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The only problems are(1) what kind of propellant,  and (2) who makes that propellant.  Designing the rocket stages themselves is no real problem,  especially if you can cartridge-load pre-made propellant grains in sleeves.  Those sleeves can double as your case insulation.  The biggest design problem to overcome is adequate vehicle attitude control,  same as von Braun faced at Peenemunde long ago.

As for what propellant,  something primitive like fireworks propellants or sugar nitrate has too low an Isp to be useful:  something in the range 80 to 180 sec Isp.  It takes a "real" propellant to do a launch job.  Even the AN composites and the plain double base propellants are too limited at 200-220 sec Isp.  It really takes either a composite-modified double base that uses AP not AN,  or an AP-composite to get enough Isp to serve.  These fall in the range of 240-260 sec sea level Isp. 

Propellants like that are very dangerous to make,  especially the double base,  because you have to handle raw nitroglycerin.  Even the AP is quite dangerous,  because all by itself it is a class 1.1 mass detonable material. The folks who do this for a business do this with remote operated equipment.  The danger is just too great.  It is not a job for amateurs. 

But,  there are folks who make such propellants for the amateur rocketry folks.  Some of it is AP-composite,  too.  High dollar stuff,  but that's what you need. 

A part of the design is chamber pressure:  both thrust coefficient and Isp are greatly improved at higher chamber pressures.  I'm unsure what pressures the amateur rocketry propellants are used at,  but it's likely under 1000 psia.  Most of the real tactical size rockets run in the 2000-4000 psia range.  It was only square-cube law effects (steel is only so strong) at giant sizes that reduced the shuttle SRB's to 900 psia.  A 20-inch diameter case made of D6ac will hold 2400 psig at a wall thickness of about 0.12 inches and 10% below ultimate tensile in hoop stress.  It'll do that maybe once to 5 times.  Smaller,  and wall thickness is thinner. 

That kind of thing isn't simple roll and weld stuff,  it's thick wall seamless tube in the annealed state cold-rolled to shape and then very carefully heat treated to strength.  According to Mil Handbook 5,  the ultimate tensile strength (if everything is done to spec) can be 220 ksi,  with 190 ksi yield.  D6ac is a semi-austenitic martensitic high-alloy stainless.  Fairly easy to form,  machine,  and weld. 

GW

Space X has just updated the data about the Falcon Heavy here: http://www.spacex.com/falcon-heavy . The payload numbers have been increased almost 20%!

Payload to LEO: 63,800 kg
Payload to GTO: 26,700 kg
Payload to Mars: 16,800 kg
Payload to Pluto: 3,500 kg

This means that two Falcon Heavies have about the same payload (127.6 tonnes) as the Saturn V (118 tonnes originally, 140 tonnes eventually). Mars Direct was designed based on a 140 tonne to LEO rocket able to throw 40-46 tonnes to TMI (depending on the delta-v needed). So the Falcon Heavy is getting rather close to those numbers, and of course it is a LOT cheaper; at 90 million per launch ($180 million for two), it can put almost as much into LEO as a Saturn V, whose launch in today's dollars would be over a billion dollars. I would not be at all surprised that a few tweaks (a larger second stage, for example) wouldn't push a Falcon Heavy's payload up to 70 tonnes. That's only another 6%!

From what I have read on VASIMR, the thruster uses microwaves to produce cold plasma, which is then accelerated magnetohydrodynamically.
This concept could work extremely well if the propellant were wastes or unprocessed natural materials, such as lunar or asteroid regolith or material gathered from the Martian moons.  This would make VASIMR a usable propulsion system for early missions to establish bases on the moon, asteroids or Mars and should result in greatly superior mission mass ratios.
A return mission to the moon could begin by launching a single VASIMR tug, which would function as a lunar transfer vehicle and a reusable SSTO lander.  Upon return from the lunar surface, the lander would carry enough lunar regolith to propel the transfer vehicle back to LEO, where it could rendezvous with the ISS and for a return trip to the moon.  The lander would remain in lunar orbit awaiting the next mission.
Subsequent missions would only need to lift the crew, food and water, propellant for the lander, spare parts and any lunar surface payload.  This would improve the economics of maintaining a lunar base.
One question would appear to be: Can VASIMR function using refractory oxides as a propellant?

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Thanks for that analysis. It is notable that the SpaceX abort system also has a similar low ISP and requires also in the range of 1,000 kg of propellant. They solve the CG/CP positioning problem by having a long trunk. And on at least the SpaceX abort test they also put fins on the trunk. This may have been to help insure a straight flight but it would also help to bring the CP further rearward.
So the simplest answer may be install a trunk on the Blue Origin capsule to bring the CP rearward.

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It takes 4 of these to meet the 80,000 lbth for 6 sec spec.  They're not very large size.  If you only ever combine 4 motors from the very same mix,  they will always have exactly the same burn time.  Given a reliable,  repeatable igniter,  they will always burn out within millisec of each other. 

There's no way to put this type of grain design into something 12 feet in diameter.  It's some other grain design.  I don't know what that might be. 

GW

Solids don't scale,  they have to be designed.  Otherwise,  somebody's expensive thrust stand is going to get blown up,  and maybe somebody killed. 

What I read just above says you want 80,000 lb thrust for 6 sec.  The total impulse of that is 480,000 lb-sec.  If the specific impulse is in the vicinity of 245 sec (a realistic goal to be exceeded,  if possible),  then the propellant mass required is about 1959 lbm. 
That is a big motor.  The as-cast volume of the propellant would be about 31,100 cu.in,  if the propellant density is 0.063 lbm/cu.in. That desity and Isp goes with an AP-HTPB-Aluminum propellant.  .2 to .7 in/sec is an achievable burn rate at 1000-1500 psia with that material. 
If I was to attempt a keyhole slot at L/D ~ 4,  cross sectional loading could be 80%.  The insulated inside volume of the case could be about 39,000 cu.in.  That would be around a 23 inch dia case ID,  and about 93 inches long inside.  Web fraction might be as high as 75%,  for a propellant web of 8.66 inch.  That must burn in 6 seconds,  for a burn rate of 1.44 in/sec.  Very few propellants are practical with burn rates that high. 
What that means is I need a longer L and a smaller ID with that keyhole.  Try L/D = 6.  Get ID = 20 inch.  Web = 7.5 inch.  Burn rate = 1.25 in/sec,  still too high.  Longer L/D is impractical with the keyhole,  as it gets too progressive. 
We could try a finocyl type design,  with which I had less experience.  To get around 0.5 in/sec burn rate,  we'll need a web of about 3 inches.  We'll need lower web fraction ~.67 and average cross sectional loading ~.65 for this type of design.  We will need about a 9-inch ID motor that is somewhere around 940 inches long,  which is not practical.  Too much void space and too small a web needed. 
I would have to think about this.  The really long multi-motor they used for the rocket sled may not be so ridiculous.  Perhaps I could go back to a high volume-packing L/D = 5 keyhole if I was to cluster multiple motors together.  Dunno yet. 
GW

Apollo LM, museum display. You realize the grey cylinder on the floor of the back part is the cover for the ascent engine.
https://www.hq.nasa.gov/alsj/LMSimulator.jpg http://farm3.static.flickr.com/2643/408 … d84685.jpg http://www.hq.nasa.gov/pao/History/SP-350/i4-4a.jpg
Still. This is for space tourism. The New Shepard abort engine is "in your face".

Wow what ever happened to this program?