1 Introduction

The minimum mission requirement for “Project Icarus” is to receive first data

from Alpha Centauri in 100 years. For maximum data-return the probe needs

to decelerate into orbit around the target star from interstellar transit speeds –

at a minimum the vehicle needs to undergo a total velocity change of 0.09c

(27,000 km/s.) The most conservative approach to achieve this is to use fusion

thrust for the whole velocity change, but this increases the required fuel

considerably, even using a two-stage rocket, as assumed in this paper.

Magnetic-Sails, or Mag-Sails, are a non-propellant means of reducing the

vehicle’s speed from its cruise speed to something closer to orbital velocity, via

creating drag against the thin gas between the stars known as the Interstellar

Medium (ISM) [Ref 1-5].

For the purposes of this paper two vehicle configurations will be studied – the

Pure Fusion and the Hybrid. The 1st Stage will be assumed to accelerate both

kinds of vehicles to cruise speed, while the 2nd Stage will perform either full

deceleration under fusion thrust, in the Pure Fusion case, or deploy a

magnetic-sail, in the Hybrid case, to decelerate to a lower speed, then use

fusion-thrust for deceleration to orbital speeds in the star-system from 1500

km/s (0.005c), at which speed the magnetic-sail becomes increasingly

inefficient as a brake. Matloff [6] has explored the possibility of braking in the

destination star’s astrosphere – the region dominated by its stellar wind – but

this may require determining the size and density of the astrosphere via as yet

undeveloped remote means before a starship can successfully brake within it,

thus it’s an option left for future studies.

2 Magnetic Sails: A Primer

Figure 1: Magnetosphere (from Janhunen [5])

A Magnetic-Sail, or Mag-Sail, for interplanetary or interstellar travel is an example of drag (or lift) creation via generating an artificial magnetosphere and interacting with the in-space medium. A Magnetosphere is created by the interaction between a near dipolar magnetic-field, such as exist around several planets in our solar system, and a large plasma flow, such as the Solar Wind (Figure 1). The interaction between the two creates a structure defined by the behavior of plasma in that region. The border between free-flowing plasma and the magnetic field is defined by equilibrium between the plasma flow’s ram pressure and magnetic pressure of the magnetic field encountered.

2.1 Ram-Pressure

Ram-pressure is defined as

(1)

…where is product of the number density and average particle mass of

the plasma flow and is the relative speed of the plasma flow to the magnetic field. Ram-pressure of the plasma in the Interstellar Medium (ISM) at the speeds in question will be much higher than the intrinsic magnetic pressure of the ISM in the Galactic magnetic field.

2.2 Magnetic Pressure

Magnetic pressure is defined (from Cole [7])...

(2)

The magnetic-field strength, B, can be be found by

where is the Dipole Moment of the magnetic loop generating the field and R is the distance from the centre. Substituting into (2) we get [Eqn.W.7, Ref. 7]…

(3)

2.3 Radius of Magnetosphere

Equating the ram-pressure, and the magnetic-pressure, lets us find the radius of the magnetosphere formed…

(4)

This forms the basis for determining performance of magnetic sails, but is an approximation and only applicable when the assumptions of the magnetohydrodynamic (MHD) plasma flow are valid [5]. Experiments with laboratory scale plasma sources and mini-magnetospheres have confirmed that the approximation is applicable to magnetic-sails [8].

2.4 Mag-Sail Dynamics

At its simplest a Mag-Sail uses a single-loop of current to produce the magnetic-field to form an artificial magnetosphere as seen in Figure 2.

Figure 2: The Magnetic-Sail (from [5])

Dipole Moment for a single loop of conductor is computed as:

(5)

Janhunen et.al. [5] derived the following equation for drag created by an artificial magnetosphere in a plasma flow:

(6)

where D is the drag, the ram-pressure of the vehicle’s motion against the interstellar medium, I the current in the loop, and magnetic-sail loop radius. The vehicle mass is Mv, thus deceleration of the vehicle is: D/Mv A useful quantity is the mass of the magnetic-sail, Ms. The minimum cross-sectional area required of superconductor, As, is computed by the ratio of the current, I, to the maximum super-current density, Jc. However in actual application bare superconducting wire will be surrounded by protective layering, a cooling system and normal conductor to allow “quenching” of the super-current in case of failure of superconductivity. Thus the engineering super-current density, Je ,is more appropriate, a fraction of the ideal. The minimum Mag-Sail mass, Ms is therefore:

Ms = 2.π.Rm.(I/ Je) For a given set of magnetic-sail characteristics, I & Rm , and vehicle mass, the variables reduce to a Deceleration constant, κ κ (7) Substituting in Equation (1) & (6) into Equation (7), and some algebra, allows the Deceleration constant to be computed as follows:

κ

(8)

This allows analysis of the magnetic-sail’s equations of motion. According to Zubrin [] the following relationship for an interstellar vehicle braking against the ISM applies, with vehicle speed, V, a function of time:

κ (9)

is the initial velocity and is the time since the start of braking. Taking the first derivative with respect to time gives acceleration as a function of time:

κ κ

(10)

At t=0, this equates to:

κ (11)

…which allows the Deceleration constant to be expressed as a function of the initial acceleration, and initial velocity, :

κ (12)

Integrating Equation (9) with respect to time allows us to derive the displacement as:

κ κ

(13)

This can be simplified by substituting equation (11) for κ

for

κ to give the displacement in terms of the initial and final

velocities

(14)

The ratio

has the dimensions of time and provides the useful variable of

the characteristic time, , of the magnetic-sail’s deceleration. Substituting in, Equation (13) becomes:

(15)

Total deceleration time, to reach a final velocity, can be computed as:

(16)

3 Two Stage Fusion Rockets

“Project Daedalus” remains the most well-defined interstellar fusion rocket,

and has been chosen as the base-line for comparison. The following

characteristics (from [11] & [12]) are germane to this discussion:

Table 1: Daedalus Two-Stage Fusion Rocket

Stage Thrust (newtons)

Dry Mass* (kilograms)

Exhaust Velocity (metres/second)

Mass-Flow Rate (kilograms/second)

Fuel-Tank Mass-Fraction*

1 7.54E+6 1.4E+6 1.06E+7 0.72 0.047 2 6.63E+5 9.31E+5 9.21E+6 0.0711 0.053 *computed from data in Ref 11 & 12

A significant fraction of the vehicle dry mass is tied up in the propellant and

propulsion system tankage. Helium-3 at 3 K has a vapor pressure of 81.3 kPa,

so the tanks require their void space (26%) to be filled to maintain constant

pressure else the pellets of frozen deuterium and liquid helium-3 would out-

gas. As the tank is emptied some kind of contracting walls would be required

to maintain constant pressure. The tank walls are titanium, with a safety factor

of 1.33. These, with 117 layers of insulation, add a considerable mass also –

212 tonnes for the 1st Stage, 28.4 tonnes for the 2nd. Over 1,700 tonnes of mass

for the first stage, and 156 for the second, are required for the helium-3

pressurization.

This limits the maximum mass-ratio that can be achieved by a single-stage

“Daedalus” class vehicle. For a given fuel-tank mass-fraction, b, then the

required propellant mass, P, to achieve a given mass-ratio, R, can be computed

for a given vehicle mass, Ms, via:

(17)

The numerator implies an upper limit to the mass-ratio of:

(18)

As is approached the propellant mass goes to infinity.

Rocket motion, at constant thrust, is described by the following equations for

the acceleration and deceleration phases (derived from Appendix A, in Orth

[9]):

(19)

(20)

…where is the exhaust velocity, is the time required to expel in exhaust

mass the equivalent of the vehicle’s dry-mass at the mass-flow rate for that

stage, R is the mass-ratio. The deceleration stage variables are primed as they

are different for the 2nd Stage of “Daedalus”.

Using the parameters from Table (1) and Equations (16), (18) & (19) the flight

time can be computed via:

(21) Where the acceleration and deceleration times, are computed via the

following relation:

(22) With taking their relevant stage values.

For various cruise velocities the results for travel-time were plotted against the

resulting vehicle total mass in Figure 3:

Figure 3 Time (years) vs Mass (tonnes)

The minimum flight time, 71.58 years, is achieved at a cruise velocity of 0.07c

and an initial mass of 281,181 tonnes. Increasing the trip-time by just 5 years

decreases the initial mass by 66%. For the same mass budget as the original

“Daedalus” star-probe (just over 54,000 tonnes) the flight to Alpha Centauri

takes 81.6 years at a cruise speed of 0.05516c, while the “Project Icarus”

minimum mission requirement of a 95.6 year flight-time, to achieve first data

reception on Earth in 100 years, requires an initial mass of 28,600 tonnes and a

0.04635c cruise speed.

0

20

40

60

80

100

120

140

160

0 500000 1000000 1500000

Time vs Mass

Time vs Mass

4 The Interstellar Medium

The Interstellar Medium (ISM) is a thin mix of mostly hydrogen and helium, in

plasma, atomic and molecular form. About 1% of the ISM is in the form of

heavier elements, called “metals” by astrophysicists, either as gas or very small

dust grains. Some fraction of the ISM is in the form of larger objects, perhaps

comet-size or larger, but so spread-out that it can be ignored for the purposes

of this discussion.

Relevant to the operation of magnetic-sails is the hydrogen/helium plasma/gas

mix. Only the charged component, the plasma (a mix of ions and electrons),

interacts directly with the artificial magnetosphere created by the Mag-Sail.

However, relative to the ISM, the magnetic-field is moving very rapidly, and

this rapid change is potentially sufficient to cause ionization of the electrically

neutral atoms and molecules in the ISM.

According to Crawford’s [10] review of the Local ISM (LISM), the nearby G

cloud might extend all the way to Alpha Centauri, with a gas density of

between 0.1-0.2 atoms/cm3. Interestingly an astrosphere has been detected

around Alpha Centauri, though its properties are yet to be determined. For the

purposes of this study, a density of 0.1 atoms/ cm3 will be assumed.

On average the particles comprising the ISM mass 2.2 x 10-27 kg (75%

hydrogen, 25% helium, by mass) which, combined with the number density

above, allows the Deceleration constant to be computed for Mag-Sails of a

given super-current and radius.

The astrospheres of the stars, and the Sun’s own heliosphere, also suggest a

possible means of protecting magnetic-sail structures against the hail of hyper-

velocity dust particles. Dust particles become charged by UV photons and

cosmic-rays removing electrons from their surfaces. Sufficiently charged dust is

thus deflected by an astrosphere and prevented from entering the inner Solar

System in the case of the Sun [10]. Possibly the artificial magnetosphere of an

operating magnetic-sail will allow dust to be deflected away.

5 Performance of the Magnetic-Sail Fusion-Rocket

The Mag-Sail equipped “Daedalus” 2nd Stage deploys a magnetic-sail to

decelerate from the cruise speed to 1500 km/s, then use fusion-thrust for

deceleration to orbital speeds in the star-system. A “Daedalus” 2nd Stage, with

450 tonnes Payload, will need an initial mass of 1097 tonnes to decelerate

from 1500 km/s. The Mag-Sail is assumed to mass 1,000 tonnes, with a size

optimized for a lower maximum magnetic-field strength for the

superconductor (assumed to be of the same density as magnesium diboride,

MgB2 ) and an engineered Deceleration constant, κ , of 2x10-11 s2/3/m1/3 for the

assumed ISM density of 0.1 atoms/cm3.

The mass of the “Daedalus” 2nd Stage has been modelled, with the same

assumed tankage fraction and exhaust velocity used for the two-stage Pure

Fusion vehicle.

Figure 4 summarises the results:

Figure 4: Initial Mass (tonnes) vs Total Trip Time (years)

The minimum is achieved at a cruise speed of 0.08015c, with a total trip-time

of 68.033 years and a mass of 73,700 tonnes. The total trip time drops under

70 years for masses between 43,000 and 126,000 tonnes. The minimum

“Project Icarus” requirement (95.6 years total travel time) is met for an initial

mass of 12,300 tonnes. This is a significant improvement over the 28,600

tonnes of the two-stage pure-fusion case.

Also due to the runaway hyperbolic rise of the propellant mass, as discussed

above, the minimum mission can’t be achieved by a single-stage vehicle based

on “Project Daedalus” vehicle characteristics.

6 Conclusions & Recommendations

Magnetic-Sails/Mag-Sails provide a means for significantly improving the

performance of fusion rockets on missions to nearby stars, possibly eliminating

the need for deceleration fuel to be carried to the destination. Future research

will be required to determine the best means of protecting such large, thin

structures against the dust component of the ISM. In particular the suggested

magnetic deflection of charged ISM dust needs to be modeled in detail.

7 References

[1] Andrews, D.G., Zubrin, R.M., "Magnetic Sails and Interstellar Travel", JBIS, Vol.43, pp.265-272, 1990.

[2] Zubrin, R.M., Andrews, D.G., "Magnetic Sails and Interplanetary Travel," AIAA-89-2441, Journal of Spacecraft and Rockets, Vol.28, pp.197, 1991.

[3] Zubrin, R.M., “Detection of Extraterrestrial Civilizations via the Spectral Signature of Advanced Interstellar Spacecraft”, Astronomical Society of the Pacific Conference Series, Volume 74. Progress in the Search for Extraterrestrial Life. 1993 Bioastronomy Symposium, held in Santa Cruz, California, August 16-20, 1993. Editor, G. Seth Shostak; Publisher, Astronomical Society of the Pacific, San Francisco, California, 1995. [ http://adsabs.harvard.edu/abs/1995ASPC...74..487Z]

[4] Zubrin, R.M., Martin, A., “NIAC Study of the Magnetic Sail”, NIAC Final Report, NASA Institute of Advanced Concepts, 2000.

[http://www.niac.usra.edu/files/studies/final_report/320Zubrin.pdf] [5] Toivanen, P. K., Janhunen, P., Koskinen, H. E. J., “Magnetospheric

Propulsion (eMPii)”, Finnish Meteorological Institute Reports 2/2004, ISBN 951-697-594-1, ISSN 0782-6079, 78 p., Finnish Meteorological Institute, 2004. [ http://www.space.fmi.fi/~pjanhune/papers/eMPii_final_1.3.pdf]

[6] Matloff, G., “Electromagnetic Starship Deceleration in a Stellar Wind”, JBIS, Vol.62, pp.66-68, 2009.

[7] Cole, G.H.A., Woolfson, M.M., “Planetary Science: the Science of Planets Around Stars”, IOP Publishing, Bristol, 2002.

[8] Ueno, K., Ayabe, T., Funaki, I., Horisawa, H.,Yamakawa, H., "Imaging of Plasma Flow around Magnetoplasma Sail in Laboratory Experiment", J. Plasma Fusion Res. SERIES, Vol. 8, 2009.[ http://www.jspf.or.jp/JPFRS/PDF/Vol8/jpfrs2009_08-1585.pdf ].

[9] Orth, C.D., “VISTA – A Vehicle for Interplanetary Space Transport Application Powered by Inertial Confinement Fusion”, Lawrence Livermore National Laboratory Report No. UCRL-LR-110500, 2003. [ http://dx.doi.org/10.2172/15015945 ]

[10] Crawford, I.A., “Project Icarus: A review of local interstellar medium properties of relevance for space missions to the nearest stars”, Acta Astronautica, v. 68, iss. 7-8, p. 691-699., 2011.

[11] Bond, A., Martin, A.R. "Project Daedalus: The Propulsion System, Part I; Theoretical Considerations and Calculations", JBIS, Supplement, pp.S44-S62, 1978.

[12] White, A.G.A., Parfitt, J.A. "Project Daedalus: The Structure", JBIS, Supplement, pp.S104-S115, 1978.