| Criterion | Points |
|---|
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Submit a complete derivation with equations, units, labeled quantities, and a short narrative interpretation for each problem. Show every step; a correct final number without a visible derivation receives partial credit at most, and quoting a lecture number without adapting it to this problem's specific inputs receives no credit for that step.
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materials/ASTR320/src/generate_astr320_content.py where referenced above.{escape(title)}
Award full marks for correct method, units, and a numeric result " f"consistent with the arithmetic shown (allow reasonable rounding). Verify that the student re-derives " f"the number for this problem's specific inputs rather than quoting a lecture example without adaptation. " f"Watch specifically for unit-conversion errors (AU/m, days or years/seconds, bar/Pa) and for comparative-" f"magnitude statements stated in the wrong direction (e.g., a tidal acceleration or greenhouse excess " f"described as a multiple when it is actually a small fraction, or vice versa) -- these are the most " f"common sources of silent error in this course.
This is a data-analysis lab using real, spacecraft-measured parameters for the
two most massive main-belt asteroids, Ceres and Vesta, both characterized in detail by NASA's Dawn mission
(2011-2018; standard published values, not independently re-verified by live fetch this session -- see
../reference-log.md). No telescope time is required; the lab tests Lecture 02's frost-line and
condensation-sequence prediction directly against two real bodies straddling the frost line.
| Body | Mass (kg) | Radius (km) | Semimajor axis (AU) |
|---|---|---|---|
| Ceres | {CERES['mass']:.3e} | {CERES['radius']/1000:.1f} | {CERES['a_au']} |
| Vesta | {VESTA['mass']:.3e} | {VESTA['radius']/1000:.1f} | {VESTA['a_au']} |
Because ρ ∝ R⁻³, the fractional uncertainty in a body's computed density is approximately three times its fractional radius uncertainty (for a fixed, precisely known mass); Dawn's radar and imaging-derived radii for Ceres and Vesta are precise to well under 1%, so this uncertainty is small compared to the roughly 60% density difference between the two bodies, meaning the observed density contrast is a real physical result, not a measurement artifact.
materials/ASTR320/reference-log.md.materials/ASTR320/src/generate_astr320_labs_psets.py.Pluto has mass {PLUTO['mass']:.3e} kg and radius {PLUTO['radius']/1000:.1f} km, and orbits at a mean " f"semimajor axis of {PLUTO['a_au']} AU, far beyond the frost line. Compute Pluto's mean density and compare it " f"to Ceres's and Vesta's densities from Lab 01 and to the icy-satellite density range from Lecture 01. " f"State whether Pluto's density is consistent with the condensation-sequence prediction for a Kuiper Belt object.
"), problem(2, "Titan's Density and the Ice/Rock Mixture", f"Titan has mass {TITAN['mass']:.4e} kg and radius {TITAN['radius']/1000:.1f} km. Compute Titan's mean " f"density. A pure water-ice body would have density ≈ 917 kg/m³; a pure rocky body would have density " f"≈ 3000 kg/m³ or higher. Estimate, using a simple linear mixing argument, what mass fraction of rock " f"(versus ice) Titan's bulk density implies, and state one reason this simple estimate is only approximate.
"), problem(3, 'Frost Line for a Fainter and a Brighter Young Star', f"Using the equilibrium-temperature-based frost-line method from Lecture 02 (setting T(r) = 150 K), compute " f"the frost-line distance for a young star with (a) half the Sun's luminosity and (b) twice the Sun's " f"luminosity. Compare both results to the Sun's own frost line (∼2.8 AU) and state, in one sentence, how " f"frost-line distance scales with stellar luminosity.
"), problem(4, 'Runaway versus Oligarchic Growth: A Conceptual Comparison', "In two to three sentences each, explain (a) why gravitational focusing makes runaway growth " "self-reinforcing for the largest body in a local planetesimal swarm, and (b) why oligarchic growth " "eventually slows this process down once a few embryos dominate their feeding zones. Your answer must " "reference the gravitational-focusing cross-section argument from Lecture 02, not just restate that " "'bigger objects grow faster.'
"), problem(5, 'Critical Core Mass and the Gas Giant/Ice Giant Divide', "Jupiter's core accretion is estimated to have reached the ∼5-10 Earth-mass critical core mass while " "the gas disk still had several Myr of lifetime remaining, while Uranus's and Neptune's cores are thought " "to have formed more slowly. Using the core-accretion argument from Lecture 02, explain quantitatively " "(with reference to the ∼90% hydrogen/helium mass fraction of Jupiter and Saturn versus the ∼10-20% " "hydrogen/helium mass fraction of Uranus and Neptune) why this timing difference alone is sufficient to " "explain the compositional gas-giant/ice-giant divide.
"), ) solutions = ( solution(1, 'Density and Composition of Pluto', f"ρ = M/(4/3πR³) = {PLUTO['mass']:.3e} / [(4/3)π({PLUTO['radius']:.4e})³] = " f"{plu_density:.0f} kg/m³. This is well below any solid rocky-planet density (>3900 kg/m³) and " f"comparable to the icy-satellite range (1000-2000 kg/m³), consistent with Pluto being a Kuiper Belt " f"object built largely from ices condensed far beyond the frost line, as the condensation sequence predicts.
", [('Correct density calculation with unit conversion', 6), ('Correct comparison to icy-body density range', 5), ('Correct qualitative conclusion tied to condensation sequence', 4)]), solution(2, "Titan's Density and the Ice/Rock Mixture", f"ρ = {TITAN['mass']:.4e} / [(4/3)π({TITAN['radius']:.4e})³] = {titan_density:.0f} kg/m³. " f"A simple linear mixing estimate, x·3000 + (1-x)·917 = {titan_density:.0f}, gives x ≈ " f"{(titan_density-917.0)/(3000.0-917.0):.2f}, i.e., roughly {(titan_density-917.0)/(3000.0-917.0)*100:.0f}% " f"rock by mass. This estimate is only approximate because real planetary ices and rock do not simply " f"average their densities when mixed (compression, porosity, and phase changes at Titan's actual internal " f"pressures all modify the true mixed density), and because Titan is now known (from Cassini gravity data) " f"to be at least partially differentiated rather than a uniform mixture.
", [('Correct density calculation', 5), ('Correct linear-mixing rock-fraction estimate', 6), ('Correctly identifies a specific limitation of the simple mixing model', 4)]), solution(3, 'Frost Line for a Fainter and a Brighter Young Star', f"r = √[L/(16πσT⁴)], so r ∝ √L. For L = 0.5 L⊙: r = " f"{frost_line_faint_au:.2f} AU. For L = 2 L⊙: r = {frost_line_bright_au:.2f} AU. Compared to the Sun's " f"own frost line (∼2.8 AU), the frost line moves inward for a fainter young star and outward for a " f"brighter one, scaling as the square root of luminosity.
", [('Correct application of the frost-line formula to both cases', 8), ('Correctly states the sqrt(L) scaling', 7)]), solution(4, 'Runaway versus Oligarchic Growth: A Conceptual Comparison', "(a) Runaway growth is self-reinforcing because an embryo's effective gravitational-focusing cross-section " "grows with its own escape velocity relative to the surrounding planetesimals' velocity dispersion (cross-" "section ∝ 1 + (v_esc/v_rel)²), so a slightly larger embryo accretes disproportionately faster, " "widening the gap with its neighbors. (b) Oligarchic growth slows this process once a handful of embryos " "dominate their local feeding zones, because each embryo's own gravity begins to stir up (increase the " "velocity dispersion of) the remaining planetesimals in its zone, reducing the gravitational-focusing " "advantage and roughly equalizing growth rates among the surviving embryos.
", [('Correctly explains runaway growth via gravitational focusing (not just size)', 8), ('Correctly explains the transition to oligarchic growth via self-stirring', 7)]), solution(5, 'Critical Core Mass and the Gas Giant/Ice Giant Divide', "Once a core exceeds the critical mass (∼5-10 Earth masses), runaway gas accretion can capture " "hundreds of Earth masses of hydrogen/helium within ∼10⁴-10⁵ years, but only if the gas disk is " "still present. If Jupiter's and Saturn's cores reached this threshold with several Myr of disk lifetime " "remaining, they could accrete gas for an extended period, explaining their ∼90%+ hydrogen/helium " "composition. If Uranus's and Neptune's cores formed more slowly (plausible given the much lower disk " "density at 19-30 AU) and only reached, or never fully reached, the critical mass shortly before the gas " "disk dispersed, they would have captured only a comparatively small hydrogen/helium envelope before the " "gas supply ran out, leaving them dominated by their icy/rocky cores -- exactly the observed ∼10-20% " "hydrogen/helium mass fraction.
", [('Correctly explains the critical-mass/runaway-accretion mechanism', 7), ('Correctly connects accretion timing to the observed compositional divide', 8)]), ) pset_html = pset_page(1, 'Solar Nebula, Condensation, and Planet Formation', 'Apply the frost-line, condensation-sequence, and core-accretion concepts from Lectures 01-02 to real dwarf planets, satellites, and formation-timing arguments.', ''.join(problems), 'OpenStax Astronomy 2e, Sections 7.2 and 8.5.') sol_html = solutions_page(1, 'Solar Nebula, Condensation, and Planet Formation', ''.join(solutions)) asmt = assessment_md(1, 'Solar Nebula, Condensation, and Planet Formation', ['Problem 1: density calculation and icy-body comparison both required for full credit.', 'Problem 2: linear-mixing estimate must show the algebra, not just state a rock fraction.', 'Problem 3: both frost-line values and the sqrt(L) scaling statement are required.', 'Problem 4: answer must invoke gravitational focusing and self-stirring specifically, not generic size arguments.', 'Problem 5: answer must connect accretion timing to the specific observed compositional percentages.'], ['Forgetting to convert radius from km to m before computing density (an extremely common unit-conversion slip).', 'Treating density differences as automatically proof of composition without considering measurement uncertainty (Problem 1).', 'Applying the frost-line formula without correctly propagating the square-root scaling with luminosity (Problem 3).']) return pset_html, sol_html, asmt # --------------------------------------------------------------------------- # LAB 02: Moment of inertia and radiogenic heat budget, Earth vs. Mars # --------------------------------------------------------------------------- def make_lab_02() -> str: sections = f"""This is a data-analysis lab combining two independent real datasets: measured
moment of inertia factors (from spacecraft tracking and precession) and measured/modeled surface heat flow, for Earth
and Mars. Standard published values, not independently re-verified by live fetch this session (see
../reference-log.md).
| Body | C/MR² | Mean surface heat flow (mW/m²) | Mean radius (km) |
|---|---|---|---|
| Earth | {EARTH_CMR2} | {EARTH_HEAT_FLOW_MWM2} | {PLANETS['Earth']['radius']/1000:.1f} |
| Mars | {MARS_CMR2} | {MARS_HEAT_FLOW_MWM2} | {PLANETS['Mars']['radius']/1000:.1f} |
| Moon | {MOON_CMR2} | 16-21 (Apollo heat-flow probes) | {MOON['radius']/1000:.1f} |
Mars's heat flow value in this lab is a representative published geophysical estimate rather than a directly measured value, because InSight's HP³ heat-flow probe did not achieve its planned burial depth; students should treat this number as having larger uncertainty than Earth's or the Moon's directly measured heat flow values, and should state this explicitly in their analysis.
materials/ASTR320/reference-log.md.materials/ASTR320/src/generate_astr320_labs_psets.py.MESSENGER measurements give Mercury's moment of inertia factor as C/MR² = {MERCURY_CMR2}. Compute " f"its percent deviation below the uniform-sphere value of 0.4, and compare this deviation to Earth's " f"({earth_deviation:.1f}% computed from C/MR² = {EARTH_CMR2}). State what this comparison implies about " f"Mercury's core, in light of Lecture 03's giant-impact hypothesis for Mercury's anomalously large core.
"), problem(2, "Radiogenic Heat Production 2 Gyr Ago", f"Using the decay constants λ = ln2/t₁⁄² for ⁴⁴K (t₁⁄² = " f"{K40_HALFLIFE_GYR} Gyr) and ²³⁶U (t₁⁄² = {U238_HALFLIFE_GYR} Gyr), compute " f"what fraction of each isotope's present-day abundance existed 2 Gyr ago (i.e., the ratio of abundance at " f"t = -2 Gyr relative to today, noting this ratio is e⁺λt for t = 2 Gyr since we are looking " f"backward in time). Explain which isotope's heat contribution changed more dramatically over this " f"interval and why.
"), problem(3, "Surface-to-Volume Scaling for Io versus the Moon", "Io (radius 1821.6 km) and the Moon (radius 1737.4 km) are nearly the same size, yet Io is the most " "volcanically active body in the solar system while the Moon is essentially geologically dead. Using the " "surface-to-volume cooling argument from Lecture 04, explain why this argument alone would predict " "similar (low) geological activity for both bodies, and then explain what additional physical mechanism " "(introduced in Lecture 04 as a preview and developed fully in Lecture 13) resolves this apparent " "contradiction.
"), problem(4, "Rayleigh Number Scaling", "The Rayleigh number is Ra = (ρgαΔTd³)/(κμ). If a planet's mantle thickness " "d were doubled while all other quantities (ρ, g, α, ΔT, κ, μ) stayed fixed, by what " "factor would its Rayleigh number change? Given that Earth's mantle Rayleigh number is estimated at " "10⁶-10⁷, vastly above the critical value of ∼1000-2000, would this change plausibly push a " "convecting mantle below the critical threshold into a non-convecting regime? Justify your answer " "quantitatively.
"), problem(5, "Comparing Bulk Silicate Earth's Heat Budget to Observed Heat Flow", "Earth's present-day radiogenic heat production is estimated at roughly 20 TW, while its total observed " "surface heat flow is roughly 47 TW. Compute the percent of Earth's total heat flow that is NOT accounted " "for by present-day radiogenic heat production, and explain, using the concept introduced in Lecture 04, " "what additional heat source is thought to make up this difference.
"), ) solutions = ( solution(1, "Mercury's Moment of Inertia and Core Size", f"Mercury: (0.4 - {MERCURY_CMR2})/0.4 × 100 = {mercury_deviation:.1f}%. Earth: {earth_deviation:.1f}%. " f"Mercury's deviation below the uniform-sphere value is substantially larger than Earth's, implying an " f"even more strongly core-concentrated interior relative to its total size than Earth's -- consistent " f"with Mercury's core occupying an unusually large fraction (∼85%) of its radius, and with the " f"giant-impact hypothesis that one or more early impacts stripped away a disproportionate share of " f"Mercury's original silicate mantle, leaving its iron core relatively more dominant than on any other " f"terrestrial planet.
", [('Correct percent-deviation calculation for Mercury', 6), ('Correct comparison to Earth\'s deviation', 5), ('Correct connection to the giant-impact hypothesis', 4)]), solution(2, "Radiogenic Heat Production 2 Gyr Ago", f"Fraction 2 Gyr ago relative to today = e⁺λt. For ⁴⁴K: e⁺({lam_k40:.4f})(2) = " f"{math.exp(lam_k40*2.0):.3f}, i.e., ⁴⁴K was about {math.exp(lam_k40*2.0):.2f} times more abundant " f"(and hence produced about {math.exp(lam_k40*2.0):.2f} times more heat) 2 Gyr ago than today. For " f"²³⁶U: e⁺({lam_u238:.4f})(2) = {math.exp(lam_u238*2.0):.3f}, only about " f"{math.exp(lam_u238*2.0):.2f} times more abundant. ⁴⁴K's much shorter half-life (1.25 Gyr versus " f"4.47 Gyr) means its heat contribution changed far more dramatically over this interval than " f"²³⁶U's, which decays much more slowly and so contributed comparably to both eras.
", [('Correct decay-law calculation for both isotopes', 8), ('Correct identification and explanation of which isotope changed more', 7)]), solution(3, "Surface-to-Volume Scaling for Io versus the Moon", "Because Io and the Moon are nearly the same size, the surface-to-volume scaling argument (heat loss/" "generation ∝ 1/R) predicts nearly identical cooling rates and hence nearly identical present-day " "geological activity for both bodies -- both should have long since radiated away their formation-era and " "radiogenic heat and become geologically quiescent. This prediction is correct for the Moon but " "dramatically wrong for Io, because Io receives an enormous additional, non-radiogenic heat source: " "tidal heating, sustained by its orbital resonance with Europa and Ganymede (Lecture 13), which has no " "analog for the Moon (whose orbit around Earth has already circularized and is not maintained in a " "forced-eccentricity resonance with another body).
", [('Correctly applies the surface-to-volume argument to predict similar activity', 7), ('Correctly identifies tidal heating/resonance as the resolving mechanism', 8)]), solution(4, "Rayleigh Number Scaling", "Since Ra ∝ d³, doubling d increases Ra by a factor of 2³ = 8. Starting from Ra ∼ " "10⁶-10⁷ and multiplying by 8 gives Ra ∼ 8×10⁶-8×10⁷, still vastly above " "the critical value of ∼1000-2000 by four to five orders of magnitude. This change would not plausibly " "push Earth's mantle into a non-convecting regime; Earth's mantle convection is so strongly super-" "critical that even an order-of-magnitude change in a single parameter would not be sufficient to " "suppress convection.
", [('Correct cubic scaling calculation', 8), ('Correct quantitative conclusion about remaining super-criticality', 7)]), solution(5, "Comparing Bulk Silicate Earth's Heat Budget to Observed Heat Flow", "(47 - 20)/47 × 100 ≈ 57%. Just over half of Earth's total observed surface heat flow is not " "accounted for by present-day radiogenic heat production alone. Lecture 04 attributes this difference to " "secular cooling: Earth's core and mantle are still releasing heat left over from formation and " "differentiation (accretional and differentiation heat, discussed as one-time heat sources at the start " "of Lecture 04), in addition to their ongoing radiogenic heat production.
", [('Correct percent calculation', 7), ('Correctly identifies secular cooling as the additional source', 8)]), ) pset_html = pset_page(2, 'Interior Structure and Heat Budgets', 'Apply moment-of-inertia interpretation, the Rayleigh-number criterion, and the radiogenic decay law from Lectures 03-04 to Mercury, Io, and Earth\'s overall heat budget.', ''.join(problems), 'OpenStax Astronomy 2e, Section 9.1-9.2.') sol_html = solutions_page(2, 'Interior Structure and Heat Budgets', ''.join(solutions)) asmt = assessment_md(2, 'Interior Structure and Heat Budgets', ['Problem 1: both percent-deviation values must be computed and correctly compared.', 'Problem 2: correct exponential-decay direction (looking backward in time uses e^{+lambda t}, not e^{-lambda t}) is required.', 'Problem 3: answer must explicitly name tidal heating/orbital resonance as the resolving mechanism, not a vague reference to "different composition."', 'Problem 4: cubic scaling must be shown explicitly (not just stated).', 'Problem 5: percent calculation and correct naming of secular cooling both required.'], ['Sign error in the decay-law direction when computing a past abundance rather than a future one (Problem 2).', 'Confusing static core-size deviation with dynamic, time-dependent heat flow as though they must always track each other (Problem 3 context).', 'Forgetting to cube the ratio in the Rayleigh-number scaling problem (a linear-scaling error).']) return pset_html, sol_html, asmt # --------------------------------------------------------------------------- # LAB 03: Crater scaling and surface age dating using real lunar mare data # --------------------------------------------------------------------------- MARE_TRANQUILLITATIS_AGE_GYR = 3.7 MARE_TRANQUILLITATIS_CRATER_DENSITY = 3.5e-3 # craters/km^2 above 1 km diameter, representative published value MARE_IMBRIUM_AGE_GYR = 3.85 MARE_IMBRIUM_CRATER_DENSITY = 2.7e-3 HIGHLANDS_AGE_GYR = 4.3 HIGHLANDS_CRATER_DENSITY = 2.0e-2 def make_lab_03() -> str: slope = (MARE_TRANQUILLITATIS_CRATER_DENSITY - MARE_IMBRIUM_CRATER_DENSITY) / (MARE_TRANQUILLITATIS_AGE_GYR - MARE_IMBRIUM_AGE_GYR) sections = f"""This lab uses real, radiometrically anchored lunar chronology calibration points from Apollo sample-return sites, following the lunar chronology function methodology introduced in Lecture 06. Representative published crater densities (craters per km² above 1 km diameter) are provided for three radiometrically dated regions.
| Region | Radiometric age (Gyr) | Crater density (craters/km², D>1 km) |
|---|---|---|
| Mare Tranquillitatis (Apollo 11 site) | {MARE_TRANQUILLITATIS_AGE_GYR} | {MARE_TRANQUILLITATIS_CRATER_DENSITY} |
| Mare Imbrium (Apollo 15 site) | {MARE_IMBRIUM_AGE_GYR} | {MARE_IMBRIUM_CRATER_DENSITY} |
| Lunar highlands (Apollo 16 site) | {HIGHLANDS_AGE_GYR} | {HIGHLANDS_CRATER_DENSITY} |
The local linear interpolation used in step 2 is explicitly an approximation, valid only over a narrow age range where the true crater-density-versus-age curve is not changing its slope too rapidly; extending it far beyond the calibration points it was fit to (as the highlands comparison in step 3 illustrates) is a common and serious methodological error in real crater-count studies.
materials/ASTR320/reference-log.md.materials/ASTR320/src/generate_astr320_labs_psets.py.The 1908 Tunguska event is estimated to have involved a stony/icy impactor roughly 60 m in diameter " f"striking at ∼15 km/s (it exploded in the atmosphere before reaching the ground, an airburst rather " f"than a cratering impact, but its kinetic energy can still be estimated the same way). Using an assumed " f"density of 2000 kg/m³, compute the impactor's kinetic energy and express it in megatons of TNT " f"equivalent (1 MT = 4.184 × 10¹⁵ J). Compare your result to the commonly cited estimate of " f"3-15 megatons for the Tunguska event.
"), problem(2, 'Olympus Mons and the Hawaiian Hotspot Track', f"The Pacific Plate moves at roughly {pacific_plate_speed_cmyr} cm/yr over the fixed Hawaiian hotspot, " f"and the resulting Hawaiian-Emperor seamount chain is roughly {hawaii_chain_length_km} km long. Estimate " f"the total time this chain took to form, in millions of years, and compare it to the estimated multi-" f"hundred-million-year construction time of Mars's single, stationary Olympus Mons ({olympus_mons_height_km} " f"km tall, {olympus_mons_diameter_km} km diameter). Explain, referencing Lecture 05's plate-tectonic " f"argument, why Mars produced one enormous volcano rather than a long chain of smaller ones.
"), problem(3, 'Crater Density and Venus\u2019s Resurfacing Event', "Venus's crater population is far below saturation and shows a narrow range of degradation states " "across its entire surface, despite Venus being 4.5 Gyr old. Using the concepts from Lectures 05-06, " "explain in a short paragraph why this observation supports a discrete, relatively recent (∼300-700 " "Myr ago) global resurfacing event rather than either (a) a surface that has simply never been struck by " "many impactors, or (b) a surface that formed recently at the same time as the planet itself.
"), problem(4, 'Simple versus Complex Crater Transition', "The simple-to-complex crater transition diameter is expected to be smaller on more massive bodies " "(because stronger gravity more effectively collapses the initial transient cavity). Mars (g = 3.71 " "m/s²) has stronger surface gravity than the Moon (g = 1.62 m/s²), so gravity alone predicts " "Mars should have a smaller transition diameter than the Moon. Yet published values give the " "Moon a transition diameter of roughly 2-4 km, while Mars's is roughly 5-10 km -- larger, the " "opposite of the gravity-only prediction. Explain, in a short paragraph, why this is not actually a " "contradiction of the underlying physics.
"), problem(5, 'Mercury\u2019s Crater-Count Volcanic History', "MESSENGER-based crater counts indicate Mercury's youngest volcanic plains are roughly 3.7-3.9 Gyr old. " "Using the surface-to-volume cooling argument from Lecture 04 and Mercury's small size relative to Earth " "and Mars, explain in a short paragraph why an early cessation of volcanism on Mercury is consistent with " "theoretical expectation, and state one specific piece of independent evidence (from Lecture 03 or 05) " "that supports the same conclusion.
"), ) solutions = ( solution(1, 'Estimating the Tunguska Event Energy', f"Mass = (4/3)π(30 m)³(2000 kg/m³) = {tunguska_mass:.3e} kg. Energy = ½mv² = " f"½({tunguska_mass:.3e})(15000)² = {tunguska_energy:.3e} J = {tunguska_mt:.1f} MT. This result " f"({tunguska_mt:.1f} MT) is broadly consistent with, though somewhat below, the commonly cited 3-15 MT " f"range, which reflects genuine uncertainty in the actual impactor size and airburst altitude, not an " f"error in the basic kinetic-energy calculation.
", [('Correct mass calculation with unit conversion', 6), ('Correct kinetic energy and MT conversion', 6), ('Correctly and honestly compares to the published range rather than overstating precision', 3)]), solution(2, 'Olympus Mons and the Hawaiian Hotspot Track', f"Time = distance/speed = {hawaii_chain_length_km} km / (0.07 km/yr) = {hawaii_age_myr:.1f} Myr ≈ " f"{hawaii_age_myr:.0f} Myr, comparable in order of magnitude to Olympus Mons's estimated multi-hundred-" f"Myr construction time. On Earth, the Pacific Plate's motion over the fixed Hawaiian hotspot spreads " f"volcanic material into a long chain of progressively older, smaller volcanic islands and seamounts. On " f"Mars, with no plate tectonics (Lecture 05), the lithosphere never moves relative to the underlying " f"mantle hotspot, so all the erupted material from repeated eruptions over a comparably long timescale " f"piles up in a single location, building one enormous volcano instead of a chain of smaller ones.
", [('Correct time calculation with unit conversion', 7), ('Correctly explains the plate-tectonics/stationary-lithosphere contrast', 8)]), solution(3, 'Crater Density and Venus\u2019s Resurfacing Event', "A surface that had simply never been struck by many impactors, or one that formed at the same time " "as the planet itself (4.5 Gyr ago) with no subsequent resurfacing, would both be expected to show a " "wide range of crater degradation states (heavily degraded ancient craters mixed with sharper younger " "ones accumulated steadily over 4.5 Gyr), because impacts would have been accumulating and being " "progressively degraded by subsequent smaller impacts and weathering throughout that entire span. " "Instead, Venus shows a low but narrow, roughly uniform range of crater degradation and density across " "its whole surface, which is the specific signature expected if nearly the entire surface was reset " "(buried by volcanism or otherwise resurfaced) at approximately the same time, comparatively recently, " "after which a much shorter interval of impact accumulation began anew.
", [('Correctly explains why continuous accumulation over 4.5 Gyr would look different', 8), ('Correctly explains why a narrow degradation range specifically implies a discrete resurfacing event', 7)]), solution(4, 'Simple versus Complex Crater Transition', "The apparent contradiction is resolved because the simple-to-complex transition diameter depends on " "target material strength as well as gravity, and Mars's surface (a mix of regolith, sedimentary, and " "volcanic rock, in some places ice-rich) does not have uniformly stronger material strength than the " "Moon's regolith/rock in a way that offsets its stronger gravity in the direction assumed here; more " "importantly, the specific transition-diameter values quoted for the Moon versus Mars in the published " "literature reflect a real, measured combination of both the gravity dependence and material-strength " "differences (including, for icy or volatile-rich near-surface material on parts of Mars, weaker " "effective strength than typical dry lunar regolith), so gravity alone is not the only variable and " "should not be used in isolation to predict the transition diameter, which is exactly the caveat this " "problem is testing.
", [('Correctly identifies that gravity is not the sole controlling variable', 8), ('Correctly explains the role of target material strength as a compensating factor', 7)]), solution(5, 'Mercury\u2019s Crater-Count Volcanic History', "Mercury, the smallest of the terrestrial planets (aside from dwarf planets), has the largest surface-" "to-volume ratio and is therefore expected, by the 1/R cooling-rate scaling from Lecture 04, to cool and " "lose active volcanism earliest among the terrestrial planets, all else equal; Mercury's crater-count-" "derived cessation of volcanism by roughly 3.7-3.9 Gyr ago is directly consistent with this expectation. " "Independent supporting evidence includes Mercury's widespread lobate scarps (Lecture 05), interpreted " "as a record of global contraction as its large core cooled and shrank -- itself a direct geological " "signature of the same rapid interior cooling implied by the crater-count volcanic-cessation age.
", [('Correctly applies the surface-to-volume cooling argument to Mercury', 8), ('Correctly cites lobate scarps as independent supporting evidence', 7)]), ) pset_html = pset_page(3, 'Impact Cratering and Comparative Surface Geology', 'Apply crater-scaling, chronology, and plate-tectonic-versus-stagnant-lid arguments from Lectures 05-06 to Tunguska, Olympus Mons, Venus, and Mercury.', ''.join(problems), 'OpenStax Astronomy 2e, Sections 9.4-9.6.') sol_html = solutions_page(3, 'Impact Cratering and Comparative Surface Geology', ''.join(solutions)) asmt = assessment_md(3, 'Impact Cratering and Comparative Surface Geology', ['Problem 1: unit conversion to megatons must be shown explicitly.', 'Problem 2: both the numeric time estimate and the plate-tectonics explanation are required.', 'Problem 3: answer must specifically address why a narrow (not wide) degradation range is the key diagnostic.', 'Problem 4: answer must identify material strength as the compensating factor, not merely restate the contradiction.', 'Problem 5: both the surface-to-volume argument and the specific independent evidence (lobate scarps) are required.'], ['Reporting kinetic energy without converting to megatons for comparison to the published range (Problem 1).', 'Treating the Hawaiian-chain formation time and Olympus Mons\'s construction time as needing to match exactly rather than being compared at order-of-magnitude precision (Problem 2).', 'Assuming gravity alone fully determines the simple-to-complex crater transition diameter (Problem 4).']) return pset_html, sol_html, asmt # --------------------------------------------------------------------------- # LAB 04: Scale height and equilibrium temperature for Titan and comparison # --------------------------------------------------------------------------- def make_lab_04() -> str: titan_g = surface_gravity(TITAN['mass'], TITAN['radius']) titan_h_n2 = scale_height_m(TITAN['teff'], N2_MASS, titan_g) titan_teq = equilibrium_temperature_k(L_SUN_W, TITAN['a_au'] * AU_M, TITAN['albedo']) sections = f"""This lab applies the hydrostatic scale-height formula (Lecture 07) and the radiative-equilibrium-temperature formula (Lecture 08) to Titan, using real Cassini-Huygens-measured parameters, and compares the results to Earth, Venus, and Mars (recomputed here rather than simply copied from the lecture table).
| Body | Mass (kg) | Radius (km) | Semimajor axis (AU) | Surface/effective temperature (K) | Bond albedo |
|---|---|---|---|---|---|
| Titan | {TITAN['mass']:.4e} | {TITAN['radius']/1000:.1f} | {TITAN['a_au']} | {TITAN['teff']} | {TITAN['albedo']} |
| Earth | {PLANETS['Earth']['mass']:.4e} | {PLANETS['Earth']['radius']/1000:.1f} | {PLANETS['Earth']['a_au']} | {PLANETS['Earth']['teq_obs']} | {PLANETS['Earth']['albedo']} |
Titan's actual atmosphere also contains a few percent methane and trace hydrocarbons, which slightly changes its true mean molecular mass from pure N₂'s 28.014 amu; students should state this simplification explicitly and note that it introduces a small (a few percent) systematic uncertainty in the computed scale height.
materials/ASTR320/reference-log.md.materials/ASTR320/src/generate_astr320_labs_psets.py.The scale-height formula H = kT/(mg) requires a bound atmosphere in hydrostatic equilibrium. " "Mercury's escape velocity is 4.25 km/s and its dayside surface temperature reaches ∼700 K. Using the " "Jeans-escape reasoning previewed in Lecture 07 and developed fully in Lecture 09, explain in a short " "paragraph why Mercury cannot sustain a substantial bound atmosphere, making a meaningful scale-height " "calculation for a thick Mercury atmosphere physically inapplicable.
"), problem(2, "Mars's Small Greenhouse Excess", f"Using the equilibrium-temperature table values from Lecture 08, Mars's greenhouse excess is " f"ΔT_gh = {mars_greenhouse:+.0f} K, compared to Venus's {venus_greenhouse:+.0f} K, despite both " f"planets having CO₂-dominated atmospheres. Compute the ratio of Venus's to Mars's surface pressure " f"(92 bar versus 6 mbar) and use it to explain, in a short paragraph, why atmospheric mass/opacity, not " f"merely the presence of CO₂, controls the size of the greenhouse excess.
"), problem(3, "Earth's Scale Height with a Different Surface Temperature", f"Recompute Earth's atmospheric scale height (using N₂ as the dominant species) for a " f"hypothetical mean surface temperature of 250 K (an ice-age-like scenario) instead of the standard 288 " f"K, holding gravity fixed. Compare your result to the standard {EARTH_H_N2/1000:.2f} km value from " f"Lecture 07 and state the percent change.
"), problem(4, 'Bond Albedo and Cloud Cover', "Venus has the highest Bond albedo (0.76) of any planet, due to its global sulfuric-acid cloud deck, " "while Mercury has the lowest (0.088), due to its dark, airless regolith surface. Using the equilibrium-" "temperature formula, compute what Venus's zero-greenhouse equilibrium temperature would be if its " "albedo were instead equal to Mercury's (0.088), holding its semimajor axis fixed, and explain in one " "sentence why this hypothetical value is higher than Venus's real computed equilibrium temperature.
"), problem(5, 'The Runaway Greenhouse and D/H Ratio', "Venus's atmospheric deuterium-to-hydrogen ratio is roughly 100-150 times Earth's. Using the " "differential-escape reasoning previewed in Lecture 08 and developed in Lecture 09 (lighter isotopes " "escape preferentially), explain in a short paragraph why this elevated D/H ratio is considered strong " "indirect evidence that Venus once had a much larger water inventory than it has today, rather than " "simply having formed with unusually deuterium-rich water.
"), ) solutions = ( solution(1, 'Why Mercury Has (Essentially) No Scale Height to Compute', "Mercury's low mass and correspondingly low escape velocity (4.25 km/s), combined with its very high " "dayside temperature (∼700 K, giving even heavy atmospheric molecules substantial thermal speeds), " "together give even relatively heavy gas species a Jeans escape parameter λ small enough for " "rapid, geologically fast atmospheric loss (the same λ = v_esc²/v_mp² reasoning applied " "quantitatively to Earth and Mars in Lecture 09). Any substantial primordial atmosphere Mercury may have " "had would therefore have been lost long ago, leaving only an extremely tenuous, transient exosphere " "(supplied by solar-wind sputtering and micrometeorite vaporization) far too thin to be usefully " "described by a hydrostatic scale-height formula.
", [('Correctly connects low escape velocity and high temperature to a small Jeans parameter', 8), ('Correctly concludes the scale-height formula is not meaningfully applicable', 7)]), solution(2, "Mars's Small Greenhouse Excess", f"92 bar / 6 mbar = 92000/6 ≈ 15,300, i.e., Venus's surface pressure (and hence total atmospheric " f"mass and infrared opacity) is roughly four orders of magnitude greater than Mars's, despite both " f"atmospheres being >95% CO₂ by composition. This confirms that the greenhouse effect's magnitude " f"depends on total atmospheric mass and column opacity to outgoing infrared radiation, not merely on " f"which gas is present: Mars's atmosphere, though chemically similar to Venus's, is simply too thin to " f"trap much outgoing thermal radiation, giving it only a small greenhouse excess despite its CO₂-" f"dominated composition.
", [('Correct pressure-ratio calculation', 6), ('Correctly explains that mass/opacity, not composition alone, controls greenhouse magnitude', 9)]), solution(3, "Earth's Scale Height with a Different Surface Temperature", f"H(250 K) = H(288 K) × (250/288) = {EARTH_H_N2/1000:.2f} × (250/288) = " f"{EARTH_H_N2/1000*(250.0/288.0):.2f} km, a decrease of {(1-250.0/288.0)*100:.1f}% relative to the " f"standard 288 K value, since H ∝ T linearly at fixed g and m.
", [('Correctly applies the linear T-scaling of the scale-height formula', 8), ('Correct percent-change calculation', 7)]), solution(4, 'Bond Albedo and Cloud Cover', f"T_eq(A=0.088) = [(1-0.088)/(1-0.76)]¹⁄⁴ × T_eq(A=0.76) = " f"[{(1-0.088)/(1-0.76):.3f}]¹⁄⁴ × {TEQ_BOND['Venus']:.1f} K = " f"{((1-0.088)/(1-0.76))**0.25 * TEQ_BOND['Venus']:.1f} K. This hypothetical value is higher than Venus's " f"real computed equilibrium temperature because a lower albedo means Venus would absorb a larger " f"fraction of the sunlight it intercepts, requiring a higher equilibrium temperature to radiate that " f"larger absorbed power back to space.
", [('Correct proportional recomputation using the (1-A)^(1/4) scaling', 9), ('Correct one-sentence physical explanation', 6)]), solution(5, 'The Runaway Greenhouse and D/H Ratio', "Because hydrogen (light) escapes far more readily than deuterium (heavier by roughly a factor of two " "in mass) via Jeans escape, any water vapor that is photodissociated high in Venus's atmosphere loses " "its hydrogen preferentially, progressively enriching the remaining hydrogen inventory in deuterium over " "time. The presently observed, strongly elevated D/H ratio is therefore best explained as the cumulative " "fossil signature of a much larger original water/hydrogen inventory having been lost over billions of " "years, rather than as evidence that Venus's water was unusually deuterium-rich from the start (which " "would require an ad hoc, unexplained difference in Venus's initial volatile delivery relative to " "Earth's, for which there is no independent supporting evidence).
", [('Correctly explains the differential-escape mechanism', 8), ('Correctly argues why this is stronger evidence than an alternative "born deuterium-rich" hypothesis', 7)]), ) pset_html = pset_page(4, 'Atmospheric Structure, Radiative Balance, and the Greenhouse Effect', 'Apply scale-height scaling, equilibrium-temperature/albedo relationships, and the greenhouse-excess concept from Lectures 07-08 to Mercury, Mars, Venus, and Earth.', ''.join(problems), 'OpenStax Astronomy 2e, Sections 10.1-10.4.') sol_html = solutions_page(4, 'Atmospheric Structure, Radiative Balance, and the Greenhouse Effect', ''.join(solutions)) asmt = assessment_md(4, 'Atmospheric Structure, Radiative Balance, and the Greenhouse Effect', ['Problem 1: must connect escape velocity AND temperature to Jeans escape, not just assert Mercury has no atmosphere.', 'Problem 2: must state the pressure-ratio order of magnitude AND correctly conclude mass/opacity (not composition) controls greenhouse magnitude.', 'Problem 3: must show the linear T-scaling explicitly, not just state a new H value.', 'Problem 4: must use the (1-A)^(1/4) proportional scaling correctly (a common error is treating T_eq as scaling linearly with (1-A)).', 'Problem 5: must explain why differential escape is favored over the "born deuterium-rich" alternative, not just describe the escape mechanism alone.'], ['Treating T_eq as proportional to (1-A) rather than (1-A) to the one-fourth power (Problem 4).', 'Confusing scale-height\'s linear dependence on temperature with the equilibrium-temperature formula\'s quarter-power dependence on flux (Problems 3 vs. 4).', 'Describing Venus\'s and Mars\'s CO2 atmospheres as fundamentally different in composition rather than in total mass/opacity (Problem 2).']) return pset_html, sol_html, asmt # --------------------------------------------------------------------------- # LAB 05: Jeans escape and magnetopause standoff comparison # --------------------------------------------------------------------------- def make_lab_05() -> str: jupiter_standoff = magnetopause_distance_m(JUPITER_B0_T, PLANETS['Jupiter']['radius'], SOLAR_WIND_RHO_KGM3, SOLAR_WIND_V_MS) earth_standoff = magnetopause_distance_m(3.1e-5, PLANETS['Earth']['radius'], SOLAR_WIND_RHO_KGM3, SOLAR_WIND_V_MS) sections = f"""This lab combines the Jeans-escape formalism (Lecture 09) with the magnetopause pressure-balance model (Lecture 10), using real measured planetary magnetic-field strengths, to compare Earth's and Jupiter's magnetospheric standoff distances quantitatively.
| Planet | Equatorial surface field B₀ (T) | Radius (km) |
|---|---|---|
| Earth | 3.1 × 10⁻⁵ | {PLANETS['Earth']['radius']/1000:.1f} |
| Jupiter | {JUPITER_B0_T:.2e} | {PLANETS['Jupiter']['radius']/1000:.1f} |
Assume typical solar-wind conditions at 1 AU (ρ_sw ≈ {SOLAR_WIND_RHO_KGM3:.1e} kg/m³, v_sw ≈ {SOLAR_WIND_V_MS/1000:.0f} km/s) for Earth; for Jupiter, note that the solar wind is roughly 27 times less dense at 5.2 AU than at 1 AU (inverse-square dilution), a correction you must apply.
Solar-wind density and speed both vary substantially with solar activity (by roughly a factor of a few), so the specific standoff-distance values computed here should be understood as representative, not exact, and should be reported with this caveat.
materials/ASTR320/reference-log.md.materials/ASTR320/src/generate_astr320_labs_psets.py.Compute the Jeans escape parameter λ for molecular oxygen (m = 32 amu) at Earth (assume " "exospheric T = 1000 K) and at Mars (assume exospheric T = 300 K, since Mars's thinner atmosphere and " "weaker greenhouse give it a cooler exosphere than Earth's). Compare both values to the hydrogen values " "computed in Lecture 09 (Earth λ_H2, Mars λ_H2) and state which species (H₂ or " "O₂) is retained more effectively on each planet, and why.
"), problem(2, "Mars's Ion Pickup Loss and MAVEN", "NASA's MAVEN mission has directly measured Mars's present-day ion-pickup atmospheric loss rate. " "Explain, in a short paragraph referencing Lecture 09 and Lecture 10, why this present-day measured loss " "rate, even if precisely known, cannot alone establish how much atmosphere Mars has lost over its entire " "4.5 Gyr history, and what additional information or assumption would be required to make that " "extrapolation.
"), problem(3, 'Comparing Uranus\u2019s and Neptune\u2019s Magnetic Fields', f"Uranus's equatorial surface field is roughly {URANUS_B0_T:.2e} T and Neptune's is roughly " f"{NEPTUNE_B0_T:.2e} T. Compute the ratio of their magnetic pressures (B²) and state what this " f"implies about their relative magnetopause standoff distances (in planetary radii), holding solar-wind " f"conditions and planetary radius ratio approximately fixed for this estimate.
"), problem(4, "Venus's Induced Magnetosphere", "Venus has no measurable intrinsic magnetic field, yet it does have a weak, induced magnetosphere " "formed by the interaction between the solar wind and Venus's ionosphere. Explain, in a short paragraph, " "why this induced magnetosphere is expected to provide much less protection against atmospheric erosion " "than Earth's intrinsic, dynamo-generated magnetosphere, referencing the dynamo requirements from " "Lecture 10.
"), problem(5, 'Escape Velocity and the Retention of Helium', "Helium (m = 4 amu) is lighter than most atmospheric gases but heavier than atomic hydrogen. Using " "Earth's escape velocity (11.2 km/s) and an assumed exospheric temperature of 1000 K, compute the Jeans " "escape parameter for helium and compare it to the hydrogen value from Lecture 09. State whether Earth's " "atmosphere should retain helium more or less effectively than hydrogen, and explain why Earth's " "atmosphere nonetheless contains only trace helium today (a fact not fully explained by Jeans escape " "alone).
"), ) solutions = ( solution(1, 'Jeans Escape of Molecular Oxygen: Earth versus Mars', f"O₂ thermal speed at 1000 K: v_mp = √(2kT/m) = √(2 × 1.381e-23 × 1000 / " f"({32*AMU:.3e})) ≈ 721 m/s; at Earth, λ_O2 = (11200/721)² ≈ " f"{earth_lambda_o2:.0f}. At Mars (T=300 K), v_mp ≈ 395 m/s; λ_O2 = (5030/395)² ≈ " f"{mars_lambda_o2:.0f}. Both λ_O2 values are far larger than the corresponding λ_H2 values " f"from Lecture 09 (∼15 for Earth, ∼3 for Mars), confirming that molecular oxygen is retained far " f"more effectively than hydrogen on both planets, because oxygen's much larger mass gives it both a " f"lower thermal speed and (through λ ∝ m) a much larger Jeans parameter at the same " f"temperature and escape velocity.
", [('Correct thermal-speed and Jeans-parameter calculations for both planets', 9), ('Correct comparison to Lecture 09\'s hydrogen values with correct physical explanation', 6)]), solution(2, "Mars's Ion Pickup Loss and MAVEN", "A present-day measured loss rate gives only a snapshot of current conditions; extrapolating it " "across 4.5 Gyr would implicitly assume the loss rate has been constant throughout Mars's history, which " "is almost certainly false, since both the solar wind's intensity (much stronger from the young, more " "active Sun) and Mars's own atmosphere (likely much thicker early on) have changed dramatically over " "time. Making a reliable historical extrapolation would require independent modeling of the young Sun's " "solar-wind and XUV output (informed by observations of young solar-type stars) combined with a model " "of how Mars's atmospheric thickness itself evolved over time, not simply multiplying today's measured " "rate by 4.5 Gyr.
", [('Correctly identifies that a present-day rate cannot simply be extrapolated linearly', 8), ('Correctly identifies what additional modeling would be required', 7)]), solution(3, 'Comparing Uranus\u2019s and Neptune\u2019s Magnetic Fields', f"Ratio of magnetic pressures (B²) = ({NEPTUNE_B0_T:.2e})² / ({URANUS_B0_T:.2e})² = " f"{neptune_standoff_ratio:.2f}. Since magnetopause standoff distance scales as B²/6 (the sixth root " f"of B², from the dipole field's r⁻³ falloff and the pressure balance derived in Lecture " f"10), a magnetic-pressure ratio of {neptune_standoff_ratio:.2f} implies Neptune's standoff distance " f"(in its own planetary radii) should be only modestly larger than Uranus's (roughly " f"{neptune_standoff_ratio**(1/6):.2f} times as large), since the sixth-root dependence strongly " f"compresses even a substantial pressure-ratio difference into a much smaller distance-ratio " f"difference.
", [('Correct magnetic-pressure ratio calculation', 7), ('Correctly applies the sixth-root scaling to standoff distance rather than a linear or squared scaling', 8)]), solution(4, "Venus's Induced Magnetosphere", "Lecture 10 established that a self-sustaining dynamo requires a convecting, electrically " "conducting fluid organized by planetary rotation into helical flow; Venus's induced magnetosphere, by " "contrast, arises passively from the solar wind interacting with Venus's ionosphere and does not require " "any internal dynamo process at all. An induced field is generally much weaker and more variable than an " "intrinsic dipole field, and it does not extend nearly as far from the planet, meaning it stands off the " "solar wind much closer to Venus's ionosphere/atmosphere than Earth's intrinsic magnetosphere does " "relative to Earth's atmosphere, providing correspondingly less protective standoff distance against " "direct atmospheric erosion.
", [('Correctly distinguishes induced from intrinsic (dynamo-generated) fields', 8), ('Correctly explains why an induced field provides less standoff protection', 7)]), solution(5, 'Escape Velocity and the Retention of Helium', f"Helium thermal speed at 1000 K: v_mp = √(2 × 1.381e-23 × 1000 / (4 × " f"1.66054e-27)) ≈ 2029 m/s. λ_He = (11200/2029)² ≈ 30.4, about twice the hydrogen " f"value (∼15) from Lecture 09, since helium is roughly twice as massive as H₂. Earth should " f"retain helium somewhat more effectively than hydrogen via Jeans escape alone. However, Earth's " f"atmosphere contains only trace helium today primarily because Earth's helium is continuously but only " f"slowly replenished by radioactive decay (alpha decay of uranium and thorium, Lecture 04) and, unlike " f"nitrogen or oxygen, has no significant chemical sink that would otherwise retain it at the surface; " f"the modest but non-negligible Jeans and non-thermal escape over geological time is therefore " f"sufficient to keep helium's atmospheric abundance low despite its relatively large λ, since its " f"supply rate is also very low.
", [('Correct thermal-speed and Jeans-parameter calculation for helium', 8), ('Correctly explains the low supply-rate resolution rather than treating Jeans escape as the full explanation', 7)]), ) pset_html = pset_page(5, 'Atmospheric Escape and Planetary Magnetic Fields', 'Apply Jeans escape and magnetopause pressure-balance reasoning from Lectures 09-10 to oxygen and helium retention, ion pickup, and the ice giants\u2019 magnetic fields.', ''.join(problems), 'OpenStax Astronomy 2e, Sections 10.3 and 10.5.') sol_html = solutions_page(5, 'Atmospheric Escape and Planetary Magnetic Fields', ''.join(solutions)) asmt = assessment_md(5, 'Atmospheric Escape and Planetary Magnetic Fields', ['Problem 1: both planets\' Jeans parameters must be computed and correctly compared to Lecture 09\'s hydrogen values.', 'Problem 2: must identify BOTH that solar activity and Mars\'s own atmospheric thickness likely changed over time, not just one factor.', 'Problem 3: must apply the sixth-root scaling, not a linear or squared scaling, to the standoff-distance ratio.', 'Problem 4: must explicitly distinguish induced from intrinsic fields using the dynamo-requirement language from Lecture 10.', 'Problem 5: must correctly resolve the apparent tension between helium\'s relatively large lambda and its low observed abundance via the low-supply-rate argument.'], ['Applying a linear or squared (rather than sixth-root) scaling to the magnetopause standoff-distance ratio (Problem 3).', 'Treating a present-day measured escape rate as directly applicable to the entire planetary history without qualification (Problem 2).', 'Concluding from a large Jeans parameter alone that a species must be abundant in the atmosphere, without considering its supply rate (Problem 5).']) return pset_html, sol_html, asmt # --------------------------------------------------------------------------- # LAB 06: Giant planet density comparison and the Roche limit for Uranus's rings # --------------------------------------------------------------------------- def make_lab_06() -> str: uranus_roche_fluid = roche_limit_fluid_m(PLANETS['Uranus']['radius'], URANUS_DENSITY, ICE_DENSITY_KGM3) epsilon_ring_m = 51149e3 # Uranus epsilon ring radius, standard published value sections = f"""This lab extends Lecture 11's giant-planet density comparison and Lecture 12's Roche-limit derivation to Uranus, using real Voyager 2 and stellar-occultation-derived ring data.
| Quantity | Value |
|---|---|
| Uranus mass | {PLANETS['Uranus']['mass']:.4e} kg |
| Uranus radius | {PLANETS['Uranus']['radius']/1000:.0f} km |
| Uranus mean density (to be computed) | — |
| Uranus's outermost major ring (epsilon ring) radius | {epsilon_ring_m/1000:.0f} km |
The exact density of Uranus's ring particles is not independently well constrained; using pure ice density is a simplifying assumption whose effect on the computed Roche limit should be explicitly stated as a limitation, not hidden, following this course's standard for honestly characterizing approximations.
materials/ASTR320/reference-log.md.materials/ASTR320/src/generate_astr320_labs_psets.py.Phobos (mass {PHOBOS['mass']:.4e} kg, radius {PHOBOS['radius']/1000:.2f} km) orbits Mars at " f"{PHOBOS['a_m']/1000:.0f} km, well outside Mars's Roche limit today, but is known to be spiraling " f"slowly inward due to tidal dissipation inside Mars. Compute Phobos's own density from its mass and " f"radius, then compute the fluid Roche limit for a body of that density orbiting Mars (mean density " f"computed from Mars's mass and radius). Compare Phobos's current orbital radius to this Roche limit " f"and state, in one sentence, what this comparison implies about Phobos's long-term fate.
"), problem(2, 'Saturn versus Jupiter: Why Does Density Differ Despite Similar Composition?', f"Saturn's and Jupiter's mean densities differ substantially (ratio " f"{saturn_jupiter_density_ratio:.3f}) despite both planets sharing a similar bulk hydrogen/helium-" f"dominated composition. Using the self-compression argument introduced for terrestrial planets in " f"Lecture 03 and revisited for giant planets in Lecture 11, explain in a short paragraph why Jupiter's " f"much stronger self-gravity is sufficient to explain this density difference without requiring the two " f"planets to have fundamentally different compositions.
"), problem(3, "Neptune's Ring Arcs", "Neptune's rings include unusual partial arcs of clumped material rather than a uniformly distributed " "ring, maintained by resonant confinement with the nearby moon Galatea. Using the shepherd-moon concept " "from Lecture 12 (developed there for Saturn's F ring), explain in a short paragraph how a single nearby " "moon's resonant gravitational influence could concentrate ring material into discrete arcs rather than " "spreading it uniformly around the full ring circumference.
"), problem(4, "Jupiter's Zonal Wind Depth and Gravity Harmonics", "Juno's gravity-field measurements indicate Jupiter's zonal winds persist to a depth of roughly 3000 " "km below the visible cloud tops before being damped out. Using the concept of increasing electrical " "conductivity with depth (Lecture 11), explain in a short paragraph why the winds should be expected to " "damp out at increasing depth, rather than persisting all the way to Jupiter's center.
"), problem(5, 'Why Ice Giants Are Not Simply Small Gas Giants', "Explain, in a short paragraph referencing both Lecture 02's core-accretion timing argument and " "Lecture 11's compositional data (∼90%+ hydrogen/helium for Jupiter/Saturn versus ∼10-20% for " "Uranus/Neptune), why classifying Uranus and Neptune as 'ice giants' rather than merely 'smaller gas " "giants' reflects a genuine compositional and formation-history distinction, not just an arbitrary size " "cutoff.
"), ) solutions = ( solution(1, "Phobos and Mars's Roche Limit", f"Phobos's density: ρ = {PHOBOS['mass']:.4e} / [(4/3)π({PHOBOS['radius']:.3e})³] ≈ " f"1876 kg/m³. Mars's mean density (from Lecture 01's table) is {DENSITIES['Mars']:.0f} kg/m³. " f"Fluid Roche limit: d = 2.44 R_Mars (ρ_Mars/ρ_Phobos)¹⁄³ = " f"{phobos_roche/1000:.0f} km. Phobos's current orbital radius ({phobos_a/1000:.0f} km) is larger than " f"this computed Roche limit but is decreasing over time due to tidal dissipation inside Mars, implying " f"that Phobos will eventually cross inside the Roche limit and be tidally disrupted, likely within the " f"next tens of millions of years.
", [('Correct Phobos density and Mars mean density calculations', 6), ('Correct fluid Roche-limit calculation', 6), ('Correct qualitative conclusion about Phobos\'s eventual fate', 3)]), solution(2, 'Saturn versus Jupiter: Why Does Density Differ Despite Similar Composition?', "Jupiter's substantially larger mass produces much stronger self-gravity throughout its interior, " "compressing its hydrogen/helium envelope (and any core material) to higher density at any given " "pressure level than Saturn's weaker self-gravity can achieve, exactly as Lecture 03 showed that Earth's " "stronger self-compression (relative to Mercury) raises its bulk density above what its material " "composition alone would suggest. This self-compression effect, not any difference in bulk hydrogen/" "helium/heavy-element composition, is sufficient on its own to explain why Jupiter is denser than " "Saturn despite both planets sharing a broadly similar protosolar-like composition.
", [('Correctly identifies self-compression, not composition difference, as the explanation', 9), ('Correctly connects this to the analogous terrestrial-planet argument from Lecture 03', 6)]), solution(3, "Neptune's Ring Arcs", "A shepherd moon in resonance with a ring can exert a gravitational torque that varies periodically " "as the moon and a given ring particle repeatedly return to the same relative orbital geometry; this " "resonant torque can trap ring material at specific longitudes relative to the moon's orbit (librating " "around stable resonance points) rather than allowing it to spread freely around the entire ring " "circumference, producing discrete, confined arcs of enhanced ring-particle density at those resonance " "points instead of a smooth, uniform ring.
", [('Correctly explains resonant trapping/libration as the confinement mechanism', 15)]), solution(4, "Jupiter's Zonal Wind Depth and Gravity Harmonics", "As depth increases inside Jupiter, pressure and temperature both rise, and hydrogen's electrical " "conductivity increases correspondingly (eventually reaching the fully metallic state described in " "Lecture 11). Once the surrounding fluid becomes sufficiently electrically conductive, magnetic forces " "(generated by Jupiter's own dynamo-produced field) become strong enough to resist and damp out organized " "large-scale zonal flow, effectively braking the winds at the depth where conductivity becomes " "significant -- roughly the depth Juno's gravity data indicate, ∼3000 km.
", [('Correctly connects increasing conductivity with depth to magnetic braking of the winds', 15)]), solution(5, 'Why Ice Giants Are Not Simply Small Gas Giants', "The gas-giant/ice-giant distinction is not simply a matter of Uranus and Neptune being smaller " "versions of Jupiter and Saturn; if Uranus and Neptune had captured hydrogen/helium gas as efficiently " "as Jupiter and Saturn did, they would be expected to show a similarly dominant (∼90%+) hydrogen/" "helium mass fraction regardless of their smaller total mass. Instead, their measured ∼10-20% " "hydrogen/helium fraction reflects a genuine difference in formation history: their cores are thought to " "have formed more slowly (in a lower-density, more slowly orbiting part of the disk, per Lecture 02's " "core-accretion argument) and so captured comparatively little nebular gas before the gas disk " "dispersed, leaving them dominated by ices and rock rather than hydrogen/helium -- a compositional " "outcome, not merely a size difference.
", [('Correctly explains why size alone would not predict the observed composition difference', 8), ('Correctly connects the composition difference to core-accretion formation timing', 7)]), ) pset_html = pset_page(6, 'Giant Planets, Rings, and Satellites', 'Apply the Roche limit, self-compression, and core-accretion timing arguments from Lectures 02, 03, 11, and 12 to Phobos, Saturn/Jupiter, and Neptune\u2019s ring arcs.', ''.join(problems), 'OpenStax Astronomy 2e, Sections 11.1-11.4.') sol_html = solutions_page(6, 'Giant Planets, Rings, and Satellites', ''.join(solutions)) asmt = assessment_md(6, 'Giant Planets, Rings, and Satellites', ['Problem 1: both the Phobos density and Mars Roche-limit calculations are required, plus a correctly reasoned conclusion about Phobos\'s fate.', 'Problem 2: must correctly attribute the density difference to self-compression, not to a claimed composition difference.', 'Problem 3: must specifically invoke resonant trapping/libration, not a vague "gravity holds it there" explanation.', 'Problem 4: must connect conductivity increase with depth to magnetic braking specifically.', 'Problem 5: must explain why size alone is an insufficient explanation before giving the core-accretion-timing resolution.'], ['Attributing Jupiter/Saturn density differences to composition differences rather than self-compression (Problem 2).', 'Using the rigid-body rather than fluid-body Roche-limit formula without justification (Problem 1), inconsistent with Lecture 12\'s stated preference for the fluid formula for rubble-pile bodies like Phobos.', 'Vague, non-mechanistic explanations for ring-arc confinement that do not reference resonance (Problem 3).']) return pset_html, sol_html, asmt # --------------------------------------------------------------------------- # LAB 07: Capstone -- tidal heating and habitability synthesis for TRAPPIST-1 # --------------------------------------------------------------------------- def make_lab_07() -> str: # Illustrative tidal-heating estimate for TRAPPIST-1e assuming an Io-like # forced eccentricity, to synthesize Lecture 13's tidal framework with the # Lecture 14 TRAPPIST-1 dataset. Uses the same tidal_acceleration function. trappist1e_a_m = TRAPPIST1_PLANETS['e']['a_au'] * AU_M # Approximate TRAPPIST-1e as Earth-sized/massed for this order-of-magnitude estimate (stated explicitly as an approximation). trappist1e_tidal_a = tidal_acceleration(TRAPPIST1_STAR['mass_msun'] * M_SUN_KG, PLANETS['Earth']['radius'], trappist1e_a_m) trappist1e_tidal_frac = trappist1e_tidal_a / EARTH_SURFACE_G * 100 sections = f"""This capstone lab combines the tidal-acceleration framework from Lecture 13 with the real, live-verified TRAPPIST-1 system data introduced in Lecture 14, synthesizing the semester's tidal and radiative-balance tools on a single exoplanet system.
| Quantity | Value |
|---|---|
| TRAPPIST-1 stellar mass | {TRAPPIST1_STAR['mass_msun']} M⊙ |
| TRAPPIST-1e semimajor axis | {TRAPPIST1_PLANETS['e']['a_au']} AU |
| TRAPPIST-1e orbital period | {TRAPPIST1_PLANETS['e']['period_d']} days |
| Io tidal acceleration fraction (Lecture 13, for comparison) | {IO_TIDAL_FRAC:.2f}% of Io's own surface gravity |
Approximating TRAPPIST-1e as Earth-sized/massed for the tidal-acceleration calculation is a genuine simplification (its true mass is less precisely measured than its radius); students must state this explicitly rather than presenting the resulting tidal-heating estimate as equally well-constrained as the equilibrium-temperature calculation, which uses only directly measured quantities (semimajor axis and stellar luminosity).
materials/ASTR320/reference-log.md.materials/ASTR320/src/generate_astr320_labs_psets.py.Kepler-186f orbits a red dwarf star with luminosity {KEPLER186F['star_lum_lsun']} L⊙ at a " f"semimajor axis of {KEPLER186F['a_au']} AU (orbital period {KEPLER186F['period_d']} days), and has a " f"measured radius of {KEPLER186F['radius_rearth']} Earth radii. Using the identical equilibrium-" f"temperature formula from Lecture 08/14, compute Kepler-186f's zero-albedo equilibrium temperature and " f"compare it to TRAPPIST-1e's ({TRAPPIST1_TEQ['e']:.1f} K, from Lecture 14) and Earth's " f"({EARTH_TEQ_ZERO_ALBEDO:.1f} K, from Lecture 08).
", points=20), problem(2, 'TRAPPIST-1b and 1h: The Hottest and Coldest Planets in the System', f"Using the equilibrium-temperature table from Lecture 14, TRAPPIST-1b's zero-albedo equilibrium " f"temperature is {trappist1b_teq:.1f} K and TRAPPIST-1h's is {trappist1h_teq:.1f} K. Compute the ratio " f"T_eq(b)/T_eq(h) and compare it to the ratio of their semimajor axes (a_h/a_b), verifying that the " f"ratio of equilibrium temperatures follows the expected inverse-square-root scaling with orbital " f"distance (T_eq ∝ a⁻¹⁄²).
", points=20), problem(3, 'Tidal Locking Timescale Reasoning', "Tidal-locking timescales decrease steeply with decreasing orbital distance and increasing host-star " "mass (roughly as a⁶ for fixed stellar/planetary parameters, from standard tidal-despinning theory, " "not derived in this course but stated here for use). Given that all seven TRAPPIST-1 planets orbit " "within 0.062 AU (compared to Mercury's 0.39 AU orbit around the much more massive Sun), explain in a " "short paragraph why essentially all TRAPPIST-1 planets are expected to be tidally locked today, even " "though Mercury itself is not (Mercury is in a 3:2 spin-orbit resonance rather than a strict 1:1 " "lock).
", points=15), problem(4, 'Detection-Method Bias in the Exoplanet Census', "Explain, in a short paragraph, why the radial-velocity method's greater sensitivity to massive, " "close-in planets caused early exoplanet surveys to discover a disproportionate number of 'hot " "Jupiters,' and why this early sample is not representative of the exoplanet population as a whole, " "referencing Lecture 14's discussion of detection bias.
", points=15), problem(5, 'Capstone Synthesis: Our Solar System in Context', "In 200-300 words, synthesize this course's comparative-planetology themes (formation, interior " "structure, atmospheres, magnetism, tides) to address the open question posed in Lecture 01: how " "representative is our solar system's architecture of planetary systems generally? Your answer must " "cite at least three specific pieces of evidence developed across the semester (e.g., the frost-line " "argument, the exoplanet mass-radius degeneracy, TRAPPIST-1's compact resonant architecture) and must " "distinguish which parts of your answer are well-supported by current data versus still genuinely " "open questions in the research literature.
", points=30), ) solutions = ( solution(1, 'Kepler-186f: A Second Real Exoplanet Comparison', f"T_eq = [L/(16πσa²)]¹⁄⁴ = [({KEPLER186F['star_lum_lsun']*L_SUN_W:.3e}) / " f"(16π×5.670374e-8×({KEPLER186F['a_au']*AU_M:.3e})²)]¹⁄⁴ = " f"{kepler186f_teq:.1f} K. This is comparable to TRAPPIST-1e's {TRAPPIST1_TEQ['e']:.1f} K and to Earth's " f"{EARTH_TEQ_ZERO_ALBEDO:.1f} K, placing Kepler-186f, like TRAPPIST-1e, within the temperature range " f"commonly associated with the habitable zone, though (as with TRAPPIST-1e) this equilibrium-temperature " f"calculation alone cannot confirm habitability without atmospheric and albedo information.
", [('Correct equilibrium-temperature calculation with correct unit conversions', 12), ('Correct, appropriately hedged comparison to TRAPPIST-1e and Earth', 8)]), solution(2, 'TRAPPIST-1b and 1h: The Hottest and Coldest Planets in the System', f"T_eq(b)/T_eq(h) = {trappist1b_teq:.1f}/{trappist1h_teq:.1f} = {trappist1b_teq/trappist1h_teq:.3f}. " f"√(a_h/a_b) = √({TRAPPIST1_PLANETS['h']['a_au']}/{TRAPPIST1_PLANETS['b']['a_au']}) = " f"{math.sqrt(TRAPPIST1_PLANETS['h']['a_au']/TRAPPIST1_PLANETS['b']['a_au']):.3f}. These two ratios agree " f"closely (both ≈ {trappist1b_teq/trappist1h_teq:.2f}), confirming the expected T_eq ∝ " f"a⁻¹⁄² scaling directly from real TRAPPIST-1 data.
", [('Correct temperature-ratio calculation', 8), ('Correct semimajor-axis-ratio square-root calculation and correct verification of the scaling law', 12)]), solution(3, 'Tidal Locking Timescale Reasoning', "Because tidal-despinning timescale falls off steeply with decreasing orbital distance (roughly as " "a⁶), even a modest decrease in orbital distance produces an enormous decrease in locking " "timescale. All seven TRAPPIST-1 planets orbit at less than one-sixth of Mercury's orbital distance " "from the Sun, and TRAPPIST-1 itself, though much less massive than the Sun, is old (∼7.6 Gyr), " "giving even a slower-acting tidal torque ample time to act; the combination of extremely small orbital " "distance (steeply reducing the timescale) and a old system age is expected to be more than sufficient " "to have tidally locked essentially every TRAPPIST-1 planet, whereas Mercury's larger orbital distance " "from a more massive star left it in a more weakly torqued regime where it settled into a 3:2 rather " "than a strict 1:1 resonance.
", [('Correctly invokes the steep (a^6-type) distance scaling of tidal locking timescale', 8), ('Correctly contrasts with Mercury\'s different regime rather than treating all close orbits identically', 7)]), solution(4, 'Detection-Method Bias in the Exoplanet Census', "Radial-velocity amplitude scales with planet mass and inversely with orbital distance and stellar " "mass, so the method most easily detects massive planets orbiting very close to their star (the 'hot " "Jupiter' population); lower-mass planets, or planets on wider orbits, produce a smaller reflex " "stellar wobble that is harder to distinguish from instrumental and stellar-activity noise. Because " "early radial-velocity surveys were the dominant discovery method before large-scale transit surveys " "(e.g., Kepler, TESS) came online, the earliest confirmed exoplanets were disproportionately hot " "Jupiters, giving an initially skewed impression of typical exoplanet system architecture that later, " "less biased transit-survey statistics have substantially corrected.
", [('Correctly explains the radial-velocity sensitivity bias', 8), ('Correctly explains how this biased early demographic conclusions', 7)]), solution(5, 'Capstone Synthesis: Our Solar System in Context', "[Model answer, 200-300 words; graded for content and correct hedging rather than for an exact " "match] A strong answer should note: (1) the frost-line argument (Lecture 02) shows our solar system's " "terrestrial/giant-planet split is a specific, well-understood consequence of where the frost line fell " "in our particular protoplanetary disk, not a universal necessity -- other disks with different " "temperature profiles or disk masses could plausibly produce very different architectures; (2) the " "exoplanet mass-radius degeneracy (Lecture 14) shows that our confidence in solar-system-like " "interior compositions elsewhere is genuinely limited by current data, an honestly open question; (3) " "TRAPPIST-1's compact, resonant, tidally locked architecture (Lecture 14) has no solar-system analog at " "all, and such compact systems appear statistically common in the broader exoplanet census, suggesting " "our solar system's wider-orbit, non-resonant architecture may not be the 'default' outcome of planet " "formation. A strong answer explicitly separates well-supported claims (the existence of common compact " "systems, the basic frost-line physics) from open questions (whether Earth-like habitability is common, " "whether TRAPPIST-1 planets can retain atmospheres) rather than presenting a falsely confident overall " "conclusion.
", [('Cites at least three specific pieces of evidence from across the semester', 12), ('Correctly and explicitly distinguishes well-supported claims from open questions', 12), ('Coherent, well-organized synthesis within the word limit', 6)]), ) pset_html = pset_page(7, 'Capstone: Exoplanet Comparative Planetology', 'Synthesize the full semester\u2019s toolkit -- equilibrium temperature, tidal locking, detection-method bias, and comparative architecture -- using real TRAPPIST-1 and Kepler-186f data.', ''.join(problems), 'OpenStax Astronomy 2e, Chapter 13.') sol_html = solutions_page(7, 'Capstone: Exoplanet Comparative Planetology', ''.join(solutions)) asmt = assessment_md(7, 'Capstone: Exoplanet Comparative Planetology', ['Problem 1: full unit-converted equilibrium-temperature calculation required, plus an appropriately hedged (not overconfident) habitability comment.', 'Problem 2: both ratios must be computed and explicitly compared to verify the scaling law, not just asserted.', 'Problem 3: must invoke the steep distance-scaling of tidal-locking timescale and correctly contrast with Mercury\'s different resonance outcome.', 'Problem 4: must explain the RV sensitivity bias mechanistically, not just state that hot Jupiters were found first.', 'Problem 5 (capstone): must cite at least three specific pieces of semester evidence and explicitly separate well-supported from open conclusions; a fluent but generic essay without specific citations or without honest hedging should not receive full credit.'], ['Presenting equilibrium-temperature agreement with Earth as proof of habitability rather than as one necessary but insufficient condition (Problem 1).', 'Treating Mercury and the TRAPPIST-1 planets as following identical tidal-locking outcomes without accounting for the steep distance/mass scaling (Problem 3).', 'Writing an overly generic capstone synthesis (Problem 5) that could apply to any course rather than citing this course\'s specific quantitative results.']) return pset_html, sol_html, asmt def main(): LAB_DIR.mkdir(parents=True, exist_ok=True) PSET_DIR.mkdir(parents=True, exist_ok=True) labs = [make_lab_01, make_lab_02, make_lab_03, make_lab_04, make_lab_05, make_lab_06, make_lab_07] for i, fn in enumerate(labs, start=1): (LAB_DIR / f"lab-{i:02d}.html").write_text(fn(), encoding='utf-8') psets = [make_pset_01, make_pset_02, make_pset_03, make_pset_04, make_pset_05, make_pset_06, make_pset_07] for i, fn in enumerate(psets, start=1): pset_html, sol_html, asmt = fn() (PSET_DIR / f"problem-set-{i:02d}.html").write_text(pset_html, encoding='utf-8') (PSET_DIR / f"problem-set-{i:02d}-solutions.html").write_text(sol_html, encoding='utf-8') (PSET_DIR / f"problem-set-{i:02d}-assessment.md").write_text(asmt, encoding='utf-8') print(f"Wrote {len(labs)} labs and {len(psets)} problem sets (with solutions and assessment instructions) " f"to {LAB_DIR} and {PSET_DIR}") if __name__ == '__main__': main()