It is difficult to believe that this will be the last space shuttle launch.

Clearly, I must do something to commemorate this event. But what? How about I look at space ships in orbit and consider the energy required. WITH GRAPHS.

How much energy does it take to get 1 kg into orbit?

First, what orbit am I talking about? Let me assume low Earth orbit — which is about 360 km above the surface of the Earth. Now, you need to realize that in order to be in this orbit, the object has to go at a certain speed. The only force acting on the mass would be the gravitational force. The acceleration that goes along with this force is the acceleration of an object moving in a circle.

Since you have to get this thing going fast, it must increase in kinetic energy. Also, since it has to increase in distance from the center of the Earth, it must increase in gravitational potential energy (technically, the Earth-mass system increase in gravitational potential energy).

I will skip all the intermediate steps and show you the change in energy need to get an object into orbit. Here are all the details if you are interested.

These are the relevant constants:

• G = 6.67 x 10^-11 N*m²/kg² (gravitational constant)
• M_E = 5.97 x 10²⁴ kg (mass of the Earth)
• R_E = 6.38 x 10⁶ m (radius of the Earth)

Using these, the energy for the 1 kg to get into low Earth orbit is 3.29 x 10⁷ Joules. If you paid for that with the electricity from your house, you would write it in kilowatt hours. That would be 9.1 kW*hrs per kg. In the U.S.A, the average kilowatt*hr costs 11.2 cents. This would just cost you about $1 — of course assuming that your electric-based rocket was 100% efficient.

Unfortunately, it costs way more to actually put 1 kg into orbit. The current estimate is more than $1,000 per kg of material. Why? First, there is the whole expensive rocket thing. Next, you have to fuel and stuff. Yes, you actually have to get some of the fuel almost all the way into orbit so that you can use it.

Why is it better to launch a spaceship near the equator?

News flash: the Earth rotates. It does. This rotation is like a bonus starting speed. How fast is this starting speed? Well, the Earth rotates at around one revolution a day (it is actually a little bit less than rotation per day). But how fast does this mean something is moving?

Imagine you are on a merry-go-round with your friend. Your friend is near the middle and you are on the edge. You both have the same rotation rate (angular velocity) but since you have a much greater distance to go (all the way around the outside), you have to go faster. If the magnitude of the angular velocity is represented by ? then the speed will be:

Where r in this case is the distance from the axis of rotation. Suppose you launch a rocket from the North Pole. In this case, the distance from the axis of rotation would be zero meters. You would get no “speed bonus”. The greatest bonus is at the equator since that is the farthest from the axis of rotation.

If you consider this speed boost, then what is the energy to get into orbit (per kg) as a function of latitude? Here you go.

Launching from Cape Canaveral (28.5°) is a 0.3% energy savings compared to the North Pole. Maybe that doesn’t seem like a big deal, but every bit helps.

Would launching from a mountain help?

Moving towards the equator gives you a little speed boost. Moving to a mountain would make the change in gravitational potential energy to get to orbit a bit smaller. Suppose the mountain has a height of s (I already used h for the orbit height). This would change my change in energy equation to:

This assumes starting the mass at rest (so no speed boost). Mount Everest is 8,850 meters above sea level. So, here is a plot of the energy needed to get 1 kg into low Earth orbit for heights starting from sea level to the top of Everest.

Launching from the top of Mount Everest would give you a 0.2% savings in energy per kg.

What about a giant mountain at the equator?

This would be the best case scenario, wouldn’t it? If there was an 8,850 meter hight mountain at sea level, it would do two things. First it would start the rocket off at a higher point. Second it would give it even more of a starting speed than at the equator. Why? Because it isn’t on the equator. It is 8,850 meters above the equator. But is that a big difference?

The speed at sea level on the equator is (using a rotation period of 23 hours and 56 min):

And the starting speed if on a mountain at sea level:

Not much of a difference. Although Mount Everest is tall, it is small in comparison to the Earth. The total energy needed to get 1 kg of mass into orbit from a mountain on the equator would be 3.276 x 10⁷ J/kg. So, not that big of a savings.

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