Relativistic motion of a magnetic sail

[qraal]

Hi All

Robert Zubrin and Dana Andrews developed the magnetic-sail for use in interplanetary flight and deccelerating from interstellar speeds.

The basic classical equation of motion is this...

V = Vo/(1 + Vo^(1/3)*k*t)^3

...where k is a constant, t is time since decceleration began, Vo is initial velocity, and V final velocity.

It's pretty straight forward to then integrate to find displacement...

s = (1/2k)*(Vo^(2/3)-V^(2/3))

...remembering that V is V(t), a function of t.

Differentiating V(t) gives...

a = -3*Vo^(4/3)*k/(1 + Vo^(1/3)*k*t)^4

...but (V/Vo)^(1/3) = 1/(1+Vo^(1/3)*k*t) so the equation simplifies to...

a = -3*V^(4/3)*k

Those are the basic equations of motion, classically. How would I go about turning them into relativistic equations of motion? Would V(t) need to be rapidity, and t become <tau>?

 

[tiny-tim]

Welcome to PF!

Hi qraal! Welcome to PF!

(have a gamma: γ and try using the X² and X₂ tags just above the Reply box )

V = Vo/(1 + Vo^(1/3)*k*t)^3
…
Those are the basic equations of motion, classically. How would I go about turning them into relativistic equations of motion? Would V(t) need to be rapidity, and t become <tau>?

That equation looks as if it comes from a more basic differential equation …

what is that? …

that's the one you'll have to convert, by changing momentum to mvγ and energy to mγ

 

[qraal]

Hi tiny-tim

Thanks for the tips on formatting. Very handy.

So the basic equation is dV/dt = -3kV^4/3 and most of the tricky stuff is in that k factor. Zubrin's initial equation is...

D/M= 0.59 (μ₀ρ²V⁴R_m/I)^1/3(J/ρ_m)

...which is the self-acceleration of the magnetic-sail in an ion flow of density ρ and relative velocity V. Thus if the sail is doing the moving then the acceleration is negative to the direction of motion. I is current, μ₀ = 4π x 10^-7, and (J/ρ_m) the maximum current density of the loop.

I think my main question is just how powerful the magnetic field needs to be at high speeds to actually deflect the ion flow as required and not just have it fly past the loop, compressing the generated magnetosphere. I have a 1990 paper by Giovanni Vulpetti on magnetic braking which I should probably study a bit more.