# ASTR 109 HOMEWORK #5 (Hamilton) Solutions

1. A Star's Life.
• a) The Sun will be a Red Giant for about 1.4 billion years. It burns Hydrogen for the first 12.25 billion years and Helium for only 120 million years. So it burns Helium for only about 0.12/12.37 ~ 1% of its life.  Event Time (billion years) Present Day Earth 4.514 Earth's Oceans Boil 7 Red Giant 11 Helium Burning 12.25 White Dwarf 12.37
• b) Large stars live out their lives much more quickly than small stars. Small stars below 9 solar masses produce planetary nebula and White Dwarfs. Larger stars with 9-20 solar masses explode as supernovas and leave behind Neutron Stars. The largest stars with more than 20 solar masses explode as supernovas and leave behind Black Holes.  Star Mass (solar masses) Total Lifetime Large Remnant Small Remnant 0.8 25 billion years Planetary Nebula White Dwarf 1.0 12.3 billion years Planetary Nebula White Dwarf 2.0 1.4 billion years Planetary Nebula White Dwarf 4.0 200 million years Planetary Nebula White Dwarf 9.0 30 million years Supernova Neutron Star 20.0 9 million years Supernova Neutron Star 40.0 4.8 million years Supernova Black Hole 85.0 3.3 million years Supernova Black Hole
2. Fueling the Sun
• a) We have E=mc2 and E and c are given. So we divide both sides of Einstein's equation by c2 to obtain m = E/c2. Plugging in numbers:
m = 3.8 * 1026 / (3*108)2 kg
= 3.8 * 1026 / 9*1016 kg
= 3.8/9 * 1010 kg
= 0.42 * 1010 kg
= 4.2 * 109 kg
= 4.2 * 106 tons (since 1 ton = 1000 kg)
= 4.2 millions tons.
The Sun converts 4.2 million tons of mass into energy every second.
• b) 2 * 1030 kg / (4.2 *109 kg/sec)
= 2/4.2 * 1021 sec
= 4.8 * 1020 sec
Now one year has 60 sec/min * 60 min/hr * 24 hr/day * 365 day/year = 31,104,000 seconds
So 4.8 * 1020 sec / (3.1 * 107 sec/year)
= 4.8/3.1 * 1013 years
= 1.5 * 1013 years
= 15000 billion years!
The Sun could last a very long time if it was careful with its Hydrogen!
• c) From Question #1, the Sun also burns Helium for fuel late in its life. It burns the most Hydrogen while it is a Red Giant. Huge amounts of energy flowing outward though the star are necessary to support it against collapse - the larger the star is, the most energy must be flowing out from its center, and the more Hydrogen it must be burning to create that energy. The unburned Hydrogen goes into an extended planetary nebula and a remnant white dwarf when the star dies. Incidentally, the long age we calculated in the last question can be used to show that only 4.5 billion / 15000 billion = 0.0003 = 0.03% of the Sun's mass has been converted from Hydrogen to Helium since the Sun formed. The Sun's composition of 71% Hydrogen, 27% Helium, and 2% other materials has barely changed since it was born! A more detailed application of this method shows us that most of the Helium in the Universe was already present when the first stars formed!