Selecting an optimal Isp / what makes an Isp too high (mostly for high Isp)

Question

When, and why, is it possible to have an Isp (usually of an ion or nuclear engine with Isp far beyond the limits of chemical fuel) that's too high? What makes the Isp too high, and (other things being equal) why wouldn't you just want the maximum possible propellant efficiency so you can have plenty of delta-V without your vehicle being a flying fuel tank?

(This is assuming that it is fundamentally possible / affordable to be capable of a very high Isp. Obviously there is no point in the expense, hazard, and minimum size of a nuclear rocket when a cold gas thruster will easily achieve the needed delta-V. This is also not about the basic case of high-Isp thrusters often having too little thrust for planetary takeoff or other high-thrust needs.)

Answer 1

The concept of "optimization" typically means "given the constraints". This question isn't really answerable definitively, but the general idea to your answer would be: "The ISP is too high if the thruster you chose has other, undesirable properties". If all else really is equal then there is no practical reason one wouldn't want the highest fuel efficiency possible.

Some of these constraints you seem to be uninterested in - an ion engine with super high ISP that cannot take off from Earth has "too high" an ISP but that's specifically because all else is not equal.

So to attempt to answer the question, I would argue that the ISP is too high if the tradeoffs you have to make mean you cannot accomplish your mission goals. Otherwise I think you need to clarify the question.

Answer 2

The tradeoff with specific impulse is energy: the higher the exhaust velocity (or, equivalently, the specific impulse), the more energy it requires. This is a direct consequence of the formula for kinetic energy: $k_e = 1/2mv^2$. The tradeoff mostly doesn't matter for chemical fuels, since those contain their own energy, but for ion thrusters and other externally-powered engines, it's critical.

With these engines, although the mass of your fuel tank goes down as your specific impulse goes up, the mass of your power source also goes up. At the extreme high end, an ideal photonic rocket has a specific impulse of 30,570,000 seconds -- and draws 300 MW of power for each Newton of thrust. A one-Newton thruster, sufficient to lift a single ISO-standard apple under Earth gravity, would need to be powered by the reactor from a Nimitz-class aircraft carrier.

Answer 3

There are several perspectives to look at this from, and a good number of different assumptions to consider -- some of which are more realistic than others, and some of which are heavily influenced by your mission design and the economic and organizational / operational setting the mission is designed in -- a realistic science-fiction-style space opera tramp freighter has very different concerns from a modern day communications satellite, which has different concerns from an expendable Mars probe, which has different concerns from an expendable probe meant to travel to the outer planets in a short period of time, which has different concerns from a minimal mass Kuiper Belt or interstellar probe, which has different concerns from a "space tug" or similar vehicle.

Several principles are dictated by the mathematics of rocketry:

• Based on the Rocket Equation, required mass ratios are small when delta-V is significantly less than V_e, reasonable (about 2 to 4) when delta-V is equal or a bit more than V_e, and impractically large when delta-V is larger than V_e, necessitating staging.

• Achievable delta-V varies linearly with Isp / V_e, while thrust power (which is both the lower limit on energy needed to drive the engine and also a good basis for heat dissipation and energy-handling ability needed) varies as the product of thrust and the square of V_e.

• Both propellant and power sources have mass and cost money, and so does fuel for generating power if your power source uses fuel (such as uranium fuel rods for a fission reactor, or deuterium for a fusion reactor.)

• Propellant probably requires tanks which have at least some mass.

• There's a limit to how much power you can get out of a given power source per unit mass. (for open-cycle devices like chemical rockets and nuclear thermal rockets, this limit can be incredibly high). In space, this also tends to be limited by heat radiators. Very often this limit is nowhere near high enough to drive the engine you would like. For some reason, practical solar panels and practical nuclear reactors both seem to be on the rough order of 10-20 kg per kilowatt.

• Our perceptions of the mathematics tend to be a bit skewed by the very low V_e of chemical rockets and also the fact that chemical rockets combine the power source and the propellant intrinsically. However, it's easy to ignore the need for a power source when considering more advanced propulsion methods such as ion engines.

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There are two basic ways that an Isp can be too high:

1. It wastes energy that you could have saved at the cost of some more propellant.

2. (relatedly) it requires too much energy to be a good fit for available power sources.

3. (also relatedly) To get a level of thrust appropriate to the mission, an infeasible amount of power density is needed.

Let's look at a notional ion-drive nuclear-powered ship with variable Isp, and a 10-ton engine (not including power reactor) which can process 10 MW of thrust power at 75 percent efficiency.

Let's also look at the reactor: if we go with 10 kg/kW, this reactor will weigh about 133 tons. Let's say it's 50 percent efficient.

Let's give the ship 10 tons of structure and tankage (probably optimistic), 20 tons of payload, and 519 tons of propellant, rounding it out with a wet mass of 692 tons and a dry mass of 173 tons.

Let's look at this operating at an Isp of 30,000 S (V_e of 294300 m/s):

Thrust: 68 N Initial Accel: 9.82E-5 m/s^2 Delta-V: 408 km/s Time to accelerate that delta-V: 71 years Time to accelerate 5 km/s: 589 days Propellant expended for 5 km/s: 12 tons Energy used for 5 km/s: 6.9E14 J Amount of uranium fuel burned for 5 km/s: 16.6 kg

That's a lot of delta-V, but it takes forever just to use a small part of it such as one would use to make a cislunar or interplanetary transfer that shouldn't even take more than 2 years! Meanwhile the vast majority of this ship's dry mass is a huge nuclear reactor -- and it burns over 16 kg of uranium fuel.

This ship makes sense if you're doing a very high delta-V, very long duration mission, probably into the Kuiper Belt, or a series of outer-planet transfers that take years at a time, without any ISRU.

If you had a reactor that was vastly lighter for the power produced, you could make this ship much faster -- but there's only so far you can do that until you reach torchship territory where you are struggling hard against the ability to handle absurd amounts of thrust power.

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Let's look at the same ship operating with an Isp of only 3000 s (29430 m/s), with the same thrust power and a much higher propellant consumption:

Thrust: 680 N Initial Accel: 9.82E-4 m/s^2 Delta-V: 40.8 km/s Time to accelerate that delta-V: 260 days Time to accelerate 5 km/s: 54 days Propellant expended for 5 km/s: 108 tons Energy used for 5 km/s: 6.24E13 J Amount of uranium fuel burned for 5 km/s: 1.5 kg

That's very different: By reducing the Isp by a factor of ten, you get a ship that can instead make a couple large interplanetary transfers without ISRU and with burns measured in weeks rather than months, and also use a factor of 10 less expensive uranium fuel for the initial 5000 m/s burn. The nuclear reactor is still very large and still the majority of the dry mass of the ship (and this is a somewhat optimistic specific power).

This isn't universally applicable -- consider that if you only need 5000 m/s and the ship can be expendable or have a high mass ratio and arrive empty, you can make this much faster and cheaper with chemical fuel. Meanwhile, if you have access to a very high specific power source with cheap or free fuel (possibly beamed power?) you might as well just use high Isp and avoid having a high mass ratio. (Remember, however, that a present-day jet airliner has a mass ratio around 2, so moderately high mass ratios don't necessarily mean flying fuel tanks). If you're doing something like transport ISRU-sourced fuel over a long delta-V chain to get it in the place needed and can take your time, you might want very high Isp ion thrusters so as to avoid a huge multiplier of the amount of fuel needed at the source. However, this should provide a pretty clear example of a case where an Isp can be much too high.

Answer 4

The ISP is too high when the top line (mission cost) busts the budget. You said “usually of an ion or nuclear engine”. While no true nuclear engine (primary propulsion) has launched (that we know of), plenty of missions have turned down ion engines because they wouldn’t pay for the power management and high- (…very) voltage apparatus. Which has knock-on effects, like testability, worker safety, lower supplier base and fewer fallback options, etc.

Meanwhile, some non-ion solution can be found for the mission in question. Since this is a technology leap, not an incremental version, it’s no mental leap to A/B test the two mission concepts before one leaves paper, and starts spending hard money. If the A/B test is still inconclusive, most humans go with their gut: no top line gambled on the technology leap.