Copyright ©2019 Adam Crowl. All rights reserved.
Notes Towards Ultra-High Acceleration Magnetic-Sail Design
Adam J. Crowl*
*Corresponding author: adam@crowlspace.com
Abstract
Ultra-High Acceleration Neutral-Particle Beam drivers have been proposed for pushing Interstellar probes
to 0.06 c or faster [1]. The basic features of a particle beam riding magnetic-sail (Mag-Sail) interstellar probe
are outlined. Critical magnetic-field intensity of superconducting materials must be taken into account when
designing a Mag-Sail able to ride the neutral particle beam to such speeds. Using the most efficient
operating mode (as per [2]) is not possible for current materials, but a modified acceleration profile at lower
energy efficiency is within current material limits.
Keywords: Magnetic-Sail, critical current, superconducting materials
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1. Introduction
Near-term interstellar missions to the nearby stars will require cruising speeds of at least several percent
the speed of light. While ultra-light payloads, massing on the order of a gram, have been proposed for the
first flyby missions by the Breakthrough Starshot Initiative, heavier payloads for more substantial missions,
such as settlement via nanotechnologically miniaturised self-replicating machinery [3] will require larger
energy commitments. Mole [3] specifically proposes to propel the settlement package, massing ~1 kg, via
intense particle beams. Benford [4] has pointed out the limitations of nominal particle-beam propulsion
concepts assumed by Mole [3] and studied by Andrews [5], demonstrating that the physics of particle beams
mean they disperse far too quickly for the proposed acceleration distances. Recently Benford & Mole [1]
have proposed the Ultra-High Neutral Particle Beam Acceleration concept, which accelerates a 1 kg
Magnetic-Sail (Mag-Sail) payload to 0.06 c in 0.001 AU at 0.1 million gees, by-passing the beam
temperature dispersion as a result.
The proposed payload is a miniature Mag-Sail, with a diameter of 260 metres [3], [1] and a total mass of
1 kg, with the self-replicating artificial intelligence engraved at a molecular level. While the Mag-Sail’s
characteristics were sketched for the above studies, its coupling to the intense plasma stream was not
compared against the known properties of currently available superconducting materials. This study is a
first step at doing so*.
1.1 Magnetic-Sails
Initially proposed to couple a vehicle to the Solar Wind via magnetic fields, the Magnetic-Sail or MagSail [see Figure 1] has been proposed as the means of choice, by multiple studies, to couple to artificial
plasma streams [6],[7],[8] for acceleration as well as the Interstellar Medium, for braking [9],[10].
Figure 1: Mag-Sail schematic [from 7]
The mag-sail was originally approximated as a mini-magnetosphere [9], which is reasonable when
modelled interacting with the Solar Wind at relatively low speeds. Gros [10] has recently argued, the
conditions for formation of an artificial magnetosphere do not apply in a high speed (>0.01 c) number dense,
plasma flow. The mini-magnetosphere model assumes applicability of the Magnetohydrodynamic
approximation, but in intense plasma streams, such as the driver beams of Benford & Mole [1], a current
carrying wire-loop is better studied as acting as a magnetic mirror. Gros [10] analyses this case to derive a
fitting function which can be used to determine the required supercurrent flowing within the loop for deflecting
a plasma stream at a given relative speed.
As will be shown the magnetic mirror model computes the required supercurrent to be of the order of a
mega-amp (MA), which has important implications for the self-generated magnetic field of the
superconducting wire.
*
Footnotes can be useful, but distracting. Thus References.
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The High Acceleration Mag-sail case I will examine assumes the following parameters from Benford &
Mole [1] :
Beam and Sail Parameters
Mercury Beam
θ
0.8 microradian
Acceleration
105 gees=106 m/sec2
Sail diameter
260 m
Sail final velocity
0.06 c, 1.8 x 107 m/s
Acceleration distance
1.6 x 105 km, 10-3 AU
Acceleration time
18 sec
Magsail mass
1 kg
Kinetic energy
4 1014 J
Beam power
1.8 1013 W, 18 TW
Table 1: Mercury Neutral Beam Driven Mag-Sail adapted from Benford & Mole[1]
The Mag-Sail mass, acceleration and cruising velocity will be used to determine the required
supercurrent for the Mag-Sail wire-loop to carry, which will help determine its operating parameters. A
comparison with the properties of present day superconducting wire will then be made, and achievable
operating parameters defined.
3. Theory and calculation
A Mag-Sail works by reflecting an ionised stream of particles and being pushed, thanks to Newton’s
Third Law, in the opposite direction to the ions. For the Ultra-High Acceleration Neutral Particle Beam
application, the required acceleration is 106 m/s2 .
The thrust from perfectly reflecting a stream of particles (in this case ions with their detached electrons
to ensure overall neutrality) of constant average mass is:
T = 2A 𝑛𝑝 𝑚𝑝 𝑣𝑟2
(1)
…where T is thrust (in newtons), A is the Effective Area, relative velocity vr , np is the number density of
the neutral plasma stream and mp is the average particle mass.
To determine the required current in the Mag-Sail wire a simple physical argument based on Magnetic
Pressure can be used, to act as sanity check on the more specific model from Gros [ref] , which will then be
used to give more precise estimates for the proposed application.
The diameter, D, of the Mag-Sail loop means the geometric Area of the Sail is:
A=
π𝐷2
4
(2)
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As Thrust is a force, and as it is a function of the effective Area, the Ram Pressure from the particle
stream in Equation (1) is thus:
P = T/A = 2 𝑛𝑝 𝑚𝑝 𝑣𝑟2
(3)
The pressure supplied by the Mag-Sail to counter the particle stream’s Ram Pressure, is Magnetic
Pressure, which is defined as:
𝐵2
𝑃𝑚 =
(4)
2μ𝑜
The exact strength and shape of a magnetic field encountering a dense plasma stream is a complex
problem, so as first approximation, for a simple wire loop, at a minimum, the magnetic pressure at the centre
of the wire loop should equal the ram pressure being resisted. Otherwise the particle stream will flow through
the middle of the loop and the effective area will be less than the geometric area of the Mag-Sail.
For a simple loop the Magnetic Field strength, B, at the centre of the loop is:
B = μ0
𝐼
(5)
𝐷
The total force exerted over the area of the Mag-Sail by the magnetic field is approximately:
𝑃𝑚 A =
𝐵2 π𝐷2
(6)
2μ𝑜 4
Substituting Equation (5) into (6) gives the following:
𝑇 = 𝑃𝑚 A = 𝐼 2
πμ𝑜
(7)
8
Which allows the required current to be computed directly from the desired Thrust:
I = [𝑇
8
πμ𝑜
]
1
2
(8)
Thus for a Thrust of 106 newtons, the current required is 1.42 MA. This is independent of size of the MagSail.
For the more complex case, when the Mag-Sail’s effective area doesn’t equal the geometric area, I’ve
used the work of Gros [10]. In his study the Mag-Sail’s effective area is directly computed from the field’s
interaction with the incoming particles, by solving numerically the Biot-Savart Law equation, over subsections of the Mag-Sail loop.
The Effective Area, from the fitting function derived by Gros [10], is defined as:
𝐼
A(𝑣𝑟 ) = 𝐴𝑐 [𝑙𝑛 (β𝐼 )]
𝑐
where β
3
(9)
= vr/c and the Area constant is equal to:
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A𝑐 = 0.081 π𝑅 2
with the critical current constant:
IC = 1.55 × 106 amps
For a given speed β there is a critical current below which the Effective Area is essentially zero. Basically
the Mag-Sail is no longer effectively reflected the stream of particles, merely deflecting them ineffectually.
As Gros [ref] acknowledges the reality is probably more complicated, but basically the Mag-Sail fails to
operate as a sail below the critical current, even if some fraction of the particle flow is deflected by the field.
From the scaling relation it can be seen that the effective area is equal to the Area Constant Ac when the
ratio of current to velocity times critical current is:
𝐼
(β𝐼 ) = e
(10)
𝑐
And that the Effective Area is equal to the area of the Mag-Sail when the ratio is:
3
𝐼
(β𝐼 ) = 𝑒 ( √1/0.081) ≈ 10
(11)
𝑐
The mass of the Star-Probe is assumed to be equal to the mass of the Mag-Sail by Benford & Mole [1],
and if Je is the engineering current density, the mass is thus:
𝐼
𝑀𝑆 = π𝐷 (𝐽 )
(12)
𝑒
The volume of a wire loop is a toroid with the small diameter, d, of the wire being set by the amount of
supercurrent to be carried:
𝐼
π
(𝐽 ) = 𝐴𝑐 = 4 𝑑 2
(13)
𝑒
Where Ac is the cross-sectional area.
For an very long wire, carrying a current I, the magnetic field at a distance x from the centre of the wire
is:
B≈
μ0 2𝐼
4π 𝑥
(14)
At the surface of the wire the self-magnetic field generated by the flowing supercurrent will be at its
maximum, which thus computed by:
𝐵𝑚 =
μ0 𝐼
π 𝑑
(15)
Nordley & Crowl [2] discussed the maximum energy efficiency for momentum exchange. This can be
derived as follows:
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The Power is a function of the Beam Velocity, VB, and the mass flow rate, itself a function of VB:
1
P(vB ) = ṁ(vB )vB2
2
(16)
To find the mass flow rate we use the Thrust, T, which is equal to the Mag-Sail mass, M, and its
acceleration, g, for a given Mag-Sail speed, v :
T = 𝑀𝑔 = 2𝑚̇(𝑣𝐵 )(𝑣𝐵 − 𝑣)
Mg
∴ 𝑚̇(𝑣𝐵 ) = 2(𝑣
𝐵 −𝑣 )
(17)
(18)
Inserting (18) into (16) we get:
1
Mg
P(vB ) = 4 (𝑣
𝐵
2
v
B
−𝑣)
(19)
Introduce the variable 𝑘 = 𝑣𝐵 /𝑣 , the ratio of the beam velocity to the Mag-Sail velocity, and substitute
into (19) giving:
𝑘2
1
P(𝑘) = Mgv (𝑘−1)
4
(20)
To find the minima, take the first derivative:
dP
dk
dP
dk
=
𝑘2
Mgv ((𝑘−1))
4
dk
1
1
d
𝑘2
2𝑘
= Mgv ((𝑘−1) − (𝑘−1)2 )
4
(22)
(23)
Then setting the derivative to zero:
dP
= 0
dk
Thus
2𝑘
(𝑘−1)
𝑘2
= (𝑘−1)2
(24)
Solving for k we find that the maximum efficiency (i.e. ratio of kinetic energy input to kinetic energy in
the Mag-Sail) is when the ratio of the beam velocity to the sail velocity, k, equals 2.
Integrating Equation (20), using the fact that the Power, P, and acceleration, g, are the first derivatives
of energy and velocity with respect to time respectively gives the total Beam Energy, EB:
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1
E𝐵 = Mv
2
8
𝑘2
(𝑘−1)
(25)
while the Mag-Sail’s kinetic energy is merely:
1
𝐸 = Mv 2
(26)
2
Allowing us to find the ratio of the Mag-Sail Kinetic Energy to the total Beam Energy expended for
any value of k, and define the efficiency thus:
η = E/E𝐵 =
−1
𝑘2
4 ((𝑘−1))
(27)
Additionally the efficiency of using a beam would be maximised by the beam “filling the sail” – the whole
of the beam is encountered by the Mag-Sail and reflected. This would at least require the effective reflective
area of the Mag-Sail’s magnetic field to be equal to the Mag-Sail’s physical area i.e. when the ratio of the
Loop current is equal to about 10 times the critical current.
With this in mind, we can compute the supercurrent required to maximise the efficiency of the Neutral
Particle Beam Pushed Mag-Sail described by Benford & Mole [1]. With β = 0.06c, the critical current needs
to be 93 kiloamps, and with the effective area set equal to the size of the Mag-Sail, the Mag-Sail current is
938 kiloamps. This is close to the earlier approximation from Equation (8).
The Mag-Sail mass is set at 1 kg, while the diameter is 260 metres. For various possible superconductor
densities (graphene, magnesium boride and copper oxides) this allows us to compute the self-field of the
current carrying wire:
Table 3: Self-field and Critical Current Density of Mag-Sail Wire on Full Load
Wire Density
(kg/m3)
Wire
diameter
(mm)
I
(kA)
B
(tesla)
Je
(MA/cm2)
1800
0.93
938
403
108
2500
0.79
938
475
150
8000
0.44
938
853
485
Present day materials have maximum critical current densities in the range of 0.1-10 MA/cm2 [11]. A ten to
fifty fold improvement might eventually be achieved by future materials, approaching the maximum flux
pinning values of thin film high-temperature superconductors, and will certainly be needed for higher
relativistic speeds, but the governing equations suggest the requirement can be modified by relaxing the
maximum efficiency criterion.
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If the effective area is also decreased, then the required supercurrent is also relaxed. Table 4 gives
some examples using relaxed values:
Table 4: Parameters Varied
Effective
Area (A)
k (vB/v)
β
I/Ic
I (kA)
Bmax
(tesla)
Jc
(MA/cm2)
η
η.A
1.0000
2.0000
0.0600
10.0866
938
403
138
1.0000
1.0000
0.0810
1.1667
0.0100
2.7183
42
18
6
0.4898
0.0397
0.1000
1.3333
0.0200
2.9235
91
39
13
0.7500
0.0750
0.2000
1.3333
0.0200
3.8636
120
51
18
0.7500
0.1500
0.3000
1.3333
0.0200
4.6983
146
63
21
0.7500
0.2250
0.4000
1.3333
0.0200
5.4899
170
73
25
0.7500
0.3000
0.5000
1.3333
0.0200
6.2614
194
83
29
0.7500
0.3750
As can be seen, the beam efficiency isn’t greatly reduced by lower maximum relative velocity between
the beam and the Mag-Sail. For the lowest value, where I/Ic = e, and the maximum relative velocity is 0.01
c, the product of the beam efficiency and effective area are ~4% - thus at least 25 (=1/0.04) times the power
of the optimal case is needed to compensate to maintain the same acceleration. The critical current density
is an achievable 6 MA/cm2 with current superconducting materials. Development of higher purity and higher
critical current density superconductors would enable higher cruise speeds and higher efficiencies, and as
these are desired for other applications already, improved materials can be reasonably expected once niche
applications like Mag-Sail propulsion become seriously funded. Their development as part of an Interstellar
Propulsion Program would, contrariwise, be an immensely valuable spin-off of such a program.
4. Energy Requirements
The energy required for a given supercurrent stored in a single loop is computed from the selfinductance, derived in [12] as:
𝐿 =
𝑑2
1
µ 𝐷[ln(8𝐷/𝑑) − 2] + µ0 𝒪 ( 𝐷 )
2 0
(19)
where the final correction term, is very small, as the ratio of the wire’s cross section (~mm) to the loop
radius (m) is small, meaning the self-inductance can be usefully approximated as:
𝐿≅
1
µ 𝐷[ln(8𝐷/𝑑)
2 0
− 2]
(20)
The magnetic energy stored in the loop is thus:
𝐸=
1
2
𝐿𝐼 2
(21)
Page 8 of 10
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Using the Mag-Sail example with a current of I = 938 kA, D = 260 metres, and for graphene a wire
diameter of 0.93 mm, the self-inductance is 2.06 henrys, and total magnetic energy is 907 megajoules.
Conceivably this could be supplied to the Mag-Sail inductively before launch.
5. Conclusion
Mag-sails using present day materials can achieve significant fractions of the speed of light, is the main
result of this study. Speeds of 0.06 c are feasible given high critical current density superconductors for
Mag-Sail applications envisaged by Benford & Mole [1].
Page 9 of 10
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References
[1] J.Benford & A. Mole, "Ultrahigh Acceleration Neutral Particle Beam-Driven Sails", 3 January 2019,
"Centauri Dreams" https://www.centauri-dreams.org/2019/01/03/ultrahigh-acceleration-neutral-particlebeam-driven-sails/ accessed 31/7/19.
[2] G.D. Nordley & A.J. Crowl, “Mass Beam Propulsion, An Overview”, JBIS 68, pp. 153166, 2015.
[3] Alan Mole, “One Kilogram Interstellar Colony Mission”, JBIS, 66, pp.381-387, 2013.
[4] James Benford, “Beam-Driven Sails and Divergence of Neutral Particle Beams” JBIS 70, pg. 449-452,
2017.
[5] Dana G. Andrews, "Cost Considerations for Interstellar Missions," paper IAA-93-706 (1993); also
presented at Conference on Practical Robotic Interstellar Flight, New York University, August 29-Sept.
1, 1994.
[6] Geoffrey A. Landis “Interstellar Flight by Particle Beam,” AIP Conference Proceedings 552, 393 (2001).
DOI: 10.1063/1.1357952
[7] Adam Crowl, "High-Speed Magnetic-Sail Interstellar Precursor Missions Enabled by Metastable Metallic
Hydrogen," IAC 2017, Paper IAC-17,D4,4,3,x41020, Adelaide Australia, September 2017.
[8] Gerald D. Nordley, "Relativistic Particle Beams for Interstellar Propulsion," JBIS 46 number 4, pp 145150,1993
[9] D.G. Andrews & R.M. Zubrin, "Magnetic Sails and Interstellar Travel", JBIS, 43, pp. 265-272, 1990
[10] Claudius Gros, “Universal scaling relation for magnetic sails: momentum breaking in the limit of dilute
interstellar media”, Journal of Physics Communications
(2017) https://doi.org/10.1088/23996528/aa927e
[11] P. Chaddah, "Critical current densities in superconducting materials", Sadhana Vol. 28, Parts 1 & 2,
February/April 2003, pp. 273–282.
[12] R. Dengler, “Self inductance of a wire loop as a curve integral”, Advanced Electromagnetics, Vol 5, No
1 (2016), pages 1-8.
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