Problem Set #3

Phys 728, Spring 2012 Homework Problem Set #3

3.1. What is the electron density in the vicinity of the Ulysses spacecraft during the observation shown in the figure below? Note that the start time of the emission is a bit later at lower frequencies (so-called frequency drift). Assuming that the spacecraft was at Jupiter at the time (5.2 AU from the Sun), use the delay of the leading edge of the emission to estimate the speed of the fastest electrons.

3.2 Use the following procedure to determine the electron density used in producing the thermal bremsstrahlung spectrum plot in the lecture:

Determine the frequency at which the brightness temperature falls by a factor 1/e. This is the frequency at which τ = 1.
Write down the appropriate expression for thermal bremsstrahlung optical depth τ = κL, set it equal to unity, and solve for the electron density.

You should be able to see that from a measured brightness temperature spectrum one can determine the electron temperature and density directly from the spectrum. More generally, since L is not known, one is actually measuring the column emission measure,

n²dl.

3.3 The above-referenced spectrum is a two-layer spectrum (chromosphere and corona). In this problem you will calculate and plot the free-free (bremsstrahlung) brightness temperature spectrum of a three-layer atmosphere consisting of:

Layer 1 (bottom layer, chromosphere): T_e = 10,000 K, n_e = 10¹³ cm^-3, L = 1000 km

Layer 2 (middle layer, hot corona): T_e = 2 x 10⁶ K, n_e = 10¹⁰ cm^-3, L = 10,000 km

Layer 3 (top layer, cooler, overlying loops): T_e = 5 x 10⁵ K, n_e = 10⁹ cm^-3, L = 70,000 km

(a) First, estimate the turn-over frequency for each case, by setting τ = 1 as in the above problem and solving for ν. You should find that the chromospheric layer has a very high (10 THz) turn-over frequency. Take a little time to think about what you might expect the spectrum to look like, before doing any calculation (you do not have to write this in your homework, although you may if you wish).

(b) Now use IDL or Python to calculate the spectrum of each layer, using equation (8) of the lecture to calculate tau, and equation (6) to calculate the corresponding brightness temperature. The T_b from a lower layer will form the T_o of the next layer up. Plot all three spectra on a log-log plot, with frequency ranging from 10⁹ - 10¹¹ Hz (1 -100 GHz), and brightness temperature ranging from 10³ - 10⁷ K. The second spectrum should look like the one in the lecture. Discuss the third spectrum in terms of how it behaves at low frequencies, and at higher frequencies. Why does the third spectrum lie slightly above the second spectrum at higher frequencies? Why does it lie considerably below the second spectrum at low frequencies?