Lecture 1 Solutions: Spoiler Alerts
Practice Problem Solutions
Student note: These are model solutions written to show every step, units, and checks. I do not expect your work to be this detailed; shorter solutions are fine if your setup, units, and reasoning are correct.
HW note: Homework uses only a subset of these problems; the full set is included for future study and self checking.
Conceptual Questions
Problem 1: Observable vs. Inferred
Restatement: Decide whether each quantity is directly measured or inferred.
Key insight: The four direct observables are brightness, position, wavelength, and timing. Most physical properties are inferred from those measurements plus physics.
Answer:
| Quantity | Measured or Inferred? | Why |
|---|---|---|
| (a) The brightness of a star | Directly measured | We count photons arriving at the detector (flux). |
| (b) The temperature of a star | Inferred | From spectrum or color using physics (hotter objects are bluer). |
| (c) The position of a star on the sky | Directly measured | We record where it appears in images (angles on the sky). |
| (d) The distance to a star | Inferred | From parallax, standard candles, or other distance methods. |
| (e) The wavelength of light from a star | Directly measured | We spread light into a spectrum and measure wavelengths. |
Common misconception: Treating temperature or distance as directly measurable just because they are listed in catalogs.
Problem 2: The Inference Pipeline
Restatement: Explain what is wrong with the claim that we know the Sun is hydrogen because we can see it, and state what we measure and infer.
Key insight: We measure spectra (wavelengths), then use atomic physics to infer composition.
Answer: We do not see hydrogen directly. We measure the Sun’s spectrum, specifically absorption lines at precise wavelengths. Those wavelengths match hydrogen’s atomic fingerprint, so the composition is inferred by comparing measurements to a physical model (atomic energy levels). The inference chain is: spectrum (measurement) -> absorption line wavelengths (measurement) -> atomic physics model -> hydrogen present (inference).
Common misconception: Confusing a measurement (spectrum) with the inferred property (composition).
Problem 3: Standard Candles
Restatement: Explain why astronomers need standard candles to measure distance.
Key insight: Brightness alone is ambiguous without the intrinsic luminosity.
Answer: Apparent brightness depends on both luminosity and distance. A faint nearby object and a bright distant object can look equally bright. Standard candles provide objects whose intrinsic luminosity is known, so once we measure the flux, the inverse square law lets us solve for distance.
Common misconception: Thinking brightness by itself directly gives distance.
Problem 4: Lookback Time
Restatement: If Andromeda is 2.5 million light-years away, when did the supernova occur?
Key insight: A light-year is a distance; light travel time equals the distance in years.
Answer: The explosion occurred 2.5 million years ago. The light takes 2.5 million years to reach us, so we are seeing the event long after it actually happened.
Common misconception: Thinking the explosion happened when we see it.
Problem 5: Four Observables
Restatement: Identify the observable in a redshifted absorption line measurement and what it implies.
Key insight: Wavelength is the measured observable; Doppler shifts let us infer radial motion.
Answer: The observable is wavelength, measured by spectroscopy. A redshift indicates the star is moving away from us. The amount of shift gives the radial velocity, and time variation could imply an orbiting companion.
Common misconception: Treating motion as a direct observable rather than an inference from wavelength shifts.
Calculations
Problem 6: Relative Brightness
Restatement: Two identical stars have fluxes that differ by a factor of 9. Find the distance ratio.
Given: - Star A appears 9 times brighter than Star B: \(F_A / F_B = 9\) - Identical stars, so same luminosity \(L\)
Find: \(d_B / d_A\)
Equation: \[F = \frac{L}{4\pi d^2} \quad \Rightarrow \quad \frac{F_A}{F_B} = \frac{d_B^2}{d_A^2}\]
Solution: \[9 = \frac{d_B^2}{d_A^2}\] \[\frac{d_B}{d_A} = \sqrt{9} = 3\]
Unit check: The ratio \(d_B/d_A\) is dimensionless, as expected.
Sanity check: If Star B is farther away, it should be dimmer. A factor of 3 in distance gives a factor of \(3^2 = 9\) in brightness, consistent.
Answer: Star B is 3 times farther away than Star A.
Problem 7: Light Travel Time
Restatement: Find the light travel time from the Sun to Earth and convert to minutes.
Given: - \(c = 3 \times 10^8\) m/s - \(d = 1.5 \times 10^{11}\) m
Find: \(t\) in seconds and minutes
Equation: \[t = \frac{d}{c}\]
Solution: \[t = \frac{1.5 \times 10^{11} \text{ m}}{3 \times 10^8 \text{ m/s}} = \frac{1.5}{3} \times 10^{11-8} \text{ s} = 0.5 \times 10^3 \text{ s} = 5.0 \times 10^2 \text{ s}\]
Convert to minutes: \[5.0 \times 10^2 \text{ s} \times \frac{1 \text{ min}}{60 \text{ s}} = 8.3 \text{ min}\]
Unit check: m divided by (m/s) gives seconds.
Sanity check: The standard value is about 8 minutes; this matches.
Answer: \(5.0 \times 10^2\) s, or about 8.3 minutes.
Problem 8: Building a Light-Year
Restatement: Compute how far light travels in one year.
Given: - \(c = 3 \times 10^8\) m/s - \(t = 3.15 \times 10^7\) s
Find: \(d\) in meters
Equation: \[d = c t\]
Solution: \[d = (3 \times 10^8 \text{ m/s})(3.15 \times 10^7 \text{ s}) = (3 \times 3.15) \times 10^{8+7} \text{ m}\] \[d = 9.45 \times 10^{15} \text{ m}\]
Unit check: (m/s) times s gives meters.
Sanity check: A light-year is of order \(10^{16}\) m, so \(9.45 \times 10^{15}\) m is reasonable.
Answer: \(9.45 \times 10^{15}\) m.
Problem 9: Inverse-Square Reasoning
Restatement: A star moves from 10 ly to 40 ly. Find the brightness factor.
Given: - \(d_{\text{final}} = 40\) ly - \(d_{\text{initial}} = 10\) ly
Find: \(B_{\text{final}}/B_{\text{initial}}\)
Equation: \[B \propto \frac{1}{d^2} \quad \Rightarrow \quad \frac{B_{\text{final}}}{B_{\text{initial}}} = \left(\frac{d_{\text{initial}}}{d_{\text{final}}}\right)^2\]
Solution: \[\frac{B_{\text{final}}}{B_{\text{initial}}} = \left(\frac{10}{40}\right)^2 = \left(\frac{1}{4}\right)^2 = \frac{1}{16}\]
Unit check: The brightness ratio is dimensionless.
Sanity check: Increasing distance should reduce brightness; 4 times farther gives \(1/16\), which is correct.
Answer: The star is 1/16 as bright (16 times fainter).
Problem 10: Lookback Time Calculation
Restatement: A galaxy is 50 million light-years away. When did the light leave it?
Given: - Distance \(= 50\) million light-years
Find: Lookback time in years
Equation: A light-year is the distance light travels in one year, so the numerical value in light-years equals the travel time in years.
Solution: \[t = 50 \text{ million years}\]
Unit check: Light-years map directly to years of light travel time.
Sanity check: Tens of millions of light-years correspond to tens of millions of years of lookback time.
Answer: The light left 50 million years ago.
Synthesis
Problem 11: The Course Thesis
Restatement: Explain the course thesis in 2 to 3 sentences.
Key elements a full answer should include: - Images are data, not just pictures - We directly measure only a few observables - Physics connects measurements to inferences - Most properties are inferred, not directly measured
Sample response: “Astronomy turns pretty images into measurements, then uses physics to infer properties we cannot directly access. We can measure only a few things like brightness, position, wavelength, and timing. Everything else we claim about stars and galaxies comes from combining those measurements with physical models.”
Grading guidance: Full credit includes all four key elements; partial credit if one or more elements are missing or vague.
Problem 12: Connecting Wavelength to Physics
Restatement: Explain why optical and radio observations of the same galaxy show different information.
Key elements a full answer should include: - Optical light traces stars and hot surfaces - Radio (21 cm) traces cold neutral hydrogen - Different wavelengths are produced by different physical processes - Same object, different wavelength reveals different components
Sample response: “Optical light comes mainly from stars, so the optical image shows the stellar disk and spiral arms. Radio observations at 21 cm trace neutral hydrogen gas, which often extends beyond the visible stars. Different wavelengths are produced by different physical processes, so the same galaxy looks different depending on what physics the telescope is sensitive to.”
Grading guidance: Full credit requires both components (stars and gas) and the physical reason different wavelengths reveal different structures.
Problem 13: Dark Matter Evidence
Restatement: Give one observational evidence for dark matter, stating what is measured and what is inferred.
Key elements a full answer should include: - A specific observation (for example, galaxy rotation curves) - What is directly measured (for example, Doppler shifts and velocities) - The model used (gravity) - The inference (extra unseen mass)
Sample response: “From Doppler shifts of spectral lines, we measure how fast stars orbit at different radii in a galaxy. Using Newtonian gravity, we compute the mass needed to keep those stars in orbit. The required mass is far larger than the visible matter, so we infer a large amount of unseen mass, which we call dark matter.”
Grading guidance: Full credit must distinguish measurement from inference and reference the gravitational model; partial credit if any piece is missing.
Problem 14: Observable -> Model -> Inference
Restatement: Use the Observable -> Model -> Inference pattern for a periodic position wobble.
Key elements a full answer should include: - Observable: position (tracked over time) - Model: gravitational two-body orbit around a center of mass - Inference: an unseen companion and orbital properties
Sample response: “The observable is the star’s position on the sky, measured repeatedly over time. The model is Newtonian gravity: if a star has a companion, both orbit their common center of mass, so the visible star wobbles. From the wobble’s period and size, we infer an unseen companion and can estimate its orbital period and a lower limit on its mass.”
Grading guidance: Full credit requires naming the observable, stating the gravity model, and making a clear inference about a companion.