Rocketplanes

LA to Paris in 90 minutes ... on paper at least.

The Lockheed NF-104A (source)

EDIT: there’s a substantial update to this post here.

The switch to reusable rockets is justified with statements like: “imagine if we threw away a 747 after every flight, then flights would be prohibitively expensive.” It’s looking likely that reusable rockets will be developed, which begs the question: what if we threw away the 747 and replaced it with a rocket? Could rocketplanes be a viable form of travel?

Strapping people to an ICBM is not the worst idea¹. Rockets can go extremely fast, reduce drag by spending most of their time out of the atmosphere, and use well-established technology. We’re going to look at a toy model of rocketplane costs and see if they can compete.

What are the key costs to reusable rocket flight? One is building the rocket itself, a large upfront cost that gets amortized over many flights. Another is the cost of fuel, which is a repeated operating cost. For rockets in particular, fuel is a big deal, composing roughly 90% of the mass of the entire rocket. So to start, let’s assume that the rocket gets enough reuse to make amortized capital costs small and that fuel costs are the bulk of ticket prices.

Fuel costs go up dramatically the faster you go, according to the Tsiolkovsky rocket equation:

\(m_x = m_p (\exp(\frac{\Delta V}{I_{sp} g}) - 1)\)

Where m_x is the mass of fuel required, m_p is the mass of the rocket plus the mass of the people, I_sp is the specific impulse of the rocket and g is Earth’s gravitational acceleration. It is extremely expensive to add more mass, so rocketplanes will have no luggage.

What’s the benefit of going fast? It saves people time. We don’t want to spend hours flying LA to Paris, a rocket can get us there in half an hour! Specifically, if the (great circle) distance from one point to another is D, and the average velocity is V then the total time is t = D/v.

In reality, the rocket’s trajectory is more complicated, it has to boost out of the atmosphere and then glide back through it to slow down. That being said, setting the distance traveled to D isn’t a terrible approximation, these rockets only have to get up 100 km or so to get out of the atmosphere, so the trip out of the atmosphere and back tacks on only ~200 km to a flight that’s thousands of kilometers long.

For the velocity, delta-V can be used to approximate the average speed². In reality, there is a brief acceleration time and the rocket decelerates through the atmosphere, but for most of the flight, it will be on a ballistic trajectory at roughly delta-V.

To turn this time of flight into a cost, we have to consider the cost of passengers time C_time. Then the overall equation becomes:

\(C_{2} = C_{time} \frac{D}{\Delta V}\)

Putting the costs together, we get:

\(C_{total} = C_{time} \frac{D}{\Delta V} + C_{fuel} \cdot m_p (\exp(\frac{\Delta V}{I_{sp} g_0}) - 1)\)

Now we just have to set our constants and minimize with respect to delta V.

• For D, I used 10000 km, the other side of the earth is 20000 km away, so this seemed like a good order-of-magnitude³.

• For C_time, I assumed that rocketplanes would be used by people who are pretty rich, high initial prices would make this true automatically. Over decades, the price would fall, but average global wealth will rise, so I think it’s safe to stick with a high estimate. Let’s say they make $100/hour ($200K/year) so the cost of their time is 2.8 cents per second.

• For the rocket equation, I based numbers off of the SpaceX Raptor engine, with an I_sp of 350 seconds (in atmosphere). The payload mass m_p is 100 kg (60 kg person + 40 kg of rocket hardware per person).

• The cost of fuel C_fuel has conflicting estimates, this article claims that methane costs $9/kg, but looking at natural gas spot prices, I get something more like $0.1-$0.2/kg. I’m going to stick with the higher number for now⁴⁵.

Minimizing the total cost equation using these numbers, we get a delta V of 1.78 km/s. At that speed, a flight from LA to Paris would take just under an hour and a half. Propellant costs would be $123. Google flights tells me that an airplane from LA to Paris typically takes 11 hours and costs $200-$400. So rocketplanes can take 7x less time for roughly the same price.

That being said, this is the most charitable possible case for rockets as a transport method. Rockets need to be extremely safe for passengers to even consider getting on board. Launch has to be automated and repeatable so that capital and labor costs are small⁶. Even if the economics worked out, no country is going to allow an intercontinental ballistic missile full of people to land on their territory. Noise ordinances alone are probably enough to destroy any on-paper advantage that rockets have⁷.

Rockets would have to compete with super- or hypersonic aircraft companies like Astro mechanica, Hermeus, Venus Aerospace, Boom, Spike, and others. These designs don’t need to carry their oxygen on board, reducing fuel mass by 5x. These planes can also take off from the same runways we have today and can achieve speeds of Mach 1-5, close to the optimized rocket speed. On the other hand, developing these planes seems challenging and they would only be able to carry a few passengers per flight; it’s not clear to me which approach will take most of the market for fast travel.

While either implementation will have major technical challenges, nothing stops us from getting around the world pretty quickly if we build the infrastructure. One day, you may be able to choose a much faster mode of transport for your international travel. But for now, there are plenty of opportunities to make airplanes better while we wait for these technologies to be commercialized.

EDIT: here’s an image of what the “depressed trajectory” might look like for a spaceplane (the lower one labeled FOBS). Notice the small altitude change relative to the Earth’s radus.

Source

EDIT: these capabilities are particularly valuable for military operations, see SpaceX's Starship: America's Secret Weapon.

EDIT: bean points out that 40 kg of rocket per person is too optimistic. Looking at Starship’s specs, 120 kg of rocket per person works better, and I get an optimal speed of 1.4 km/s (1 hr 50 minutes flight time) and a propellant cost of $164.

1. Technically, the worst idea would probably be along the lines of transporting people via large cannons.

2. This doesn’t account for the fact that gravity losses and atmospheric drag will lower the speed for a given amount of fuel.

3. For shorter distances, rockets make no sense as their trajectory becomes significantly longer than a normal flight.

4. When I use something like $1/kg the rocket speed increases to about 4 km/s with fuel costs falling to $45. At $0.1/kg, optimum is about 7.5 km/s with fuel costing $16. I wonder about this in the context of companies like Terraform Industries, who could potentially produce fuel at the launch site with cheap solar energy.

5. Note that liquid oxygen is much cheaper, and methane is only 20% of the overall propellant mass.

6. Because high speed isn’t as important, solid rocket propellants like those used in ICBM’s might be more repeatable and safe than methalox rockets.

7. Though I do wonder about rockets for moving goods. The speed is only valuable for arbitraging price differences, perhaps rockets could be used to transport LNG to parts of the world with transient high prices?