Mathematical Applications

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 = mc² 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 10⁸ m/s

This formula tells you how much mass is needed to produce a certain amount of energy, with all of the mass being converted into energy. The reverse also holds true, if you want to make a bit of mass, you need a certain amount of energy to do it. This formula does tell us how these transitions occur, it just provides the amount of the quantities involved.

Typical Problems

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

Solution: Plug into the formula
E = mc²
E = 5 x (3.0 x 10⁸)² = 5 x (9.0 x 10¹⁶)
E = 4.5 x 10¹⁷ 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 = mc²
2 x 10 ^-14 = m(3.0 x 10⁸)²
2 x 10 ^-14 / (9.0 x 10¹⁶) = 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

This formula is fairly simple, since you put in only one thing usually, the mass of the star, and get out how long it spends on the Main Sequence.

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 10⁸ 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 10¹⁴ years)
If you wanted, you could have just leave it as 100,000 billion years or 100 trillion years.