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The time a star spends with core hydrogen nuclear fusion as its principal energy source. If the usable fuel fraction $f$, conversion efficiency $ϵ$, and composition are comparable, the main-sequence lifetime scales as $M/L$. With a local mass-luminosity relation $L\propto M^{\alpha }$, the lifetime scales as $M^{1-\alpha }$. This is a scaling estimate: convective core size and changing luminosity affect detailed ages.

TODO: assign the physical differences to the actual two panels once the missing evolved-star sequences are recovered.

The main intrinsic parameters for old globular clusters are age, metallicity, helium abundance and the distribution of envelope masses after red giant branch mass loss. Distance and foreground interstellar extinction alter the appearance and must first be removed before assigning intrinsic differences.

Age controls the main-sequence turnoff: a more massive star has a shorter main-sequence lifetime, approximately fuel divided by luminosity. An older coeval population therefore has a smaller turnoff mass and a fainter, generally redder main-sequence turnoff. The separation in astronomical magnitude between the main-sequence turnoff and horizontal branch is useful because a common distance modulus cancels.

At comparable age, larger metallicity increases stellar-envelope opacity and atmospheric line blanketing. This generally makes the red giant branch and main sequence redder; metal-poor populations tend to have hotter, bluer red giant branches. Metallicity also affects hydrogen burning through the CNO cycle, so a quantitative age inference needs composition-dependent stellar models.

The horizontal branch is especially sensitive to envelope mass: at similar helium-core mass, a thinner hydrogen envelope gives a higher effective temperature and hence a bluer location. Red giant branch mass loss, age and helium abundance all influence that envelope and produce the familiar “second parameter” complexity. Enhanced helium also changes mean molecular weight, luminosity and evolutionary lifetime, and can shift main sequences and horizontal branches.

Interstellar extinction dims the diagram, while wavelength-dependent interstellar extinction reddens $B-V$; distance primarily supplies a vertical distance modulus shift. A systematic bend or different horizontal branch colour distribution cannot be explained by distance alone. Because the source hides the evolved branches, it is not possible to assign “older”, “more metal rich”, or “bluer horizontal branch” to a particular panel reliably. The preceding mechanisms give the warranted physical interpretation without inventing the lost observational differences.

For ordinary single stars in these old globular clusters, no current core-collapse supernovae are expected, and no new black holes are expected from the normal plotted population. Their main-sequence turnoff masses are typically around $0.8M_{\odot }$, far below the roughly $8M_{\odot }$ initial-mass scale for advanced burning and core collapse. More massive original stars exhausted their much shorter main-sequence lifetimes long ago. The precise transition mass depends on composition and mass loss, but this large separation makes the conclusion robust.

The low-mass stars now leaving the main sequence evolve through the red giant branch, helium ignition, horizontal branch and Asymptotic giant branch. They lose their envelopes and leave mainly carbon-oxygen white dwarfs, typically around $0.5$–$0.6M_{\odot }$ for current old-cluster progenitors, supported by electron degeneracy pressure. A visible planetary nebula is possible if the exposed core heats before the ejecta disperse; it is not guaranteed for every low-mass remnant. Lower-mass main-sequence stars repeat this evolution only much later, and the very lowest-mass isolated stars can ultimately leave helium-rich degenerate remnants without igniting helium.

The clusters may already contain neutron stars and black holes made by their original massive stars; these remnants need not appear on an optical colour-magnitude diagram. Binary interactions can instead produce stripped helium white dwarfs, mergers or a possible Type Ia supernova; such exceptions are distinct from normal single-star core collapse. No such special progenitor can be established from the damaged plots. In particular, a luminous red giant is not by itself evidence for a massive core-collapse supernova progenitor.

Adopt the approximate mass-luminosity relation $L/L_{\odot }\simeq (M/M_{\odot }{)}^3$ and equal available fuel fractions. The main-sequence lifetime then scales as $M/L$, givingUsing an exponent $3.5$ instead gives about $5.4\times 10^7$ and $3.1\times 10^8$ years; these are model-dependent estimates. At $r=3r_s=6GM/c^2$ the Newtonian gravitational potential energy released from infinity is $GMm/r=m{c}^2/6$. Half remains as kinetic energy in a circular orbit, so the radiatable binding energy is $GMm/(2r)$:Using the whole potential energy loss would give $1/6$, which does not subtract the final kinetic energy. A fully relativistic calculation for a nonrotating black hole gives $1-\sqrt{8/9}\simeq 5.7\%$. When the surviving star exhausts hydrogen burning, it expands off the main sequence and can fill its Roche lobe. Binary mass transfer to the black hole can form an accretion disk and an X-ray binary; unstable binary mass transfer may instead produce a common envelope. This is conditional on the binary star remaining bound after the first remnant forms. The supplied black hole endpoint is an assumption, not a universal outcome for an isolated $8M_{\odot }$ star.

A free-fall timescale estimate follows from $R/t^2\sim GM/R^2$:Falling to a central point mass gives the more precise factor $\pi /(2\sqrt{2})$, or $1.77\times 10^3\mathrm{s}$; a distributed star changes an order-one coefficient. For any spherical mass distribution with radius $R$,The inequality is strict for an extended star; equality is approached by a thin shell at $R$. For an monatomic gas the virial theorem is $2U+\Omega =0$, so half the gravitational potential energy released by contraction heats the star and half can be radiated. The Kelvin-Helmholtz cooling time is therefore of orderA uniform-density model gives $3G{M}^2/(10RL)\simeq 9.4\times 10^6\mathrm{yr}$. The given bound fixes a scale, rather than an exact contraction history; both estimates are far shorter than the Sun's main-sequence lifetime.

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.

Adopt the approximate mass-luminosity relation $L/L_{\odot }\simeq (M/M_{\odot }{)}^3$ and equal available fuel fractions. The main-sequence lifetime then scales as $M/L$, givingUsing an exponent $3.5$ instead gives about $5.4\times 10^7$ and $3.1\times 10^8$ years; these are model-dependent estimates. At $r=3r_s=6GM/c^2$ the Newtonian gravitational potential energy released from infinity is $GMm/r=m{c}^2/6$. Half remains as kinetic energy in a circular orbit, so the radiatable binding energy is $GMm/(2r)$:Using the whole potential energy loss would give $1/6$, which does not subtract the final kinetic energy. A fully relativistic calculation for a nonrotating black hole gives $1-\sqrt{8/9}\simeq 5.7\%$. When the surviving star exhausts hydrogen burning, it expands off the main sequence and can fill its Roche lobe. Binary mass transfer to the black hole can form an accretion disk and an X-ray binary; unstable binary mass transfer may instead produce a common envelope. This is conditional on the binary star remaining bound after the first remnant forms. The supplied black hole endpoint is an assumption, not a universal outcome for an isolated $8M_{\odot }$ star.

A free-fall timescale estimate follows from $R/t^2\sim GM/R^2$:Falling to a central point mass gives the more precise factor $\pi /(2\sqrt{2})$, or $1.77\times 10^3\mathrm{s}$; a distributed star changes an order-one coefficient. For any spherical mass distribution with radius $R$,The inequality is strict for an extended star; equality is approached by a thin shell at $R$. For an monatomic gas the virial theorem is $2U+\Omega =0$, so half the gravitational potential energy released by contraction heats the star and half can be radiated. The Kelvin-Helmholtz cooling time is therefore of orderA uniform-density model gives $3G{M}^2/(10RL)\simeq 9.4\times 10^6\mathrm{yr}$. The given bound fixes a scale, rather than an exact contraction history; both estimates are far shorter than the Sun's main-sequence lifetime.

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 radiation registered in the stated solar luminosity includes two Positron annihilations but excludes the two escaping neutrinos. Thus the visible energy per completed helium-producing chain isThe signs in this energy ledger matter: adding the neutrino energy would overestimate the energy available to sustain the measured radiation. Using $1\mathrm{MeV}=1.602176634\times 10^{-13}$ J, the chain rate and hydrogen-consumption rate areWith a year taken as $365.25$ days, the stated elapsed time is $1.4516496\times 10^{17}$ s. Consequently,At the same fixed luminosity, the remaining time would beOnly about $5.93\%$ of the stipulated initial reservoir has been consumed under these assumptions. This is a calculation for an entirely accessible hydrogen reservoir with constant luminosity. It is not a prediction of the Sun's remaining main-sequence lifetime: only part of the hydrogen participates in core burning, and stellar structure and luminosity evolve.