Appendix 4: Worked Example

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  Winchell Chung jr. after Jon C. Rogers

Here is a rambling example of how I go about computing an atomic rocket. Beware that I am prone to amateurish mistakes in arithmetic so double check the math before you use the figures.

As a model we will use the classic atomic rocketship the Polaris from the Tom Corbett Space Cadet books.

Actually, the Polaris is so classic (i.e., 1952) that I fear most younger readers have never heard of it. Tom Corbett Space Cadet was an action packed science fiction series aimed at the juvenile demographic, appearing in hardback novels, comic books, Sunday newspaper comic strips, radio serials, and a TV series. Not to mention coloring books, punch-out books and View-Master reels. What is really sad is that the youngest of the younger readers have never heard of radio serials, View-Master reels, and punch-out books either. Sunday newspapers and hardback novels are on the way out, and I'm sure it is just a matter of time before comic books, coloring books and TV series follow them. But I digress.

Anyway, the Tom Corbett books were "inspired" by Robert Heinlein's classic novel Space Cadet (if you like this website you'll probably like this novel). If you are vaguely interested in the the Tom Corbett novels they are out of copyright and are available as free ebooks (but if you are expecting deathless prose you will be sadly disappointed.).

The technical adviser was the legendary Willy Ley who did his best to keep things scientifically accurate but was often over-ruled. As near as I can figure the Polaris in the novels had some sort of closed-cycle gas core nuclear thermal rocket engine.

According to the novels the Polaris is 61 meters tall (200 feet) and 181 metric tons of mass (200 US tons). However, the indispensable Spaceship Handbook, scaling from images from the TV show, say it is closer to 43 meters (140 feet). That seems more reasonable to me.

I'll keep the ship's dimension, but I'm not going to try and keep the Polaris at 181 metric tons. Instead I'll see what mass is implied by the calculations.

Examining the blueprints in the Handbook, the ship has enough of a torpedo shape that trying to figure the volume using the formula for a cylinder probably won't work. The lower part, maybe, but not the top. Using the information from the blueprint about the ogive curves, I'll model the upper part as a cone 5 meters diameter at the base and 23 meters high. The lower part (i.e., to the base of the engine, not to the base of the fins) is approximated as a cylinder 5 meters in diameter and 16 meters wide.

Volume of a cone = 1/3πr²h = 1/3π2.5²23 = 164 m³.

Volume of a cylinder = πr²h = π2.5²16 = 340 m³.

So the total interior volume of the Polaris is 164 + 340 = 504 m³.

We better figure the surface area as well, so we can add armor. Figure the surface area of the cone, then subtract the surface area of a disc with a radius of 2.5 meters (i.e., the base, which is inside the ship). Figure the surface area of the cylinder, then subtract two discs (i.e., the top and the bottom, the top is inside, the bottom has the exhaust). Add the two together, I get a surface area of 433 m²

Now for the Mission. I'll cheat and examine the Mission Chart. The line for the year 2090 is attractive. The next year entry is the start of those huge deltaV Brachistochrone capable engines. Just to keep the Polaris closer to current capabilities, I'll opt for the 2090 engine performance. This means a month and a half transit time to travel to Mars, and seven and a half months to the Asteroid Belt, but that's not too unreasonable. The 2090 specifies a NSWR using a 22% uranium tetrabromide solution (i.e., mostly water). Exhaust velocity of 182,000 m/s, thrust of 13,000,000 N, and 10 metric tons per engine.

I will mandate that the Polaris will have to be capable of acceleration up to 10 g (98 m/s) (so as to reduce gravity drag), and have a mass ratio of 3 (because that is what the Mission Chart assumes.).

Let's look at the important part:

The table makes it clear that there is a trade-off between volume and mass. If you only look at the available volume and mass, I suppose one could use calculus to make a min-max function and find the perfect balance (my knowledge of calculus is not equal to the task, alas). Of course, the other factors are important as well, the accountants will be interested in how much it costs to fill the propellant tanks with uranium tetrabromide.

For no particular reason, I'm going to go with the 50% option. This gives me 46,143 kilograms and 252 m³ of payload to play with.

The available payload volume and mass has to hold everything else. Heat radiators, air, food, water, radar gear, lifeboats, air ducting, sewage treatment, damage control replacement parts, ship's surgery, space suits, crew members, atomic torpedoes, laser cannon turrets, hammocks, periscopic sextant, toothbrushes, toilet paper, everything!. And don't forget the tail-fins.

• 
  From Outrim by J. Mauloni. Another scientifically accurate webcomic. Note the Polaris at the red arrow. Clockwise from the top: Beowulf class free trader from the Traveller RPG, the Serenity from Firefly, the Polaris from Tom Corbett Space Cadet, a Klingon battle cruiser from Star Trek TOS, the DY-100 "Botany Bay" from the Star Trek TOS episode "Space Seed", the Starship Enterprise (NCC-1701, no bloody A, B, C, or D) from Star Trek TOS, the Jupiter II from Lost in Space, the Millenium Falcon from Star Wars, the Outrim Queen from Outrim, and in the center an Imperial Star Destroyer from Star Wars.

The air won't mass too much. The 50% propellant option has a payload volume of 241 m³. Air at one atmosphere of pressure has a density of 1.2 kg/m³, so the mass of air required to pressurize the entire payload section is 241 * 1.2 = 289 kg or 0.3 metric tons. That is just to pressurize the section, more will be required as the crew consumes oxygen.

The air, food, and water for four crew members (Tom, Roger, Astro, and Captain Strong) isn't too bad, even for a 16 month (480 day) round-trip to Ceres. Actually, we should have five crew members, using Raymond McVay's stripped down version of the Mission Control Model. The physical bodies of the crew will take up about 340 kg and 0.34 m³

Each crew member requires 10 litres of water, which is recycled. 0.25 litres will be lost each day due to inefficiencies in recycling, so each crew member will require 10 litres + (0.25 litres * 480 days) = 130 litres = 0.13 m³ of water, which will mass 0.13 metric tons. Multiply by 5 crew members and the grand total is 0.65 m³ and 0.65 metric tons.

Each crew member requires 48 litres of air per day. 48 litres * 480 days = 23,040 litres. 23,040 litres * 5 crew = 115,200 litres or 115 m³. Air is stored at 250 bar, so the actual volume is 115 / 250 = 0.46 m³. Air has a density of 1.2 kg/m³ so the mass is 115 * 1.2 = 138 kg or 0.1 metric tons.

Each crew member requires 2.3 kg of food per day (except for Astro, who can eat enough for three people). 2.3 kg * 480 days = 1,104 kg. 1,104 kg * 5 crew = 5,520 kg or 5.5 metric tons. Food has a density of roughly 0.375 kg per litre, so 5,520 / 0.375 = 14,720 litres or 15 m³. This is the bare minimum, increase to raise the crew's morale.

The grand total for consumables is 0.65 metric tons water + 0.1 metric tons air + 5.5 metric tons food = 6.25 metric tons total. 0.65 m³ water + 0.46 m³ air + 15 m³ food = 16.11 m³ total.

• 
  Artist unknown. Tales from the White Hart by Sir Arthur C. Clarke (1957)
• 
  From the Worlds Fair Futurama exhibit. (1938)

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  TransHab module
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  TransHab modified for artificial gravity
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  TransHab first floor
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  TransHab second floor
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  TransHab third floor

I belatedly realized that I am an idiot, overlooking a resource that I can plagiarize ... er, ah ... research. Back on the Advanced Design page was a section talking about a NASA report on TransHab which included a detailed break-down of the mass budget for a habitat module. It was for a six crew member mission with a duration of 18 months, which is close enough for government work to the Polaris' five crew member 16 month mission. I can cut and paste it into this example and see where it gets me. So ignore all the calculation in the "Life Support" section above. Refer to the original report for more details on each habitat module item.

The paper totals up all the cubic meters under "Crew Accomodations" and uses that as a first approximation for available living space. It comes to about 168 m³. Divide by 5 crew members and you get 33.7 m³, which is about twice the spartan bare minimum of 17 m³. You will also note that each crew member has been alloted 30 kg/0.6 m³ for personal and recreational items.

I'm not sure what the power budget for the Polaris will be. With the severely limited payload allowance, I'm sure it will be impossible to fit in a massive power hog system like a Free-electron laser. Until I can figure out something better, I'll just assume it needs one megawatt. Later I might try to figure out the surface area of the Polaris' heat radiators and do some pointless calculations on the maximum power size.

On the Basic Design page, it describes a 1 megawatt reactor that has a mass of 493 kg. I have no idea what its volume is, though.

We need a shadow shield to protect the crew from the deadly radiation from the reactor. The SPAD has a typical shadow shield as about 3,500 kilograms per square meter of shield. Yikes! For first approximation the shield is a disc with a radius equal to the reactor core. Somewhere (I'll look it up, I promise!) I saw a NASA paper on an ion-drive spacecraft with a 1 MW reactor with a radius of 18 centimeters (0.18 meters). This would make the shield 356 kg. Not sure about the volume, but at those densities it cannot be much.

The heat radiators are external, so their volume does not have to be subtracted from the payload volume. At the link, the quote from Tremolo says "Our current light water reactors have about a 35% efficiency for conversion to electric power." So if the reactor is putting out 1 MW of usable power, it is putting out 1.9 MW of waste heat. Optimistically a radiator is about 0.01 kilograms per waste kilowatt dissipated. So the radiator will be about 83 kilograms.

After deducting the crew, habitat module, and reactor from the available payload we find that there is 22,715 kg and 24.3 m³ with which to customize the Polaris.

The Tom Corbett Space Cadet novels mention that the Polaris has a couple of "space boats." A couple of fully fueled Project Orion 2-person space taxis will cost 2896 kg and 4 m³. Throw in one refuel for each for an additional 1650 kg and 1.65 m³.

Armor is a problem, it really eats into your payload mass allotment. One 5 g/cm² (50 kg/m²) over a ship with a surface area of 433 m² is 21,650 kg. This would eat up almost all of our remaining payload. I'll scrimp and only use 1 g/cm². That is only 4,330 kg, but the question arises is such a tin-foil like layer of armor even worth it?

Finally, we have to have some kind of weapon. But we cannot afford much. A single Trident missile is a whopping 58,500 kg, which is way over budget. Casaba howitzer charges are low mass (we conjecture, it's still classified). My best guess is each charge is about 1,150 kg and 0.4 m³. We can just about afford six of them, for a total of 6,900 kg and 2.4 m³

And just for the heck of it, issue each of the crew members a Gyrojet rocket pistol with 18 rounds (3 magazines). Hey, if a rocket gun is good enough for Buck Rogers, it is good enough for the Polaris crew. Each pistol is 0.4 kg, and each round is 0.009 kg.

This will leave a paltry 6,939 kg and 16.25 m³ for random cargo.

With such a limited payload, what is the Polaris useful for besides transporting five people? This probably means that the Polaris design should be scaled up a bit. The habitat and reactor mass is fixed, scaling up the ship will give more remainder payload to add goodies.

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  Poster preview, click for larger image

Shameless plug, I made a poster of the Polaris, you can purchase it here.

The details are more or less the ones calculated for the 43 meters tall Polaris. This is the shorter of the two Polaris versions, but as you can see it is still freaking tall. The door on the tail fin at the top of the ladder is 2 meters tall, just to give you some idea of scale. The basic outline is as drawn by master researcher and blueprint draftsman Jon Rogers. The ship's skeleton was designed by an engineer who goes by the handle JanJaap. Additional engineering expertise was supplied by Rob Davidoff. The crew is based on the budget version of the Mission Control Model proposed by Rick Robinson and developed by Ray McVay. The structural girders are painted a splendid bilious zinc chromate yellow-green. They are also full of circular lightening holes. There is an over-sized astrodome for navigation purposes. The life support deck has huge glowing green tanks filled with spirulina algae. The magazine contains six deadly missiles with Casaba howitzer warheads, ready to skewer hostile warships with swords of pure nuclear flame. The hangar bay carries two space taxis. And the propulsion system is Zubrin's outrageous continuously-detonating Nuclear Salt-Water Rocket.

It is amazing what one can create with an open-source CGI editor application like Blender, and a lot of work. Plus a little help from your friends, and from the website you wrote.

This gives you an idea of the detail. The 300DPI shows how fine the detail is, the 85DPI is about the actual size it will appear on the printed poster.

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  View looking down towards the tail
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  Astrodome with navigational telescope and protective metal shutters

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  View towards the algae tanks on the life support deck
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  Detail of the girder structure of the ship decks

This is a selection of the ship scenes in the lower left.