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The smaller galaxy is treated as a test particle released from rest at $r_0$. Conservation of energy in the fixed gravitational potential givesThe galactic gas is stationary in its hydrostatic model, so this is the relative speed squared. Write $\Delta (d)=d^{-1/2}-r_0^{-1/2}\ge 0$ on the inward path. The astrophysical ram pressure is the incident normal momentum flux $\rho (d){\dot{d}}^2$, givingIt vanishes at release and increases during infall. This ram pressure convention is $\rho v^2$, not the $\rho v^2/2$ sometimes called dynamic pressure. Gas drag and the object's response are neglected consistently with the prescribed ballistic motion.

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

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

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

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

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

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

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

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

Take $v>0$ to mean the magnitude of the inward radial velocity. In the Gaussian units used for the magnetic pressure, approximate the surface density of a disk by $\Sigma \simeq 2H\rho$. The mass accretion rate and free-fall speed then giveAt the magnetospheric truncation radius, balance the magnetic pressure of the magnetic dipole field with the specified ram pressure:ThusDiscarding the convention-dependent numerical factor givesThe magnetic pressure grows inward as $r^{-6}$, faster than the ram pressure $\propto r^{-5/2}$, so the field disrupts the flow inside this radius. The numerical coefficient depends on the vertical averaging and on treating the inflow as radial free fall, as stipulated here; a detailed thin disk boundary model need not have the same coefficient.

For an isolated approximately spherical mass distribution outside its luminous extent, retain the gravitational force $GMm/r^2$ and use the specified low-acceleration inertial law. circular motion then givesThe radius cancels when enclosed luminous mass is effectively constant. This is the asymptotic modified Newtonian dynamics result associated with the baryonic Tully-Fisher relation; an arbitrary extended disk cannot everywhere be replaced exactly by a point mass.

At the disk edge,Using $G=6.67\times 10^{-11}{\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$ givesThis is an order-of-magnitude isolated, low-acceleration estimate. The edge acceleration is below $a_0$, but not extremely small compared with it; a full interpolation law, disk geometry and an external gravitational field can affect a more precise mass fit.

For the spherical mass profile for a flat rotation curve, assume spherical symmetry where the halo dominates. A flat rotation curve gives $M(<r)=v^{2}r/G$, and differentiating $M(<r)=4\pi {\int }_0^r\rho (s)s^{2}ds$ yieldsHere $\rho$ is the density of the spherical gravitating model; it is not an exact deprojection of a thin luminous disk or a proof that the central density must diverge. At the outermost stated measured radius,This gives the requested galaxy-plus-halo mass estimate if the model is truncated at the outermost observed radius $2R$. The observation itself constrains enclosed mass within $2R$; any additional outer halo remains unconstrained. Extending $\rho \propto r^{-2}$ to infinity would give divergent mass. Subtracting the MOND-inferred luminous mass to obtain a Newtonian halo mass would mix independent model-dependent estimates without an independently measured luminous mass.

In the Bullet Cluster collision, most observed baryonic mass is hot gas. ram pressure slows and shocks the gas, while the galaxies pass through much more freely. weak gravitational lensing maps the projected gravitational mass without requiring that the gas be in hydrostatic equilibrium. The dominant lensing peaks lie closer to the galaxies than to the displaced X-ray gas. This supports a dominant weakly collisional dark matter component under the usual gravity model. It is stronger evidence than a mass-to-light discrepancy alone because distinct components separate during the same encounter. The inference needs background-source shape/redshift calibration, projection control and uncertainty in the lensing reconstruction; it does not identify a WIMP species or by itself exclude every modified-gravity theory.^[ref]

For dark-matter direct detection, consider XENON100 at Gran Sasso. A halo WIMP can undergo elastic scattering from a xenon atomic nucleus, leaving a recoil of only a few to tens of keV. The expected rate is very low, so a large target, low threshold and long exposure must be combined with background rejection. The nuclear-recoil kinematics of WIMP scattering gives $E_{R,\max }=2{\mu }_{\chi N}^2{v}^2/m_N$ in the nonrelativistic limit, where ${\mu }_{\chi N}$ is the reduced mass and masses and velocities use consistent units. This explains why low-energy response calibration matters.

The dual-phase xenon detector measures prompt particle scintillation S1 and delayed electroluminescence S2 from ionization electrons extracted into gas. Their time separation gives interaction depth; the S2 light pattern supplies transverse position. An inner fiducial volume uses xenon's self-shielding to suppress external backgrounds. The S2/S1 distribution statistically separates nuclear recoils from electron recoils; it is not perfect event-by-event identification. Underground siting reduces cosmic-ray backgrounds, while passive shielding, low-radioactivity components, xenon purification and a surrounding active detector veto reduce other events.^[ref]

Neutrons can also make nuclear recoils, so charge-to-light discrimination alone cannot eliminate them. Multiple-scatter rejection, shielding and background modelling remain necessary. Gamma-source and neutron-source calibrations map electron- and nuclear-recoil distributions, energy scale and selection acceptance. Position-dependent light collection and charge losses need correction; photon-counting fluctuations and uncertain low-energy yields limit threshold sensitivity. Blind event selection and independent sidebands guard against tuning cuts to apparent candidates. The 224.6-live-day, 34 kg search found two benchmark candidates against about one expected background event, consistent with background. Its interaction limit therefore depends on the assumed halo distribution, WIMP mass and nuclear response; it is not a detection or an exclusion of all dark matter.^[ref]

Corotation reaches the speed of light at $R_c=c/\Omega$. Extrapolating a dipole field gives $B_c\sim B_{\ast }(R_{\ast }/R_c{)}^3$ and magnetic energy density ${ϵ}_c\sim B_c^2/{\mu }_0$, suppressing order-one factors. An outward electromagnetic energy flux of scale ${ϵ}_{c}c$ through area of scale $R_c^2$ therefore givesThis assumes a dipole-dominated field tied to the rotating star and an effective outgoing Poynting flux near the light cylinder. The dipole extrapolation, angular pattern and surface-field convention affect the coefficient. A strictly aligned vacuum dipole would not radiate, whereas an oblique vacuum dipole does. A plasma-filled magnetosphere opens field lines and can spin down even an aligned rotator, so the light-cylinder estimate is a scaling argument, not an exact vacuum emission calculation or literal rigid motion at $c$. Force-free calculations explicitly demonstrate this distinction.^[ref]

The energy comes primarily from stellar rotation. With constant moment of inertia $I$, $E_{\mathrm{rot}}=I{\Omega }^2/2$, and conservation of energy gives $I\Omega \dot{\Omega }=-P$. ThusFor constant $K$, differentiating gives $\ddot{\Omega }=-3K{\Omega }^2\dot{\Omega }$, hence $n=\Omega \ddot{\Omega }/{\dot{\Omega }}^2=3$. Evolving field strength, inclination, moment of inertia or additional torques invalidate the constant-$K$ prediction; generally $n=3+(\Omega /\dot{\Omega })(\dot{K}/K)$ in this parametrization.

For radial inflow onto mass $M$, take $v_{\mathrm{ff}}\sim (GM/r{)}^{1/2}$ and $\rho \sim \dot{M}/(4\pi r^2{v}_{\mathrm{ff}})$. magnetic pressure decreases as $B_{\ast }^2{R}_{\ast }^6/({\mu }_0{r}^6)$, while ram pressure scales as $\rho v_{\mathrm{ff}}^2\propto \dot{M}(GM{)}^{1/2}{r}^{-5/2}$. Balancing them yieldsNumerical factors and disc geometry are not determined by this spherical estimate. The displayed proportionality holds $M$ fixed and requires a magnetospheric boundary outside the stellar surface.

In a disc, inflowing matter near $R_M$ has specific angular momentum approximately $\sqrt{GM{R}_M}$. If it is transferred to a slower star, the material torque is $N\sim \dot{M}\sqrt{GM{R}_M}$ and $I\dot{\Omega }\sim N>0$. The corotation radius is $R_{\mathrm{co}}=(GM/{\Omega }^2{)}^{1/3}$, distinct from the light-cylinder radius. As stellar spin approaches the inner disc's Keplerian frequency, net spin-up weakens. If $R_M$ lies outside corotation, the field rotates faster than the disc and can centrifugally expel matter in the propeller regime. A characteristic equilibrium isfor fixed mass and magnetic-coupling convention. A complete magnetic torque model can change the coefficient; a star also cannot rotate past its mass-shedding limit, of order $\sqrt{GM/R_{\ast }^3}$.

For opacity per unit mass $\kappa$, an isotropic radiative force balances gravity at $L_E=4\pi GMc/\kappa$. Efficient release of surface gravitational binding energy gives $L_{\mathrm{acc}}\sim GM\dot{M}/R_{\ast }$ and therefore the surface accretion Eddington rateThe radius dependence matters: it uses efficiency $\eta \sim GM/(R_{\ast }{c}^2)$, not an arbitrarily fixed reference efficiency. Substituting this maximal rate into the equilibrium period givesmass, opacity and geometry factors are held fixed. This is the classical radiatively efficient Eddington-limited scaling, not a universal upper bound on every mass supply rate: beaming, advection, outflows and opacity changes can alter it. For sufficiently small field, $R_M\lesssim R_{\ast }$ and the magnetic scaling must be replaced by surface-disc and mass-shedding physics.

A dipole's magnetic pressure scales as $B_{\ast }^2{R}_{\ast }^6/({\mu }_0{r}^6)$. Spherical free fall with $\rho =\dot{M}/(4\pi r^{2}v)$ and $v\propto (GM/r{)}^{1/2}$ has ram pressure proportional to $\dot{M}(GM{)}^{1/2}{r}^{-5/2}$. Equating them gives the seventh-root scaling. Disc geometry changes the coefficient; the estimate ceases to describe an exterior magnetosphere if the resulting radius lies inside the star.