Homework 5

Masses from Motion + Magnitudes/HR Synthesis (Midterm 1 Review)

Assignment Info

Due	Tuesday, March 3, 2026 at 11:59 pm
Grade Memo	Friday, March 6, 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 binary-star dynamics to infer stellar masses from observables.
• Connect radial velocity amplitudes to mass ratio and interpret what is directly measured versus inferred.
• Apply the magnitude system and distance modulus to move between apparent brightness, absolute brightness, and distance.
• Use ratio-based stellar scalings to connect mass, luminosity, and main-sequence lifetime.
• Synthesize multiple observables into a coherent observable → model → inference chain.

Concept Throughline

• You do not measure stellar mass directly. You infer it from motion.
• Brightness is not luminosity. Distance and logarithmic magnitudes matter.
• The HR diagram is a map of physics, not just a plot.
• Scaling + sanity checks should guide every quantitative step.

Prerequisites

• Module 1 tools: unit handling, scaling, interpretation checks
• Kepler/Newton orbital reasoning from Module 1 and Module 2
• Doppler/radial velocity basics
• Logarithms and scientific notation

Relevant Sources (Module-Based)

• Module 2 (reading): The Last Piece — Weighing Stars
• Module 2 (reading): The HR Diagram — Finding Patterns, Needing Models
• Module 1 review: Gravity and Orbits
• Module 1 review: Light as Information

Note

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

Tip

HW5 note: This is your last pre-midterm assignment. Tool hints are not shown. Choose methods deliberately and include one-line sanity checks.

Note

Use these constants and relations unless a problem states otherwise:

• Solar-unit Kepler form (for binary total mass): \[ \frac{M_1 + M_2}{M_\odot} = \frac{\left(a/\text{AU}\right)^3}{\left(P/\text{yr}\right)^2} \]
• SB2 radial-velocity amplitude relation: \[ \frac{M_2}{M_1} = \frac{K_1}{K_2} \] Reminder: the more massive star has the smaller radial-velocity amplitude (\(K\)).
• Distance modulus: \[ m - M = 5\log_{10}\!\left(\frac{d}{10\,\mathrm{pc}}\right) \]
• Main-sequence scaling (approx.): \[ \frac{L}{L_\odot} \approx \left(\frac{M}{M_\odot}\right)^{3.5} \]
• Lifetime scaling (approx.): \[ t_{\text{MS}} \propto \frac{M}{L} \propto M^{-2.5}, \quad t_{\odot} \approx 10\,\text{Gyr} \]

Note

Required reporting format:

• Every numeric answer must include units.
• Use scientific notation where appropriate.
• Include a one-line sanity check for each problem.

Tip

Sanity-check scalings:

• \(M_{\text{tot}} \propto a^3/P^2\)
• \(K_1/K_2 = M_2/M_1\)
• \(m - M\) increases with distance
• \(L \propto M^{3.5}\) and \(t_{\text{MS}} \propto M^{-2.5}\)

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Problems (10 total)

Part A — Masses from Motion + Magnitudes

Problem 1 — Why “Weighing a Star” Is an Inference

A student says: “Astronomers can’t really know stellar masses because nobody can put a star on a scale.”

• 1. Identify one direct observable used to infer stellar mass in binaries.
• 2. Name the physical model that turns that observable into mass.
• 3. In 2–4 sentences, explain why this is still a valid measurement in scientific practice.

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Problem 2 — Total Mass from a Visual Binary

A visual binary has orbital period \(P = 8.0\,\text{yr}\) and semi-major axis \(a = 4.0\,\text{AU}\).

• 1. Compute the total system mass \((M_1 + M_2)\) in solar masses.
• 2. If the two stars have equal mass, what is each mass?
• 3. Is this system total mass larger or smaller than the Sun’s mass? Give a one-line interpretation.

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Problem 3 — Mass Ratio from SB2 Velocities

For a double-lined spectroscopic binary (SB2), you measure radial-velocity amplitudes:

• \(K_1 = 30\,\text{km/s}\)
• \(K_2 = 45\,\text{km/s}\)

The total mass is known from independent orbital data to be \(M_1 + M_2 = 3.0\,M_\odot\).

• 1. Compute the mass ratio \(M_2/M_1\).
• 2. Solve for \(M_1\) and \(M_2\) in solar masses.
• 3. Which star is more massive, and how can you tell from the velocity amplitudes?

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Problem 4 — Inclination and the Hidden-Mass Problem

Two binaries have identical measured periods and identical radial-velocity amplitudes, but one system is eclipsing and the other is not.

• 1. Why does the eclipsing system usually allow a better mass determination?
• 2. What is the role of orbital inclination in turning observed radial velocity into true orbital speed?
• 3. Explain why a nearly face-on system can hide large true masses.

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Problem 5 — Required Capstone: Eclipsing SB2 Full Inference Chain

Note

Required capstone: This is the longest problem on HW5 and is intentionally integrative.

An eclipsing SB2 system has:

• Period: \(P = 2.00\,\text{yr}\)
• Semi-major axis: \(a = 2.00\,\text{AU}\)
• Velocity amplitudes: \(K_A = 40\,\text{km/s}\) and \(K_B = 20\,\text{km/s}\)

Assume both stars are main-sequence and use \(L/L_\odot \approx (M/M_\odot)^{3.5}\). Assume \(a\) is the true semi-major axis (not a projected/angular separation).

• 1. Compute \(M_A + M_B\).
• 2. Compute the mass ratio and then \(M_A\) and \(M_B\).
• 3. Which star is more massive? Which star moves faster in orbit?
• 4. Estimate the luminosity ratio \(L_B/L_A\).
• 5. In 3–5 sentences, summarize the full observable → model → inference chain used in this problem, including at least one assumption.

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Problem 6 — Apparent vs. Absolute Magnitude Misconception

A student says: “Star X has \(m=4\) and star Y has \(m=9\), so star X is intrinsically brighter.”

• 1. Explain why this conclusion is not always valid.
• 2. State what additional quantity you need to compare intrinsic brightness.
• 3. In one sentence, define apparent magnitude and absolute magnitude.

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Problem 7 — Distance Modulus Practice

• 1. A star has \(m = 11.2\) and \(M = 1.2\). Compute its distance in parsecs.
• 2. A cluster is at \(d = 250\,\text{pc}\). A member star has apparent magnitude \(m = 14.0\). Compute its absolute magnitude \(M\).
• 3. Two stars differ by 5.0 magnitudes in apparent magnitude. By what factor do their observed fluxes differ?

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Part B — HR Synthesis + Cumulative Midterm Bridge

Problem 8 — Main-Sequence Fitting by Magnitude Offset

A reference cluster at \(d_{\text{ref}} = 100\,\text{pc}\) has a calibrated main sequence. A target cluster’s main sequence is shifted fainter by \(\Delta m = +3.0\) mag at the same colors.

• 1. Use distance-modulus reasoning to compute \(d_{\text{target}}/d_{\text{ref}}\).
• 2. Compute \(d_{\text{target}}\) in parsecs.
• 3. If uncorrected extinction adds \(0.5\) mag of dimming, would your distance estimate from part (b) be too large or too small? Explain briefly.

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Problem 9 — Bridge to Module 3: Mass, Luminosity, and Lifetime

Use the main-sequence scalings: \[ \frac{L}{L_\odot} \approx \left(\frac{M}{M_\odot}\right)^{3.5}, \qquad \frac{t_{\text{MS}}}{t_\odot} \approx \left(\frac{M}{M_\odot}\right)^{-2.5} \] with \(t_\odot = 10\,\text{Gyr}\).

• 1. Estimate the main-sequence lifetimes of a \(2.0\,M_\odot\) star and a \(5.0\,M_\odot\) star.
• 2. A cluster’s turnoff mass is about \(2.0\,M_\odot\). Estimate the cluster age.
• 3. Why does this scaling imply that very massive stars are rare in old clusters?

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Problem 10 — Cumulative Inference: One System, Multiple Models

You observe a binary system in a cluster and measure:

• Orbital period: \(P = 4.0\,\text{yr}\)

• Semi-major axis: \(a = 4.0\,\text{AU}\)

• Component velocity amplitudes: \(K_1 = 25\,\text{km/s}\), \(K_2 = 20\,\text{km/s}\)

• Cluster distance modulus: \(m - M = 10.0\)

• Combined apparent magnitude of the binary: \(m = 8.0\)

• 1. Compute the total mass and the individual masses.
• 2. Compute the combined absolute magnitude of the binary.
• 3. In 3–6 sentences, describe what is measured directly versus inferred in this workflow.
• 4. Name one additional observation that would reduce uncertainty in the physical interpretation, and explain why.