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!
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