We looked in this chapter at the one star we can study in detail, the Sun.
Most of what we see in the Sun is a relatively thin layer, the photosphere. In studying radiative transfer, we saw how we can see to different depths in the photosphere by looking at different wavelengths. The distance we can see corresponds to about one optical depth.
The photosphere doesn't have a smooth appearance. Instead, it has a granular appearance, with the granular pattern changing on a time scale of several minutes. This suggests convection currents below the surface. Supergranulation suggests even deeper convection currents.
The chromosphere is difficult to study. In the chromosphere the temperature begins to increase as one moves farther from the center of the Sun. This trend continues dramatically into the corona. The best studies of the corona have come from total solar eclipses or from space.
We saw how sunspots appear in place of intensified magnetic fields, as evidenced by the Zeeman effect. The number of spots goes through a 22-year cycle, which includes a complete reversal in the Sun's magnetic field. The structure of the magnetic field is also related to other manifestations of solar activity, such as prominences.
6.1. In Fig. 6.3, what would be the effect on the total cross section of (a) doubling the density of particles, (b) doubling the length of the tube, (c) doubling the radius of each particle?
*6.2. (a) What does it mean to say that some sphere has a geometric cross section of 10~16 cm2? (b) Suppose we do a quantum mechanical calculation and find that at some wavelength some atom has a cross section of 10~16 cm2. What does that mean?
6.3. In studying radiative transfer effects, we let t be a measure of where we are in a given sample, rather than position x. Explain why we can do this.
6.4. What does it mean when we say that something is optically thin?
6.5. Suppose we have a gas that has a large optical depth. We now double the amount of gas. How does that affect (a) the optical depth,
(b) the amount of absorption?
6.6. Why do we need the "mean" in "mean free path"?
6.7. (a) Explain how we can use two different optical depth spectral lines to see different distances into the Sun. (b) Why is Ha particularly useful in studying the chromosphere?
6.8. What other situations have you encountered that have exponential fall-offs?
6.9. Explain how absorption and emission by the H~ ion can produce a continuum, rather than spectral lines?
6.10. Explain why we see a range of Doppler shifts over a spectral line.
6.11. How does Doppler broadening affect the separation between the centers of the Ha line and the Hp line in a star?
6.12. What do granulation and supergranulation tell us about the Sun?
6.13. If the corona has T = 2 X 106 K, why don't we see the Sun as a blackbody at this temperature?
6.14. Why can't you see the corona when you cover the Sun with your hand?
6.15. What advantages would a coronagraph on the Moon have over one on the Earth?
6.16. (a) Why does the F-corona still show the Fraunhofer spectrum? (b) Would you expect light from the Moon to show the Fraunhofer spectrum?
6.17. Why is the low density in the corona favorable for high levels of ionization?
6.18. (a) Why are collisions important in cooling a gas? (b) Why does the cooling rate depend on the square of the density? (c) How would you expect the heating rate to depend on the density?
6.19. Explain why charged particles drift parallel to magnetic field lines.
6.20. How are the various forms of solar activity related to the Sun's magnetic field?
6.1. Appendix G gives the composition of the Sun, measured by the fraction of the number of nuclei in the form of each element. Express the entries in this table as the fraction of the mass that is in each element. (Do this for the ten most abundant elements.)
6.2. Calculate the effective temperature of the Sun from the given solar luminosity, and radius, and compare your answer with the value given in the chapter.
6.3. Assume that for some process the cross section for absorption of a certain wavelength photon is 10-16 cm2, and the density of H is 1 g/cm3. (a) Suppose we have a cylinder that is 1 m long and has an end area of
1 cm2. What is the total absorption cross section? How does this compare with the area of the end? (b) What is the absorption coefficient (per unit length)? (c) What is the mean free path? (d) How long a sample of material is needed to produce an optical depth of unity.
6.4. Suppose we have a uniform sphere (radius R©) of 1 M© of hydrogen. What is the column density through the center of the sphere?
6.5. How large must the optical depth through a material be for the material to absorb: (a) 1% of the incident photons; (b) 10% of the incident photons; (c) 50% of the incident photons; (d) 99% of the incident photons?
6.6. If we have material that emits uniformly over its volume, what fraction of the photons that we see come from within one optical depth of the surface?
*6.7. Suppose we divide a material into N layers, each with optical depth dr = r/N, where r is the total optical depth through the material and dr V 1. (a) Show that if radiation I0 is incident on the material, the emergent radiation is
(b) Show that this reduces to I = I0e-r (equation 6.19) in the limit of large N. (Hint: You may want to look at various representations of the function ex.)
6.8. For what value of x does the error in the approximation ex = 1 + x reach 1%?
*6.9. Suppose we have a uniform sphere of radius R and absorption coefficient k. We look along various paths, passing different distances p from the center of the sphere at their points of closest approach to the center. (a) Find an expression for the optical depth r as a function of p. (b) Calculate dr/dp, the rate of change of r with p. (c) Use your results to discuss the sharpness of the solar limb.
*6.10. Consider a charge Q near a neutral object. If the object is a conductor, charge can flow within it. The presence of the charge Q induces a dipole moment in the conductor, and there is a net force between the dipole and the charge. (a) Show that this force is attractive. (b) How does this apply to the possible existence of the H- ion?
6.11. What is the thermal Doppler broadening of the Ha line in a star whose temperature is 20 000 K?
6.12. We observe the Ha line in a star to be broadened by 0.05 nm. What is the temperature of the star?
6.13. Compare the total thermal energy stored in the corona and photosphere.
6.14. (a) At what wavelength does the continuous spectrum from sunspots peak? (b) What is the ratio of intensities at 550 nm in a sunspot and in the normal photosphere? (c) What is the ratio of energy per second per surface area given off in a sunspot and in the normal photosphere?
6.15. How long does it take before material at the solar equator makes one more revolution than that at 40° latitude?
6.16. Calculate the energy per second given off in the solar wind?
*6.17. (a) What is the pressure exerted on the Earth by the solar wind? (Hint: Calculate the momentum per second on an object whose cross sectional area is that of the Earth.) (b) How large a sail would you need to give an object with the mass of the space shuttle an acceleration of 0.1 g at the distance of the Earth from the Sun?
*6.18.To completely describe the radiative transfer problem, we must take emission into account as well as absorption. The source function S is defined so that S dj is the increase in intensity due to emission in passing through a region of optical depth dj. This means that
(a) If S is a constant, solve for I vs. j, assuming an intensity I0 enters the material. (b) Discuss your result in the limits j V 1 and j W 1.
the radiative transfer equation should be written dl/dj = -I + S Computer problems
6.1. Consider the situation in Fig. 6.4 with 1000 layers. 6.2. Estimate the Doppler broadening for the Ha lines
Draw a graph of the fraction of the initial beam from the atmospheres at the mid-range of each emerging from each layer, for total optical depths spectral type (e.g. 05, B5, etc.). (Hint: Scale from the
(a) 0.1, (b) 1.0, (c) 10.0. Show that the fraction result in example (6.1).) emerging from the final layer agrees with equation (6.18).
Was this article helpful?