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TODO: numerical endpoints require the unreadable source diagram. Let ${\ell }_{V,\max }$ and ${\ell }_{V,\min }$ be the maximum and minimum main sequence visual luminosities, in units of the Sun's visual luminosity. At a distance $d$, the inverse-square law and the magnitude difference relation giveThus, using $m_{V,\odot }=-26.77$,At the naked-eye detection threshold $m_V=6$, the greatest distances for the intrinsically brightest and faintest main sequence stars would respectively beThe historical $m_V=1$ endpoint is a brightness reference, not the visibility threshold: if that endpoint were intended instead, the corresponding coefficient would be $1.736$ rather than $17.36$.

The source Hertzsprung-Russell diagram has lost the stellar loci and luminosity scale except for a surviving top tick; neither endpoint can be read reliably. If its scale was bolometric luminosity rather than visual luminosity, bolometric corrections would also be needed, especially for very hot and very cool stars. Equating the two luminosity ratios would then be an explicitly approximate assumption. These missing data are not replaced by invented endpoint values. Interstellar extinction is neglected in the displayed distance relation.

TODO: the plotted location of Procyon B is missing from the damaged source diagram. The Stefan–Boltzmann law would determine the stellar radius from its effective temperature and bolometric luminosity:A visual luminosity alone must first be converted with the appropriate bolometric correction. Procyon B is a white dwarf, so its small stellar radius and hot surface place it below the main sequence, rather than among giant stars or supergiants. This classification does not supply the missing numerical coordinates. A source-based numerical stellar radius cannot be claimed from the surviving image.^[ref]

The Stefan–Boltzmann law gives $L=4\pi R^2\sigma T_{\mathrm{eff}}^4$ for bolometric luminosity. HenceWith the central values above, this is approximately $1.51$ for the first geometry and $0.67$ for the second. If only one-band luminosities are measured, bolometric corrections are needed before using this formula for effective temperatures.

Two direct methods are fitting the continuum stellar spectrum or several photometric colors to a stellar atmosphere model, and measuring temperature-sensitive absorption lines or line ratios in a stellar spectrum. The former includes the approximate black-body radiation estimate ${\lambda }_{\max }T=b$ from Wien's displacement law; the latter uses ionization and excitation populations. Observations during total eclipse or spectral disentangling help separate the two stellar spectra.

For comparable composition and accessible fuel fractions, the main-sequence lifetime is proportional to available nuclear fuel divided by luminosity, $t_{\mathrm{MS}}\propto M/L$. Estimate the masses from a calibrated mass-luminosity relation $L\propto M^{\alpha }$ in the appropriate mass range. ThenFor illustration $\alpha =3.5$ gives about $1.27$ and $5.6$ for the two graphical luminosity ratios. The common main sequence approximation is additional physics; eclipse depths alone do not determine masses or ages. A main sequence consistency check would also favor the second geometry because ordinary main-sequence stars have radius and effective temperature increasing with mass.

The trigonometric parallax gives $d=1/0.02=50\mathrm{pc}$. Neglecting extinction, the distance modulus and given bolometric correction giveUsing the specified solar bolometric magnitude,The mass estimate uses the cubic scaling normalized to solar values. A lifetime estimate additionally needs the fraction of fuel accessible to core burning. Take a core-accessible fraction $f_c=0.1$ and initial hydrogen mass fraction $X=0.7$. The supplied fusion energy givesThe alternative common solar-lifetime normalization $t\simeq 10^{10}(M/M_{\odot })/(L/L_{\odot })$ years gives $1.69\times 10^9$ years. Thus this simplified turnoff argument gives an age of order $10^9$ years, with its fuel fraction and mass-luminosity assumptions explicitly stated. It does not use a modern fitted Hyades age or a full stellar isochrone. Unresolved binaries, extinction and deviations from the homology law affect the estimate.

At D, $M_V=20.3-15.09=5.21$. Ignoring differential bolometric corrections, $L/L_{\odot }\simeq 10^{-0.4(5.21-4.74)}\simeq 0.65$, and $L\propto M^4$ gives $\boxed{M_D\simeq 0.90M_{\odot }}$. If accessible fuel fraction is similar, lifetime scales as $M/L\propto M^{-3}$. Normalizing to a solar main-sequence lifetime about 10 Gyr gives an age near $\boxed{14\mathrm{Gyr}}$.

This is much older than the Sun's current approximately 4.6 Gyr and close to the cosmological age scale. The approximate estimate is not evidence for a cluster older than the Universe: bolometric corrections, abundance, accessible hydrogen fraction and the crude mass-luminosity law limit precision. The stated 26.7 MeV conversion and an assumed core fuel fraction can supply the order-of-magnitude solar lifetime; they do not by themselves specify the burned fraction or composition precisely.

The distance modulus at $100\mathrm{pc}$ is five magnitudes. Thus the two approximate main-sequence endpoints read from the original figure, $m_V\simeq 0,20$, correspond to $M_V\simeq -5,15$. With the given solar reference,Use the figure's approximate endpoint effective temperatures $25,000$ K and $2500$ K. If $V$-band luminosity is used as a rough bolometric proxy, the Stefan–Boltzmann law givesTaking $T_{\odot }\simeq 5800$ K givesThese are schematic estimates, not precision radii. The diagram supplies $V$ magnitudes rather than bolometric luminosities, and hot and cool endpoints need different bolometric corrections. The exact range cannot be recovered without those corrections; the quoted values state the intended approximation explicitly.