Terminus: talking past a flaring red star

Proposal section 6: the loudest thing in the sky

Someday, generations from now, a curious mind in the twilight band will build the first radio receiver from copper wire and patience, sweep it across the dial, and make a discovery that would break a terrestrial radio pioneer's heart: the sky is shouting. Their gentle red sun, the fixed lamp on the horizon, is not quiet at radio wavelengths. Red dwarfs of its kind crackle with flares and, worse, emit organized, coherent radio bursts — brightest across roughly 1 to 3 gigahertz, the very range where radio is easiest to build. On Earth those frequencies carried the first satellite phones and navigation beacons. Here they belong to the star, which sits on every horizon, forever, transmitting.

Every link in our architecture — terminal to wheel, wheel to anchor — is a radio or laser beam sharing the sky with that voice. This section chooses the frequencies, and the choice starts with understanding just how faint a satellite whisper is.

The arithmetic of a whisper

A radio transmission spreads like paint on an expanding balloon: by the time the wavefront reaches a satellite, the energy that left the antenna is smeared across a sphere thousands of kilometers wide, and the receiver catches only the fleck that lands on it. This section has to put a number on that fleck, and the number is so small that ordinary notation gives up first. So, two tools before the physics: a unit built for multiplying, and a derivation short enough to check by hand.

The unit. A decibel (dB) is ten times the base-ten logarithm of a power ratio: 10 dB is a factor of ten, 20 dB a factor of a hundred, and 3 dB — near enough — a factor of two. Ratios are scored in logarithms because a radio link is a chain of multiplications — spread by this much, focus by that much, lose a little to rain — and logarithms turn the chain into a sum. The whole trade study below is additions and subtractions of dB. One convention to watch: when the thing being doubled is a distance or a frequency rather than a power, it enters the power ratio squared, so its term reads 20·log₁₀ instead of 10·log₁₀. Both appear shortly.

Step one: the balloon. Strip the antennas away and let an idealized bare wire radiate its power P equally in every direction. At distance d that power has been smeared over a sphere of area 4πd², so the power crossing each square meter of sky is P / 4πd². Twice the distance, a quarter the flux; three times, a ninth. This is geometry and nothing else. Frequency has not entered.

Step two: the catch. A receiving antenna takes power from the passing wave in proportion to the area it presents — its effective aperture. The reference receiver that every path-loss table assumes is that same idealized wire, and its effective aperture turns out to be λ²/4π. That constant is worth deriving rather than accepting, because it is the only place frequency enters the whole derivation — and the route to it passes through the one thing a wave cannot help doing.

Send a transmission out through any face of width D — a dish, a slit in a wall — and it refuses to stay collimated: the beam fans out by roughly λ/D radians. Narrower faces and longer wavelengths fan wider. That is diffraction, the signature move of everything wavelike, and it sets a floor under how tightly any antenna can focus. Run the bookkeeping over both directions of fan-out, and a face of area A paints a patch of sky whose solid angle is λ²/A: smaller face or longer wave, bigger patch.

Now compare that patch with the balloon. The whole sky, in the same units, is 4π steradians, so a face that squeezes its power into λ²/A of it has concentrated the signal by

G  =  4π / (λ²/A)  =  4πA / λ²

— the antenna's gain. And antennas are reciprocal: the face that focuses this well on transmit catches exactly as well on receive, so gain and effective aperture are locked in a fixed exchange rate, A = G · λ²/4π, the same for every antenna ever built.

The idealized wire is the antenna with no focus at all — gain exactly 1, by definition. Feed that into the exchange rate, and its equivalent catching area is λ²/4π: a disc of radius λ/2π, about as far as the wire's own fields reach into the passing wave. It shrinks as the square of the wavelength. At L-band this reference catcher's mitt is 28 cm² of equivalent area; at Ka-band, 0.08 cm².

Step three: divide. Received power over transmitted power is the flux at the receiver times the aperture that catches it:

Pr / Pt  =  1/(4πd²) · λ²/4π  =  ( λ / 4πd )²

Flip the fraction over, substitute λ = c/f, and the loss between two idealized wires is (4πdf/c)² — in decibels, the free-space path loss, exactly as the simulator states it:

/// Free-space path loss in dB over `distance` (m) at `frequency` (Hz).
pub fn fspl_db(distance: f64, frequency: f64) -> f64 {
    20.0 * (4.0 * PI * distance * frequency / SPEED_OF_LIGHT).log10()
}

From a terminal at the worst edge of a footprint to its satellite, 3,642 km away, this formula charges about 193 dB at our primary band: the received power is the transmitted power divided by ten, nineteen times in a row and then some.

Nobody talks across that with bare wire. You focus. An antenna with a face — a dish is the familiar shape, though not the only one — is a light-gatherer for radio: it concentrates the transmit power into a beam instead of a balloon, and gathers arriving energy across its whole face. Its gain, scored in dBi — decibels over that idealized wire — is step two's exchange rate run forward: a dish of diameter D offers an area of η·πD²/4 (η, the efficiency, is the fraction of the face that pulls its weight — 0.6 here), and dividing by the reference's shrinking λ²/4π leaves η·(πD/λ)²:

/// Gain (dBi) of a parabolic dish: 10·log₁₀(η·(πD/λ)²).
pub fn dish_gain_dbi(diameter: f64, frequency: f64, efficiency: f64) -> f64 {
    let lambda = SPEED_OF_LIGHT / frequency;
    let x = PI * diameter / lambda;
    10.0 * (efficiency * x * x).log10()
}

Spreading is geometry

One beam, three distances. The rays pass through the corners of every grid because the growth is exact: double the distance and the same power crosses four tiles, triple it and nine. The violet cell is a receiver face of fixed size — it catches P, then P/4, then P/9. Frequency appears nowhere in this panel.

The f² is the reference antenna

FSPL is scored against a fictitious bare-wire antenna whose collecting area is λ²/4π — the violet squares, to scale. Triple the frequency and that reference keeps only a ninth of its area; from L to Ka it shrinks 352×. The face you actually bolt to a roof does not shrink at all.

Now read the derivation and the figure together, and a counterintuitive law falls out. The f² in free-space path loss never came from the balloon — step one had no frequency in it. It came from step two: the reference antenna shrinks as frequency rises. Bolt a real face of fixed size to each end of the link instead, and the penalty not only vanishes — it reverses. Double the frequency, and FSPL rises by 6 dB, but each antenna's gain rises by 6 dB too, and there is an antenna face at both ends. Six lost, twelve gained: for fixed antenna sizes, the higher band wins. Intuition says higher frequencies are "harder"; for point-to-point links between focused antennas, the opposite is true.

The menu, priced

cargo run -p terminus-orbits --example frequency_plan

band     f (GHz)   FSPL (dB)  beam (deg)  vs L-band (dB)  stellar 1-3 GHz?
L            1.6       167.8        26.2            +0.0   IN STELLAR BAND
S            2.5       171.6        16.8            +3.9   IN STELLAR BAND
X            8.4       182.2         5.0           +14.4             clear
Ku          14.0       186.6         3.0           +18.8             clear
Ka          30.0       193.2         1.4           +25.5             clear

The fourth column is the law at work: with half-meter faces on both ends, Ka-band at 30 GHz delivers 25.5 dB more link than L-band — a three-hundredfold power advantage — despite paying the most spreading loss. And the last column is the star having its say. The two bands at the top of the table, the cheap, forgiving, beginner-friendly bands every young radio civilization masters first, are exactly the bands their sun broadcasts in. No filter removes interference that lives inside your own channel. We do not fight the star for 1–3 gigahertz. We concede it, completely, as the RFP requires.

The third column is the quieter bonus — step two's λ/D fan-out read out in degrees (about 70·λ/D for a real dish, measured between the half-power points). A half-meter antenna at L-band throws a 26-degree flood of a beam — pointed at a satellite 25 degrees above the horizon, the beam drags its skirts across the horizon itself, where the star sits. The same half-meter at Ka throws a 1.4-degree pencil. Our coverage rules already guarantee every serving satellite stands at least 25 degrees up; a pencil beam aimed there passes nowhere near the star. Geometry and frequency, working the same side of the problem: the star is loud, but with a narrow enough beam and a high enough band, you are never looking at it.

Every strength buys a weakness

Ka's fine print is weather, and this fine print rewards reading in full, because it is quantum mechanics you can watch from a porch.

A radio photon is a feeble thing. At 30 GHz it carries about an eighth of a thousandth of an electronvolt — thousands of times too weak to rearrange the electron orbitals of any atom (that chemistry lives up at visible-light energies) and far too weak even to set a molecule vibrating. What such a photon can do is nudge the two gentlest quantized motions a molecule has: its tumbling, and its spin. A water molecule is bent, with its charge slung unevenly — the oxygen end slightly negative, the hydrogen end slightly positive — so a tumbling water molecule is a tiny rotating antenna. Quantum mechanics permits it only certain tumbling rates, and one of the gaps between them costs exactly the energy of a 22.235 GHz photon: a radio wave near that frequency crosses humid air paying a toll at every molecule it meets. Oxygen offers no such electric handle — the O₂ molecule is symmetric — but it hides two unpaired electrons whose spins make it faintly magnetic, and their permitted flips cluster in a thicket of lines between 50 and 70 GHz. At sea-level pressure the thicket smears into a single wall near 60 GHz that charges 15 dB per kilometer: dead air, standing just above our primary band.

The sky’s price list, 1–100 GHz

Two walls and a basement. Clear air charges almost nothing below 15 GHz, a small toll at the 22.235 GHz water-vapor line, and 15 dB/km at the 60 GHz oxygen wall — quantum energy levels, drawn as geography. Rain is the violet curve: negligible at X, 4.7 dB/km at Ka in a heavy storm. And the shaded band is the star’s own voice, 1–3 GHz, conceded outright.

Rain is the same molecule in a rougher mood. In vapor, each water molecule tumbles freely and absorbs at its one sharp line. In a droplet, every molecule is gripped and jostled by its neighbors; the neat line smears into a broad continuum, and the drop as a whole soaks up microwaves across the band — the same mechanism a kitchen microwave oven exploits. Worse, storm drops grow to millimeters, a meaningful fraction of Ka's one-centimeter wavelength, so they scatter the beam as well as absorb it. X-band's 3.6-centimeter wavelength steps over the drops — too long to scatter off them, too far from any line to absorb:

cargo run -p terminus-orbits --example atmospheric_attenuation

point       f (GHz) clear (dB/km)       5 mm/h       25 mm/h  note
L             1.600         0.007         0.00          0.00  IN STELLAR BAND
S             2.500         0.007         0.00          0.00  IN STELLAR BAND
X             8.400         0.011         0.04          0.38  diversity band
Ku           14.000         0.022         0.23          1.36
H2O line     22.235         0.172         0.59          2.97  vapor rotation
Ka           30.000         0.087         1.05          4.70  primary band
O2 wall      60.000        15.088         2.90          9.79  spin-flip complex

Read the Ka row against a storm. At 25 mm/h — a proper downpour over a settlement — a beam crossing a 5 km rain cell pays about 23.5 dB: the weather can spend the entire 25.5 dB advantage the aperture bought. (The absorption arithmetic is the ITU's, fitted from a century of Earth's weather — ITU-R P.676 for the gas lines, ITU-R P.838 for rain — and that is fair play: the survey's working assumption is air of Earth-like pressure and composition, which it can hardly help being when the settlers breathe it and it rains on them. If the mix differs in detail, the lines shift in strength but not in kind. Any breathable sky is drawn by the same two molecules.)

This is where the second band earns its place: X-band at 8.4 GHz gives up 11 dB of the aperture advantage, but shrugs at weather — 1.9 dB through the same storm cell — and rides out elevated noise. So the plan is a pair, not a champion:

• Ka-band, 30 GHz — primary. The broadband workhorse for clear skies, which is most skies, most of the time.
• X-band, 8.4 GHz — diversity. The all-weather fallback that keeps every session alive, slower but unbroken, through storms and flares.

When the star does flare — raising the noise floor even in the clear bands, agitating the ionosphere — the response is written into the requirements: rate drops, band falls back, and the session never drops. A child mid-question during a flare gets slower words, not silence. The machinery that makes that graceful — coding the same conversation across both bands at once, repair packets on X healing losses on Ka — is rich enough to deserve its own volume of this proposal, and it gets one. The decision itself is recorded now as ADR-0005.

From blankets to beams

One honest confession closes the section. This entire analysis priced a single antenna face talking to a single antenna face, and quietly let you picture two dishes — but a satellite serving a 5,000-kilometer-wide footprint with one beam is throwing a blanket over a hundred settlements, dividing its power and capacity among all of them, indiscriminately. Modern satellites do something far more interesting: they hold hundreds of electronically steered pencil beams at once, painting each town with its own private spot of radio — and once a beam covers only one small spot, strange and wonderful simplifications follow for the terminals underneath it. That is next: beams, not blankets — why our parachuted terminals will never have to hunt for their signal at all, and what the antenna on the terminal's own roof turns out to be, given that nothing in this sky ever holds still.