Haute Lumière

Commerce · VII.10 · MMXXVI · daylight

La Bourse  /  Volume VII  /  Nº VII.10  /  Quiz, reflection, essays

ASSESSMENT · Chapter VII.10 — The Cosmological Frame

Three instruments: a ten-point quiz, eight reflection questions, five essay prompts. The quiz checks comprehension rather than recall. The reflections are private and first-person. The essays are arguable from more than one side.

---

THE QUIZ — ten points

Four on recall.

1. State, in one sentence each, what enters and what leaves the Earth system in a year of equilibrium, and what the difference between them is.

The same quantity of energy enters and leaves — 122,526 TW absorbed, 122,526 TW emitted. The difference is not energy but temperature: it arrives from a source at 5,772 K and leaves at 255.1 K. One mark for "energy in equals energy out", one for naming the temperature difference as the thing that changes. An answer that says energy is consumed has missed the entire chapter.

2. What is Landauer's limit, and what exactly does it charge for?

k_B · T · ln 2 per bit — 2.87 zJ at 300 K. It charges for erasure, not for computation. Bennett (1982) showed that logically reversible computation has no lower bound; the cost attaches to discarding information because erasure reduces the number of accessible states.

3. Give humanity's position on the Sagan continuous Kardashev scale, and the formula that produces it.

K = (log₁₀ P − 6) / 10 with P in watts. At 19.6 TW, K = 0.729. Credit an answer that also notes the whole Sun is 2.058 and the galaxy 3.089 on the same scale.

4. Name the atmospheric window and say why a surface facing it behaves differently from one facing sideways.

Roughly eight to thirteen micrometres, where the atmosphere is nearly transparent to infrared. A surface with a clear upward view through that band is radiating to deep space rather than exchanging heat with the local air, so it can sit below ambient temperature with no power input.

Four on application.

5. A colleague argues that a fully renewable grid removes any physical limit on how much energy a civilisation can use, since the Sun supplies thousands of times what we consume. What is missing?

The sink. Every watt dissipated, from any source, must leave the planet as infrared, and the planet can only emit more by getting warmer: dT/T = (1/4)(dP/P). Dissipation of 1,921 TW adds a whole kelvin irrespective of greenhouse gases. Full marks require both halves: the source really is effectively unlimited and the sink is the term that binds first — at two percent growth, 231 years against 441.

6. Someone proposes that your organisation adopt an internal carbon-style charge on waste heat, justified by the two-hundred-and-thirty-one-year threshold. Diagnose the proposal.

The physics is right and the business case is empty. At a three percent discount rate a payoff at 231 years has a present value of 0.000978 — one part in a thousand — so the threshold cannot justify any allocation today. The stronger answer rescues the proposal: the same charge is defensible on near-term grounds (electricity cost, chiller capacity, siting), and should be argued there. The cosmological term belongs in the reasoning, never in the paper.

7. Why is today's climate problem not a waste-heat problem, and what is the number that shows it?

Human direct dissipation warms the planet by about 0.010 K. Earth's energy imbalance is 459 TW against a human energy system of 19.6 TW — 23.4 times larger. The forcing is radiative gating, a change in what the atmosphere lets out, not energy released. Credit any answer naming the error bar too: the uncertainty on the imbalance is 204 TW, 10.4 times the whole human energy system.

8. A radiative-cooling vendor offers panels at a site where the commercial electricity tariff is 0.12 USD/kWh. On the chapter's assumptions, what do you tell them?

The break-even tariff on those assumptions is 0.20 USD/kWh — 0.07 × 100 / 35.0. At 0.12 the aperture does not clear a seven percent hurdle and no thermodynamic argument changes that. The stronger answer asks which assumption is site-specific — duty cycle, chiller COP, installed cost — and recomputes before refusing, rather than refusing on the headline figure.

Two that require the arithmetic to be done.

9. From a total solar irradiance of 1,361 W/m² and a Bond albedo of 0.294, compute the effective radiating temperature. Show every step.

Spread the beam over the sphere: 1,361 / 4 = 340.25 W/m². Absorbed: 340.25 × (1 − 0.294) = 240.2 W/m². Invert Stefan-Boltzmann: T = (240.2 / 5.670374419e-8)^0.25 = 255.1 K, which is -18.0 C. Credit any method reaching 255 K. The point of the question is that three inputs and one constant fix the sink temperature of the whole planet, and a student who has done it once will never again treat it as a claim to be taken on trust.

10. Using T_sun = 5,772 K and your answer to question 9, compute the ratio of entropy exported to entropy received, and state how many infrared photons leave for each visible photon that arrives at equal energy.

5,772 / 255.1 = 22.6. The four-thirds blackbody factor cancels in the ratio, so it holds on either convention. Mean photon energy scales with temperature, so about twenty-three infrared photons leave per visible photon arriving. The stronger answer gives the absolute export too — 6.12 × 10^14 W/K on the four-thirds convention, 4.59 × 10^14 on the bare Q/T convention — and names which convention it used.

---

REFLECTION — eight questions, for one person and a pen

These are not for a room. Write the answers by hand if you can; the slowness is the point.

1. What is the largest frame you personally use when deciding something — a quarter, a career, a lifetime, a lineage? Write down the honest one, not the admirable one.

1. Where have you used a very large idea to feel resolved about something without changing any behaviour? What would have had to change for it to count?

1. This chapter says the scarce good is cold, not heat. Where else in your life have you been carefully managing the supply of something while ignoring where it has to go afterwards?

1. Think of a piece of knowledge you are certain will still be true in two hundred years. What do you do differently because of it — and if the honest answer is nothing, is that a failure or is it correct?

1. What resource are you currently using that costs nothing and belongs to nobody? What would it take to make your access to it secure before somebody else notices it?

1. Where are you counting quantity when you should be counting grade — hours rather than attention, headcount rather than capability, revenue rather than margin? Name one, specifically.

1. When you last found yourself moved by an argument about the very long run, what were you actually deciding? Was the argument doing work, or was it supplying a feeling that let you decide something else?

1. Write the one sentence you would want a person opening your records in two hundred years to find still checkable. Then write down what you would have to do this month for that sentence to be there.

---

ESSAY PROMPTS — five

Each is arguable from more than one side. Each requires at least one source the chapter cites and at least one it does not.

1. The discount rate against the physics. The chapter concedes that at a three percent discount rate nothing in it binds a decision this century, and then argues the frame is worth holding anyway. Argue either that this concession is fatal — a frame that cannot change an allocation is decoration — or that discount rates are the wrong instrument for irreversible planetary quantities and the concession understates the case. Use Murphy (2022), and one source on discounting for long horizons that the chapter does not cite.

2. Is the waste-heat limit real, or is it a limit on a scenario nobody is in? Two hundred and thirty-one years at two percent energy growth assumes two percent energy growth. Global energy use has grown more slowly than output for decades. Argue whether the thermal constraint is a genuine feature of the long run or an artefact of extrapolating a growth rate through a period in which the relevant elasticity changed. Engage the chapter's intensity identity directly, and one empirical source on energy–GDP decoupling that it does not cite.

3. Kardashev, and what a scale does to the thing it measures. The chapter treats the Kardashev ladder as accounting and rejects it as a ranking. Argue the other side: that a scale which orders civilisations by energy throughput encodes a value judgement — that more power is more advanced — and that adopting it as an accounting frame smuggles the judgement in. Use Kardashev (1964) and Sagan (1973), and one source from the history or sociology of science on metrics and their effects that the chapter does not cite.

4. The floor under computation, and what happens when it is reached. Landauer's limit is roughly six and a half orders of magnitude below current practice, with perhaps six decades of headroom at Koomey's post-2000 rate. Write the economic history of the sixty years after the halvings stop: what happens to the cost structure of information services, to the location of computation, to the value of reversible and analogue approaches. Use Landauer (1961) and Bennett (1982), and one source on the economics of computing costs that the chapter does not cite. State your assumed bit-operations-per-FLOP factor and show how your conclusion moves with it.

5. Owning the sky. The chapter proposes a recorded sky-access easement drafted from solar-access precedent, so that a radiative cooling aperture cannot be shaded out by a neighbouring development. Argue either that this is a straightforward and overdue extension of existing property doctrine, or that creating enclosable rights in the upward view is a new enclosure of a commons with distributional consequences worth refusing. Use Raman et al. (2014) for the physics, and one source on property rights in light, air or solar access that the chapter does not cite.