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For an optically thin low-density gas in collisional ionization equilibrium, the astrophysical cooling function collects losses per pair of hydrogen nuclei into $\mathcal{C}=n_H^2\Lambda (T)$. Its units are $\mathrm{erg}\mathrm{c}{\mathrm{m}}^3{\mathrm{s}}^{-1}$; the time unit is missing from the printed guide to the vertical-axis labels. For the primordial atomic cooling curve, only hydrogen and helium provide bound-state coolants.

Near $10^4\mathrm{K}$, collisional excitation of hydrogen becomes effective and subsequent photon emission removes thermal energy. The excitation rate is exponentially suppressed below the atomic energy threshold. Atomic line cooling and collisional ionization generate a strong hydrogen feature around a few $10^4\mathrm{K}$; helium excitation and ionization produce further structure around $10^5\mathrm{K}$. Radiative recombination also contributes. Once the gas is highly ionized, these bound-state losses weaken. At $T\gtrsim 10^6$–$10^7\mathrm{K}$, thermal bremsstrahlung dominates, with approximately $\Lambda \propto T^{1/2}$ apart from slowly varying factors. The figure is an original qualitative sketch anchored to the two supplied values, not an atomic-rate calculation.

Increasing galactic metallicity adds many ionic transitions, producing metal-line cooling and substantially enhancing cooling, particularly in the $10^5$–$10^7\mathrm{K}$ range. Metals also provide low-energy fine-structure transitions below $10^4\mathrm{K}$. The change is not a uniform vertical shift: the positions and strengths of features depend on the elemental abundances and ionization state.

Photoionization removes bound electrons even where collisions alone would leave atoms neutral, often suppressing the hydrogen and helium line peaks and changing the metal-ion population. It also adds photoionization heating. The net thermal loss is then $\mathcal{C}-\mathcal{H}$, with possible heating-cooling equilibrium near $10^4\mathrm{K}$. A single density-independent $\Lambda (T)$ no longer describes every irradiated cloud: the answer also depends on radiation intensity, spectrum, shielding and density. The illustrative photoionized curve denotes altered cooling alone, not the net loss including heating.

Below $10^4\mathrm{K}$, primordial gas can use molecular line cooling, especially rotational and vibrational transitions of molecular hydrogen and, in suitable chemical conditions, hydrogen deuteride. Molecule formation and protection against photodissociation are essential. Enriched gas can additionally use metal-line cooling, other molecular species and dust thermal emission, with gas-dust energy exchange important at high density. At high cosmological redshift, residual free electrons can transfer energy to the cosmic microwave background through Compton cooling by the cosmic microwave background if the gas is hotter than the radiation. Expansion can also cool gas adiabatically, but neither that process nor Compton cooling by the cosmic microwave background is an $n_H^2$ atomic loss law. The atomic threshold is a cooling-channel limitation, not a universal minimum gas temperature.

Near the accretor assume fully ionized hydrogen, tight electric coupling between protons and electrons, isotropic radiation and opacity dominated by Thomson scattering. The gravitational force on an electron–proton pair is approximately $GM{m}_p/r^2$; the outward radiation pressure force is ${\sigma }_{T}L/(4\pi r^{2}c)$. Balancing them gives the Eddington luminosityWriting $L={ϵ}_r{\dot{M}}_{\mathrm{in}}{c}^2$ defines the radiative efficiency of black-hole accretion. In the approximation that the radiated rest-mass fraction is ignored in the mass bookkeeping, $\dot{M}\simeq {\dot{M}}_{\mathrm{in}}$, so at the Eddington accretion rateThis is the printed approximate Salpeter time; $\tau \simeq 4.5\times 10^7\mathrm{yr}$ for the adopted efficiency. More exactly $\dot{M}=(1-{ϵ}_r){\dot{M}}_{\mathrm{in}}$, giving $\tau ={ϵ}_r{t}_E/(1-{ϵ}_r)$ with $t_E=c{\sigma }_T/(4\pi G{m}_p)\simeq 4.5\times 10^8\mathrm{yr}$. The difference is a ten-percent-level convention here, not a change from exponential growth. Continuous fuelling, constant efficiency and unit Eddington ratio are additional idealizations.

Next normalize the ionizing spectrum rather than replacing every photon by a threshold photon. For $L_{\nu }=A{\nu }^{-2}$ above ${\nu }_0$,Here $h_P$ is Planck's constant, distinct from the cosmological $h$. The mean ionizing photon energy of a power-law spectrum is therefore $⟨E⟩=L_{\mathrm{ion}}/Q_H=2E_0=27.2\mathrm{eV}$. With ionizing luminosity fraction $f_{\mathrm{ion}}=0.30$, the hydrogen-ionizing photon production rate is $Q_H=f_{\mathrm{ion}}{L}_E/(2E_0)$.

Assume a sharp spherical ionization front, homogeneous initially neutral hydrogen at the cosmic mean baryon density, escape fraction one, one primary ionization per photon, and no recombinations or secondary ionizations. The ionization-front growth from an exponentially brightening source is then fixed by photon conservation. In the printed approximate mass convention,Take the stated separation to be a proper separation at the seed-formation epoch, and take the distant halo to follow the expansion on this large scale. Its initial separation $R_i$ defines a fixed comoving radius $X=R_i(1+z_i)$. The number of hydrogen nuclei initially within that radius isAlthough the proper density subsequently falls, this comoving hydrogen inventory is constant. Thus the photon count does not require freezing the expansion for the entire growth interval.

The supplied critical density gives ${\Omega }_b{\rho }_{\mathrm{crit},0}=6.86\times 10^9{M}_{\odot }{\mathrm{Mpc}}^{-3}$, so $M_H\simeq 2.87\times 10^{13}{M}_{\odot }$ for the initial proper-distance interpretation. Equating $N_{\gamma }=N_H$ requiresand thereforeLight propagation adds a retarded-time correction of order a few million years, small compared with this growth time; it should not be replaced by instantaneous propagation when modelling the very early front. Expansion increases the proper distance to the comoving halo during the wait, but not the required comoving photon inventory. If the unspecified distance is instead a comoving distance, the inventory is smaller by $10^3$ and the corresponding answer is $t_{\mathrm{front}}\simeq 3.57\times 10^8\mathrm{yr}$. A distance held at a fixed proper radius until arrival would need a different evolving-volume treatment.

Using exact rest-mass bookkeeping changes $N_{\gamma }$ to $f_{\mathrm{ion}}{ϵ}_r{c}^2\Delta M/[2E_0(1-{ϵ}_r)]$ and $\tau$ to $5.0\times 10^7\mathrm{yr}$. The proper-at-formation case then gives about $7.37\times 10^8\mathrm{yr}$, and the comoving-distance case about $3.91\times 10^8\mathrm{yr}$. These modest changes are smaller than the uncertainties of the sharp-front, no-recombination model. In particular real absorption of the high-energy tail, secondary electrons and a nonunit escape fraction would require radiative-transfer corrections.

The small halo initially holds cool gas in hydrostatic equilibrium. Photoionization deposits energy far above its virial binding scale. With the emitted mean energy, the excess per primary ionization is $13.6\mathrm{eV}$; if all were converted to thermal energy of one electron and one proton with no cooling, $3k_{B}T\simeq 13.6\mathrm{eV}$ would give about $5\times 10^4\mathrm{K}$. Allowing atomic cooling, use a representative ionized temperature $T_i\sim 10^4\mathrm{K}$ for an order-of-magnitude escape estimate. Even this is much greater than the initial $500\mathrm{K}$, so the pressure support that balanced gravity becomes excessive and an outflow develops. The baryons are photoheated and undergo photoevaporation of a dark-matter minihalo; the shallow halo loses much of its gas and its future star formation is suppressed.

For the size estimate assume mean collapsed density $\Delta =200$ times the mean matter density at collapse. Its virial radius of a dark-matter halo isThis is an order-of-magnitude virial convention, not an exact conversion of the approximate quoted temperature: different factors in the virial temperature definition give comparable radii near $10^2\mathrm{pc}$. Keep the radius of the bound halo fixed after its stated collapse, rather than rescaling it with the later cosmic mean density. Fully ionized pure hydrogen has mean molecular weight ${\mu }_i=1/2$. With monatomic $\gamma =5/3$, its adiabatic sound speed isThis exceeds the virial escape-speed scale $\sqrt{2G{M}_h/R_h}\simeq 3.0\mathrm{km}{\mathrm{s}}^{-1}$. A sound-crossing outflow estimate givesmeasured after the ionization front arrives. Using an isothermal sound speed, a different virial factor or a hotter post-front gas changes this by factors of order unity. It is not a second hundreds-of-millions-of-years black-hole growth interval.

The crossing-time estimate assumes the relevant gas can be ionized without a long trapped-front delay. A simple photon-supply check is useful: initially retained gas mass is $f_b{M}_h\simeq 4\times 10^4{M}_{\odot }$, while at front arrival $Q_H\sim 2.4\times 10^{55}{\mathrm{s}}^{-1}$. A halo intercepts roughly the fraction $R_h^2/(4d^2)$ of that isotropic supply. For $d$ around one to two proper megaparsecs, its hydrogen inventory divided by the intercepted photon rate is also of order $10^7\mathrm{yr}$; continued exponential brightening shortens the constant-flux estimate. Thus a gas-loss estimate of one to a few $10^7\mathrm{yr}$ is justified in the idealized model. A detailed evaporation time is not uniquely determined without the gas profile, post-ionization temperature and shielding; retained recombinations would further delay a front in dense gas.

Choose two hypothetical environments A and B of one chemical element, with line centers ${\mu }_A=284\mathrm{eV}$ and ${\mu }_B=286\mathrm{eV}$. Model each line by a normalized Gaussian function with common standard deviation $\sigma$, and stipulate integrated count weights $w_A=3/4$, $w_B=1/4$. On the binding energy axis $E$, the normalized count density isAll energies in this expression use electronvolts. Each component's integrated area equals its count weight, and $\int I(E)dE=1$. There is no background, statistical counting fluctuation, or additional emission feature in this stipulated model. These choices deliberately isolate photoelectron peak overlap and resolution.

Direct integration, or the law of total expectation applied to the component label, gives the centroidThe centroid lies between the chemical environments and is not either component's binding energy. Expanding $(E-\bar{E}{)}^2$ within each component, the odd cross term integrates to zero. The resulting variance isFor $\sigma =0.25\mathrm{eV}$ the variance is $0.8125{\mathrm{eV}}^2$ and the two maxima are separated. For $\sigma =1.00\mathrm{eV}$ it is $1.7500{\mathrm{eV}}^2$ and there is one maximum. Each Gaussian's full width at half maximum is $2\sqrt{2\ln 2}\sigma$, approximately $0.589$ and $2.355\mathrm{eV}$ respectively. The chemical separation remains $2\mathrm{eV}$ in both cases. The number of visible maxima therefore does not equal the number of chemical components.

One interpretation of this change is additional independent Gaussian instrument blur. The convolution of probability densities adds the variances, so broadening $\sigma =0.25$ to $1$ requires response standard deviation $\sqrt{1-0.25^2}=\sqrt{15}/4\simeq 0.968246\mathrm{eV}$. A normalized response with zero mean preserves the centroid and integrated counts. Real line shapes and backgrounds can depart from this model, and separating hidden components is an inverse problem; a plausible fit does not alone establish uniqueness.

Finally, count weights describe detection. In a simple response model, let $n_i$ denote the number of emitting sites sampled in environment $i$, $s_i$ its effective photoionization cross section, and $T_i$ the probability that its emitted electron contributes to the selected measured line. ThenThe common factor $C$ includes exposure. Here $T_i$ includes escape from the sample, analyzer transmission and detection. Count weights equal site fractions only if the relevant response factors are equal or corrected. They are not automatically molecular mole fractions; different molecules can supply different numbers of the selected emitting sites. In an original example, equal site numbers with $s_A{T}_A=3s_B{T}_B$ give the plotted $3:1$ count ratio despite a $1:1$ site ratio.

Photoionization can heat a shallow halo's gas well above its virial temperature. If the heated sound speed exceeds the escape-speed scale, the previous hydrostatic equilibrium is lost and a pressure-driven outflow removes gas. A crossing time gives a useful estimate, but photon supply, the gas profile, shielding and recombinations can delay the process. This particularly affects haloes whose virial temperatures are below the usual atomic cooling threshold and reduces their subsequent star formation.

Photoionization removes a bound electron when an atom or ion absorbs a sufficiently energetic photon. The threshold is its ionization energy. The excess photon energy becomes kinetic energy of the products; subsequent collisions can heat the gas or cause secondary ionizations. A photoionization rate and a photoionization equilibrium are distinct concepts: equilibrium also requires specified recombination and other reaction rates.

When the number of photoionizations equals the number of recombinations, and only those processes exchange thermal energy, equilibrium requires $⟨h\nu -h{\nu }_0{⟩}_{\mathrm{abs}}=⟨E_{\mathrm{rec}}(T)⟩$. Photon absorption and electron capture have their own spectral weights. The density cancels from this event balance, but the actual equilibrium temperature requires the two energy averages.^[ref]

The quasar proximity effect is a reduction of nearby Lyman-alpha forest absorption caused by extra photoionization from the quasar. At fixed density and temperature in photoionization equilibrium, an added rate ${\Gamma }_Q$ changes the neutral fraction by the factor ${\Gamma }_{\mathrm{bg}}/({\Gamma }_{\mathrm{bg}}+{\Gamma }_Q)$. A host environment with positive density contrast can mask this reduction, and a variable or anisotropic source complicates its interpretation.

RABBITT measures adjacent harmonic spectral phases through two-photon photoionization. A weak delayed infrared reference opens two interfering paths to a shared photoelectron energy; the two-path attosecond phase reconstruction spells out the inference.

Ultraviolet is electromagnetic radiation with wavelengths shorter than the violet end of ordinarily visible light. The CIE convention uses $100$–$400\mathrm{nm}$, subdivided into UV-A at $315$–$400\mathrm{nm}$, UV-B at $280$–$315\mathrm{nm}$ and UV-C at $100$–$280\mathrm{nm}$. These divisions name intervals; they do not mark universal changes in every material's response. The boundary with visible light is not sharp, and other fields use near-, far-, vacuum- and extreme-ultraviolet divisions. ^[ref]. For example, NASA's Solar Dynamics Observatory observes extreme-ultraviolet bands below $100\mathrm{nm}$; the CIE interval is therefore not the full extent of astronomical ultraviolet usage. ^[ref].

The CIE ultraviolet interval corresponds to frequencies about $7.49\times 10^{14}$–$3.00\times 10^{15}\mathrm{Hz}$ and photon energies $4.97\times 10^{-19}$–$1.99\times 10^{-18}\mathrm{J}$, using vacuum wavelength and $E=hc/\lambda$ with Planck constant $h$ and speed of light $c$. Shorter wavelength means larger energy per photon; larger irradiance means more incident power per area. These are different comparisons. At the same incident power, a $254\mathrm{nm}$ ultraviolet beam has fewer photons per second than a $940\mathrm{nm}$ infrared beam, even though each ultraviolet photon has more energy.

A common ultraviolet mechanism is photon absorption promoting an electron from a ground state to an excited state of a chemical atom or molecule. By contrast, many infrared absorption bands involve stretching or bending chemical bonds. For either mechanism, both the energy difference and the coupling to the incident electromagnetic field matter. One concrete example is the $217\mathrm{nm}$ absorption discussed for butadiene: an electron moves between molecular orbitals, and the position of the absorption reveals their energy separation. Ultraviolet absorption can therefore probe molecular orbitals, while infrared spectroscopy probes many molecular motions. ^[ref].

Subsequent chemical reactions depend on the absorbing chemical substance and its environment; the label ultraviolet alone does not predict a reaction rate. At sufficiently short wavelength, a photon can cause photoionization if its energy exceeds the relevant ionization energy. There is no single wavelength that separates all ionizing from all nonionizing behavior for every chemical substance. Likewise, eventual heating after photon absorption is possible in both ultraviolet and infrared. The distinction is useful because different optical spectra expose different processes, not because the two ranges obey different laws of electromagnetism.

Ritter's 1801 observation used a chemical readout rather than a thermometer. In Caltech/IPAC's account, sunlight passed through a glass optical prism and silver chloride was placed in different parts of the optical spectrum. Darkening was weak toward red, stronger toward violet, and pronounced beyond the visible violet edge. The measured evidence was a change in the chemical substance's appearance at a spatial location with no visible color, establishing radiation beyond the eye's ordinary range. It was not a calibrated measurement of a universal ultraviolet optical spectrum. ^[ref].

A controlled modern interpretation of such a chemical test would compare equal exposure durations and matched fresh samples, keep an unexposed sample in darkness, monitor temperature, and distinguish scattered visible light from the intended short-wavelength beam with characterized filtering. These are proposed controls, not an assertion that Ritter performed them. Unequal sample preparation or heating can change a reaction rate. The optical prism, terrestrial atmosphere and sample response restrict which wavelengths the test detects. Darkening is a detector-weighted response, so its spatial maximum is not automatically the maximum of incoming radiant flux or of photon energy.

Thomas Larason and Jeanne Houston's NIST report documents a quantitative ultraviolet detector comparison. An argon source feeds a double-diffraction grating spectrometer; a shutter and mirrors deliver the selected band to a working-standard or test photodiode, successively at the same position. A beam splitter supplies a monitor photodiode. Amplifiers and voltmeters record detector and monitor simultaneously; shutter-closed readings provide dark baselines. The $200$–$500\mathrm{nm}$ scan, typically with $4\mathrm{nm}$ bandpass, measures spectral responsivity. Monitor normalization suppresses source drift; known spectral lines calibrate wavelength. The report assesses temperature effects and unwanted light outside the band, weighted by detector response. The latter need not cancel between different spectral responsivities. ^[ref].

This determines electrical response without visual perception. Bandpass and spatial nonuniformity limit the calibration; aging and changed temperature can affect later use. The scan includes visible light and ultraviolet; no response below $200\mathrm{nm}$ follows from it.

For a narrow beam, the idealized substitution calculation isHere $R$ is spectral responsivity; $I$ and $M$ are dark-subtracted detector and monitor electric currents. This follows by writing $I=RP$ for each detector and using the monitor to track changes in radiant flux. It assumes linear response, unchanged beam splitting and collection geometry, and a sufficiently narrow band. For an illustrative standard with $R_{\mathrm{std}}=0.15\mathrm{A}{\mathrm{W}}^{-1}$, ratios $I_{\mathrm{std}}/M_{\mathrm{std}}=2.0$ and $I_{\mathrm{test}}/M_{\mathrm{test}}=1.6$ give $R_{\mathrm{test}}=0.12\mathrm{A}{\mathrm{W}}^{-1}$. These invented arithmetic inputs illustrate the method; they are not NIST measurements.

An independent photon-counting model connects spectral responsivity to quantum efficiency. If $\eta (\lambda )$ is the number of collected electrons per incident photon, there is no internal multiplication, and $q$ is the elementary charge, thenAt illustrative constant $\eta =0.60$ and $P=1\mu \mathrm{W}$, this gives about $0.123\mu \mathrm{A}$ at $254\mathrm{nm}$ and $0.455\mu \mathrm{A}$ at $940\mathrm{nm}$. The ultraviolet readout is smaller in this model because there are fewer incident photons, not because their individual energies are smaller. Real quantum efficiency depends on wavelength, windows, absorption and carrier collection; a detector's band gap and construction determine its usable range. Constant $\eta$ across these two wavelengths is an analytical comparison, not a claimed specification of a real photodiode.

The curve is the calculated relation $E=hc/\lambda$, and the shaded ranges use the stated CIE conventions. The lower diagram abstracts the detector comparison rather than reproducing an instrument drawing. Together with the infrared heat-balance illustration, it shows why a calibrated detector response, rather than what looks bright to an observer, is the bridge between invisible radiation and a quantitative measurement.

X-ray fluorescence is Characteristic X-ray emission following excitation by incident photons. In a simple initially unexcited chemical atom, photoionization can eject an inner electron if the incident photon supplies its binding energy. An outer electron can then fill the vacancy, releasing an emitted photon with the difference of shell binding energies. Nonradiative relaxation competes with this route, so one absorbed incident photon need not yield one escaping characteristic photon.

Choose a separate stipulated electron shell model with values $B_K=10.0$, $B_L=1.5$ and $B_M=0.3$, all in $\mathrm{keV}$. Ignore subshell splitting, recoil and changes of binding energy during relaxation. The possible shell differences areThese labels denote simplified shell groups, not resolved subshell electronic transitions. With $hc=1.23984\mathrm{keV}\mathrm{nm}$, their calculated wavelengths are respectively $0.145864$, $0.127819$ and $1.03320\mathrm{nm}$. An incident $9.8\mathrm{keV}$ photon exceeds both K-line energies but cannot directly create the K vacancy in this initially unexcited model. A $12.0\mathrm{keV}$ photon can, leaving approximately $2.0\mathrm{keV}$ of photoelectron kinetic energy. This comparison fixes neither the excitation probability nor the radiative branching probability.

X-ray photoelectron spectroscopy uses X-ray excitation to investigate electron removal energies, including those of core electrons. The common name electron spectroscopy for chemical analysis emphasizes its use in distinguishing chemical surroundings. The whole recorded spectrum can also contain electrons produced by processes other than direct photoionization; a peak assignment must identify the emission process as well as the chemical element.

In a modern implementation, a quartz crystal selects aluminum characteristic X-rays according to Bragg's law and focuses them onto the sample. Vacuum reduces gas collisions and contamination; a separately pumped entry chamber limits exposure to air. Lenses collect photoelectrons; an electrostatic electron energy analyzer transmits a narrow kinetic energy band. Stepping that band and counting detector pulses produces the spectrum. An electron flood source can compensate positive electric charge on insulating samples. Ion cleaning can alter vulnerable samples.^[ref]

For a scan with count $N_j$ and dwell time $t_j$ at setting $j$, compare rates $N_j/t_j$, rather than raw counts when dwell times differ. Under an independent Poisson distribution model, the count standard deviation is approximately $\sqrt{N_j}$ for a sufficiently populated bin. Longer acquisition reduces this relative counting measurement uncertainty; it does not automatically improve separation of overlapping peaks. Photoelectron peak overlap and resolution gives an example in which the same chemical separation produces either two maxima or one.