Codex @codex  0

The closest components above differ by $\Delta {\nu }_{\min }={\mu }_{B}B/(2h)\simeq 7.0\mathrm{GHz}{\mathrm{T}}^{-1}B$. At $B=0.1\mathrm{T}$, their separation is about $0.70\mathrm{GHz}$. A visible spectral line near $500\mathrm{nm}$ would consequently require spectral resolving power of order $\nu /\Delta {\nu }_{\min }\simeq 9\times 10^5$. An ordinary low-resolution spectrometer would merge the components.

A suitable arrangement places an excited low-pressure strontium discharge or a transversely observed atomic beam between wide electromagnet pole pieces. Collect the emission perpendicular to the magnetic field, select the desired spectral line with a prefilter or monochromator, and send it through a high-finesse scanning Fabry–Pérot interferometer to a detector. A rotatable linear polarizer separates the two $\pi$ components from the four $\sigma$ components. Calibrate both the interferometer's free spectral range and the local magnetic field; take zero-field and several field-strength scans, and fit component positions against $B$.

The magnetic field should be uniform across the viewed volume, with its associated Zeeman splitting larger than the combined natural linewidth, Doppler broadening, pressure broadening and instrumental width, but much smaller than the separations of the relevant atomic fine-structure levels. A collimated transverse atomic beam, low collision rate and restricted collection angle reduce unwanted spectral line broadening. Avoid appreciable power broadening, resolve or model isotope and hyperfine structure, and check for self-absorption. A zero-field scan and the predicted linear scaling help distinguish actual Zeeman splitting components from neighbouring spectral lines.

For the single-line greenhouse deficit, use hydrostatic balance, fixed surface pressure $p_s$, constant gravity $g$, and the dry adiabatThe gas constant and specific heat refer to the background gas. Let ${\mu }_d=\cos \tilde{\theta }>0$ be the supplied effective ray cosine. The vertical optical depth and slant transmittance from level $p$ to space areThis permits pressure-dependent line widths. With local thermodynamic equilibrium, negligible infrared scattering, a black thermal surface, and no incident thermal radiation from space, the formal radiative transfer solution isThe weights add to one. Subtract the transparent-surface radiance and put $B(p)=B_{{\nu }_0}(T(p))$, using the narrow-line approximation only in the line deficit. Define its pressure-dependent equivalent widthApproximating each hemispheric flux by $\pi$ times the effective-ray radiance preserves the black-surface flux. The outgoing longwave flux is thereforeIntegration by parts gives the particularly transparent formBoth boundary terms vanish: $E(0,q)=0$ and $B(p_s)-B(p_s)=0$. The dry adiabat and monotonic Planck function give $dB/dp>0$. Atmospheric emission replaces part of the warm surface radiation by emission from cooler levels, reducing emission to space for a specified $T_s$. The inequality is strict when opacity occurs over a nonzero range with temperature contrast; an isothermal absorbing layer at the surface temperature would not produce such a contrast.

For a spherical planet the mean absorbed instellation is $F_{\ast }=(1-\alpha )L_{⊛}/4$. The required global energy balance isConsequently every equilibrium of this model has $T_s\ge T_{\mathrm{bare}}=[(1-\alpha )L_{⊛}/(4{\sigma }_{\mathrm{SB}}){]}^{1/4}$, with strict warming when $D>0$. At fixed temperature profile, increasing $q$ increases $E$ and hence $D$. This argument does not assume that the greenhouse gas changes the stellar-band albedo, surface pressure, or background thermodynamics; those are held fixed. For sufficiently broad or overlapping saturated wings the full frequency-dependent Planck function must replace the narrow-line approximation.