Mass of conventional multi-stage rocket

I wanted to complete my thoughts on suborbital refueling, but found that I needed a robust way of calculating the mass of conventional rockets in order to show potential savings. I got carried away and turned what could have been some simple one-off calculations into a whole package for optimizing conventional rockets. (see rocket_sizing.m).

One can estimate the mass of rocket components based on historic data. For example, the tank mass ratio (empty mass over full mass) can be calculated from an existing rocket, and applied to rockets with similar levels of technology and propellant density. The problem is that you don’t know the mass of propellant needed until you know the total empty mass of the rocket, which depends on the mass of the propellant. Thus, an iterative solution must be performed. This is a nice classroom problem because the iteration converges rapidly. The Mathematica FixedPoint function handles the problem nicely. The new package also makes a simple calculation for the optimal staging velocity that gives the lowest lift-off mass. When trying this approach against real rocket data, I found that it didn’t match well. I believe this is because I’m not taking aerodynamic and gravity losses into account. Typically most of these losses take place during the first-stage burn; requiring a larger first stage than my function suggests. Overall, I think the package will serve my purpose of giving a baseline lift-off mass for different sets of assumptions.

The CDF version of the study and users guide includes a really cool widget that calculates the mass distribution for rockets with a selector for the number of stages. The widget will work if you have the Mathematica CDF player installed. I tried to embed the widget into a web page, also part of the new Wolfram technology. However, after a couple of hours of frustration I figured out that the CDF browser add-on prevents the widget from working because of possible security problems – my code uses functions (such as ToString) that could also be used in malicious code. So for now, download the CDF document. When the document is opened, it will explicitly ask for permission to run the dynamic content. You can look at the code that builds the widget if you download the notebook version and open the cell directly above the widget.

Suborbital Refueling

Till now I’ve been writing about concepts that have been proposed elsewhere. My contributions have been in doing some detailed calculations and making the calculation methods available. And perhaps thinking on a larger scale. This time, we’re off on some new territory. I’ve seem one figure on a website that looks similar, but now I cannot find that reference.

Suborbital refuel is a concept that combines a conventional rocket launch, with a gun-launch system. The idea is to place a “delicate” payload on top of a conventional rocket. The rocket boosts to about 1/2 of orbital velocity and then follows a ballistic, suborbital trajectory. For mass efficiency, the vehicle may drop the empty propellant tanks. Meanwhile, an earth-based accelerator (or “gun”) launches a projectile that contains more propellant and a docking mechanism. The gun launch is also for 1/2 of orbital velocity and along the identical trajectory. The two vehicles quickly dock while well above the atmosphere. The docking mechanism includes fuel lines, so the combined modules can reignite the rocket engine and continue to orbit. The advantage of refueling is that the mass fraction (final mass to initial mass) of the rocket is about four times better than a single-stage-to-orbit rocket. There are many other combinations possible. For example, the gun launched module could include a rocket engine so that hooking up propellant lines can be avoided.

It is important to realize that the vehicles are above the atmosphere during docking, and they are following what is really an orbit. It’s just that the orbit eventually intersects the earth. The docking is therefore similar to a conventional orbit docking, except that everything has to be accomplished in a few minutes.

There is an alternative version of this system that uses oxygen coming down from orbit instead of propellants being launched from a gun. The idea is to use the atmospheric harvesting idea to gather oxygen in orbit. A propellant laden vehicle then partially reenters the atmosphere, bleeds velocity, and skips back up to match trajectory with the rocket from earth.

Schematics of the concept and an initial study are available on the main website. The only study to-date is a calculation of the time available for docking for various initial trajectory assumptions. We also look at the required gun launch angle. I had held off publishing this information because I had meant to do a study of the vehicle masses to demonstrate the potential cost savings. But it’s been so long since I posted I thought it best to put up what I’ve got and work incrementally. Too many fall projects on the old house.