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Instruments and Methods

133

Sky temperature resolution by microwave radiometry
A.G. Kislyakov
State University, Nizhny Novgorod, Russia

Introduction
The threshold sensitivity of a radiometer is determined by well known equation Karlov & Manenkov, 1966
"

hE i =

h

hn i +1 hn i + N t


G0 , 1 G0

1=2

:

1

Here we denote hE ias the minimum detectable increment of radiation spectral density E at a frequency . N is the mode number of the front end device detector or ampli er having the gain G0 within operating frequency bandwidth of ; t is the integration time; hn i is the mean number of photons in input radiation. The angle brackets imply the ensemble average. Equation 1 suggests that the radiometer sensitivity is limited by external thermal radiation uctuations so called "ideal" radiometer case. By de nition Karlov & Manenkov, 1966; Kislyakov, 1997, the ideal radiometer sensitivity does not essentially depend on the presence of input ampli er; so that hereafter we put G0 =1. Besides, for the limit temperature resolution to be achieved such broadband input device as detector is preferable. Taking into account that the radiation power incrementP = hE iN the following expression can be derived from Eq. 1 Kislyakov, 1997:
n h q P = h hti hn i +1N ,! t N hn i: hn i!0

s

2

It was assumed that t ' 1. The transition hn i ! 0 implies the seldom photons in external radiation. As it follows from Eq. 2, the radiometer sensitivity is unlimited if hn i ! 0. However, the photon number in every electrodynamic system has a lower limit because of zero- eld uctuations with energy spectral density of N h =2. On the other hand, the zero- eld photons are absent in a radiation ux they are not involved in energy transport. Including the zero- eld uctuations into consideration we obtain a new equation, valid under hn i 1, instead of Eq. 2: P '
u h u hn i N t t N v

1 h + 4 t ,! N 2t hn i!0

!

3

in accordance with physically evident expression P = h = , where is the time constant, under N 1. Equation 3 gives a correction to the expression for limit detector sensitivity Kislyakov, 1997.


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Instruments and Methods

Temperature resolution
The di erential temperature resolution by thermal emission measurements can be estimated using the expression T = 2shz=2 q
z T N


t

:

4

d derived from Equation 1 with its left part substituted by hE i = h dT hn iT T , ,1 , where z = h =kT . Equation 4 is valid assuming G0 = 1, and hn iT = expz , 1 under T = kT =h. Under constant T , the value T gives approximately the width of thermal spectrum. Evidently, the radiometer frequency bandwidth must be close to T in order to minimize the value T . Total thermal spectral sensitivity TS of a detector radiometer can be derived from its detection capability DC given by

q = T = F z N t; 5 T T where F z = z 2shz=2 ,1 . Making both sides of Eq. 5 squared, letting = d = T dz and integrating the right part of Eq. 5 over z we obtain the expression for integrated DC: 2 3
D

Z1 4 1 + az 2 F 2 z dz 5 DT = T t q
0

1=2

= 'T T t;

q

6

where a = 2Sn=2 , Sn is the detector operating area and T = ch=kT . The mode T number N give rise to the factor 1 + az2 in Eq. 6. If N 1, then a = 0 and 'T 3;29. It is easy to see that TS is determined by ratio T=DT , therefore,
0 TT

= T =q
D
T

T 'T

T

t

:

7

Figure 1


Instruments and Methods

135

Fig. 1 presents two curves calculated from Eq. 7 under a = 0 the single-mode case, dashed line and for Sn = 0;1 cm2 multi-mode case. In both cases t = 1 s. For p T while the multisingle-mode detector, the TS grows monotonically so that TT0 mode detector has a maximum in TS under T 1 K. Obviously, the T -position of this maximum is strongly dependent on Sn. Only right wing of the curve TT0 corresponds to multi-mode operation of detector.

Nonideal detector
One should take into account that equations Eqs. 2 and 3 present the radiometer sensitivity with an ideal detector as front end. It means that thermal and shot noise of a detector were neglected. However, under T ! 0 both noise sources will be of importance as the input thermal noise will be negligibly small. The detector radiometer sensitivity in the case of detector thermal noise taken into account equals to Kislyakov, 1997

where M = = Y0 is the detector gure of merit; is detector e ectiveness; Y0, T0 are di erential conductivity and physical temperature of a detector; f is the output frequency bandwidth of radiometer. Eq. 8 is valid under hf kT0 . The detector e ectiveness can be formally described by expression Kislyakov, 1997 = 1 classic detector : h = qe =kT0
qe h

p

z 2 2 T = e z, z1 MN T0 f: 2e k

s

8

"

1 , exp ,



h kT0

!

:

9

Eq. 9 incorporates two extreme cases:
kT
0

2 quantum detector : h = qe=h

kT

0

10

where qe is the electron charge. Eq. 9 shows that unlimited increasing of = qe=kT0 in classic case under T0 ! 0 is impossible. Analogously to TS calculations presented above, one can obtain the following expression in classic case
TT00 In = q T T; T : e T 0

11

Here we introduce the detector noise current In =
T; T
0

q

4kT0Y0f and some new function

=

Z1 zez 1 + az 2
0

ez , 1

2

1 , e,bz dz;

where b = T=T0 . Fig. 2 presents the single- top curves and multi-mode functions 00 TT T .

All the curves at Fig. 2 correspond to Y0,1 = 102 , f = 1 Hz and Sn = 0;1 cm2. The multi-mode functions change their slope at the point T = T1 , where the input noise transformed by detector begin to dominate over its thermal noise. The magnitude of T1 is strongly depended on detector operating temperature T0 . Therefore, the curves TT0 T


136

Instruments and Methods

Figure 2 and TT00T in multi-mode case are coincident under T T1. Analogously, the singlemode curves should change their behavior under T T2 T1 . Notice that T2 has p the 00 T T same sense for single-mode case as T1 in multi-mode one. If T T2 then TT in accordance with dashed curve at Fig. 1. It is worth to note that the quantum detector case can be considered using for the 2 formula from 10 and replacing the expression for In with corresponding equation for shot noise current see Kislyakov, 1997.

Sky temperature resolution
Under sky e ective temperature Ts measurements, one should take into account the antenna angle resolution ' =2D single-mode case or ' d=D multi-mode case, where D is antenna diameter and d the detector crossection. Assuming gaussian beam shape and gaussian distribution of Ts over sky with the angle correlation radius of , the following equations for minimum detectable temperature variations Ts can be obtained Kislyakov &Shvetsov, 1974 T T
s1

= T

s

T

1+ d2 2 ; 12 sN = TT p D d 2 where Ts1 and TsN correspond to single- and multi-mode cases, respectively. Both quontities are frequency dependent, so that we derive, as previously, two expressions for sky temperature inhomogenieties detectability:
2

2 1+ 4D22 ; s

D D

s1

= =

q q

2Z1 3 p2 ,1 2 T t 4 1 + 2 F z dz 5
0

1=2

z

;

13 sh z=2 0 where p = T =2D. Fig. 3 presents the functions Ts1T = T=Ds1 top curve and TsN T = T=DsN calculated using 13 under following parameters: t = 1 sec, d =
sN



T



d t T

1 +

d2 , D2 2

1=2

r 2Z1 1+ az 2 31=2 4 dz 5 ; 2

2


Instruments and Methods

137

0:3 cm, D = 10 m and  =3 10,3. It is easy to see that these parameters allow to realize the multimode operation under T 1 K. As it could be expected, Ts1T and TsN T behave very similarly to corresponding dependences on Fig. 1. The next plots on Fig. 4 give an impression on angular dependences of Ts1 and TsN .

Figure 3

Figure 4 One can see here the sequence of antenna angular resolution e ects. The curves of Fig. 4 were calculated using the same values of such parameters as t, d and D. As it was mentioned above, the multimode antenna resolution is limited by ratio d=D thus explaining the sharp increase in TsN  function under  ! 0, and in this case a single-mode telescope is preferable. However, under  !1, the TsN  is proportional to decreasing factor of d=D thus leading to TsN  Ts1 . The lack of site does not allow to analyze the nonideal detector case.


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Instruments and Methods

Final remarks
As it follows from consideration presented above, the choice of appropriate radiometer and telescope type should be made in accordance with average temperature and brightness spatial distribution of a source. For instance, the operating waverange of a telescope under T =2:7 K the "Big Bang" black body radiation temperature should be close to =1:5 mm and the receiver bandwidth should be of the order of T ' 58 GHz. Depending on the spatial spectrum to be investigated, the single-mode or multi-mode operation for a given D may be preferred. It is worth to note that in the case of metagalactic background radiation the spatial scale of cosmological interest is rather extended: from 90 quadrupole anisotropy up to few sec of arc see, for instance, Partridge, 1986. Another example, is the Galaxy di used matter thermal emission with the expected brightness temperature of about 10 K. In this case the maximum of Planck curve is close to =0:3mm and the corresponding T ' 200 GHz. The spatial scales of galactic dark nebulae lie approximately within 10 1000. The curves on Fig. 4 help to choose the telescope scheme in this case. This contribution is an attempt to establish some limits in sky temperature resolution under microwave measurements as dependent on source parameters. The only parameter of a telescope has been taken into account the diameter. All other reasons restricting the receiver sensitivity and antenna angular resolution say, the Earth atmosphere in uences or hard ware problems were neglected. The detailed analysis taking all possible factors into account can lead to other conclusion on the proper telescope scheme. However, the limit values of Ts1 and TsN will remain as a quality indicator of a telescope.

References
Karlov N.V. and Manenkov A.A. Quantum Ampli ers. Advances of Science. Radio zika. M., 1966 Kislyakov A.G. Izv. VUZov, Radio zika. 1997. V.40 , No 7. PP. 824 835 Kislyakov A.G. and Shvetsov A.A. Izv. VUZov, Radio zika. 1973, V.16 , PP. 1846 1852 Partridge R.B. In "Highlights of Astronomy". 1986, V.7, P.307