As a target of study, NASA has identified the asteroid 1999AO10 as the 2025 destination for human exploration. We've heard that NASA plans to build a giant heavy lift vehicle to make the trip, but is it really necessary?
I previously described a human asteroid mission, but I assumed the logical choice of asteroid with the lowest known delta-v (and included analysis for the second lowest too), but for some reason this isn't as interesting to NASA, so let's consider how one might do the trip to their preferred target, using existing SpaceX hardware.
The reference numbers are: Earth departure stage 3291m/s, and storable propellant 3939m/s of total delta-v. We could improve this by carefully measuring the boiloff of LOX in Falcon 9 upper stages and analyzing the required insulation to do the arrival rendezvous using non-storable propellants, but I don't really have that information handy, so I'll just go with the storables.
Like last time, we'll use the Dragon spacecraft as our crew vehicle. Unlike last time we'll actually have a look at the thrusters, it uses 18 Draco thrusters which are similar to the Aerojet 445 in performance with 309s of ISP. The Dragon is carried to orbit on a Falcon 9 with 3000kg of payload in the "trunk", 3000kg of payload pressurized (that includes the astronauts), and 1290kg of storable propellant, giving a dry mass of around 1710kg. To provide 4710kg (dry mass + pressurized payload) with a total of 3939m/s of delta-v requires a gross mass of 19544kg, which means the external tank will be 13544kg when filled.
The Earth departure stage will be the Falcon 9 upper stage, with the Merlin 1C vacuum performance of 342s ISP. The total initial mass in low Earth orbit is therefore 56710kg. Meaning 37166kg of that is LOX/RP-1. With a mixture ratio of 2.56, that means 10440kg of RP-1 and 26726kg of LOX. (btw, if we had a Raptor stage the IMLEO would be 46328kg, and presumably mass-to-LEO of each flight would be bigger, but the boil-off analysis would be completely different, as you'll see).
Ok, so we have all the numbers we need, now we just have to decide what order to launch it all in to reduce the total mission risk. Storable propellants are called that because they can sit without being used for long periods of time and be ready to go when needed. Also, they don't suffer from boil-off when stored on-orbit, or at least not so much as we need to care in this kind of analysis when compared to cryogenics like LOX. As such, I'm of the opinion that the best strategy is to launch the storable propellants in the external tank, and the RP-1, first.
This is a total of 23984kg and includes some of the mass for the tanks to contain the propellants. So we're looking at two Falcon-9 flights with a shortfall of 3084kg. If an initial parking orbit is chosen well, these first two flights can be spread out over as long a time period as desired, limited only by orbital decay.
We now need to deliver 26726kg of LOX, along with the remaining 3084kg of non-cryogenic propellants, for a total of 29810kg. These three Falcon-9 flights will deliver 1540kg of excess LOX which should account for boil-off if the flights are launched without delays. At 0.1% per day boiloff, the launch campaign must be complete in 50 days. But we have an ace up our sleeve.
The final flight of the launch campaign is the manned Dragon. It will be carrying the crew, with all their provisions, on-board propellant, some of which will be used for rendezvous and orbital assembly activities, and 3000kg of LOX in the trunk. Carrying cryogenics in the trunk may seem risky, but it has such an awesome payoff on cryogenic boil-off that it's worth it.
As described in the NASA concept, the mission departs the low Earth parking orbit, and arrives at the asteroid 3.5 months later. The astronauts stay on the rock for 2 weeks, then return to Earth a month later. The entire trip is 5 months. Radiation exposure is similar to a year long stay on the ISS, about half of an astronaut's lifetime limit.
Because of the suboptimal choice of destination, the new mission has five tanking flights at $56M each, and the manned Dragon flight which I estimate at $150M, for a grand total of $430M. This is about 100 times less than what I expect NASA will spend trying to do the same thing.