Thermal Inversion Formulas Overview

• Thermal inversion formulas are analytical techniques that recover hidden thermal structures from indirect observables in diverse physical systems.
• They integrate methods from LTE/NLTE spectral modeling, kinetic theory, conformal field theory, and backprojection for applications across astrophysics and tomography.
• These formulas offer practical diagnostic tools for mapping atmospheric inversions, non-equilibrium plasmas, and tomographic reconstructions through precise quantitative criteria.

“Thermal inversion formulas” denotes a heterogeneous but technically coherent family of constructions used either to diagnose temperature profiles that increase with altitude or dilution, or to recover thermal structure from indirect observables. In the literature surveyed here, the term spans LTE and NLTE inversion of solar Ca II spectra into temperature stratifications, radiative criteria for atmospheric inversions in irradiated planets, kinetic formulas for anticorrelated density–temperature profiles in collisionless plasmas and long-range interacting systems, finite-temperature OPE inversion formulas in conformal field theory, and explicit backprojection formulas in thermoacoustic/photoacoustic tomography (Beck et al., 2012, Zilinskas et al., 2020, Teles et al., 2015, Petkou et al., 2018, Antipov et al., 2011).

1. Terminological scope and core diagnostics

In planetary-atmosphere work, a thermal inversion is diagnosed by a temperature increase with altitude. In pressure variables this is written as

$$\frac{dT}{dz} > 0 \qquad\text{or equivalently}\qquad \frac{dT}{d\ln p} < 0,$$

within the photospheric layers (Zilinskas et al., 2020). The same criterion is used for ultra-hot Jupiters, where an inversion in the line-forming region implies that temperatures rise toward smaller optical depth and can generate emission features through an Eddington–Barbier argument (Rajpurohit et al., 2020).

In kinetic theories of non-equilibrium matter, the defining signature is instead an anticorrelation between temperature and density. For a phase-space distribution $f(x,v,t)$, the local fields are

$$\rho(x,t)=\int f(x,v,t)\,dv, \qquad T(x,t)=\frac{1}{d}\frac{\int v^2 f(x,v,t)\,dv}{\int f(x,v,t)\,dv},$$

and inversion may be expressed as $dT/d\rho<0$ or by a negative Pearson correlation $C=\mathrm{Corr}[T(x),\rho(x)]<0$ (Teles et al., 2015). In gravitationally confined collisionless plasmas, the same phenomenon is written as $\partial_z T(z)>0$, $\partial_z n(z)<0$, and $\nabla n\!\cdot\!\nabla T<0$ (Barbieri et al., 2024).

In solar spectroscopy, “thermal inversion” refers not to a temperature inversion in the atmosphere, but to an inverse problem: the retrieval of a temperature stratification $T(\tau)$ from an observed spectrum. The forward relation is the 1D radiative transfer equation,

$$\frac{dI_\nu}{d\tau_\nu}=I_\nu-S_\nu,\qquad S_\nu=B_\nu(T),$$

and the inversion uses response functions $f(x,v,t)$0 to map spectral residuals into corrections of $f(x,v,t)$1 (Beck et al., 2012).

A formally different usage appears in finite-temperature CFT. There, the inversion formula reconstructs spectral data $f(x,v,t)$2 from a thermal two-point function, with poles of $f(x,v,t)$3 encoding operator dimensions and thermal OPE coefficients (Petkou et al., 2018). In thermoacoustic/photoacoustic tomography, the inverse problem is to reconstruct a function from spherical means measured on the boundary of a ball in Euclidean, spherical, or hyperbolic geometry (Antipov et al., 2011). This suggests that the common thread is not a single physical mechanism but a shared mathematical motif: explicit formulas that map thermal or thermally generated observables into hidden structure.

2. Solar spectral inversions of thermal stratification

For Ca II H, the LTE inversion framework of Beck et al. is based on the SIR code and a two-step procedure. The forward model assumes a 1D plane-parallel atmosphere, complete frequency redistribution, and LTE source function $f(x,v,t)$4. Emergent intensity is written as

$f(x,v,t)$5

and small perturbations satisfy

$f(x,v,t)$6

The inversion first matches the observed spectrum against an archive of about $f(x,v,t)$7 pre-calculated LTE Ca II H profiles, using

$f(x,v,t)$8

with $f(x,v,t)$9, doubled weight in the core window 396.81–396.89 nm, and masked strong blends (Beck et al., 2012).

The archive is parameterized by perturbations to a modified HSRA atmosphere: Gaussian temperature perturbations in $$\rho(x,t)=\int f(x,v,t)\,dv, \qquad T(x,t)=\frac{1}{d}\frac{\int v^2 f(x,v,t)\,dv}{\int f(x,v,t)\,dv},$$0, global offsets $$\rho(x,t)=\int f(x,v,t)\,dv, \qquad T(x,t)=\frac{1}{d}\frac{\int v^2 f(x,v,t)\,dv}{\int f(x,v,t)\,dv},$$1 K, and additional straight-line slopes. Its depth grid spans 75 points from $$\rho(x,t)=\int f(x,v,t)\,dv, \qquad T(x,t)=\frac{1}{d}\frac{\int v^2 f(x,v,t)\,dv}{\int f(x,v,t)\,dv},$$2 to $$\rho(x,t)=\int f(x,v,t)\,dv, \qquad T(x,t)=\frac{1}{d}\frac{\int v^2 f(x,v,t)\,dv}{\int f(x,v,t)\,dv},$$3, and macroturbulence is fixed at $$\rho(x,t)=\int f(x,v,t)\,dv, \qquad T(x,t)=\frac{1}{d}\frac{\int v^2 f(x,v,t)\,dv}{\int f(x,v,t)\,dv},$$4 to reproduce observed blend widths. The second stage refines the wing fit through a response-based temperature update,

$$\rho(x,t)=\int f(x,v,t)\,dv, \qquad T(x,t)=\frac{1}{d}\frac{\int v^2 f(x,v,t)\,dv}{\int f(x,v,t)\,dv},$$5

followed by a smooth fourth-order polynomial fit in $$\rho(x,t)=\int f(x,v,t)\,dv, \qquad T(x,t)=\frac{1}{d}\frac{\int v^2 f(x,v,t)\,dv}{\int f(x,v,t)\,dv},$$6 and repeated synthesis. Convergence typically occurs in 2–3 iterations, and no Levenberg–Marquardt step is used (Beck et al., 2012).

This LTE Ca II H workflow achieves rms deviations of about $$\rho(x,t)=\int f(x,v,t)\,dv, \qquad T(x,t)=\frac{1}{d}\frac{\int v^2 f(x,v,t)\,dv}{\int f(x,v,t)\,dv},$$7 of $$\rho(x,t)=\int f(x,v,t)\,dv, \qquad T(x,t)=\frac{1}{d}\frac{\int v^2 f(x,v,t)\,dv}{\int f(x,v,t)\,dv},$$8 in the wings and about $$\rho(x,t)=\int f(x,v,t)\,dv, \qquad T(x,t)=\frac{1}{d}\frac{\int v^2 f(x,v,t)\,dv}{\int f(x,v,t)\,dv},$$9 of $dT/d\rho<0$0 in the core. It reproduces quiet-Sun line cores and, after refinement, the wings of both quiet-Sun and active-region spectra, but it remains limited by 1D LTE, fixed disk-center $dT/d\rho<0$1 mapping, and an empirical post-facto velocity treatment. LOS velocities are assigned to fixed optical-depth anchors using Cr I 396.37 nm, Fe I 396.61 nm, Fe I 396.93 nm, and the Ca II H core, then interpolated to form $dT/d\rho<0$2 (Beck et al., 2012).

A related but faster LTE archive method was developed for Ca II 854.2 nm. That code uses about 240,000 precomputed spectra, minimizes the same nearest-neighbor $dT/d\rho<0$3 mismatch over the observed wavelength grid, and returns the associated $dT/d\rho<0$4 without solving for nodes during inversion. On a 4-core 2.4 GHz machine, the reported performance is about $dT/d\rho<0$5 s per SPINOR profile and about $dT/d\rho<0$6 s per IBIS profile; on a 32-core 2.7 GHz server, about $dT/d\rho<0$7 s per profile is reported for a single job (Beck et al., 2014).

Because LTE increasingly underestimates chromospheric temperatures above $dT/d\rho<0$8, the same study derives an empirical multiplicative correction

$dT/d\rho<0$9

with $C=\mathrm{Corr}[T(x),\rho(x)]<0$0 up to $C=\mathrm{Corr}[T(x),\rho(x)]<0$1, rising to $C=\mathrm{Corr}[T(x),\rho(x)]<0$2 at $C=\mathrm{Corr}[T(x),\rho(x)]<0$3 in the quiet Sun and to $C=\mathrm{Corr}[T(x),\rho(x)]<0$4 at $C=\mathrm{Corr}[T(x),\rho(x)]<0$5 in the umbra. This first-order correction was inferred by comparing the LTE archive inversion to NICOLE NLTE inversions of the same SPINOR spectra (Beck et al., 2014).

3. Atmospheric and interfacial thermal inversion criteria

For irradiated nitrogen-rich super-Earths, the central inversion criterion is the shortwave-to-longwave opacity ratio

$C=\mathrm{Corr}[T(x),\rho(x)]<0$6

Within the HELIOS 1D radiative–convective equilibrium framework, CN and, at the hottest temperatures with sufficient hydrogen, CH enhance $C=\mathrm{Corr}[T(x),\rho(x)]<0$7 in the visible–near-IR. Consistent with Hubeny/Guillot-style reasoning, inversions occur when $C=\mathrm{Corr}[T(x),\rho(x)]<0$8 in irradiated layers with $C=\mathrm{Corr}[T(x),\rho(x)]<0$9. The RCE solutions show that for $\partial_z T(z)>0$0 K and $\partial_z T(z)>0$1, CN can raise $\partial_z T(z)>0$2 above unity through the photosphere and produce inversions extending from about $\partial_z T(z)>0$3–$\partial_z T(z)>0$4 bar, and in carbon-rich, hydrogen-poor cases even to the surface (Zilinskas et al., 2020).

The associated heating term is written as

$\partial_z T(z)>0$5

with inversion expected when $\partial_z T(z)>0$6 over the relevant pressure range. A practical abundance criterion extracted from the models is that photospheric $\partial_z T(z)>0$7–$\partial_z T(z)>0$8 favors inversion. Spectroscopically, the inverted cases show near-blackbody behavior at high temperature, enhanced shortwave emission below about $\partial_z T(z)>0$9–$\partial_z n(z)<0$0m, and modeled inverted–non-inverted contrasts exceeding $\partial_z n(z)<0$1 ppm over $\partial_z n(z)<0$2–$\partial_z n(z)<0$3m in some cases (Zilinskas et al., 2020).

For WASP-19b, the PHOENIX BT-Settl line-by-line, spherically symmetric calculations diagnose a dayside inversion above about $\partial_z n(z)<0$4 and at pressures near $\partial_z n(z)<0$5–$\partial_z n(z)<0$6 bar. The analytic guide is again the ratio

$\partial_z n(z)<0$7

with large $\partial_z n(z)<0$8 implying shortwave absorption above the IR photosphere. In the modeled dayside atmosphere, TiO/VO and alkalis provide optical absorption, CO remains the dominant IR opacity, and $\partial_z n(z)<0$9 partially dissociates at pressures below $\nabla n\!\cdot\!\nabla T<0$0 bar. The predicted observational consequence is a CO emission feature near $\nabla n\!\cdot\!\nabla T<0$1m in the secondary-eclipse spectrum (Rajpurohit et al., 2020).

A distinct fluid-mechanical inversion problem arises for evaporation and condensation between two parallel plates. The vapor energy flux is

$\nabla n\!\cdot\!\nabla T<0$2

and the interfacial temperature jumps are governed by kinetic-theory boundary conditions. The paper proves that the jumps always have the same direction as the externally imposed wall-temperature difference:

$\nabla n\!\cdot\!\nabla T<0$3

Its rederived inversion criterion is

$\nabla n\!\cdot\!\nabla T<0$4

which implies vapor-temperature inversion. For water vapor near ambient conditions, $\nabla n\!\cdot\!\nabla T<0$5, so inversion is always predicted in the regime studied (Chen, 2023).

4. Collisionless and long-range systems

In long-range interacting systems, temperature inversion is a nonequilibrium property of quasi-stationary states generated after an impulsive perturbation. The collisionless dynamics is governed by the Vlasov equation

$\nabla n\!\cdot\!\nabla T<0$6

and the local fields are defined by velocity moments of $\nabla n\!\cdot\!\nabla T<0$7. In the Hamiltonian Mean Field model,

$\nabla n\!\cdot\!\nabla T<0$8

a short external kick lowers the magnetization and drives the system into an inhomogeneous Vlasov-stationary QSS. The same mechanism appears in a two-dimensional self-gravitating system. In both cases, inversion is attributed to resonant wave–particle interaction, Landau damping, and velocity filtration: suprathermal particles preferentially populate dilute regions, so $\nabla n\!\cdot\!\nabla T<0$9 becomes higher where $T(\tau)$0 is lower (Teles et al., 2015).

The confined-plasma formulation of Barbieri et al. makes this mechanism explicit for a loop-like gravitational atmosphere with a fluctuating thermal boundary. In dimensionless loop variables $T(\tau)$1, the coarse-grained stationary distribution for species $T(\tau)$2 is

$T(\tau)$3

with

$T(\tau)$4

because the stationary self-consistent electrostatic field vanishes, $T(\tau)$5 (Barbieri et al., 2024).

After Gaussian momentum integration, one obtains closed profiles

$T(\tau)$6

and

$T(\tau)$7

In height coordinates $T(\tau)$8, these are rewritten in terms of Laplace-type transforms $T(\tau)$9 and $$\frac{dI_\nu}{d\tau_\nu}=I_\nu-S_\nu,\qquad S_\nu=B_\nu(T),$$0. For any $$\frac{dI_\nu}{d\tau_\nu}=I_\nu-S_\nu,\qquad S_\nu=B_\nu(T),$$1 and any $$\frac{dI_\nu}{d\tau_\nu}=I_\nu-S_\nu,\qquad S_\nu=B_\nu(T),$$2 with support on $$\frac{dI_\nu}{d\tau_\nu}=I_\nu-S_\nu,\qquad S_\nu=B_\nu(T),$$3, the model predicts $$\frac{dI_\nu}{d\tau_\nu}=I_\nu-S_\nu,\qquad S_\nu=B_\nu(T),$$4 and $$\frac{dI_\nu}{d\tau_\nu}=I_\nu-S_\nu,\qquad S_\nu=B_\nu(T),$$5 (Barbieri et al., 2024).

The same framework is applied to the solar corona in a semicircular loop model with intermittent heating at the chromospheric footpoints. There the stationary distribution is a convex mixture of Boltzmann factors at $$\frac{dI_\nu}{d\tau_\nu}=I_\nu-S_\nu,\qquad S_\nu=B_\nu(T),$$6 and $$\frac{dI_\nu}{d\tau_\nu}=I_\nu-S_\nu,\qquad S_\nu=B_\nu(T),$$7, transported by Liouville dynamics in the gravitational potential. With solar-like parameters, the reported profiles rise from a base temperature near $$\frac{dI_\nu}{d\tau_\nu}=I_\nu-S_\nu,\qquad S_\nu=B_\nu(T),$$8 K to coronal values near $$\frac{dI_\nu}{d\tau_\nu}=I_\nu-S_\nu,\qquad S_\nu=B_\nu(T),$$9 K, while density decreases strongly through the transition region and more gently higher up. A stated implication is that a million-Kelvin solar corona can be produced without local heat deposition in the upper atmosphere (Barbieri et al., 2023).

5. Finite-temperature CFT and thermal OPE inversion

In odd-dimensional CFT on $f(x,v,t)$00, the thermal inversion formula reconstructs operator spectrum and OPE data directly from the thermal two-point function. Writing the Euclidean thermal correlator as $f(x,v,t)$01, one introduces a spectral function $f(x,v,t)$02 whose poles in complex $f(x,v,t)$03 encode operator dimensions and residues. The Euclidean inversion formula is

$f(x,v,t)$04

with

$f(x,v,t)$05

A Lorentzian-like analytic continuation then splits the answer into discontinuity and arc terms,

$f(x,v,t)$06

which is the thermal analogue of Caron-Huot-style inversion logic (Petkou et al., 2018).

Odd $f(x,v,t)$07 admits closed kernels in terms of Bessel polynomials and leads to explicit gap equations for thermal masses. In the bosonic case,

$f(x,v,t)$08

and in the fermionic case,

$f(x,v,t)$09

These equations implement the cancellation of residues associated with towers such as $f(x,v,t)$10 or $f(x,v,t)$11, thereby enforcing the absence of an infinite set of operators from the thermal spectrum (Petkou et al., 2018).

The reported solution pattern is dimension-dependent and often complex. For bosons, $f(x,v,t)$12 yields the real solution

$f(x,v,t)$13

whereas $f(x,v,t)$14 yields a complex-conjugate pair $f(x,v,t)$15. For fermions, $f(x,v,t)$16 gives purely imaginary solutions $f(x,v,t)$17. The paper interprets the pattern of real and complex thermal masses as an indicator of the large-$f(x,v,t)$18 vacuum structure at zero temperature, with real solutions associated with stable thermal vacua and complex solutions with metastable extrema carrying decay widths (Petkou et al., 2018).

6. Inversion formulas in thermoacoustic and photoacoustic tomography

A separate but mathematically related line of work studies inversion formulas for thermoacoustic/photoacoustic tomography on spaces of constant curvature. The inverse problem is: reconstruct a function $f(x,v,t)$19 supported in an $f(x,v,t)$20-dimensional ball $f(x,v,t)$21 if the spherical means of $f(x,v,t)$22 are known over all geodesic spheres centered on $f(x,v,t)$23. In Euclidean space,

$f(x,v,t)$24

with analogous formulas on $f(x,v,t)$25 and $f(x,v,t)$26 (Antipov et al., 2011).

The spherical means satisfy the Euler–Poisson–Darboux equation

$f(x,v,t)$27

where $f(x,v,t)$28 is $f(x,v,t)$29, $f(x,v,t)$30, or $f(x,v,t)$31 and $f(x,v,t)$32 is the curvature-adapted radial function. The paper’s central device is analytic continuation in a parameter $f(x,v,t)$33 for a family of operators $f(x,v,t)$34, combined with a backprojection operator $f(x,v,t)$35, evaluated at the critical value $f(x,v,t)$36 (Antipov et al., 2011).

In Euclidean odd dimensions, the resulting filtered backprojection formula is

$f(x,v,t)$37

where

$f(x,v,t)$38

In even dimensions, the inversion acquires a logarithmic kernel,

$f(x,v,t)$39

Corresponding explicit formulas are given for $f(x,v,t)$40 and $f(x,v,t)$41, with geometric weights such as $f(x,v,t)$42 in the spherical case and $f(x,v,t)$43 in the hyperbolic case (Antipov et al., 2011).

Here the adjective “thermal” enters through thermoacoustic/photoacoustic tomography rather than through a temperature inversion of the medium itself. Even so, the formal resemblance to the other usages is notable: an explicit inversion operator acts on thermally generated wave data, reconstructing otherwise inaccessible structure from boundary measurements (Antipov et al., 2011).