Spread the same power over a sphere
Let luminosity L be emitted energy per second in a specified band, measured in watts. Let F be received energy per second per square metre in the same band. Use an isotropic source in transparent Euclidean space, with distance d in metres. Each sphere around the source receives the same total power. Dividing by its surface area gives the flux.
Double the distance and the area becomes four times as large. Each square metre receives a quarter of the flux. An angular parallax provides an independent nearby distance: a one-astronomical-unit baseline subtends one arcsecond at one parsec. Inverting a noisy measured parallax needs an error model; this elementary relation describes the ideal geometry.
Reading: Riess et al. · Reid
Subtract magnitudes to obtain a distance modulus
A magnitude assigns a smaller number to brighter light. For two fluxes measured with the same zero point, their magnitude difference is minus 2.5 times the base-ten logarithm of their flux ratio. Define the absolute magnitude M as the apparent magnitude the same source would have at ten parsecs, with extinction removed. The apparent magnitude m used here is corrected to the same passband convention.
Divide the flux at distance d by the flux at ten parsecs, then substitute the inverse-square ratio into the magnitude difference. The factor two from distance squared multiplies 2.5 into 5. The resulting distance modulus is an additive shift shared by all stars at one galaxy distance. For cosmological sources, luminosity distance takes the place of Euclidean distance and passband K corrections belong in the brightness model.
Reading: Riess et al.
The pulsation period selects a mean luminosity
Cepheids are useful because their period predicts a mean luminosity after population and observational corrections. We use period P in days and center its logarithm at ten days. This centering makes the zero point the mean absolute magnitude of a ten-day star. A negative slope means that longer-period stars are brighter.
A Wesenheit magnitude combines several measured bands with a fixed color coefficient to reduce extinction sensitivity. It is a constructed brightness measure, not a new detector band. Residual extinction-law differences, crowding and metallicity can still matter. Let Z be metallicity relative to the chosen reference in dex, and let gamma describe its fitted brightness dependence.
Reading: Riess et al. · Yuan et al.
Worked example from one prepared catalogue row
Take M101 star 54672, original matrix row 0. Its period is 6.853491 days. The published matrix first subtracts a base period slope of−3.285 magnitudes per dex; its stored value 23.15228653 is therefore a prepared value, not a raw H-band magnitude. Restoring the base period term gives the Wesenheit value 23.69131604. Do not confuse this with its separately listed, rounded F160W magnitude 24.04.
For a simple numerical example, assume a ten-day zero point−5.80, slope−3.30 and ignore metallicity for this one calculation. The period offset is−0.16408813 dex, so the predicted absolute magnitude is−5.25850917. Subtracting that from 23.69131604 gives modulus 28.94982521 and distance 6.165 Mpc. These assumed calibration values serve the arithmetic example; the later eight-parameter fit estimates them jointly.
| Quantity | Saved or calculated value |
|---|---|
| Period | |
| Prepared observation | |
| Restored Wesenheit value | |
| Assumed modulus and distance |
Reading: PantheonPlusSH0ES DataRelease · Riess et al.
How a geometric anchor fixes the zero point
NGC4258 contains water masers whose angular positions, Doppler velocities and accelerations support a geometric disk-distance model. The 2019 result is 7.576 Mpc with separate 0.082 statistical and 0.076 systematic Mpc uncertainties. Riess et al. use a reference distance modulus 29.398 with about 0.032 magnitude uncertainty. The selected matrix retains its exact released anchor constraint.
At a known distance, move the distance term to the other side of the Cepheid relation and fit the luminosity zero point. A red-giant-tip calibration uses the same logic with the sharp edge of the luminosity function in an old population. The stellar physics and sample selection differ; the geometric starting distance can be shared. Keeping that shared dependence is essential when the two routes are compared.
Reading: Reid · Riess et al.
An old stellar population supplies a second calibrated ruler
The tip of the red giant branch uses a sharp upper edge in the brightness distribution of old red giants. Its calibration depends on the selected band, color and stellar population. In the F814W convention discussed by Reid et al., the NGC4258 anchor gives an absolute tip magnitude near. A sufficiently resolved population can supply this edge even where young Cepheids are scarce.
For a calculated example, hold that color and population calibration fixed and adopt the reference modulus29.398. Add the calibrated absolute magnitude to the modulus: the predicted extinction-corrected apparent tip is25.388. This is a calculation from the stated calibration, with both terms in the same photometric convention. It shows how a different stellar indicator can share the same distance anchor.
Reading: Reid · Riess et al.