# The cost of putting an object into Low Earth Orbit

Suppose that you want to put an object in orbit around the Earth --- at roughly the same altitude as the International Space Station, or many other satellites. We'll use an altitude of H = 400 km for this example.

For simplicity, let's pick a simple, small object: a textbook with a mass of m = 1 kg..

#### Step 1: raise it from the ground

The first thing to do is to compute the work done by gravity on the textbook as it moves from the surface of the Earth to the height H.

How much work must our rocket perform on the book in order to raise the book against the pull of gravity?

Now, suppose that we do manage to raise the book from the surface of the Earth to the desired height H. What will happen when we open the rocket's payload bay and release the book?

```

AAaaaaaaaaaa.........          *thump*
```

#### Putting the book into a circular orbit

Hmmmm. Apparently, simply raising the book up above the surface isn't enough. We also need to give the book enough speed that it circles the Earth in a stable orbit.

1. What is the speed the book must have to remain in a circular orbit at this position?
2. How much kinetic energy must the book have?
3. If the book is placed at this location, but initially is at rest, how much work must we do on the book to give it that final velocity?

Which is larger?

```

Work to raise book to orbit         =

Work to give book orbital velocity  =

```

#### How much should it cost (simplified version)?

So, we know much energy it takes to raise the book to orbital height, and then give it orbital velocity. How much should it cost?

Suppose we could use an electric motor and a really, really long extension cord. The cost of electrical energy from RG&E is pretty low:

```

1 kilowatt-hour   =    \$ 0.10

```
1. How much does one Joule of energy cost?
2. How much would it cost to put the book in orbit?

Does this look ... fishy to you?

#### How much does it REALLY cost?

There are several companies which will put an object into orbit for you, using rockets. One of them is SpaceX , which was founded by Elon Musk, who made his fortune in an Internet startup company.

SpaceX has several models of rocket. Let's look at the Falcon-9.

1. What is the total payload delivered to LEO (in kg)?
2. How much does it cost per launch?
3. What is the cost per kg of payload?
4. How does this compare to your earlier calculation?

Golly, that's a LOT more expensive.

#### Don't forget about air resistance

Well, one of the things we haven't taken into account is the air resistance on the rocket as it makes its way into orbit. To a very rough approximation, we can compute the air resistance on the rocket with a formula like this:

We'll simplify again, but it will give us a good idea for the order of magnitude. Let's use the following values:

```

ρ  =

CD  =

A    =

v    =    100 m/s

```

1. What is the force of air resistance on the Falcon-9 rocket under these conditions?
2. How does it compare to the force of gravity?
3. What if the speed goes up to 300 m/s?

Note that the air resistance force goes up sharply as one approaches and crosses the speed of sound ...

.... so the real force of air resistance will be even greater than we have calculated.

What sort of work does air resistance do on the rocket? Why?

#### The really big problem -- fuel

But the major reason that it costs SO MUCH to put even a small payload into orbit is because one has to carry much of the fuel along for the ride. If you want to put a 1-kg textbook into space, you have to use a pretty big machine to do it. The walls and motors and especially the fuel have to go up into the air, too, greatly increasing the amount of stuff you are lifting off the ground.

For example, consider the Falcon-9 again.

1. How much payload does it take into LEO?
2. What is the mass of the rocket and fuel?
3. What is the ratio of payload mass to other mass?

This is the reason that travel to orbit is so expensive. In just a few weeks, we'll take a more quantitative look at the ratio of payload to fuel when we talk about momentum and rockets.