Nuclear Power in the Sun

CAP Astronomy Module (Level 2-3)
Written by Dr. Craig Tyler, Fort Lewis College

Mathematical concepts: Basic algebra, nuclear energy, E = mc2

Objective: Study energy generated in the Sun

 

Warm-up Questions

1.         How bright is the Sun?

2.         Where does the Sun get all the energy to shine so brightly?

3.         What does the Sun use for fuel, and how does it consume that fuel differently, for example, than a car engine?

4.         Will the Sun ever run out of fuel?

 

Introduction

Sunlight shines at a rate of 3.9 ´ 1026 W (“W” stands for “Watts”). For comparison, a typical light bulb is around 100 W. Because the Sun’s power output is so enormous, the Sun must possess an equally enormous power source. That source is nuclear fusion.

 

The word “nuclear” refers to the energy stored in the nucleus at the center of an atom. In non-nuclear processes – like electricity or fire – the nuclei (plural of nucleus) of atoms never change. This is true element by element. For example, consider burning a fuel, like gasoline. Most fuels are made from “hydrocarbons” which contain atoms of hydrogen and carbon. After the fuel is burned, there is no more fuel. But there are by-products, like ash, smoke, and/or water vapor, and and these by-products still contain just as many hydrogen atoms and carbon atoms as the fuel initially contained.

 

But that’s for a non-nuclear process. In a nuclear process, it is possible to create and/or destroy individual elements like hydrogen. In nuclear fusion, the nuclei of two or more elements come together and fuse, or merge together, into a different nucleus, of a different element. Inside the Sun, the main power source is the nuclear fusion of four hydrogens into one helium. The mechanism for this is quite complicated, but the result is simpler – the production of a great deal of power, which ultimately emerges as sunlight.

 

In a region at the center of the Sun, called the core, the following reaction is taking place right now:

            4 Hydrogen ® 1 Helium + Energy (which becomes sunlight) .

In this module, you will see how individual atoms (actually, just the nuclei of atoms!) manage to generate all of the power output of the entire Sun.

 

Using Math to Solve Problems

First, let’s verify that the Sun really needs nuclear power – that some lesser power source would be inadequate. To do this, we’ll need a version of the famous equation “rate ´ time = distance”. But we’re interested in the rate of energy production, not the rate of distance traveled. The rate of energy production is called power, and is measured in Watts. But one Watt equals one Joule per second, where one Joule (written “J”) is an amount of energy (in metric units). So power (Watts) represents the rate (per second) of energy (Joules) produced, and the equation becomes “power ´ time = energy.” For example: a 100 W light bulb running for one minute produces (100 J/sec) ´ (60 sec) = 6000 J of energy.

 

1.  The mass of the Sun is 2 ´ 1030 kg, and roughly the inner 10% of that mass, the core, is responsible for generating all of the Sun’s power. On average then, how many Watts of power does each kilogram of core matter produce?

 

2.  The Sun has been shining at approximately its current rate for 4.6 billion years, ever since it first formed. During that time, from the Sun’s birth until now, how many Joules of energy has each kilogram of core matter produced? Your answer should be approximately 300 trillion Joules (per kilogram).

 

Burning one kilogram of gasoline produces about 50 Joules of energy; the Sun’s power source somehow produces about 300 million Joules. Therefore, the relatively simple calculation you just did shows that the Sun can’t operate on “conventional” power, like burning fuel with fire. The Sun requires much more power (6 million times more!), and astronomers have confirmed that that power source is nuclear fusion.

 

Now that you’ve demonstrated that the Sun really does need nuclear power, we can move on to consider a little bit about how that nuclear power works. Nuclear theory is quite advanced, but the basic idea is not. You know that nothing is free. In terms of energy, a car ride requires gasoline, and your body requires air and food; photosynthesis in plants requires sunlight. In all of these cases, something gets used up (gas, food, oxygen, sunlight). Nuclear energy requires mass; mass gets used up. As the Sun performs nuclear fusion to produce energy, some of its mass gets used up. The sun is getting just slightly lighter every second.

 

3.  When the Sun converts four hydrogen nuclei into one helium nucleus, how much mass is used up? The mass of a hydrogen nucleus is 1.6736 ´ 10-27 kg, and the mass of a helium nucleus is 6.6466 ´ 10-27 kg.

 

4.  The way nuclear energy works, whatever mass you lose gets converted into energy. The conversion rate is Einstein’s famous equation – probably the most famous equation there is – “E = mc2.” That’s “(energy produced) = (mass lost) ´ (the speed of light)2.” (The speed of light, which is 3 ´ 108 meters/second, is relevant because this equation is part of a larger theory of Einstein’s, called “relativity,” which involves fast speeds.) How much energy is produced in the Sun each time a helium nucleus is formed?

 

5.  Your answer to #4 may not sound like much energy, but that’s just what you get from 4 individual hydrogen nuclei (the central part of 4 atoms). But one nucleus at a time, matter is very light! It takes 6.02 ´ 1026 hydrogen atoms to amount to 1 kg of hydrogen, and the nucleus is by far the heaviest part of an atom, so it’s fine to treat atoms and nuclei as the same thing in this context. How much energy is one kilogram of core matter from the Sun capable of producing each second? What about the whole core? What if you consider the entire lifetime for the Sun, until now, in order to express the last answer in terms of power? Based on all these numbers you’re calculating, can you determine whether or not nuclear power is capable of powering the Sun?

 

6.  At this point, you’ve already demonstrated the success of nuclear fusion in powering the entire Sun. But how quickly is this process using up the Sun’s “nuclear fuel,” hydrogen? To answer this, calculate the rate at which the Sun uses up the hydrogen in its core (by converting it to helium and energy) in kilograms per second. Use that to calculate how many kilograms of hydrogen the Sun has already “spent.” Compare these numbers you calculate to the combined mass of all the people on Earth, and to the total mass of the Sun. (Note: 1 kg = 2.2 pounds.)

 

7.  Lastly, since the Sun formed, how much mass has it lost completely and forever – not just turned into helium, but completely gone?

 

Follow-up Question

Do you think that the Sun might burn out? If so, when?