Moving matter

A few posts ago we noted that if we want to use the interstellar medium, then to get a mass comparable to our ship we'll need to excavate a tube roughly 200 km in diameter during our voyage. Presumably we'll need to move material from the outside of that cone to some central structure. A physical funnel would be far too heavy, so we'll need to use electric and magnetic fields. Can we do this with reasonable values for these fields?

Our vessel is moving through space and needs to move material 200 km from the edge of our tube to our processing facility in the center. How long do we have to do that? The ship is moving past the interstellar medium at some velocity V_ship in the range from 0 to about 0.2 c. Let's characterize the length of the ship, including the apparatus we have to move the matter by some length L_ship. This is likely comparable to the width. If so we have about

    Δt = Lship/vship

to move. If we look at accelerations we need to get

    1/2 a Δt2 = Lship (where we are assuming the width and length of the ship are comparable).

So

    1/2 a (Lship/vship)2 == Lship

Or

    a = 2 vship2 / Lship

The material we want to move are the protons of ionized hydrogen. Their acceleration in a magnetic field is given by

    a = q/m v X B

where q and m are the charge and mass of the proton and v and B are the vector values of the particle velocity and the magnetic field. Here we're only interested in the magnitudes we can just write this as

    a = q/m vshipB

where we've assumed that the nearly relativistic velocity of the ship is much larger than the thermal or bulk motions of the ISM. [For a 10⁶ Kelvin ISM, the thermal velocity of the protons is just a bit over 100 km/s.] Even for a medium at a temperature of 10⁶ K, this should be true except at the very beginning and end of the trip.

We can use these two equations to get an estimate our our needed magnetic field.

    2 vship2 / Lship = q/m vshipB

or

   B = m/q vship / Lship

In MKS units the ratio m/q is about 10^-8. If we pick 0.1c as our average velocity and 200 km for the size we get

   B = 10-8 3x107 / 2x105 Tesla = 1.5x10-6 T.

The response to the electric field depends upon the gradient of that field, but we need to impart an acceleration on protons sufficient to get them moving as fast as the ship. That means the electric field must impart about 1/2 mv² energy in the particles within L. But if v_ship is about 0.1c that implies that we need to impart about 1/2 m(.1c)² or about 1/200 of the mass energy of the proton. Protons have the same charge as electrons (apart from the sign) so we need to give protons a kick of 1/200 of their mass which is about 10⁹GeV. Here an electron volt is just the energy that a particular with the charge of an electron gains across a volts electric field. So we need a 1 GeV/200 or 5 MeV voltage differece between the edge of our tube and the the processing apparatus. That's about 5 megavolts/200,000m or just about 25 volts/m.