Problems

#07

The solar lifetime is 10 billion years. Each year has about 30 million seconds. The solar luminosity is $4\times 10^{26}$ Watts (or Joules/sec). Hence the radiant energy output of the sun over its entire lifetime is the product of these three numbers, or about $10^{44}$ Joules. A SN generates about 10% of this energy in light during its explosion.

#09

It has been expanding for about $2004-1054 = 1950$ years. A radius of 1 parsec is $3\times 10^{13}$ km. Recalling that there are 30 million seconds in a year, the average speed of expansion is $3\times 10^{13}/1950/ 30\times 10^6 \approx 500$ km/s. The gas ejected in a SN explosion comes out at about 10,000 km/s, but this quickly slows down as it plows into gas in the interstellar medium. Even though the gas was traveling much faster at first, for most of the last 2000 years, a constant value of 500 km/s is fairly accurate.

#13

This is a statistical problem. Our galaxy is circular as seen from above, with a radius extent of 30 kpc. Our view from Earth is limited to about 5 kpc. (Draw a large circle. Draw a smaller circle inside it that is 5/30 = 1/6 as large. This is the representative geometry that we are talking about.)

Now SNe occur randomly throughout the galaxy about once every 30 years. If we had an unlimited view of the galaxy, then we should see a SN every 30 years, of course. But our view is limited. What are the odds that a SN will go off within 5 kpc from the Sun. The odds are the ratio of areas for 5 kpc versus 30 kpc, or $(5/30)^2 = 25/900 = 0.028$, which is 2.8%. So for every 100 SNe, we should see about 3. It will take about 3000 years for 100 SNe to explode. Since we would only see 3 in that time, we can expect to see a SN about every 1000 years. Or put another way, the frequency with which we see SNe inside our 5 kpc viewing zone is $30/0.028 = 1080$ years.