Homework 3

Parallax + Surface Flux & Radiation Inference

Assignment Info

Due	Tuesday, February 17, 2026 at 11:59 pm
Grade Memo	Friday, February 20, 2026 at 11:59 pm
Est. Time	4–6 hours
Submit via	Canvas; upload a single clearly scanned PDF or a legible PDF export

Learning Objectives

• Use parallax geometry to infer stellar distance and avoid common interpretation errors.
• Apply the inverse-square law to connect observed flux, luminosity, and distance.
• Propagate uncertainty along the chain: parallax → distance → luminosity.
• Use Wien’s law to infer effective temperature from peak wavelength/color.
• Use Stefan-Boltzmann scaling to infer surface flux and stellar radius.
• Synthesize the full observable → model → inference chain for a star.

Concept Throughline

• Measured angular shift gives distance.
• Distance plus measured flux gives intrinsic luminosity.
• Color gives temperature.
• Luminosity plus temperature gives radius.
• Every step has assumptions; good inference means checking them explicitly.

Prerequisites

• Scientific notation and proportional reasoning
• Unit conversion (arcsec, pc, cm, nm, K)
• Module 1 light basics (Wien + Planck ideas)

Relevant Sources (Module-Based)

• Module 2 (slides): Distance & Parallax
• Module 2 (reading/source set): Surface Flux & Colors of Stars
• Module 1 review: Light as Information

Note

Before you start: Review the Homework Guidelines for required format and tools.

Tip

HW3 note: Tool hints are not shown. Part of the goal is deciding which model and scaling apply.

Use ratios whenever possible to avoid unnecessary constants.

Note

Use these constants unless a problem states otherwise:

• \(1\ \text{pc} = 206{,}265\) AU
• \(1\ \text{AU} = 1.496 \times 10^{13}\) cm
• \(1\ \text{pc} = 3.086 \times 10^{18}\) cm
• \(T_\odot = 5800\) K
• \(b = 2.898 \times 10^6\) nm·K

Note

Required reporting format:

• Every numeric answer must include units.
• Use scientific notation where appropriate.
• Include a one-line sanity check for each problem (e.g., farther should look dimmer; hotter should peak at shorter wavelength).

Tip

Sanity-check scalings:

• \(d(\mathrm{pc}) = 1/p('')\)
• \(F \propto Ld^{-2}\)
• \(T = b/\lambda_{\text{peak}}\)
• \(L \propto R^2T^4\)

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Problems (10 total)

Part A — Distance, Parallax, and Luminosity

Problem 1 — Parallax Is Not Angular Size

A student says: “This nearby star has a large parallax, so it must have a large angular size.”

• Be explicit: parallax is an apparent positional shift against distant background, not an apparent size.
• 1. Explain why this statement is incorrect.
• 2. Define parallax angle and angular size in one sentence each.
• 3. Give one observational signature that distinguishes the two measurements.

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Problem 2 — Parallax to Distance

• Do the factor comparison in pc first; convert to cm afterward.
• 1. Star A has \(p = 0.50''\). Find \(d\) in pc and in cm.
• 2. Star B has \(p = 0.020''\). Find \(d\) in pc and in cm.
• 3. Which star is farther, and by what factor?

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Problem 3 — Inverse-Square Flux Scaling

Two identical stars have the same luminosity. Star X is at 20 pc and Star Y is at 50 pc. Assume no extinction/reddening.

• 1. Compute \(F_X/F_Y\) using inverse-square scaling.
• 2. Which star appears brighter at Earth?
• 3. In one sentence, explain why equal luminosity does not imply equal observed flux.

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Problem 4 — Luminosity from Flux + Distance

A star has measured parallax \(p = 0.10''\) and bolometric flux at Earth \(F_\star = 2.0 \times 10^{-11}\,F_\odot\), where \(F_\odot\) is specifically the Sun’s bolometric flux measured at 1 AU.

• 1. Compute distance in pc and in AU.
• 2. Use \[\frac{L_\star}{L_\odot} = \frac{F_\star}{F_\odot}\left(\frac{d_\star}{1\,\text{AU}}\right)^2\] since \(F_\odot\) is measured at 1 AU.
• 3. State whether the star is more or less luminous than the Sun, and by what factor.

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Problem 5 — Same Flux, Different Parallax

Two stars have the same measured bolometric flux at Earth. Their parallaxes are:

• Star A: \(p_A = 0.100''\)

• Star B: \(p_B = 0.025''\)

• 1. Which star is farther, and by what factor?
• 2. Infer \(L_B/L_A\) using a ratio method.
• 3. In 2–3 sentences, explain why equal observed flux does not imply equal intrinsic luminosity.
• 4. Include a one-line equation showing the scaling you used at fixed observed flux.

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Part B — Radiation, Temperature, and Radius

Problem 6 — Same Luminosity, Different Color

Two stars have the same luminosity. One appears blue, the other red.

• 1. Which star is hotter?
• 2. Which star has the larger radius?
• 3. Justify your answer using Stefan-Boltzmann scaling, not just verbal intuition.

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Problem 7 — Temperature from Wien’s Law

A star’s spectrum peaks at \(\lambda_{\text{peak}} = 725\) nm.

• 1. Compute the star’s effective temperature from the peak wavelength (report to 2 significant figures).
• 2. Compare this temperature to the Sun’s by computing \(T/T_\odot\).
• 3. State whether this star is hotter or cooler than the Sun.

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Problem 8 — Planck Limits and Physical Meaning

Goal: show how Planck’s law reduces to Rayleigh-Jeans behavior at long wavelength and becomes exponentially suppressed at short wavelength.

Consider the Planck function \[B_\lambda(T) = \frac{2hc^2}{\lambda^5}\frac{1}{e^{hc/\lambda k_B T}-1}.\]

• 1. Define \(x = hc/(\lambda k_B T)\). In the long-wavelength limit (\(x \ll 1\)), use an appropriate approximation to show the leading scaling of \(B_\lambda\) with \(T\) and \(\lambda\). (Hint: \(e^x \approx 1+x\) for \(x\ll1\).)
• 2. In the short-wavelength limit (\(x \gg 1\)), show the leading scaling behavior of \(B_\lambda\) and identify which term suppresses emission. (Hint: \(e^x-1 \approx e^x\) for \(x\gg1\).)
• 3. Explain in words why these two limits resolve the ultraviolet-catastrophe issue from classical physics.
• 4. Connect your result to the statement: “photons are expensive when \(h\nu \gg k_B T\).”

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Problem 9 — Radius from Luminosity and Temperature

A star has luminosity \(L = 16\,L_\odot\) and temperature \(T = 2\,T_\odot\).

• 1. Solve for \((R/R_\odot)^2\).
• 2. Solve for \(R/R_\odot\).
• 3. Interpret whether the star is physically larger or smaller than the Sun.

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Problem 10 — Full Chain: Parallax + Flux + Color

You observe a star with:

• Parallax: \(p = 0.050''\)

• Bolometric flux at Earth: \(F_\star = 1.5 \times 10^{-12}\,F_\odot\)

• Peak wavelength: \(\lambda_{\text{peak}} = 500\) nm

• 1. Compute distance in pc and in AU.
• 2. Infer luminosity in units of \(L_\odot\) using a ratio method.
• 3. Compute temperature from the peak wavelength.
• 4. Compute radius in solar units using the appropriate radiation relation.
• 5. Assume the parallax is measured as \(p = 0.050'' \pm 0.003''\). Estimate the fractional uncertainty in distance, and then in luminosity (assuming flux uncertainty is negligible). (Use fractional uncertainty: if \(d \propto p^{-1}\), then \(\delta d/d \approx \delta p/p\).)
• 6. In 2–4 sentences, summarize the full observable → model → inference chain you used.