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Assume the stellar yield is common to the compared stellar populations and compare their mass-weighted stellar metallicities. Inverting the metallicity distribution in a leaky box result givesFor example, with $y=Z_{\odot }$, a solar-metallicity massive elliptical galaxy requires $\nu \simeq 0$, whereas $⟨Z{⟩}_{\ast }<0.01Z_{\odot }$ in a dwarf galaxy requires $\nu >99$. A supersolar stellar metallicity requires a correspondingly supersolar stellar yield: outflow alone cannot raise $⟨Z{⟩}_{\ast }$ above $y$. If $y=2Z_{\odot }$, a massive elliptical galaxy with $⟨Z{⟩}_{\ast }$ between $Z_{\odot }$ and $2Z_{\odot }$ has $0\le \nu \le 1$, while the same low-metallicity dwarf galaxy needs $\nu >199$.

For a yield of order solar, the required mass loading runs from order zero or unity in massive spheroids to order one hundred or more in faint dwarfs. Without a specified stellar yield, the robust comparison isThis explains the sense of the galaxy mass--metallicity relation; galactic gas inflow, incomplete depletion, and changes in the initial mass function can also change observed stellar metallicity.

The escape-speed scaling of wind mass loading makes the proposed trend dynamically plausible: the shallower Newtonian gravitational potential of a dwarf galaxy lets stellar feedback expel more gas per unit stellar mass. If coupled feedback supplies specific energy $e_{\ast }$, escape requires $\nu v_{\mathrm{esc}}^2/2\lesssim e_{\ast }$; fixed coupling suggests $\nu \propto v_{\mathrm{esc}}^{-2}$ for an energy-driven outflow. Fixed specific momentum $p_{\ast }$ instead gives $\nu v_{\mathrm{esc}}\lesssim p_{\ast }$, or $\nu \propto v_{\mathrm{esc}}^{-1}$ for a momentum-driven outflow.

Since escape velocity has $v_{\mathrm{esc}}^2\sim GM/R$, galaxies whose characteristic radius grows more slowly than their gravitating mass have increasing binding energy per unit mass with increasing mass. For $R\propto M^a$ with $a<1$, the energy-driven outflow estimate gives $\nu \propto M^{a-1}$. Equivalently, greater velocity dispersion in massive spheroids implies greater escape energy and more effective retention of metals.

For scale, one core-collapse supernova of $10^{51}\mathrm{erg}$ per $100M_{\odot }$ formed supplies about $5\times 10^{15}\mathrm{c}{\mathrm{m}}^2{\mathrm{s}}^{-2}$ before coupling losses. At ten percent coupling the energetic upper estimate $2e_{\ast }/v_{\mathrm{esc}}^2$ is about $110$ for $v_{\mathrm{esc}}=30\mathrm{km}{\mathrm{s}}^{-1}$, but about $0.4$ for $500\mathrm{km}{\mathrm{s}}^{-1}$. Efficient loss from shallow potentials and retention in deep potentials can span the required range. These are illustrative energetics rather than a unique prediction: cooling, entrainment, fallback and the actual initial mass function determine the escaping mass-loading factor.

The differential gravitational field of the Milky Way removes loosely bound outer material by tidal stripping, giving globular clusters and satellite dwarf galaxies a characteristic tidal radius. For a low-mass satellite of mass $m$ on a circular orbit of radius $R$ in a spherical host, the Jacobi tidal radius is $r_t\simeq R\{m/[(3-d\ln M_h/d\ln R)M_h(<R)]{\}}^{1/3}$; eccentric orbits often make the strongest restriction near pericentre. Tidal limitation explains the sharp outer falloff, although escaping stars can form extended tidal tails, so a fitted luminosity cutoff is not an exact boundary of all associated material.

A flux-limited galaxy selection samples a much larger volume for bright galaxies: in the nearby Euclidean limit $d_{\max }\propto \sqrt{L}$, hence $V_{\max }\propto L^{3/2}$, so very numerous faint dwarf galaxies are usually detectable only nearby. With a Schechter function $\varphi (L)\propto L^{{\alpha }_S}{e}^{-L/L_{\ast }}$, detected counts per logarithmic luminosity interval scale as $L^{{\alpha }_S+5/2}{e}^{-L/L_{\ast }}$ and, for ${\alpha }_S\sim -1$ to $-1.3$, peak near $L\sim (1.2\text{–}1.5)L_{\ast }$ rather than at the numerous faint end. The competition between increasing survey volume and the bright-end cutoff concentrates detected luminosities near $L_{\ast }$, with the precise width depending on the selection and survey depth.

Equating the satellite's mean density inside $r_t$ with the host's mean density inside orbital radius $R$ givesThus a measured tidal radius and independently estimated satellite mass infer $M(<R)\simeq m(R/r_t{)}^3$. Measurements from satellites over a range of $R$ trace the host's enclosed-mass profile and hence its gravitational potential. Globular clusters are more often tidally limited: their stellar extent can approach the Jacobi boundary, whereas dwarf galaxies commonly occupy extended dark-matter haloes and their observed stars may substantially underfill it. Orbital eccentricity, mass loss, and nonequilibrium structure complicate this inference.

Let the separation vector point from the host to the satellite. Subtracting the two Newton's second law equations givesThe relative orbit is therefore a one-body Kepler orbit with gravitational parameter $G(M+m)$. A circular orbit of separation $R_0$ hasThe centre-of-mass condition gives $x_m=M{R}_0/(M+m)$ for the satellite's distance from the barycentre.

Consider the inner collinear equilibrium a distance $r_J$ toward the host from the satellite. Taking the positive axis from the host toward the satellite, differentiation of the gravitational plus centrifugal effective potential givesSince $r_J\ll R_0$,The constant host-gravity term cancels ${\Omega }^2{x}_m=GM/R_0^2$. The remaining equation isFor a low-mass satellite, the term proportional to $m{r}_J$ is negligible compared with $3M{r}_J$, and the Jacobi tidal radius isThe outer collinear point gives the same leading result.

Stars that cross the two nearby Lagrange points are no longer bound to the satellite. Small energy and angular-momentum offsets place them on slightly different host orbits, producing one leading and one trailing tidal tail. Differential orbital frequency stretches these streams around the host, while epicyclic motion can create density clumps. The tail's position and velocity structure retain information about the satellite orbit and host potential.

The definition of the virial radius of a dark-matter halo immediately gives the virial mass of a dark-matter haloIn an Einstein-de Sitter universe, spherical collapse gives $v=18{\pi }^2\simeq 178$, conventionally rounded to about $200$.

For the Navarro--Frenk--White profile, put $x=r/r_s=cs$. Direct integration givesUsing $r_s=r_v/c$ and ${\delta }_{\mathrm{char}}=v{c}^{3}g(c)/3$ therefore yieldsAt small radius the bracket is $(cs{)}^2/2+O((cs{)}^3)$, so $M(r)\propto r^2$, consistent with the central $\rho \propto r^{-1}$ cusp. At large radius it is $\log (cs)-1+o(1)$, so the mass diverges logarithmically unless the halo is truncated.

The primordial free streaming of warm dark matter erases small-scale density fluctuations and lowers the central phase-space density, tending to replace the smallest, earliest cold-dark-matter cusps by shallower central profiles or cores.

Since $V_v^2=G{M}_v/r_v$, the spherical potential that vanishes at infinity isIndeed, $rd\Phi /dr=GM(r)/r$, and normalization at $s=1$ requiresThe circular speed is consequentlyIt rises as $s^{1/2}$ near the centre, peaks at $r\simeq 2.16r_s$, and then declines approximately as $\sqrt{\log r/r}$. Increasing the concentration of a dark-matter halo moves the peak inward in units of $r_v$ and raises it relative to $V_v$. Thus an NFW curve can be fairly broad but is not exactly a flat galaxy rotation curve; stellar and gas contributions matter when comparing with an observed galaxy rotation curve.

Lower-mass haloes typically collapse earlier, when the cosmic background density is larger. Their characteristic inner densities are consequently larger relative to the present virial density, producing the mass-concentration relation of dark-matter haloes in which concentration decreases weakly with mass.

For a mass smaller by $1024=2^{10}$,and $c=101024^{0.1}=20$. HenceA halo of roughly $10^9{M}_{\odot }$ and kiloparsec scale naturally hosts a dwarf galaxy, possibly an extremely faint one if star formation is inefficient.