Formula - Stellar Lives

Special Relativity Formula - The most famous formula from Einstein, which tells us how much energy you can get from matter and vice-versa.

Formula: **E = mc ^{2}** where:

- E = amount of energy produced by a given amount of mass, measured in Joules
- m = mass that is converted into energy, measured in kg
- c = speed of light, in this case 3.0 x 10
^{8}m/s

**Typical Problems**

1. How much energy could you get out of 5 kg of material?

Solution: Plug into the formula

E = mc^{2}

E = 5 x (3.0 x 10^{8})^{2} = 5 x (9.0 x 10^{16})

**E = 4.5 x 10 ^{17} Joules.**

And just so you know, this amount of energy is about the equivalent of a 100 megaton bomb.

2. How much mass can you get from a typical gamma-ray (energy = 2 x 10 ^{-14} J

Solution: This will require a little manipulation

E = mc^{2}

2 x 10 ^{-14} = m(3.0 x 10^{8})^{2}

2 x 10 ^{-14} / (9.0 x 10^{16}) = m

**2.2 x 10 ^{-31} kg = m**

Which is pretty small....

Main Sequence Life-time formula - used to approximate how long a star will live on the Main Sequence. Not a really accurate formula since it isn't very accurate at the very high or very low mass ends of the scale, but it's a good first guess.

Formula: **time = (1/M ^{2.5}) x 10 billion years** where:

- time = time spent on the main sequence in years
- M = mass of the star in solar masses

**Typical Problems**

1. If a star is 6 times the mass of the Sun, how long will it be on the Main Sequence?

Solution: Pop it into the formula

time = (1/M^{2.5}) x 10 billion years

time = (1/6^{2.5}) x 10 billion years

time = (1/88) x 10 billion years, since 6^{2.5}=88

time = 0.011 x 10 billion years

time = 0.11 billion years.
**time = 110 million years (1.1 x 10 ^{8} years)**

If you wanted you could have just leave it as 0.11 billion years.

2. If a star is 0.025 the mass of the Sun, how long will it be on the Main Sequence?

Solution: Pop it into the formula

time = (1/M^{2.5}) x 10 billion years

time = (1/0.025^{2.5}) x 10 billion years

time = (1/9.9 x 10^{-5}) x 10 billion years, since 0.025^{2.5}=9.9 x 10^{-5}

time = 10,000 x 10 billion years

time = 100,000 billion years.

**time = 100 trillion years (1.0 x 10 ^{14} years)**

If you wanted, you could have just leave it as 100,000 billion years or 100 trillion years.