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The fraction of nuclei in a specified ionized state. For pure singly ionized hydrogen, $x=n_e/(n_H+n_e)$ by charge neutrality; a different species or normalization requires a different definition.

Take fully ionized hydrogen with electron and proton number densities both $n$, fixed local heat input $\Gamma$ per unit volume, and volumetric cooling $3n^2{\alpha }_B(T)k_{B}T/2$. The ideal gas law is $P=2n{k}_{B}T$. At fixed pressure, net cooling becomes $\mathcal{L}=3P^2{\alpha }_B(T)/(8k_{B}T)-\Gamma$. At an equilibrium $T_0$, differentiation gives the displayed derivative.

The linearized thermal equation is ${\rho }_0{c}_p\dot{\delta T}=-{\mathcal{L}}_T\delta T$, so equilibrium is stable exactly when the logarithmic slope of ${\alpha }_B$ exceeds one. A constant or decreasing coefficient gives instability. A different heating prescription must be differentiated too; constant incident radiation does not generally imply a density-independent local heating rate.

Across a thin steady transition with negligible external impulse, mass conservation and momentum conservation give the displayed relations. If the gas stays isothermal while neutral atomic hydrogen becomes molecular hydrogen, the mean particle mass doubles and $a_2^2=a_1^2/2$ for its isothermal sound speed. Let $J=u_1+a_1^2/u_1$. The downstream subsonic branch has $u_2=[J-\sqrt{J^2-2a_1^2}]/2$, ${\rho }_2/{\rho }_1=u_1/u_2$, and $p_2-p_1={\rho }_1{u}_1(u_1-u_2)>0$.

For $u_1\ll a_1$, the leading behavior is $u_2\simeq u_1/2$, ${\rho }_2\simeq 2{\rho }_1$ and $p_2\simeq p_1$, at unchanged temperature. Radiation of the molecular binding energy prevents that released energy from necessarily heating the gas; it does not, by itself, establish an isothermal thermal closure. If temperature evolution is allowed, an energy balance and a cooling model are required to determine all four changes uniquely.

On 10 July 1908, Heike Kamerlingh Onnes's Leiden laboratory circulated purified helium through liquid air and evaporatively cooled liquid-hydrogen precooling, a regenerative heat exchanger, and an expansion valve. Returning helium cooled incoming helium; liquid collected in a transparent vacuum vessel. Reflected light revealed its liquid meniscus, crossed by thermocouple wires. Subsequent mass density and spectroscopy checks tested contamination.^[ref]

A constant-volume helium gas thermometer below the valve supplied the temperature indication. An oxygen boiling reference check disagreed with its expected reading; the real gas correction to thermodynamic temperature remained provisional. A temperature plateau suggested immersion before the liquid meniscus was recognized.^[ref]

The measurement principles separate two questions. A visible liquid meniscus establishes an interface; a gas thermometer estimates temperature through an equation of state. A stationary temperature trace can also result from a steady state without phase coexistence, so it cannot replace the interface observation. Conversely, seeing a liquid does not calibrate a thermometer. Thermometer calibration, the gas thermometer dead-space correction, departures from the ideal gas law and incomplete thermal equilibrium all matter when assigning an absolute temperature. A check far from the intended measurement range tests the instrument but does not by itself bound every low-temperature error.

In 1947, Willis Eugene Lamb Jr. and Robert Curtis Retherford dissociated hydrogen in a tungsten oven, crossed its atomic jet with electrons for collisional excitation, and detected surviving $2S$ atoms by electron ejection from metal. An electrometer and galvanometer read the current. They interrupted fixed-frequency microwaves while sweeping the static magnetic field; a flip coil calibrated the field. Their Zeeman splitting comparison placed $2S_{1/2}$ about $1000$ $\mathrm{MHz}$ above $2P_{1/2}$; the plotted $1000$ $\mathrm{MHz}$ displacement was explicitly not a best fit.^[ref]

The 1955 lecture's apparatus cross-section identifies movable slits, an electron bombarder, a radiofrequency transmission line with beam slots, a d.c. quenching electrode, metal target, electron collector, and magnet poles. A $2S\to 2P$ transition followed by rapid spontaneous emission leaves a ground-state atom unable to eject detector electrons. The scan holds radiofrequency and amplitude fixed. Natural linewidth and unresolved hyperfine structure broaden resonances, whose analysis yields zero-field separations.^[ref]

Static electric field quenching supplied a control. The 1947 report identifies natural linewidth, hyperfine structure, excessive microwave intensity, and field inhomogeneity as width contributions, and flags imperfect magnetic calibration. It does not specify an absolute microwave-frequency calibration procedure.^[ref]

The surname is Robert Curtis Retherford's: “Retherford” and “Rutherford” name different people. The historical Lamb-Retherford experiment should retain the former spelling.^[ref]

In 1918, Irving Langmuir measured adsorption using freshly split mica strips, handled at their edges, separated by iron spacers and held with brass fasteners. A glass bulb contained the bundle; a similar blank contained matching iron and brass but no mica. Evacuated glass connections, Mercury seals, liquid air traps, and McLeod gauges provided controlled gas handling. Calibrated compartment volumes and pressure readings before and after expansion gave free-gas amounts; their deficit indicated uptake. Lowering pressure and measuring released gas tested desorption.^[ref]

Hydrogen blank runs calibrated cooling-related pressure differences. Irving Langmuir pre-cooled the bulbs to reduce thermal delay; he suggested slow diffusion between strips could control the remaining approach to equilibrium. Critically, blank uptake was not subtracted: he estimated the mica-bulb results were 10–30% too high. Liquid air temperature drift was unrecorded. These limits constrain inference of intrinsic binding from the uptake curve. The paper's pressure unit called “bar” was $1\mathrm{dyn}\mathrm{c}{\mathrm{m}}^{-2}=0.1\mathrm{Pa}$, unlike the modern bar used above.^[ref]

Ernest Orlando Lawrence and Milton Stanley Livingston reported repeated ion acceleration in 1932. Their evacuated brass chamber used one driven hollow cyclotron dee; the grounded box and slotted divider supplied the other electrode. Filament electrons ionized hydrogen near the gap. A perpendicular magnetic field bent the ions; a high-frequency oscillator drove the gap. Deflecting plates and slits selected particles for a Faraday cup, read by an electrometer.^[ref]

Scanning the magnetic field produced narrow resonance peaks. The optimum deflecting voltage tracked the kinetic energy predicted from field and orbit radius, supporting the reported $1.22\mathrm{MeV}$ protons. Source width and slit geometry limited this check; it was not a direct energy measurement of every particle.^[ref]

This combination distinguishes producing a circulating electric current from establishing its kinetic energy. The electrometer measures collected electric charge per unit time; it cannot by itself tell whether the collected particles are fast. The separate deflection selection supplies a speed-dependent test. Agreement of two consequences of the model, the cyclotron resonance condition and the cyclotron energy and orbit radius relation, makes the inference stronger than either a current signal or a predicted orbit alone.

Milton Stanley Livingston built the early cyclotrons with Ernest Orlando Lawrence; the 1939 Nobel Prize in Physics went to Ernest Orlando Lawrence. His lecture, The evolution of the cyclotron, was delivered on 11 December 1951.^[ref][ref]

Before calculating electronic energy, ask a simpler question: where can the available valence electrons go? A Lewis structure makes that budget visible, and nitrate shows why a correct budget can still require several resonance structures.

A Lewis structure records which chemical atoms are connected in a molecule or ion and how its valence electrons are allocated. One line represents one bond pair, two lines two bond pairs, and three lines three bond pairs; two dots beside a chemical atom represent one lone pair. The marks are a counting language, rather than measured electron positions. See the IUPAC Gold Book, Lewis formula.

For water, ${\mathrm{H}}_2\mathrm{O}$, the budget is $6+2\times 1=8$ valence electrons. Two O–H lines use four electrons; the other four form two lone pairs on oxygen. For ammonia, ${\mathrm{NH}}_3$, $5+3\times 1=8$ valence electrons give three N–H lines and one lone pair on nitrogen. Both molecules contain four electron pairs, but their allocations differ. Each shared bond pair is counted once in the total budget, even though both endpoints count it when checking the octet rule.

A practical construction method is:

For carbon dioxide, $4+2\times 6=16$ valence electrons. Starting with two single C–O connections leaves twelve electrons, enough for three lone pairs on each oxygen, but only four electrons around carbon. Converting one lone pair from each oxygen into a second bond pair gives $\mathrm{O}=\mathrm{C}=\mathrm{O}$. Eight electrons are in the four bond pairs, and eight are in four lone pairs, two on each oxygen. Carbon has no lone pair; all three chemical atoms satisfy the octet rule, and all three formal charges vanish.

For the nitrate ion, ${\mathrm{NO}}_3^{-}$, the valence electron budget isNitrogen contributes five, the three oxygen chemical atoms contribute eighteen, and the negative electric charge adds one electron. Start with nitrogen connected to each oxygen by one bond pair: those three lines use six electrons. Put the remaining eighteen electrons into three lone pairs on each oxygen. Each oxygen now satisfies the octet rule, but nitrogen counts only six electrons. Reassign one oxygen lone pair as a second bond pair on its N–O connection. This changes the allocation without adding an electron; it applies the construction procedure in Flowers et al., Chemistry 2e, 7.3: drawing Lewis structures.

The resulting nitrate Lewis structure has one N=O connection and two N–O connections. Its four bond pairs use eight electrons, and its eight lone pairs use sixteen: $8+16=24$. The doubly connected oxygen has two lone pairs; each singly connected oxygen has three; nitrogen has none. All four chemical atoms satisfy the octet rule. Any of the three oxygen chemical atoms can supply the additional bond pair, giving the three equivalent resonance structures below.

Why not draw three N=O connections to make the ion look symmetric? With two lone pairs on each oxygen, that drawing still spends $12+12=24$ electrons and has the right total formal charge, but places twelve electrons around nitrogen. It fails the octet rule for this second-period chemical element. Three equivalent resonance structures express the symmetry without that invalid local count.

The economy is beautiful: a handful of dots and lines rules out many impossible counts and identifies lone pairs that can participate in acid-base chemistry. A Lewis structure helps choose rational candidate connections before a detailed quantum chemistry calculation. It does not specify a complete many-electron wavefunction, actual electron density, or molecular geometry. Adding the separate VSEPR model, described in Chemistry 2e, 7.6: molecular structure and polarity, predicts bent water and pyramidal ammonia; the layouts above alone make no such prediction. Molecular orbital theory supplies a richer electronic description, including the unpaired electrons of molecular oxygen that an all-paired Lewis structure misses.

In 1916 Gilbert N. Lewis proposed two-dot notation for the shared electrons of a chemical bond and fuller dot formulae for the outer electrons, explicitly relating a shared pair to the existing single-line notation. See Lewis, The Atom and the Molecule, JACS 38, 762–785 (1916), pp. 777–778, available as an ^[ref]; the relevant passage is also readable in a ^[ref]. The modern quantum mechanics interpretation should be distinguished from his historical model of fixed outer electrons.

The limiting reagent is the reactant exhausted first if the specified chemical reaction proceeds to the maximum conversion permitted by stoichiometry. With initial values of amount of substance $n_{i,0}$ for the reactants and positive consumption coefficients $a_i$, the largest forward extent of reaction is ${\xi }_{\max }={\min }_i(n_{i,0}/a_i)$. Compare these quotients, rather than simply choosing the smallest initial quantity. This calculation predicts the maximum reaction product quantity and identifies which reactant supply must increase to raise it. See Flowers et al., Chemistry 2e, limiting reactants.

Start with $3\mathrm{mol}$ of molecular hydrogen, $1\mathrm{mol}$ of molecular oxygen and no water. Assume complete conversion by the displayed chemical reaction, without competing reactions. ThenThe molecular oxygen is the limiting reagent. Consumption of $2\mathrm{mol}$ of molecular hydrogen and $1\mathrm{mol}$ of molecular oxygen forms $2\mathrm{mol}$ of water, leaving $1\mathrm{mol}$ of molecular hydrogen. Initially there are $1.806642228\times 10^{24}$ molecular hydrogen molecules and $6.02214076\times 10^{23}$ molecular oxygen molecules; after conversion there are $1.204428152\times 10^{24}$ water molecules and $6.02214076\times 10^{23}$ leftover molecular hydrogen molecules.

As a check, the initial inventory has $6\mathrm{mol}$ of hydrogen chemical atoms and $2\mathrm{mol}$ of oxygen chemical atoms. The final water contains $4\mathrm{mol}$ of hydrogen chemical atoms, and leftover molecular hydrogen supplies the remaining $2\mathrm{mol}$. The atom totals balance even though the total amount of substance of molecules falls from four to three moles.

A lone pair is an electron pair assigned to one chemical atom without a connecting line to another chemical atom in a Lewis structure. It is usually shown as two dots. Water has two lone pairs on oxygen, accounting for four valence electrons; ammonia has one lone pair on nitrogen, accounting for two. In $\mathrm{O}=\mathrm{C}=\mathrm{O}$, each oxygen has two lone pairs, and carbon has none.

A lone pair can be reactive. In protonation of ammonia, the nitrogen lone pair becomes a new bond pair to a proton:The proton supplies no electron, so the eight valence electrons are retained: three bond pairs plus one lone pair become four bond pairs. The resulting nitrogen has formal charge $5-0-8/2=+1$; each hydrogen has formal charge zero. This is why identifying an available lone pair helps explain the chemical base behaviour of ammonia, rather than merely decorating its formula.

Lewis already used the ammonia to ${\mathrm{NH}}_4^{+}$ example in The Atom and the Molecule (1916), pp. 777–778. In modern terms the dots identify an allocation of electrons, not two stationary particles or an isolated patch unaffected by the rest of the molecule. A molecular orbital description can spread electronic amplitude over several chemical atoms while a localized Lewis structure remains useful for acid-base chemistry.

A neutral hydrogen atom absorbs a photon near the Lyman-alpha resonance by excitation from $1s$ to $2p$. Absorption lines in a quasar spectrum can therefore locate hydrogen along the sightline by their redshift, subject to peculiar-velocity and line-profile effects.

Radiative decay from the hydrogen $2p$ level to $1s$ produces a Lyman-alpha photon. In an optically thin collisionally excited gas, escape of this photon removes the excitation energy deposited by an electron collision. The excitation threshold is responsible for the rapid temperature sensitivity of this cooling channel.

The Lyman-alpha forest is the multitude of foreground Lyman-alpha absorption features in a distant quasar's spectrum, at wavelengths shorter than its own redshifted Lyman-alpha emission. The quasar supplies a bright background; intervening neutral hydrogen atoms remove photons near their local Lyman-alpha line resonance. Many dips can therefore come from one atomic transition encountered at different redshifts, rather than from many different chemical elements. Some features are instead transitions of other ions, so a dip alone does not identify Lyman-alpha absorption. This makes faint foreground gas detectable through the light it removes, even when its own emission would be difficult to see. ^[ref].

Ignoring peculiar velocity for the moment, an absorber at cosmological redshift $z_a$ puts its Lyman-alpha line atThe rest wavelength is in the ultraviolet; at $z_a=2$ it appears at $3647.01\text{A˚}$ in the optical spectrum. In a toy sightline to a quasar at $z_Q=3$, absorbers at $z_a=2,2.35,2.7$ appear at $3647.01,4072.4945,4497.979\text{A˚}$, all shortward of the quasar's Lyman-alpha emission position $4862.68\text{A˚}$. The absorbing atoms see the selected photons at their own resonance; the observed dip's redshift belongs to the foreground gas. Peculiar velocity adds a Doppler shift, so the inferred $z_a$ is not a pure distance coordinate. The rest wavelength and resonance interpretation are given in Rauch (1998), section 1.

The diagram uses invented optical depths and 26 Doppler cores; three selected foreground regions are marked. Their drawn positions and sizes are schematic. The dashed comparison imposes a uniform optical depth $\tau =2$, giving transmission $e^{-2}\simeq 0.135$ throughout the chosen foreground band. It is an idealized smooth-absorption model, not a measured spectrum or a prediction of the actual intergalactic medium.

The smooth comparison explains the Gunn-Peterson effect: neutral hydrogen distributed throughout an expanding intergalactic medium can absorb continuously over a wavelength interval, because different parts of the path resonate with different observed wavelengths. The resolved Lyman-alpha forest instead has strong transmission fluctuations and distinguishable absorption lines. These need not represent isolated compact clouds: diffuse sheets and filaments in the cosmic web can generate peaks in optical depth along one line of sight. A continuously varying gas distribution can look like a collection of lines. ^[ref]. ^[ref].

An early observation makes the identification concrete. Lynds's 1971 paper analyzed six spectrograms of the quasar 4C 05.34, with emission redshift $2.877$, taken in spring 1970 using an image-tube spectrograph on Kitt Peak's 84-inch telescope. Five covered $3200$–$7500\text{A˚}$ with approximately $10\text{A˚}$ resolution; the most informative covered about $3500$–$6000\text{A˚}$ at $5\text{A˚}$ resolution. Its reproduced spectrogram shows the strong Lyman-alpha emission flanked mostly on the short-wavelength side by numerous absorption lines. Lynds tabulated their wavelengths and estimated strengths, identified five absorption redshift systems, and proposed that many remaining unidentified lines were the same Lyman-alpha line at additional redshifts. He explicitly warned about blending, an incomplete census of weak lines, and wavelength errors around $0.5$–$1\text{A˚}$. The result established a striking spectral pattern; the paper's proposed interpretation did not measure a separate physical cloud for every dip. ^[ref].

A contrasting observation demonstrates smooth, very strong absorption. Becker and collaborators (2001) used the Echelle Spectrograph and Imager on Keck II to observe the quasar SDSSp J103027.10+052455.0 at $z_Q=6.28$. Sky lines gave an instrumental full width at half maximum of about $1.8\text{A˚}$ near $8500\text{A˚}$, or spectral resolving power $R\simeq 4700$. After sky subtraction and flux calibration, the mean continuum-normalized transmission over $8450$–$8710\text{A˚}$ was $0.0038\pm 0.0026$, consistent with zero; this corresponds to absorption redshifts approximately $5.95$–$6.16$. They identified a Gunn-Peterson trough. Crucially, resonant Lyman-alpha absorption is so sensitive that even a small neutral fraction can make transmission undetectable: the trough alone does not establish a mostly neutral intergalactic medium or uniquely date reionization. ^[ref].

What a spectrum constrains is a radiative transfer problem. For an ideal foreground screen with negligible emission into the observed beam, the intrinsic continuum-normalized transmission isHere $F$ is spectral radiative flux, $F_{\mathrm{cont}}$ is its unabsorbed continuum, $\tau$ is optical depth, $n_{\mathrm{HI}}$ is neutral hydrogen number density, and $N_{\mathrm{HI}}$ is neutral hydrogen column density along proper path length $\ell$. A Voigt profile model constrains redshift, column density and a velocity width together. A nearly black line has lost much of its sensitivity to further increases in optical depth; blending and an uncertain continuum can also admit different fitted components. Finite spectral resolving power and noise limit which widths and weak features can be measured. ^[ref]. ^[ref].

For a purely thermal Doppler core, use the velocity convention $\tau (v)\propto \exp [-(v-v_0{)}^2/b^2]$. The Maxwell-Boltzmann velocity distribution then giveswhere $k_B$ is the Boltzmann constant, $T$ is temperature and $m_{\mathrm{H}}$ is a hydrogen atom's mass. This $b$ is $\sqrt{2}$ times the Gaussian distribution's standard deviation. Higher temperature spreads the same column density over more velocities. An instrument also broadens a line: locally, for a nearly flat continuum, the recorded transmission is approximately the convolution of $\mathcal{T}$ with the normalized instrumental response, not the exponential of a convolved optical depth. A Gaussian response with full width at half maximum $c/R$ has standard deviation $c/[R\sqrt{8\log 2}]$, where $c$ is the speed of light and $R$ the spectral resolving power.

The upper model fixes the integral $\int \tau (v)dv$, hence the neutral hydrogen column density for this transition, while changing temperature. A cooler line blurred by an instrument resembles a warmer intrinsic line. The lower model has two equal Doppler cores separated by $24\mathrm{km}{\mathrm{s}}^{-1}$: reducing spectral resolving power merges them into one apparent feature. Both panels use invented profiles, Gaussian instrumental responses, and no noise or damping wings. They illustrate an inverse problem, rather than measurements of temperature or a universal relation between a dip and a cloud.

Better observations provide the inputs to that inverse problem. Kim, Cristiani and D'Odorico (2001) analyzed three quasars with the VLT's UVES spectrograph. For HE2217–2818, Table 1 records $z_Q=2.413$ and $16200\mathrm{s}$ of exposure in the $3050$–$3860\text{A˚}$ setup in October 1999. Isolated thorium-argon lamp lines measured $R\simeq 45000$; the data were resampled into $0.05\text{A˚}$ bins, with signal-to-noise ratios about $45$–$50$ per pixel over $3550$–$4050\text{A˚}$. They fitted a local polynomial continuum, fitted Voigt profiles, and compared statistics of the transmitted flux. Their analysis excluded the region near the quasar to avoid the quasar proximity effect and used the interval between its Lyman-beta and Lyman-alpha emission to reduce confusion with other Lyman-series absorption. Features below $N_{\mathrm{HI}}=10^{12.5}\mathrm{c}{\mathrm{m}}^{-2}$ became confused with noise, setting their adopted lower column density limit. Figure 1 shows the measured spectrum, fitted profiles and residuals. This was a calibrated absorption measurement, with explicit selection limits. ^[ref].

Even after instrumental correction, a width need not be purely thermal Doppler broadening. Hubble flow across extended gas, peculiar velocity structure and unresolved blends can contribute. If an isolated line's intrinsic width obeys $b^2=b_{\mathrm{th}}^2+b_{\mathrm{nt}}^2$ with an independent Gaussian nonthermal contribution $b_{\mathrm{nt}}^2\ge 0$, then $T\le m_{\mathrm{H}}{b}^2/(2k_B)$; $b=25\mathrm{km}{\mathrm{s}}^{-1}$ gives a computed upper bound about $3.79\times 10^4\mathrm{K}$. A complex blend need not satisfy this single-component model. Noise can also bias the fitted widths of weak lines. This is why the thermal cutoff of Lyman-alpha line widths, calibrated against gas models, is more informative than assigning a temperature to every dip. ^[ref].

Finally, neutral hydrogen column density is not total gas mass density. In a pure hydrogen approximation, with gas optically thin to ionizing radiation and nearly fully ionized in photoionization equilibrium, the Lyman-alpha forest photoionization scaling giveswhere $n_{\mathrm{H}}$ counts all hydrogen nuclei, $\Gamma$ is the photoionization rate, $\alpha (T)$ is the radiative recombination coefficient and $x_{\mathrm{HI}}=n_{\mathrm{HI}}/n_{\mathrm{H}}$ is the neutral fraction. At fixed temperature and $\Gamma$, doubling $n_{\mathrm{H}}$ quadruples $n_{\mathrm{HI}}$; doubling $\Gamma$ halves it. These computed examples explain both the sensitivity to density contrast and the ambiguity: stronger absorption can mean denser gas, weaker ionizing radiation, a different temperature, or more path length. The Jeans-length ionization correction for the Lyman-alpha forest adds a size model rather than an independently measured depth. Optically thick Lyman limit systems and damped Lyman-alpha systems require different treatment. ^[ref]. ^[ref].

The beauty of the Lyman-alpha forest is that a spectrum converts otherwise faint foreground gas into a sequence of markers through cosmic time. With a model of radiative transfer, velocities, ionizing radiation and the temperature-density relation of photoionized intergalactic gas, its transmission statistics probe the cosmic web and gas thermal history. A single line of sight remains one narrow sample, not a unique three-dimensional density map. ^[ref].

In optically thin, almost fully ionized hydrogen in photoionization equilibrium, $n_{\mathrm{HI}}\Gamma =n_e{n}_p\alpha (T)$ and $n_e\simeq n_p\simeq n_H$. Thus the neutral fraction is approximately $\alpha (T)n_H/\Gamma$, while the neutral density is proportional to $n_H^2$. At fixed overdensity, cosmic dilution gives $n_H\propto (1+z{)}^3$, hence $n_{\mathrm{HI}}\propto \alpha (T)(1+z{)}^6/\Gamma$. Gas density, temperature, ionizing radiation and path length are all required to infer a baryon density from Lyman-alpha absorption.

The Lyman-alpha line is the hydrogen $2p\leftrightarrow 1s$ electric-dipole transition, with photon energy about $10.2\mathrm{eV}$ and vacuum wavelength about $1216$ angstroms. Its emission cools collisionally excited gas, while absorption traces neutral hydrogen along a background-source sightline.

The conventional addition orientation in which the incoming hydrogen goes to the alkene carbon already bearing more hydrogens, in the ordinary applicable unsymmetrical-alkene setting.

For a specified age and irradiation, a hydrogen–helium sequence rises at low mass, has radii near Jupiter's across giant planets and brown dwarfs, and declines as electron degeneracy pressure becomes important. Above the hydrogen-burning minimum mass, sustained fusion supports a rising low-mass stellar branch.

Let a projectile have mass $m$, incident speed $v$, and incident kinetic energy $E=m{v}^2/2$. The stationary target has mass $M$. In a head-on elastic collision, the signed outgoing projectile velocity $v^{\prime }$ and target velocity $u$ obey momentum conservation and conservation of energy:Factor the second equation and eliminate $Mu=m(v-v^{\prime })$. The nontrivial collision branch has $u=v+v^{\prime }$, henceThis is the maximum over scattering angles, not the energy of every recoil. To see the bound without assuming a head-on event, set the recoil momentum magnitude to $q$ and its laboratory angle to the incident direction to $\theta$. Conservation of energy with outgoing projectile momentum $\mathbf{p}-\mathbf{q}$ givesThus $T=T_{\max }{\cos }^2\theta$ for a nonzero recoil, and the endpoint occurs at $\theta =0$.

For an original numerical example, stipulate $E=4\mathrm{MeV}$ and $m=m_p$, with $m_p$ the proton mass. Approximate hydrogen, helium and nitrogen target masses by $M=m_p$, $4m_p$ and $14m_p$ respectively. Their maximum fractions and recoil energies areThese stipulated values illustrate the approximation $m_n=m_p$; they are not measurements from 1932. With $m_p{c}^2\simeq 938\mathrm{MeV}$, the incident ratio $E/(m_p{c}^2)\simeq 0.0043$ gives $v/c\simeq 0.092$, consistent with the nonrelativistic limit used here. A million electronvolts is an energy unit, whereas a mass expressed in $\mathrm{MeV}/c^2$ requires the factor $c^2$ when inserted into an energy equation.

The hydrogen endpoint permits essentially complete transfer because the projectile and target masses match. For a very heavy target, $M\gg m$, the fraction approaches $4m/M$, which is small: the target can take substantial momentum with little kinetic energy. The exact inequality follows from $(m-M{)}^2\ge 0$, which implies $(m+M{)}^2\ge 4mM$.

Consider a nonrelativistic particle of mass $m$ undergoing an elastic collision with a stationary free target of mass $M=Am$, where $A\ge 1$. Assume the outgoing direction is uniformly distributed over solid angle in the center of mass frame; this assumption concerns the scattering angle distribution and need not hold for every target or energy. Let $\mu$ be the cosine of that frame's scattering angle. Transforming the outgoing velocity back to the laboratory givesThe uniform distribution of $\mu$ on $[-1,1]$ makes $x$ uniform on $[\alpha ,1]$. For $A>1$, integration yieldsFor equal masses, take the limit $\alpha \to 0$: $\xi =1$. The expected value of the fractional energy after a collision is instead $\mathbb{E}[x]=(1+\alpha )/2$. These are different averages: averaging a logarithm does not give the logarithm of an average.

For independent collisions with the same distribution, the logarithms of successive energy ratios add. ThusHere $E_{\mathrm{geom}}$ is the geometric-mean energy, and $n_{\log }$ is a characteristic collision count for reaching an energy scale $E_{\ast }$. It is not an exact first-passage expected value. Individual histories fluctuate, and neither the time between collisions nor absorption is specified by this model.

In an original chosen example, set $E_0=10^6\mathrm{eV}$ and $E_{\ast }=0.025\mathrm{eV}$. Approximate hydrogen by $A=1$ and carbon by $A=12$. ThenThese are illustrative model values, not measured collision counts from 1934. The energy scale marks where neglected thermal motion matters; the calculation does not assert continued cooling to zero energy.

Metallic hydrogen is a high-pressure conducting state of hydrogen. Its fluid form in the deep interiors of gas giants supplies a conducting medium for a planetary dynamo, unlike the iron-alloy conducting cores of terrestrial planets.