Sailing the Proton Winds
An Investigation into the Principles of Magnetic Sailing

Charles Danforth
Johns Hopkins University, Baltimore, MD 21218

Magnetic sails are related to the more familiar lightsails in that they both utilize ambient energy for propulsion. While lightsails use photons, magnetic sails use charged particles. They can operate in two orientations, with rather different results. A magnetic sail is designed using current technology and modeled to see its behavior and to determine net thrust. Advantages and disadvantages are discussed and some intriguing possibilities are put forward.

1 - Introduction:
The concepts of sailing a boat on a lake and riding a rocket to Mars are rarely associated. It appears, however, that intuition does not rule in this case as it is indeed quite possible to sail to Mars or indeed anywhere else.
There are several ways to do this. The first, lightsails, are a well known and comfortably mundane entity to science-fiction fans and cutting edge aerospace inventors. Just like a sail on a terrestrial boat uses wind to push it in another direction, a solar sail uses light pressure from the sun or a laser to push it in the opposite direction. Typically constructed of titanic sheets of very thin metallic foil spread over thousands of square kilometers, they trap the feeble momentum of photons. Though only one small test model was build to date, these objects are relatively well understood and well-treated elsewhere. Solar sails will not be dealt with here except in comparison.
The second way to sail in outer space is the magnetic sail. Instead of light pressure, the magnetic sail uses the redirected momenta of charged particles to power its flight.
The structure of the magsail is simple and in fact could be retrofitted onto almost any spacecraft. All that is required is a loop of wire. The spacecraft simply dumps this loop of wire overboard and sets up a current in it. Lenz law says that flux will attempt to remain constant. This will cause the area of the loop to increase until it forms a circle with little or no structural help required. All that is then required is to attach the loop to the spacecraft with a few shroud lines to couple the two (Fig 1.)
Assuming a mono-directional flux of charged particles, we can look at the two extremes of magsail orientation (angle of attack). The simplest case is the symmetric or axial case--so called because the particles approach along the axis of the magnetic field and scatter symmetrically. Particles entering at any other impact parameter than directly along the axis will experience a lateral acceleration thereby imparting part of their initial z-momentum to the field and hence to the spacecraft.
The other configuration, the asymmetric or normal orientation, is a little less obvious. With the dipole moment oriented in z (loop is in the xy plane) and the particles entering with V=Vx, they will be bent by the field into giving a y component to the velocity. Thus a component of x and y momentum is transfered to the sail. Thinking of this as a craft in a circular orbit around the sun, we now have the equivalent of outward thrust and lateral thrust, a very useful situation indeed. Thrust can be further vectored by varying the current and orientation of the loop. Indeed very complicated maneuvers are possible as related by (Zubrin and Andrews,91).

2 - Modeling a Magsail

Design parameters
Since no magsail has ever been built, nor does the need for one yet exist, we need to determine the operating parameters of our sail from scratch. At present, no work exists on double-loop or more complicated configurations of sails, and there is no reason to believe that such a structure would be more efficient than a single loop of current.
Clearly we wish to maximize the scattering cross section while minimizing mass. The effectiveness of the sail is related to the strength of magnetic field it can produce. Calling the radius of the sail R, the effective cross section scales as R2 while the magnetic field strength scales as R-2. The mass of the sail itself will also scale with R. Thus, let's choose an intermediate value for sail area, say 10 kilometers. (Note for those unfamiliar with the field: 10km is tiny compared to most usable light sail designs.)
Field strength also scales as I, so we should attempt to get maximum current through the cable. To minimize current loss, it would be a good idea to make the loop out of some sort of superconductor. While it has certain advantages in not needing a huge generating capability on board (solar panels, radio-thermal generator, batteries, etc), this brings with it the inherent difficulty of critical temperature and current beyond which superconducting properties break down.
Recently discovered high-temperature superconductors will make this easier to accomplish. Ba2Cu3O7Y, a new ceramic-type superconductor has a critical current of about 1010 Amps/m2 (Zubrin&Andrews, 91) and a critical temperature of 90K (CRC). While in interstellar space, the background temperature of ~3K poses no threat, direct solar illumination in interplanetary space can quickly raise the temperatures above critical. With a thin layer of reflective insulation (doubling as protection from debris punctures), superconductivity can be maintained. The shroud lines need be nothing fancy at all as the accellerations experienced by the craft are not large.
The density of our superconductor is presumably something on the order of 5grams/cm3. If we give it a diameter of 12mm--about the diameter of a good mountaineering rope--it could carry a megampere of current and still maintain superconductivity. The insulation can be assumed to be fairly light (~0.5 grams/cm3) and fairly thin (~3mm). This gives a total mass for the current loop and insulation of 36,000 kg or 36 tons.
Table 1: Design Parameters
sail radius = 10 km
wire radius = 6 mm
max. current= 106 Amps
crit. temp = 90 Kelvin
center field=6.28x10-5 Tesla
sail mass = 36,000 kg

Modeling: Behavior
To get some idea of the behavior of a magnetic sail, a computer code was written (see Appendix) to find the trajectories of particles through the magnetic field. The particles in question are supplied by the solar wind which is composed mostly of free protons and electrons. Protons were choosen for our study because they carry about 1000 times the momentum of electrons and thus are much more efficient at transferring momentum to the spacecraft. The incident velocity was choosen to be 500 km/s, a good average velocity for particles at 1AU from the sun. In fact, the velocity varies between about 300 and 700 kilometers per second depending on solar conditions and the locations of coronal holes (Zelik and Smith).
Defining some geometry (fig 3), the vector from the origin (placed for convenience at the center of the loop) to the loop r' was written in terms of the equatorial angle phi

The infinitesimal current vector Idl was found by taking Rxez since the loop is known to lie in the x-y plane and be perpendicular to the radius
Defining r' to be the vector from the current element to the particle P and r to be the vector from the origin to P,
Unlike the modified dipole approximation used by Andrews and Zubrin (q.v.), I used the full Biot-Savart solution solved for component by component. This seemed more accurate as the protons are often close to the radius of the current loop where the dipole approximation becomes invalid.

Numerically integrating over the loop, the components of B are found for the proton's location. The accelleration components follow directly from the field components by the Lorentz force law

The velocities and positions are then updated and the process iterated. Data is output to Mathematica, interpolated and plotted parametrically. Simulated particle tracks for the two orientations at 50 kiloamps and 1 megamp are shown in Figure 4. All plots cover 250ms of time.
Figure 4a: Axial configuration for a current much less than Imax. Particles are incoming with speed of 500 km/s from the bottom (-z). The current loop is drawn to scale. Only those particles passing very close to the loop suffer large deflections.
Figure 4b: Normal orientation at low current. Particles are incident from the left in the xy plane. There are no deflections into the z direction. By and large, flow is uninterupted.
Figure 4c: Axial configuration for I~Imax. Trajectories are much more disrupted now. Note the symmetry of the scattering.
Figure 4d: Normal orientation for near-critical current. Particles never even come close to the current ring! Again, there are no scatterings out of the plane. Notice the extremely turbulent wake behind the sail.
Modeling: Results
After the particle tracking program was perfected, it was modified to serve as a Monte-Carlo simulation engine. A large number of particles with random starting coordinates were input into the system and allowed to run. Four different situations were modelled (the four represented in Fig 4a-d). For each particle the change in velocity was found. With a large number of particles, the average change in velocity and the incident area allowed us to find the cross section.

Table 2: Cross Section, Monte-Carlo Results
Config.I(amps)deltaV/#part.#part.Xsection(m2)Fractional Xsection

So current makes a tremendous difference in the scattering cross section of the magsail. This is, however, only a very rough estimate. (Because of computer time restrictions, larger numbers of particles could not be run. At 5 seconds of computer time per particle, accuracy begins to take a very long time!)
To complete the model, let's talk about the capabilities of this craft. In the paragraphs above we calculated the mass of the magsail to be 36 tonnes. At the radius of the earth, the particle density is between 0.4 and 80 million particles per cubic meter (again, closely tied to the solar weather (Zelik and Smith)). Choosing an intermediate value of 2x107 protons/m3 and using the usual value of 500km/sec as the particle velocity, we can see that the momentum flux is
The force on the magsail is then
=4.15 Newtons
for the maximum current axial configuration. This is comparable to the result generated by Zubrin and Andrews of 19.8 Newtons. Dividing by the mass of the sail we get an accelleration of 0.0001 m/s2. Using the sail at normal orientation would increase this to 0.009 m/s2. These results are fairly consistant with those calculated for a similar model independantly by (Zubrin and Andrews, 91). The differences arise from differences in field approximations and construction materials.

3 - Conclusions
Clearly, the big advantage of magnetic sails is their ability to operate with "ambient" power sources and no heavy on-board propellant. The technology is relatively simple and is essentially possible today. Furthermore, it is a technology which can be retrofitted onto existing, or resonably near-future, spacecraft with little trouble.
One of the greatest hazards to humans in space, especially for long duration flights away from the earth's sheltering magnetosphere, is the harsh radiation and high-energy particle environment of space. While ionizing radiation is not affected by the magnetic sail, charged particles are. The sail's magnetosphere deflects any high energy charged particle from the center of the current loop. While this helps a bit, more conventional sheilding (such as tanks of water) must be used to absorb radiation and neutral particles.
One would think that the entire spacecraft would need to be shielded from the strong magnetic fields. However, due to the sail's large radius, the field strength at the center of a 10km sail is only 6x10-11 tesla per amp of loop current. For a typical operating current of a megamp, this still produces a field of only 63 microtesla or 0.63 Gauss, comparable to the magnetic field of the Earth. This poses no hazard to humans or machinery.
While deployment of 2-dimensional structures (such as light sails) is a very complicated matter involving special folding techniques, one-dimensional structures (such as the loop of a magsail) can simply be wound around spools and deployed at a moment's notice. It should also be possible to vary the size of the loop ("reefing" in nautical terminology) providing another navigational degree of freedom.
There are two disadvantages inherent to magnetic sailing. The first, one which it shares with light sailing, is that the density of the momentum-bearing medium falls off as r-2. There is a bright side to this, however. The solar wind tends to accellerate as it moves outward so, even though its density falls, the momentum per particle rises.
The other major disadvantage, one unique to magsails, is the unpredictability of the "propulsion system". Traditionally, timing is crucial when dealing with spacecraft trajectories. Orbits are calculated to the nth order years ahead of time. The magsail's thrust, and hence the mission time, depends largely on the velocity and number density of incoming protons. As has been pointed out previously in this paper, solar wind velocities and densities vary wildly depending on conditions on the surface of the sun. Fortunately, there is a way to make this navigational uncertainty much less debilitating. Since sails can vary their thrust and in effect hover, they can travel over a standard transfer orbit and then loiter at the destination until the target body arrives. Velocity matching is simply a matter of adjusting current and angle of attack. Details of these calculations do not come into this paper, but are breifly outlined in (Zubrin and Andrews 91).
While it is doubtful that magnetic sails could serve to accellerate a rocket ship at a high enough rate to make interstellar travel feasible, they could function nicely as a decellerating factor for interstellar vehicles. In this case, it is not the charged particles which enter the magnetic field but rather the magnetic field sweeping through a volume of particle-filled space. Interstellar space is filled with gas, some of it ionized and some not. However, the shock of crossing the magnetospheric boundary of the magsail (what is referred to as a Bow Shock in hydro- and magnetodynamics) would serve to ionize the medium very effectively. Furthermore, since the relative velocity of the medium is so much higher than is encountered in interplanetary space, the efficiency of the magsail as brake rises. One can easily picture interstellar probes unfurling magnetic wings as they brake to investigate distant stars...

4 - References

5 - Technical Appendix

The core Fortran code is presented below. It follows the path of one particle through the magnetic field given a set of initial coordinates and velocities, calculates the magnetic field, and finds the total delta-v of the particle. A data file is written giving the timesteps and the (x,y,z) coordinates and velocities of the proton at each step. It can be easily modified for many particles or for a Monte-Carlo simulation. The reader is assumed to have a working knowledge of basic Fortran77.

      program magtracer
c     by Charles Danforth for JHU 171-304
c     4-24-97 last modified 4-29-97

c     Variable declarations
      real b(3)                       (magnetic field
      real p,dp                       (integration angle, interval 
      integer t,tmax,dt               (time, max, interval
      real rr2,g,current,const        (radius squared, loop rad
      real r(3,10000)                 (particle coordinate(time)
      real v(3,10000)                 (particle velocity(time)
      real a(3,10000)                 (particle accelleration(time)
      real radius                     (scalar dist from origin
      real dv(3),deltav               (deltaV vector and scalar
      real Pi,mu0,q,m                 (physical constants (SI units)

c     Constant Definitions

c     Open data file

c     &&& multiparticle loop start &&&

c     positions in meters (1,2,3)---(x,y,z)
c     velocities in meters/second

c ---- PARAMETERS ----
c     Set number of time steps TMAX in units of dt
c     Set dp increment (radians)
c     Set mag field scale factors current=I(amps), loop radius=g(meters)

c     write initial condition in the data file

c ### Start moving the particle ##############
      do t=2,tmax

c ^^^^^^ FIND MAGNETIC FIELD at current particle location ^^^^^^^
c     Code offers three options for magnetic field.  The first is the
c     full Biot-Savart law which should be universally applicable.
c     Second is the magnetic dipole approximation which is very similar 
c     to the first but takes much less time to run since it involves no
c     numerical integration routines.
c     Thirdly, for comparison, a uniform field is presented.

c     &&&&& BIOT-SAVART LAW FOR CURRENT LOOP (radius 1) &&&&&
         do i=1,3
         end do

c     Numerically Integrate around loop, dl X r / r^3 
       do p=0,6.28,dp
       end do

c     &&&&& DIPOLE FIELD (dipole approx) &&&&&
c     >> Current Loop in xy plane, dipole in z direction <<
c         if( then
c            b(1)=0
c            b(2)=0
c            b(3)=bo
c         else
c            b(1)=bo*3*r(1,t-1)*r(3,t-1)/(radius)**5
c            b(2)=bo*3*r(2,t-1)*r(3,t-1)/(radius)**5
c            b(3)=bo*(2*r(3,t-1)**2-r(1,t-1)**2-r(2,t-1)**2)/(radius)**5     
c         end if
c     &&&&& CONSTANT FIELD &&&&&
c     >> Constant field in z direction <<
c         b(1)=0
c         b(2)=0
c         b(3)=bo

c     Find the accelleration for this step (componentwise Lorenz Force Law)

c     Move the particle in r and v
         do i=1,3
         end do

c     Print everything to a file and screen

      end do

c     Calculate delta-v of particle (and hence of the field)
      do i=1,3
      end do
      print*,'DELTA V=',dv(1),dv(2),dv(3),deltav

c     &&& End of multiparticle loop &&&