Abstract
Jumping longer distances costs exponentially more fuel. In a hypothetical scenario where any amount of fuel can be spent on one jump, there exists a high amount at which outfitting more fuel will harm the jump range, due to the cost of carrying that fuel being greater than the diminishing returns from spending it. This limit is a linear proportion which depends only on the FSD class and the starship mass while main fuel is empty.Result
Where F is the starship main fuel in tons, mₑ is the starship mass while the main fuel is empty and p is a constant depending on the FSD class, the fuel limit is:The reference values for p are:
| FSD class | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| p | 2.00 | 2.15 | 2.30 | 2.45 | 2.60 | 2.75 | 2.90 |
This fuel limit is independent of FSD optimal mass and of any related engineering effects, and the effect of using a Guardian FSD booster is limited to the mass of the module itself, although both of these still affect the range of that maximum-fuel jump. The actual range is beside the main point and can involve some complicated interaction, so a description and example has been moved to Appendix A.
Analysis
We begin with the Hyperspace Fuel Equation, reproduced here with some adjustments for later clarity. Where F is fuel in tons, r is a linear constant depending on FSD rating and SCO model, x is the jump distance in light-years, mₛ is the starship mass including fuel, m₀ is the FSD optimal mass, and p is a power constant depending on the FSD class:Normally, this calculates the fuel needed to perform a jump of distance x. For the purpose of a theoretical all-fuel jump, F is equal to the total starship main fuel which can be adjusted via Outfitting, and the distance x becomes the result. However, adding fuel does not simply traverse that exponential curve, because it also adds to the starship mass mₛ.
This mass requires decoupling; let mₑ be the starship mass while the main fuel is empty, such that mₛ = mₑ + F. Note that the power plant reserved fuel cannot be used for the jump, thus instead is part of the starship mass. Substitute in that separated mass, and separate further the fuel mass F from the distance x:
This we can draw; using values of r, p, mₑ and m₀ for a stock Type-9 Heavy with a 6A FSD, it looks like this:
Clearly there is a turning-point around 700 T where increasing fuel starts to harm the jump range when spending it all. The point where this occurs is when distance change per fuel change reaches zero, which we can examine by using the first derivative. The complete closed-form dx/dF solution is an inelegant expression and has been moved to Appendix B; for the purpose of finding a turning-point, we can use instead a variable-separated derivative:
We seek a point where adding fuel gives no extra distance, i.e. where dx = 0 despite dF > 0 (equivalently, where dx/dF is zero). Only one factor involving F achieves this; it happens only when:
As fuel in terms of mass:
With that scale factor (p − 1)⁻¹, we find that there is a fuel limit directly proportional to the empty starship mass, at which adding more fuel no longer affects the distance of a jump which uses it all. Indeed, because F·p rises faster than mₑ + F and causes the above factor to become negative while all others remain positive, we see that the all-fuel jump range only ever reduces beyond this limit.
The scale factor depends only on the FSD class:
| FSD class | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| p | 2.00 | 2.15 | 2.30 | 2.45 | 2.60 | 2.75 | 2.90 |
| Scale (p − 1)⁻¹ | 1.0 | 0.8696 | 0.7692 | 0.6897 | 0.625 | 0.5714 | 0.5263 |
Using the previous stock Type-9 example, this single-jump fuel limit is 702.5 T, indeed as expected approximately around 700 T. If upgraded to use the stock 6A FSD at no extra mass, this massive jump expenditure reaches 67.188 Ly, or 72.390 Ly with 6A SCO.
Appendix A
Guardian FSD Booster modules are a complicated consideration; they are specified in terms of a jump range increase when the FSD maximum fuel is spent. Because the Hyperspace Fuel Equation is inviolable, the actual means is therefore an equivalent modification to the mass ratio which would cause the stated range increase when that maximum fuel is spent. This notion becomes meaningless in a scenario where the FSD limit no longer exists.The best one can do is to speculate a bit and suppose that the extra range becomes defined in terms of the actual fuel spent, and ignore the problem of small jumps violating the Hyperspace Fuel Equation by reaching zero fuel. For example, consider that a very high-range Mandalay with a Guardian FSD booster now can reach 99.38 Ly for one low-fuel jump. Without the booster, the empty starship mass becomes 263.7 T and its unrestricted fuel limit becomes 181.862 T, which is achievable narrowly to produce a single all-fuel jump range of 229.496 Ly.
With the class 5 booster, the starship mass becomes 265 T for which the unrestricted fuel limit is 182.759 T. This is no longer achievable due to the occupied module slot, although if it were, the base range would be 228.829 Ly and the boosted range would become 239.329 Ly. In practice, module space limits the fuel capacity to 154 T to give a base range of 228.029 Ly and a boosted range of 238.529 Ly, that the booster is still very much worth taking rather than up to 32 T fuel.
Appendix B
In two slightly different forms, each of which look displeasing in different ways, the actual change in distance per change in fuel is:Alongside the stock Type-9 with 6A FSD example and magnified with factor 400, it looks like this:
The negative rate then proceeds to approach zero from below—as it must, otherwise the jump range eventually would reach zero, which would violate the Hyperspace Fuel Equation. Spending a single speck of hydrogen fuel always provides some finite jump distance!
