Radiometer equation
Also: integration time · radiometer sensitivity · ΔT = Tsys / √(Bτ)
The relationship between system temperature, bandwidth, integration time and the smallest temperature change you can detect. It tells you how long to integrate to see a line of a given strength.
Noise is random, so averaging it reduces its fluctuations. A receiver measuring in a channel of bandwidth B collects B independent samples of the noise per second. After τ seconds it has averaged Bτ of them, and the noise fluctuation falls to T_sys divided by the square root of that. That fluctuation, ΔT, is the smallest change you can hope to see. To detect a line of temperature T_line at a chosen signal-to-noise ratio, rearrange for τ.
Worked examples
With T_sys = 115 K (the example in system temperature), 10 kHz channels (about 2.1 km/s; see spectral resolution) and a target SNR of 5:
- a 1 K line: (5 × 115 / 1)² / 10,000 = 33 s
- a 0.2 K line wing: (5 × 115 / 0.2)² / 10,000 = 827 s, about 14 minutes
What changes it:
- Halve T_sys → quarter the time. System temperature enters squared, which is why fixing the front end beats everything else.
- Halve the channel width → double the time. Finer velocity resolution costs integration.
- Halve the line strength → four times the time. Faint features get expensive fast.
Why the real number is longer
The equation assumes the noise is perfectly random and your receiver perfectly stable. Neither is true. Gain drifts as the LNA and SDR warm up, the bandpass has ripple, and the reference spectrum you subtract never quite matches the on-source one. In practice these set a floor that more integration doesn’t break through. LAB-009 recommends planning on one to five minutes per spectrum and repeating, whatever the formula says.
What the equation does tell you honestly is relative cost. If a change to your chain doubles the predicted time, it roughly doubles the real one. The receive-chain noise calculator applies it to your own chain, and if you’re doing a drift scan, check the result against how long the target stays in your beam.
Work it out
Where it comes up
Related terms
- System temperature (T_sys)The total noise of a radio telescope expressed as a temperature: receiver noise plus sky, plus ground picked up by spillover and sidelobes. Every sensitivity calculation starts from it.
- Spectral resolution and channel widthHow finely a spectrum is divided in frequency. At the hydrogen line every channel width corresponds to a velocity width, so it decides how much detail you can see in the gas motions.
- Brightness temperature and antenna temperatureRadio astronomers measure power in kelvin. Brightness temperature describes the sky; antenna temperature is how much of it actually reaches your receiver through your antenna's beam.
- Drift scanObserving with the antenna held still while the Earth's rotation sweeps the sky through the beam. It's how most small radio telescopes, and every fixed horn, work.
Part of the hydrogen-line field referencefrom Sixmile Run. Found an error? Tell us.