Problems

#12

We are told there are two arms. At 8 kpc, the circumference of the sun's orbit is $2\pi r \approx 50$ kpc. The distance between the two arms is 25 kpc. Both the sun and arm pattern are moving. The sun is going faster, so it is constantly overtaking, passing through and moving beyond the arms. An arm ``sees'' the sun moving at $220-120=100$ km/s, and so at this relative speed, it takes the sun a time $t=25\;{\rm kpc}/100$ km/s to move from one arm to the next. This time is $t=25000 \times 3\times 10^13/100/3.1\times 10^7 = 240$ million years. If the sun is 4.6 billion years old (or 4600 million years), it has passed through a spiral arm $4600/240\approx 19$ times.

#13

The lifetime of an O star, being 10 million years, is smaller than the time required to move from one arm to the next. It is smaller by the factor $10/240=0.042$. The distance between arms is 25 kpc, so at most the O star can be about 1 kpc away.