Owing to my attempts to analyse the Lensman books, I'm wondering when and if the drag equation breaks down, and if there's anything that can be used to replace it. In particular, what happens at the extremely low densities of interstellar space (on the order of 1 atom per 1-10cm^3), and what happens as you approach the speed of light (or even exceed it)?
For those wondering why I'm even asking this, in the Lensman books the ships use an inertialess reaction drive (don't ask, it just works ), in which the top FTL speed of a ship is said to be limited solely by, in effect, its terminal velocity (ie when thrust from engines = drag). Now whilst I can and have used the drag equation to work out how powerful the engines must be to get a ship zipping around at 90 parsecs an hour, that's only helpful if the equation actually holds true at such velocities.
I know that such an FTL drive will mean all sorts of trouble when it comes to little, insignificant things like how the hell people manage to live whilst inertialess, but if Doc Smith's willing to just handwave such things away then so will I (for this thread at least).
You can probably calculate the drag ab initio for a low-density homogeneous medium by calculating the number of collisions per second and using conservation of momentum without resorting to using the drag equation and praying that it works in the setting you're applying it.
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If we're dealing with a very low density medium we should be able to get away with just saying that our ship just bashes each particle individually and brings them up to its speed.
If the ship with front cross sectional area A is moving at speed v with respect to a low density, d, medium, then in a time dt it will sweep out a volume of v*A*dt, which contains a mass of d*v*A*dt. If it then accelerates that (small) mass to its speed then that will be a momentum transfer of v*(d*v*A*dt). That means there is a momentum exchange of dAv^2 in time dt, and therefore a drag force of dAv^2.
That should work pretty well in low density, high speed situations. Its not very accurate normally, but in these situations it should actually work better as we dont have to worry about pressure effects and waves as the medium is so low density.
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Possibly, but I don't think I can use your method in this case Steel, given I'm actually dealing with FTL speeds. What I'm more concerned with is whether (for example) your random hydrogen atom in the way will collide & pass through the ship - I believe that there was a mention of something similar to this in a SW / SW v ST thread here recently.
Steel, accoring to that analysis, the requisite engine thrust should be proportional to ργ²v². There is a Lorentz factor coming from relativistic momentum and another one from either the increased rate of incident particles in the ship frame (due to time dilation, dt = γdτ) or, equivalently, the increased number density in the ship frame (Lorentz-Fitzgerald contraction reduces volumes by γ, thus increasing densities by a corresponding amount). The generalization to FTL is physically problematic; although I suppose that one can simply say by fiat that |v|>c corresponds to -1 < γ² < 0, one might as well be making up completely new physics.
Given a perfect fluid with mass-energy density ρ, pressure p, and four-velocity u, the stress-energy tensor is
[1] T = (ρ + p/c²)u⊗u + pg,
In STR, the metric is Minkowski, g^{αβ} = diag(-1/c²,+1,+1,+1) in Cartesian (t,x,y,z)-coordinates, so that if the motion is in the x-direction only with velocity v, then u^α = [γ;γv;0;0], and the mass-energy flux through a surface with a normal in the x-direction is
[2] T^{tx} = (ρ+p/c²)(u^t)(u^x) + pg^{tx} = (ρ+p/c²)γ²v.
For a photon gas, p = ρc²/3, giving T^{tx} = (4/3)(ργ²v). The equation of state is ρc² = aT^4, where a = 4σ/c = 7.5658E-16 J/(m³K^4) and this T is the cosmic background temperature (2.725K), so that the engine thrust needs to be F = (Ac)(T^{tx}) = (4/3)(aT^4γ²)(Av/c), where A is the scattering cross-section. This will generally be an underestimate if the physical cross-section area is used, since re-radiation will be non-isotropic.
The drag due to cosmic radiation is particularly easy to calculate because photon momenta and energies are identical (in units of c = 1). For matter fluids, if Steel's suggestion is generalized by supposing that the ship leaves behind, "on average," a fluid of identical density and pressure in its own reference frame, then the force can be calculated from T^{xx}, the flux of x-momentum through a surface with a normal in the x-direction (i.e., x-component of pressure):
[3] T^{xx} - p = (ρ+p/c²)(u^x)(u^x) + pg^{xx} - p = (ρ+p/c²)γ²v²
The downside is that there does not seem to be any physically compelling reason for the interaction to be such, so the above probably can't be more than a rough scaling law at best. Then again, unlike the CBR, the interstellar medium has a pressure much less than its density, so treating as a dust shouldn't introduce much error compared to, e.g., ignoring the geometry of the ship altogether,
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), in which the top FTL speed of a ship is said to be limited solely by, in effect, its terminal velocity (ie when thrust from engines = drag). Now whilst I can and have used the drag equation to work out how powerful the engines must be to get a ship zipping around at 90 parsecs an hour, that's only helpful if the equation actually holds true at such velocities.
