega$; for solid-body rotation $\kappa =2\Omega$. Conservation of $j$ also gives $\dot{\phi }\simeq {\Omega }_0(1-2x/r_0)$, explaining the accompanying azimuthal epicycle rather than a purely radial motion in the inertial plane.

A protostar forms when a self-gravitating cloud fragment can contract faster than pressure support and energy loss can arrest it. A useful local scale is the Jeans length, ${\lambda }_J\sim c_s/\sqrt{G\rho }$, with characteristic Jeans mass $M_J\sim c_s^3/(G^{3/2}{\rho }^{1/2})$ and free-fall timefor an initially uniform sphere. Geometry, turbulence and magnetic support change the detailed threshold; none of these estimates asserts that every cloud contracts as a uniform sphere.

While cooling is efficient, compression is nearly isothermal. Then $M_J$ decreases as the density rises, allowing fragmentation. Rotation becomes increasingly important during contraction because a parcel conserving specific angular momentum $j$ encounters a centrifugal scale $r_c\sim j^2/(GM)$. The resulting accretion disc must transport angular momentum before matter can reach the central object. Infall, a dense accreting interior and an outer envelope can therefore coexist; a protostar is not initially an isolated fixed-mass star.

Increasing optical depth eventually slows radiative cooling. The central gas then becomes less compressible and can form a pressure-supported first core. Molecular hydrogen dissociation absorbs compressional energy and can trigger a second collapse before a compact stellar core is established. Subsequent accretion supplies a luminosity of orderThe retained fraction depends on shock and disc radiation, so this expression is an energy scale rather than a guaranteed luminosity all emerging from the stellar surface.

A central accreting core can be nearly hydrostatic on its short sound-crossing time while its mass, entropy and radius evolve on longer times. A low-mass object is often largely convective because of its opacity and steep radiative temperature gradient; deuterium burning can temporarily contribute to support and moderate contraction. Neither convection nor the timing of deuterium ignition is universal across all masses and accretion histories.

When the main accretion phase ends, gravitational contraction remains an energy source. The virial theorem explains how contraction both heats the interior and supplies radiation: for an ideal-gas-supported object with negligible surface-pressure correction, $2U+W\simeq 0$, so only part of the released binding energy appears as increased thermal energy. The corresponding Kelvin-Helmholtz time isFully convective pre-main-sequence evolution approximately follows a Hayashi track, with decreasing radius and luminosity at comparatively weakly changing effective temperature. Objects developing radiative interiors can follow Henyey tracks. Eventually sustained core hydrogen burning balances the luminosity and the star reaches the main sequence. Massive objects can ignite hydrogen while substantial accretion continues, so “accretion ends, then hydrogen ignites” is not a universal ordering.

A useful structural distinction is therefore between an accretion-powered protostar and a largely fixed-mass pre-main-sequence contraction model. An infrared-bright envelope or disc traces obscured formation and reprocessing of radiation; a schematic evolutionary track should not replace the actual mass- and entropy-dependent stellar-structure problem.

An encounter between galaxies or clusters mainly couples their collective gravitational fields; it need not involve physical impacts between individual stars. Direct stellar collision rates depend on number density, relative speed and gravitational focusing. For two stars the collision cross-section is approximatelyIn an ordinary galaxy, the small geometrical stellar filling factor makes such impacts rare even while the galaxy undergoes a major encounter. Dense stellar cores are an important exception, not a reason to treat all galaxies as collisional fluids.

Tidal forces depend on the gradient of the perturber's field. For a distant fast encounter of mass $M_p$, impact parameter $b$ and speed $V$, the impulse scale for a star at displacement $r$ isA more specific result illustrates the geometry. Put the straight-line perturber trajectory in the $(x,y)$ plane along $y$, with closest approach at $x=b$. In the tidal and impulse limits,for an initially spherical, nonstreaming target with randomly sampled internal phases. The cross term averages to zero, and $⟨x^2+z^2⟩=2⟨r^2⟩/3$. Heating, expansion and tidal stripping preferentially affect weakly bound outer stars. A slow encounter can be adiabatically shielded rather than impulsive; a penetrating or strongly resonant encounter needs more than this distant straight-line approximation.

Disc rotation and orbital orientation matter. Prograde passages can drive stronger coherent distortions than corresponding retrograde passages. Tidal bridges, long tails, asymmetric arms and shells can record the encounter. Restricted gravitational calculations already reproduced prominent bridge and tail structures without requiring individual stellar collisions; full self-consistent models additionally follow the gravity of the responding discs and haloes.^[ref]

A first high-speed passage need not bind the pair. Capture or merging requires orbital energy and angular momentum to be redistributed into internal motions, escaping material, gas radiation or the surrounding halo. Dynamical friction transfers orbital energy to a gravitational wake and can bring a bound satellite inward. In a major merger, the potential changes on orbital times and violent relaxation redistributes stellar energies; phase mixing then smooths fine orbital structure. These are collective collisionless processes, distinct from slow two-body relaxation.

Gas behaves differently from stars: gas streams can shock, dissipate energy, lose angular momentum and fuel a central starburst or active nucleus. Gas-poor mergers are correspondingly more nearly dissipationless. Outcomes depend on mass ratio, orbital angular momentum, relative velocity, halo structure and gas fraction: they range from a modest flyby and tidal debris to disruption of a satellite or a merged remnant. Slow encounters in a group are generally more favorable to merging than otherwise comparable very fast unbound passages, but no single speed or mass-ratio threshold follows without an explicit structural model.

Galaxy formation combines gravitational assembly of dark-matter haloes with dissipative baryonic physics. Primordial density fluctuations supply a statistical seed distribution. In linear theory, each Fourier mode evolves through a growth factor and a scale-dependent transfer function; the power spectrum describes the variance by scale. Radiation pressure, free streaming and the time of horizon entry influence the small-scale transfer, so one should not replace the spectrum by a single universal growing power of time.

When fluctuations become nonlinear, overdense regions collapse, virialize and build larger structures through accretion and mergers. Cold dark matter permits a broad hierarchy of bound haloes. A spherical-collapse threshold and a mass-variance prescription, such as the Press-Schechter formalism, organize an approximate halo abundance; departures from spherical collapse and correlated environments limit its literal interpretation.

Gas falling into a halo is compressed and may be shock-heated toward a virial scaleUnlike collisionless dark matter, it can radiate away thermal energy. A rough cooling time iswhere the denominator specifies a chosen optically thin astrophysical cooling function convention. Cooling is effective for condensation when this time is short compared with the relevant dynamical or supply time. Atomic transitions, molecular processes, ionization state, density and metal abundance matter. A stable virial shock is not obligatory for all accretion: cold dense inflows can supply gas in regimes where shock-heated gas would cool rapidly.

The distinction between a gravitationally assembled massive halo and a radiatively condensing luminous component is fundamental. In the classic two-stage picture, non-dissipative clustering builds potential wells while cooling allows gas to concentrate much further than the halo component.^[ref]

Angular momentum inherited from tidal torques and later accretion helps gas settle into a rotationally supported disc after cooling; disc size depends on angular-momentum retention and transport. Locally self-gravitating gas fragments and forms stars. Stellar radiation, winds and supernovae then heat gas, drive turbulence, enrich it with heavy elements and can expel it from shallow potential wells. Metal enrichment can also accelerate later cooling, so feedback is more than a single positive or negative energy term.

Galaxy growth is usefully expressed as a gas-budget competition:Here $\mathcal{R}$ is the promptly returned stellar mass fraction; unlike the simplification in Question1, a general evolutionary model need not set it to zero. The equation separates newly supplied, converted, expelled and recycled gas rather than assuming that halo mass uniquely fixes stellar mass.

Accreting central black holes can provide radiative and mechanical feedback. Heating a halo, ejecting gas or maintaining a hot atmosphere can reduce fresh fuel; removal of a satellite's gas supply and ram pressure in a surrounding medium provide additional environmental routes. Quiescent galaxy assembly can still add stellar mass through dry mergers, so cessation of star formation is not identical to cessation of all mass assembly.

Mergers and disc instabilities can rearrange morphology and trigger bursts, while smooth accretion can rebuild discs. Cosmological expansion changes the supply and merger environment; accelerated late-time expansion also slows linear growth. A predictive model must connect these mechanisms to the stellar mass function, gas fractions, star-formation histories, sizes, chemical abundances and clustering. Those observables test different pieces of the assembly-and-feedback picture; none alone implies a one-to-one instantaneous relation between galaxy luminosity and total halo mass.

The detector is intended for arcsecond-scale astronomical seeing, rather than the telescope's diffraction limit. A circular aperture of this area has diameter $1.13\mathrm{m}$, giving a Rayleigh criterion angle about $0.11$ arcsec at the observing wavelength. One-arcsecond pixels conventionally sample an astronomical seeing profile of roughly two arcseconds across with two samples; a one-arcsecond profile would be undersampled. The pixel scale alone does not determine an exact astronomical seeing width.

Use the supplied surface brightness as radiative flux integrated over the photometric passband: its units contain no inverse hertz. Treat $h\nu$ as the representative photon energy, and assume unit throughput and quantum efficiency. A genuinely frequency-specific brightness would require a bandwidth. With angles measured consistently in arcseconds, an annulus has area $2\pi rdr$ square arcseconds, soThis is flux per collecting area; collected energy is $AtF$. There is no extra conversion to steradians when both brightness and angular area use square arcseconds.

For Poisson photon counting, a pixel with expected count $N$ has energy $e=Nh\nu$ and standard deviation ${\sigma }_e=h\nu \sqrt{N}$. Thus the local four-sigma rule gives $N=4\sqrt{N}$, or 16 photons at the boundary. With pixel area $\Delta \Omega =1{\mathrm{arcsec}}^2$,Consequently $r_m=(1-16/32.6991)/k=10.2138$ arcsec andIntegrating to the model's outer radius $R=1/k=20$ arcsec instead givesThe pixel cut retains 51.6% of this total. It is an imposed isophotal photometry rule, not a fundamental limit: averaging annuli or fitting a justified profile can recover signal below the per-pixel cut.

In practice, sky background photons add shot noise even after subtraction of the mean sky. read noise, dark current, imperfect flat-field correction, throughput calibration and atmospheric extinction also matter. At large radius the dominant uncertainty can be a tiny error in the sky level, scattered light or the wings of the point spread function, rather than source photon statistics. Mask foreground stars and background objects, compare blank-sky regions and dithered exposures, and check profile recovery by inserting artificial extended sources. astronomical seeing and finite pixels convolve the intrinsic profile; model that response when extrapolating the total.

The luminosity convergence of an outer surface-brightness profile follows directly: $I\propto r^{-p}$ contributes $dF\propto r^{1-p}dr$, hence $F(<R)\propto R^{2-p}$ at large radius for $p<2$. For either shallow index considered, ever larger annuli dominate: extending the law to infinity gives an unphysical divergent luminosity. Such a law must eventually steepen or truncate. Over a finite range, however, substantial light can lie below a surface-brightness selection threshold and be mistaken for sky. This biases galaxy luminosities, the luminosity function and the inferred diffuse light budget; distant diffuse galaxies become especially incomplete because bolometric cosmological surface-brightness dimming scales as $(1+z{)}^{-4}$. An assumed finite-aperture total is therefore a model-dependent quantity.

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.

An astronomical interferometer combines light from collectors separated by projected interferometric baseline $\mathbf{B}$. For a distant, narrowband, spatially incoherent source, the Van Cittert-Zernike theorem gives normalized complex visibilityThe resolving scale is $\lambda /B$, while the individual collector and beam combiner restrict the field of view. Delay lines compensate geometric path differences. To see broadband fringes, residual optical path length mismatch must lie within the coherence length of order ${\lambda }^2/\Delta \lambda$; fringe tracking follows rapidly changing atmospheric piston in interferometry. Baseline changes and Earth-rotation aperture synthesis sample additional spatial frequencies.

For an ideal uniform circular star of full angular diameter $\Theta$,Real limb darkening changes the conversion. Several baselines constrain binaries, stellar radii and circumstellar structure; orbital angular separations combined with radial velocities can give physical orbital scales. calibration stars test contrast losses, alongside wavelength, photometric imbalance, polarization and bandwidth controls. For three collectors the closure phase cancels telescope-based phase errors: ${\varphi }_{12}+{\varphi }_{23}+{\varphi }_{31}$ retains source asymmetry information. Sparse baseline coverage still leaves image/model ambiguities.

In stellar intensity interferometry, thermal-light correlations instead give $g^{(2)}(0)-1=|g^{(1)}{|}^2$ for an ideal single-polarization resolved temporal mode. The Sirius experiment correlated separate detector outputs rather than optically combining the beams.^[ref] Pure atmospheric phase shifts do not alter local intensities, but finite detector bandwidth dilutes the correlation and photon statistics limit sensitivity; the two-detector observable also loses the complex phase.

annual stellar parallax uses the known Earth-orbit baseline, with $d(\mathrm{pc})=1/\varpi (\mathrm{arcsec})$. Repeated positions separate parallax from proper motion; calibration offsets, binarity and noisy-parallax selection require checks. Nearby stars calibrate absolute luminosities. Extinction-corrected distance modulus then givesmain-sequence fitting compares a cluster sequence with nearby calibrated stars; age, metallicity, reddening and unresolved binaries limit the match. Cepheid variables extend the range to nearby galaxies: measure periods and mean multiband magnitudes, calibrate the Cepheid period-luminosity relation, and correct for extinction, metallicity, crowding and pulsation mode. RR Lyrae stars and the tip of the red giant branch provide complementary population-II routes with their own composition and calibration dependence.

Some anchors bypass intermediate rungs. An eclipsing-binary distance starts from a double-lined eclipsing binary: velocities give $a\sin i$; eclipses give inclination $i$ and relative radii $R/a$, hence physical radii. An independently calibrated surface-brightness/colour relation gives angular radii and distance; alternatively corrected bolometric flux satisfies $F_{\mathrm{bol}}=\sigma T_{\mathrm{eff}}^4(R/d{)}^2$ for a component. temperature, extinction, third light and surface-brightness calibration remain systematic errors. Eight late-type binaries gave a 2.2%-accuracy LMC anchor in 2013.^[ref]

A maser-disk geometric distance uses water masers orbiting in the nearly edge-on nuclear disk of NGC 4258. very-long-baseline interferometry, Doppler velocities, proper motions and accelerations constrain its orbit scale and distance. In an ideal circular, matched-radius example $R=v^2/a_{\mathrm{orb}}$ and $d=R/\theta$; real fits account for disk warp and different maser locations. This geometric method calibrated an extragalactic distance independently of stellar luminosities.^[ref] These overlapping anchors cross-check the nearby-galaxy ladder; no Hubble-flow fit is needed here.

Stellar reflex motion. A planet and its host orbit their common barycentre. The star's line-of-sight velocity shifts its absorption lines by $\Delta \lambda /\lambda \simeq v_r/c$ at low speeds. A time series can be fitted withHere $\nu (t)$ is true anomaly, $\varpi$ the argument of periastron and $\gamma$ an arbitrary systemic velocity offset. At small companion mass and known stellar mass, the orbit supplies $M_p\sin i$, not the true planet mass without an inclination constraint. Long monitoring helps distinguish a reproducible Keplerian signal from stellar oscillations, magnetic activity, spots and instrumental trends.

The Lick reference-cell implementation. Starlight passes through a temperature-controlled cell of molecular iodine before entering the Hamilton high-resolution echelle spectrograph. The cell imprints a dense pattern of reference absorption lines on the stellar spectrum, predominantly in the approximately $500$–$630\mathrm{nm}$ interval. These lines share the same exposure and downstream optical path as the stellar light. A calibrated high-resolution Fourier-transform iodine spectrum and an iodine-free stellar template allow the wavelength solution and instrumental line-spread function to be fitted along with the stellar Doppler shift. The method corrects shifts and line-shape changes that a separate calibration exposure would not necessarily follow.^[ref]

A useful forward model for one short spectral segment isfollowed by pixel integration, continuum scaling and a fitted wavelength calibration. Here $S$ is the intrinsic stellar spectrum, $T_{{\mathrm{I}}_2}$ the cell transmission and $\mathcal{P}$ the instrumental line-spread function. The spectra multiply before convolution; simply dividing a recorded blended spectrum by the laboratory iodine spectrum is not equivalent to this model. The iodine pattern is stationary in the observatory frame while stellar lines move relative to it. Separate spectral segments accommodate wavelength dependence of the instrumental profile.

Earth's rotational and orbital velocities must then be removed with a barycentric correction using accurate timing, ideally weighted by photon arrival rate. Many stellar lines constrain a shift smaller than one instrumental resolution element; spectral resolution alone is not the velocity error. Photon statistics, template quality, instrumental-profile modelling and intrinsic stellar variability limit the precision. The method measures changes relative to a template and does not by itself provide a perfectly absolute centre-of-mass velocity. No particular discovery, modern precision floor or historical observing performance is assumed here.

Measure counts in electrons so that photon and detector noise can be combined directly. For source electrons $S$, $n$ aperture pixels, total time $t$, sky rate $b$ per pixel, dark-current rate $d$ per pixel, $M$ reads with rms read noise $r$ per pixel, and an independently estimated mean background per pixel of variance ${\sigma }_{\hat{B}}^2$, an ideal aperture-sum budget isThe last term represents the propagated calibration uncertainty if it can be treated as a random variance; systematic residuals can instead bias the measurement and need an explicit error model. For a background estimated from $n_b$ independent, equally exposed sky pixels, one approximation is ${\sigma }_{\hat{B}}^2=[(b+d)t+M{r}^2]/n_b$. Correlated resampling, detector cross-talk or a spatially structured sky invalidates the independent-pixel simplification.

Photon counting and sky. Source photon shot noise has variance $S$ and cannot be calibrated away. Longer integration or greater throughput improves its fractional error as $S^{-1/2}$. Sky photons, including moonlight, airglow and scattered light, add their own Poisson fluctuations. A dark site, appropriate observing time and filter, good baffling and a smaller or optimally weighted source aperture reduce sky contributions. Better seeing or adaptive optics can concentrate a point source into fewer noise-bearing pixels. Sky subtraction removes the mean background but not the random sky photons already recorded, and the sky estimate adds uncertainty of its own.

Dark and read noise. Thermal dark current contributes a mean and shot noise. Cooling reduces its rate; well-matched dark frames remove a reproducible mean and map hot pixels, but do not undo the science exposure's random dark-current fluctuations. Each read adds electronic noise. At fixed total integration, many short exposures add more read variance than fewer long exposures, so use exposures long enough to become sky limited when saturation, cosmic rays and temporal changes permit. Multiple non-destructive reads can help on detectors that support them. Subtracting a bias frame corrects an electronic offset, not the stochastic read noise.