Caution!! Although I am happy to share my research notes below with all visitors to my webpages, these pages are mainly designed for my own use and subject to change without warning. I do not guarantee the correctness of all contents as well.

Instruments -- general issues

1. Conversion of votage counts into antenna temperature.
2. Antenna beam Omega_A and diameter D.
3. Gaussian main beam Omega_mb and main beam size theta_b
4. Conversion of K into Jy.
5. Sidelobes and stray light.
6. How to measure aperture efficiency eta_A and main beam efficiency eta_mb.
7. How to measure different telescope efficiencies.
8. How to measure antenna beam shape.
9. How to measure antenna error pattern.
10. Frequency dependence of aperture efficiency.
11. Skydip.
12. Tsys and Trms.
13. Autocorrelation.
14. Crosscorrelation.
15. Velocity conventions.
16. Vane/chopper calibration.

Conversion of votage counts into antenna temperature 

(from ARO-HHT document, page 294): First observe a hot load of room temperature (e.g., 280K), the count is C_hot. 
Then observe a cold load (e.g., ~ 80K for liquid nitrogen immersed absorbers),the count is C_cold. Then the convertion ratio Kelvin/counts is simply: 

R =  (280K - 80K) / (C_hot - C_cold).

Because counts are usually proportional to input flux (expressed in temperature). The ratio R is expected to be constant for all observations. Therefore, multiplying R to the counts derived towards real source, we get the measured antenna temperature at once. (? Is the counts for zero K really 0 count? If not, we should take the cold or hot load as reference point for converting counts in to Kelvin. Then, we have formula 

R = (T_source - 80K) / (C_source - C_cold).

Therefore, the formula for T_source is

T_source = 80K + R * (C_source - C_cold)  

right?) 

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Antenna beam Omega_A and diameter D: 

The effective antenna beam solid angle is 

Omega_A = lambda^2 / A_e = lambda^2 / A_p / eta_A, 

where A_e is the effective antenna area, A_p = 0.25*PI*D^2 is the physical antenna area, eta_A is the aperture efficiency.

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Gaussian main beam Omega_mb and main beam size theta_b: 

The main lobe of the antenna beam pattern (power pattern) is called the main beam. The main beam usually can be well approximated by a gaussian function (normalized to a maximum of 1). It is called a gaussian main beam. The gaussian beam is frequently used in formula deduction for single dish. The size of a gaussian beam is characterized by FWHM of the power pattern: theta_b. Then solid angle of such a gaussian beam is

Omega_mb = int{exp[-4ln2 * theta^2/theta_b^2] * 2*PI*theta}d(theta) = PI/4/ln2 * theta_b^2 =~ 1.133 * theta_b^2.

The last equation has accuracy of about 5%. An accuracy of 1% can be obtained by using the 0.1 power beam size theta_0.1:

Omega_mb = int{exp[-4ln10 * theta^2/theta_0.1^2] * 2*PI*theta}d(theta) = PI/4/ln10 * theta_0.1^2 =~ 0.3411 * theta_0.1^2.

(from K. Rohlfs and T.L. Wilson, 'Tools of Radio Astronomy', third revised and enlarged edition, 1996, Springer, page 190-191.)  

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Conversion of K into Jy: 

Under Reyleigh-Jeans approximation, flux density can be computed from antenna temperature T'_A (corrected for antenna efficiencies eta_r, eta_fss and eta_rss) or main beam temperature T_mb. According to the definition of T'_A, the total amount of received radiation flux is equivalent to what we can get by pointing the same antenna to an infinitly large and homogeneous blackbody with R-J temperature T'_A. Hence, the flux is

  S_nu = B_nu(T'_A)*Omega_A = (2k*T'_A/lambda^2) * (lambda^2/A_e) = (2k/A_e)T'_A = 3520 * T'_A[K]/eta_A/D^2[m^2]  [Jy/K].

In the last equation, D is physical diameter of the antenna aperture. According to the definition of T_mb, the total amount of received radiation flux is equalent to what we will get by point a Gaussian beam with beam solid angle of Omega_mb to an infinitely large and homogeneous blackbody with R-J temperature T_mb. Omega_mb is related to whole antenna beam Omega_A through main beam efficiency eta_mb = Omega_mb/Omega_A. Hence, the flux is

  S_nu = B_nu(T_mb)*Omega_mb = (2k *T_mb/ lambda^2) * (lambda^2*eta_mb/ A_e) = (2k/A_e)T_mb*eta_mb.

Actually the above two sets of formulas can be directly related to each other through eta_mb = T'_A/T_mb. ;) Therefore, once we know the effectvie antenna area (or physical diameter together with aperture efficiency), we can convert antenna temperature in K into flux density in Jy. Note that beam filling factor has no role in this Jy/K conversion factor.

Alternatively, if we know the angular diameter of the effective beam (or main beam) phi("), then (from Dr. Tatsuhiko Hasegawa):

S_nu = 2k/lambda^{2 *} int{ T_B(Omega) } d Omega

    = 2k/lambda² * T_mb(obs) * Omega_{mb (definition of mainbeam temperature)}

Inversely,

T_mb = Snu * lambda²(cm) / phi²(") * 1.360 X 10³        [K] 

    = Snu / freq²(GHz) / phi²(") * 1.222 X 10⁶        [K] 

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Sidelobes and Stray light

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How to measure aperture efficiency eta_A and main beam efficiency eta_mb: 

(partly from ARO_HHT document, page 295-296) We assume the main beam diameter theta_mb is already known. Then we can observe a planet that is smaller than the antenna main beam size but not too small to determine the two telescope efficiencies. Let's assume angular diameter theta_p and surface blackbody temperature T of the planet is known (from ephemeris prediction, see, e.g., the SMA webtool or the JPL HORIZONS System). The we start from the definition of the two efficiencies:

eta_A = A_e / A_p,

eta_mb = Omega_mb / Omega_A,

where A_e and A_p are the effective and physical area of the antenna aperture while Omega_mb and Omega_A are the main beam and total beam solid angle, respectively. First, we need to determine eta_A by observing the planet. The observed flux density, under Rayleigh-Jeans approximation, is 

S_obs = 2k * T'_A / A_e,

where T'_A is the observed antenna temperature.  Assuming the antenna main beam is a gaussian beam, the planet is a homogeneous disk with angular diameter theta_p, then according to gaussian beam-disk coupling formula

S_obs = int{ J(T) * Pn(r,phi)}dOmega 
      = J(T) * PI/4/ln2 * theta_mb^2 [1-exp(-4ln2 * theta_p^2/theta_mb^2)),

where Pn(r,phi) is the normalized gaussian beam pattern (the main beam), PI/4/ln2 * theta_mb^2 = Omega_mb. Then we get 

eta_A = (2k*T'_A / A_p) * (4ln2 * theta_p^2/theta_mb^2) * 1/{pi*theta_p^2*J(T)*[1-exp(-4ln2 * theta_p^2/theta_mb^2)]}

Then, with the relationship lambda^2 = A_e * Omega_A, we get

eta_mb = (A_p * Omega_mb / lambda^2) * eta_A

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How to measure different telescope efficiencies: 

(from Kutner, Mundy and Howard, 1984ApJ...283..890K)

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How to measure antenna beam shape: 

Total power drift scan mapping of planet such as Jupiter and Venus at different frequencies to characterize the beam shape at these frequencies. (The planets are usually taken as homogeneous disk sources.) N.B.: Each mapping measurements should be performed in a small hour angle range, otherwise, the telescope rotates when tracking the planet and so smear out the true beam shape.

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How to measure antenna error pattern: 

Drift scan mapping of the Sun at a representative frequency to characterize the error pattern shape.  N.B.: The mapping measurements should be performed in a small hour angle range, otherwise, the telescope rotates when tracking the planet and so smear out the true beam shape.

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Frequency dependence of aperture efficiency: 

(from Baars J.W.M., 1973, IEEE transactions on Antenna and propergation, AP-21, No. 4, 461; Ruze R., 1966, Proc. IEEE, 54 No. 4, 633) (The formula from the two literature was cited by Kutner, Mundy and Howard, 1984ApJ...283..890K)

eta_A / eta_A0 = exp(-16PI^2<eps^2> / lambda^2), 

in which eta_A0 is the aperture efficiency of a perfect reflector (infinite wavelength limit), <eps^2> is the mean squared reflector deviation due to random surface errors. This is also one of the possible ways to calculate the reflector surface errors, if a serials of eta_A values have been known at different lambda.

Skydip: 

The purpose is to determine zenith optical depth tau0. (from ARO 12M online manual, Chapter 6) When an antenna is pointed to a empty sky area, the antenna temperature is

T_A(z) = Trx + T_atm*eta_l*[1-exp(-tau0*A(z))] + T_amb*(1-eta_l) + Tbg*eta_l*exp(-tau0*A(z)). 

Here, Trx is receiver noise temperature, T_atm is Rayleigh Jeans temperature of the atmosphere, Tamb is ambient Reyleigh Jeans temperature, Tbg is the background emission from outside of the atmosphere, eta_l is feed efficiency (=eta_r*eta_rss*eta_fss), tau0 is zenith optical depth, A(z) is air mass at zenith angle z (A(z)=1/cos(z) for idea weather condition). With vane calibration, the antenna temperature when the vane is put before the receiver is

T_A(vane) = Trx + Tvane.

The difference between sky and vane temperatures is

dT_A(z) = T_A(vane)-T_A(z) = Tvane -  T_atm*eta_l*[1-exp(-tau0*A(z))] - T_amb*(1-eta_l) - Tbg*eta_l*exp(-tau0*A(z))

In practice, we usually roughly assume Tvane = T_atm = T_amb, Tbg = 0 and eta_l = 1.0, so we have

ln(dT_A(z)) = ln(T_amb) - tau0 * A(z)

Use the telescope to observe several empty sky positions with different z, will allow to determine tau0. This is the so called skydip. A taumeter is just a smaller telescope specially used to determine tau0 frequently.　

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Tsys and Trms: 

With the chopper wheel (or vane) method for spectral line calibration, the effective system temperature that includes corrections for atmospheric attenuation and antenna spillover, blockage, and ohmic losses but not for error pattern losses (as the error beam will couple differently to different sources) is

    T^*_sys = (1+Gi/Gs) * [Trx + T_A(sky)] / eta_l / eta_fss / exp(-tau0*A).

Here Gi and Gs are the gain of the image and signal sidebands, Trx is the receiver temperature, T_A(sky) is the sky noise (including the signal from objects), eta_l and eta_fss are telescope efficiencies, tau0 is the zenith atmospheric opacity, A is airmass. The T_A(sky) is composed of three parts:

    T_A(sky) = Tsky + Tant + Tcmbr 
             = eta_l * Tm * (1-exp(-tau0*A)) + (1-eta_l) * Tspill + eta_l * Tbg * exp(-tau0*A).

Here, Tm is the mean atmospheric temperature, Tspill is the spillover temperature and, Tbg is the cosmic background radiation temperature. Note that all T's are Rayleigh-Jeans temperatures J(nu,T). Then the rms noise temperature for beam or position switch observation with half time on source half time off source is 

    T_rms = 2 * T^*_sys / eta_spec / sqrt(dnu * t_scan)

Here, eta_spec is the spectrometer efficiency (e.g., it is 0.81 for MAC and 1.0 for FB at the ARO 12M telescope), dnu is frequency width in Hz, t_scan is total exposure time in sec.

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Autocorrelation: 

(see Wiki entry) It is a cross-correlation of a signal with itself. The definition for a continum signal f(t) is 
    R_ff(tau) = f^conj(-tau)*f(tau) = Int^inf_-inf {f(t)f^conj(t-tau)dt}
in which tau is correlation steps, f^conj is the conjugation of f, the '*' sign means convolution. The definition for a discrete signal series x(n) is
    Rxx(j) = Sum_n{x(n)x^conj(n-j)}
in which j is correlation steps, x^conj is the conjugation of x. 

Wiener-khinchin theorem states that the power spectral density of a wide-sense-stationary random process is the Fourier transform of the corresponding autocorrelation function. That is to say,
    R(tau) = Int^inf_-inf {S(f)e^j2PIftaudf},
    S(f)   = Int^inf_-inf {R(tau)e-^{j2PI f tau}dtau}.

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Cross-correlation: 

(see Wiki entry) The definition for two continum signals f(t) and g(t) is 
    f&g(tau) =  Int^inf_-inf {f^conj(t)g(t+tau)dt}
in which tau is correlation steps, f^conj is the conjugation of f, the '&' sign represent smoothing. The definition for two discrete signal series f(n) and g(n) is
    f&g(j) = Sum_n{f^conj(n)g(n+j)}
in which j is correlation steps, f^conj is the conjugation of f. 

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Velocity conventions (RAD,OPT,REL): 

There are usually three types of velocity conventions used in radio astronomy:
           radio velocity: f_sky (RAD) = f_rest * (1 - (V_obj-V_antenna) / c)
         optical velocity: f_sky (OPT) = f_rest / (1 + (V_obj-V_antenna) / c)
    relativistic velocity: f_sky (REL) = f_rest * sqrt[(1 - (V_obj-V_antenna)/c)/(1 + (V_obj-V_antenna)/c)]. 
Here f_sky is the sky frequency corresponding to the given rest frequency, V_obj and V_antenna are velocity of the object and antenna w.r.t. LSR. The different conventions of RAD and OPT velocity originate from different definition of velocity by optical and radio astronomers:

    V(OPT) = c*z = c*(lambda-lambda0)/lambda0 = c*(f0-f)/f
    V(RAD)       = c*(f0-f)/f0 = c*(lambda-lambda0)/lambda != V(OPT) !! 
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Vane/chopper calibration: 

First of all, a calibration scale temperature TC will be derived by alternatively 
observating the vane (or chopper) and cold sky:

    TC = [T(vane) - T_A(sky)] / eta_l / eta_fss / exp(-tau0*A)

Typical value of TC = 400 K for SSB and 800K for DSB at the ARO 12m telescope (these are values used in practical observations). But it will vary with efficiency factors (eta_l and eta_fss), temperature (T(vane)), atmospheric opacity (tau0) and airmass (A). Then, a gain array will be computed for each spectral channel.

For position switched or frequency switched observation, the gain array is

    Ci = (Rci-Zi) / (Sci-Rci) * TC.

Ci is the effective system temperature for spectral channel i, Rci, Sci and Zi are receiver response to the calibration reference (cold sky), calibration signal (vane) and no signal. The average of Ci over all channels is the shown system temperature in the data header:

    Tsys = <Ci>

Note that, if a two-channel receiver is in use, Tsys is an average of both receiver channels. Finally, the calibrated antenna temperature T*_R is

    (T*_R)i = (Si-Ri)/(Ri-Zi) * Ci.

Here Si and Ri are the receiver response at the on-source and off-source positions. (my comment: This formula is correct only when the effective system temperature Ci during the vane calibration is equal to the that during the off-position observation of the program star. Otherwise, extra uncertainty will be introduced.)

For beam switched observation, the gain array is defined differently:

    Ci = TC / (Sci-Rci) 

where the terms are defined above. Then the antenna temperature is given by

    (T*R)i = (Si-Ri)*Ci 

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