Seeking chart of nebula particle density vs. ship velocity

Question

The traditional Sci-Fi nebula is thick as ketchup (I'm lookin' at you Star Trek!) but the reality of nebulae we know about is that a pilot wouldn't even notice that they're in one due to particle sparsity unless the computer told them (which means the closer you get to a nebula, the more it vanishes... that's really cool if you think about it). I suspect most people don't realize that the seemingly opaque and beautifully colored nebulae we look at are that way because one is looking through lightyears' worth of volume.

So here's the question. Can someone provide a graph of particle density vs. maximum ship velocity so worldbuilders have an idea of what they can do with a nebula?

• The ship is a sphere measuring 1 km in diameter.
• The Y axis is particle density. Non-relativistic particle density. The kind of solid dust that makes up nebulae in the first place. Energy passing through the nebulic space is not a part of this question. (We're starting to strain at gnats, folks.)
• The X axis is maximum ship velocity.
• The ship has no active shields. The leading hemisphere of the ship is 10-meter-thick titanium.
• The fact that real nebulae have variable particle densities is to be ignored.

One assumes that there comes a time where either particle impact damage or heat from the passage will prohibit greater velocity. Based on this chart, worldbuilders should be capable of deriving nebula opacity at various distances. (Think "visibility" from an airplane pilot's perspective.) However, providing that kind of reference is not part of this question.

Answer 1

Your ship's frontal area is actually big enough that a real nebula would actually create meaningful drag! I tried to make a graph in Wolfram Alpha, but I had to remove all of the units, so I'll show my work and let you check if I kept my units correct.

For starters, let's begin with the ram pressure equation, which is:

P = 0.5 * ρ * V^2

P is the ram pressure (in pascals or N/m²),
ρ (rho) is the density of the fluid (in kg/m³),
V is the velocity of the fluid relative to the object (in m/s).

For a nebula with 10000 particles per cubic centimeter, assuming each particle is a proton, the density is 1.672621911×10^-17 kilograms per cubic meter. Velocity is our x variable, and if we multiply that by the frontal area of your 1km diameter ship, which is 785398 square meters, we can get a drag force, which would be y.

If you want to vary particle density with a second variable, I have no clue how to graph it, but it is a linear term, at least.

Here is Wolfram Alpha finally making a graph if I take out ALL of the units.

Here is Wolfram Alpha graphing with a gamma correction, note the asymptote at c.

Anyway, if you wanted to see drag in newtons which your ship would have to match with its own thrust to maintain constant velocity, as a function of it's velocity in meters per second, here you go, sans units:

Please note, however, that technologies exist which can make the plasma of space useful, that being the so-called "q-drive" or "plasmadyne". To briefly summarize, it lets you make a plasma magnet windmill to create power to drive an ion engine or other electric rocket without needing the weight of a reactor, and only needs you to have a high relative velocity to the surrounding medium to work. This lets you not only maintain speed in a nebula, but increase it, or even brake or turn using the plasma sail (a big plasma magnet, basically) and propeller (a whistler wave antenna that needs to be tuned to the group velocity of the surrounding plasma per its particular properties, that provides thrust not unlike a propeller.) The details are extremely technical, and might not suit all settings. Also, it is NOT reactionless, you do need propellant for it to continue to work - the key is that you don't need a heavy reactor, and braking & using the propeller is propellantless.