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The trace-free mass quadrupole moment is diagonal:The same constant factors multiply its third derivatives, giving ${\stackrel{...}{Q}}_{ij}{\stackrel{...}{Q}}_{ij}=(2/3)({\stackrel{...}{I}}_{zz}{)}^2$. The quadrupole formula and the preceding infall result therefore giveFor infall from infinity, $P=m^5/(15z^5)$ and $|\dot{z}|=\sqrt{m/(2z)}$. To find the emitted energy, integrate power over time, using $dt=|dz|/|\dot{z}|$:The total initial mass is $2m$, so the radiated fraction is $1/840\simeq 1.19\times 10^{-3}$. The Sun would emit that fraction of its rest energy inunder the stipulated constant luminosity. This is the head-on quadrupole radiation from equal masses prediction with the prescribed stopping rule. At the endpoint $|\dot{z}|=1/2$ and $m/z=1/2$, so extending the slow-motion weak-field formula that far is an extrapolation, not a controlled strong-field prediction.

The conventional Kelvin-Helmholtz cooling time is of order $G{M}^2/(RL)$. For the monatomic uniform-density stellar model, its precise accessible total-energy reservoir is $|E|=3G{M}^2/(10R)$, soUsing $M_{\odot }\simeq 1.9885\times 10^{30}\mathrm{kg}$, $R_{\odot }\simeq 6.957\times 10^8\mathrm{m}$ and $L_{\odot }\simeq 3.828\times 10^{26}\mathrm{W}$, the usual order-of-magnitude normalization without the structural factor is $t_{\mathrm{KH}}\simeq 3.1\times 10^7\mathrm{yr}$. The full gravitational binding reservoir $|\Omega |/L$ is about $1.9\times 10^7\mathrm{yr}$, but that overcounts the radiatable energy in virial equilibrium.

The Sun and the Solar system are approximately $4.6\times 10^9\mathrm{yr}$ old, much older than any of these contraction estimates. Gravitational contraction cannot sustain the Sun's long-lived present power. The dominant long-term source is stellar nuclear fusion. Contraction can still supply transient luminosity, especially in a pre-main-sequence star. The timescale comparison rules out a contraction-only explanation over the observed age; it does not assert that all gravitational energy release is absent.

For typical well-mixed structures, the dominant processes are:

Representative temperatures must specify the level: Earth has about $288\mathrm{K}$ at the surface (about $255\mathrm{K}$ effective emission temperature); Jupiter has about $165\mathrm{K}$ near one bar (about $125\mathrm{K}$ effective temperature); hot Jupiters commonly have photospheric temperatures of order $1000$–$2500\mathrm{K}$; and the Sun's photosphere is about $5800\mathrm{K}$. Upper layers, nightsides, deep interiors and the solar corona have different temperatures. These are characteristic values, not constant temperatures throughout each atmosphere.

A natural interpretation of the short-wavelength peak is reflected starlight, while the longer-wavelength peak is planetary thermal radiation. A reflected spectrum approximately follows the stellar spectrum multiplied by the wavelength-dependent geometric albedo; thermal radiative flux approximately follows the planet's Planck function, modulated by molecular opacity. Thus two peaks need not represent two planetary surface temperatures.

For a quantitative estimate assume both are broad $F_{\lambda }$ peaks, reflection has a slowly varying geometric albedo, and star and planet have approximately blackbody spectral envelopes. Wien's displacement law then givesThe stellar estimate is compatible at order of magnitude with an old, relatively small main-sequence star; equal age does not mean equal temperature to the Sun. Assume the far-infrared signal is thermal and sufficiently long-wavelength that the Rayleigh-Jeans law applies to both bodies. The planet-star radius estimate in the Rayleigh-Jeans limit givesand thereforeThis is a small volatile-rich-planet-sized estimate, not a Jupiter-sized one. It is conditional: peaks caused by molecular windows, strongly chromatic reflection, or a spectrum expressed as $F_{\nu }$ rather than $F_{\lambda }$ do not support those two Wien estimates. Without $T_p/T_{\ast }$, the far-infrared ratio only fixes $R_p^2{T}_p/T_{\ast }$.

For a close-in hot planet around a small star, transit-based atmospheric observations are the natural route if the orbital geometry allows them. Exoplanet transmission spectra gain from the small stellar radius, with a limb signal scaling as $2R_{p}H/R_{\ast }^2$; they probe composition at the day-night terminator. The stated $10^{-3}$ thermal contrast also makes exoplanet secondary eclipse measurements a particularly useful route to the dayside exoplanet emission spectrum. Resolving such a close-in small planet by exoplanet direct imaging is much harder. Transit and secondary-eclipse spectroscopy are favoured for a transiting close-in interpretation; the supplied spectrum alone does not determine the orbital geometry or a unique best technique.