Chapter 18, Problem 18.83CP

CP Dark Nebulae and the Interstellar Medium. The dark area in Fig. P18.83 that appears devoid of stars is a dark nebula, a cold gas cloud in interstellar space that contains enough material to block out light from the stars behind it. A typical dark nebula is about 20 light-years in diameter and contains about 50 hydrogen atoms per cubic centimeter (monatomic hydrogen, not H₂) at about 20 K. (A light-year is the distance light travels in vacuum in one year and is equal to 9.46 × 10¹⁵ m.) (a) Estimate the mean free path for a hydrogen atom in a dark nebula. The radius of a hydrogen atom is 5.0 × 10⁻¹¹ m. (b) Estimate the rms speed of a hydrogen atom and the mean free time (the average time between collisions for a given atom). Based on this result, you think that atomic collisions, such as those leading to H₂ molecule formation, are very important in determining the composition of the nebula? (c) Estimate the pressure inside a dark nebula. (d) Compare the rms speed of a hydrogen atom to the escape speed at the surface of the nebula (assumed spherical). If the space around the nebula were a vacuum, would such a cloud be stable or would it tend to evaporate? (e) The stability of dark nebulae is explained by the presence of the interstellar medium (ISM), an even thinner gas that permeates space and in which the dark nebulae are embedded. Show that for dark nebulae to be in equilibrium with the ISM, the numbers of atoms per volume (N/V) and the temperatures (T) of dark nebulae and the ĨSM must be related by

$\frac{{(N/V)}_{\text{nebula}}}{{(N/V)}_{\text{ISM}}}=\frac{T_{\text{ISM}}}{T_{\text{nebula}}}$

(f) In the vicinity of the sun, the ISM contains about 1 hydrogen atom per 200 cm³. Estimate the temperature of the ISM in the vicinity of the sun. Compare to the temperature of the sun’s surface, about 5800 K. Would a spacecraft coasting through interstellar space burn up? Why or why not?

Figure P18.83