Computational Astronomy Tut 4 ==================================== Due date: 1600 Thu 15 Apr. The following table records measurements made during recent tests of the KAT-7 antennas. The numbers show the total detected noise temperature T as a function of zenith angle z. z (deg) T (kelvin) ====================================== 0 57.38 5 57.62 10 57.37 15 57.64 20 57.48 25 57.59 30 58.46 35 57.98 40 58.32 45 58.93 ====================================== Each measurement was the average from an integration duration of 1 min. The bandwidth was 1 kHz. Task 1: Calculate and record the uncertainty of each temperature measurement. The total noise temperature as a function of z is expected to follow the relation T(z) = T_inst + T_zenith / cos(z). Here T_inst represents the noise power solely due to the detector and T_zenith is the noise power from the troposphere (the troposphere is the lower, denser, neutral part of the Earth's atmosphere) at z=0. Task 2: Plot a graph of these T measurements + uncertainties. Transform either or both T and z coordinates such that the plotted points would be expected to lie along a straight line. Task 3: Fit a straight line to the points in this transformed scale. You may use any means in the public domain. Record the values of T_inst and T_zenith you obtain. Task 4: Say whether the physical origin of the radio waves from the troposphere, at decimetre wavelengths, arises (in significant measure) from one or more of the following: - Syncrotron radiation. - Bremsstrahlung. - Thermal radiation. - An atomic or molecular transition. Give reasons. Task 5: Estimate the polarisation fraction of these tropospheric-origin radio waves at decimetre wavelengths. The true, 'thermometer' temperature of the troposphere is about 270 kelvin. The measured noise temperature is, as you have seen, very much lower. Task 6: Use these two values, ie the effective temperature and the true temperature, to calculate the optical depth of the troposphere at the zenith. Task 7: Express T_zenith as a power spectral density (units watts per herz). Hint: you will need to make use of Boltzmann's constant, which is 1.38e-23 J/K. Task 8: Convert this last figure to a brightness value (units janskys per steradian) at 21 cm. Task 9: Calculate the collecting area of the KAT-7 12-m diametre dishes (assume 100% efficiency). Task 10: Calculate the approximate area in steradians of the 'beam' this antenna creates on the sky at 21 cm. Task 11: What will be the respective changes in the collecting area and beam area if the antenna diameter were doubled? Task 12: Use this last result to explain why, if the same measurement is made with an antenna of a different diameter, the power spectral density recorded from the atmosphere will remain at about the same value. A source is observed at 21 cm. The following measurements of the Stokes parameters are obtained: Flux density (mJy) Uncert (mJy) ========================================================= I 38.1 0.1 Q -1.16 0.02 U 4.04 0.02 V -0.03 0.02 ========================================================= Task 13: Say whether the amount of circular polarisation in the source is significantly different from zero. Task 14: Calculate the fraction of the radiation which is polarised, and the uncertainty in this fraction. Task 15: Calculate the position angle (PA) (in degrees) of the linear polarized light using the formula PA = 1/2 arctan(U/Q). Task 16: Calculate the uncertainty in this angle (in degrees). Task 17: If the Faraday rotation measure is -0.16 degrees cm^-2, what is the PA (in degrees) of the linear polarized light at the point where it is just leaving the source? The Doppler relation between frequency shift and recession velocity v is f_rest - f_observed v --------------------- = ---. f_observed c Task 18: Calculate d/dv(f_observed). Ie, the derivative of f_observed with respect to v. Task 19: Hence estimate the bandwidth over which to expect a significant amount of HI signal from a cloud of HI with a recession velocity of 1500 km/s and a velocity spread around this of 120 km/s. (Hint: where delta_f/f << 1, you can approximate delta_F by the the differential df.) Task 20: Use the current best estimate of Hubble's constant to estimate the distance of this HI cloud, and hence its mass, if its average flux density is 1.5 Jy. The appropriate formula is D = 2e-3 sqrt(M//delta_v) where D is the distance in Mpc, M is the HI mass in solar masses, is the average flux density in janskys and delta_v is the velocity spread in km/s.