Interstellar Ionized Nebulae

Abstract

This chapter discusses ionized interstellar nebulae, in particular the photoionized H II regions and planetary nebulae. It is shown that the ionized gas is confined in the so-called Stromgren sphere. The effects of grains are analyzed, and an example is given of the heating and cooling processes in H II regions.

Exercises

• 8.1 Table 8.2 relates effective temperatures (T _eff) with radius (R _*) of main sequence hot stars. Consider these data and estimate the number of ionizing photons emitted per second by the stars. Use blackbody fluxes and compare the results with the values given in the table, which were obtained from atmosphere models.

• 8.2 H free–free emission (bremsstrahlung) can contribute to the cooling function in H II regions. For an ion with density n _i and charge Z _i, the energy loss per cm³ per second is given by

  $$ {\varLambda_{\mathrm{ ff}}}=\frac{{{2^5}\pi{e^6}Z_{\mathrm{ i}}^2}}{{3\sqrt{3}h{m_{\mathrm{ e}}}{c^3}}}{{\left[ {\frac{{2\pi k T}}{{{m_{\mathrm{ e}}}}}} \right]}^{1/2 }}{n_{\mathrm{ i}}}{n_{\mathrm{ e}}}{g_{\mathrm{ ff}}}. $$

  (a) Estimate Λ_ff/n _p n _e for an H II region of pure H with T = 10⁴ K. Take the Gaunt factor to be g _ff ≃ 1. (b) Estimate Λ_ff/n _p n _e for an H I region with H and He. Assume that both are ionized once and consider a normal abundance for He. (c) Compare the results with the equivalent values obtained for cooling due to collisional excitation.

• 8.3 Measurements of oxygen and sulfur line intensities in planetary nebula NGC 6302 give the following results, already corrected from interstellar extinction, on a scale where I(Hβ) = 100.0. [O III]: I(4959) = 361.4, I(5007) = 1,352.0, I(4363) = 17.0, [O II]: I(3729) = 51.1, I(3726) = 53.9, [SII]: I(6716) = 11.3, and I(6731) = 11.1. (a) Estimate the nebula temperature and density using the O II and O III lines. (b) How would the above results be affected with the sulfur lines inclusion?

• 8.4 A main sequence star with 2.0 M _⊙ reaches the giant branch where it maintains a mass loss rate of 10⁻⁶ M _⊙ per year during a 10⁶-year period. At the top of the asymptotic giant branch (AGB), it ejects a planetary nebula with mass M _pn, whose central star evolves into a 0.7 M _⊙ white dwarf. (a) What is the mass of the planetary nebula? (b) Supposing the timescale of the planetary nebula to be 20,000 years, what is the mean mass loss rate necessary to form the nebula? Neglect mass loss during the main sequence phase.

• 8.5 The ionized mass of a planetary nebula may be written M _i = (4/3)π R _i ³ μ n _e εm _H, where R _i is the ionized radius; μ is the mean molecular weight; n _e is the electron density; ε is the filling factor, which takes into account the ionized gas distribution in the nebula; and m _H is the H atom mass. It can be shown that the electron density is proportional to F ^1/2 ε ^−1/2 R _i ^−3/2 d, where F is the observed free–free flux at 5 GHz and d is its distance. (a) Show that the ionized mass may be written in the form

  $$ {M_{\mathrm{ i}}}=\mathrm{ constant}\times {F^{1/2 }}{\varepsilon^{1/2 }}{\theta^{3/2 }}{d^{5/2 }}, $$

  where θ = R _i/d is the nebula angular radius. (b) Show that the distance to the nebula may be written as

  $$ d=K{F^{-1/5 }}{\varepsilon^{-1/5 }}{\theta^{-3/5 }}M_{\mathrm{ i}}^{2/5 }, $$

  where K is a constant. This is the Shklovsky method for determining distances to planetary nebulae. (c) Constant K ≃ 50, if the flux at 5 GHz is in mJy, θ is in arcseconds, M _i is in solar masses, and d is in kpc. Determine the distance to the nebula NGC 7009 with the following data: F ≃ 700 mJy, ε ≃ 1, θ ≃ 15″, and M _i ≃ 0.2 M _⊙ (1 mJy = 10⁻²⁹ W m⁻² Hz⁻¹).