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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.
We'll take Tom Corbett's Polaris. According to the novels it 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.
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.).
How to decide on the interior tankage? Well, I decided to try and sneak up on the problem.
The Polaris has an interior volume of 504 m3. If the entire ship was totally filled with propellant, that would be the upper limit, correct? Uranium tetrabromide solution is basically salt water. Water has a density of 1000 kg/m3. So a waterlogged Polaris would mass 504 * 1000 = 504,000 kg or 504 metric tons. This will be the upper limit of the Polaris' mass, set by volume. (Always be aware that things are simplified in this example since water is one ton per m3. Thing are a tad more complicated with liquid hydrogen, at only 71 kg/m3)
I'll do the calculations for a Polaris that is 30%, 50% and 60% propellant, by volume. Then the most attractive results will be chosen.
First the 50% option. 504 * 0.5 = 252 m3 propellant tankage. 252 * 1000 = 252,000 kg or 252 metric tons. Since the mass ratio is 3, the dry mass is 126 metric tons, for a total mass of 252 + 126 = 378 metric tons.
The Polaris is mandated to have an acceleration of 10 g (98.1 m/s). One NSWR has a thrust of 13,000,000 N. This would result in an acceleration of 13,000,000 / 378,000 = 34.4 m/s. Convert to gs: 34.4 / 9.81 = 3.5 g. Not good enough. Three NSWR have a thrust of 3 * 13,000,000 N = 39,000,000 N. 39,000,000 / 378,000 = 103 m/s. 103 / 9.81 = 10.5 g. That will do. 3 * 10 metric tons per engine = 30 metric tons total engine mass. This will come out of the dry mass capacity.
Now to figure the structural mass. The Polaris has a density of (M/1000) / V = (378,000 /1000) / 504 = 0.75 tons/m3.
The structural volume required to support the spacecraft is = (V4/3 * Apg0 * D) / (1000 * Thm) = (504 1.333 * 10.5 * 0.75 ) / (1000 * 2.86) = 11 m3.
The structural volume needed avoid buckling is = (V1.15 * (Apg0 * D)0.453) / 300 = (504 1.15 * (10.5 * 0.75 )0.453) / 300 = 11 m3.
Since both are the same, the actual structural volume is 11 m3.
We'll make the hull out of titanium. The density of titanium is 4,507 kg/m3 (compared to 7,850 kg/m3 for steel and 1,738 kg/m3 for magnesium) so the structural mass is 11 * 4,507 = 49,580 kg = 50 metric tons.
The available payload (i.e., mass and space for everything that isn't propellant or structure) is 126 mton dry mass - 30 mton engine mass - 50 mton structural mass = 46 metric tons available payload mass. 504 m3 total volume - 254 m3 propellant volume - 11 m3 structural volume = 241 m3 available payload volume.
So the specs for the 50% propellant volume spacecraft are:
The results for the other options are:
33% propellant volume spacecraft:
66% propellant volume spacecraft:
Let's look at the important part:
| Percent Propellant by volume |
Available Payload Volume |
Available Payload Mass |
| 33% | 329 m3 | 23 metric tons |
| 50% | 241 m3 | 46 metric tons |
| 66% | 153 m3 | 50 metric tons |
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.
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.
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. Keeping in mind that four is a little too few for such a long trip.
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 4 crew members and the grand total is 0.52 m3 and 0.52 metric tons.
Each crew member requires 48 litres of air per day. 48 litres * 480 days = 23,040 litres. 23,040 litres * 4 crew = 92,160 litres or 92 m3. Air is stored at 250 bar, so the actual volume is 92 / 250 = 0.4 m3. Air has a density of 1.2 kg/m3 so the mass is 92 * 1.2 = 110 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 * 4 crew = 4,416 kg or 4.4 metric tons. Food has a density of roughly 0.375 kg per litre, so 4,416 / 0.375 = 11,776 litres or 12 m3. This is the bare minimum, increase to raise the crew's morale.
The grand total for consumables is 0.52 metric tons water + 0.1 metric tons air + 4.4 metric tons food = 5 metric tons total. 0.52 m3 water + 0.4 m3 air + 12 m3 food = 13 m3 total.
From this point, you know as much as I do. Do your own research on the volume and mass requirements for other vital pieces of equipment. You can also make an assumption on the separation between decks, and use that to slice the spacecraft into decks. A bit of geometry will give you the diameter of each deck, and you can try to draw some floor plans. Remember to keep them radially symmetric or the ship will tumble.
