Surface Flux & Colors of Stars

Useful constants: $b = 2.898 \times 10^6$ nm·K $= 0.2898$ cm·K; $\sigma = 5.67 \times 10^{-5}$ erg cm⁻² s⁻¹ K⁻⁴; $1$ pc $= 3.086 \times 10^{18}$ cm; $L_\odot = 3.828 \times 10^{33}$ erg/s; $T_\odot = 5800$ K; $R_\odot = 6.96 \times 10^{10}$ cm $\approx 696{,}000$ km; $1$ km $= 10^5$ cm.

Show explicit units, and for each result run a sanity check. Worked solutions are released after the homework due date.

Conceptual

Problem

⭐ Two stars, same luminosity, different colors. Star A is blue; Star B is red; both have the same luminosity.

(a) Which has the higher effective temperature?
(b) Which must have the larger radius? Explain using $L = 4\pi R^2 \sigma T^4$ .
(c) Which surface radiates more power per cm²? How do you know?

Problem

⭐ Why Betelgeuse is huge. Betelgeuse has $L \sim 10^5 L_\odot$ and $T \sim 3{,}500$ K (~60% of the Sun’s). Use Stefan-Boltzmann to explain qualitatively why it must be so large despite being cooler than the Sun. Reference $F_* = \sigma T^4$ and surface area.

Problem

⭐⭐ Peak wavelength, temperature, and radius. A star peaks at $\lambda_{\text{peak}} = 1000$ nm; the Sun peaks at 500 nm.

(a) Hotter or cooler than the Sun, and by what factor? (Wien as a ratio.)
(b) At the same luminosity, which has the larger radius, and by what factor? ( $R \propto T^{-2}$ .)
(c) Which has the greater surface flux $F_* = \sigma T^4$ , and by what factor?

Problem

⭐⭐ Surface brightness independence. Explain why you cannot determine a star’s distance from how bright its disk appears per unit solid angle, given that $\text{SB} = F/\Omega$ is distance-independent. Does this apply most directly to unresolved point sources or to resolved objects?

Calculation

Problem

⭐ Sun’s temperature from Wien’s law. The Sun peaks at $\lambda_{\text{peak}} = 500$ nm.

(a) Convert 500 nm to cm.
(b) Use $T = b/\lambda_{\text{peak}}$ with $b = 0.2898$ cm·K; show the unit cancellation.
(c) Compare to the known 5800 K.

Problem

⭐ Stellar temperature from color. A star peaks at 700 nm (red).

(a) Calculate its effective temperature.
(b) Hotter or cooler than the Sun, and by what factor?

Problem

⭐⭐ From received flux to luminosity. A star at $d = 10$ pc has bolometric flux $F = 8.0 \times 10^{-6}$ erg s⁻¹ cm⁻².

(a) Convert $d$ to cm.
(b) Use $L = 4\pi d^2 F$ ; show units at every step.
(c) Express in $L/L_\odot$ . Does it remind you of a worked-example star?

Problem

⭐⭐ Stellar radius (ratio method). A star has $L = 4 L_\odot$ , $T = 2 T_\odot$ .

(a) Show $(R/R_\odot)^2 = (L/L_\odot)/(T/T_\odot)^4 = 4/16 = 0.25$ .
(b) Raise to the $1/2$ power: $R/R_\odot = ?$
(c) Larger or smaller than the Sun? Convert to km.

Problem

⭐⭐ Rigel’s radius (full calculation). Rigel has $L = 120{,}000\,L_\odot$ and $T = 12{,}100$ K.

(a) Compute $T_{\text{Rigel}}/T_\odot$ (verify dimensionless).
(b) Raise to the fourth power.
(c) Compute $(R/R_\odot)^2$ , then the $1/2$ power for $R/R_\odot$ .
(d) Express $R_{\text{Rigel}}$ in km.
(e) Compare to Betelgeuse ( $\approx 870\,R_\odot$ ) — why is one so much larger?

Synthesis

Problem

⭐⭐ Complete inference chain. A red giant: $F = 1.28 \times 10^{-6}$ erg s⁻¹ cm⁻², $\lambda_{\text{peak}} = 1{,}000$ nm, $d = 50$ pc.

(a) Temperature from Wien (show units), then $T/T_\odot$ .
(b) Convert $d$ to cm; use $L = 4\pi d^2 F$ ; express as $L/L_\odot$ .
(c) Radius in solar units: $(R/R_\odot)^2$ then the $1/2$ power.
(d) Convert to km; compare to the Sun and to Betelgeuse. Does cool + large make sense?

Problem

⭐⭐ HR diagram reasoning. Three stars: A ( $T = 6{,}000$ K, $L = 1 L_\odot$ ); B ( $T = 3{,}000$ K, $L = 100 L_\odot$ ); C ( $T = 12{,}000$ K, $L = 1 L_\odot$ ).

(a) Radius of each in solar radii ( $(R/R_\odot)^2$ first, then $1/2$ power).
(b) Largest? Smallest?
(c) Sketch on an HR diagram ( $T$ increasing leftward) with lines of constant radius. What do they tell you about giants and white dwarfs?

Problem

⭐⭐⭐ Extinction’s effect on temperature inference. Dust reddens starlight. An observer finds $\lambda_{\text{obs}} = 700$ nm; the true (dust-corrected) peak is $\lambda_{\text{true}} = 500$ nm.

(a) Apparent temperature from the reddened color (show units).
(b) True temperature from the corrected color.
(c) Bias factor $T_{\text{apparent}}/T_{\text{true}}$ .
(d) Using the reddened temperature at the correct luminosity, would the inferred radius be too large or too small, and by what factor? ( $R \propto T^{-2}$ .)