Radio power spreads over an ever-larger sphere as it travels, so received power falls with the inverse square of distance: doubling the range cuts the signal to a quarter. By the time a signal like Voyager 1's reaches Earth from beyond 160 AU, it has weakened by roughly 20 log₁₀(d) decibels — a barely perceptible trickle of energy.
P_rx(dBm) = P_rx@1AU − 20·log₁₀(distance)
N(dBm) = N₀(dish) + 10·log₁₀(dataRate)
SNR(dB) = P_rx − N (link viable if SNR ≥ margin)
- Distance slider — logarithmic, from Earth orbit (1 AU) out past Voyager 1's current range (~166 AU) to 220 AU.
- Auto-adapt — solves for the fastest data rate that still clears the SNR margin at the current distance, exactly as real DSN mission controllers do; the link stays ACTIVE forever, just ever slower.
- Fixed data rate (auto-adapt off) — a wider signal (more bits/second) needs a wider receiver bandwidth, which lets in more noise. Hold the rate high and the link drops to LOST once distance erodes the margin below zero.
- Dish size — the Deep Space Network's 70 m dishes collect roughly 4x the area of the 34 m ones, buying extra margin — the same trick as raising a telescope's aperture.
Real-world relevance: this is why Voyager 1's telemetry rate fell from 160 bit/s at Jupiter to about 40 bit/s today — the spacecraft never sped up its transmitter, ground controllers just kept trading speed for reach.