Relativistic motion of a magnetic sail

In summary: This summary is provided by a third party. In summary, Robert Zubrin and Dana Andrews developed a magnetic-sail for use in interplanetary flight and deccelerating from interstellar speeds. The basic classical equation of motion is Vo = 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)), but differentiation V(t) gives a=-3*Vo^(4/3)*k/(1 + Vo
  • #1
qraal
790
3
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>?
 
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  • #2
Welcome to PF!

Hi qraal! Welcome to PF! :smile:

(have a gamma: γ and try using the X2 and X2 tags just above the Reply box :wink:)
qraal said:
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 :smile:
 
  • #3
Hi tiny-tim

Thanks for the tips on formatting. Very handy.

So the basic equation is dV/dt = -3kV4/3 and most of the tricky stuff is in that k factor. Zubrin's initial equation is...

D/M= 0.59 (μ0ρ2V4Rm/I)1/3(J/ρm)

...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/ρm) 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.
 

What is a magnetic sail?

A magnetic sail, also known as a magsail, is a proposed propulsion system that uses the magnetic field of a planet or star to accelerate a spacecraft. It consists of a large, superconducting loop that interacts with the plasma in the interstellar medium, creating a drag force that can propel the spacecraft forward.

How does a magnetic sail work?

A magnetic sail works by taking advantage of the magnetic field of a planet or star. The superconducting loop of the sail generates a magnetic field that interacts with the charged particles in the interstellar medium, creating a drag force. This force can be used to accelerate the spacecraft in the opposite direction, allowing it to travel through space without the need for traditional fuel-based propulsion systems.

What is relativistic motion?

Relativistic motion refers to the movement of an object at speeds close to the speed of light. At these speeds, the effects of special relativity, such as time dilation and length contraction, become significant and must be taken into account. In the context of a magnetic sail, relativistic motion is important because the sail will be traveling at high speeds, and its performance will be affected by these relativistic effects.

What are the potential applications of a magnetic sail?

A magnetic sail has the potential to revolutionize space travel by providing a highly efficient and sustainable means of propulsion. It could be used for long-distance interstellar travel, as well as for shorter trips within our own solar system. It could also be used for spacecraft that need to make frequent adjustments to their trajectory, such as those in orbit around a planet or moon.

What are the challenges of developing a magnetic sail?

One of the main challenges of developing a magnetic sail is the need for a large and powerful superconducting loop. This requires advanced technology and materials that are not currently available. Additionally, the sail must be able to withstand the extreme conditions of space, including intense radiation and high speeds. There are also logistical challenges, such as how to deploy and maneuver the sail in space. Further research and development are needed to overcome these challenges before a magnetic sail can be successfully implemented for space travel.

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