Separate emitted power from received flux

Move the observer without changing the source

Keep temperature at 6000 K and radius at R₀. Compare distances 5, 10 and 20 pc. Predict what changes at the source when only the observer distance changes. The surface temperature, spectrum emitted per square meter and total luminosity should remain the same.

Received flux decreases with distance because the same outward power is distributed over a larger sphere. The geometry diagram uses separate not-to-scale views for source radius and observer distance. Their actual size ratio is far too large to draw usefully on one page, and the diagram must not suggest that the star physically shrinks or cools as the observer moves away.

On paper, draw concentric receiving spheres at d and 2d. Label their areas 4πd² and 16πd². The second sphere has four times the area across which the same luminosity passes.

Calculate the inverse-square dilution

Received bolometric flux is F = L/(4πd²). Substituting L = 4πR²M gives F = M(R/d)². Both radius and distance must be expressed in compatible spatial units; the model uses meters.

A parsec is a distance unit, not a time interval. It is defined as (648000/π) astronomical units, about 3.08568 × 10¹⁶ m. The astronomical unit is exactly 149,597,870,700 m. See the IAU distance-unit notes and JPL astronomical-unit definition. This lab does not measure parallax; the distance is a prescribed input.

At the default 10 pc, received bolometric flux is about 3.73559 × 10⁻¹⁰ W/m². At 20 pc it is one quarter as much, about 9.33898 × 10⁻¹¹ W/m². At 5 pc it is four times the default, about 1.49424 × 10⁻⁹ W/m².

Calculate those factors before copying the model’s decimals. The distance ratios alone are sufficient for this controlled experiment. Keep the unchanged luminosity beside every row so the lower received flux is not mislabeled as reduced emitted power.

Read the spectrum at a different distance

At fixed temperature and radius, every wavelength’s received spectral flux scales by the same inverse-square factor. The wavelength peak and peak-normalized shape do not move. The absolute spectrum shifts in amplitude, while the dimensionless shape curve remains coincident with its previous version.

The absolute spectrum uses a logarithmic vertical axis. A factor of ten corresponds to one full decade; a factor of four is less than a decade. Equal vertical distances on this plot represent equal ratios, not equal additive W m⁻² nm⁻¹ differences. Use the spectral ledger when you need a numerical comparison at 300, 500, 1000 or 2000 nm.

Bolometric received flux integrates the whole spectrum. The plotted 200–3000 nm window leaves radiation outside its boundaries. Neither the curve’s peak height nor its apparent area on a logarithmic graph is the all-wavelength received power per area.

State the conditions behind the relation

The model assumes an isolated steady spherical source, isotropic outward power at large distances and transparent vacuum between source and observer. There is no extinction, reddening, detector response, redshift or unresolved companion. Under those assumptions, distance alone changes amplitude but not shape.

For real observations, apparent brightness can also differ because intrinsic luminosity differs or light is altered along its path. A brightness ratio therefore becomes a distance ratio only when the relevant intrinsic luminosities and propagation assumptions are established independently.

Write two sentences after your controlled comparison: one explaining the model result when distance alone changes, and one naming additional evidence needed when two unknown sources appear different. This separates a forward prediction from an inverse inference.

Transfer from a brightness ratio

An unknown source appears four times brighter than another in calibrated bolometric flux. Is it necessarily half as far away? State a conditional explanation and a competing possibility.

Check the stellar-distance transfer

It would be half as far away if both sources had the same luminosity and unobscured isotropic emission under the model assumptions. It could instead be four times as luminous at the same distance, or differ in both properties. Independent luminosity or distance evidence is needed.