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Put ${\mu }_J=G{M}_J$. At the periapsis distance, the radial velocity vanishes. Insert $E=m{v}_0^2/2$ and $J=mb{v}_0$ into a:The positive root isRationalizing gives the numerically stable alternative $r_{\min }=b^2{v}_0^2/[\sqrt{{\mu }_J^2+b^2{v}_0^4}+{\mu }_J]$. Attraction makes $r_{\min }<b$. This is a point-planet scattering calculation; a noncolliding physical encounter must also have $r_{\min }$ outside Jupiter's surface. At $b=0$, the limiting formula gives a radial collision, not a flyby.

For the single-line greenhouse deficit, use hydrostatic balance, fixed surface pressure $p_s$, constant gravity $g$, and the dry adiabatThe gas constant and specific heat refer to the background gas. Let ${\mu }_d=\cos \tilde{\theta }>0$ be the supplied effective ray cosine. The vertical optical depth and slant transmittance from level $p$ to space areThis permits pressure-dependent line widths. With local thermodynamic equilibrium, negligible infrared scattering, a black thermal surface, and no incident thermal radiation from space, the formal radiative transfer solution isThe weights add to one. Subtract the transparent-surface radiance and put $B(p)=B_{{\nu }_0}(T(p))$, using the narrow-line approximation only in the line deficit. Define its pressure-dependent equivalent widthApproximating each hemispheric flux by $\pi$ times the effective-ray radiance preserves the black-surface flux. The outgoing longwave flux is thereforeIntegration by parts gives the particularly transparent formBoth boundary terms vanish: $E(0,q)=0$ and $B(p_s)-B(p_s)=0$. The dry adiabat and monotonic Planck function give $dB/dp>0$. Atmospheric emission replaces part of the warm surface radiation by emission from cooler levels, reducing emission to space for a specified $T_s$. The inequality is strict when opacity occurs over a nonzero range with temperature contrast; an isothermal absorbing layer at the surface temperature would not produce such a contrast.

For a spherical planet the mean absorbed instellation is $F_{\ast }=(1-\alpha )L_{⊛}/4$. The required global energy balance isConsequently every equilibrium of this model has $T_s\ge T_{\mathrm{bare}}=[(1-\alpha )L_{⊛}/(4{\sigma }_{\mathrm{SB}}){]}^{1/4}$, with strict warming when $D>0$. At fixed temperature profile, increasing $q$ increases $E$ and hence $D$. This argument does not assume that the greenhouse gas changes the stellar-band albedo, surface pressure, or background thermodynamics; those are held fixed. For sufficiently broad or overlapping saturated wings the full frequency-dependent Planck function must replace the narrow-line approximation.

Assume a plane-parallel atmosphere in local thermodynamic equilibrium, negligible infrared scattering, a black surface, and surface temperature equal to the adjacent air temperature. For a specified ray, let $\mathfrak{T}$ be its transmittance from a level to space, increasing from ${\mathfrak{T}}_0$ at the surface to one at the top. The formal radiative transfer solution isUsing ${\mathfrak{T}}_0+{\int }_{{\mathfrak{T}}_0}^{1}d\mathfrak{T}=1$ yieldsThis expresses absorption and re-emission together: opacity does not simply remove surface radiation without adding atmospheric radiation. An absorber close to a thermally coupled surface has nearly the same Planck function as that surface, so $B-B_0$ is small. Its emission nearly compensates the surface radiation it absorbs, giving weak radiance sensitivity to near-surface concentration. In an exactly isothermal atmosphere at surface temperature, $L=B_0$ for every optical depth, a concrete example of complete concentration ambiguity in this thermal measurement. Finite surface emissivity, reflection, scattering or a surface-air temperature discontinuity would require additional terms.

After injection, the radiopharmaceutical circulates and accumulates according to its biological targeting and clearance. Technetium-99m undergoes an isomeric nuclear transition, producing approximately $140\mathrm{keV}$ gamma rays; its roughly six-hour half-life permits useful counting with limited persistence. Some photons undergo absorption or Compton scattering in the patient. A lead collimator in the gamma camera accepts a narrow range of directions so the accepted event retains positional information.

An accepted gamma ray deposits energy, commonly in a NaI(Tl) scintillator, by photoelectric effect or Compton scattering. The resulting visible particle scintillation light is collected by photomultipliers. Their relative pulse sizes determine an event position and their summed pulse estimates its energy. A pulse-height analyser accepts an energy window around the photopeak to reject many scattered events. Position and uniformity corrections are applied; accepted events increment a two-dimensional pixel histogram. After acquisition the computer displays this count map as a planar projection of tracer activity. Finite counting statistics, attenuation, scattering and the collimator limit the image's quantitative accuracy and spatial resolution.^[ref]

Start with a fast massive projectile following an approximately straight path. Its time-dependent electric field perturbs bound absorber electrons. At large impact parameter, treating the perturbation to first order and summing excitation and ionization transition probabilities gives an energy transfer weighted by the absorber's oscillator strengths. The oscillator-strength sum rule reduces the leading coefficient to the number of target electrons, while their logarithmically weighted excitation energies define $I$.

The classical impulse picture explains the logarithm: a transverse impulse scales as $1/(bv)$, so transferred energy scales as $1/(b^2{v}^2)$; multiplying by the number of encountered electrons in $2\pi bdbdx$ gives $db/b$. Quantum scattering replaces ad hoc short- and long-distance cutoffs by recoil kinematics and bound-state excitation. Relativistic projectile kinematics supplies $T_{\max }$ and the $-{\beta }^2$ term. Summation over many independent encounters gives the mean stopping power, rather than the energy-loss straggling of individual tracks.

The leading approximation assumes $M\gg m_e$, a sufficiently fast projectile for perturbation theory, a largely fixed projectile charge, and weak correlations between collisions. At low velocity, electron binding, charge exchange and nuclear stopping become important; at high velocity, polarization screens distant interactions and gives the density correction. The projectile's bremsstrahlung is suppressed by its large mass and is excluded from the collision stopping power.

Between about 1880 and 1920 a recognizable investigation of subatomic entities emerged from work on electrical discharge, radiation and atomic model. Calling it particle is retrospective: it was not yet the later high-energy specialization with accelerators, and boundaries with atomic physics, nuclear physics and chemistry remained fluid. Its formation nevertheless had both scientific and institutional causes.

Better vacuum pumps, discharge tubes, high voltages, photographic plates and sensitive electrical instruments made new phenomena experimentally reproducible. Cathode rays could be deflected by electric and magnetic fields, letting investigators compare their charge-to-mass ratio under different conditions. In 1897 Joseph John Thomson argued that their carriers were common constituents of matter with far smaller mass than an atom. This made a divisible atom an experimental research problem rather than simply a speculative possibility.^[ref]

Wilhelm Röntgen's X-rays, Henri Becquerel's spontaneous uranium radiation and the work of Marie Curie and Pierre Curie on strongly radioactive substances revealed additional emissions and transformations. Their classification by penetration, charge and deflection separated physically different entities. Radioactive decay introduced characteristic lifetimes and suggested that atoms could change identity. Chemical separation and quantitative electrical measurements were therefore as important as a purely mechanical picture of particles.^[ref]

Rutherford scattering made the atom's internal structure accessible to an incisive quantitative probe. Hans Geiger and Ernest Marsden measured rare large deflections, and Ernest Rutherford's 1911 analysis connected them to a concentrated nuclear charge rather than a diffuse one. The experimental task involved a collimated source, thin target foils and detection of individual scintillations at different angles; changing thickness and angle tested explanations of single versus multiple scattering. This style of inferring invisible structure from statistical distributions became characteristic of the developing field.^[ref]

Specialization also depended on laboratories capable of maintaining apparatus, obtaining rare sources, training assistants and students, and circulating standardized measurements. Research schools sustained linked problems across many experiments; new journals, meetings and prizes gave their practitioners a recognizable identity. Industry supplied pumps and electrical equipment, while medical interest in radiation supplied additional demand and resources. These conditions help explain why the specialization consolidated in that period rather than emerging automatically from one discovery.

The date range should not imply completion. By 1920 the electron, nuclear electric charge, radioactivity and early nuclear transformations had established a new scale of questions, but the neutron, mature quantum mechanics and accelerator-based high-energy research still lay ahead. The emerging specialty was the joint product of experimentally identifiable new objects, a distinctive toolkit and institutions able to make their investigation a continuing career.

Photon diffusion approximates the spreading of radiation after many scattering events have largely randomized propagation directions. Individual photons travel along paths; their distribution can obey a diffusion equation on scales much larger than a transport mean free path. The longer arrival times arise from longer, repeatedly redirected paths.

For homogeneous, conservative isotropic scattering, the photon diffusion coefficient for isotropic scattering is $D=v\ell /3$, where $v$ is the transport speed and $\ell$ the scattering mean free path. Directional persistence replaces that scale by a transport mean free path in an appropriate transport model. Crossing a characteristic thickness $L$ then takes a time of order $L^2/D$, with geometry-dependent factors, which can greatly exceed the direct travel time $L/v$.

A diffuse photon time-of-flight experiment sends a short pulse into a scattering sample and measures the delayed transmitted or returning light. A pulsed source, a fast photodetector and time-correlated single-photon counting can build an arrival-time histogram from repeated pulses. The detector's timing response must be accounted for before attributing all measured broadening to the sample.^[ref]

The 1989 study by M. S. Patterson, Britton Chance and B. C. Wilson used a slab diffusion model to relate picosecond-pulse reflectance and transmittance to absorption and scattering, comparing predictions with preliminary measurements and Monte Carlo simulations.^[ref] The earliest nearly unscattered light, nearby boundaries and strongly directional transport can require a fuller radiative transfer description.

For conservative photon transport at constant speed $v$, suppose successive scattering directions are independent and isotropically distributed, and free-path lengths follow an exponential distribution of mean $\ell$. In three spatial dimensions, the resulting long-time diffusion coefficient isHere $\ell$ is the scattering mean free path. These assumptions describe homogeneous, memoryless interactions without absorption.

The velocity autocorrelation function gives a direct derivation. An initially isotropic ensemble has zero mean velocity. During a lag $s$, the probability of no scattering is $e^{-vs/\ell }$. No scattering preserves the direction; one or more independent isotropic scattering events give zero expected directional correlation. ConsequentlyThe three-dimensional Green-Kubo formula for diffusion now yieldsThe factor $1/3$ converts the dot-product correlation into one Cartesian component under isotropy.

The same correlation gives the full mean squared displacement:At short times this is $v^2{t}^2$; at long times its leading term is $6Dt$. The model therefore connects direct flights to the diffusive many-scattering limit.

The free-path distribution matters. A fixed-step isotropic random flight with every length equal to $\ell$ has mean-square displacement $n{\ell }^2$ after $n$ steps and elapsed time $n\ell /v$, giving $D=v\ell /6$. Replacing exponential flights by fixed-length flights thus changes the coefficient by a factor of two. For anisotropic scattering, directional persistence instead leads to a transport mean free path; strong absorption and boundaries require additional modeling.

Conservative cloud scattering produces no material heating when each interaction preserves photon energy and the model includes no absorption or thermal emission. This global constraint does not imply that an approximation retaining only one scattering event predicts the correct outgoing angular distribution.

For example, a thick cloud can redirect a photon many times before escape. A single-scattering calculation can miss that redistributed intensity or violate the total energy budget if scattered light is simply removed after its first interaction. Conversely, a method can enforce total energy conservation yet still predict a poor specific intensity in a chosen direction. Conservation and directional accuracy are separate tests of a radiative transfer calculation.

Radiative transfer follows how emission, absorption and scattering change specific intensity. Along a ray, the radiative transfer equation has the formwhere $j_{\nu }$ adds radiation and ${\alpha }_{\nu }$ removes it from the ray. Scattering into the ray can couple this equation to other directions.

The optical depth measures attenuation along a path; the source function compares emission with extinction. Their variation explains why spectral lines probe different depths of a stellar atmosphere rather than a single emitting surface.^[ref]

Robert Hofstadter investigated nuclear charge distributions through energetic electron scattering. His 1956 review compares $400\mathrm{MeV}$ electrons scattered from helium gas with predictions for point charges and several extended charge profiles. Beyond about $70^{\circ }$, the elastic scattering was over a hundred times smaller than the point-charge prediction. The extracted nuclear charge form factor distinguishes candidate spatial profiles, connecting measured angular counts with a model of nuclear size and shape.^[ref]

A physicist.^[ref]

Stanisław Ulam helped develop the modern computational use of Monte Carlo methods. The firsthand account by Nicholas Metropolis describes Ulam's proposal to revive statistical sampling with electronic computers and its subsequent development with John von Neumann.^[ref]

In particle-transport calculations, one samples successive events—scattering, absorption and branching—to produce many possible histories. Averaging their outcomes estimates a macroscopic quantity. The link between a physical random process, probability distributions and an executable algorithm made difficult calculations accessible without enumerating every possible trajectory.^[ref]

Transillumination examines internal structure using light transmitted through a body region or specimen. The source illuminates one side and the observer or detector records light emerging after propagation through the material. Different optical absorption and scattering properties can create image contrast.^[ref]

A simple physical demonstration illuminates a fingertip from behind and observes the diffuse transmitted glow. Its appearance depends on wavelength, path length and the tissue's optical properties. Strong scattering redistributes the paths, so a brightness difference cannot generally be interpreted as a sharply localized anatomical boundary without an additional model.