SpaceX CEO claims he can send you on a round-trip to Mars for $500K

First, there is the rocket equation. That little gem tells us the amount of fuel it takes to achieve a given change in velocity for a given energy density of fuel. Then there are some results from orbital mechanics and aerodynamics that tell us what that change in velocity is. For the curious, escape velocity is just sqrt(2) times the circular speed at any given altitude.

I've laid this out before, but here it goes again. To put a pound of anything into orbit has a fuel cost of a little over $20. "Incredible!," you say, "It costs $10k/lb on the Space Shuttle! How can that be?!" Like so. Typical mass fractions are on the order of 2%. That is, 2% of the stuff on the pad, fuel, structure, payload, everything, actually ends up in orbit. About 12% of that mass is structure, things like tanks and engines and the like. That leaves 86% of the thing as fuel. 86:2 is 43:1. 43 lbs of fuel for every pound of payload. Assuming that propellant is roughly as dense as water and roughly the price of milk, both easily verified, that's under 6 gallons of propellant for every pound of payload, which will run you $21 at $3.50/gallon.

Multiply sqrt(2) by $21/lb and you have something like $30/lb. If you and your capsule weigh 2,000 lbs, That's $60,000 for a one way ticket. A little over 8 times that price may be a reasonable number. So, what makes up the difference in cost for current launch systems, or even for antiquated and clunky systems like the Shuttle? Low safety margins and their concomitant need for enormous administrative costs for each part, disposable launch systems where that administration cost burns up in the atmosphere or splashes down in the Pacific, and enormous system complexity driven by a lowest-flight-weight-results-in-the-cheapest-vehicle mentality.

We can begin to address, based on SpaceX's design philosophy and planned vehicle, how they may be able to make these claims without deserving to be "laughed out of the room."

First, SpaceX has reduced engineering and integration costs by reusing common components and simplifying designs at every step. they were (and I think, still are) using a pintle injector which is much less susceptible to catastrophic combustion oscillations than the more typical injector-face solution, at a cost of some performance. The tanks for all of their stages are the same diameter, allowing them to engineer and build one capital-intensive jig rather than two or three, and they get more experience with that hardware since all their work is done on it. They're using a pneumatic stage sep mechanism rather than a pyrotechnic one to eliminate material-handling, static, and other safery concerns related to pyrotechnics. Rather than relying on one or a few very large engines to power the first stage, they've chosen to use 9 smaller engines on the first stage and isolate each one in its own cato-proof container, again allowing them to gain more experience with a single system, prove its reliability, and leverage that experience and track record to perform a larger job.

Second, they have plans for full reusability of the launch system based on incremental changes to their existing systems. Yes, there is a fuel and performance penalty for going this route, but the savvy armchair aerospace critic will note that those penalties are expressed in tens of dollars per pound, whereas 100% disposal is measured in thousands to tens of thousands of dollars per pound. That is to say, even if reusability results in a 10-fold increase in fuel cost but allows vehicle cost to be amortized to negligibility, we're still approaching Musk's $500k/flight number. As to his actual plan, the fuel cost to land a booster segment is tiny compared to the cost of launching a vehicle. The first stage will simply reenter without having to retro-burn, and the second stage will need to retro-burn just enough to enter the atmosphere to achieve the rest of the braking. After that, the delta-v required is on the order of 100m/s, hardkly the 10km/s needed for orbit. You seem to know what you're doing, so I leave the derivation of that penalty, using the rocket equation, to you.

tl;dr: You're absolutely wrong in the most irrelevant way, and had you addressed SpaceX's achievements and plans in anything like a rigorous way, you could easily have answered your own question.

EDIT: The fuel cost for escape will not be sqrt(2) times the cost for circular speed. The real factor will be something more like 2 or 3, not 1.414... Still, we're in the range of $60/lb, not $6,000/lb.

-Antoine de Saint-Exupéry

I think personalities like Musk are behind the above quotation. A dreamer with demands, and the demands are concrete in the respect that they may be technically possible while still being dreams - be they Mars at 500K or yielding 10x the mechanical properties. I think people are drawn to such personalities because it seems to be the most encouraging way you can possibly frame a dream.

I would actually say, don't underestimate perseverance. Might sound like he is an optimist, but I think you have to be with anything to do with spaceflight. At least at this day an age. I would also consider him an expert in the subject, and what are you?

The idea of producing rocket fuel on Mars is not only a good one, but quite likely the only way such a mission would even be viable. The idea of hauling all that extra rocket fuel for the return trip to Mars and back simply doesn't work.

In essence, let's assume you've made your own fully-shielded space capsule for one, and that you need, for example, a pound of material for every square inch of its surface, and that you've made it into a sphere to economize on material. Wise choices, all of them. A 2m diameter capsule would come in at 19,500 lbs, or so, for that many square inches.

Congratulations. You've just found out why we don't shield small spacecraft. Now, however, let's get an estimate of what it would cost to shield 200 people, providing them each with a volume-equivalent of your capsule, or 4.2ish cubic meters per person. Really packing them in. This is a 11.6-meter diameter sphere, with roughly 3,300 pounds of shielding per person. At 2000 people, packed in like sardines, we're at 25.2 meters in diameter and 1550 lbs of shielding per person. A 54-meter sphere packs 20,000 sardines, and requires a mere 717 lbs of shielding per person.

A factor of 10-20 better. Obviously, we want to relax the space constraints a bit. However, the form factor of shielding in the larger craft is going to be much better suited to the reutilization of supplies as shielding. In the limiting case, of course, there is effectively no required shielding per occupant. Long before that, magnetic shielding schemes become a viable option, too.

A more practical limit would be to consider a transport module made up of re-entry craft seating 1-10 persons, with the ablative heat-shields and supplies facing outward. In this case, assume we can get down to 5 times your cross-sectional area in shielding, and guess that that's about 2.5 square meters. That's 4000/lbs per person. So, still a factor of 5 better than the solo case.

Additionally, this is a lot of arithmetic to ask of google and http://www.calculatorsoup.com/calculators/geometry-solids/sp..., so the numbers may be off. Also, the Dragon capsule checks in at 5.8 m^2 of heat shield per astronaut, which may be a practical limit. And I just made up the 1lb/in^2 shielding requirement.

(100 rem = 1 sievert)

Doesn't that completely depend on the thickness and material of the walls of the spacecraft? I.e., isn't it just a question of making the walls thick enough?

The limits for nuclear workers are extraordinarily conservative, so it may not be as much of a problem as you'd think. Given the safety track record of manned spaceflight, the radiation is probably the least of your worries.

Could this theoretically stop radiation, and protect them? Lead blocks most radiation, specifically gamma.

edit- Parachutes are probably more practical. They work if you make them big enough.

Then send these to Mars, have them remotely operated and autonomous and build a habitable setup there first without humans.

Then ship fuel pods to orbit mars which would allow you to go there with an empty lander - fill it up in orbit then land.

I think we need to take logical steps before we shove humans in a pringles can and launch them to mars.

Well the good thing is that you won't have to do the driving part..

Imagine if some one told you Shangri La actually exists and you have to bear some pain for an year to get there. What would you do? Will you take some pain to get there and live at Shangri La for some days?

I definitely would. And I'm sure many would.

http://en.wikipedia.org/wiki/Olympus_Mons

Mars also has an atmosphere suitable for flying around in with helium or hydrogen balloons. You could mount a camera on a bunch of balloons and get some amazing pictures.

If you don't mind my saying so, "The view is better" doesn't sound like your real reason for preferring the moon.