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A brown dwarf is a self-gravitating body that does not sustain ordinary core hydrogen burning. Contraction initially heats it, but growing electron degeneracy pressure limits further compression before long-lived hydrogen fusion can establish the thermal balance of a main-sequence star. It subsequently cools and becomes fainter.

Some brown dwarfs temporarily burn deuterium; the deuterium-burning mass is a composition-dependent classification scale rather than a unique formation test. Their radii can be close to Jupiter's across a broad mass range. Consequently, exoplanet transit photometry alone cannot distinguish a giant planet from a brown dwarf. The transit and Doppler spectroscopy measurements of CoRoT-3b illustrate why independent mass information is decisive for characterizing such compact companions.

Hydrogen burning is the net conversion of hydrogen into helium through nuclear fusion. The proton–proton chain and CNO cycle provide different sequences with the same principal fuel and final helium product. In the CNO cycle, carbon, nitrogen and oxygen act as catalysts rather than being consumed as the principal fuel.

The released nuclear binding energy supplies the long-term luminosity of a main-sequence star, with some energy escaping in neutrinos. Core exhaustion ends that evolutionary phase even if the outer envelope still contains abundant hydrogen. A hydrogen-burning shell can subsequently sustain an evolved star. The word burning denotes nuclear reactions here; it does not require combustion with molecular oxygen.

For main-sequence stars, luminosity increases steeply with mass. A local power-law approximation uses an exponent that varies with the stellar mass range and composition; $\alpha \simeq 4$ is a useful rough near-solar estimate. Evolved stars need not follow that same calibration.

An approximate relation between the mass and radius of main-sequence stars in a specified composition and mass range. It is not determined by hydrostatic equilibrium alone. For gas-pressure stellar homology with Kramers' opacity law, radiative diffusion gives $L\propto {\mu }^{15/2}{M}^{11/2}{R}_{\ast }^{-1/2}$. Substitution of $R_{\ast }\propto M^q$ gives $L\propto M^{(11-q)/2}$ at fixed composition, explaining precisely which extra radius assumption produces any claimed exponent.

A main-sequence star is a star powered predominantly by sustained hydrogen burning in its core. Hydrostatic equilibrium balances gravity with a pressure gradient, while thermal equilibrium approximately balances the nuclear energy supply with losses. The Sun is an example; a red dwarf is a cooler, lower-mass example.

Membership in the main sequence describes an evolutionary state, rather than simply small size or low luminosity. A white dwarf can be hotter and smaller while lacking sustained core hydrogen burning. Within the main sequence, a steep mass-luminosity relation makes massive stars consume their accessible fuel faster, so a larger fuel supply does not necessarily give a longer main-sequence lifetime.

Hydrostatic equilibrium and dimensional scaling give characteristic stellar pressure $p\sim G{M}^2/R^4$ and mass density $\rho \sim M/R^3$. For a fixed-index stellar polytrope with $p=K{\rho }^{1+1/n}$, eliminating density gives the polytropic mass-radius relationThe dimensionless coefficient is determined by the corresponding Lane-Emden equation and boundary conditions. Both the index and $K$ must be held fixed to infer this mass dependence.

Nonrelativistic electron degeneracy has $n=3/2$, hence $R\propto M^{-1/3}$ for cold fixed-composition white dwarfs: adding mass makes the object smaller. Relativistic electron degeneracy approaches $n=3$; the mass scale becomes proportional to $(K/G{)}^{3/2}$ independently of the radius, giving the ideal Chandrasekhar limit. The limiting model signals loss of the ordinary stable cold sequence; it does not determine a unique radius at that mass.

For a gas-pressure-supported star, the virial theorem instead gives characteristic temperature $k_{B}T\sim G\bar{m}M/R$. Specifying a common characteristic temperature gives $R\propto M$, but stellar temperatures are not generally identical. A main-sequence star also requires nuclear energy generation and radiative or convective transport. Consequently there is no universal mass-radius exponent applicable to every main-sequence mass and composition. An isentropic convective model and a cold degeneracy-supported model can share index $3/2$ while having quite different physical meanings and values of $K$.

On the main sequence, central hydrogen burning converts four protons into helium, positrons, neutrinos and released energy. The net reaction is $4p\to {}^4\mathrm{He}+2e^{+}+2{\nu }_e$, with positron annihilation contributing to the total roughly $26.7\mathrm{MeV}$ release before subtracting escaping neutrino energy. In the low-mass main-sequence stars of an old globular cluster, the proton–proton chain generally dominates. The CNO cycle makes the same net conversion and becomes important in hotter burning regions.

After central hydrogen exhaustion, a subgiant contracts its helium core and establishes a hydrogen-burning shell. On the red giant branch, that shell supplies most of the luminosity, commonly through the temperature-sensitive CNO cycle, while the inert helium core grows and becomes supported mainly by electron degeneracy pressure. The envelope expands and its surface cools. The helium core is not yet powered by helium nuclear fusion.

Helium ignition in a sufficiently low-mass degenerate core produces the helium flash, removing the strong electron degeneracy pressure dominance. A horizontal-branch star subsequently burns helium in its core through the Triple-alpha process, $3{}^4\mathrm{He}\to {}^{12}\mathrm{C}+\gamma$, and through ${}^{12}\mathrm{C}(\alpha ,\gamma ){}^{16}\mathrm{O}$. A surrounding hydrogen-burning shell can continue simultaneously.

After central helium exhaustion, the Asymptotic giant branch has an inert carbon-oxygen core with a helium-burning shell and an outer hydrogen-burning shell. Alternation between quiescent burning and helium-shell thermal pulses drives mixing and substantial mass loss. Subsequent post-asymptotic giant branch evolution exposes a hot core that may ionize its expelled envelope as a planetary nebula. Finally a white dwarf cools mainly by loss of stored thermal energy, rather than sustained central nuclear fusion.

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.

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.

A red dwarf is a cool, low-mass main-sequence star supported by sustained core hydrogen burning. Its small radiating area and low effective temperature make it faint compared with the Sun. A brown dwarf may have a similar radius but does not sustain that same long-lived hydrogen-burning state.

The lowest-mass red dwarfs are approximately fully convective stars. Mixing can bring fresh hydrogen into the burning region instead of leaving most fuel isolated in an envelope. Their low luminosity relative to available fuel therefore permits long main-sequence lifetimes. A small stellar radius also makes a given exoplanet block a larger fraction of the disc, improving the geometric signal in exoplanet transit photometry.

Sirius is a nearby binary star containing a luminous main-sequence star and a much fainter white dwarf. Their coexistence makes a useful lesson in stellar appearance: a hot object can be faint because its radiating area is small. The resolved Hubble Space Telescope image separates the compact companion from the bright primary's surrounding light.^[ref]

The common orbit provides dynamical information, whereas spectral classification and luminosity constrain the two bodies' physical states. The white dwarf records prior stellar evolution; its current faintness does not mean it was always the less massive or less luminous member. Sirius therefore connects binary star measurements with white-dwarf cooling and the interpretation of the Hertzsprung-Russell diagram.

Spectral classification groups stars by the patterns of absorption lines, emission lines and molecular bands in their spectra. The Harvard spectral classification orders the familiar O, B, A, F, G, K and M types mainly by effective temperature. Changes in ionization and excitation make a given element's lines strongest over only part of that sequence; weak hydrogen lines do not directly imply little hydrogen.

Surface gravity also affects line shapes and ratios, supplying a separate luminosity class. Together the temperature type and luminosity class help distinguish a cool red giant from a cool main-sequence star. Composition, stellar activity, rotation and unresolved companions complicate the inference, so classification is an observational guide to a physical model rather than a complete measurement of the stellar structure.

A star is a self-gravitating astronomical body whose luminous life is governed by the competition between gravity, internal pressure, energy loss and nuclear fusion. A main-sequence star sustains core hydrogen burning; a pre-main-sequence star is still contracting toward that stage, while evolved stars can burn different fuels or in shells. The name therefore includes more than the currently hydrogen-burning phase.

Hydrostatic equilibrium relates the inward pull of gravity to an outward pressure gradient. Radiative transfer and convection carry energy through the interior, and the photosphere supplies much of the observed spectrum. Mass, composition, age and interactions determine the resulting stellar radius and luminosity. The Sun offers a nearby example in which resolved observations and solar neutrino detection test different parts of the same physical picture.

Stellar nuclear fusion converts light atomic nuclei into more tightly bound products, releasing energy from the change in nuclear binding energy. In a main-sequence star, the net conversion of hydrogen to helium proceeds mainly through the proton–proton chain or the CNO cycle, depending on core conditions. The high temperature supplies energetic collisions, while quantum tunnelling allows reactions through the repulsive electrical barrier.

The released energy heats the interior and replaces energy lost through the photosphere and by neutrinos. A stellar thermostat arises because extra heating can expand the star, reduce core temperature, and lower the reaction rate. Later stellar evolution changes the available fuel and core structure, enabling helium burning and, in sufficiently massive stars, further burning stages.

The Sun is the star at the centre of the Solar system and a nearby example of a main-sequence star. Its effective temperature is approximately $5772\mathrm{K}$, its stellar radius about $6.96\times 10^8\mathrm{m}$, and its luminosity about $3.83\times 10^{26}\mathrm{W}$. These describe different aspects of the same object: radiating area and effective temperature combine through the Stefan–Boltzmann law to give solar luminosity.^[ref]

Core hydrogen burning supplies the long-term energy budget. Energy reaches the photosphere through radiative transfer and convection, while solar neutrino detection samples the nuclear processes more directly. Resolved solar granulation and starspots reveal that the visible disc is structured and time-dependent. Comparing these accessible phenomena with unresolved stars helps calibrate wider stellar astrophysics.