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A post-helium-ignition stage of low-mass stellar evolution with core helium nuclear fusion and usually a hydrogen-burning shell. The branch spans a broad range of effective temperatures at comparatively similar luminosities, largely because envelope masses vary.

Removing helium from a hydrogen-rich envelope lowers He/H and raises its reciprocal H/He. A helium-rain explanation of atmospheric depletion must keep those two ratio conventions distinct. Settling helium-rich material can release gravitational energy while changing atmospheric composition; excess luminosity alone does not uniquely determine the interior process.

With carbon-ion thermal energy $U=M{k}_B{T}_c/(8m_u)$ and luminosity $L=A(M/M_{\odot })T_c^{7/2}$, the time between two core temperatures is $k_B{M}_{\odot }(T_f^{-5/2}-T_i^{-5/2})/(20m_{u}A)$. Late cooling is slow in this approximation. Extrapolating the formula does not prove a categorical observational absence of ultracool remnants, since atmosphere, crystallization and heat-capacity physics alter the simple model.

The time integral of the incident-beam overlap flux, with inverse-area units. For a fixed production cross section, the mean produced event count is $N={\mathcal{L}}_{\mathrm{int}}\sigma$. A selected count includes efficiency and acceptance, often integrated over final-state kinematics. This is distinct from astrophysical radiative luminosity, which has power units.

A swept gas shell of fraction $f$ at radius $R$ in a galaxy of one-dimensional dispersion $\sigma$ has mass $2f{\sigma }^{2}R/G$. Interior non-gas gravity supplies weight $4f(1-f){\sigma }^4/G$. Fully intercepted single-scattering Eddington momentum overcomes it when $M_{\mathrm{BH}}>f(1-f){\sigma }^4{\sigma }_T/(\pi m_p{G}^2)$. This neglects shell self-gravity, central-hole gravity and stored outward kinetic energy; weaker coupling raises the required luminosity.

For a negligible-mass ideal-gas envelope above a core of mass $M_c$, with constant luminosity $L$ and Kramers' opacity law, division of hydrostatic equilibrium by radiative diffusion gives $PdP/dT=16\pi acG{M}_c\mathcal{R}{T}^{15/2}/(3{\kappa }_{0}L\mu )$. Taking the outer pressure constant to be negligible yields $P=C{T}^{17/4}$, where $C^2=64\pi acG{M}_c\mathcal{R}/(51{\kappa }_{0}L\mu )$, and $\rho =(\mu C/\mathcal{R})T^{13/4}$. Hydrostatic equilibrium then gives $dT/dr=-4\mu G{M}_c/(17\mathcal{R}{r}^2)$, so $T=B/r+T_0$ with $B=4\mu G{M}_c/(17\mathcal{R})$. The radiative-zero choice $T_0=0$ is exact for a vanishing-temperature boundary at infinity and a deep-envelope approximation for a very extended finite star. It is not implied by small surface temperature alone at a specified finite radius.

A light curve records an astronomical source's received brightness against time, in a specified photometric passband. Its measured form is a time series of timestamps, brightness estimates, and uncertainties; a smooth $F(t)$ is a model of that record. Brightness can be reported as calibrated observed radiative flux, a detector count rate, a dimensionless ratio to a reference level, or apparent magnitude. These quantities require different units and calibration conventions. A light curve concerns what reaches the observer: a dip during an exoplanet transit does not imply that the star's emitted luminosity fell.

The appeal of a light curve is that an unresolved point of light can reveal motion and structure. In exoplanet transit photometry, repeated dips constrain the orbital period; under the uniform-disc, dark-planet, full-overlap assumptions, the exoplanet transit depth is the blocked area fraction $(R_p/R_{\star }{)}^2$. A toy one-percent dip therefore corresponds to a radius ratio of $0.10$. Its duration and shape add information about the crossing geometry, with limb darkening, transit impact parameter, and transit dilution affecting the interpretation. See exoplanet transit photometry for the geometry. The first figure instead follows the measurement: an instrumental trend and sky contribution can obscure a small physical dip.

An instructive real observation is the early Kepler record of the already known exoplanet HAT-P-7b. Welsh and colleagues analysed 33.5 days from 13 May to 15 June 2009, containing 15 complete exoplanet transits at an orbital period of about $2.20473$ days. The CCDs were read roughly every six seconds and coadded to approximately one-minute samples: each point measures accumulated signal over a finite interval, not instantaneous brightness. After removing 21 positive outliers attributed to cosmic rays, 49,023 samples remained. The authors estimated a scatter of about $150$ parts per million per one-minute sample. A gradual $0.13\%$ count decrease, suspected to arise from focus drift with a fixed extraction aperture, was removed with a cubic polynomial fitted while masking the exoplanet transits. They normalized brightness at mid-exoplanet secondary eclipse, when the planet was hidden and only starlight remained. ^[ref].

What does the instrument actually measure? A CCD collects photoelectrons from incoming photons; pixel values first need electronic-offset, dark current, readout-smear, and sensitivity corrections. ^[ref]. Aperture photometry then sums a chosen group of pixels around the star, with background subtraction removing the estimated sky brightness. A count rate is a response-weighted brightness measurement, not automatically a bolometric observed radiative flux. Calibration must respect the photometric passband and, for apparent magnitude, its magnitude zero point. Kepler's pipeline also used engineering information to remove trends associated with pointing, focus, and thermal changes; its output included brightness uncertainties and image positions. ^[ref].

Calibration and interpretation need separate checks. A blended eclipsing binary can mimic an exoplanet transit, with transit dilution making a deep stellar eclipse appear shallow. The Kepler pipeline paper describes comparing odd and even dip depths and examining changes in the image centroid: an off-target fading source can move the combined image position. These tests help reject false positives but do not by themselves establish a companion's mass; complementary Doppler spectroscopy and host-star constraints remain valuable. ^[ref].

Even a real exoplanet's signal can have several contributors. Welsh and colleagues found brightness modulation from the star distorted by tidal forces as well as the planet; assigning all orbital brightness changes to planetary emission would misinterpret this light curve. ^[ref]. Starspots, limb darkening, and transit dilution likewise belong in the relevant physical model. A flexible trend fitted through a dip could absorb that dip; the masked fit in the real example illustrates why the fitting procedure is part of the measurement. Exposure-averaged light curves and phase folding of a light curve make two further limitations explicit.

For a circular stellar disk whose specific intensity is independent of azimuth, its mean specific intensity over projected area isThe polar coordinates element is $d{A}_{\mathrm{proj}}=2\pi rdr$ after angular integration, and $rdr=-R_{\star }^2\mu d\mu$. Substitution of the linear limb-darkening law givesFor a distant observer at distance $D\gg R_{\star }$ receiving light normally to the detecting plane, the observed radiative flux isIt is spectral irradiance, with units $\mathrm{W}{\mathrm{m}}^{-2}\mathrm{H}{\mathrm{z}}^{-1}$ when $I_{\nu }$ is per hertz. The local outward radiative flux per actual surface area is instead $F_{\nu }=2\pi {\int }_0^1{I}_{\nu }(\mu )\mu d\mu =\pi {\overline{I}}_{\nu }$, a spectral radiant exitance. In a spherical model with the same stellar atmosphere everywhere, the luminosity is $L_{\nu }=4\pi R_{\star }^2{F}_{\nu }$ and $f_{\nu }=L_{\nu }/(4\pi D^2)$, consistent with the inverse-square law. Compare disks at fixed $I_c$ or fixed total observed radiative flux explicitly: increasing $u$ lowers the total in the first comparison, while the second requires a larger $I_c$.

Luminosity is emitted radiative power: the energy carried away by electromagnetic radiation per unit time, measured in watts. Bolometric luminosity includes all wavelengths; a measurement through one photometric passband captures only part of it. Spectral luminosity records how that power is distributed over frequency. Received observed radiative flux has different units, watts per square metre, because it measures what crosses an observer's collecting area. ^[ref].

For example, an ideal source with luminosity $10^{26}\mathrm{W}$ emits $10^{26}\mathrm{J}$ each second regardless of whether it is viewed from $20$ or $40$ parsecs. Its observed radiative flux changes with distance, while its luminosity remains fixed. Combining stellar parallax with observed radiative flux therefore lets us compare the actual outputs of stars rather than their apparent brightnesses.

A stellar spectral category based on gravity-sensitive line widths and ratios, distinguishing supergiants, giants and dwarfs at comparable spectral temperature. Examples are I for supergiants, III for giants and V for main-sequence dwarfs. It is inferred spectroscopically and is not a direct substitute for measuring absolute luminosity in every individual star.

The luminosity per unit spatial volume. For a spherical source, its enclosed luminosity is $L(r)=4\pi {\int }_0^r\lambda (s)s^{2}ds$, and its projected surface brightness is $\mu (R)=2{\int }_R^{\infty }\lambda (r)rdr/\sqrt{r^2-R^2}$.

At fixed positive redshift, intrinsic luminosity and present Hubble constant, a flat dust-vacuum model with $0<{\Omega }_{\Lambda 0}<1$ has luminosity distance between those of an Einstein-de Sitter universe and pure positive vacuum. The Hubble parameter of the dust-only model is largest at every positive integration redshift; that of pure vacuum is smallest. Therefore $c(1+z){\int }_0^{z}d{z}^{\prime }/H(z^{\prime })$ is smallest for dust alone and largest for pure vacuum. Received radiative flux has the reverse ordering. This compares physical distances rather than arbitrarily normalized comoving coordinates.

Coherent intrinsic luminosity variation on observed time $\Delta t_{\mathrm{obs}}$ bounds the characteristic emitting radius by a light-crossing argument. relativistic beaming shortens the observed interval by Doppler factor $D$, while cosmological redshift lengthens it by $1+z$. This bound assumes that the variation samples the emitting region causally; geometry, independent subregions and propagation-driven variation can change its interpretation.

The classical parallax-calibration problem combines measurement errors with the greater volume available at larger distances. A locally uniform spatial population has $p(r)\propto r^2$, which transforms under $r=1/\varpi$ to $p(\varpi )\propto {\varpi }^{-4}$ on the valid distance interval. Parallax cuts and the nonlinear conversion to luminosity can bias calibration samples. A correction is not universal: the population model, selection information and inferential target must be specified. Distinguish the repeated-sampling bias of a selected sample from inference for a particular star using a justified prior.^[ref]

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.

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.

The main-sequence turnoff is the location in a coeval population's Hertzsprung-Russell diagram where stars are leaving the main sequence after exhausting their core hydrogen. More massive stars generally have larger luminosity per unit mass and shorter nuclear lifetimes, so the most massive stars still on the main sequence constrain the population's age. In a simple model, equating the stellar nuclear timescale $t\sim qM/L(M)$ to the common age determines a turnoff mass; detailed calibration requires composition and stellar models.

If birth counts obey $dN/dm\propto m^{-s}$ and individual luminosity scales as $m^q$, multiply by mass or luminosity and by $dm=md\ln m$ to obtain the displayed logarithmic weights. With $s=2.35$ and $q=4$, birth mass has weight $m^{-0.35}$ while light has weight $m^{2.65}$. Finite mass cutoffs are essential; stellar lifetimes also alter the luminosity weighting of an evolved population.

For a steady point-like astronomical source with isotropy of emission, in transparent space with negligible relativistic or cosmological effects, conservation of energy spreads its luminosity over a sphere of surface area $4\pi d^2$. Its bolometric observed radiative flux isThis inverse-square law reflects geometric dilution: the same emitted power crosses increasingly large spheres. It does not require light to lose energy on the journey. For a fixed luminosity, doubling Euclidean distance quarters the observed radiative flux. ^[ref].

With $L_0=10^{26}\mathrm{W}$ and $d_0=20$ parsecs, the result is $F_0\simeq 2.09\times 10^{-11}\mathrm{W}{\mathrm{m}}^{-2}$. At $40$ parsecs, the same source gives $F_0/4\simeq 5.22\times 10^{-12}\mathrm{W}{\mathrm{m}}^{-2}$. An ideal collecting area of $1{\mathrm{m}}^2$ at $20$ parsecs intercepts approximately $2.09\times 10^{-11}\mathrm{W}$, before detector losses.

PASP 85, 573–578. Primary classical treatment of parallax-based luminosity calibration under a spatial-population model. Its assumptions do not justify a universal correction to every modern individual-star distance inference. Bibliographic details are also verified in the references of An eclipsing-binary distance to the Large Magellanic Cloud accurate to two per cent by G. Pietrzyński and others.