The thermal model, Eq. (37), expresses the unbound Gibbs equation (10) through a combination of equivalent mass and kinetic energy. We construct the thermal model at the time of last scattering, or more simply “last scatter”, with a finite difference method using the radius r of a sphere as the variable. A popular value of the cosmic redshift at last scatter is z = 1089 and we use this. A spreadsheet is used for the calculations. This is less satisfactory than an analytic derivation, but it does give solace in that expressions herein describe thermal loss ΔU_i precisely. It is only in the partition of ΔU_i between kinetic gain and gravity loss where error accrues. We treat the gas as a single species with a mean atomic weight Ж. The BBN estimate for baryons [30] gives a mass proportion of about 75% hydrogen : 25% helium, so Ж = 1.24 x 10^-3 kg/mol. Hydrogen was monatomic and nonrecombinant to diatomic form, absent catalysis through aggregation. Atomic collisions are all treated as elastic. The initial parameters for our model are given in Table 1 and were derived by the author from table 6 of Planck 2020 [9]. The baryon temperature T at z = 1089 will be set to 2971K, the CMB’s extrapolated value. From our chosen H₀ in Table 1, the baryon density ρ_b at last scatter was 5.46 x 10^-19 kg/m³. This is very low and we can easily treat the gas as ideal.

The observed tiny wiggles [9], in the otherwise perfect Boltzmann³ distribution of the CMB, are telling. There is a metric, η_slip, found from the wiggles. It describes variance in the behavior of light between Einstein’s and Newton’s⁴ Universes at last scatter. If η_slip = 1, then there was no variance. The value of η_slip was found to be 1.004 ± 0.007. The text in [9] proclaims that such a value of η_slip justifies an assumption of minimal gravitational anisotropy. Gravitational stress tensors in Einstein’s field matrix vanish. This allows us to say that at last scatter, the entire instant Universe was effectively Euclidean.⁵ Thermal energy $U_i$ is nonrelativistic at 2971K, so atomic momentum tensors were Newtonian, which allows the gas laws to be applied to their collective behavior. The wiggles do indicate minor density variance, attributable in part to baryon acoustic oscillation, i.e. “sonic waves”. This variance may have been negligible back then, but these acoustic waves began to overlap within a colder and colder Universe, creating permanent regions of high gas density whose formation was abetted by the cold. Within these regions of higher density, acoustic resonance gave occasional antinodes that were dense enough to cause gravitational collapse into stars via a process first described by James Jeans (1877-1946) [31]. Treatment of baryon mass using all of general relativity becomes necessary after that, and we will accept without reservation the consistency of myriad observations of the behavior of energy in all its forms with the predictions of general relativity. That said, the Euclidian approximation at last scatter remains accurate. We can also treat the intergalactic medium (IGM) today as Euclidean. The Universe is widely regarded as mathematically “flat”. A flat Universe is Euclidean ad infinitum, and the IGM today comprises about 90% of the Universe’s volume. Accreted matter is only a local perturbation within this larger and much more massive volume. Progress of time in the IGM has been linear ever since last scatter. Its density was and is so low that negligible gravitational time dilation has occurred from then to now.

The thermal model is constructed using monatomic gas expressions found in many introductory engineering textbooks and Wikipedia. Most of the z values in the thermal model (9→1089) cover the dark age [31,32]. The effect of ionizing radiation on the thermal model at z ≥ 9 is practically nil.

The thermal model (37) can be expressed with the inverse scale factor $z^{\prime }$:

Eq. (38) is developed in Appendix C. The term $T^{\prime }$ = 0.0025K is a constant, expressed in Kelvins.¹¹ The term ${\mathrm{ɱ}}^{\prime }$ expresses relative mass density, has a range of 0.757 ≤ ${\mathrm{ɱ}}^{\prime }$ ≤ 1, and is given by (39):

The $\Omega$’s are given in Table 1. When $z_1^{\text{'}}$ = 1090, we get the pure thermal model:

Eq. (40) gives < 0.1 ppm deviance from (37)’s manually calculated $H_g$’s for all z = 0→1089 (see Supplemental Material). It’s exact for any input H₀ and is a perfect expression of gas thermal behavior at scale. We just need to find the pure endpoint from (26) to get thermal $H$ values at lower z from (40).

When we eliminate CMB energy from (40) we get Newton’s Universe. The term ${\mathrm{ɱ}}^{\prime }=1.000$ for all z, giving $H_g^{\text{'}}$:

Here, the “pure endpoint” $r_{e_{1090}}$ is Newtonian. Eq. (41) also gives < 0.1 ppm deviance from (37). Since “$r_{e_{1090}}$” has increased, $H_g^{\text{'}}<H_g$.

This is the Newtonian version of the Friedmann¹² equation. It describes the behavior of expanding rocks and CDM in a vacuum, in Newton’s Universe, at their comoving critical density. The thermal (41) and Friedmann (42) equations give an invariant $H_g^{\text{'}}/H_{\Lambda }^{\text{'}}=1.09$ for all z. We can bring $H_g^{\text{'}}/H_{\Lambda }^{\text{'}}$ down to 1.00 with an accretion term.

Accretion is expressed with a mass partition ${\rho }^{*}$≤ 1, giving accreted thermal models. The mean gas density ${\rho }_g$ is divided by the mean baryon density ${\rho }_b$:

The mass partition removes accreted mass, its kinetic energy, and its proportion of CDM mass from (24)’s endpoint calculation. CMB energy, if present, is unaffected. This twinned “Universe” is both unaccreted and has a lower baryon density, so it expands more slowly. We connect it with its denser, mass-partitioned twin through ${\rho }^{*}$:

The pure endpoint must be used for ${\rho }^{*}$ to be properly expressed as a second independent variable. Eq. (44) again gives <0.1 ppm deviance from its less-dense twin’s manually found values (which have larger endpoints). Since we use the pure endpoint, the accreted baryons aren’t really missing from (44). They’re coasting alongside the expanding gas from which they formed, and exhibit Friedmann behavior.

When ${\rho }^{*}$ = 0.8418, for the entire domain $z$ = 0 to 1089, the thermal and Friedmann equations give identical results:

In other words, expanding rocks which are forever slowing to an eventual halt behave exactly a freely expanding gas forever pushing itself apart. The only thing the CMB does is increase the push, and that mostly happens way back near last scatter. The “rocks”, of course, are stars. In today’s intergalactic medium, entropic pressure makes its gas expand, separating the tendrils of the cosmic web. About 90% of our Universe’s present volume is occupied by this expanding gas, which contains 84% of all baryon mass. After around z = 9 or so, accretion into the cosmic web seems to have stabilized at 16% of the total. This 84/16 proportion probably hasn’t changed much since then; however, mass loss from nuclear fusion remains unaddressed.

Eqs. (41) – (45) deal with Newton’s Universe, where the CMB’s equivalent mass isn’t included. In Einstein’s Universe, they underestimate $H_g$. Also, CMB mass is unaffected by ${\rho }^{*}$.

The pure endpoint $r_{e_{1090}}$ re-includes CMB energy. The ${\mathrm{ɱ}}^{\prime }$ term now includes ${\rho }^{*}$:

Where $z_1^{\text{'}}$ = 1090. For ${\rho }^{*}$ = 0.8418, its ${\mathrm{ɱ}}^{\prime }$ range is 0.757 ≤ ${\mathrm{ɱ}}^{\prime }$ ≤ 1.046. We again find that (46) gives < 1ppm deviance from the manually found values (37) for our input H₀’s (see Supplemental Material).

The term (1-${\rho }^{*}$) is the accretion parameter. It’s the fraction of (baryon + CDM) mass that is gravitationally bound: Stars, planets, black holes, bound gases, etc. At scale, these accreted baryons act like rocks and comprise the cosmic web of galaxies with all of its gravity-bound behavior. How did accretion evolve? We may be able to answer this question through observation of stars at cosmic dawn.

For z > 9, “dark” energy is minimal (Ω_Λ < 0.002), and ${\rho }^{*}$ (43) might be estimable from the H values of any luminous bodies we are fortunate enough to see. The accuracy of ${\rho }^{*}$ depends on a proper value of H₀ for which the range of distance-ladder estimates remains large. What we can do is determine if observed H/H_Λ values deviate upwards as we go back in time from z = 9→50 or so. This in turn depends on whether or not star formation and accretion are coevolving phenomena. If accretion ends before starlight begins, then ${\rho }^{*}$= 0.84 applies and the value of H/H_Λ will remain close to one. However, if accretion and starlight coevolve, then H/H_Λ > 1 may be significant enough to measure. For example, the newly found galaxy JADES-GS-z13-0 [35,36] has a spectroscopic redshift of 13.2, so Ω_Λ = 0.0007, a minimal dark energy value. If ${\rho }^{*}$ = 0.84 at this redshift then $H_{g^{*}}$/H_Λ = 1.002, not significant. If ${\rho }^{*}$ = 0.9 then $H_{g^{*}}$/H_Λ = 1.037 which if true ought to be detectable. The theoretical upper limit of $H_{g^{*}}$/H_Λ at z = 13.2 is 1.09, shown in Figure 2. These ratios don’t change in z for any H₀, but if H₀ = 67.70 km/sec/Mpc is in fact a low estimate then observed ratios using it should start high (H/H_Λ = 1.139 for ${\rho }^{*}$=0.9) and get higher as z increases (if the Universe coevolves). This author believes that star formation and accretion do coevolve, and I predict upwardly-trending deviance of H from H_Λ in the z > 9 domain now accessible from the James Webb space telescope. If we can get reliable luminosity-based distance estimates from these faint bodies, and can find the proper H₀, we should be able to follow the progress of accretion through (1-${\rho }^{*}$).

Figure 2 gives a complete picture of the thermal model over the entire domain z = 0 to 1089. There are two domains of variance, overlapping at cosmic dawn, where both dark and CMB energies aren’t large. For z > 9, predicted variances $H_{g^{*}}$/H lie somewhere in the area between the pure and accreted curves in Figure 2. For z < 9, the variance of $H_{g^{*}}$/H due to added repulsive force is clearly shown and discussed in section 5 below.

Two numbers are worthy of mention. The dark-matter-to-baryon ratio ${\Omega }_c/{\Omega }_b$ is 5.311. The accretion ratio ${\rho }_g/\left(1-{\rho }_g\right)$ for z < 9 is 5.321. These differ by less than half a percent. Is this just a coincidence, or is there a causal connection?

None of the above expressions come any closer to explaining the source of the repulsive “dark energy” term Ω_Λ in the ΛCDM model. I propose that suprathermal kinetic energy in the IGM is responsible. It arises mostly from Compton¹³ scattering, reliant in turn on energetic photon flux $E_{\gamma }$. There are direct partial flux estimates available at the high end [37,38] but the process of connecting these and other sources to produce a definitive suprathermal model is an undertaking of considerable magnitude. The present paper is merely an introduction.

The suprathermal model is developed in Appendix D and expressed by (48):Where $U_{{\beta }_s}$ is electron suprathermal energy in the adiabatic sphere, $U_b$ is nucleon thermal energy, and $U_{{\beta }_s}/U_b$ is the suprathermal ratio.

Eq. (48) presupposes thermal plasma, a safe bet for a reionized Universe. Accretion is presumed complete after z = 9 so we use ${\rho }^{*}$ = 0.84. The suprathermal ratio $U_{{\beta }_s}/U_b$ is, like $H_{g^{*}}$, zero-order in T,¹⁴ so (48) is completely temperature-independent. We fit the suprathermal model to the ΛCDM model by convergence of $U_{{\beta }_s}/U_b$ around H/H_Λ = 1 for each datum. These results are shown in Figure 3. A ln-ln line is found¹⁵ for z = 0→2, giving (49):

At z = 0, $U_{{\beta }_s}/U_b$ = 2.2397 gives H/H_Λ = 1; the regressed value is 2.236. This is close to the ratio ${\Omega }_{\Lambda }/\left({\Omega }_b+{\Omega }_c\right)$ in the ΛCDM model, 2.235, and a simple restatement of the source of “dark energy” repulsion. A modest deviance from linearity in the plot occurs above z = 2; these are shown in Figure 3 but not included in the regression. The line is still very good out to z = 4. At z = 5 there’s deviance and $U_{{\beta }_s}/U_b$ drops to 0.01. The crossover to $U_{{\beta }_s}$dominance is found at z = 0.309. Eq. (49) derives from ΛCDM and accordingly gives the suprathermal ratio as proportionate to ${\left(z^{\prime }\right)}^{-3}$.

Inserting (49) into (48) gives:

Using the same data, a ln-ln regression of r_e vs. z’ gives (51):¹⁶

Where r₀ = r_e at z = 0. From (51) we see that the adiabatic volume V_e/V₀ varies as ${\left(z^{\prime }\right)}^{\frac{-9}{2}}$, not ${\left(z^{\prime }\right)}^{-3}$. A constant “dark energy density” expressed as $\left[U_{{\beta }_s}/U_b\right]/{\left(z^{\prime }\right)}^3$ might be accurate, but within GCDM, isn’t proper.

From (48) – (51) we arrive at an expression of H for z = 0 to 2:When H₀ = 74.40 km/sec/Mpc, the thermal $H_{g_0^{*}}$ = 1.339 x 10^-18 sec^-1 (at z = 0). Eq. (52) gives 100.00%→99.90% of the ΛCDM value (19) for z = 0 to 2. The exponents in (49) – (51) are rounded to the nearest fraction. The rounding excises CMB density from r_e/r₀ (51), giving some of the observed -0.1% deviance at z = 2. When (50) is properly expressed with (46), the error drops considerably, only reaching 0.1% at z = 9. (see Supplemental Material).

From (26), (46), (47), and (50), a full expression of H can be given:Which contains two independent variables: a (=1/[z+1]), and ρ* ≤ 1. For z < 9, ρ* = 0.84, and Eq. (53) agrees with ΛCDM (19) (±0.1%). For z > 9, ρ* rises, giving tension with (19). At z = 1089, ρ* = 1, and (53)/(19) = 1.25.

At z = 0, the models converge:

If we include thermal free electron energy $U_{{\beta }_t}$ in the denominator of (54), we get $\frac{U_{{\beta }_s}}{\left(U_b+U_{{\beta }_s}+U_{{\beta }_t}\right)}=$ 0.53, still more than half of all kinetic energy in the IGM today.

The suprathermal model gives a “pumped Universe” scenario. In a pumped Universe, the intergalactic medium is continually fed with suprathermal energy $U_{{\beta }_s}$. This energy persists and accumulates. Most of it comes from electrons which in turn derive their energy from photons produced by nuclear fusion within the cosmic web of galaxies. Compton scattering imparts kinetic energy to the electrons. Much of that energy is well above the nucleons’ thermal range given by the Boltzmann curve, so the Universe isn’t just getting hotter. Its kinetic energy is increasing above and beyond simple thermal heat. This adds a tremendous amount of entropic force d(TS)/dr.

We should be able to connect suprathermal energy generation with the small number of known sources which are capable of producing $E_{\gamma }$ photons: Type “O” stars and active galactic nuclei. Maybe additional scattering of free electrons by the lower-energy photons emitted by all stars today gives net addition enough to account for the “dark energy” effect. If $U_{{\beta }_s}$ persists indefinitely, then $d\left(U_{{\beta }_s}\right)/dt$ can be estimated from stellar photon flux and lookback. Those calculations involve several other assumptions not covered here and lie outside the scope of the present paper.

By z ≈ 6 the Universe was fully reionized, so Compton scattering from neutral IGM atoms isn’t a significant source of suprathermal energy after that. A second source is Compton scattering of free electrons in the IGM, which would taper off to a steady state if the energy profiles of the electrons and photons were to converge. Cosmic radiation $U_{b_s}$ from the web is a third source. The reader is asked to consider a fourth source, kinetic energy of electron escape from the web, as a present-day contributor to the IGM’s suprathermal content. Any resulting IGM charge buildup is presently untreated. This author estimates these electrostatic effects as minor compared to $U_{{\beta }_s}$.

Suprathermal electrons do not behave like thermal baryons, and their kinetic gain hasn’t been properly addressed. Thermal free electrons are treated as a gas (appendix D) which seems proper, but I’m not so sure if it’s accurate. This author is unaware of any treatments involving electron kinetic gain. Electron thermal energy may partly comprise $U_{{\beta }_s}$. What we can say is that electron kinetic energy in general only minimally affects nucleon endpoint values. The presumption that v_e/v_i = K remains constant with added suprathermal energy is conjecture, but it appears to work well. For special effects, any relativistic mass increase would cause a decrease in r_e. The entropic partition (2/3) and K may also change with introduction of highly relativistic kinetic energy. I believe such changes in K and r_e aren’t large. Even if K/r_e is theoretically shown to change by as much as 1%, the suprathermal model still allows us to use known and conserved energy sources, in compliance with (1), to account for H.

The question of Universal curvature is historically important and extensively treated in modern introductory texts [39,40]. That debate is largely settled now. Most of us believe in a flat Universe, so the Friedmann equation can be simply expressed:

The Friedmann equation overlooks the fact that most Universal baryon mass is gaseous. Instead of the thermal model’s perpetual entropic pressure, we have ΛCDM’s rocks at a “miraculous” critical density. While the Friedmann equation (43) may not properly express entropic gain, it can be shown as identical to thermal behavior (45) through ρ*.

Friedmann employed Einstein’s field equation which makes no provision for entropic gain. Einstein invented the Λ force to offset gravity, as the Universe was thought to be static at the time, and he had to do something to prevent its collapse. Both Friedmann’s and Einstein’s entropic omissions were benign. It just didn’t occur to either author to include entropy. Einstein’s later doubts about Λ are well known [41].

The fluid equation is a different story. It actively excludes entropic gain. We start, as do the texts, with the bound Gibbs equation (5) which describes e.g. gas in a vessel:

Over time, this is:When applied to the unbound Universe, dE/dt = 0. We include rest mass equivalence E = Mc². Nuclear fusion. The unbound Gibbs equation (10) is now:

At last scatter there was no fusion, so M was unchanged, $d\left(M{c}^2\right)$ = 0, and (57) = (10). Photon energy from the CMB isn’t included here and is discussed in more detail below (section 6.3.2). It’s unimportant for now.

The fluid equation’s development continues with heat flow:

This relates thermal change dQ with the entropy change dS in a vessel. If there’s no vessel, we get the Universe. We will assume there’s no heat flow in or out of the Universe. This assumption gives rise to a philosophical discussion about the meaning of the first law vs. unbound thermal loss. We will sidestep that entire debate and can properly treat lost kinetic energy as entropic gain.

The second law (2) is clear about entropy: dS/dt > 0 always. At scale, we can get a rough estimate of gas dS/dt from the thermal model (37) at a constant T and P by setting dE/dt = 0 in (56). For H₀ = 74.40 km/sec/Mpc and z = 1089, this gives:

= 106 (J/K)/sec/kg

The fluid equation sets Universal dS/dt = 0 despite the second law. Baryon mass is treated as a pressureless perfect fluid, called a “dust”, which is inconsistent with its actual existence as a gas having entropic pressure. Entropic gain is eliminated, and all energy change is isoentropic. Eq. (56) now looks like this:Which describes gas in an adiabatic vessel. The term -dE/dt = -d(U_i)/dt describes its rate of thermal loss. Setting dS = 0 means that all the loss is reversibly stored, and in the unbound Universe, there’s only one place this author knows of to store it: work against gravity. I’ve shown that only a third of this thermal loss is stored that way. The remaining two-thirds vanishes, and requires entropic power d(TS)/dt to properly describe its fate. Entropic power gives over time an irreversible loss of thermal energy, and a reduction in density which acts as a thermodynamic repository or “sink”. If entropy is excised from the Gibbs equation, the lost thermal energy cannot be accounted for, giving inconsistency with (1).

There is another significant issue with the fluid equation which only became relevant after 1998. Internal energy change dE gets improperly redefined in the fluid equation’s development. It isn’t thermal anymore, it’s total, and now includes rest mass dE = d(Mc²). Eq. (60) is then used to treat the energy change of rest mass d(Mc²) as a thermal variable. Rest mass does not change thermally, it stays constant.¹⁷ From (1), dE/dt in (60) should be zero. Conflation of the dE terms can thus appear to create an enormous amount of what is fictitious repulsive energy from what is actually a much smaller amount of suprathermal energy. In this author’s opinion, that’s what happened when the distance ladder line was found to be a curve by the seminal observations of the two research groups who first published their findings in 1998 [4],[6]. They accounted for the distance-ladder curvature by populating Λ with fictitious negative mass. The reader may wish to consider the alternative, suprathermal energy, and the possibility that the source of Λ cannot be identified because it doesn’t exist.

Further development of the fluid equation is in the texts. The result is the same for both Einsteinian and Newtonian versions, which differ only by c². The Newtonian expression of the fluid equation is:In (61), “${\rho }_{M^{\prime }}$” includes ${ϵ}_{\Lambda }/c^2$. The other terms in “P/c²” are equivalent mass density of energy not at rest, so if we exclude Λ, (61) is an attractive term. The energy density term “P” is thus also attractive and labelled as “pressure”:

The w terms are dimensionless numbers whose values are expressed using a: w_b << 1, w_λ = 1/3, and w_Λ = -1. This definition of “pressure P” is known as the equation of state. It’s the third leg of ΛCDM and is discussed below. The equation of state inverts the meaning of pressure. Most of us think of positive pressure as repulsive, like in a balloon. Not here. Repulsive energy density $w_{\Lambda }{ϵ}_{\Lambda }$ in (62) is called “negative pressure”.

When we talk about the meaning of pressure, the Jeans model of star formation [30] is relevant. Both (62) and the Jeans model operate concurrently within any given volume. The Jeans model treats P as repulsive, an offset against gravitational collapse. The fluid and state equations treat “P” as attractive, which is inconsistent with the Jeans model. The GCDM model treats P as repulsive, which is consistent with the Jeans model’s treatment of P.

The equation of state (62) is combined with the Friedmann (55) and fluid (61) equations to complete the ΛCDM model (19).

An .XLSX workbook containing the model and its output is available.

We connect atomic movement to kinetic gain on a small scale, where work against gravity d(PV) can be neglected.

We start with the increment radial velocity $v_{s_{\Delta r}}$:

We follow $v_{s_{\Delta r}}$ as a function of r, using a fixed increment $\frac{\mathsf{\Delta }{r}_i}{r}$ = 10^-9. Below the cutoff radius r_c = 0.003 r_e, loss to gravity is negligible, and all these small spheres have the same $E_{k_{\mathsf{\Delta }r}}/M$ value to within 5 ppm (Figure B1):

Here we used dE =PdV. Combining (27) and (B1) gives an initial radial velocity v_i:

We can determine if energy is conserved within (B2) by examining the partition error (B3):

We increment a small sphere (r₁ = 1 x 10¹² m) giving P₂ and T₂. The pressure drop from thermal loss of our gas is given by:

And the temperature drop by:

Which for (B2) gives a large error. Development in (B1) uses the ideal gas law and not thermal energy. We rearrange (A1):

Substituting $\frac{U_i}{M}$ for $\frac{E_k}{M}$ in (B2) gives (29):

By use of (29) the partition error (B3) is only 2 x 10^-8.

The redshift z₂ can be obtained from any starting value z₁ by enlarging the increment $\frac{\Delta r_i}{r}$:

The temperature is found from (B4) and (B5) using (C1) to get V₂. Given T_(z’=1090) = 2971K, we find from the spreadsheet that $T_{z^{\prime }}=T_{\left(z+1\right)}$ is exactly:²²

Where the root temperature $T^{\prime }$ = 0.002500631 is expressed as degrees K. If treated as an abrupt decoupling at z = 1089, then at z = 60, T = 9.3K; at z = 10, T = 0.3K. These are lower than other estimates [42] but since H_g is temperature-independent, dark thermal heating by the CMB doesn’t affect H. We just need the root temperature.

We insert (C2) into (29), giving:

The endpoint r_e is adjusted for both T and ρ.

For the T adjustment, the thermal radius change $r_{e_2}/r_{e_1}$ vs. T at constant ρ and Ж is found exactly:

For the ρ adjustment, the thermal radius change $r_{e_2}/r_{e_1}$ vs. ρ at constant T and Ж is found exactly:

For nonrelativistic mass,

Combining these gives:

Inserting (C3) and (C7) into the thermal model (37) gives Newton’s thermal model (C8):

We adjust for relativistic mass with ${\mathrm{ɱ}}^{\prime }$ (39):

A linear dependence of H on $\mathrm{ɱ}\text{'}$ is found from the spreadsheet, giving the thermal model in Einstein’s Universe (38):

When $z_1^{\text{'}}$ = 1090, the pure endpoint $r_{e_{1090}}$ applies, and we get the pure thermal model (40).

Suprathermal kinetic energy adds entropic pressure to the IGM:Where $P_{V_t^{\text{'}}}$ and $P_{V_s^{\text{'}}}$ are thermal and suprathermal pressures respectively.

The thermal model (37) has three terms: K, v_i, and r_e. If we want to express suprathermal content within that framework, we need to increase v_i or K, decrease r_e, or some combination. We express v_i (29) as a sum:Where $U_{i_t}$ and $U_{i_s}$ are thermal and suprathermal kinetic energies in the adiabatic sphere.

The total nucleon kinetic energy U_b is:Where $U_{b_t}$ is the thermal value of U_b, and $U_{b_s}$ is cosmic radiation.

The total electron kinetic energy $U_{\beta }$ is:Where $U_{{\beta }_b}$ is the thermal energy of atomically bound electrons, $U_{{\beta }_t}$ is the thermal energy of free electrons, and $U_{{\beta }_s}$ is their suprathermal energy.

Inserting (D3) and (D4) into (D2) gives:

We neglect cosmic radiation $U_{b_s}$ for now as its omission doesn’t substantially affect the logic of the following expressions. This means $U_{b_t}=U_b$, so:

We examine the thermal energies $U_{{\beta }_b}$ (bound) and $U_{{\beta }_t}$ (free). Thermal free electrons are held to behave at very low densities as a monatomic gas. Treatment as such reduces the mean atomic weight Ж. The thermal model is independent of both Ж and T and dependent only on the mean gas density ρ_g. The result of thermal ionization is thus an increase in both v_i and r_e without affecting H or K. If v_i is doubled, so is r_e, as is the case with pure hydrogen plasma which will serve as our example.

In a thermal system with no ionized H₁: so $U_i=U_b$ is reasonably accurate. When H₁ is 100% ionized at e.g. 4000K, the number of gas particles is doubled, the atomic weight halved, and energy equipartitioned: $U_{{\beta }_b}=0,$ $U_{{\beta }_t}=U_b$, and Ж’ = ½Ж, where Ж’ is the mean atomic weight of the plasma. Making these substitutions into (D6) with no $U_{{\beta }_s}$ gives:

The added $U_{{\beta }_t}=U_b$ gives twice the old value of v_i from (29); more generally, added $U_{{\beta }_t}$ gives a linear increase in v_i and we can expect the same for r_e. This means that for thermal plasmas, the thermal model (37) is more properly expressed using the nucleon kinetic energy $U_b$ alone:Where v_i and r_e have their non-ionized values. The denominator term associated with r_e in (D9) is inserted to comply with the thermal model’s zero-order dependencies. In (D9), v_i remains close to (29): U_b ≈ 0.9995U_i, so the effect of thermal ionization on H is minimal, and we can exclude $U_{{\beta }_t}$ from the thermal model.