This answer is a hard-science expansion of this answer. Please read that other answer to get a description of the system I am proposing, as well as justification of its technical feasibility. That post also has lots of reference links for various design decisions. I will summarize the system here and numerically address the questions posed.
System summary
The power source is a pebble bed fission reactor. The fuel source is uranium nitride pellets coated in a pyrolitic carbon moderator. These fuel pellets are held in molybdenum 'pins' in a geometry that will make them supercritical if a neutron reflector is placed outside the reactor. Heat exchange is done directly with the working fluid to save mass.
The working fluid is helium, which is passed through the reactor core. Electrical power is generated through a Brayton-cycle turbine similar to a marine gas turbine used on ships, except replacing the combustion chamber with the reactor core. The helium is compressed by a compressor coupled to the gas generating turbine into the core, and then allowed to expand over the gas generating and power turbines. Exhaust will still be at ~700 K, and will then be run over various auxiliary systems to utilize this extra energy. The exhausted gas will then have its remaining energy bled off into space through heat exchangers and then fed back into the compressor. The rotational power generated by the power turbine is then coupled to an electrical dynamo to generate power for the vessel.
The main propulsion system is a magnetoplasmadynamic Lorentz Force Accelerator (LFA) arcjet thruster. Lithium fuel is ionized and fed into an acceleration chamber, where a combination of magnetic and electrical fields are applied. The induced current in the plasma, once the input power is in the MW range, will help maintain the magnetic field in the plasma while will then induce an electric current in a tungsten-barium cathode.
System Specifications
The reactor must produce 300 MW of heat energy. This is possible from a pebble bed reactor, the Chinese are building a pair of production 250 MW pebble bed reactors at Shidao Bay. From this thermal energy, gas generating turbines produce an output of 100 MWe at 33% efficiency. This is equivalent to the power output of 4 GE LM2500 marine gas turbines, which is the same energy source as an Arleigh Burke-class destroyer. The LM2500 has efficiency of about 40%, but we are losing efficiency due to the reactor core being cooler than a typical combustion chamber (our core is ~1750 K compared to ~2250 K in a marine gas turbine). The overall system mass estimate for the power generation portion is 0.4 kg/KWe (based on a NASA estimate), or 40,000 kg.
The size of the MPD thruster is much more conjectural, as no thruster of nearly the size required has been built. I have estimated the characteristics from the information available at the EPPD laboratory at Princeton. This design calls for a single 7.5 kN thruster at a fuel usage rate of 0.5 kg/s with an ISP of 15 km/s. There is an available high ISP mode where thrust drops to 1 kN at 0.01 kg/s with and ISP of 100 km/s. The mass of the thruster unit is 10,000 kg. I honestly do not have an good basis for this estimate, but it is needed to proceed.
Reactor Safety
The pebble bed fission power system is inherently safe. There are several avenues for a nuclear accident, the two most significant being an overpower casualty (Chernobyl) and a loss of coolant casualty (Three Mile Island, Fukushima).
An overpower casualty is not physically possible for a pebble bed reactor. The fuel source will use low-enriched Uranium, enough to achieve critical mass, but low enough that there are significant interactions between U-238 and neutrons in the core. As temperature of the fuel pellets increases, U-238 is affected by doppler broadening, causing it to absorb more neutrons. This lowers the number of neutrons available to cause fissions in U-235,thereby lowering the reaction rate and reducing power input. Therefore, the core is naturally moderated at an upper temperature controlled by the U-235/U-238 ratio, which will be engineered at 1750 K. At temperatures below this, with the reflectors (to be discussed later) in place, the temperature will increase to 1750 K. As fluid flow over the core is increased and heat removal increases, the reaction rate will increase to keep temperature stable, and this power output is naturally controlled by demand. At temperatures above 1750 K, power output will decrease due to U-238 absorption until temperate settles back at 1750 K. Therefore, there is no human or computer based control of the reactor. Once started it simply outputs energy at the rate heat is removed from the core, moderating itself at 1750 K. This effect is trustworty; computer modeling in Strydom, 2004 indicates that the uncertainty band during a loss of forced cooling casualty will amount to less than 100 C even for a reactor shutting down from full power.
As an aside, we should discuss the way that the reactor is started and stopped. In the core's state as built, it is sub-critical. The core will be undergoing fission at a very low rate, but too many neutrons will be lost passing out of the core for a chain reaction to occur. This is changed by surrounding the core with beryllium reflectors. Once these reflectors are positioned in place, they reflect neutrons back into the core, as well as helping to moderate the high energy neutrons produced by fission. As a result the core will be super-critical and increase temperature until the upper limit described in the last paragraph. By removing the beryllium reflectors, the core can be shut down.
A loss of coolant casualty is the most dangerous remaining one. However, and simplest strategy for this risk is to ignore it. On Earth, reactor casualties are costly because they leave radiation that no one wants to deal with. In space, probably no one cares. Sure, you lose the ship, but people shipped plenty of things in the Age of Sail while the risks of losing the ship were great. Transportation in space has more in common with the Age of Sail, what with month long travel times and low cargo capacities, than it does with modern shipping.
System complexity
As described above, there is no requirement for control systems for the reactor itself, only the activation of one safety system in case of emergency (removing the reflector for shutdown). The emergency heat removal system will be self activating.
The Brayton cycle gas generators will be designed to operate continuously for the duration of a mission. Already, ships at sea using marine gas turbines operate for 1 year + without the turbine enclosure or electrical generator enclosure being opened. The conditions at sea are far more challenging than space, what with salt and water both present. Long term maintenance can be performed at a (space)port between missions. Furthermore, the advantage of operating multiple turbine units in parallel is that the thruster will still be able to fire (if at a reduced power level) if turbine are offline, even when only one turbine is operational.
The MPD thruster is, again, the least developed part of this plan and the most conjectural, so I cannot make any statements about its reliability. However, it does have the advantage of no moving parts; power is generated and transferred through the movement of gas, current, and electromagnetic fields.
Power and Fuel Efficiency
Given the above specifics, we can calculate some burn times and travel times. Here is a list of delta-v needed for various Hohmann transfers.
Tsiolkovsky's rocket equation is solved for fuel mass, $m_f$, by
$$m_f = m_0\left(\exp{\left(\frac{\Delta v}{v_e}\right)}-1\right).$$
Our parameters are $m_0$ (mass without fuel) is 50,000 kg plus cargo size; and, $v_e$ is either 15,000 m/s or 100,000 m/s depending on operating mode of the thruster.
The burn time can then be calculated by dividing fuel expended by mass flow rate. The mass flow rates are given as 0.5 kg/s or 0.01 kg/s, depending on the operating mode of the thruster.
Below is a table for required fuel mass and burn times for various configurations. A 3.0 delta-V will get you to Mars or Venus, 8.8 delta-V to Jupiter, and 12.3 anywhere in the Kuiper belt:
Cargo (tons) deltaV (km/s) V_e(km/s) Fuel(tons) Burn(days)
1000 3.0 15 232 5
1000 3.0 100 32 37
1000 8.8 15 838 19
1000 8.8 100 97 112
1000 12.3 15 1334 31
1000 12.3 100 137 159
10000 3.0 15 2225 52
10000 3.0 100 306 354
10000 8.8 15 8020 186
10000 8.8 100 924 1070
10000 12.3 15 12769 296
10000 12.3 100 1315 1522
100000 3.0 100 3047 3527
100000 8.8 100 9203 10652
100000 12.3 100 13095 15156
A few things to note. The optimal burn profile (how long to burn thrusters in which mode) is still an open question. I posted a question about that using similar numbers to this answer, but didn't get a great answer. I might take a stab at that question again later. The reason you have to calculate the optimal burn profile is that fuel has a cost. If you are moving 100,000 tons of raw lithium from Mars orbit to Earth orbit, not only does your burn take 10 years, but you also burn 13,000 tons of refined lithium doing it! That makes it seriously questionable whether moving bulk cargoes is going to be profitable in your solar system. Also note that the above calculations use a 100% fuel burn; you aught to leave at least something in reserve, which cuts further into your fuel efficiency.
I didn't post the scores for using the 15 km/s mode with cargos of 100,000 tons, because the fuel usage is ridiculous. As it is, those numbers are in tons of lithium fuel. Keep in mind world lithium reserves are estimated at about 34 million tons, so you can see how you'd burn through that quickly.
A big open question with this process is the availability of lithium for fuel. If it can be mined in commercial quantities from space rocks, then that sort of operation would be the equivalent of petro-states here on Earth. It may be possible to use alterative propellants, though there would likely be a loss in efficiency. Neon, Argon and Xenon are not very common, either, but hydrazine is another possible propellant. It could be that hydrazine refining in the orbit of the gas giants is the oil refining of your near-future solar system.
Conclusion
Here is a system for space propulsion that provides a reasonable ability to traverse the solar system using technology mostly already demonstrated today. The big exception is scaling up the magnetohydrodynamic propulsion system to kN power levels.
Most burns that you might imagine for a sublight space opera set in the solar system are feasible. Cargo capacity is relatively low, with the 100,000 tankers (roughly the size of large container ships today) being probably unfeasible for fuel cost reasons. Taking 1000 tons of cargo from Earth to the Kuiper Belt isn't that inefficient; you must burn 14% of your cargo mass in fuel, and the burn takes half a year, but what is half a year compared to the decade or more it will take to coast there?
Meanwhile, a quick hop to mars could be done in relatively fast time. If you skip a Hohmann transfer orbit and try something else, you could burn more fuel to get somewhere faster. For example, a max burn from Earth orbit with 1000 tons of cargo and 1000 tons of fuel in the high thrust mode can get you to Mars orbit in a matter of days. Of course, the problem is you have to stop. The point I'm trying to make is that for the lower delta-V transfers at lower distances, this spaceship is powerful enough to ignore Hohmann transfers and attempt some other orbital transfer that requires more energy. Now what that transfer might be sounds like the subject of a future post :)
