# An Analysis of the LEGO City Deep Space Rocket

The LEGO City Deep Space Rocket and Launch Control is a “modular, multi-stage rocket with cockpit, booster and payload storage modules.” Prominently featured on the page are images of the launch control tower, launchpad, and various extra equipment such as a lunar rover. But can this rocket really fly?

LEGO helpfully provides an “Explore in 3D” option for us, which provides us a good measure of the exact size of the rocket. We will first assume that:

- LEGO City is on Earth
- A minifigure, 3 blocks high, is repesentative of an average American adult, at 1.753 meters tall (US Dept. of Health and Human Services; et al.)
- The two boosters are similar to the 4-segment solid rocket boosters used on the space shuttle
- The first stage uses a cryogenic rocket engine like the RS-25

Given these, we can find these measurements for each of the respective components:

- Main stage: 20 blocks (11.69m) tall, 6 blocks (3.51m) wide
- SRB: 13 blocks (7.60m) tall, 4 blocks (2.34m) wide
- Payload stage: 12 blocks (7.01m) tall, 6 blocks (3.51m) wide
- Space Capsule: 18 blocks (10.52m) tall, 8 blocks (4.68m) wide, cone-shaped

An interesting aspect of this design is that it removes the commonly-used second stage booster and replaces it with a payload. While this increases cargo space, it also increases the total mass that needs to be put into orbit, which makes it harder to accelerate as well.

Using these figures, we can start calculating the amount of fuel in our main stage. Let’s assume that that the main stage contains fuel and oxidizer similar to that in the space shuttle’s external fuel tank. For comparison, the shuttle’s external fuel tank holds 735,601 kilograms of fuel (both LOX and LH2), in a volume of 2,050,798 litres. That’s roughly 0.36 kg of fuel per litre. As our main stage has volume (assuming it’s a cylinder) 113,115 litres, it can hold about 40,573 kilograms of fuel.

As for our solid rocket boosters, they hold 500,000 kg of rocket fuel in a volume of about 491.43 cubic meters (assuming a fully cylindrical booster), or about 1017 kg of fuel per cubic meter. Knowing that our SRB has volume 27.94 cubic meters, we can assume that each of our SRBs hold 28,410 kg of propellant.

For simplicity, let’s just assume that our space capsule is similar in mass to the Orion crew module, or 10,400 kg. Let’s assume that the payload stage has the same mass as our crew module for simplicity. We now have the information we need to use the rocket equation to examine its motion:

We know our initial mass (m_{0}) to be 107,793 kg (we’re assuming the stages are 100%
propellant), and we know our final mass (m_{f}) to be 10,400 kg (that’s just the crew
module). Standard gravity (g_{0}) is just 9.81 m/s^2.

We’ll now have to calculate our specific impulse (I_{sp}). Since these engines also have
different mass flow, we’ll need to get final I_{sp} with this equation. Plugging
in those numbers to Mathematica gives us:

Our final I_{sp}, which is 263.1 seconds. Plugging this back into the rocket equation, we
get our final delta-v:

4246.43 meters per second, or 4.25 km/s. That’s not high enough for even LEO, which requires about 9.4 km/s, though it would definitely break the Karman line. This raises troubling questions for LEGO City leadership. Why does the set include a rover and a grappling arm, if it will never reach the moon? What’s the satellite used for if it doesn’t have the delta-v to reach even low-earth orbit? LEGO, we need answers!