By Rhett Allain
The primary role of the Automated Transfer Vehicle is to bring supplies to the International Space Station. Supplies include food, water, oxygen, scientific equipment and candy bars. Yes, I listed food twice. Candy is food, but I listed it separately so that we could look at candy in space.
Here is the question: Suppose an astronaut requests an extra candy bar to be sent up on the ATV. How much extra energy is required to get this candy into orbit? That is the question.
Editor's note: In addition to having a knack for science communication, Rhett Allain is Associate Professor of Physics at Southeastern Louisiana University. He writes regularly for Wired's Dot Physics blog and is a bit of a physics fanatic who spends more time than many pondering how daily life intersects with science. His recent posts have looked at the physics of a (fake) broken swing image, why doesn’t the Moon crash into Earth (gravity's involved!), Star Wars blaster speeds (answer: 34.9 m/s) and how well-made are Lego blocks (really well made). With the recently announced development of ATV in cooperation with NASA for Orion, we're delighted to feature a few posts from the far side of the Atlantic. Enjoy! – DGS
To start, let’s get some initial values. Some of these will be estimates. What about the ISS? It has an orbit that is about 420 km above the surface of the Earth and it moves with a speed of 7,700 m/s. Both the altitude and speed are important in the calculation of the energy needed to get supplies there.
There really is just one other important piece of information. We need the location of the ATV launch pad, which is in Kourou, French Guiana. In case you aren’t familiar with the location, here it is in Google Earth.
Kourou is only 5 degrees above the equator. There is a reason for this as we will soon see.
Oh, we need one more thing. What about the mass of a candy bar? I don't wish to single out any particular brand of candy, so I will just assume an average chocolate candy bar. Let's say it has a mass of 50 grams with 250 calories (that's food calories, which are different than chemistry calories – just to be clear).
Physics and Energy
Now for a little bit of physics. Why does it even take energy to get anything to the ISS? Well, there are two things you need to do to a candy bar in order for an astronaut in space to eat it. First you have to lift the candy up to the height of the ISS. Second, you have to increase the speed of the candy bar so that it is going at the same speed as the ISS. Let me look at these two things separately.
Suppose you find this 50-gram candy bar on the ground and you lift it about 1 meter up to place it on a table. This requires that you do some work on the candy to change its energy. But how much energy would it take? One way to look at this is via the change in gravitational potential energy. On the surface of the Earth, the change in gravitational potential can be calculated as:
Now, what if I want to increase the height of the candy bar all the way up to the ISS? Can I just do the same calculation but change the height from 1 meter to 420 km? No, I can’t. The above model for gravitational potential energy assumes that the gravitational force on the object is constant. This is a good assumption near the surface of the Earth, but not so good as you get higher (though at the ISS height it isn't the worst approximation you could ever make).
If we use a better model for the change in gravitational potential, it would be this:
Here, G is the universal gravitational constant. The two masses in the expression are the mass of the candy bar and the mass of the Earth. The values on the bottom of the expression are the distances from the centre of the Earth. So, the candy bars ends at the altitude of the ISS (I call this h) and starts at the radius of the Earth.
If you put in the values for G and the radius and mass of the Earth, you would find that it takes 1.93 x 105 Joules of energy to get that candy bar up to the right altitude.
But that’s not all of the energy for the candy bar. If you put that much energy into the candy and let it go, it would just fall back to the Earth. The other kind of energy the candy needs is kinetic, i.e. moving, energy. This has an expression of:
Since we know the speed of the ISS, shouldn’t this be easy to calculate? If I put in the mass of candy (0.05 kg) and the speed of 7,700 m/s, I get a kinetic energy of 1.48 million Joules. Actually, this is the wrong answer. Why? It assumes that we took that candy and increased its speed starting from rest. The only problem is that before the launch, the candy is already moving. It is moving because it is on a rotating Earth.
Let's say the Earth rotates once every 24 hours (which it doesn’t actually – that is the time for the Sun to get back into the same position but this value is close enough for us). This means that the speed of the candy before launch can be calculated as:
But what is r in this expression? It is the radius of circle in which the object moves due to the rotation of the Earth. At the equator, r is the radius of the Earth. At the North pole, the radius would be zero. If I put in the radius of the Earth (and convert hours to seconds), I get a starting candy bar speed of 464 m/s. That might be small compared to the ISS speed, but every little bit helps. And this is why ESA launches the ATV from Kourou. It is very close to the equator.
OK, so what about the new change in kinetic energy of the candy? Launching from the equator, you would need about 1.47 million Joules.
The total energy to get this candy bar to the ISS is just the sum of the two values we've now calculated: the change in kinetic and change in gravitational potential energy. This works out to 1.66 million Joules. That is over a million Joules of energy for just that one tiny candy bar and it assumes a perfectly efficient method for getting things into orbit without any energy losses. This is why we don’t all live in space. It's just expensive.
Energy to orbit using candy
It's pretty hard to get a feeling for an energy of 1 million Joules. What about a comparison to the energy in the candy bar? If you consume this candy, it can produce 250 food calories. One food calorie is 1000 calories which is 4,180 Joules.
Let’s go backwards. If it takes 1.66 million Joules to get candy into orbit, how many food calories is that? This is a pretty straight-forward unit conversion problem. Remember that the trick to unit conversions is to always multiply by a fraction that is equivalent to 1.
Well, that's not so bad. It takes a little more than 1 candy bar of energy to get a candy bar to the ISS!
OK, one more thing: What if we wanted to get all of the ATV cargo to the ISS just using candy as energy? The ATV can carry a payload of about 20 tonnes, or 20,000 kilograms. If this payload comprised just candy bars, that would be 400 thousand bars of candy; the energy needed to get into orbit is the same energy you would get from consuming 640,000 candy bars.