## By The Numbers

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.

## Dimensions and Volume

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 = 1/3πr^{2}h = 1/3π2.5^{2}23 = **164 m ^{3}**.

Volume of a cylinder = πr^{2}h = π2.5^{2}16 = **340 m ^{3}**.

So the total interior volume of the ** Polaris** is 164 + 340 = 504 m

^{3}.

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 m^{2}

## Mission

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.)*.

## Mass

### Propellant

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 m

^{3}. 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/m

^{3}. So a waterlogged

**would mass 504 * 1000 =**

*Polaris***504,000 kg or 504 metric tons**. This will be the upper limit of the

**mass, set by volume.**

*Polaris'**(Always be aware that things are simplified in this example since water is one ton per m*

^{3}. Thing are a tad more complicated with liquid hydrogen, at only 71 kg/m^{3})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 m^{3} 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.

### Structure

Now to figure the structural mass. The ** Polaris** has a density of (M/1000) / V = (378,000 /1000) / 504 =

**0.75 tons/m**.

^{3}The structural volume required to support the spacecraft is = (V^{4/3} * A_{pg0} * D) / (1000 * T_{hm}) = (504 ^{1.333} * 10.5 * 0.75 ) / (1000 * 2.86) = **11 m ^{3}**.

The structural volume needed avoid buckling is = (V^{1.15} * (A_{pg0} * D)^{0.453}) / 300 = (504 ^{1.15} * (10.5 * 0.75 )^{0.453}) / 300 = **11 m ^{3}**.

Since both are the same, the actual structural volume is **11 m ^{3}**.

We'll make the hull out of titanium. The density of titanium is 4,507 kg/m^{3} *(compared to 7,850 kg/m ^{3} for steel and 1,738 kg/m^{3} for magnesium)* so the structural mass is 11 * 4,507 =

**49,580 kg = 50 metric tons**.

### Payload

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 m^{3} total volume - 254 m^{3} propellant volume - 11 m^{3} structural volume = **241 m ^{3} available payload volume**.

### Specifications

So using the same techniques above, here are the 33% and 60% propellant volume spacecraft compared to the 50% option:

33% | 50% | 66% | |
---|---|---|---|

Total mass | 252 metric tons | 378 metric tons | 504 metric tons |

Propellant mass | 168 metric tons | 252 metric tons | 336 metric tons |

Dry mass | 84 metric tons | 126 metric tons | 168 metric tons |

Mass ratio | 3 | ||

Number of NSWR engines | 2 | 3 | 5 |

Fully loaded acceleration | 10.5 g | 10.5 g | 13.1 g |

Structural volume | 9 m^{3} | 11 m^{3} | 15 m^{3} |

Structural mass | 41 metric tons | 50 metric tons | 68 metric tons |

Available payload volume | 329 m^{3} | 241 m^{3} | 153 m^{3} |

Available payload mass | 23 metric tons | 46 metric tons | 50 metric tons |

## Deciding on an option

Let's look at the important part:

Percent Propellant by volume | Available Payload Volume | Available Payload Mass |
---|---|---|

33% | 329 m^{3} | 23 metric tons |

50% | 241 m^{3} | 46 metric tons |

66% | 153 m^{3} | 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.

For no particular reason, I'm going to go with the 50% option. This gives me 46,143 kilograms and 252 m^{3} 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.

## Life Support

The air won't mass too much. The 50% propellant option has a payload volume of 241 m^{3}. Air at one atmosphere of pressure has a density of 1.2 kg/m^{3}, 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 m^{3}

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 m ^{3}** of water, which will mass

**0.13 metric tons**. Multiply by 5 crew members and the grand total is

**0.65 m**.

^{3}and 0.65 metric tonsEach 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 m^{3}. Air is stored at 250 bar, so the actual volume is 115 / 250 = **0.46 m ^{3}**. Air has a density of 1.2 kg/m

^{3}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 m ^{3}**. 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 m^{3} water + 0.46 m^{3} air + 15 m^{3} food = **16.11 m ^{3} total**.

*From Outrim by J. Mauloni. Another scientifically accurate webcomic. Note the*at the red arrow.*Polaris*

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.

*From the Worlds Fair Futurama exhibit. (1964)**Artist unknown.***Tales from the White Hart**by Sir Arthur C. Clarke (1969)

## Habitat Module

*TransHab module**TransHab modified for artificial gravity**TransHab first floor**TransHab second floor**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.

System | Mass (kg) | Volume (m^{3}) |
---|---|---|

Power System | ||

Batteries | 485 kg | 0.44 m^{3} |

Internal power wiring | 396 kg | 16.40 m^{3} |

Power management | 625 kg | 1.05 m^{3} |

Avionics | ||

Comm | 169 kg | 0.16 m^{3} |

Voice Peripherals | 4 kg | 0.01 m^{3} |

DMS | 35 kg | 0.50 m^{3} |

INS | 39 kg | 0.05 m^{3} |

Attitude Initialization | 6 kg | 0.01 m^{3} |

Displays & Controls | 14 kg | 0.01 m^{3} |

Video | 8 kg | 0.01 m^{3} |

Wiring | 121 kg | 0.25 m^{3} |

Enviro Control & Life Support | ||

Atmosphere control | 1133 kg | 4.67 m^{3} |

Atmosphere revitalization | 1021 kg | 3.25 m^{3} |

Temperature and humidity | 113 kg | 6.32 m^{3} |

Fire detection/supression | 13 kg | 0.05 m^{3} |

Water recov/management | 2199 kg | 6.02 m^{3} |

Waste management | 550 kg | 11.19 m^{3} |

Thermal Control System | ||

Internal thermal control | 135 kg | 0.34 m^{3} |

External thermal control | 167 kg | |

thermal control radiators | 274 kg | |

Crew Accommodations | ||

storm cellar | 1500 kg | |

galley and food | 8063 kg | 91.00 m^{3} |

Wardroom | 194 kg | 6.78 m^{3} |

waste collection system | 327 kg | 8.83 m^{3} |

personal hygeine | 283 kg | 8.83 m^{3} |

clothing | 438 kg | 1.91 m^{3} |

Rec and personal store | 150 kg | 3.00 m^{3} |

Housekeeping | 215 kg | 3.61 m^{3} |

Op supplies/restraints | 120 kg | 0.01 m^{3} |

Maintenance | 1092 kg | 5.91 m^{3} |

Sleep accommodations | 120 kg | 2.82 m^{3} |

Other | 987 kg | 21.81 m^{3} |

EVA Systems | ||

Space Suits | 690 kg | 4.15 m^{3} |

Vehicle support EVA | 291 kg | 0.40 m^{3} |

EVA translation aids | 123 kg | 3.36 m^{3} |

EVA tools | 132 kg | 0.20 m^{3} |

Airlock | 377 kg | 8.18 m^{3} |

Medical Operations | ||

Human research facility | 289 kg | 2.50 m^{3} |

Crew health care | 759 kg | 3.67 m^{3} |

Habitat Total | 22,156 kg | 227.7 m^{3} |

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 m^{3}. Divide by 5 crew members and you get 33.7 m^{3}, which is about twice the spartan bare minimum of 17 m^{3}. You will also note that each crew member has been alloted 30 kg/0.6 m^{3} for personal and recreational items.

## Power Reactor

System | Mass (kg) | Volume (m^{3}) |
---|---|---|

1 MW reactor | 493 kg | ?? m^{3} |

0.18 m dia shadow shield | 356 kg | ?? m^{3} |

reactor radiator | 83 kg | |

Reactor Total | 932 kg | ?? m^{3} |

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

**heat radiators and do some pointless calculations on the maximum power size.**

*Polaris'*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.

## Goodies

System | Mass (kg) | Volume (m^{3}) |
---|---|---|

Payload allowance | 46,143 kg | 252 m^{3} |

Minus | ||

x5 Crew | 340 kg | in habitat allowance |

Habitat Module | 22,156 kg | 227.7 m^{3} |

Reactor | 932 kg | ?? m^{3} |

Remainder | 22,715 kg | 24.3 m^{3} |

Minus | ||

x2 Space Taxis | 2,896 kg | 4 m^{3} |

x2 Space Taxi refuel | 1,650 kg | 1.65 m^{3} |

1 g/cm^{2} armor | 4,330 kg | |

x17 Casaba howitzer | 1,955 kg | 2.4 m^{3} |

x5 Gyrojet pistols +90 rounds | 2.8 kg | |

Remainder (cargo) | 11,881 kg | 16.25 m^{3} |

After deducting the crew, habitat module, and reactor from the available payload we find that there is 22,715 kg and 24.3 m^{3} with which to customize the ** Polaris**.

The Tom Corbett Space Cadet novels mention that the ** Polaris** has a couple of "jet boats." A couple of fully fueled Project Orion 2-person space taxis will cost 2896 kg and 4 m

^{3}. Throw in one refuel for each for an additional 1650 kg and 1.65 m

^{3}.

Armor is a problem, it really eats into your payload mass allotment. One 5 g/cm^{2} *(50 kg/m ^{2})* over a ship with a surface area of 433 m

^{2}is 21,650 kg. This would eat up almost all of our remaining payload. I'll scrimp and only use 1 g/cm

^{2}. That is only 4,330 kg, but the question arises is such a tin-foil like layer of armor even worth it? In the RocketCat universe I have the

*Polaris*armored in muon-metal, which is very dense but very unobtainium.

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 m~~ Just got the latest inside scoop from Scott Lowther. He estimates each Casaba Howitzer charge is only about 115 kg and 0.14 m^{3}. We can just about afford six of them, for a total of 6,900 kg and 2.4 m^{3}^{3}. So **60** of the the deadly little things will fit in 6,900 kg, but only 17 charges will fit into 0.4 m^{3}. So that is **x17 Casaba howitzer charges at 1,955 kg and 0.4 m ^{3}**

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~~ 11,884 kg and 16.25 m^{3} for random cargo.

Now if the Polaris was going to war, it could use the random cargo allotment for 40 more Casaba Howitzer charges. That would use up the remaining 16.25 m^{3} of cargo volume. The ship has enough spare cargo mass allotment to add about 100 more charges, but there isn't enough volume. It ran out of volume allotment before it ran out of mass allotment. In the terminology, it *bulked-out* before it *massed-out*.

With such a limited payload, what is the ** Polaris** useful for besides transporting five people? This probably means that the

**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.**

*Polaris*## Final Design

Dimensions | ||
---|---|---|

Height | 43 m | |

Surface Area | 433 m^{2} | |

Num Crew | x5 | |

Life Support | 3,240 person-days (5 crew for 22 months) | |

Power Reactor | 1 MW | |

Performance | ||

Initial Acceleration | 103 m/s (10.5 g) | |

Engine Type | NSWR | |

Fuel | 22% Enriched UTB | |

Propellant | 2% UTB aqueous solution | |

Exhaust Velocity | 182,000 m/s | |

I_{sp} | 18,600 sec | |

Mass Ratio | 3.0 | |

ΔV | 200,000 m/s | |

Thrust/engine | 13,000,000 N | |

Engine mass | 10,000 kg | |

Num Engines | x3 | |

Mass Schedule | ||

Item | Mass | Volume |

Structure | 50,000 kg | 11 m^{3} |

1 g/cm^{2} Armor | 4,330 kg | |

Habitat+airlock +5 crew | 22,156 kg | 227.7 m^{3} |

Power Reactor | 932 kg | ? |

x2 Space Taxis | 2,896 kg | 4 m^{3} |

x2 Taxis refuel | 1,650 kg | 1.65 m^{3} |

x17 Casaba Howitzer | 1,955 kg | 2.4 m^{3} |

x5 Gyrojet pistols +90 rounds | 2.8 kg | |

Cargo | 11,881 kg | 16.25 m^{3} |

DRY MASS | 126,000 kg | 252 m^{3} |

PROPELLANT | 282,000 kg | 252 m^{3} |

WET MASS | 378,000 kg | 504 m^{3} |

## Polaris Boost and Gravity Turn

*Byron Coffey did an analysis of the Polaris performing a lift-off from Terra.*

I had a bit more time to play launch vehicles, and I managed to track down some MATLAB code that will do gravity turns and basic booster analysis. I decided to plug in your Polaris, and played with the numbers until it went close to being in a reasonable circular orbit:

Trial 1 | |
---|---|

Initial flight path angle | 27 deg |

Pitchover altitude | 130 m |

Burn time | 81.7309 s |

Final speed | 7.15147 km/s |

Final flight path angle | 6.42764 deg |

Altitude | 41.666 km |

Downrange distance | 278.328 km |

Drag loss | 1.30739 km/s |

Gravity loss | 0.160685 km/s |

Apoapsis | 214.15 km |

Circularization Delta-V | 855.4 m/s |

Gravity Drag Factor | 0.200 |

Total Delta-V | 9475.0 m/s |

However, this is grossly overpowered. To avoid going into a ludicrously high orbit, I had to dial the initial angle way down, which is undesirable because of what it does to the drag number, and the fact that you're flying low to the ground with a NSWR. Also, there's going to be more drag after burnout, so the delta-V here is quite low. It turns out that flying a lower T/W gives better overall performance. It looks like T/W 3-4 on launch is about perfect.

I suspect there's still going to be a reasonable amount of drag, but it's not totally outrageous.

Also, lower T/W would mean you need less engine, and can have more other stuff onboard. I'm still trying to work out a way to get the code I have into a form people who don't have MATLAB can use.

T/W = 4: | |
---|---|

Initial flight path angle | 57 deg |

Pitchover altitude | 130 m |

Burn time | 214.544 s |

Final speed | 7.88897 km/s |

Final flight path angle | 0.728657 deg |

Altitude | 64.4822 km |

Downrange distance | 784.973 km |

Drag loss | 0.321213 km/s |

Gravity loss | 0.40909 km/s |

Apoapsis | 193.60 km |

Circularization Delta-V | 54.8 m/s |

Gravity Drag Factor | 0.194 |

Total Delta-V | 8674.4 m/s |

T/W = 3: | |
---|---|

Initial flight path angle | 72 deg |

Pitchover altitude | 130 m |

Burn time | 286.058 s |

Final speed | 7.7951 km/s |

Final flight path angle | 0.914701 deg |

Altitude | 109.762 km |

Downrange distance | 1008.53 km |

Drag loss | 0.123772 km/s |

Gravity loss | 0.700486 km/s |

Apoapsis | 163.61 km |

Circularization Delta-V | 76.0 m/s |

Gravity Drag Factor | 0.250 |

Total Delta-V | 8695.6 m/s |

I can do better than that. I played with the numbers a bit more, looking at a ~200 km orbit, and I've included all three outputs. The code also outputs a couple of plots, but it's going to take a little while to get all three sets of data on the same axes and in a form that might be postable.

T/W = 10: | |
---|---|

Initial flight path angle | 27 deg |

Pitchover altitude | 130 m |

Burn time | 81.5492 s |

Final speed | 7.13266 km/s |

Final flight path angle | 6.43016 deg |

Altitude | 41.5207 km |

Downrange distance | 277.046 km |

Drag loss | 1.30679 km/s |

Gravity loss | 0.160488 km/s |

Apoapsis | 209.71 km |

Circularization Delta-V | 871.8 m/s |

Gravity Drag Factor | 0.201 |

Total Delta-V | 9471.8 m/s |

T/W = 4: | |
---|---|

Initial flight path angle | 57.18 deg |

Pitchover altitude | 130 m |

Burn time | 214.067 s |

Final speed | 7.87434 km/s |

Final flight path angle | 1.05931 deg |

Altitude | 68.8326 km |

Downrange distance | 781.083 km |

Drag loss | 0.306026 km/s |

Gravity loss | 0.419502 km/s |

Apoapsis | 208.23 km |

Circularization Delta-V | 73.1 m/s |

Gravity Drag Factor | 0.200 |

Total Delta-V | 8673.1 m/s |

T/W = 3: | |
---|---|

Initial flight path angle | 72.25 deg |

Pitchover altitude | 130 m |

Burn time | 285.422 s |

Final speed | 7.75005 km/s |

Final flight path angle | 1.55674 deg |

Altitude | 121.122 km |

Downrange distance | 998.612 km |

Drag loss | 0.120114 km/s |

Gravity loss | 0.728703 km/s |

Apoapsis | 206.83 km |

Circularization Delta-V | 133.9 m/s |

Gravity Drag Factor | 0.260 |

Total Delta-V | 8733.9 m/s |

I did a bit more tinkering, and managed to get the code to run each version out to 8 minutes, which meant it was able to account for all of the drag. I've attached the revised plots, and the results. I compensated the gravity drag values for this, so they're only the drag values during the burn, and the factor is only relative to the burn.

Note that these are not necessarily the most efficient trajectories to fly. All involved using 8600 m/s for the initial boost, and then aiming for a ~200 km orbit, with circularization delta-V being the only variable. I'm going to play with lowering the initial delta-V (which was originally a rectal extraction anyway) and seeing if I can find more efficient flight paths with more of the burn made outside the atmosphere. This should improve things some, although I expect that the T/W = 10 will still do poorly.

T/W = 10: | |
---|---|

Initial flight path angle | 27.91 deg |

Pitchover altitude | 130 m |

Burn time | 85.6267 s |

Final speed | 6.93375 km/s |

Final flight path angle | 0.240133 deg |

Altitude | 200.673 km |

Downrange distance | 2987.26 km |

Drag loss | 1.28702 km/s |

Gravity loss | 0.167672 km/s |

Apoapsis | 200.90 km |

Circularization Delta-V | 850.4 m/s |

Gravity Drag Factor | 0.200 |

Total Delta-V | 9450.4 m/s |

T/W = 4: | |
---|---|

Initial flight path angle | 57.2 deg |

Pitchover altitude | 130 m |

Burn time | 214.067 s |

Final speed | 7.82175 km/s |

Final flight path angle | 1.05118 deg |

Altitude | 108.759 km |

Downrange distance | 2837.89 km |

Drag loss | 0.309448 km/s |

Gravity loss | 0.420664 km/s |

Apoapsis | 202.34 km |

Circularization Delta-V | 73.7 m/s |

Gravity Drag Factor | 0.200 |

Total Delta-V | 8673.7 m/s |

T/W = 3: | |
---|---|

Initial flight path angle | 72.23 deg |

Pitchover altitude | 130 m |

Burn time | 285.422 s |

Final speed | 7.71036 km/s |

Final flight path angle | 1.17619 deg |

Altitude | 155.236 km |

Downrange distance | 2470.3 km |

Drag loss | 0.120399 km/s |

Gravity loss | 0.726431 km/s |

Apoapsis | 201.17 km |

Circularization Delta-V | 128.7 m/s |

Gravity Drag Factor | 0.259 |

Total Delta-V | 8728.7 m/s |

I did some tests at reduced initial delta-V. I didn't plot any of them yet, but they should provide some idea of what sort of trajectories are possible. I should add that I was playing with the total time to get the final altitude in the ballpark I wanted it (~150 km). Also, it should be noted that the gravity drag figures are not reliable, as I did not compensate them like I did those above. The drag magnitude is for the entire period of integration, which varied depending on the initial delta-V.

DV = 8000 m/s initial | |
---|---|

T/W = 10: | |

Initial flight path angle | 29.05 deg |

Pitchover altitude | 130 m |

Burn time | 79.7831 s |

Final speed | 6.5801 km/s |

Final flight path angle | 2.84436 deg |

Altitude | 181.712 km |

Downrange distance | 1688.19 km |

Drag loss | 1.04799 km/s |

Gravity loss | 0.371957 km/s |

Apoapsis | 201.60 km |

Circularization Delta-V | 1231.3 m/s |

Gravity Drag Factor | 0.475 |

Total Delta-V | 9231.3 m/s |

DV = 8000 m/s initial | |
---|---|

T/W = 4: | |

Initial flight path angle | 58.9 deg |

Pitchover altitude | 130 m |

Burn time | 199.458 s |

Final speed | 7.20215 km/s |

Final flight path angle | 3.12853 deg |

Altitude | 148.619 km |

Downrange distance | 1376.37 km |

Drag loss | 0.228781 km/s |

Gravity loss | 0.56919 km/s |

Apoapsis | 201.76 km |

Circularization Delta-V | 650.0 m/s |

Gravity Drag Factor | 0.291 |

Total Delta-V | 8650.0 m/s |

DV = 8000 m/s initial | |
---|---|

T/W = 3: | |

Initial flight path angle | 72.95 deg |

Pitchover altitude | 130 m |

Burn time | 265.944 s |

Final speed | 7.07216 km/s |

Final flight path angle | 3.08721 deg |

Altitude | 158.703 km |

Downrange distance | 1078.22 km |

Drag loss | 0.111336 km/s |

Gravity loss | 0.816371 km/s |

Apoapsis | 201.05 km |

Circularization Delta-V | 767.3 m/s |

Gravity Drag Factor | 0.313 |

Total Delta-V | 8767.3 m/s |

At this point, delta-V for the T/W = 3 case is going up, so I dropped it from further tests.

DV = 7500 m/s initial | |
---|---|

T/W = 10 | |

Initial flight path angle | 30.1 deg |

Pitchover altitude | 130 m |

Burn time | 74.8988 s |

Final speed | 6.2009 km/s |

Final flight path angle | 2.39828 deg |

Altitude | 190.956 km |

Downrange distance | 1605.23 km |

Drag loss | 0.895938 km/s |

Gravity loss | 0.403182 km/s |

Apoapsis | 200.88 km |

Circularization Delta-V | 1597.7 m/s |

Gravity Drag Factor | 0.549 |

Total Delta-V | 9097.7 m/s |

DV = 7500 m/s initial | |
---|---|

T/W = 4 | |

Initial flight path angle | 59.66 deg |

Pitchover altitude | 130 m |

Burn time | 187.247 s |

Final speed | 6.66838 km/s |

Final flight path angle | 3.41618 deg |

Altitude | 170.08 km |

Downrange distance | 1314.43 km |

Drag loss | 0.209062 km/s |

Gravity loss | 0.622781 km/s |

Apoapsis | 201.24 km |

Circularization Delta-V | 1158.6 m/s |

Gravity Drag Factor | 0.339 |

Total Delta-V | 8658.6 m/s |

T/W = 4 has started to go up, so it has also been dropped.

DV = 7000 m/s initial | |
---|---|

T/W = 10 | |

Initial flight path angle | 31.27 deg |

Pitchover altitude | 130 m |

Burn time | 70.001 s |

Final speed | 5.82418 km/s |

Final flight path angle | 4.18311 deg |

Altitude | 178.836 km |

Downrange distance | 1175.76 km |

Drag loss | 0.771064 km/s |

Gravity loss | 0.404803 km/s |

Apoapsis | 200.74 km |

Circularization Delta-V | 1994.6 m/s |

Gravity Drag Factor | 0.589 |

Total Delta-V | 8994.6 m/s |

DV = 6000 m/s initial | |
---|---|

T/W = 10 | |

Initial flight path angle | 34.1 deg |

Pitchover altitude | 130 m |

Burn time | 60.1649 s |

Final speed | 4.94637 km/s |

Final flight path angle | 3.67476 deg |

Altitude | 192.213 km |

Downrange distance | 1022.86 km |

Drag loss | 0.574641 km/s |

Gravity loss | 0.478994 km/s |

Apoapsis | 201.32 km |

Circularization Delta-V | 2854.2 m/s |

Gravity Drag Factor | 0.812 |

Total Delta-V | 8854.2 m/s |

At this point, the model is starting to break down slightly. I'm not certain that burns as big as this one can be assumed to be instant, which is the basis for the calculations. However, I will ignore this, as the approximation should still be pretty good.

DV = 5000 m/s initial | |
---|---|

T/W = 10 | |

Initial flight path angle | 37.9 deg |

Pitchover altitude | 130 m |

Burn time | 50.2747 s |

Final speed | 4.05972 km/s |

Final flight path angle | 8.33846 deg |

Altitude | 175.16 km |

Downrange distance | 615.022 km |

Drag loss | 0.424973 km/s |

Gravity loss | 0.51526 km/s |

Apoapsis | 200.65 km |

Circularization Delta-V | 3782.7 m/s |

Gravity Drag Factor | 1.045 |

Total Delta-V | 8782.7 m/s |

DV = 4000 m/s initial | |
---|---|

T/W = 10 | |

Initial flight path angle | 43.7 deg |

Pitchover altitude | 130 m |

Burn time | 40.3299 s |

Final speed | 3.04832 km/s |

Final flight path angle | 9.61331 deg |

Altitude | 184.193 km |

Downrange distance | 475.527 km |

Drag loss | 0.301192 km/s |

Gravity loss | 0.650448 km/s |

Apoapsis | 200.67 km |

Circularization Delta-V | 4786.0 m/s |

Gravity Drag Factor | 1.644 |

Total Delta-V | 8786.0 m/s |

By now, T/W = 10 is going up as well. I assume that the ideal location might be somewhere around 4500 m/s, so I ran that as a last test

DV = 4500 m/s initial | |
---|---|

T/W = 10 | |

Initial flight path angle | 40.45 deg |

Pitchover altitude | 130 m |

Burn time | 45.3091 s |

Final speed | 3.56485 km/s |

Final flight path angle | 8.8907 deg |

Altitude | 180.035 km |

Downrange distance | 548.172 km |

Drag loss | 0.360414 km/s |

Gravity loss | 0.574759 km/s |

Apoapsis | 200.67 km |

Circularization Delta-V | 4273.0 m/s |

Gravity Drag Factor | 1.293 |

Total Delta-V | 8773.0 m/s |

So it turns out that the absurdly high acceleration didn't cost as much as you might think, and is in fact very close to the other T/Ws, although the circularization burn is larger than the initial burn. Overall, though, you're better off with the smaller engine.

-- Byron Coffey

## Poster

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 *(in the Tom Corbett novels they called these "jet boats")*. 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.

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

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

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