Cosmology · Light, redshift and first distance clues

Brightness, intrinsic luminosity and distance

A faint source may be distant, weak or obscured. Build the inverse-square relation from a spreading energy budget, then identify the additional information needed for a useful distance estimate.

Preparation: ratios, powers of ten and ordinary arithmetic. Physical quantities and notation are introduced as they are needed.

Back to the chapter

After reading, you can

Distinguish source luminosity from received energy flux.
Use the inverse-square law under its stated geometric and propagation assumptions.
Explain the luminosity-distance definition without confusing it with a simultaneous cosmic separation.

What belongs to the source and what belongs to the measurement?

Two lamps can look equally bright even though one emits more power. The stronger lamp can be farther away, and a screen can dim either one. The apparent brightness at your position therefore does not uniquely identify either the source's power or its distance.

Luminosity L is energy emitted per unit time. Its SI unit is the watt, W, equal to one joule per second. Here we begin with bolometric luminosity, which includes the electromagnetic power integrated across all wavelengths. Received energy flux F is power crossing a unit collecting area, expressed in watts per square metre, W/m².

A detector does not directly read the source's total L. It intercepts a small part of the arriving radiation and measures a signal through its own response. Specifying energy flux also differs from counting photons: different photon energies can give different energy fluxes at the same photon arrival rate. Ordinary 'brightness' needs this context before it enters a calculation.

Spread a conserved power across a larger area

Take an ideal steady point source emitting equally in all directions, with a transparent stationary Euclidean surrounding region. Neglect absorption, scattering, gravitational focusing and cosmological effects for this local example. The source's luminosity crosses a sphere of radius d and area 4πd².

Dividing the same emitted power among that area gives F = L/(4πd²). If d doubles, the area quadruples and the flux falls to one quarter. The decrease comes from geometric spreading; it does not require each individual photon to lose energy along this idealized stationary vacuum path.

Check the units: W divided by m² gives W/m². The numerical constant 4π is dimensionless. The radius is the distance from the effective point source, not the diameter of the sphere and not the detector's own collecting radius.

This formula applies to an unresolved source treated with the stated geometry. An extended source's surface brightness and a resolved image require additional distinctions. Directional emission also needs its angular emission pattern. A real star or galaxy is not guaranteed to be a perfect isotropic standard lamp merely because the relation is useful.

Same ideal source at chosen distances; normalized to the flux at d
Distance	Area relative to 4πd²	Flux relative to F at d
d	1	1
2d	4	1/4
3d	9	1/9
d/2	1/4	4

An analytic synthetic curve shows normalized received flux F/F₀ = (d₀/d)^2 for distance ratios d/d₀ from one to three, on linear axes. Hollow markers at distance ratios 1, 2 and 3 have flux ratios 1, 1/4 and 1/9. F₀ is the flux at the chosen reference distance d₀. The domain deliberately excludes zero distance. This assumes a steady isotropic point source in transparent stationary Euclidean geometry, without attenuation, focusing or cosmological effects. Original Ark teaching schematic.

A flux measurement becomes a distance only with a luminosity

If L is independently supplied, rearrange the local relation to d = √[L/(4πF)]. The square root matters. Halving the measured flux does not double the inferred distance; at fixed luminosity it multiplies the distance by √2.

Use a deliberately artificial isotropic source with L = 400π W. At d = 10 m, F = (400π W)/(4π × 100 m²) = 1 W/m². At 20 m the flux is 0.25 W/m². These chosen values make the geometry visible; they are not a model of a household lamp's detailed spectrum or of a particular astronomical object.

Now leave L unknown. A source four times as luminous at twice the distance gives exactly the same F in this model. One flux measurement cannot distinguish those possibilities. To use a standard candle, we need a justified estimate or calibrated relation for the luminosity, with its scatter, selection effects and possible evolution included.

Uncertainty in the source scale propagates into the inferred distance. At fixed flux, a luminosity estimate four times larger produces a distance twice as large. This is why calibrating the source physics is part of the distance method, not an optional detail after the measurement.

Retain the relation by defining luminosity distance

At cosmological scales, expansion changes photon energies and arrival intervals, while geometry relates the emitting and receiving areas. We should not silently substitute a present-time proper separation into the stationary local formula.

Instead define luminosity distance Dₗ through the bolometric relation F = L/(4πDₗ²), using the appropriate rest-frame luminosity and received bolometric flux. Dₗ is the distance-like quantity that makes this relation hold in the specified propagation model. It can differ from present proper, emission proper and angular-diameter distances.

A definition is not an independent observation of Dₗ. Inferring it still requires L and treatment of attenuation, focusing and calibration. Relating Dₗ to redshift requires a cosmological model. Later distance lessons derive those relationships; here the essential habit is to retain the definition attached to the quoted length.

A filter measures a band, not every emitted wavelength

An astronomical detector often measures through a limited band. As redshift changes, that observed band corresponds to different rest-frame wavelengths at the source. Comparing two objects' numbers without accounting for this can mix different parts of their spectra.

A band correction therefore depends on the source spectrum as well as the instrument response. Do not apply the bolometric formula directly to a single-band number while pretending that it contains all emitted power. Likewise, dust can reduce and reshape the arriving radiation rather than simply move the source farther away.

A useful distance statement records the source luminosity calibration, wavelength coverage, propagation corrections and distance definition. The received flux is the starting measurement. The distance is an inference whose reliability depends on those additional pieces.

Worked reasoning

Construct a brightness ambiguity

In the transparent stationary point-source model, source A has luminosity L at distance d. Source B has luminosity 9L. At what distance would B give the same energy flux?

Set 9L/(4πdᵦ²) = L/(4πd²).
Cancel common factors to obtain dᵦ² = 9d².
Take the positive distance: dᵦ = 3d.

B can be three times farther away and have exactly the same flux as A in this model.

Try the reasoning

At fixed supplied luminosity, a source has twice the measured flux of another. What is its inferred distance relative to the other in the ideal local model?
Show the answer and reasoning
1/√2, about 0.707 times the distance.
Distance varies as the inverse square root of flux. This inference assumes the same intrinsic luminosity and propagation conditions.
A screen transmits one quarter of the flux. If its attenuation is ignored, by what factor is a known-luminosity source's distance overestimated?
Show the answer and reasoning
By a factor of 2.
Replacing F by F/4 multiplies √[L/(4πF)] by √4 = 2. This chosen screen is an idealized attenuation model.
Does a cosmological luminosity distance automatically equal the source's present proper separation?
Show the answer and reasoning
No.
Luminosity distance is defined through luminosity and received flux. A simultaneous proper separation uses a different definition in the spacetime model.

What to carry forward

Luminosity is emitted power; received energy flux is power per collecting area.
The inverse-square law follows from spreading an isotropic source's power over spherical area under stated conditions.
Flux alone cannot distinguish a weak nearby source from a stronger distant one.
Cosmological luminosity distance and single-band corrections require explicit definitions and models.

References

Sources supporting the explanations, physical distinctions and examples in this lesson.