Astronomy 10 Homework SolutionsChapter 9

Review Questions:

1. The temperature of a gas is a measure of how fast, on the average, the moleculesin the gas are moving. If the temperature of the gas is high, the molecules move fastand push each other away, which acts to impede the gravitational contraction of thecloud. If the temperature is low, gravity can overcome the low speeds of the mole-cules.

2. The Sun has been shining as a main sequence star for five billion years, and willcontinue to do so for another five billion years. At the end of this time, the hydrogenfuel in the core of the Sun will be exhausted, and the core of the Sun will begin tocontract. This contraction will heat the shell of gas around the core sufficiently tocause hydrogen fusion in this region, which will cause the Sun to expand into a redgiant. Eventually, the core of the Sun will get hot enough to cause helium, throughthe triple alpha process, to fuse into carbon and oxygen.

3. Only the largest and most luminous stars in a young cluster will have evolved offthe main sequence because only such stars evolve fast enough for this to occur.Thus, in a very young cluster, the turn off point, the point at which stars in thecluster move off the main sequence, will be in the range of O and, perhaps, B typestars. In an older cluster, A and, perhaps, F type stars might have time to evolve offthe main sequence. In an older cluster, the turn off point will be in this range ofstars. As the age of a cluster increases, the turn off point moves down along themain sequence.

Problems:

1. Recall the mass-luminosity relation:

where the luminosity L is measured in units of the luminosity of the Sun and themass M is measured in solar mass units. A star with a mass of 1.5 solar masses willhave a luminosity of

which means that such a star is 4.13 times as luminous as the Sun. Similarly, a 15-solar mass star will have a luminosity of

or 13,100 times the luminosity of the Sun. The ratio of the second of theseluminosities to the first gives the number N of 1.5-solar mass stars needed toachieve the luminosity of a 15-solar mass star:

2. We now use the relation that gives the main sequence lifetime of a star in termsof its mass:

where, again, the quantities in this equation are measured in terms of solar units.The lifetime of a 1.5-solar mass star is, then,

which means that this star will last 0.363 times as long as the Sun or 3.63 billionyears. For a 15 solar mass star we find,

or 0.00115 times as long as the Sun, 11.5 million years. The ratio of the lifetime ofthe low-mass star to the high-mass star is

Thus, the low-mass star will last 316 times longer than the high-mass star.