Homework 2

Gravity & Orbits II + Light as Information

Author

Dr. Anna Rosen

Assignment Info

DueTuesday, February 10, 2026 at 11:59 pm
Grade MemoThursday, February 12, 2026 at 11:59 pm
Est. Time4–6 hours
Submit viaCanvas; upload a single clearly scanned PDF or a legible PDF export

Learning Objectives

  • Interpret the sign of orbital energy and connect it to bound vs. unbound motion.
  • Use Newton + Kepler to move between periods, masses, and orbital sizes.
  • Apply escape velocity and orbital velocity as scaling tools with unit checks.
  • Use the virial theorem to connect motions to binding energy.
  • Use spectra and Doppler shifts to infer velocities and physical properties.
  • Treat equations as stories with units, not just recipes.

Concept Throughline

  • Measurement → model → inference remains the core chain.
  • Energy sets orbit type (bound, parabolic, unbound).
  • Light is information: spectra and Doppler shifts let us infer motion and composition.
  • Scaling beats big numbers when the physics is clear.

Prerequisites

  • Scientific notation and exponent rules
  • Basic algebra and proportional reasoning
  • Unit conversion (cm, km, s; g, kg; nm)

Relevant Readings

Note

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

Tip

HW2 note: Tool hints are not shown. Part of the goal is recognizing which tools apply.

Problems (10 total)

Part A — Gravity & Orbits

Problem 1 — Energy Signs and Orbital Fate

The total mechanical energy is \(E = \tfrac{1}{2}mv^2 - GMm/r\).

    1. Why is gravitational potential energy negative (with zero at infinity)?
    1. What does \(E < 0\), \(E = 0\), and \(E > 0\) mean physically?
    1. A comet approaches the Sun with \(E > 0\). A student says, “It has positive energy, so it’s gaining energy as it approaches.” Correct this.

Problem 2 — Orbital Velocity Practice

Using \(v_{orb} = \sqrt{GM/r}\):

    1. Calculate Earth’s orbital velocity around the Sun using \(M_\odot = 2 \times 10^{33}\) g and \(r = 1\) AU \(= 1.5 \times 10^{13}\) cm. Express in cm/s and km/s.
    1. Show explicitly that \(\sqrt{GM/r}\) has units of velocity.
    1. Mars orbits at 1.52 AU. Using ratios, find Mars’s orbital velocity.
    1. Sanity check: should Mars move faster or slower than Earth?

Problem 3 — Escape Velocity Comparison

Using \(v_{esc} = \sqrt{2GM/R}\):

    1. Calculate Earth’s escape velocity. Use \(M_\oplus = 6 \times 10^{27}\) g and \(R_\oplus = 6.4 \times 10^8\) cm.
    1. The Moon has \(M_{Moon} = 7.3 \times 10^{25}\) g and \(R_{Moon} = 1.7 \times 10^8\) cm. Calculate its escape velocity.
    1. By what factor is Earth’s escape velocity larger than the Moon’s?
    1. Interpret: why did Apollo astronauts need a smaller rocket to leave the Moon?

Problem 4 — Kepler III with Newton

Newton’s version is \(P^2 = \dfrac{4\pi^2}{G(M+m)}a^3\).

    1. For Earth, \(m \ll M_\odot\). Show the simplified form.
    1. Verify it gives \(P \approx 1\) year for \(a = 1\) AU.
    1. Confirm both sides have units of s\(^2\).
    1. A binary star has two equal \(1\,M_\odot\) stars. How does its Kepler constant compare to the Sun-Earth system?

Problem 5 — Complete Workflow: Weighing Jupiter

Io orbits Jupiter with period \(P = 1.77\) days and semi-major axis \(a = 4.22 \times 10^{10}\) cm.

    1. Use Newton’s Kepler III to solve for Jupiter’s mass.
    1. Unit check: verify \(\frac{a^3}{GP^2}\) has units of mass.
    1. Express your answer in grams and in solar masses.
    1. What observable quantities did you need?

Problem 6 — The \(\sqrt{2}\) Factor

Escape velocity is \(v_{esc} = \sqrt{2}\,v_{orb}\) at any radius.

    1. Derive this relationship from \(v_{orb} = \sqrt{GM/r}\) and \(v_{esc} = \sqrt{2GM/r}\).
    1. A spacecraft in circular orbit wants to escape. By what factor must it increase its speed?
    1. Explain the energy interpretation: why does escape require adding kinetic energy equal to the orbital kinetic energy?

Problem 7 — Virial Theorem Capstone

A star cluster has total kinetic energy \(K = 1.0 \times 10^{50}\) erg in stellar motions and is in virial equilibrium (\(2K + U = 0\)).

    1. Use the virial theorem to find its gravitational potential energy \(U\).
    1. Compute the total energy \(E = K + U\) and state whether the cluster is bound.
    1. A tidal interaction injects an additional \(0.6 \times 10^{50}\) erg of kinetic energy without changing the cluster size. What happens to the sign of \(E\)? Does the cluster remain bound?

Part B — Light as Information

Problem 8 — Kirchhoff’s Laws in Action

You see three spectra from three different sources:

    1. A smooth rainbow with dark lines
    1. A smooth rainbow with no lines
    1. A dark background with bright lines

For each, identify the physical source (hot dense object, hot thin gas, or cool gas in front of a hot source). Explain your reasoning.

Problem 9 — Why Skies Are Red at Sunset

Use Rayleigh scattering to explain why the Sun looks red near the horizon but not at noon. What changes about the path through the atmosphere, and how does \(\lambda^{-4}\) make the effect strong?

Problem 10 — Photon Energy from Wavelength

An orange photon has \(\lambda = 600\) nm.

    1. Convert this wavelength to cm.
    1. Compute its frequency.
    1. Compute its energy in erg using \(E = hc/\lambda\) with \(h = 6.626 \times 10^{-27}\) erg·s.