The Red Coast Link Budget

The Mirror is the Sun, Earth and Jupiter in three dimensions, placed from the JPL approximate elements for any day from 1969 to 1973 with the orbits and the ecliptic grid drawn from the same numbers. Two routes are drawn from Jupiter, straight to Earth in cyan and via the Sun in gold, and a clock runs two wavefronts along them at the speed of light: the burst itself and the echo the novel's mirror would send. Chips carry the three distances, the detour and its light-time, the book's sixteen minutes and forty-two seconds with the days of the year on which it was possible, and the year's ceiling at opposition; a timeline along the bottom marks both arrivals. A second mode runs the autumn 1971 shout the other way, Earth to the Sun and the Sun outward, and marks the minute it crosses Jupiter's orbit. Four cameras, a year row, the two burst dates and the opposition as buttons.

The Shout is a ladder of radiated power from a megawatt to a Type III: Red Coast's transmitter, the Arecibo message's EIRP, Earth in 1973, Kardashev's 1964 anchors, the book's Type I band, II and III, the Sun, and the shout itself moving with the gain dial, with Sagan's K on the right edge and the gain a Type II would need drawn as a bracket. Beside it a ledger: the shout on Sagan's scale, the mirror expressed as a dish, the flux and the janskys at Alpha Centauri or any distance, Arecibo at the same distance, who hears it and how far, the quiet Sun's own 12 GHz, and whether the shout is the brightest star. Below, the brightest-star bar: the stars' spectral luminosities at centimetre wavelengths with the shout's marker moving as the bandwidth dial moves.

The Oven is the cover story: a beam from the peak to a satellite, the spot it makes, the artillery cone beside it for scale, and a ledger of gain, EIRP, air, flux against the required 0.1 to 1 W/cm², the dish the story would need, the beam's width and crossing time and the pointing budget; below it the three bands side by side. The File runs fourteen cards, the report, the note in the journal, the ceiling, the dates, the mirror, the ladder, Alpha Centauri, Arecibo, the brightest star, the oven, the report's sums, the clock, the light-hours and the wave past Jupiter, then two verdicts at equal size.

This site's signed review of The Three-Body Problem (Cixin Liu, 2008; Ken Liu's translation) says, at the sophon: "I will note that the one claim in the book with numbers attached is the sun as amplifier, and that by the novel's own figures a hundred-million-fold gain on twenty-five megawatts comes to a few times ten to the fifteenth watts, which is the book's own Kardashev Type I, not the Type II it announces. The plot does not turn on it. This shelf prices numbers stated with confidence, fiction included, and that one is now on the list." This bench is that entry. Red Coast gives more than a gain: chapter 13 carries a founding report with three bands, 10 to 25 megawatts, a 200-light-year transmit sphere, a 1,000-light-year listening sphere and two tables of transmission times; chapter 22 carries a measurement, solar interference sixteen minutes and forty-two seconds after two Jovian radio bursts, the triangle Ye Wenjie draws on the blackboard; chapter 12 carries a cover story, a tenth of a watt to a watt per square centimetre cooking enemy satellites; chapters 23 and 24 carry an eight-year clock; chapter 33 a listening post's reach and a fleet's timetable. Each is a sentence with a number in it.

It is here because those sentences can be computed, and computing them does not require believing in the mirror. The triangle is real geometry: how much further a burst travels from Jupiter via the Sun to Earth than straight, given by the JPL elements for any date, with a ceiling, twice Earth's distance from the Sun, reached only at opposition; the book's number sits a few percent under the ceiling and is attached to two dates nineteen and thirty-nine days after it. The link budget is the inverse square and an aperture: a gain of a hundred million is 80 dBi, the gain of a hundred-metre dish at 12 GHz, and the Kardashev scale Ye quotes is Sagan's 1973 interpolation, which calls Earth Type 0.7 and calls this shout Type 0.94. At Alpha Centauri the shout is about twelve thousand janskys in a kilohertz, audible to a 25 m dish in a second; "the brightest star in the Milky Way at 12,000 MHz" holds for stars if the band is a kilohertz and fails if it is a megahertz, and the book gives no band. This is the fourth bench on the SETI line: the Arecibo message bench prices the one message humanity actually sent, the Wow! signal bench the one reply-shaped thing it actually heard, and the Fermi paradox bench has credited Liu for the dark forest since the day it opened; this one prices the novel's shout with the same ledger. It also sits on the fiction-pricing line after Project Hail Mary: the novel is quoted as premise, the numbers are priced, and the story is not reviewed; the review is next door, linked, not duplicated.

Everything on the bench is arithmetic on declared inputs: a date, a power, a gain, a bandwidth, a distance, a dish. The book's figures are entered as they stand and the script that ships with the site re-derives every displayed number before the build passes.

detour = d_JS + d_SE − d_JE · Δt = detour / c · ceiling = 2 r_E · P_iso = P·G · K = (log₁₀ P_iso − 6) / 10 · S = P_iso / 4πd² · G = η(πD/λ)² · d_hear = √(P_iso A_eff √τ / 4π·10·kT_sys√B)

The triangle. Earth (the Earth-Moon barycentre) and Jupiter are placed by the JPL approximate Keplerian elements, valid 1800-2050, with Kepler's equation solved by Newton's method; the elements are good to arcminutes, which is seconds of light-time here. The detour is d_JS + d_SE − d_JE, converted at 499.0 seconds per AU. By the triangle inequality d_JE ≥ d_JS − d_SE, so the detour can never exceed 2 r_E, and it reaches that only at opposition. The bench scans each year for the longest detour and for the run of days on which the book's 2.008 AU was possible.

The budget. The mirror is taken as the novel describes it, a hundred-million-fold amplifier re-emitting in every direction, so the shout is P_iso = P·G isotropic. Sagan's interpolation, which is the ladder the book quotes, places it at K = (log₁₀ P_iso − 6)/10. The flux at a distance d is P_iso/4πd², per hertz over the bandwidth dial, in janskys of 10⁻²⁶ W/m²/Hz. The dish whose gain equals the mirror's comes from G = η(πD/λ)² at η 0.6. A listener with effective area A_eff and system temperature T_sys, integrating for τ seconds over the shout's band B, detects it at SNR 10 out to d = √(P_iso A_eff √τ / (4π·10·kT_sys√B)). The Sun's own emission at 12 GHz is a blackbody at the chromosphere's brightness temperature over the Sun's solid angle, 2kT_bν²Ω/c², and its spectral luminosity 4π AU² times that.

The oven. The same aperture formula gives the dish's gain at each of the report's three bands; the EIRP over 4πh², less the clear-air loss divided by the sine of the elevation, is the flux at the satellite, compared with the cover story's 0.1 to 1 W/cm² and solved for the dish that would meet it. The half-power beam is 1.2 λ/D; times the altitude it is the spot; the spot over the orbital speed √(GM/(R+h)) is the crossing time; half the beam is the pointing budget, compared with a milliradian.

The file. Star counts are 4/3 π r³ times 0.09 per cubic parsec, a cubic light-year being (LY/PC)³ of a cubic parsec. Bit rates use 1,024-byte kilobytes over the report's minutes. The clock takes the parallaxes' 4.37 and 4.25 light-years, doubles them, and adds them to 15 October 1971. The Trisolaran hour is 8.6 years × 8,766 hours over 85,000. The shout's timing is Earth's distance from the Sun on the send date plus Jupiter's, in light-time.

Nothing is fitted. The only choices are the dish, the bandwidth, the receiver, the listening time, the send date and the artillery milliradian, each a dial and each declared; the book's figures are entered as written and what follows is computed.

A year, a day, two burst buttons and an opposition button on the triangle; a clock with a play button; a power, a gain, a bandwidth and a listening time on the shout; a target and a distance; a dish; a band, an altitude, a requirement and an elevation on the oven. Between them they draw every number the book could mean.

• The date. 1969 to 1973 by row, any day by slider, with 12 June, 2 July and the year's opposition as buttons. 12 June 1971 by default.
• The clock. Zero to 75 light-minutes; play runs the fronts at six light-minutes a second. The timeline below the scene marks both arrivals and the detour between them.
• Transmitter power. 10 to 25 MW, the founding report's range; 25 by default.
• The mirror's gain. 10⁶ to 10¹⁰ on a log dial; the book's hundred million by default. The shout's rung, its K and every downstream number move with it.
• The shout's bandwidth. 10 Hz to 1 MHz; 1 kHz by default. The book gives none. The janskys, the hear range and the brightest-star bar depend on it.
• Listening time. 1 second to about three hours; the ledger also quotes the 224 minutes the code system takes at 12 GHz.
• The target. Alpha Centauri A/B, Proxima, or any distance from 1 to 1,000 light-years.
• The dish. 10 to 300 m; 100 m by default, the size whose gain is the mirror's. Sets the in-beam EIRP on the ladder and everything on the oven.
• The oven. 2,800, 12,000 or 22,000 MHz; 200 to 36,000 km of altitude; 0.1 or 1 W/cm²; 5° to 90° of elevation.

The novel's figures, from the mirror to the fleet, with where each lands when the arithmetic is run.

1. Start on the triangle. Press play and watch the burst reach Earth, then the echo leave the Sun and arrive sixteen minutes and twenty-four seconds later. Read the chip: the book says forty-two.
2. Press 'Opposition'. The detour is now the year's longest, 16 min 51 s. Press '2 July': 15 min 03 s. Move the year to 1970 and press the dates again.
3. Switch to the shout mode and press play. The wave reaches the Sun at eight minutes and crosses Jupiter's orbit at fifty-three; the chapter's last sentence is true from then.
4. Go to the Shout. Read the shout's rung against the book's Type I band and the bracket to its Type II. Turn the gain to 10¹⁰: still Type 1.14.
5. Turn the bandwidth to 1 MHz. The brightest-star marker falls below the RS CVn flare and the ledger's last row changes its answer.
6. Go to the Oven and set the dish to 25 m. Read what 1969's largest plausible dish puts on a satellite. Slide it up until the flux row turns green, then read the pointing row.
7. End on the File. Fourteen cards and two verdicts at equal size.

The honest line between what is reported, what is measured, what is modelled on them, and what is a reading:

In one line: The elements, the scale's two versions, the parallaxes, the Sun's brightness and the stars' are the record's; the detour, the janskys, the K, the dish and the minutes are arithmetic on them at declared dials; the mirror is the book's and says so. The reading is yours.

• Cixin Liu, The Three-Body Problem (三体), tr. Ken Liu, Head of Zeus paperback, 2024 (Tor Books, 2014; Chongqing Publishing Group, 2008): chapters 12, 13, 22, 23, 24, 27 and 33, quoted as premise; the site's signed review for the reading.
• E. M. Standish, "Keplerian Elements for Approximate Positions of the Major Planets", JPL Solar System Dynamics: the 1800-2050 element table and rates the bench places Earth and Jupiter with.
• N. S. Kardashev, "Transmission of Information by Extraterrestrial Civilizations", Soviet Astronomy 8 (1964), 217-221: Types I, II and III at 4×10¹⁹, 4×10³³ and 4×10⁴⁴ erg/s.
• Carl Sagan, The Cosmic Connection: An Extraterrestrial Perspective, Anchor Press, 1973: the interpolation K = (log₁₀ P − 6)/10 and Earth as Type 0.7, the form of the scale the novel quotes.
• P. Kervella et al., "Close stellar conjunctions of α Centauri A and B until 2050", Astronomy & Astrophysics 594 (2016), A107: the 747.17 mas parallax; Gaia DR3 for Proxima's 768.07 mas.
• The Arecibo message of 1974 as this site's INST-44 prices it: 450 kW into a 305 m dish at 2380 MHz, η 0.35, 73.1 dBi, 9.1 TW of EIRP, ten bits a second.
• A. O. Benz, "Radio emission of the quiet Sun", in Landolt-Börnstein Group VI, Vol. 4B (2009): the quiet Sun's centimetre-wave brightness temperatures; the F10.7 index for scale.
• M. Güdel, "Stellar Radio Astronomy: Probing Stellar Atmospheres from Protostars to Giants", Annual Review of Astronomy and Astrophysics 40 (2002), 217-261: the radio luminosities of flare stars and active binaries; P. C. Gregory et al., Nature 239 (1972), 440, for the Cygnus X-3 outburst.
• RECONS, the 10-parsec census: about 0.09 objects per cubic parsec near the Sun, the density the star counts are checked against.
• ITU-R Recommendation P.676, Attenuation by atmospheric gases: the clear-air order of magnitude at 2.8, 12 and 22 GHz, including the 22.235 GHz water line.
• This site: the signed review of The Three-Body Problem; the Arecibo Message, Wow! Signal, Fermi Paradox and Drake Equation benches on the SETI line; the Dyson Swarm bench for the Type II the book invokes; the Project Hail Mary and Thermoptic Camouflage benches on the fiction-pricing line.