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A cored power-law density profile is finite at its centre and approaches a power law outside a core radius. A simple continuous example is $n(r)=n_0$ for $r\le r_c$ and $n(r)=n_0(r/r_c{)}^{-\beta }$ for $r>r_c$. The normalization $n_0$ is the core density. Smoothing the transition changes integrated coefficients but not convergence thresholds fixed by the outer slope.

The mean number of other galaxies within distance $R$ of a randomly chosen galaxy is the integral of its conditional number density. A random location instead has mean count $4\pi \bar{n}{R}^3/3$. For a power law $\xi (r)=(r_0/r{)}^{\gamma }$, the extra count is $4\pi \bar{n}{r}_0^{\gamma }{R}^{3-\gamma }/(3-\gamma )$ when $\gamma <3$. The central galaxy is excluded; including it adds one. This distinction matters even when the unconditional mean number density is spatially uniform.

For an isolated two-body problem in Newtonian gravity, the relative elliptic orbit's orbital period $T$ obeysHere $a$ is the relative semi-major axis, and $M+m$ is the sum of the two masses. The separation $r$ varies during an elliptic orbit, so replacing $a$ by $r$ works only for a circular orbit, where $r=a$. Likewise, either body's individual semi-major axis about the centre of mass must first be converted to the relative one. When $m$ is negligible compared with $M$, the denominator can be approximated by $GM$.

For fixed gravitational parameter, the orbital period depends on the semi-major axis and is independent of orbital eccentricity. Our $a=1$, $\mu =1$ example gives $T=2\pi$ both for $e=0.6$ and for a circular orbit with $e=0$. The eccentric body's rapidly changing speed compensates for its different path. A compact way to see the cancellation uses Kepler's second law: divide the ellipse's area $\pi ab$ by its areal velocity $h/2$. With $b=a\sqrt{1-e^2}$ and specific angular momentum $h=\sqrt{\mu a(1-e^2)}$, the result is $2\pi \sqrt{a^3/\mu }$.

Compare two separate Kepler orbits with the same gravitational parameter, $a_1=1$ and $a_2=4$:With $\mu =1$, their orbital periods are $2\pi$ and $16\pi$; measured in units of $T_1$, these are $1$ and $8$. Quadrupling the size therefore takes eight times as long to complete a circuit. This power law makes comparisons possible without first choosing physical units.

The same relation can be inverted to weigh a system: $M+m=4{\pi }^2{a}^3/(G{T}^2)$. An observed relative semi-major axis and orbital period determine the total mass, while additional information is needed to separate the two masses. This is useful for binary stars, as discussed in NASA, Orbits and Kepler's Laws: measuring masses. Geometry and a clock together become a weighing instrument.

For a two-dimensional Fermi gas of $N$ spin-one-half particles with Fermi energy $E_F$, the constant density of states makes the low-temperature internal energy correction and heat capacity at constant areaThus its low-temperature heat capacity obeys a power law with exponent one.

If stars are neither created nor lost and all follow quadratic accretion in a central Keplerian potential, conservation of number gives $F(M,t)dM=F_0(M_0)d{M}_0$. Inverting the trajectory gives $M_0=M/(1+\lambda tM)$ and its Jacobian determinant is $(1+\lambda tM{)}^{-2}$. This proves the displayed mass-density-in-mass-space transformation. In a broad high-mass range with $\lambda tM\gg 1$, whose inverse images remain in a narrow interval where $F_0$ varies little, the power law is $F\propto M^{-2}$. A strictly monochromatic initial population remains monochromatic. Finite initial support imposes a finite upper cutoff before the earliest formal blow-up, so the power law is an intermediate asymptotic subject to the accretion model's limits.

The galaxy-conditioned neighbour count is the integral of the conditional number density $\bar{n}[1+\xi (r)]$, rather than of $\bar{n}$ alone. For a power law $\xi =(r_0/r{)}^{\gamma }$ with $\gamma <3$,With the specified $\gamma =2$, $R=10h^{-1}\mathrm{Mpc}$, $r_0=5h^{-1}\mathrm{Mpc}$ and $\bar{n}=0.01(h^{-1}\mathrm{Mpc}{)}^{-3}$, the factors of $h$ cancel andA uniformly chosen point in space is not conditioned to lie on a galaxy. Its expected value is simply the mean number density times the volume:The difference $10\pi \simeq 31.42$ is the clustering excess. The first count means other galaxies; including the chosen central galaxy adds one. Although the assumed galaxy two-point correlation function diverges at $r=0$, its contribution is integrable here because $r^2\xi (r)$ is constant.

In the dominant central Newtonian gravitational potential, conservation of energy for gas falling from rest at infinity gives $u^2/2=G{M}_c/R$, hence $u=a{R}^{-1/2}$ with $a=(2G{M}_c{)}^{1/2}$. The characteristic stellar speed has the same radial scaling. For random stellar directions, the cross term in $|\mathbf{u}-\mathbf{v}{|}^2$ averages to zero, so the typical relative speed is also $V=b{R}^{-1/2}$. With the stated kinetic-energy estimate, an rms convention gives $V^2\simeq 4G{M}_c/R$; another convention changes only the constant $b$.

In the steady spherical inflow approximation, mass conservation gives $A=4\pi R^2\rho u$. ThusThis leading calculation treats the inward gas flux as essentially undepleted over the region used. Substantial distributed removal by stars would make the local flux a function of $R$, an important limitation discussed below.

Insert these scalings in the Bondi--Hoyle--Lyttleton accretion rate. The factors $R^{3/2}$ from $V^{-3}$ and $R^{-3/2}$ from $\rho$ cancel:This is quadratic accretion in a central Keplerian potential. Write its fixed coefficient as $\lambda =\Lambda$. Integrating $d(1/M)/dt=-\lambda$ givesThe cancellation of radius means that orbital variation of $R$ alone does not change this coefficient, provided the assumed steady gas profile, central mass and relative-velocity scaling remain fixed.

Conserve the number of stars along these deterministic mass trajectories. Inverting the map and taking its Jacobian determinant givesThe mass-spectrum transport under quadratic accretion is thereforeFor a narrow but nonzero initial mass interval, the mapping stretches the high end especially strongly. If its endpoints are $m_1<m_2$, their evolved ratio is $(m_2/m_1)(1-\lambda m_{1}t)/(1-\lambda m_{2}t)$, which can become large as $t$ approaches $1/(\lambda m_2)$ from below. In an intermediate high-mass range with $\lambda Mt\gg 1$, mapped back into a portion of the initial interval where $F_0$ changes little,Here $M_{\ast }$ is near $1/(\lambda t)$ on the populated side of the initial interval. Finite initial support gives a finite upper mass cutoff at every time before the earliest blow-up; this is an approximate broad-range power law, not a literal infinite tail of a finite pre-blow-up population. A strictly monochromatic initial population remains monochromatic under identical accretion coefficients and does not generate a spectrum.

The limitations of this accretion model are substantial. The star cluster must remain centrally dominated, and both the central mass and gas supply must remain effectively fixed. Stellar capture depletes the gas flux and eventually invalidates the $R^{-3/2}$ profile. Gas pressure, angular momentum, shocks, radiative feedback and winds can reduce capture relative to the cold Bondi–Hoyle accretion estimate; a broad relative-speed distribution is not faithfully represented by one typical speed, especially because capture scales as $V^{-3}$. A growing star can perturb the surrounding potential, overlap capture regions, change its orbit through momentum loading, or compete with its neighbors for the same gas. Finally, number conservation excludes mergers, new star formation and stellar losses. The formal finite-time mass divergence cannot persist under a finite available gas mass, so the inferred mass-spectrum slope applies only while these approximations remain adequate.

A useful description of the low-column Lyman-alpha forest at $z\simeq 3$ is a power law $f_X(N,z)=A(z)N^{-\beta }$, with $\beta$ around $1.4$–$1.5$ over the ordinary unsaturated forest range. The same column dependence holds for $f_z$ at a fixed redshift. It is not a universal unbroken power law: saturation, increased neutral fraction and changing absorber populations produce breaks. Lyman limit systems become optically thick at $N\simeq 10^{17.2}{\mathrm{cm}}^{-2}$ because $N{\sigma }_{912}\simeq 1$ with ${\sigma }_{912}\simeq 6.3\times 10^{-18}{\mathrm{cm}}^2$. Damped Lyman-alpha systems have $N\gtrsim 2\times 10^{20}{\mathrm{cm}}^{-2}$ and strong radiative damping wings. They are rare in number but contain substantial neutral mass. The highest-column distribution steepens or cuts off rather than extending indefinitely.

This sketch shows representative shapes and transition scales, not an observational fit; its ordinate is distribution per unit column, not counts per logarithmic bin. In an Einstein-de Sitter universe, $H=H_0(1+z{)}^{3/2}$, so a fixed comoving population with fixed physical cross-section above a fixed column threshold has $d\mathcal{N}/dz\propto (1+z{)}^{1/2}$. ThusThe observed redshift evolution of the Lyman-alpha forest is much stronger for commonly selected forest lines. For example, Kim, Cristiani and D'Odorico's absorption survey finds an index about $2.19$ in $d\mathcal{N}/dz\propto (1+z{)}^{\gamma }$ for $10^{13.64}<N/{\mathrm{cm}}^{-2}<10^{16}$ over $1.5<z<4$, corresponding to an increase by approximately $2^{2.19}\simeq 4.56$. This number is selection-dependent. Expansion changes the gas density, and the evolving cosmic ionizing background changes its neutral fraction through photoionization equilibrium, moving systems across a fixed neutral-column threshold. Structure also evolves. A non-evolving population with fixed physical cross-section is therefore not an adequate physical model of the observed forest.

No measured normalization of the neutral-hydrogen column-density distribution, ionizing field or gas temperature is supplied, so the paper's formulas do not determine numerical densities on their own. With observational data and suitable ionization corrections, the high-redshift Lyman-alpha forest gives a total gas density of order a few $10^{-2}$ relative to today's critical density, comparable to the cosmological baryon density ${\Omega }_b\sim 0.04$–$0.05$ for $H_0\sim 70\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$. The conclusion is that a large fraction of baryons at $z\simeq 3$ reside in diffuse ionized gas, rather than that every baryon has been individually accounted for. Schaye's analysis and Kim and collaborators' survey support that interpretation under their stated gas and ionizing-background assumptions.

The neutral-gas reservoir traced predominantly by damped Lyman-alpha systems is much smaller, of order $10^{-3}$ in the same convention. It is broadly comparable in order of magnitude to today's luminous stellar matter, whose cosmic stellar mass density is a few $10^{-3}$, but exact comparisons depend on cosmology, helium correction and which stellar components are counted. Storrie-Lombardi, McMahon and Irwin's neutral-gas census finds especially close comparison with present stellar disks. Most high-redshift gas is ionized; the smaller neutral reservoir has a mass of roughly the same order as today's visible stellar population. This comparison does not imply that the measured neutral reservoir alone simply becomes all present-day stars: gas accretion, recycling, ionization changes and outflows continue during galaxy evolution.

The neutral gas density from an absorption distribution is comparatively robust for high columns where the hydrogen is mostly neutral, but saturated lines, dust-obscured quasars, incomplete surveys and small numbers of rare high-column systems affect the integral. The ionized gas density from an absorption distribution has additional systematic uncertainty in the cosmic ionizing background, temperature, absorber depth, helium treatment and small-scale density inhomogeneity. At low columns, line blending and incompleteness matter; at high columns, attenuation makes an optically thin ionization correction invalid. A single power law extrapolated outside the measured column range can also produce a misleading mass integral. Hot gas, molecular gas and gas missed by the selected absorption population are not included automatically. These are observational and modelling qualifications to a baryon census, not failures of the column-mass accounting itself.

A partition function sums the Boltzmann weights of all states: $Z={\sum }_s{e}^{-{\beta }_{T}H(s)}$ for discrete states, or the corresponding integral for continuous fields. It normalizes equilibrium probabilities. Its physical Helmholtz free energy is $\mathcal{F}=-k_{B}T\log Z$, and its free-energy density is $\mathcal{F}/V$. With a momentum ultraviolet cutoff ${\Lambda }_0$, a scalar-field example isThe normalization of the functional measure and field-independent energy terms must be tracked when discussing absolute free energy.

Choose a smaller cutoff $\Lambda$ and split the field into retained modes ${\varphi }_{<}$ with $|q|<\Lambda$ and eliminated modes ${\varphi }_{>}$ with $\Lambda <|q|<{\Lambda }_0$. Define the Wilsonian coarse-grained statistical Hamiltonian bySubstitution into the partition function proves $Z=\int \mathcal{D}{\varphi }_{<}{e}^{-{\beta }_T{H}_{\Lambda }[{\varphi }_{<}]}$ for every intermediate cutoff. A source retained in this definition similarly preserves long-wavelength connected correlation functions and response. Thus the coefficients in a Hamiltonian density must run with $\Lambda$: the dependence induced by eliminated modes compensates the dependence of the remaining field integral. Holding the original bare coefficients fixed while changing the cutoff would generally change the system. Exact integration generates all interactions allowed by symmetry, including higher powers, derivative terms and an additive constant; a local derivative expansion or a truncation to a few couplings is an extra approximation.

Cutoff independence makes arbitrary intermediate coarse-graining scales unphysical. It allows calculations at the scale of the observable, avoids repeatedly resolving irrelevant short-distance details and exposes the renormalization-group fixed points responsible for universality. A microscopic cutoff, such as a Bravais lattice spacing, can still label a physical microscopic model; taking it to infinity as a continuum limit requires tuning appropriate bare couplings. This is distinct from lowering an intermediate Wilsonian cutoff in a fixed model.

To interpret the zero-cutoff limit, define $M=V^{-1}\int \varphi (x)d^{D}x$, the spatially uniform mode of the order parameter. Keep this mode fixed and integrate the others, rather than integrating over $M$ too. Write $\varphi =M+\eta$, with $\int \eta =0$, and define the constrained order-parameter free energy byIn a finite periodic box, once $\Lambda$ is below the smallest nonzero momentum, all nonconstant modes have been eliminated. The remaining effective energy is $H_{\Lambda }[M]=V{U}_{\Lambda }(M)$, so its defining integral is exactly $Z_M$. Therefore, with consistent normalization,When $\mathcal{H}(\Lambda ,M)$ denotes this uniform effective Hamiltonian density, this is the stated identification of $F(M)$ with its zero-cutoff limit. It requires a free-energy density constrained at fixed $M$, not the unconstrained extensive free energy; all integrated fluctuations and additive constants are included, the zero mode in field theory is retained, and a density for the uniform configuration is well defined. The finite-volume argument avoids an interchange of the thermodynamic and cutoff limits without justification. On a homogeneous pure-phase branch one may use a local potential description, while a thermodynamic limit allowing phase separation can convexify the constrained potential. A bare nonconvex Landau polynomial is accordingly a homogeneous-branch approximation, not a claim that the exact equilibrium Legendre potential stays nonconvex.

In a field $h$, the remaining integral is $Z(h)=\int dM{e}^{-{\beta }_{T}V[F_c(M)-hM]}$, up to the consistently chosen zero-mode measure. The thermodynamic saddle then gives $f(h)={\inf }_M\{F_c(M)-hM\}$ and selects the equilibrium magnetization. Landau theory assumes that the relevant uniform potential can be approximated by a regular analytic expansion in $M$, with symmetry-constrained coefficients, and minimizes it. At tree level the integrated fluctuations are neglected, recovering the familiar bare polynomial. With fluctuations retained, the coefficients run and can become singular near criticality; the zero-cutoff identification of constrained free energy alone does not justify mean-field critical exponents.

For an ordinary short-range scalar thermodynamic critical point, the leading interaction is quartic. Absorb inverse temperature and the gradient coefficient into field normalization and writeScale $x=b{x}^{\prime }$ and $\varphi (x)=b^{-(D-2)/2}{\varphi }^{\prime }(x^{\prime })$ to leave the gradient term invariant. The quartic term changes to $g{b}^{4-D}\int {\varphi }^{\prime 4}/4!$, so its engineering scaling exponent is $4-D$. It is irrelevant at the Gaussian fixed point for $D>4$, relevant for $D<4$, and marginal for $D=4$. HenceThe loop expansion makes the breakdown concrete. A quartic-vertex correction is proportional to $g^2{I}_2(r)$, whereFor $0<D<4$, rescaling $q=\sqrt{r}k$ gives $I_2(r)\sim C_D{r}^{(D-4)/2}$, diverging as $r\to 0$. At $D=4$, radial integration giveswhich is logarithmically divergent. Above four dimensions its infrared limit is finite; the leading cutoff-dependent constant can be absorbed into a coupling renormalization. The effective expansion parameter is thus of order $g{\xi }^{4-D}$ below four, or $g\log (\Lambda \xi )$ at four, since $\xi \sim r^{-1/2}$. It eventually becomes large for any fixed bare interaction in the lower-dimensional critical limit, defeating the uncorrected mean-field expansion. The Ginzburg criterion gives the same threshold by comparing correlation-volume fluctuations with the squared mean order parameter.

Below four dimensions an interacting renormalization-group fixed point can replace the Gaussian fixed point and change critical exponents, provided a continuous transition exists. At four dimensions the interaction is marginal and produces logarithmic corrections: the leading power exponents can remain their mean-field values, so “breakdown” there means that pure mean-field power laws omit those corrections, rather than necessarily that every power exponent changes. Above four the stabilizing quartic interaction is dangerously irrelevant for the ordered phase, explaining why mean-field thermodynamic exponents need not obey naive hyperscaling with the spatial dimension.

Particles settle toward the bed under their submerged weight. A concentration increasing downward gives an upward turbulent diffusive flux, which can oppose that settling. If turbulence cannot sustain the particles, they enter the weakly mixed near-wall region and can deposit. Sediment resuspension requires bed forces strong enough to entrain grains, not merely a specified diffusivity far above the bed.

For noncohesive grains, an appropriate criterion uses the Shields parameterwhere $d$ is grain size, ${\rho }_s$ its density and ${\Theta }_c$ the sediment entrainment threshold expressed dimensionlessly. The threshold depends on grain geometry, contacts and viscous effects; cohesion changes the criterion. Sustained suspension additionally requires sufficiently vigorous upward turbulent motions relative to $W_s$. Bed motion and well-mixed suspension are therefore not identical conditions.

At sedimentation-resuspension equilibrium, the total vertical particle flux is constant. Set ${\kappa }_T=Kz$ with $K=C/2$. The stationary equation givesSince $W_s>0$ and $\varphi \to 0$ at infinity, necessarily $J=0$. Settling and upward diffusion then balance pointwise. Taking a reference height $z_1=1$ in the specified length units gives the equilibrium settling-diffusion profileIt is nonnegative and decays to zero. The profile describes the region $z\gg \delta$; it does not assert that this power law continues through the viscous layer down to the physical bed. The supplied reference concentration summarizes the bed exchange and near-wall physics that the outer diffusion closure does not resolve.

For a steady upward plume with prescribed $N^2=-G^2<0$, the kinematic plume fluxes satisfyThe positive sign in the last equation expresses the gain of buoyancy flux as the plume rises through the prescribed unstable ambient. Seek a positive power law solution $Q=q{z}^p$, $M=m{z}^r$, $F=f{z}^s$, with $z>0$. Matching powers givesThe last relation gives $s=p+1$ and the first $r=2p-2$. Substitution into the middle relation yields $4p-5=2p+1$, henceThe amplitude equations areFrom the first relation $m=9q^2/(4{\alpha }^2)$, while the last two give $m^2=G^2{q}^2/16$. Taking positive amplitudes gives $m=Gq/4$, and consequentlyReconstructing the radius, speed and reduced gravity gives the top-hat plume in constant unstable stratification:As a direct check, $M_z={\alpha }^2{G}^2{z}^3/9=FQ/M$ and $F_z={\alpha }^2{G}^3{z}^3/9=G^{2}Q$, while $Q_z={\alpha }^{2}G{z}^2/3=2\alpha \sqrt{M}$.

The corresponding ambient gradient is ${\rho }_{0z}={\rho }_r{G}^2/g$. Thus $\rho (z)={\rho }_0(z)-{\rho }_r{G}^{2}z/(4g)$; for a linear ambient this is ${\rho }_0(0)+3{\rho }_r{G}^{2}z/(4g)$. The plume is locally lighter than the ambient even though both densities increase upward in this prescribed unstable stratification.

The coordinate $z$ can be measured from a virtual origin. This exact power law branch has fluxes tending to zero there and describes buoyancy generated by the maintained unstable ambient; it does not impose arbitrary nonzero source fluxes at that origin. The given idealization holds the ambient quiescent despite its static instability. Finally, the Boussinesq approximation limits the modeled height range to small relative density changes, here $G^{2}z/g\ll 1$. Within these assumptions, the displayed positive branch proves the requested existence.

Define $g(\mathbf{x},h,t)dh$ as the fraction of horizontal ocean area covered by sea ice of thickness in $[h,h+dh]$. Allow a Dirac delta mass at $h=0$ for open water, so the whole-area distribution has $\int gdh=1$; its first moment $H=\int hgdh$ is ice volume per ocean area, rather than mean thickness conditional on ice being present. Let $f=dh/dt$ be thermodynamic growth, $\mathbf{v}$ the horizontal ice velocity and $\psi$ the mechanical redistribution of sea ice.

Conservation in a horizontal cell and a thickness interval gives horizontal flux $\mathbf{v}g$ and thickness-space flux $fg$. For $h_1<h_2$, their contribution is $-\nabla \cdot {\int }_{h_1}^{h_2}\mathbf{v}gdh-[fg{]}_{h_1}^{h_2}$; adding ${\int }_{h_1}^{h_2}\psi dh$ and shrinking the interval yieldsThis is the sea-ice thickness distribution equation. Its distributional formulation must include creation and removal of open water when characteristics reach or leave $h=0$. On a positive-thickness smooth portion it is equivalentlyIf a nonconservative form uses a redistribution function $\tilde{\psi }$, then $\tilde{\psi }=\psi -g\nabla \cdot \mathbf{v}$; the two definitions must not be interchanged.

With the open-water boundary flux accounted for consistently, whole-area normalization requires $\int \psi dh=\nabla \cdot \mathbf{v}$. Mechanical rearrangement preserves ice volume locally, so $\int h\psi dh=0$. The volume moment consequently obeyswhere formation of new sea ice from the open-water category is included in the thermal term. Compression raises local volume per area through convergence, even though ridging does not create ice mass.

The three processes have distinct effects on the curve. Pure advection imports the distributions found upstream, so a fixed location may acquire a different mode or thick tail. Along an incompressible material trajectory, advection alone does not alter its distribution. Convergence and divergence also change area fractions through the horizontal flux, with the redistribution and open-water terms ensuring normalization.

Thermodynamic growth moves probability along thickness characteristics $\dot{h}=f(h,t)$, with $g(h,t)dh=g(h_0,0)d{h}_0$ when mechanics and horizontal divergence vanish. For example, winter conductive growth $f=\kappa /h$ gives $h^2=h_0^2+2\kappa t$ andThin ice grows faster, so the mode moves to larger thickness and neighboring thin-ice characteristics converge, sharpening the level-ice peak. Summer ablation moves classes left and creates open water when ice disappears; if melt rate depends on thickness, the shape changes as well.

Mechanical redistribution of sea ice removes area from thin or level ice and populates thick rafted or classes containing sea-ice pressure ridges, broadening the upper tail. It can also create sea-ice leads. A simple volume-conserving example, stacking a fraction $r$ of an initial distribution into double-thickness ice, isThe stacked component has area $r/2$ and unchanged volume $r\int h{g}_0(h)dh$; the freed area $r/2$ becomes open water. Thus mechanical thickening is not merely a translation of the entire curve.

The figure uses illustrative distributions, not measured data. The advection panel compares local ice with incoming thicker ice; the growth panel applies the thickness-characteristic formula; the mechanical panel uses $r=0.4$ and shows its open-water mass separately from the continuous density.

For the thick tail, take the common-shape sea-ice pressure ridge idealization to mean triangular profile sections with a common effective side slope $s=\tan \delta$, while peak thicknesses vary. Let $n(H)dH$ count sea-ice pressure ridge peaks per track length with maximum thickness in $[H,H+dH]$. Above a level-ice baseline $h_0$, a sea-ice pressure ridge with $H\ge h$ contributes two flank lengths $dh/s$ at thickness $h$; it contributes nothing at $h>H$. For a representative transect, summing these lengths gives the triangular ridge occupation identitySuppose $n(H)=B{e}^{-\lambda H}$ for $H\ge h_0$. ThenThus an exponential ridge-draft model, expressed in the same vertical coordinate as the profile, generates an exponential occupation tail with the same decay parameter. The peak-count distribution and the area-weighted thickness distribution are different distributions, so their amplitudes carry essential information.

If $\mu ={\int }_{h_0}^{\infty }n(H)dH$ is ridge line density and $p={\int }_{h_0}^{\infty }{g}_{\mathrm{tail}}(h)dh$ is the tail's area fraction, the same result becomesThe decay rate alone does not fix slope: sea-ice pressure ridge frequency or tail amplitude is also needed. If side slopes vary independently of peak height, replace $1/s$ by $⟨1/s⟩$, giving an effective harmonic-mean slope. Measured along-track slopes differ from slopes normal to a ridge if the transect is oblique. Overlap, irregular shape and porosity limit this simple geometric model. Literally identical complete ridge cross-sections would have one height, so “common shape” here necessarily permits different ridge sizes.

Sea-ice pressure ridge spacings are positive and broadly distributed. A spatial Poisson process gives an exponential distribution of gaps, a useful simple approximation. Observed sea-ice pressure ridge spacings, however, often fit a shifted lognormal distribution better, with few very small spacings because of ridge geometry, identification rules and finite beam width. This differs from the successful exponential model for peak drafts. Observed ridge-spacing and draft distributions establish this distinction.

Sea-ice lead-width distributions contain very many narrow openings and a broad tail. A power law often describes widths over a resolved range better than one negative exponential; its exponents and apparent cutoffs depend on the lead definition, resolution and count-versus-area weighting. Sea-ice lead spacings may be approximately exponential over an intermediate range, but clustering produces excess close pairs and large gaps, so there is no universal independent-spacing law. Both behaviors were found in Greenland Sea and Eurasian Basin lead statistics. Advection imports thickness classes, thermodynamics transports them in $h$, and mechanics exchanges thin-ice area for ridged ice and leads; common-slope exponential ridge populations reproduce the exponential thick tail.

In a homogeneous ambient fluid, $N=0$ and $R={\rho }_0$ is constant. Use kinematic fluxes or, equivalently, the primitive form of the unsteady top-hat line-plume balances:Here $g^{\prime }=g({\rho }_0-\rho )/R>0$ is the positive buoyant reduced gravity. Seek a nonsteady separable power-law ansatz with both time differentiation and vertical transport retained. Matching powers in the volume equation forces $b\propto z$ with no time dependence. Matching the two momentum terms then gives $w\propto z/\tau$, and the buoyancy force has $g^{\prime }\propto z/{\tau }^2$, where $\tau =t+t_{\ast }>0$ is a shifted age. Thus writeSubstitution determines the constants rather than just the exponents. The volume equation gives $2BA=\alpha A$, so for a nonzero buoyant plume $B=\alpha /2$. The momentum equation gives $C=-A+3A^2$, while the buoyancy equation gives $C(-2+3A)=0$. The buoyant nontrivial branch has $A=2/3$ and $C=2/3$. Consequently the separable decaying top-hat line plume isThe Boussinesq approximation requires $2z/(3g{\tau }^2)\ll 1$. The corresponding mass density-weighted fluxes, to this order, areFor a direct check, $Q^2/M=R\alpha z/2$ is time independent and $FQ/M=R\alpha z^2/(3{\tau }^2)$. Hence $(Q^2/M{)}_t+Q_z=2R\alpha z/(3\tau )=R\alpha M/Q$, $Q_t+M_z=R\alpha z^2/(3{\tau }^2)=FQ/M$, and $(FQ/M{)}_t+F_z=0$. This verifies all three balances and the required width coefficient.

This nonsteady branch is not the steady, continuously supplied, constant-flux line plume. The latter has constant $w$ and half-width $b=\alpha z$ in the same top-hat convention. Also, the unshifted power law has $Q,F\to 0$ as $z\to 0$ and is singular at $\tau =0$. It is therefore an interior separable solution, not by itself a solution with a nonzero continuously operating point-source flux at the origin. A finite-width source boundary, a virtual origin, or a matching region is needed to connect it to physical source and initial data. For example, replacing $z$ by $z+z_{\ast }$ supplies a finite base width and time-varying positive base fluxes when $z_{\ast }>0$. No source or initial flux history is specified in the question to select that matching; the requested power-law width follows from the local balances above.

Both background components are pressure-free, so their densities scale as $a^{-3}$ and ${\Omega }_H$ is constant. Since $\dot{a}/a=2/(3t)$ and ${\bar{\rho }}_C=(1-{\Omega }_H)/(6\pi G{t}^2)$, a power law $\delta =t^{\alpha }$ satisfiesThusFor $0\le {\Omega }_H<1$ the plus sign gives a growing mode and the minus sign a decaying mode. At ${\Omega }_H=1$ the growing exponent becomes zero, since no gravitating clumping component remains.

When ${\Omega }_H=2/3$, ${\alpha }_{+}=1/3$, so $\delta \propto t^{1/3}\propto a^{1/2}$. The redshift relation gives $a_{\mathrm{today}}/a_{\mathrm{eq}}=1+z_{\mathrm{eq}}\simeq 10^5$. Even an entirely growing initial mode therefore givesThe perturbation remains far below order unity, so it cannot produce the observed nonlinear gravitational clumping within the stated model. A decaying admixture does not provide the missing sustained growth.

Match the cutoff exponent to get $2\beta =1/2$, so $\beta =1/4$. Independently, the low-mass exponent is then $\beta -2=-7/4$, as required. ConsequentlyMatching the coefficient of the power-law Press-Schechter stellar mass function givesThusAbout $5.1\%$ of the halo baryons form stars in this model, corresponding to $1.0\%$ of total halo mass. The independent stellar-density integral gives ${\rho }_{\ast }^{\mathrm{com}}/{\bar{\rho }}_{m,0}={ϵ}_{\ast }$, checking the mass-function normalization.

The index $-3/2$ is an effective slope of the processed matter spectrum on these mass scales, not the primordial index. The cosmological transfer function suppresses modes that enter the horizon during radiation domination, altering the approximately primordial $k$ law. Around and below galaxy scales the dimensional matter spectrum has a much more negative local slope, eventually tending toward the approximately $k^{-3}{\ln }^{2}k$ small-scale limit. A local power law with $n=-3/2$ is a useful idealization over a limited interval; there is no reason for it to equal the large-scale primordial value or to hold on every scale.

Let $L_k={\dot{m}}_k^{-c}$ be the mass processed by catastrophic impacts in logarithmic bin $k$. Steady mass conservation requires the same downward mass flux across every interior size threshold. A scale-independent fragment redistribution function makes the transfer kernel depend only on the number of logarithmic bin steps. Consequently a self-similar interior steady collisional cascade has $L_k=L$, independent of size: a constant processed mass per bin supplies the constant flux, with the same kernel-dependent proportionality at every bin. This is the constant mass flux in a collisional cascade result, away from injection and removal cutoffs. Size-independent fragmentation does not eliminate boundary waves at the very ends of a finite distribution.

For equal-density planetesimals, body mass is proportional to $D^3$ and a fixed logarithmic bin contains a number proportional to $Dn(D)$. The mass per logarithmic size bin is therefore $m_{\log }\propto D^{4-\alpha }$. With the given catastrophic planetesimal collision rate, $Q_D^{\ast }\propto D^b$ givesConstant mass flux in a collisional cascade sets the exponent to zero, yielding the strength-dependent steady collisional-cascade slopeThe time to change an order-one fraction of a bin's mass is $m_k/(m_k{R}_k)=1/R_k$. Its collisional-cascade relaxation time is thus $t_c(D)\sim 1/R(D,Q_D^{\ast })$. The contemporaneous projectile population must be used when a different part of the distribution has already evolved.

For the primordial differential-number index ${\alpha }_0=7/2$, the collisional-cascade relaxation time isIt has asymptotic logarithmic slope $1/12$ in the strength regime and $7/4$ in the gravity regime. It increases monotonically: its logarithmic derivative is $1/2+(5/6)d\log Q_D^{\ast }/d\log D$, ranging from $1/12$ to $7/4$. In particular the minimum of the catastrophic disruption threshold at $D_w$ is not a minimum of the collision time. The strength-gravity disruption transition obeysSmall sizes start evolving first. Before $t\sim t_{c0}(D_{\min })$, the whole distribution is nearly primordial. At intermediate times, a transition diameter $D_t$ with $t_c(D_t)\sim t$ separates processed small sizes from mostly primordial larger sizes. The steady strength-regime index is ${\alpha }_s=41/11$, while the gravity-regime index is ${\alpha }_g=3$. Thus a plot of mass per logarithmic size bin has slopes $3/11$ and $1$ in the two evolved regimes, compared with the primordial slope $1/2$. Once $D_t\gg D_w$, both steady slopes appear below the remaining primordial tail. The transition moves to larger sizes, and the normalization eventually decays as the largest bodies are depleted. Sharp joined power laws are a schematic description; detailed kernels produce smooth transitions and possible waves.

For the uniform depletion, the binary-collision evolution operator $F$ is quadratic in all bin masses: $F(cm)=c^{2}F(m)$. Let $m(t)$ be the undepleted trajectory and suppose $m_d(T)=m(T)/f$. Differentiating verifies the exact scaling within this fixed-kernel collision model,Immediately after depletion the shape is unchanged and the catastrophic planetesimal collision rates are reduced by $f$. Further evolution therefore proceeds $f$ times more slowly on the original trajectory. The processed-to-primordial transition initially remains at $D_t$, then advances on the stretched collision clock.

A collision-only model starting with the smaller initial normalization $m(0)/f$ has trajectory $\tilde{m}(u)=m(u/f)/f$. Matching the observed depleted distribution requires $u/f=T+(t-T)/f$. Hence the collisional age after uniform dynamical depletion isThe lower observed normalization would make it appear that slow collisions had taken much longer to produce the existing break. Shape and normalization alone cannot determine $f$ without independent information about age, initial mass, or depletion history.

Finally, count all impacts exceeding the strength threshold $Q_s=Q_a{D}^{-1/2}$, including impacts which also disperse the target. This is the source's inclusive rubblising convention; a bound-remnant rubblising collision alone would exclude the dispersing impacts. In the evolved gravity-regime projectile approximation, ${\alpha }_g=3$, so the given threshold-rate law gives the rubblising-to-dispersal collision-rate ratioAt the collisional front $D_t$, the age is of order one catastrophic collision time. Thus the expected number of strength-shattering impacts has the large-$D_t/D_w$ scalingThis is the requested scaling. It is not an exact unit-coefficient identity from the stated assumptions: using the true minimum diameter gives $R_{\mathrm{rub}}/R_{\mathrm{cc}}\simeq 3^{-2/3}(D_t/D_w{)}^{4/3}$. The actual cumulative number is ${\int }_0^T{R}_{\mathrm{rub}}(D_t,t)dt$, not automatically $T{R}_{\mathrm{rub}}(D_t,T)$, and its coefficient depends on evolution of the projectile population. The $4/3$ estimate also assumes the relevant projectiles sample the gravity-regime slope; sampling the strength or primordial slope changes the exponent. The steady-cascade framework and evolving transitions are developed by Wyatt, Clarke and Booth.

The flat matter-dominated Friedmann equation gives $\rho =1/(6\pi G{t}^2)$ when $a=t^{2/3}$. With $P=C{\rho }^{4/3}$,For linear density modes with polytropic pressure, substitution gives the Euler-Cauchy equationA power law has exponent satisfying $n(n-1)+4n/3+{\bar{c}}_s^2{k}^2-2/3=0$. Thus, for distinct roots,For $k\ll k_J$, the two modes approach $t^{2/3}$ and $t^{-1}$: gravity produces a growing linear cosmological density perturbation and a decaying mode. For $k\gg k_J$, put $\nu =\sqrt{{\bar{c}}_s^2{k}^2-25/36}$; the real solution isThese are pressure-supported oscillations with a decaying envelope. For the repeated-root cosmological density mode $k=k_J$, the complete solution is $t^{-1/6}(A_k+B_k\log t)$. The specified $k_J=5/(6{\bar{c}}_s)$ separates real exponents from logarithmic oscillations; the growing exponent actually crosses zero at $k=\sqrt{2/3}/{\bar{c}}_s$, a slightly smaller value.