Questions: Stellar Effective Temperature and Color Index

Question 1

An astronomer measures two stars: Star A has a B−V color index of −0.3, and Star B has a B−V color index of +1.5. Which star has the higher effective temperature?

• A. Star B — a larger B−V value indicates greater total energy output and therefore higher temperature
• B. Star A — a more negative B−V means more flux at blue wavelengths relative to visual, indicating a hotter blackbody spectrum
• C. They are equally hot — color index measures apparent brightness, not temperature
• D. Star B — red stars burn more slowly and therefore maintain higher core temperatures

Answer: (B)

B−V color index measures the difference in brightness between the blue (B) and visual (V) filter bands. A hot star peaks at shorter wavelengths and emits more flux in B than V, giving a small or negative B−V. A cool star peaks at longer wavelengths, emitting relatively more in V than B, giving a large positive B−V. Star A (B−V = −0.3) is blue and hot; Star B (B−V = +1.5) is red and cool. This is a direct observational implementation of Wien's displacement law.

Question 2

Two stars have the same spectral type (both are G2, like the Sun) but one is 100 times more luminous. What explains the luminosity difference?

• A. Nothing — stars of the same spectral type must have the same luminosity by definition
• B. The more luminous star must have a higher effective temperature, since luminosity depends only on T_eff
• C. The more luminous star has a larger radius — since L = 4πR²σT_eff⁴, same T_eff with larger R produces higher luminosity
• D. The more luminous star is closer to Earth, making it appear brighter

Answer: (C)

The Stefan-Boltzmann relation L = 4πR²σT_eff⁴ shows that luminosity depends on both radius and effective temperature. Two stars with identical spectral type have nearly the same T_eff (their spectral classification is based on temperature), so the luminosity difference must come from different radii. This is exactly the distinction between main-sequence stars and giants/supergiants: an A-type supergiant and an A-type main-sequence star have similar colors and T_eff but vastly different luminosities because the supergiant is enormously larger. Spectral type alone does not determine luminosity.

Question 3

A star's effective temperature is the actual temperature measured at a specific physical layer of the star, such as the photosphere.

• T. True
• F. False

Answer: False

Effective temperature is a model quantity, not a direct physical measurement. It is defined as the temperature of a hypothetical blackbody that would emit the same total power per unit surface area as the star: L = 4πR²σT_eff⁴. Real stellar photospheres have temperature gradients — they are hotter deeper in and cooler at the top. T_eff is a single representative number capturing the star's overall radiative output, derived from the Stefan-Boltzmann law using the observed luminosity and radius. It is an extremely useful abstraction, but it is not 'the temperature at a layer.'

Question 4

The spectral classification sequence O-B-A-F-G-K-M is fundamentally a temperature sequence, with O stars being the hottest and M stars the coolest.

• T. True
• F. False

Answer: True

The sequence was originally organized by hydrogen line strength, but it was later recognized that hydrogen line intensity peaks at intermediate temperatures (A-type stars) because the lines require a specific balance of excitation and ionization. Reordered by temperature, the sequence runs hot to cool: O (>30,000 K), B (~10,000–30,000 K), A (~7,500–10,000 K), F (~6,000–7,500 K), G (~5,200–6,000 K), K (~3,700–5,200 K), M (~2,400–3,700 K). The different absorption features that define each class arise because temperature controls which atomic states and molecular species exist in stellar atmospheres.

Question 5

Why does a star's color indicate its surface temperature? Explain the physical principle connecting them.

Model answer: Stars approximate blackbodies, which emit radiation with a spectral peak determined by temperature via Wien's displacement law: λ_peak = b/T (where b ≈ 2.9 × 10⁻³ m·K). Hotter stars peak at shorter (bluer) wavelengths; cooler stars peak at longer (redder) wavelengths. Astronomers measure this through color indices: by comparing a star's brightness in blue (B) versus visual (V) filters, the ratio of fluxes at two wavelengths constrains the shape of the spectral energy distribution and thus the effective temperature. A blue star has most of its flux at short wavelengths (high T_eff); a red star has most at long wavelengths (low T_eff).

The connection is direct and quantitative: color is a measurement of the spectral peak's location, and Wien's law links the peak location to temperature. Interstellar dust can redden starlight (shifting observed colors toward the red), requiring correction before inferring T_eff — but this is a practical complication of the same underlying physics, not an exception to it.