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A Cepheid variable is a pulsating evolved star whose regular changes in stellar radius and temperature produce a repeating light curve. The kappa mechanism supplies driving in a suitable part of the instability strip. Large stellar radius gives a long dynamical time, connecting the pulsation period to the star's mean mass density.

The practical importance is the calibrated Cepheid period-luminosity relation. Measuring a period predicts an absolute magnitude within a specified population and spectral band; comparison with apparent magnitude then constrains distance. Extinction, composition and unresolved companions affect that comparison. Thus the variable brightness acts as a physical ruler only after calibration, rather than every pulsating star having one universal luminosity at a given period.

annual stellar parallax uses the known Earth-orbit baseline, with $d(\mathrm{pc})=1/\varpi (\mathrm{arcsec})$. Repeated positions separate parallax from proper motion; calibration offsets, binarity and noisy-parallax selection require checks. Nearby stars calibrate absolute luminosities. Extinction-corrected distance modulus then givesmain-sequence fitting compares a cluster sequence with nearby calibrated stars; age, metallicity, reddening and unresolved binaries limit the match. Cepheid variables extend the range to nearby galaxies: measure periods and mean multiband magnitudes, calibrate the Cepheid period-luminosity relation, and correct for extinction, metallicity, crowding and pulsation mode. RR Lyrae stars and the tip of the red giant branch provide complementary population-II routes with their own composition and calibration dependence.

Some anchors bypass intermediate rungs. An eclipsing-binary distance starts from a double-lined eclipsing binary: velocities give $a\sin i$; eclipses give inclination $i$ and relative radii $R/a$, hence physical radii. An independently calibrated surface-brightness/colour relation gives angular radii and distance; alternatively corrected bolometric flux satisfies $F_{\mathrm{bol}}=\sigma T_{\mathrm{eff}}^4(R/d{)}^2$ for a component. temperature, extinction, third light and surface-brightness calibration remain systematic errors. Eight late-type binaries gave a 2.2%-accuracy LMC anchor in 2013.^[ref]

A maser-disk geometric distance uses water masers orbiting in the nearly edge-on nuclear disk of NGC 4258. very-long-baseline interferometry, Doppler velocities, proper motions and accelerations constrain its orbit scale and distance. In an ideal circular, matched-radius example $R=v^2/a_{\mathrm{orb}}$ and $d=R/\theta$; real fits account for disk warp and different maser locations. This geometric method calibrated an extragalactic distance independently of stellar luminosities.^[ref] These overlapping anchors cross-check the nearby-galaxy ladder; no Hubble-flow fit is needed here.

The cosmic distance ladder starts by calibrating a physical baseline. Radar echoes from planets or spacecraft give a round-trip light time and hence a distance $d=c\Delta t/2$, after accounting for orbital motion and propagation delays. Combining this with orbital geometry fixes the Earth-Sun orbital scale; directly timing a reflected pulse from the solar surface is not required.

That baseline supplies annual stellar parallax: for small parallax angle $p$, $d\simeq b/p$, or $d$ in parsecs equals $1/p$ when $p$ is in arcseconds and $b$ is one astronomical unit. Nearby parallax distances calibrate stellar absolute magnitudes. Main-sequence fitting extends the scale to star clusters, and the calibrated Cepheid period-luminosity relation gives luminosities for Cepheids visible in nearby galaxies.

Cepheid-host galaxies then calibrate Type Ia supernovae. After light-curve and colour standardization, their high luminosities permit distances to much more remote galaxies. For a source of calibrated luminosity $L$ and observed bolometric flux $F$,with consistent bandpass definitions and corrections. Interstellar extinction, composition, stellar-population differences and calibration uncertainty must be propagated through the steps. Redshift can subsequently provide a distance within a fitted cosmological model; using it alone cannot independently calibrate the Hubble constant.

Interpret the quoted values as the paper's 2006-era rounded cosmological benchmark. The Hubble constant is the present slope of the Hubble law, $v\simeq H_{0}d$ at sufficiently small redshift. A cosmic distance ladder begins with geometrical distance anchors, calibrates the Cepheid period-luminosity relation of Cepheid variables, and uses those distances to calibrate secondary indicators such as Type Ia supernovae. Distances to objects far enough away that peculiar velocities are a small fraction of recession velocity then determine the slope. Extinction, metallicity effects, distance-anchor calibration, sample selection and peculiar velocities contribute to its uncertainty.

The Hubble Space Telescope Key Project final result was $H_0=72\pm 8\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$, combining random and systematic uncertainty, supporting the rounded value $70$. Independent approaches include distances inferred from gravitational-lens time delays and from cluster X-ray and Sunyaev-Zeldovich effect measurements; each brings different model uncertainties. Joint fits to the cosmic microwave background and large-scale structure also constrain $H_0$ within an assumed cosmological model. Thus the evidence for a value near $70$ combines calibrated distances with independent cross-checks; the approximate precision in the question should not be read as an identical uncertainty for every method.