Use continuum shape as a conditional thermometer

Start with the shape, not the apparent brightness

Set the temperature to 6000 K and choose radius 1 nominal solar radius at 10 parsecs. This is the fixed teaching reference as well as the default selected object. Its ideal continuum peaks near 482.962 nm when radiation is expressed as energy per unit wavelength. Predict what happens to that peak when temperature is halved, before selecting 3000 K.

The 3000 K peak moves to about 965.924 nm, twice the wavelength. At 10000 K the peak moves to about 289.777 nm. A hotter blackbody has its wavelength-density peak at a shorter wavelength. That is a statement about a broad continuum, not the identification of an atomic transition from the preceding spectrum investigation.

The emitter here is an ideal uniform spherical blackbody. It is not a recorded stellar spectrum or a stellar-atmosphere calculation. NASA’s thermal-radiation teaching material provides context for continuum radiation and the distinction between radiance and flux. Real spectra contain lines and atmospheric effects omitted here.

Use the wavelength peak as a conditional thermometer

The model follows λpeak = b/T, where T is absolute temperature in kelvin and b is approximately 0.002897772 m K. A nanometer is one billionth of a meter, so the same relation is λpeak in nm = 2,897,772/T. At 6000 K, division gives about 483 nm.

For a paper exercise, make a table for 3000, 6000 and 10000 K. Calculate each peak, then mark it on a shared wavelength axis before checking the curves. Label 400–700 nm as an approximate visible interval, not the complete radiation output.

This peak convention matters. Energy per unit frequency and photon number have different peak definitions. Converting a frequency-density peak with c divided by frequency does not produce this wavelength-density peak. A peak also does not mean that the star emits just one color: a perceived color depends on a broad spectrum and the observer or detector response.

Keep normalized shape separate from absolute power

The shape chart divides each spectrum by its own analytic peak value. Both maxima are therefore one. This dimensionless normalization makes shapes comparable but intentionally removes amplitude information. It is not a probability density and its area is not required to equal one.

Compare 3000 and 6000 K in the same radius-and-distance case. The normalized curves both reach one, while total emitted and received powers differ by a factor of sixteen. Equal normalized maxima therefore do not imply equal luminosity or equal apparent brightness.

The second chart preserves absolute received spectral flux in W m⁻² nm⁻¹. Its vertical axis is logarithmic: one decade represents a factor of ten, and there is no zero on the scale. Do not estimate integrated power from the visual area beneath that logarithmic curve. Use the bolometric ledger for the all-wavelength result.

Respect wavelength coverage and physical assumptions

The plotted window is 200–3000 nm. Radiation outside that window is not zero; the ultraviolet and infrared tails continue. The numerical bolometric values integrate all wavelengths analytically rather than treating the plotted window as the entire spectrum.

All continuum wavelengths in this investigation are vacuum wavelengths. The earlier atomic references were explicitly air wavelengths. They should not be overlaid as numerically identical without a conversion. The source is also assumed unobscured: dust reddening, redshift, blended binaries and detector bandpasses are absent.

For a real star, effective temperature is defined through total surface flux and need not equal a directly inferred color temperature without atmospheric modeling. Here the prescribed temperature controls both because the emitter is exactly a blackbody. The temperature-radius choices are mathematical comparisons, not an evolutionary sequence.

Transfer to an incomplete measurement window

A hypothetical detector measures only 400–700 nm, and the recorded continuum keeps increasing toward its 700 nm edge. A learner calls 700 nm the true peak and reports T = 2,897,772/700. What should be checked first?

Check the stellar-continuum transfer

The largest value inside a limited window need not be the full-spectrum peak. Extend or otherwise constrain the wavelength coverage before applying the peak relation as an equality. A true vacuum wavelength-density peak at 700 nm would imply about 4140 K under the blackbody assumptions, but an edge maximum alone does not establish that peak.