What is the approximate influence of atmospheric drag on the cost of rocket launches? Is it beneficial to have launch sites located at higher altitudes?
Cape Canaveral is at sea level, but I've noticed that Chinese launch sites are located on quite high altitudes (although they could still be higher if located in Tibet).
I'll give you the numbers. I'm breaking this up into 3 different terms. There's atmospheric drag, what I'll call the "hover" term, and the gravitational potential climb. I will more or less assume a flight directly up. You're welcome to use whatever term for velocity you want, as none of them will be representative. I'll take the Shuttle's speed at halfway to max Q. This is 1000 ft/s, or about 300 m/s.
You would think atmospheric drag would be very difficult. It's actually not. In any case, you would probably use the v^2 relationship for drag. But if you think about where that comes from, it basically assumes that all the air in front of you is accelerated to the speed of your craft (minus any departure from unity in the drag coefficient). So for a good approximation, just take the mass-thickness (I call mu) for the entire atmosphere, and multiply by the metric for velocity.
Also, I'll use the numbers for Falcon 9, which is a diameter of 3.66 meters and launch mass of 333,400 kg. Yes, a lot of these numbers change over the course of the flight, but in ways that are fairly obvious if you changed this to do numerical integration.
$$\Delta V (drag) = 1/2 \mu C_d A v / M $$ $$= (0.5)(10 \text{ tonnes} / m^2)(0.5)\pi(3.66/2 m)^2(300 m/s)/(333.4 \text{ tonnes}) $$ $$= 23.7 m/s$$
Wow. That is not much. Maybe velocity should be higher. But still, out of 10 km/s total, this is a tiny amount. Atmospheric drag complicates launches, but not much because of its Delta v value.
Next, the "hover" term. This represents the gravity drag. Again, I'm forced to assume a pretty much upward launch. I'll also compare sea-level to Mt. Everest, at a height of 8,848 m. Not that you'd set up a launchpad there, but we need this to answer the question.
$$\Delta V = g h / v = (9.8 m/s^2)(8,848 m)/(300 m/s) = 298 m/s$$
Now this is much more significant. This isn't all of the gravity drag either. It's still sucking your delta v budget after you're out of the atmosphere, until you get to full orbital velocity.
Let's move on to the gravitational potential itself.
$$\Delta V = \sqrt( g h ) = \sqrt( (9.8 m/s^2) (8,848 m) ) = 294.5 m/s$$
The sum of all these is a ballpark estimate of the benefit you would get from changing your launch location from sea-level to Mt. Everest. Honestly though, you save a comparable amount just by moving it down to the equator, where the Earth's rotation gives you a bigger boost.
Anyway, this is 616.7 m/s out of a total budget of 10 km/s. So it would be less than 10%. By the rocket equation, this can still make a difference. But then again, actual costs are complicated.
Let's compare two rockets with somewhat similar specifications, but one very large difference.
Falcon 1- Carries about 670 KG to LEO (See the User's Guide) Mass 38555 KG (Wikipedia). Launched from sea level.
Pegasus- Carries about 450 KG to LEO (See Wikipedia). Mass 18,500 KG. Launched from 40,000 feet.
This makes a lot of assumptions, but let's just assume you could scale linearly the mass of the Pegasus. That would give the rocket a mass of 27000 KG to lift the payload of the Falcon 1. That is a difference of 40% roughly. Why the difference?
Falcon 1 is a LOX/RP liquid propellant rocket whereas Pegasus is solid rocket based. The technologies have vastly different thrust/weight ratios, specific impulse, and mass fractions. Solid rockets tend to have higher mass fractions and T/W because the engines are less complex (i.e. no pressurization systems, or plumbing, or turbo-pump machinery. But solids are typically more expensive, and typically have fixed impulse burns (i.e. they cannot be turned off). This is less of an issue for early stages, but the reason the Pegasus added the optional HAPS (hydrazine) final stage for precision orbit insertion.
Because it is air-launched Pegasus can fly a different trajectory. Instead of a low angle-of-attack ascent followed by a gravity turn, Pegasus has wings. It flies at positive angle of attack and uses lift to aid in ascent.
Engines at altitude can use a more efficient engine design (i.e. rocket nozzle expansion ratios tuned to atmospheric pressure at drop vice sea level.
There is a small velocity gain from the air launch. This isn't significant (~2% of orbital velocity), but is there.
The Pegasus doesn't have to worry about changing inclinations, as the Falcon 1 does. But the specified numbers for Falcon 1 don't take in to account a varied inclination.
There is considerably less air density at 40,000 feet, which results in lower integrated drag losses.
You are 10 km higher at altitude. This probably isn't significant.
Bottom line is, raising your altitude of your launch site will give you an increase in performance, both by increasing your engine efficiency, and reducing your drag. These numbers wouldn't be as drastic for a launch site of 10,000 feet, but they still would represent a measurable change in performance.
I like the accepted answer, but it is a little abstract and I want to add to it with some visuals and more representative numbers. I created a crude launch sim and tracked the drag, gravity, and steering losses (for a sea level launch):
Edit: original version of this graph (and table) has some bad math on the steering losses, this has been fixed.
This shows that drag is indeed a very small loss factor, even for this crude simulation where the drag losses are amplified (by ~50% is my guess [1]) because I did not program a throttle down (Max q be damned).
Here is a comparison with a higher elevation launch:
| Launch Site Elevation (m) | Final Orbit | $\Delta V$ Expended | Drag Loss | Gravity Loss | Steering Loss |
|---|---|---|---|---|---|
| 0 | 252 km x 204 km | 9339 m/s | 165 m/s (1.8%) | 1316 m/s (14.1%) | 47.3 m/s (0.51%) |
| 5000 | 258 km x 203 km | 9218 m/s | 97.5 m/s (1.1%) | 1262 m/s (13.7%) | 42.1 m/s (0.46%) |
| Reduction at higher elevation launch site | -- | 1.3% | 41% | 4.1% | 11% |
So a little over 1% for the first question and for the second; the physics says yes, higher is beneficial, but the economics and logistics say no.
The flight path of a rocket is barely focused on 'up'. It is easy to get high enough to where orbit is possible. Consider the case of sounding rockets, which even smaller organizations or amateur rocket groups can achieve.
The hard part is going fast enough to orbit. I.e. Getting up to speed.
Thus a rocket typically heads straight up quickly to get out the thick part of the atmosphere and then turns and primarily accelerates to orbital speed.
Starting from a higher altitude would be minorly helpful (I cannot quantify the number, alas) for the very first few moments of flight, but after that would be of little benefit.
As noted in the comments the extra cost of logistics for getting fuel, oxidizer, parts, and payloads to the higher altitudes would mostly likely not be worth it.
