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The Hill radius compares planetary gravity with the differential stellar tide. Expanding the stellar tide near a circular orbit gives $G{M}_p/r_H^2\simeq 3{\Omega }^2{r}_H$, soIt sets the size of the planet's tidal neighbourhood, controls encounter and feeding zone scales, and limits the region within which material can remain bound against stellar shear. It is not itself a solid surface or a guarantee of capture.

The Bondi radius compares planetary potential energy per unit mass with the gas thermal scale,This convention uses the ambient sound speed; definitions based on escape speed can differ by a factor of two. Inside $r_B$, gravity can substantially perturb thermal gas. Gas still needs a way to dissipate energy and angular momentum to accrete; a geometric boundary alone does not supply a gas-capture rate.

For an embedded hydrostatic envelope a useful outer scale is $r_{\mathrm{out}}\sim \min (r_B,r_H)$, subject to disk thickness and gas recycling. Let $h=H/a=c_s/(a\Omega )$ and $q=M_p/M_{\star }$. The Bondi-Hill crossover in a protoplanetary disk obeysSmall cores are usually thermally limited, whereas larger ones become tidally limited. These radii also organize pebble accretion: gas drag can turn an encounter into capture, with drift-dominated and shear-dominated regimes depending on particle stopping and encounter times. There is no universal accretion efficiency fixed by $r_H$ or $r_B$ alone.^[ref]

Type I migration is orbital angular-momentum exchange between an embedded low-mass planet and its disk without a deep gap. The planet excites spiral density waves near Lindblad resonances; the disk's reaction torques act back on the planet. The outer wake usually supplies a negative torque and the inner wake a positive one, with their small imbalance often driving inward migration. Gas on horseshoe orbits in a planetary disk near corotation adds a corotation torque that depends on entropy and vortensity gradients.

For a thin disk, define $q=M_p/M_{\star }$, $h=H/a$ andThe coefficient $C$ is signed and depends on disk gradients, thermodynamics and diffusion; it is not a universal constant. For a circular orbit with fixed planet mass, $L_p=M_p{a}^2\Omega \propto a^{1/2}$. The Type I migration timescale from disk torque follows from ${\dot{L}}_p=\Gamma$:Thus a more massive embedded core generally migrates faster within this regime, unlike a picture in which migration starts only after giant-planet formation.

In a simple locally isothermal smooth disk the net torque is commonly negative, potentially removing cores before they acquire gas. In a non-isothermal disk an unsaturated entropy-related corotation torque can balance or exceed the negative wave torque, giving a planet trap or outward migration. Viscosity and thermal diffusion help maintain the background gradients across the horseshoe region; without adequate replenishment corotation-torque saturation can suppress that contribution. The linear low-mass scaling requires a small disturbance, roughly $q\ll h^3$; larger planets, gap formation and strong disk feedback require a different treatment. Type I migration is therefore neither universally inward nor literally viscous advection of the planet.^[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.

Oligarchic growth is the stage in which a relatively small number of planetary embryos dominate the dynamics of many smaller planetesimals. It follows runaway planetesimal growth. The embryos remain separated, grow to broadly comparable masses within similar local disk conditions, and regulate their accretion by stirring the population that supplies their material. The primary numerical study describes this embryo–planetesimal structure and typical separations of about ten individual Hill radii.^[ref]

The key feedback is gravitational focusing. In a dispersion-dominated encounter with approach speed $v$ and a small impactor, conservation of energy and conservation of angular momentum give the collision cross-sectionAt fixed internal mass density, $R\propto M^{1/3}$ and $v_{\mathrm{esc}}^2\propto M^{2/3}$. If the background random speed initially is controlled by small planetesimals and varies little with embryo mass, strong focusing gives ${\sigma }_{\mathrm{coll}}\propto M^{4/3}$. For a disk with planetesimal thickness of order $v_z/{\Omega }_K$ and a fixed ratio $v/v_z$, the associated estimate isup to geometric factors. With nearly fixed $v$, this gives $\dot{M}\propto M^{4/3}$ and $\dot{M}/M\propto M^{1/3}$: the largest bodies have the shortest fractional growth times, the essence of runaway growth.

As embryos grow, their gravitational encounters increasingly excite the random eccentricities and inclinations of nearby planetesimals. This is viscous stirring of planetesimals, not molecular fluid viscosity. Higher encounter speeds lower the focusing factor. The largest local embryo heats its feeding population most strongly and thereby suppresses its own relative growth advantage. Gas drag and dissipative collisions oppose stirring; dynamical friction tends to keep the more massive embryos' random motions relatively small while transferring random energy to lighter bodies. The balance between these processes controls the accretion rate.

For an illustrative self-regulated scaling $v\propto M^{1/3}$, the ratio $v_{\mathrm{esc}}/v$ is approximately fixed. The previous growth estimate becomesLarger embryos now grow more slowly in fractional terms, allowing neighboring embryos in comparable environments to approach similar masses. This scaling illustrates the feedback; it is not a universal exact exponent for every planetesimal size distribution, damping law or disk surface-density profile. In particular, a two-body focusing cross-section must be modified in the shear-dominated regime, where encounters are governed by the stellar tidal field.

The relevant gravitational interaction scale is the Hill radius,Close embryo encounters and orbital repulsion keep adjacent embryos separated by several Hill-scale lengths. The mutual Hill radius uses their combined mass; for equal neighboring masses it is $2^{1/3}{R}_H$. Stating which definition is used matters when quoting a spacing coefficient. The spacing is a dynamical scale, not an exact universal integer.

Each feeding zone of a planetary embryo contains a finite solid reservoir. To estimate its isolation mass, let the full annular width be $\Delta a=B{R}_H$, where the dimensionless coefficient $B$ includes the stated choice of width and Hill-radius convention. For local solid surface density ${\Sigma }_s$,Solving givesThus denser solid disks and wider feeding annuli allow larger embryos. If ${\Sigma }_s\propto a^{-p}$, the estimate scales as $M_{\mathrm{iso}}\propto a^{3-3p/2}$ at fixed host mass and $B$. This is a local exhaustion estimate, not an absolute maximum planet mass: radial delivery of solids, pebble accretion, migration and embryo mergers can replenish or enlarge a feeding zone. No solid surface-density normalization is given, so a numerical isolation mass would require an invented disk model.

As the local supply is depleted, growth slows further. After gas damping weakens, embryo orbits may cross and giant collisions can assemble larger terrestrial planets. Farther out, sufficiently massive cores may instead acquire gas before protoplanetary disk dispersal. Oligarchic growth explains why a few comparable embryos share a disk while many smaller planetesimals remain: gravitational stirring weakens focusing and makes accretion self-regulating.

For smooth inviscid flow of constant mass density under a conservative body force, Kelvin's circulation theorem states that the circulation around any closed material curve is conserved:The constant-mass density assumption is the usual homogeneous incompressible flow setting here. Merely requiring $\nabla \cdot \mathbf{u}=0$ in a fluid of varying mass density would not by itself make the pressure term below a gradient.

Parameterize the material curve by a periodic particle label $s\in [0,1]$, writing ${\mathbf{r}}_t(s,t)=\mathbf{u}(\mathbf{r}(s,t),t)$. Differentiate its line integral, keeping $s$ fixed:The second term vanishes by periodicity. If the conservative body force per unit mass is $-\nabla \Omega$, the Euler momentum equation givesIts line integral around a closed curve is zero, proving the claim. The velocity field must remain smooth on the moving curve, which cannot pass through a line vortex singularity during this argument. No assumption of irrotational flow was needed.

At the rigid bed, the no-penetration boundary condition givesThe displaced surface is material, so taking the material derivative of $z-\eta (x,t)$ gives the exact kinematic boundary condition for a free-surface graph:With upward vertical coordinate $z$, the gravitational potential per unit mass is $gz$. The Unsteady Bernoulli equation, evaluated on the surface with the prescribed pressure $p=B{\eta }_{xxxx}$, givesHere the spatially uniform time-dependent Bernoulli term has been absorbed into the velocity potential. The pressure is measured relative to the constant exterior pressure. This dynamic boundary condition uses the specified massless bending-cover model, with no ice inertia or tension.

inside-out collapse of a singular isothermal sphere starts from an idealized static envelope,with constant isothermal sound speed $a_s$. Collapse begins centrally after formation of a central mass. An expansion wave moves outward through the initially static envelope at the sound speed:“Expansion” describes the outward-moving information front; the gas behind it moves inward. The exterior remains close to its original $r^{-2}$ equilibrium until the wave reaches it.

In the inner region, the central mass dominates and pressure becomes less important:The self-similar accretion scale is ${\dot{M}}_{\ast }=m_0{a}_s^3/G$, where the dimensionless coefficient $m_0$ follows from the dynamical similarity solution, not from counting material swept by the front. Indeed, the initial mass enclosed by the front grows at $2a_s^3/G$, and some of that material remains in the infalling envelope. For fixed mean particle mass, ${\dot{M}}_{\ast }\propto T^{3/2}$. The original similarity calculation establishes the static outer and free-fall inner regimes.^[ref]

The model assumes spherical symmetry, isothermality and negligible rotation or magnetic stresses. An asymmetric optically thick spectral line alone does not establish infall: an optically thin systemic-velocity tracer and spatial maps help distinguish inward motion from rotation, foreground absorption and outflow.

galactic feedback couples star formation and black-hole accretion back to the gas supply. Stellar radiation, winds, supernovae and cosmic rays inject energy and momentum; active-galactic-nucleus feedback adds radiation, jets and winds powered by accretion. Heating, turbulence or gas ejection can suppress later star formation, while compression can sometimes promote it. Negative feedback permits self-regulation rather than unrestricted gas consumption.

Define the mass-loading factor $\eta ={\dot{M}}_{\mathrm{out}}/{\dot{M}}_{\ast }$. If the launch speed scales with the escape speed and the injected energy per stellar mass is fixed, conservation of energy suggests $\eta \propto v_{\mathrm{esc}}^{-2}$; fixed injected momentum instead suggests $\eta \propto v_{\mathrm{esc}}^{-1}$. Coupling efficiencies, radiative losses and recycling change these ideal scalings. Thermal energy injected into dense gas can radiate away before driving an outflow, so an energy budget alone does not demonstrate effective feedback.

Radio-filled X-ray cavities provide a mechanical diagnostic. For a cavity containing a relativistic fluid with $\gamma =4/3$, its enthalpy is $\gamma PV/(\gamma -1)\simeq 4PV$; dividing by an estimated age gives a mechanical-power estimate. Chandra X-ray Observatory observations of the Perseus cluster revealed fronts and ripples associated with radio bubbles, interpreted as weak shocks and pressure waves that can transport jet energy into the cooling atmosphere.^[ref]

The test requires ambient density and temperature estimates, radio/X-ray spatial comparison, projected cavity volumes and an age model. Brightness edges must be distinguished from absorption and cool rims; wave energy is not automatically thermalized locally. These controls determine whether inferred mechanical heating can actually offset radiative cooling.

The normalized cycloid hasThe initially expanding perturbation reaches maximum radius at $\eta =\pi$, $t=t_m$, then formally collapses to zero radius at $\eta =2\pi$, $t=2t_m$. A finite virialized object is assigned the conventional formation epoch $t_f=2t_m$; random motions replace the pressureless point-collapse singularity. The cycloid alone does not describe that relaxation.

For the standard top-hat virialization estimate, write $U=-C_{G}G{M}^2/R$ and retain the same gravitational structure factor $C_G$ at turnaround and final equilibrium. The coherent kinetic energy vanishes at turnaround, whereas the virial theorem gives $2K_v+U_v=0$ in the final isolated equilibrium. conservation of energy givesThe final density is eight times the turnaround density. Meanwhile $\bar{\rho }\propto a^{-3}\propto t^{-2}$, so $\bar{\rho }(t_f)=\bar{\rho }(t_m)/4$. The Einstein-de Sitter virial overdensity follows by combining these factors with the supplied turnaround contrast:The value 200 is a convenient rounded overdensity convention, not exactly the Einstein-de Sitter result.

The virial radius of a dark-matter halo follows from $M_v=4\pi R_v^3{\rho }_v/3$:The hydrostatic temperature of an inverse-square halo follows from hydrostatic equilibrium and the ideal gas law:Since ${\rho }_g\propto r^{-2}$ and $M(r)=M_{v}r/R_v$, cancellation of ${\rho }_g/r$ gives $2k_B{T}_v/\mu =G{M}_v/R_v$. ThereforeHere $\mu$ is a dimensionful mean particle mass, as specified in the paper; a dimensionless mean molecular weight would need multiplication by a particle-mass unit. The isothermal interior is a further halo idealization, not an exact continuation of the uniform top-hat profile used in the virial-overdensity estimate. Changing its gravitational structure factor in a literal energy comparison would change the turnaround-radius relation.

The formation redshift from halo mean density uses the cluster data, which imply ${\rho }_v=3M_v/(4\pi R_v^3)=5.37148\times 10^{13}{M}_{\odot }{\mathrm{Mpc}}^{-3}$. Thus, using the requested rounded convention,Using $18{\pi }^2$ instead gives $z_f\simeq 1.00$. In the Einstein-de Sitter time scaling the rounded-convention result is $t_f/t_0=(1+z_f{)}^{-3/2}\simeq 0.376$. Directly from the mass and radius,For the estimate, take mean particle mass $\mu =0.6m_p$, appropriate approximately to a hydrogen-helium mixture with complete ionization. This gives $T_v\simeq 1.41\times 10^8\mathrm{K}$, or $k_B{T}_v\simeq 12.1\mathrm{keV}$. The composition choice is additional input, not a value supplied by the examination; the boxed expression retains the result for any $\mu$.

heating and cooling balance in the interstellar medium depends on density, ionization, radiation and shielding. Write net cooling per volume as $\mathcal{C}=n^2\Lambda (T)-n\Gamma$. Ultraviolet grain photoelectric heating is important in diffuse neutral gas; photoionization heats ionized regions, cosmic rays penetrate shielded gas, and shock waves from stellar winds and supernovae supply hot gas. Cooling includes collisionally excited atomic lines, fine-structure metal lines at lower temperatures, molecular lines in shielded clouds, and continuum radiation and dust emission where appropriate. The heating efficiency and cooling abundances matter as much as a nominal temperature.

A multiphase interstellar medium can contain a cold neutral medium near $10^2\mathrm{K}$, a warm neutral medium and warm ionized medium near $10^4\mathrm{K}$, a hot ionized medium near $10^6\mathrm{K}$, and still colder molecular clouds. Cold and warm atomic equilibrium branches can coexist at similar thermal pressure over a finite pressure interval; this is a calculated heating/cooling balance, not a requirement that all galactic gas have the same pressure.^[ref]

For net cooling per unit mass $\mathcal{L}=\mathcal{C}/\rho$, an isobaric perturbation at equilibrium obeys $c_p{\partial }_t\delta T=-(\partial \mathcal{L}/\partial T{)}_p\delta T$. Thus a negative constant-pressure cooling derivative is thermally unstable: a cool perturbation cools further and condenses. Sound crossing must be fast enough to maintain pressure balance; thermal conduction suppresses sufficiently short wavelengths. Supernova-driven motions, magnetic pressure, finite cooling times and interfaces make the actual medium dynamic, so equilibrium branches describe useful phases rather than a static collection of perfectly separated components.

A dense cloud core first collapses when self-gravity overwhelms its effective support. While radiation escapes readily, compression can remain approximately isothermal and the Jeans mass decreases as density rises, encouraging gravitational fragmentation. Angular momentum usually leads to a disk rather than unrestricted spherical infall.

As the central optical depth increases, cooling becomes slower than compression and pressure produces a first hydrostatic core. molecular hydrogen dissociation near $2000\mathrm{K}$ absorbs compressional energy and lowers the effective adiabatic response, permitting a second collapse of a protostar. After dissociation the gas stiffens and a much more compact stellar core forms. An accretion shock and surrounding disk then convert infall energy into radiation. A small central core forms before most of the envelope arrives, as demonstrated in radiation-hydrodynamic collapse calculations.^[ref]

The isothermal inside-out collapse of a singular isothermal sphere model provides one idealized supply law, $\dot{M}\sim a_T^3/G$, but rotation, magnetic fields and an unsteady reservoir modify real accretion. Its luminosity scale is $L_{\mathrm{acc}}\sim GM\dot{M}/r_{\ast }$. Dust can absorb the emerging short-wavelength radiation and re-emit it in the infrared, so an embedded source need not have a directly visible stellar photosphere.

As envelope accretion declines, gravitational contraction becomes important on the Kelvin-Helmholtz cooling time $t_{\mathrm{KH}}\sim G{M}^2/(r_{\ast }L)$. deuterium-burning energy-generation rate can temporarily influence the radius; sustained core hydrogen burning eventually establishes the main sequence if the final mass is sufficient. A low-mass convective object approaches along a Hayashi-like track, while more massive objects need not share that track. accretion, contraction and nuclear burning are distinct energy sources, and a protostar is not generally a fixed-mass object.

mass and energy signs. The enclosed mass is $M(r)={\int }_0^{r}4\pi s^2\rho (s)ds$, so $dM/dr=+4\pi r^2\rho$. The negative sign printed in that equation cannot describe positive enclosed mass. Use the physical sign and define gravitational potential energy by

Boundary-pressure virial theorem. Multiply hydrostatic equilibrium by $4\pi r^3$ and integrate. The left side is $[4\pi r^{3}p{]}_0^{r_0}-3{\int }_0^{r_0}4\pi r^{2}pdr$, while the right side is $\Omega$. The central boundary contribution vanishes when $r^{3}p\to 0$ and the outer pressure is $p(r_0)=p_0$. ThusA finite external pressure is compatible with a gravitationally self-contained cloud, but is not the zero-surface-force notion of isolation. External ram stress or mass flow would require additional terms.

radiation budget of monatomic cloud collapse. For an ideal gas with fixed $\gamma$, the specific internal energy is $u=p/[\rho (\gamma -1)]$. Therefore the pressure integral is $3(\gamma -1)U$. Cold monatomic hydrogen has $\gamma =5/3$ when only translational degrees of freedom matter. With negligible final surface pressure, $2U+\Omega =0$, so $E_f=U+\Omega =\Omega /2$. If the initial cloud is sufficiently extended and cold that $E_i\simeq 0$, the emitted energy isBeing initially at rest alone does not set its binding energy to zero. If the initial total energy is appreciable, retain $E_{\mathrm{rad}}=E_i-\Omega /2$; a different equation of state also changes the half-energy result.

Inside-out accretion. The combination of $a_T$ and $G$ with dimensions of mass per time iswhere dimensional analysis leaves the dimensionless coefficient $C$ undetermined. An isothermal gas keeps $a_T$ constant, giving constant supply in the self-similar idealization. A front at radius $a_{T}t$ and an isothermal mass scale $M(r)\sim a_T^{2}r/G$ give the same scaling. The front's swept mass is not automatically the accreted central mass; the coefficient requires a dynamical similarity solution.^[ref]

constant-radius accretion energy budget. If all available accretion power is radiated at the fixed radius, $L_{\mathrm{acc}}=GM\dot{M}/r_0$. Neglecting the seed mass,This equals $-\Omega /2$ precisely whenThe positive sign in the final printed energy condition is inconsistent with the earlier use of negative gravitational potential energy. Equivalently its positive right side must denote the binding-energy magnitude $|\Omega |$. A general structural coefficient $\Omega =-qG{M}^2/r_0$ leaves a mismatch unless $q=1$ under these other idealizations. Radiation trapped in the object, outflows or release over a different radius also change the budget.

Which temperature? With $M=M_s+\dot{M}t$ and fixed $r_0$, $L_{\mathrm{acc}}\propto M$. If this power emerges thermally from area $4\pi r_0^2$, the accretion-powered effective-temperature scaling iswhen the seed mass is negligible. This is the radiating surface's effective temperature, inferred from the preceding accretion-luminosity calculation. A homologous ideal-gas interior instead has virial temperature $T_c\sim \mu m_{H}GM/(k_B{r}_0)\propto t$. Thus the surface-effective and interior-virial scalings are $t^{1/4}$ and $t$, respectively, under their stated assumptions. Inferring a detailed interior temperature profile would additionally require a structural and transport model.

virial pressure-radius curve with uniform-sphere binding energy. Put $c_s^2=k_{B}T/\mu$. Isothermality makes $\int (p/\rho )dM=c_s^2{M}_0$, so the virial theorem givesThe curve crosses zero at $r_z=B/A=G{M}_0/(5c_s^2)$, rises to its maximum atthen decreases toward zero at large radius. Since $p_0^{\prime }=(4B-3A{r}_0)/r_0^5$, a cloud displaced outward at fixed external pressure has a restoring pressure deficit only on the descending branch. Thus the homologous, isothermal radial mode is stable for $r_0>r_c$, unstable for $r_z<r_0<r_c$, and marginal at $r_c$. The negative-pressure branch is not a physical equilibrium with a positive confining pressure. This conclusion uses the stipulated gravitational-energy approximation, not the exact isothermal density profile.

critical mass from a virial pressure turning point. At the turning point, $B=3A{r}_c/4$; henceUsing the cloud's mean density $\bar{\rho }=3M_0/(4\pi r_0^3)$, the condition $r_0>4G{M}_0/(15c_s^2)$ becomesUsing the boundary density instead changes the numerical coefficient, not this scaling. The coefficient should not be identified with the exact Bonnor-Ebert sphere threshold because the assumed potential energy is different.

mass and luminosity weighting of a stellar initial mass function. Impose finite birth-mass cutoffs $m_l,m_u$. The stellar birth mass and main-sequence luminosity scale asFor a broad mass interval, the low-mass end controls the first integral and the high-mass end the second. Per logarithmic mass interval the weights are respectively $m^2\xi \propto m^{-0.35}$ and $mL\xi \propto m^{2.65}$. Finite lifetimes remove massive stars from an aging population, so the luminosity conclusion applies while those stars remain on the main sequence.

exponential closed-box gas consumption. The gross formation rate is $\Psi =ϵM_g$, but the permanent locking rate is $\alpha \Psi$. The instantaneous recycling approximation and mass conservation give ${\dot{M}}_g=-\alpha ϵM_g$, ${\dot{M}}_{\ast }=\alpha ϵM_g$. ThereforeHere $M_{\ast }$ means surviving locked stellar mass in the galactic gas fraction, not the cumulative gross mass that has passed through star formation.

locked-mass and returned-mass yield conventions. Pre-existing metals are locked at rate $\alpha Z\Psi$; the stipulated newly produced metal source is $b(1-\alpha )\Psi$. ThusFor $\alpha >0$ and initially pristine gas,The effective yield $y$ is per unit permanently locked mass; it is not the source parameter $b$. For $\alpha =0$, $M_g=M_0$ and the direct time equation gives $Z=bϵt$. For $\alpha =1$, the explicitly stipulated metal source vanishes, so this particular enrichment model gives $Z=0$. Constant yield is a trace-enrichment approximation; the formal logarithmic divergence as gas is exhausted must not be interpreted as a physical metal fraction exceeding one.

backward extrapolation of exponential closed-box star formation. Denote the present age of the universe by $t_{\mathrm{now}}$. With the last paragraph's $\alpha =1$,Adopting a present age of the universe $t_{\mathrm{now}}=13.8\mathrm{Gyr}$ for the estimate gives about $365M_{\odot }\mathrm{y}{\mathrm{r}}^{-1}$, requiring $7.3\times 10^{11}{M}_{\odot }$ of gas at that epoch in the same model. The boxed expression makes the dependence on the adopted present age explicit. This is an extreme history for a present Milky-Way-like system, although intense high-redshift star formation by itself is not forbidden.

The extrapolation forces a fixed gas reservoir to decline exponentially over many gas-depletion times. Gas accretion, outflows, evolving gas density and star-formation efficiency, delayed recycling, and galaxy assembly break that assumption. The empirical disk-averaged relation is not a universal constant-efficiency law for all gas: the original measurements combined Balmer series H-alpha, atomic gas, carbon monoxide and infrared data, with tracer calibration and dust corrections entering the inference.^[ref] Cosmological calculations explicitly support continuing cold-gas supply to rapidly forming high-redshift galaxies, providing a physical alternative to this closed-box backward extrapolation.^[ref]

A cold self-gravitating core initially contracts on a dynamical scale $t_{\mathrm{ff}}\sim (G\rho {)}^{-1/2}$ if cooling prevents pressure from halting collapse. Increasing optical depth eventually traps radiation and forms a pressure-supported first hydrostatic core. Dissociation of molecular hydrogen absorbs compressional energy, softens the equation of state and triggers a second protostellar collapse to a compact protostar. Rotation produces a disk; angular-momentum transport permits continuing accretion, while outflows remove some mass and angular momentum.

An accreting object's luminosity includes a contribution of order $L_{\mathrm{acc}}\sim GM\dot{M}/R$, with a coefficient set by where accretion energy is radiated or retained. After accretion subsides, gravitational contraction supplies internal luminosity over the Kelvin-Helmholtz cooling time $t_{\mathrm{KH}}\sim G{M}^2/(R{L}_{\mathrm{int}})$. Hydrostatic contraction heats the interior even while energy is radiated, as follows from the virial theorem and the negative heat capacity of a virialized gravitational system. accretion luminosity must not be substituted for $L_{\mathrm{int}}$ in the contraction time without an energy-budget justification.

Low-mass, nearly fully convective pre-main-sequence stars contract down a Hayashi track at roughly constant effective temperature and decreasing radius and luminosity. A developing radiative core can lead onto a Henyey track toward higher effective temperature; the route depends on mass and opacity. The deuterium-burning energy-generation rate can temporarily moderate contraction, but sustained core hydrogen burning marks arrival on the main sequence. Massive objects may begin hydrogen burning while still accreting, and substellar objects never establish long-lived hydrogen burning. An infrared spectral class reflects envelope and disk emission as well as inclination and interstellar extinction, so it is not a unique stellar age or interior evolutionary track.

Stellar radiation, stellar winds and supernovae return energy, momentum, mass and metals to surrounding gas. Heating, ionization, turbulent support and gas removal can reduce subsequent star formation; swept-up shells can also compress gas locally. Whether thermal energy drives an outflow depends on its radiative gas cooling time relative to expansion: radiative losses can remove energy while leaving substantial momentum transfer. Thus deposited energy alone does not specify the dynamical effect.

The galactic gas regulator has mass budget ${\dot{M}}_g={\dot{M}}_{\mathrm{in}}-(1-R+\eta )\Psi$, where $R$ is the returned fraction and $\eta ={\dot{M}}_{\mathrm{out}}/\Psi$ is the mass-loading factor. In a quasi-steady state, $\Psi \simeq {\dot{M}}_{\mathrm{in}}/(1-R+\eta )$. This connects feedback to the available gas supply. A wind's kinetic power ${\dot{M}}_{\mathrm{out}}{v}_w^2/2$ and momentum flux ${\dot{M}}_{\mathrm{out}}{v}_w$ must fit the source budget and gravitational binding cost. Gas that fails to escape may form a galactic fountain and later return; preferential metal loss changes chemical evolution.