---
title: "From inverse-square light to a calibrated stellar distance"
description: "Derive magnitudes, distance modulus and the Cepheid period–luminosity relation, with worked numbers and explained calibration exercises."
canonical_url: "https://sandboxphysics.com/stories/distance-ladder/from-light-to-distance/"
language: "en"
page_type: "story"
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  zh-CN: "https://sandboxphysics.com/zh/stories/distance-ladder/from-light-to-distance/"
last_modified: "2026-10-28"
---

[How the cosmic distance ladder acquires a scale](https://sandboxphysics.com/stories/distance-ladder/)

L1 · Cosmic distance ladder

# From inverse-square light to a calibrated stellar distance

Sandbox Physics · 2026-10-28

A calibrated stellar luminosity converts measured flux into distance. The calculation begins with a sphere: the same emitted power crosses a larger area as light travels outward. Logarithmic magnitudes turn that inverse-square dependence into an additive distance modulus.

## Before you begin

High-school powers and logarithms. Begin with Why a nearby distance changes the scale of the universe. We define flux, luminosity, magnitude and distance modulus here.

## 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.

$$
F=\frac{L}{4\pi d^2},\qquad d\,[\mathrm{pc}]=\frac{1}{\varpi\,[\mathrm{arcsec}]}
$$

Luminosity and received flux must refer to the same passband. The parallax relation uses angular semiamplitude and a one-AU baseline. The full peak-to-peak angular excursion corresponds to the two-AU orbit diameter.

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.

$$
m_1-m_2=-2.5\log_{10}\!\frac{F_1}{F_2},\qquad \mu\equiv m-M=5\log_{10}\!\frac{d}{10\,\mathrm{pc}}
$$

The flux ratio is dimensionless. Increasing distance raises the apparent magnitude and the distance modulus.

$$
d\,[\mathrm{Mpc}]=10^{(\mu-25)/5}
$$

One megaparsec contains a million parsecs, so the ten-parsec reference contributes the constant 25 in these units.

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.

$$
m^W=\mu+M^W_{10}+b\left[\log_{10}\!\left(\frac{P}{\mathrm{day}}\right)-1\right]+\gamma Z
$$

The quantities mW, the distance modulus and the ten-day zero point are magnitudes. The slope b is magnitudes per dex in period; gamma is magnitudes per dex in metallicity. Random scatter and covariance enter the next chapters.

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 | $6.853491\,\mathrm{day}$ |
| Prepared observation | $23.15228653$ |
| Restored Wesenheit value | $23.69131604$ |
| Assumed modulus and distance | $28.94982521,\quad6.165\,\mathrm{Mpc}$ |

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$-4.01\pm0.04$. 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.

$$
m_{\mathrm{tip},0}=M_{\mathrm{tip}}+\mu=-4.01+29.398=25.388
$$

The subscript zero marks extinction correction. The numerical example holds the source calibration convention fixed.

Reading: Reid · Riess et al.

## Try the observation

### [Check the sign of a distance calibration](https://sandboxphysics.com/labs/supernova-distance-expansion/)

1. Keep Distance ladder selected. Increase the adopted anchor modulus by 0.10 and predict the sign of every inferred-distance change before reading it.
2. Explain why the photographed Cepheid periods and brightnesses remain fixed when an adopted distance changes.

## Exercises and explained answers

A star is moved to twice its distance. Find the flux ratio and magnitude change.

The flux becomes one quarter. The magnitude rises by 2.5 log 10(4), or 1.50515 magnitudes. A positive magnitude change correctly means fainter light.

A calibration makes the assumed absolute magnitude 0.10 magnitude brighter. At fixed apparent magnitude, what happens to distance?

Brighter means the absolute magnitude decreases by 0.10. The distance modulus therefore increases by 0.10, and distance is multiplied by 10 to the power 0.02, or 1.04713.

A student uses the released row 0 value as an ordinary F160W magnitude and fits the full−3.3 period slope again. Diagnose both mistakes.

The prepared value already combines the Wesenheit/color convention and a subtracted base period slope. It must be modeled with a slope correction, or restored before fitting a full slope. F160W alone and Wesenheit are different observables.

[Previous: Why a nearby distance changes the scale of the universe](https://sandboxphysics.com/stories/distance-ladder/why-distance-scales-travel/) [Next: Join Cepheids and supernovae in one distance model](https://sandboxphysics.com/stories/distance-ladder/joining-the-rungs/)

## Sources and calculation scope

1. [Riess et al., A Comprehensive Measurement of the Local Value of the Hubble Constant, arXiv:2112.04510 v 3 (2022)](https://arxiv.org/abs/2112.04510v3)

   Sections 2.1 and 4 define the joint model and calibrations. Section 5, Table 5 and Figures 14–16 report the full fit and its correlations; sections 6–7 discuss variants and comparisons. All paper-specific claims here refer to this fixed version.

2. [PantheonPlusSH0ES DataRelease, commit c447f0fea703fcd0fff57de5000947b5ca81286b](https://github.com/PantheonPlusSH0ES/DataRelease/tree/c447f0fea703fcd0fff57de5000947b5ca81286b/SH0ES_Data)

   The ally, alll and allc FITS arrays contain 3492 prepared observations,47 parameter columns and the full covariance. The public tutorial retains 87 observations and every nonzero model column they use. The table 2.README explicitly warns against refitting from its diagonal errors.

3. [Reid, Pesce & Riess, An Improved Distance to NGC4258 and its Implications for the Hubble Constant, arXiv:1908.05625 v 2 (2019)](https://arxiv.org/abs/1908.05625v2)

   The geometric maser distance is 7.576 Mpc, with statistical uncertainty 0.082 Mpc and systematic uncertainty 0.076 Mpc. The paper also calibrates the tip of the red giant branch with this anchor, including a discussion of common starting and ending rungs.

4. [Yuan et al., A New Cepheid Distance to the Maser-Host Galaxy NGC4258, arXiv:2203.06681 v 1 (2022)](https://arxiv.org/abs/2203.06681v1)

   Sections 2–4 describe HST observations, Cepheid selection and period–luminosity fitting. The discovery catalogue and the infrared-selected calibration sample have different sizes; the matrix release contains 443 NGC4258 Cepheids.