11.2 Plots of spectra
Problem 11.21. (a) The spectral flux density in wavelength units of some source varies as the inverse fourth power of the wavelength, Sx = K4, where K is a constant. What is Sv, expressed as a function of v? See if you can do this from first principles without reference to the text. Give the units of S in both forms. (b) Develop an expression for the specific intensity in wavelength units, /x = /(k,T), for blackbody radiation. Start with the blackbody spectrum (23). Give the units of h.
Problem 11.22. An x-ray astronomer measures the spectrum of the diffuse x-ray background over the range 2-60 keV and finds it to have an exponential shape. He reports the energy specific intensity to be
/E = 3.6 x 104 exp (--keV s—1 m—2 keV—1sr—1
where hv is given in keV. (a) Convert this to a photon number specific intensity /p(v) with units s—1 m—2 Hz—1 sr—1 and with the coefficient that gives the correct quantitative values. (b) This radiation is believed to come from the active galactic nuclei (AGN) of many distant galaxies, not from an isothermal optically thin plasma as might be inferred from its spectical shape; see Section 3. If it were such a plasma, what would be its temperature in kelvin? [Ans. a /E/v; ~108 K]
Problem 11.31. Consider the sketches of thermal bremsstrahlung spectra on a log-log plot in Fig. 3c. The curves are for two identical plasmas, with constant identical Gaunt factors, except that their temperatures differ. Suppose that one is three times hotter than the other, T2 = 3T1. (a) At what photon energy hv do the curves for T1 and T2 cross. Express your answer in terms of kT1. (b) Make a quantitative log-log plot similar to Fig. 3c showing three thermal bremsstrahlung spectra (/ vs. v) for temperatures T, 2T and 3T, drawn properly to scale, again for identical plasmas with identical constant Gaunt factors. [Ans. (a) ~0.8 kT1]
Problem 11.32. (a) Demonstrate that the Rayleigh-Jeans law (24) follows from the blackbody intensity (23) in the limit of low frequency. What is the condition on the frequency for this expression to be valid? (b) Find the temperature of the CMB radiation from the value of the fitted curve at 10 GHz in Fig. 9a. Would it have been easier to use Fig. 9b because the ordinate is a linear scale? Explain. [Ans. (b) -3 K]
Problem 11.33. (a) Write an expression for the radio portion of the spectrum I(v) of the Crab nebula as presented in Fig. 2. Give your answer in the forms S = K va and S = K' (v/v0)a where v0 is some convenient frequency (e.g., some integer power of 10) in the region. Hint: find the latter form first. Include numerical values for a, K and K'. (b) Repeat for the x-ray/gamma-ray portion of the spectrum. [Ans. a ~v-0 25; a ~v-15]
Problem 11.34. (a) Integrate graphically under the curve for the spectral flux density S given in Fig. 2 for the Crab nebula to find the flux density F summed over all energy bands, from log v = 6.5 to log v = 22.5. Take small slices along the abscissa, (one decade of frequency) to minimize errors due to the logarithmic scale. Interpolate over regions with no data. (b) The Crab nebula is ~6000 LY distant. What is its luminosity, from 106 5 Hz to 1022 5 Hz? Compare to the luminosity of the sun. (c) Comment on the relative fluxes in the several frequency bands, radio, optical, etc. [Ans. ~1031 W]
Problem 11.41. (a) What is the approximate equivalent width (in units of nm) of the prominent absorption line shown at X & 485 nm toward the left in Fig. 10a? (b) Estimate the equivalent width (in eV) of the Ne X emission line at ~ 1022 eV in the Capella spectrum of Fig. 5. [Ans. ~1 nm; ~100 eV]
Problem 11.42. Calculate roughly the time for collisional de-excitation of a single oxygen (O III) atom in a metastable state if it resides in an emission nebula (H II region). The de-excitations take place because fast electrons collide with the relatively large and slow oxygen atom. The approximate time between collisions is the required answer. Let the density of electrons in the nebula be n = 1 x 108m-3, the temperature of the electrons be T & 7000 K, and the size (diameter) of the oxygen atom be d & 0.3 nm. Compare your answer to the natural or spontaneous decay time of ~ 100 s for the metastable states of Table 1 (assuming no collisions). Hints: (i ) Consider the size of the electron to be negligible when estimating the cross section for the collision. (ii) The speed of the electron may be obtained from the relation 3kT/2 = m v2/2. (iii) The oxygen atom may be considered to be a stationary target (why?). [Ans. ~3 days]
Problem 11.43. (a) For the damping profile (30), find the frequency where the maximum amplitude occurs, the value of k1 /k10 at this frequency, and (v - v0)hm , the half width of the curve at one-half the maximum amplitude (HWHM). (b) Based on these results, make rough sketches of the damping curves for y = 0.5, 1, and 2. (c) Repeat part (a) for the Doppler distribution (28). (d) Consider Fig. 15. What is the value of the parameter y for the damping expression given there? Find an expression for the Doppler response k1 that has unit amplitude and HWHM twice that of the HWHM of the damping curve. Compare to the Doppler expression given in the figure. [Ans. v0,4/y, y/2;_; v0, 1, a (2 ln 2)1/2; 1, same]
11.5 Formation of spectral lines (radiative transfer)
Problem 11.51. Consider a stellar atmosphere where Is varies with depth in the cloud as Is = a + bT where a is a positive constant and b is a constant that can be positive or negative. (In the text, we took Is to be constant throughout the cloud.) Assume that conditions of local thermodynamic equilibrium are satisfied, and that the observer views the atmosphere head on, as in Fig. 17. (The variation in Is arises through a variation the volume emissivity with position (38) which in turn is a consequence of temperature variation within the atmosphere. (a) Find the solution I (t) of the equation of radiation transfer (39) for this situation. (b) Evaluate the solution for the case of no background source, I0 = 0, with t < 1 and with t » 1. (c) Explain why spectral lines will or will not be formed in each of these two cases. If they are, what are the condition(s) on b that result in emission or absorption lines? In the t < 1 case, how would you constrain b so that only emission lines occur in the region t < 0.1, in the context of your approximations? [Ans. (b) I(t < 1) « aT + (b — a)(T2/2)]
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