[amsat-bb] Re: Converting SFU to Kelvin

[James R Miller <g3ruh@xxxxxxxxxxxxxxxxxxxx>]

On Jan 12  Lee wrote:

> As a sanity check, has anyone else tried converting Solar Flux Units to a
> Noise Temp in Kelvin?
> 
> I believe the formula comes out as:
>
>   < science fiction  omitted >
> 
> Lee-KU4OS
______________________________________________________________________________



                     Converting SFU to Kelvin
                     ------------------------
                       by James Miller G3RUH
                     
                            2004 Jan 15
                         
The Brightness of a "black-body" is given by the Plank law.  At radio
wavelengths and 'low' temperatures this simplifies to the Rayleigh-Jeans law:

        2 k Ts
  B  =  ------
         L^2
         
where:  B  = brightness, W/m^2/Hz/rad^2
        k  = Boltzmann's constant ( = 1.38E-23 W/Hz/K)
        Ts = object's temperature, K
        L  = wavelength, m
      
The power flux S we experience from this depends on the angular size of the
object:

  S = B Ws
  
where:  S  = flux density, W/m^2/Hz
        B  = brightness, W/m^2/Hz/rad^2
        Ws = solid angle subtended by source, rad^2
        
S is commonly expressed in solar-flux-units for the Sun:

       SFU = S x 10^22   (SFU)
       
and for radio astronomy   S x 10^26 Jy  (Jansky).

Thus, for the Sun, we have:

               ============================
                        2 k Ts  
                 SFU =  ------ Ws x 10^22
                         L^2   
               ============================

I believe this is what you were looking for.

You should be aware, however that unlike the Moon, which *is* thermally
a black-body (Ts=225 K), the Sun only behaves this way at frequencies above
30 GHz, whence Ts=6000 K.  Below this the relationship departs significantly
from the wavelength^2 dependency.  In addition the Sun's apparent diameter
increases somewhat as frequency decreases, and the brightness becomes very
non-uniform across the disc. [1]

Fortunately NOAA    http://sec.noaa.gov/ftpmenu/lists/radio.html  do all
the hard work for us, and daily measurements of solar flux are readily
available for a selection of frequencies, several times/day.

Thus, for a given SFU and frequency you can calculate an equivalent black-body
temperature using the equation above, assuming the Sun's diameter is
a constant 0.5 deg (Ws = 6E-5 rad^2).  This may be instructive.

Related Things
--------------
We detect this flux quantitatively using small aperture devices, such as
a radio antenna and receiver.

Small Antennas
--------------
When its gain is less than ~53 dBi the beamwidth is larger than the Sun;
in most amateur cases it is substantially so.  The power received by the
antenna from the source is:

      Pr =  S Ae / 2
  
where:  Pr = received power from object, W/Hz
        S  = flux density, W/m^2/Hz
        Ae = antenna effective area, m^2

For a dish antenna Ae is typically 55% of the physical area.  The factor
1/2 in the equation allows for the polarisation mismatch; the noise is
(usually) randomly polarised; an antenna is not.

This power is observed as a rise in noise level at the receiver detector.
The increase is equivalent to a raised system temperature dT of:

              Pr
        dT = ---
              k
              
where:  dt = Equivalent noise temperature due to source, K
        Pr = received power from source object, W/Hz
         k = Boltzmann's constant ( = 1.38E-23 W/Hz/K)
         
Now, the gain of an antenna G can be expressed in two related ways:

             4 pi Ae      4 pi
        G  = -------   =  ----
               L^2         Wa
              
where: G  = antenna gain (dimensionless)
       L  = wavelength, m
       Wa = antenna beamwidth as solid angle, rad^2
       
By substitution of  Ae = L^2 / Wa  into the previous equations we find:

                 Ws
       dT =  Ts  --- 
                 Wa

That is, the noise increase is just the object's noise temperature scaled
by the ratio of the source beamwidth to the antenna beamwidth, which confirms
a hopefully intuitive expectation ...

By another rearrangement to eliminate Ae, we find:

          8 pi k dT
      G = --------- 
            S L^2
            
This shows that if we can measure the noise increase expressed as a
temperature dT, and if we have the flux density S (i.e. the solar flux
SFU x 10^-22 if using the Sun), then the antenna gain can easily be
calculated.  The object could also be the Moon or a radio star such as
Cassiopeia A.

For most of us, an absolute measure of noise temperature is difficult to
achieve.  What is easy though, is the *relative* increase in noise over the
background noise.  That is, we can easily measure the noise power ratio:

          Tr + dT
   Y  =   -------      whence    dT = (Y-1) Tr
             Tr
             
where:  Y  = noise power increase ratio (dimensionless)
        Tr = receiver+cold_sky noise temperature,   K
        dT = noise increase due to source, K
        
Eliminating  dT  we have the classic figure of merit equation [2] for a radio
system:
                  8 pi k
   G/Tr = (Y-1) ---------        ( units K^-1)
                  S L^2
                  
G/Tr  is usually what we need to know; the more G, and the less Tr the better.

Large Antennas
--------------
The above technique is only valid when the source diameter is smaller than
the antenna beamwidth.  If this is not the case, and the object overfills the
beam, we have the situation where the whole antenna "sees" the object at
temperature Ts.   If you increase the antenna size, it still sees just Ts:

        dT = Ts
        
This measurement only tells you the receiver noise temperature, for:

    Tr = dT/(Y-1)
    
       = Ts/(Y-1)
       
Ts is well defined for the Moon (225 K), at least at frequencies below
about 5 GHz. Above 5 GHz there is a small monthly variation due to
cylic temperature changes.



References
----------
1. Kraus J.D. "Radio Astronomy", 2nd ed., Cygnus Quasar Books, 1986.
   ISBN 1-882484-00-2.  Publisher's address : PO Box 85, Powell, Ohio 43065,
   USA.  The author is of course W8JK.  http://www.cqbooks.com/
   
2. ARRL UHF/Microwave Experimenter's Manual, ARRL, 2000, ISBN 0-87259-312-6
   Pages 7-58 to 7-64

-------------

<moan>
These notes have taken 4 hours to write.  Something like them has appeared at
least 10 times on amsat-bb in the last, oh decade or two. Perhaps the Amsat
'FAQ maintainer' could add these notes to the litany ;-)
<moan/>

73 de James G3RUH


-- 
==========================================================================
    James R Miller      WWW/PGP:     http://www.jrmiller.demon.co.uk/
  Cambridge, England   Stardate:        2004 Jan 15 [Thu] 1955 utc
==========================================================================
----
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