Radiometer Equation Calculator

Estimates the sensitivity of a pulsar detection system given system parameters.  A number of strong pulsars can be selected for analysis. Please read the instructions, notes and caveats below.

Pulsar Radiometer Equation Calculator
   
Antenna Gain (dBi):
Receiver Bandwidth (MHz):
Observation Frequency (MHz):
System Noise Temperature (Ko):
Integration Time (s):
Number of Time Bins:   Auto
Pulsar Period (s):
PW50 (ms):
SPI (estimated):
   
Disable Alerts
   
 System Sensitivity Smin: mJy
 Estimated S at Obs. Freq.: mJy
 Estimated Output S/N: (dB±3)
   
Effective Aperture: m2
Equivalent Dish Size (Eff=50%): m
Estimated 3dB Beamwidth: o
Fixed Antenna 3dB Transit Time: h
   


Target Pulsar
   
Set Pulsar:
   

Instructions: Via the drop-down 'Target Pulsar' menu provided, set the target pulsar to one of the relevant pulsar candidates for amateur observers. (Note: other pulsars can be added if requested via the email address in 'Contact').

Enter relevant system parameters in fields labelled in green.

Only fields marked with green labels are editable. [Note that an exception is the "Number of Time Bins" field - which is 'read only' while the 'Auto' option is checked].

The recommended "Number of Time Bins" is automatically calculated by default from the selected pulsar's period and PW50.  If 'Auto' is unchecked and the number of time bins is entered manually, an alert will be displayed if the entered number is too low (invalid), or too high (degrades S/N) by a margin of more than 10% from the recommended value. [The alerts can be disabled via the checkbox next to the 'Calculate' button].

"SPI" is the estimated 'Spectral Index' as calculated from S values @ 400 MHz and 1400 MHz (given in the  ATNF Pulsar Database).

"System Sensitivity Smin (mJy)" is an estimate of the sensitivity limit of the system for an output S/N = 1 (0 dB). In practice, other noise sources (RFI, system imperfections, etc) will decrease the actual sensitivity from the calculated value.

"Estimated S at Obs. Freq. (mJy)" is the mean flux density calculated from the estimated SPI, assuming a smooth increase in S as the frequency decreases. This is not true for all pulsars. For example, the Vela pulsar has a 'turn-over' point near 600 MHz and falls away below this frequency.  Consequently the estimated S for Vela at observation frequencies between 400 MHz and 1400 MHz will be lower than actual - while for observation frequencies outside that range the estimate will be higher than actual.

"Calculate" shows the result for the set parameters.

Typically, the output S/N needs to be at least 4 linear (+6 dB) for a valid positive detection.

Notes and Caveats:

  • The output S/N assumes that de-dispersion has been done.  Care should be taken in an un-de-dispersed system to ensure that the bandwidth used does not result in dispersion more than ~ PW50.
     
  • Given the number of assumptions and approximations, the accuracy of the estimated output S/N is not expected to be better than ±3 dB.  The 0.1 dB display precision of the value for the estimated output S/N is provided to allow comparisons between small changes in input parameters, and is not an indication of accuracy.
     
  • The 'Smin (mJy)' result is highlighted with a background colour -  green  when the pulsar is stronger by at least 6 dB than the estimated system sensitivity (minimum detection limit), and  red  when the pulsar is weaker than the estimated system sensitivity.  In between, the highlight is  orange .
     
  • The input parameter 'System Noise Temperature (Ko)' is made up of the antenna temperature and the first LNA noise figure equivalent temperature (see conversion table below).  This assumes sufficient LNA gain to overcome following losses - 30 dB should suffice in most practical systems. A good amateur system will have a system temperature around 100 Ko to 150 Ko.  Excellent systems do better.
     
  • Poor sidelobe performance (especially rearward facing sidelobes) of the antenna will allow ground noise to enter the system raising antenna temperature.  The cosmic sky noise in the direction of the pulsar adds to the apparent antenna temperature. A pulsar in a 'hot' part of the sky will be harder to detect than one in a 'cold' part of the sky.  The temperature at 400 MHz in the direction of B0329+54 is around 60 Ko and covers a patch of the sky filling the beamwidth of a typical small antenna (3 m dish equivalent).  In contrast, for Vela (B0833-45), the sky temperature ranges from nearly 300 Ko in the immediate angular vicinity - dropping in directions away from Vela.  In the beamwidth of a typical small antenna @ 400 MHz the average Vela sky noise is about 100 Ko to 150 Ko.
     
  • 'Pulsar Period' and 'PW50' are representative values only - drawn from the  ATNF Pulsar Database.  A 1000 ppm error in either would only result in a 0.005 dB error in radiometric calculations and so they do not need to be updated for decades.  However, the value of 'Pulsar Period' should not be used for topocentric period calculations where much higher accuracy is critical.  For that calculation use the 'Estimate of Topocentric Pulse Frequency ' calculator.
     
  • There are auxiliary calculated estimates for antenna aperture and beamwidth.  Also there is a calculation for the estimated time it would take the pulsar to transit the 3 dB beamwidth of an antenna which is fixed, i.e., no tracking mechanism.  The observation time, in such a fixed antenna situation, should be equal to, or less, than the estimated transit time.  For situations where the antenna has a tracking mechanism, the observation time is only limited by the area of unobstructed sky accessible as the pulsar rises, transits and sets.

Basic Radiometer Calculation Derived from: Code supplied by Marcus Leech, Canada (CCERA).

Time Bin Factor Calculation: Derived from analysis by Peter East, UK.

Other refinements made with the help and guidance of members of the NSG.

Pulsar Data: ATNF Pulsar Database (Manchester, R. N., Hobbs, G.B., Teoh, A. & Hobbs, M., AJ, 129, 1993-2006 (2005)) - except where later epoch data is available.

Conversion from Noise Figure to Temperature: