Problem Set #3

Phys 728, Spring 2003 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 in the lecture:

Determine the frequency at which the brightness temperature falls by a factor 1/e. This is the frequency at which t = 1.
Write down the appropriate expression for thermal bremsstrahlung optical depth t = kL, 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 expression for the field strength of a magnetic dipole of dipole-moment m is (Jackson, 2nd edition, pg. 182):

B(x) = [3n(m.n)-n] / |x³|,

where n is a unit vector in the x direction. A sunspot can be modeled roughly as a dipole at some distance of order 12,000 km below the surface of the Sun. What is the scale-length L_B = B/grad(B), for such a buried dipole when on the axis of the dipole (i.e. m.n = m), in km? What is the scale-length when perpendicular to the axis (i.e. m.n = 0)? Note that the result does not depend on the value of m. What is the corresponding scale-length appropriate to gyroresonance opacity, as given in the lecture, for a coronal temperature of 1 million K, when the layer is viewed from a typical angle (q = 45^o)? This demonstrates the thinness of the gyroresonance layer. In practice, at coronal heights the magnetic field appears not to expand as quickly as a dipole, leading to a somewhat larger value for L ~ 100 km, but it is still very thin!