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Showing 1021–1050 of 1709 reference entries

Earth & cosmosStellar radial velocity and companion-mass inference

In a Newtonian isolated two-body Keplerian model, the host star's line-of-sight orbital velocity has semi-amplitude K=(2πG/P)^(1/3) Mp sin i/[(Mstar+Mp)^(2/3)√(1−e²)].

MethodEarth & cosmosInitial mass functions and stellar population accounting

For a specified birth population, an initial mass function ξ(m) defines dN=ξ(m)dm on a stated initial-mass interval.

DefinitionEarth & cosmosWhite-dwarf mass–radius relation and ideal limiting mass

Cold noninteracting electron support and spherical Newtonian hydrostatic balance give a nonrelativistic n=3/2 polytrope with R proportional to μₑ^(−5/3)M^(−1/3).

ModelEarth & cosmosRotation-powered pulsars and spin-down inference

Neutron-star pulsars have rotating magnetized stellar remnants whose emission can sweep past the observer.

ModelEarth & cosmosCore collapse and thermonuclear stellar explosions

Core-collapse supernovae involve loss of support in an evolved stellar core, rapid contraction and an explosion whose energy transfer involves dense matter and neutrinos.

ModelEarth & cosmosAccretion energy, efficiency and angular-momentum transport

Matter moving inward in a gravitational field can release binding energy if energy is dissipated.

ModelEarth & cosmosEddington luminosity as a conditional force balance

For isotropic luminosity L and a constant flux-weighted opacity per mass κ, outward radiative acceleration is κL/(4πr²c).

ModelEarth & cosmosRoche geometry and binary mass transfer

In a circular two-point-mass binary's corotating frame, the effective potential per mass is Φ_eff=−GM₁/r₁−GM₂/r₂−Ω²(x²+y²)/2, with origin at the barycentre and Ω²=G(M₁+M₂)/a³.

ModelEarth & cosmosAccretion-powered galactic nuclei and observational inference

Active galactic nuclei can radiate strongly as matter accretes around a central supermassive black hole.

ModelEarth & cosmosPrimordial nuclear freeze-out and the deuterium bottleneck

In hot Big-Bang nucleosynthesis, neutron–proton conversion initially approaches chemical equilibrium.

ModelEarth & cosmosRecombination, last scattering and reionization

Atomic recombination binds many previously free electrons and lowers photon-scattering opacity.

ModelEarth & cosmosIonized hydrogen regions and the Strömgren photon budget

An ionization-bounded pure-hydrogen region in steady balance equates absorbed external ionizing photons to net recombinations: Q_H=∫α_B nₑnₚ dV.

ModelEarth & cosmosJeans instability of a local gas model

Linear perturbations in a uniform Newtonian gas model obey ω²=c_s²k²−4πGρ₀.

ModelEarth & cosmosThermal free–free radio emission

Electron–ion acceleration produces continuum emission without capture.

ModelEarth & cosmosSynchrotron emission and electron energy loss

A magnetically deflected electron radiates P=2σ_TcU_Bβ²γ²sin²α.

ModelEarth & cosmosThomson inverse-Compton electron loss

An electron in an isotropic soft-photon field loses net power P_IC=(4/3)σ_TcU_radβ²γ² in the Thomson limit.

ModelEarth & cosmosRepresentative stellar fusion isotope pathways

The ppI branch builds deuterium and helium-3 before combining two helium-3 nuclei.

ModelEarth & cosmosConserved fluxes across a normal gas shock

A steady planar normal shock conserves mass, momentum and total-enthalpy flux across its thin transition: [ρu]=[ρu²+P]=[ρu(h+u²/2)]=0.

ModelEarth & cosmosSolar wind and magnetospheric energy transfer

The solar wind carries charged particles, including protons and electrons, and solar magnetic field into interplanetary space.

ModelEarth & cosmosCosmic-ray composition and secondary-particle evidence

In charged-particle usage, cosmic rays include energetic nuclei and electrons from space.

Empirical findingEarth & cosmosClassical event horizon and Schwarzschild radius

A classical black-hole event horizon is a causal boundary: signals from its interior cannot reach distant future observers in the exterior.

DefinitionEarth & cosmosPoint-mass deflection and Einstein angular radius

A point mass in the weak-field geometric-optics limit deflects light by magnitude α̂=4GM/(c²b).

MethodEarth & cosmosPhoton–baryon acoustic structure in the CMB

Before recombination, scattering tightly coupled photons and baryons.

ModelBiologyCohort survival and net reproduction

A cohort life table follows individuals born together.

ModelBiologyEuler–Lotka growth and age structure

In a fixed density-independent age model, the growth factor λ satisfies Σ_{x≥1}λ^(−x)l_xm_x=1.

ModelBiologyLevins patch occupancy and persistence

A simple internally colonized metapopulation tracks the occupied fraction p of equivalent habitat patches.

ModelBiologyOccupancy and imperfect species detection

Failure to detect a species does not establish absence.

MethodBiologyIndividual-based rarefaction and richness sampling

For a collection containing N individuals with N_i in each observed species, the expected species count in a uniformly drawn subset of n individuals without replacement is E[S_n]=Σ_i[1−C(N−N_i,n)/C(N,n)].

MethodBiologyPerturbation, rescue and gene-function evidence

A genetic perturbation tests a causal prediction when the intended change and relevant phenotype are measured against appropriate controls.

MethodChemistryGreen chemistry design principles and lifecycle scope

Green chemistry evaluates product and process design to reduce hazardous inputs, releases and waste over a stated lifecycle.

Method
ModelSource checked

Ionized hydrogen regions and the Strömgren photon budget

Earth & cosmos

An ionization-bounded pure-hydrogen region in steady balance equates absorbed external ionizing photons to net recombinations: Q_H=∫α_B nₑnₚ dV.

Key knowledge

An ionization-bounded pure-hydrogen region in steady balance equates absorbed external ionizing photons to net recombinations: Q_H=∫α_B nₑnₚ dV. Case B excludes direct ground-state recombinations whose ionizing photons are locally reabsorbed. A uniform spherical fully ionized model has R_S=[3Q_H/(4πα_B nₑnₚ)]^(1/3). This is a photon budget, not an assertion that every nebula is spherical or in thermodynamic equilibrium.

Values and equations

Q_H=∫α_B nₑnₚ dV; R_S=[3Q_H/(4πα_B nₑnₚ)]^(1/3)

Q_H
absorbed external hydrogen-ionizing photon rate s⁻¹
α_B
case-B recombination coefficient at the specified temperature m³ s⁻¹
nₑ,nₚ
proper electron and proton number densities m⁻³
V
ionized gas volume m³
R_S
uniform spherical equilibrium radius m

Conditions and scope

  • Fixed source, uniform temperature/coefficient in the spherical estimate, negligible dust absorption or escape, negligible collisional ionization and the case-B local reabsorption approximation. For pure fully ionized hydrogen nₑ=nₚ; other species require adjusted electron and photon accounting.
  • Density-bounded gas, clumping, expansion, variable sources, leaking radiation and dust alter the result. Photoionized low-density gas need not obey the LTE Saha relation; no universal region radius, temperature or star-formation conversion is inferred.
Sources · 3

References

Sources retain their recorded scope. A source link alone does not establish verification of every statement.

  1. University of Maryland — J. P. Harrington teaching notes2 sources
    1. Physics of Ionized Gasses in the Interstellar Medium, slides3–6: case A/B and rate balanceUniversity of Maryland — J. P. Harrington teaching notesReference guide · PDFpages.astro.umd.eduRead source
    2. Strömgren Spheres and Recombination Lines (2019), slides1–3: photon integral, uniform radius and limitsUniversity of Maryland — J. P. Harrington teaching notesReference guide · PDFpages.astro.umd.eduRead source
  2. Chapter4 §4.2, equations4.3 and4.6–4.11 and density-bounded distinctionNRAO — Essential Radio AstronomyReference guidecv.nrao.eduRead source

Evidence and review

Read actual UMD case-B ground-state recycling versus case-A escape, source photon integral and uniform balance/radius body, plus NRAO ionization/density-bounded definitions and volumetric budget. The volume integral is the disclosed extension of the uniform local rate. Excluded source malformed c² prefactor in a blackbody substitution, numerical table examples and universal dust-free observational claims.

2026-10-10 · Independent AI editorial review: All44 exact root-authored compact records, equations/variable units/conditions and140 origin edges accepted after independently reading final6/18 with bounded primary bodies and matching prior38/122 scientific fingerprints unchanged. Exact claim/edge review is distinct from combined inventory scope and separate CLE26 approval; no entire cited books or old-corpus experimental revalidation is asserted. Receipt: cosmos-completion-independent-content-and-inventory-review.1.

Related reference entries

Explore relationships within this area of study and its connections to other areas.

Showing 1–6 of 6 relationships

Scientific relationships · 6 / 6

The relationship type describes the connection. Its linked statements, sources and conditions define the scope.

  1. Related to
    Relationship sources · 2

    Linked statements

    • Atomic recombination binds many previously free electrons and lowers photon-scattering opacity. Photon last scattering describes the probabilistic final scattering along an observed path; it is related to recombination but is not an instantaneous atomic event. Later ultraviolet sources reionize intergalactic gas, creating free electrons that can scatter some CMB photons again. For homogeneous nonrelativistic Thomson scattering, optical depth from time t to now is τ(t)=∫ₜᵗ⁰ cσ_T nₑ(t′)dt′, with no-further-scattering probability e^(−τ).

    Ionizing sources and recombination also require a photon budget.

  2. Related to
    Relationship sources · 3

    Linked statements

    • An ionization-bounded pure-hydrogen region in steady balance equates absorbed external ionizing photons to net recombinations: Q_H=∫α_B nₑnₚ dV. Case B excludes direct ground-state recombinations whose ionizing photons are locally reabsorbed. A uniform spherical fully ionized model has R_S=[3Q_H/(4πα_B nₑnₚ)]^(1/3). This is a photon budget, not an assertion that every nebula is spherical or in thermodynamic equilibrium.

    Ionized regions are one component of interstellar material.

  3. Uses definition
    Relationship sources · 3

    Linked statements

    • An ionization-bounded pure-hydrogen region in steady balance equates absorbed external ionizing photons to net recombinations: Q_H=∫α_B nₑnₚ dV. Case B excludes direct ground-state recombinations whose ionizing photons are locally reabsorbed. A uniform spherical fully ionized model has R_S=[3Q_H/(4πα_B nₑnₚ)]^(1/3). This is a photon budget, not an assertion that every nebula is spherical or in thermodynamic equilibrium.

    Ionization requires photons above the relevant energy threshold.

  4. Related to
    Relationship sources · 3

    Linked statements

    • An ionization-bounded pure-hydrogen region in steady balance equates absorbed external ionizing photons to net recombinations: Q_H=∫α_B nₑnₚ dV. Case B excludes direct ground-state recombinations whose ionizing photons are locally reabsorbed. A uniform spherical fully ionized model has R_S=[3Q_H/(4πα_B nₑnₚ)]^(1/3). This is a photon budget, not an assertion that every nebula is spherical or in thermodynamic equilibrium.

    Photoionization rate balance and LTE ionization are distinct models.

  5. Related to
    Relationship sources · 3

    Linked statements

    • An ionization-bounded pure-hydrogen region in steady balance equates absorbed external ionizing photons to net recombinations: Q_H=∫α_B nₑnₚ dV. Case B excludes direct ground-state recombinations whose ionizing photons are locally reabsorbed. A uniform spherical fully ionized model has R_S=[3Q_H/(4πα_B nₑnₚ)]^(1/3). This is a photon budget, not an assertion that every nebula is spherical or in thermodynamic equilibrium.

    Using ionizing emission to infer total star formation requires a population model.

  6. Related to
    Relationship sources · 1

    Linked statements

    • Electron–ion acceleration produces continuum emission without capture. For nonrelativistic Maxwellian electrons in the radio limit hν≪k_BT, the emissivity scales as j_ν∝Z²nₑnᵢT^(−1/2)g_ff, with a frequency-dependent correction factor. A thermal particle distribution alone does not make an optically thin continuum a blackbody.

    Ionization determines electron and ion populations.

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uni-cosmos-hii-photon-budget
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