Notes

• All antenna diameters D_t and D_r are physical reflector diameters in meters.
• The default frequency is the neutral-hydrogen line at 1420.405751 MHz, corresponding to a wavelength λ of approximately 0.211061 m.
• Transmitter power P_t is the total RF power delivered to the transmitting antenna, expressed in kilowatts.
• Aperture efficiencies η_t and η_r account for illumination, spillover, surface accuracy, blockage, and other practical losses. Typical values range from 0.5 to 0.8.
• The power-transfer factor L is assumed to be close to unity. Its influence is discussed separately in Propagation Through a Dispersive and Scintillating Channel.
• The acquisition threshold ρ_acq is the required excess-power significance relative to the rms radiometric noise fluctuation. Values of about 5–7 are typical for a blind search over approximately one million independent frequency cells under Gaussian-noise assumptions. Larger values may be required in the presence of interference, non-Gaussian noise, receiver instability, or additional search dimensions.
• k_B is Boltzmann's constant.
• System temperature T_sys includes receiver noise, sky background, atmospheric emission, ground pickup, and spillover. Typical values are about 30 K for a space radio telescope and about 300 K for an uncooled terrestrial receiver.
• Integration time t_int is the total noncoherent averaging time. The minimum detectable power decreases as t_int^−1/2, so maximum range increases as t_int^1/4.
• Detector bandwidth B is the noise-equivalent bandwidth of one energy-detector cell. A noncoherent range calculation requires both B and t_int.
• The result is reported in parsecs and light-years and assumes free-space propagation, correct pointing, stationary Gaussian noise, and a signal remaining within one detector cell.