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The electromagnetic rocket gun
Acta Astronautica Vol. 12. No 3. pp. 155-161~ 1985
0094-5765/85 $3.00+.00 Pergamon Press Ltd.
Printed in Great Britain.
THE ELECTROMAGNETIC ROCKET GUN F. W I N T E R B E R G
Desert Research Institute, University of Nevada System, Reno, NV 89507, U.S.A. (Received 12 September 1983; in revised form 25 June 1984) A b s t r a c t - - A novel propulsion concept is proposed which has the potential of accelerating large masses to
velocities substantially higher than what is possible with chemical rockets. The novel concept is an electromagnetic gun, where the projectile is a rocket. The proposed concept solves the old problem of magnetic propulsion, which is the resistive dissipation of the induced electric currents into heat which will vaporize the projectile long before it can reach a high velocity. As in a rocket, where the propellant cools and thereby prevents the rocket from burning up, the same happens in the proposed concept where the propellent also cools the projectile and prevents its vaporization. The propellant, however, not only cools the projectile but in addition is resistively heated by the magnetic field and ejected from the projectile with high velocity. The resulting recoil produces an additional thrust which is approximately as large as the thrust exerted by the magnetic field alone. The energy to drive the jet is externally supplied, making the specific impulse much larger than for chemical rockets.
1. INTRODUCTION For the exploration of our solar system by instruments, the attainment of at least 10 times larger velocities than attainable with chemical rockets would be highly desirable. With chemical multistage rockets - 10 times higher velocities are in principle possible, but only with unrealistic large mass ratios. With the advent of nuclear propulsion in general and nuclear fusion propulsion in particular, much larger velocities are possible in principle, but most likely only for large spacecraft and hardly in the near future. For the more limited goal of exploration by small deep-space probes, a velocity in the range of 30-100 km/sec would be very interesting. Such a velocity would cut down the time to receive data about 10-fold, from several years to a few months. A technical means by which masses could be moved within the solar system at velocities in the range of 3 0 100 km/sec would have also other potential applications, such as the recovery of rare minerals, or for the supply of research stations positioned on other planets. As an alternative to rockets, electromagnetic guns or launchers have been proposed over the years as a means of reaching very high velocities. A proposal to launch a ferromagnetic projectile by a travelling magnetic wave can already be found in the works of Oberth[l]. An electromagnetic launcher, using a superconducting projectile, was proposed by Maissonnier[2] and the author many years ago[3, 41. These proposals were motivated by the concept of impact fusion, predicting the controlled release of thermonuclear energy by the impact of small projectiles (macrons) having velocities o f - 2 0 0 km/sec[5]. Because impact fusion would only need small cm-size projectiles, accelerators of this kind were called macroparticle- or macron-accelerators. More recently, O'Neill and Kolm[6] reinvented the superconducting projectile accelerator as a means of launching large masses. It is for this reason that they called it a mass driver.
Instead of superconducting projectiles one may also use projectiles made up of high-saturation field strength ferromagnets[7], for example the rare earth metals gadolinium or holmium. However, these substances are much too expensive and their use for magnetic propulsion is therefore limited to the acceleration of small projectiles for basic research purposes, such as equation of state studies. Furthermore, neither superconductors nor ferromagnets can be accelerated with the maximum magnetic force derived from the tensile strength of those materials. For superconductors, but also for ferromagnetic substances, the maximum magnetic field is about 5 - 1 0 times smaller than the tensile strength would permit, resulting in a - 2 5 - 1 0 0 times smaller acceleration than would be otherwise theoretically possible. A quite different situation arises if, instead of superconductors or ferromagnets, a good ordinary conductor is used. Currents set up in a good conductor can also lead to a magnetic body force, and if the currents are large enough, the magnetic body force can reach the limit set by the tensile strength. However, another problem now arises, because these large electric currents heat the conductor by resistive losses with the result that the conductor vaporizes before reaching a large velocity. This problem, which is disastrous for small cm-size macroparticles to be accelerated to velocities needed for impact fusion, is also severe for large bodies, if high velocities shall be reached. An estimate of the maximum attainable velocity is given in Appendix 2. To take advantage of the maximum magnetic body force which can act only on a conducting metallic projectile, but to avoid the problem of its burning up by resistive heating, we propose here a novel concept which we shall call the electromagnetic rocket gun. In this concept, the projectile to be accelerated by the magnetic forces is a rocket. As in a rocket, where the propellant cools the rocket and thereby prevents it from burning up, the same happens here, except that the heat is produced by resistive energy losses in the projectile, not by 155
156
F. WINTERBERG
a chemical reaction. In addition, in serving as a heat sink, the propellant of the electromagnetic rocket gun itself is also resistively heated by the magnetic field pushing against the projectile. The resistive heating of the propellant results in an arc burning behind the projectile. This arc heating leads to a large exhaust velocity of the propellant, adding to the thrust of the magnetic body force pushing the projectile. However, unlike a rocket, the propellant receives its energy externally. For a given magnetic field strength the rate of the resistive arc heating goes up in proportion to the projectile velocity and the exhaust velocity also goes up with the projectile velocity. The recoil momentum is transmitted by the magnetic force, acting on the arc-heated plasma, to the gun barrel. Because of both these effects, first the increase in the exhaust velocity with the projectile velocity, and second the momentum transfer to the massive gun barrel, the acceleration is much more efficient than for an ordinary rocket where the exhaust velocity is constant and where the recoil momentum goes into a tenuous jet. The propulsion of the projectile is caused by the same kind of mechanism as in hybrid arc-magnetogasdynamic rocket propulsion. We refer in particular to a paper by Peters[8]. The thermal arc heating alone would produce an exhaust velocity % given by
From eqn (5) it follows that w~ - w~. The total exhaust velocity w is given by w ~ = w~ + w~ ~ 2w~,
(2)
However, it was shown by Maecker[9] that for an archeated plasma expanding in a magnetic field there is in addition to the pure gasdynamic force a magnetohydrodynamic body force. It alone would lead to a plasma velocity w, given by w~ ~ l j l p c 2,
(3)
where 1 and j are the total electric current and current density in the arc plasma, both measured in electrostatic cgs units. With the help of Maxwell's equation (4g/c)j = curl H, where H is the magnetic field, eqn (3) takes the form w~ ~ H 2 / 4 ~ p .
(4)
The arc occurs because the magnetic field diffuses into the plasma which results in resistive heating. If the magnetic field has completely penetrated the plasma, the enthalpy per volume p c , T of the plasma is equal to the magnetic energy density H2/8jr, hence pcpT = H2/8x.
(8)
(l)
where h is the enthalpy of the arc-heated gas. For an ideal gas of specific heat at constant pressure c , , heated by the arc to the temperature T, one has h = ct, T, and hence w~ = 2cpT.
(7)
which shows that the thrust is twice as large if only one of the forces would act alone. Before entering the region where the arc burns, the gas is preheated within the projectile by the transfer of the heat produced by the resistive losses inside the projectile. If not cooled by the propellant, the projectile would burn up as a result of this resistive heating. We therefore may assume a projectile temperature of - 1 0 ~ K, just below the melting point of the projectile material. If, for example, the propellant is hydrogen, it would at such a temperature be expelled from the projectile with a velocity of a few km/sec, for example v,, = 3 x 105 cm/sec. In entering the magnetic field, an electric field E is induced inside the gas moving with the velocity v,, into the magnetic field, which is given by (measured in electrostatic cgs units) E = (v,,Ic)H,
w~ - 2h,
(6)
(5)
where H is the magnetic field strength measured in gauss. If, for example, H ~ l0 s G, it would follow that E 1 ESU = 300 V/cm. This voltage is more than enough to start by electric breakdown an arc in the gas entering the magnetic field. We mention that an alternative concept to reach high projectile velocities, which to some degree avoids the vaporization problem, has been advanced by Rashleigh and Marshall[10]. In their concept a nonmetallic projectile is accelerated by a hot plasma which in turn is propelled by magnetic forces. In particular, they studied the acceleration of a plastic projectile with a magnetic rail gun, where an arc plasma drives the projectile. This concept has two drawbacks. First, it does not permit the direct acceleration of metallic objects where the magnetic body force can become very large. Second, for large accelerations the hot arc plasma ablates at least part of the projectile.
2. DIFFERENT WAYS TO REALIZE THE NEW LAUNCH CONCEPT
The principle of the proposed novel launch concept, called electromagnetic rocket gun, is applicable to all electromagnetic propulsion concepts. As a first example, we illustrate this principle for a railgun. As shown in Fig. l, a rocket-like projectile P is positioned between two conducting rails R~ and R2. The projectile is hollowed out to make space for a propellant P and is equipped with a nozzle. The ideal propellant is liquid hydrogen, but one could also use other propellants, for example lithium-hydride, or even a combustible fuel-propellant
157
Electromagnetic rocket gun
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Ra
P
PL
Fig. 1. Electromagnetic rocket gun principle applied to a railgun. R,, R2: rails which carry current 1 passing through rocket-like projectile. P: propellant, PL: payload, J: exhaust jet.
mixture. Hydrogen is preferred from the standpoint that at a given temperature T, it has the largest energy per mass.t If a current passes through R,, the projectile P and Rz, the projectile is accelerated as in any railgun by the resulting magnetic body force, which is proportional to the product of the current and magnetic field strength. However, unlike in a railgun propelling a nonrockettype conducting projectile, the heat produced by resistive heating is now removed by the evaporation of the propellant inside the projectile. After the propellant has passed through the nozzle, it enters the strong magnetic field positioned behind the projectile. It is thereby resistively heated to high temperatures by an electric arc burning behind the projectiles between the two rails. As a result, it is thereafter ejected at a high velocity in a direction opposite to the direction the projectile moves. As a second example we take a projectile accelerated by a travelling magnetic wave, shown in Fig. 2, where in the projectile azimuthal currents are induced by the travelling magnetic wave, giving it a magnetic moment. The travelling magnetic wave is generated by magnetic field coils which are turned on as the projectile passes
by. The magnetic force acting on the projectile is here proportional to the product of the induced magnetic moment and the magnetic field gradient. Again, as in the case of the railgun configuration, the propellant acts both as a heat sink and as an additional source of thrust to accelerate the projectile. In this configuration an electrodeless toroidal arc bums behind the projectile. One advantage this configuration has in comparison to the railgun is the axial magnetic field. It has the form of a magnetic mirror and therefore can radially confine the exhaust after it has been transformed into a high-temperature plasma. Finally, in Fig. 3 a configuration is shown where even the encasement holding the propellant is evaporated, becoming part of the exhaust. In this configuration, there is no nozzle, a function taken over by the confining magnetic mirror field. This configuration has the advantage of permitting very large mass ratios and therefore very large final velocities. As before, one has here an electrodeless toroidal arc. At high velocities the projectile must be kept away from the wall, because frictional losses would become otherwise disastrous. The projectile can be kept away
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{
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\ Fig. 2. Electromagnetic rocket gun principle applied to a travelling magnetic wave gun. P, PL and J as in Fig. 1; C: magnetic field coils, H: magnetic lines of force. tThis, of course, is also the reason why hydrogen is chosen as a propellant in Nerva-type nuclear rocket reactor propulsion engines.
F. WINTERBERG
158
IVIIk;
Fig. 3. Electromagnetic rocket gun principle applied to projectile in travelling magnetic wave gun, where the part P of the projectile holding the propellant F also vaporizes and together with the propellant becomes part of the jet J; PL: projectile payload, MC: magnetic field coils, H: magnetic lines of force. from the wall by magnetic feedback control, with magnetic or optical sensing of the projectile position. This, of course, is not possible for the railgun configuration where a contact with the wall is needed to let the current pass through the projectile.
3. T H E D Y N A M I C PROBLEM
The magnetic body force density f acting on the projectile is given (in electrostatic units) by f = ( l / c ~ x H,
(10)
we obtain 1
f = - --H 4ze
x curlH.
3nkT = H2/8n,
(9)
where j is the current density vector, H the magnetic field strength and c the velocity of light. From Maxwell's equation (4zc/c)j = curl H,
encasement holding the propellant in case of the configuration shown in Fig. 3. After the vaporized and preheated propellant has left the rearside of the projectile, it is strongly heated through resistive dissipation in entering the region occupied by the strong magnetic field. Because the magnetic field is very strong the propellant becomes a plasma. If the atomic number density of the plasma is n, and if all the magnetic energy, with an energy density H 2 / 8 ~ , is converted into heat, one has (see Appendix 1)
(11)
(15)
valid for a singly ionized plasma, for example a fully ionized hydrogen plasma. Equation (15) is a special case of eqn (5). After it is heated to the temperature given by eqn (15), the plasma becomes a jet. The jet is radially confined by the wall of the gun barrel, and in case the projectile is accelereated by a travelling magnetic wave as shown in Figs. 2 and 3, also by the magnetic field. The jet will produce a recoil force acting on the projectile which is given by F~
=
- vdm/dt,
(16)
With the vector identity H x curl H = ( 1 / 2 ) V H 2 - ( H ' V)H,
(12)
where v is the jet velocity relative to the projectile and the mass ejected per unit time. I f p is the density of the jet, one has
dm/dt
and in case VIH[ is perpendicular to the direction of H, making (H • V) H = 0, we have f =
-
V(H2/87r).
(13)
In eqn (13) H2/8re is the usual expression of the magnetic pressure in the Maxwell stress tensor. For the railgun configuration, but also for the travelling magnetic wave configuration, H ± 17IHI, it follows that for a projectile of cross section A the maximum magnetic force F , acting on its rear side is given by FH = A H 2 / 8 z e .
(14)
The resistive heat produced in the projectile vaporizes the propellant which thereafter is ejected at a moderate speed through the nozzle, in case of the configuration shown in Figs. 1 and 2, and together with the metallic
-Apv,
(17)
Fr~ = A p v 2.
(18)
dm/dt
=
and hence
But since 3 n k T = pC-
= HZ/8zc,
(19)
if one neglects the electron mass against the ion mass, one also has Fr~ = A H 2 / 8 r c = FH.
(20)
The total force F acting on the rear side of the projectile is therefore F = Fu + F~c = 2Fr~.
(21)
159
Electromagnetic rocket gun With the mass flow rate given by eqn (17) one has F = -
with the asymptotic expression z
(22)
2vdm/dt.
We now make the special assumption that in each moment during the acceleration of the projectile the jet velocity v is equal to the projectile velocity. We show below that this assumption can be satisfied by a properly programmed evaporation rate of the propellant. If the jet velocity is always equal to the projectile velocity, then, as seen from a frame at rest with the magnetic gun, the jet comes to rest and no kinetic energy is lost into the jet. Under this special mode of operation the efficiency of the rocket drive is always maximized. The equation of motion of the projectile is now given by
Eliminating the time t from eqn (27) and (29) gives us a relation between v and the acceleration length z: z = ( 2 / 3 ) ( m o v ~ / F ) l ( v / v o ) 3'2 -
II.
(31)
For v > > vo this is asymptotically z
~ (2/3)(rnovJo/Z/F)v 3'2.
(32)
The time dependence of the rocket mass, rn = m ( t ) is given by eqn (25) if we insert v = v(t) given by eqn (27). The result is m/mo
mdv/dt
(30)
~ (l/12)(F2/m~,Vo)t 3.
= [1 + ( F / 2 m o V o ) t ] - ' ,
(33)
(23)
= F = - 2vdm/dt,
with the asymptotic form resulting in the differential equation m/mo
~ (2movo/F)t
~.
(34)
(24)
mdv = -2vdm.
From eqn (33) we find for the rate the rocket loses mass: By integration of eqn (24) we find
dm/dt
(25)
v/vo = ( m o / m ) 2,
where rno is the initial projectile mass at the time t = 0, and where v = vo is the initial projectile velocity. Comparing eqn (25) with the rocket equation v = c l n ( m o / m ) , where c is the constant exhaust velocity, we see that with the electromagnetic rocket gun much higher velocities can be reached than with rockets. If, for example, vo = 1 k m / s e c a n d m o / m = 10, thenv = 100km/sec is possible. Of course, the electromagnetic rocket gun needs a gun barrel and the affordable length of this gun barrel sets a practical upper limit for the attainable velocity. If we assume that the total force F = A H ~ / 4 n , acting on the rocket-like projectile during its acceleration is constant, the equation of motion of the projectile can be integrated in closed form. We have dv/dt
= F/m
(26)
= ( F / m o ) (v/vo) z,
from which we obtain the first integral
= -
(F/2v,,)[1 + (F/2moVo)t] -2,
(35)
with the asymptotic limit din~dr
~ (2m~,vo/F)t -2.
(36)
The power needed to drive the projectile is given by P = F v = A(H2/4~z)v.
(37)
Finally, the propulsion efficiency is given by (1/2)mv 2 q = (l/2)m,,v~, + fo' Pdt"
(38)
With P d t = F v d t = - 2v2dm, eqn (38) can be brought into the form q = [(moV2o/mV2) + (4/mv2) f~o v2dm]_~.
(39)
With the help of eqn (25) we then find (27)
v = vo[l + (F/2moV,,)t]%
q = 3/[4 - (m/m,,i3].
(40)
with the asymptotic expression for large t: v
(28)
~ (1/Vo) ( F / 2 m o ) 2 t "-.
If z is the distance the projectile has moved down the magnetic gun barrel, with z = 0, and v = v,, at t = 0, a second integral is obtained from eqn (27), after putting v = dz/dt: z = v,,
In the limit m / m o ~ 0 one has ~/--~ 3/4 = 75%. For larger mass ratios, r/is even larger. The electromagnetic rocket gun has a lower efficiency than a pure electromagnetic gun, but the price in a lower efficiency is well paid because it prevents the vaporization of the projectile. 4. THE MAXIMUM POSSIBLE ACCELERATION
Io
[1 + (F/2moVo)t]2dt
= (2/3)(moV~,/F)[(! + ( F / 2 m o v o ) t ) 3 -
(29) 1],
The maximum possible acceleration is determined by the tensile strength of the rails or driver coils, but also by the tensile strength of the projectile.
160
F. WINTERBERG
If the tensile strength of the rails or coils is tr. equating the magnetic shear stress H2/4~z with tr~ leads to H
=
Hma x :
(41)
4~/4~.
For ~rs -~ 10 ~° dyn/cm 2, which is typical for steel but also for other high tensile strength materials, one finds Hma~ = 3 x 105 G. The same value of H f o l l o w s for the projectile, if its tensile strength is equal to that of the rails or the coils. The maximum possible acceleration is not only determined by the tensile strength, but also by the length of the projectile. The acceleration a produces inside the projectile a pressure gradient, with the maximum pressure given by Pmax :
where lo is a characteristic length over which the propellant is heated. We thus can write for eqn (49):
pal,
ep = poV,,CpTo/lo.
Continuity requires that poV,, = p v ,
(52)
e.p = p v c p T o / l o .
(53)
and therefore also
To vaporize the propellant one must have ~ -< e. This leads to a condition for a:
(42)
a = ( c - / 2 g c p T o ) (I,,v/l-).
where p is the density of the projectile material and I its length. Equating Pmax with a~ leads to (43)
a .... = a , / p l .
The acceleration is also given by a = F/pAl
= (H~-/4n)/pl, Hma x :
(54)
Equation (54) is the condition for the evaporation rate to match the final jet velocity v. Equation (54), derived under the validity of eqn (52), therefore satisfies the assumption made following eqn (22), which was that the jet velocity is equal the projectile velocity. Typically one has c),To ~ 10~ erg/g, hence
(44) a -- 1011 l , , v / l 2.
and from which again follows
(51)
(55)
4~/~'-~G~. By order of magnitude l,, ~ I and hence
5. T H E R M O D Y N A M I C CONSIDERATIONS
The heat per unit volume generated in the projectile by resistive dissipation is (45)
e = j2/a,
where a is the conductivity of the part of the projectile which carries the current. From Maxwell's equation (10) we have j = (c/4zt)(H/l),
(46)
where l is a characteristic length and which by order of magnitude is equal to the length of the conducting projectile. We thus have C2
H 2
2~zcrl2 8~z'
(47)
With the help of eqn (19) this can be also written as follows e = (c2/2~za)(pv2/12).
a-
10 jl v / l .
(56)
For example, if v - 3 × 106 cm/sec and l - - 30 cm, one would have ~r - 10Wsec, which is in line with the conductivity of steel. After its evaporation the propellant enters the region behind the projectile where the magnetic field rapidly rises. In entering this region the propellant is shock heated as in a theta pinch and ionized. The presence of a strong magnetic field is likely to result in a collisionless shock with an anomalous resistivity. The shock speed is of the order H / ~ v / - ~ p , and which is according to eqn (19) of the same order as the final jet velocity. This suggests a rapid thermalization of the magnetic energy inside the jet. The final density and temperature of the jet are determined by eqn (19) with the result that p = H U 8 ~ v 2,
(57)
T = (a/3R)v
(58)
2.
(48)
The heating rate e; of the propellant of initial density 6. AN E X A M P L E
Po is determined by the equation e~, = poc~,OT/Ot.
(49)
If To is the vaporization temperature and v,, the evaporation velocity one has O T / a t ~- v,,T,,/l,,,
(50)
For a small, high-velocity deep-space probe we may take the following example: v = 100 km/sec, v,, = l km/sec and m = 10 kg. It then follows from eqn (25) that mr = 100 kg. For the initial projectile length we may assume that 1 - 1 m. With a tensile strength ~r, = 10 L°dyn/cm 2 p - 1 g/cm 3 (average density of hydrogen
161
Electromagnetic rocket gun propellant a n d nonpropellant c o m p o n e n t s ) , we find from eqn (43) that a = ama~ = 108 c m / s e c 2. A c c o r d i n g to eqn (44) F / A = a l p = o's. F o r A = 102 c m we thus find F = 10 ~2dyn. T h e length o f the accelerator is given by e q n (32) and we find z = 750 m. At v = 107 c m / s e c one c o m p u t e s from e q n (57) p --~ 5 × 10 -5 g / c m 3, or n = 3 × 10 '9 c m -3, and from eqn(58) T--4 × 105K. To reach for the s a m e payload a final velocity o f 300 k m / s e c w o u l d m a k e the accelerator ~ 10 k m long.
netic body force density in the conductor given by f = (l/c)j × H,
(A.2.1)
where j is the current density vector. With Maxwell's equation, (4n/c)j = curl H, one obtains from eqn (A.2.1) f = ( 4 n l / c 2 ) j 2,
(A.2.2)
where I is a length which, up to a factor of order unity, is equal to the linear dimension of the conductor. The rate of the resistive energy dissipation in the conductor is t = j2/a,
(A.2,3)
APPENDIX 1
Here we describe the formation of the arc burning behind the projectile. If a plasma of finite conductivity mixes with a magnetic field, eddy currents dissipated within the plasma heat up the plasma until the magnetic field energy becomes equal to the internal plasma energy. This fact, expressed by eqn (15), can be easily proved. The mixing and resistive heating can occur in a number of ways. The most important modes of dissipation are turbulence and shock waves, the first one for subsonic and the second one for supersonic flow of the propellant relative to the magnetic field. Furthermore, in the presence of magnetic fields, collisionless shock waves are possible which imply an anomalous high resistivity of noncollisional origin. In the second and third projectile acceleration mode, an axial magnetic field radially implodes the propellant with the heating of the propellant likely to resemble the heating of a plasma by an imploding theta pinch. In the first configuration, that is the railgun configuration, the propellant passes perpendicular through the magnetic field. There turbulent heating is probably preceded by a plane shock parallel to the magnetic field. Using electrostatic cgs units, the resistive dissipation rate per unit volume is given by e = j2/a,
(A.I.I)
where o- is the conductivity. Due to this resistive heating, the conductor melts after the time to given by pcpTm,x = tto,
where p is its density, cp its specific heat at constant pressure and T~0~ its melting temperature. The acceleration of the conductor is given by a = f/p
c2
0t - 4rco-
= 4~rljZ/pc 2,
(A.2.5)
and therefore the maximum velocity Vm,x = ato = (4rcal%T~,~)/c 2.
(A.2.6)
At temperatures near the melting point the conductivity is comparatively low, even if it was high at normal temperatures. We may therefore assume that a ~ 10 '6 sec-'. Then, for the example cpT,,,~ ~ 2 × 109 erg/g, and l ~ 10 cm, we find v.,,~ ~ 20 km/sec. In the reality the maximum velocity which can be reached is about 10 times smaller, since a space probe to be accelerated is likely to tolerate only temperatures which are ~ 10 times smaller than the melting temperature. For large projectiles, vm,~goes up in proportion to l, but only at the price of an accelerator rapidly increasing in its length. The minimum length L~m of the accelerator is given by
where j and a are the plasma current density and conductivity. The diffusion of the magnetic field into the plasma is given by 0H
(A.2.4)
L,,i. = v2.,,~/2a ....
(A.2.7)
where V-'H,
(A.I.2) a,,,x = fm,~/P = (1/pc)j.,,,H~,x ~ HZ.,J4z~pl,
which leads to the characteristic diffusion time r = 41~a22/c 2,
(A. 1.3)
where 2 is a characteristic length over which the magnetic field changes. In case of turbulent dissipation 2 is a measure for the linear dimension of a turbulent cell and in case of a shock wave of the shock thickness. The total energy which can be dissipated is e = er = 4~z(j2/c) 2.
(A.I.4)
With the help of Maxwell's equation (41t/c)j = curl H, we find that e ~ H2/8~z.
(A. 1.5)
A rigorous albeit less transparent treatment of the same problem gives e = H2/87~. For a singly ionized hydrogen plasma e = 3 n k T and we thus obtain eqn (15).
APPENDIX 2
To compute the maximum velocity attainable by the magnetic acceleration of an ordinary conductor, we start with the mag-
(A.2.8)
where we made use of Maxwell's equation (4n/c)jma, = Inserting the expressions for Vma~and a~,~ into eqn (A.2.7) we find OHmJ3Z ~ HmJl.
Lmin = 327r3(ffCpTm,x/C¥1max)2pl 3,
which shows that
L m i n o~
(A.2.9)
13.
REFERENCES
1. H. Oberth, W e g e z u r R a u m s c h i f f a h r t . R. Oldenbourg, Munich (1929). 2. C. Maissonnier, N u o v o C i m e n t o 42B, 332-340 (1966). 3. E Winterberg, J. Nucl. E n e r g y Part C 8, 541-553 (1966). 4. D. Anderson, S. Claflin and E Winterberg, Z. N a t u r f o r s c h 26a, 1415-1424 (1971). 5. E Winterberg, P l a s m a P h y s i c s 10, 55-77 (1968). 6. H. Kolm, P. Mongeau and F. Williams, I E E E Trans. M a g netics 16, 719-721 (1980). 7. R. L. Garwin, R. A. Muller and B. Richter, A t o m k e r n e n e r g i e / K e r n t e c h n i k 35, 300-301 (1980). 8. Th. Peters, Z.f. Na!fg. 19a, 1129-I 130 (1964). 9. H. Maecker, Z.f. Phys. 141, 198-216 (1955). 10. S. C. Rashleigh and R. A. Marshall, J. Appl. P h y s . 49, 2540-2542 (1978).
0094-5765/85 $3.00+.00 Pergamon Press Ltd.
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THE ELECTROMAGNETIC ROCKET GUN F. W I N T E R B E R G
Desert Research Institute, University of Nevada System, Reno, NV 89507, U.S.A. (Received 12 September 1983; in revised form 25 June 1984) A b s t r a c t - - A novel propulsion concept is proposed which has the potential of accelerating large masses to
velocities substantially higher than what is possible with chemical rockets. The novel concept is an electromagnetic gun, where the projectile is a rocket. The proposed concept solves the old problem of magnetic propulsion, which is the resistive dissipation of the induced electric currents into heat which will vaporize the projectile long before it can reach a high velocity. As in a rocket, where the propellant cools and thereby prevents the rocket from burning up, the same happens in the proposed concept where the propellent also cools the projectile and prevents its vaporization. The propellant, however, not only cools the projectile but in addition is resistively heated by the magnetic field and ejected from the projectile with high velocity. The resulting recoil produces an additional thrust which is approximately as large as the thrust exerted by the magnetic field alone. The energy to drive the jet is externally supplied, making the specific impulse much larger than for chemical rockets.
1. INTRODUCTION For the exploration of our solar system by instruments, the attainment of at least 10 times larger velocities than attainable with chemical rockets would be highly desirable. With chemical multistage rockets - 10 times higher velocities are in principle possible, but only with unrealistic large mass ratios. With the advent of nuclear propulsion in general and nuclear fusion propulsion in particular, much larger velocities are possible in principle, but most likely only for large spacecraft and hardly in the near future. For the more limited goal of exploration by small deep-space probes, a velocity in the range of 30-100 km/sec would be very interesting. Such a velocity would cut down the time to receive data about 10-fold, from several years to a few months. A technical means by which masses could be moved within the solar system at velocities in the range of 3 0 100 km/sec would have also other potential applications, such as the recovery of rare minerals, or for the supply of research stations positioned on other planets. As an alternative to rockets, electromagnetic guns or launchers have been proposed over the years as a means of reaching very high velocities. A proposal to launch a ferromagnetic projectile by a travelling magnetic wave can already be found in the works of Oberth[l]. An electromagnetic launcher, using a superconducting projectile, was proposed by Maissonnier[2] and the author many years ago[3, 41. These proposals were motivated by the concept of impact fusion, predicting the controlled release of thermonuclear energy by the impact of small projectiles (macrons) having velocities o f - 2 0 0 km/sec[5]. Because impact fusion would only need small cm-size projectiles, accelerators of this kind were called macroparticle- or macron-accelerators. More recently, O'Neill and Kolm[6] reinvented the superconducting projectile accelerator as a means of launching large masses. It is for this reason that they called it a mass driver.
Instead of superconducting projectiles one may also use projectiles made up of high-saturation field strength ferromagnets[7], for example the rare earth metals gadolinium or holmium. However, these substances are much too expensive and their use for magnetic propulsion is therefore limited to the acceleration of small projectiles for basic research purposes, such as equation of state studies. Furthermore, neither superconductors nor ferromagnets can be accelerated with the maximum magnetic force derived from the tensile strength of those materials. For superconductors, but also for ferromagnetic substances, the maximum magnetic field is about 5 - 1 0 times smaller than the tensile strength would permit, resulting in a - 2 5 - 1 0 0 times smaller acceleration than would be otherwise theoretically possible. A quite different situation arises if, instead of superconductors or ferromagnets, a good ordinary conductor is used. Currents set up in a good conductor can also lead to a magnetic body force, and if the currents are large enough, the magnetic body force can reach the limit set by the tensile strength. However, another problem now arises, because these large electric currents heat the conductor by resistive losses with the result that the conductor vaporizes before reaching a large velocity. This problem, which is disastrous for small cm-size macroparticles to be accelerated to velocities needed for impact fusion, is also severe for large bodies, if high velocities shall be reached. An estimate of the maximum attainable velocity is given in Appendix 2. To take advantage of the maximum magnetic body force which can act only on a conducting metallic projectile, but to avoid the problem of its burning up by resistive heating, we propose here a novel concept which we shall call the electromagnetic rocket gun. In this concept, the projectile to be accelerated by the magnetic forces is a rocket. As in a rocket, where the propellant cools the rocket and thereby prevents it from burning up, the same happens here, except that the heat is produced by resistive energy losses in the projectile, not by 155
156
F. WINTERBERG
a chemical reaction. In addition, in serving as a heat sink, the propellant of the electromagnetic rocket gun itself is also resistively heated by the magnetic field pushing against the projectile. The resistive heating of the propellant results in an arc burning behind the projectile. This arc heating leads to a large exhaust velocity of the propellant, adding to the thrust of the magnetic body force pushing the projectile. However, unlike a rocket, the propellant receives its energy externally. For a given magnetic field strength the rate of the resistive arc heating goes up in proportion to the projectile velocity and the exhaust velocity also goes up with the projectile velocity. The recoil momentum is transmitted by the magnetic force, acting on the arc-heated plasma, to the gun barrel. Because of both these effects, first the increase in the exhaust velocity with the projectile velocity, and second the momentum transfer to the massive gun barrel, the acceleration is much more efficient than for an ordinary rocket where the exhaust velocity is constant and where the recoil momentum goes into a tenuous jet. The propulsion of the projectile is caused by the same kind of mechanism as in hybrid arc-magnetogasdynamic rocket propulsion. We refer in particular to a paper by Peters[8]. The thermal arc heating alone would produce an exhaust velocity % given by
From eqn (5) it follows that w~ - w~. The total exhaust velocity w is given by w ~ = w~ + w~ ~ 2w~,
(2)
However, it was shown by Maecker[9] that for an archeated plasma expanding in a magnetic field there is in addition to the pure gasdynamic force a magnetohydrodynamic body force. It alone would lead to a plasma velocity w, given by w~ ~ l j l p c 2,
(3)
where 1 and j are the total electric current and current density in the arc plasma, both measured in electrostatic cgs units. With the help of Maxwell's equation (4g/c)j = curl H, where H is the magnetic field, eqn (3) takes the form w~ ~ H 2 / 4 ~ p .
(4)
The arc occurs because the magnetic field diffuses into the plasma which results in resistive heating. If the magnetic field has completely penetrated the plasma, the enthalpy per volume p c , T of the plasma is equal to the magnetic energy density H2/8jr, hence pcpT = H2/8x.
(8)
(l)
where h is the enthalpy of the arc-heated gas. For an ideal gas of specific heat at constant pressure c , , heated by the arc to the temperature T, one has h = ct, T, and hence w~ = 2cpT.
(7)
which shows that the thrust is twice as large if only one of the forces would act alone. Before entering the region where the arc burns, the gas is preheated within the projectile by the transfer of the heat produced by the resistive losses inside the projectile. If not cooled by the propellant, the projectile would burn up as a result of this resistive heating. We therefore may assume a projectile temperature of - 1 0 ~ K, just below the melting point of the projectile material. If, for example, the propellant is hydrogen, it would at such a temperature be expelled from the projectile with a velocity of a few km/sec, for example v,, = 3 x 105 cm/sec. In entering the magnetic field, an electric field E is induced inside the gas moving with the velocity v,, into the magnetic field, which is given by (measured in electrostatic cgs units) E = (v,,Ic)H,
w~ - 2h,
(6)
(5)
where H is the magnetic field strength measured in gauss. If, for example, H ~ l0 s G, it would follow that E 1 ESU = 300 V/cm. This voltage is more than enough to start by electric breakdown an arc in the gas entering the magnetic field. We mention that an alternative concept to reach high projectile velocities, which to some degree avoids the vaporization problem, has been advanced by Rashleigh and Marshall[10]. In their concept a nonmetallic projectile is accelerated by a hot plasma which in turn is propelled by magnetic forces. In particular, they studied the acceleration of a plastic projectile with a magnetic rail gun, where an arc plasma drives the projectile. This concept has two drawbacks. First, it does not permit the direct acceleration of metallic objects where the magnetic body force can become very large. Second, for large accelerations the hot arc plasma ablates at least part of the projectile.
2. DIFFERENT WAYS TO REALIZE THE NEW LAUNCH CONCEPT
The principle of the proposed novel launch concept, called electromagnetic rocket gun, is applicable to all electromagnetic propulsion concepts. As a first example, we illustrate this principle for a railgun. As shown in Fig. l, a rocket-like projectile P is positioned between two conducting rails R~ and R2. The projectile is hollowed out to make space for a propellant P and is equipped with a nozzle. The ideal propellant is liquid hydrogen, but one could also use other propellants, for example lithium-hydride, or even a combustible fuel-propellant
157
Electromagnetic rocket gun
/ Ri I
~
~
~
~
~
I
~
. . . . . . . . . . . . . . . . . . . . . .
~
~
~
$
i!iiiii i;i i!iiiili!!iiii i!':iii',',l? •
I
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/
Ra
P
PL
Fig. 1. Electromagnetic rocket gun principle applied to a railgun. R,, R2: rails which carry current 1 passing through rocket-like projectile. P: propellant, PL: payload, J: exhaust jet.
mixture. Hydrogen is preferred from the standpoint that at a given temperature T, it has the largest energy per mass.t If a current passes through R,, the projectile P and Rz, the projectile is accelerated as in any railgun by the resulting magnetic body force, which is proportional to the product of the current and magnetic field strength. However, unlike in a railgun propelling a nonrockettype conducting projectile, the heat produced by resistive heating is now removed by the evaporation of the propellant inside the projectile. After the propellant has passed through the nozzle, it enters the strong magnetic field positioned behind the projectile. It is thereby resistively heated to high temperatures by an electric arc burning behind the projectiles between the two rails. As a result, it is thereafter ejected at a high velocity in a direction opposite to the direction the projectile moves. As a second example we take a projectile accelerated by a travelling magnetic wave, shown in Fig. 2, where in the projectile azimuthal currents are induced by the travelling magnetic wave, giving it a magnetic moment. The travelling magnetic wave is generated by magnetic field coils which are turned on as the projectile passes
by. The magnetic force acting on the projectile is here proportional to the product of the induced magnetic moment and the magnetic field gradient. Again, as in the case of the railgun configuration, the propellant acts both as a heat sink and as an additional source of thrust to accelerate the projectile. In this configuration an electrodeless toroidal arc bums behind the projectile. One advantage this configuration has in comparison to the railgun is the axial magnetic field. It has the form of a magnetic mirror and therefore can radially confine the exhaust after it has been transformed into a high-temperature plasma. Finally, in Fig. 3 a configuration is shown where even the encasement holding the propellant is evaporated, becoming part of the exhaust. In this configuration, there is no nozzle, a function taken over by the confining magnetic mirror field. This configuration has the advantage of permitting very large mass ratios and therefore very large final velocities. As before, one has here an electrodeless toroidal arc. At high velocities the projectile must be kept away from the wall, because frictional losses would become otherwise disastrous. The projectile can be kept away
/C
/
,//
'H
I
::ii~iii~:~:~::::~.::::::=:~:~:~:~:~:~:~:~:~:~:i:i:i:~:i:!:!:~:i:i:i:i:i:i:i:::::i:i:i:i:i:!:!:i:!:i:i:i::.:.:.:..... _
{
\
\ Fig. 2. Electromagnetic rocket gun principle applied to a travelling magnetic wave gun. P, PL and J as in Fig. 1; C: magnetic field coils, H: magnetic lines of force. tThis, of course, is also the reason why hydrogen is chosen as a propellant in Nerva-type nuclear rocket reactor propulsion engines.
F. WINTERBERG
158
IVIIk;
Fig. 3. Electromagnetic rocket gun principle applied to projectile in travelling magnetic wave gun, where the part P of the projectile holding the propellant F also vaporizes and together with the propellant becomes part of the jet J; PL: projectile payload, MC: magnetic field coils, H: magnetic lines of force. from the wall by magnetic feedback control, with magnetic or optical sensing of the projectile position. This, of course, is not possible for the railgun configuration where a contact with the wall is needed to let the current pass through the projectile.
3. T H E D Y N A M I C PROBLEM
The magnetic body force density f acting on the projectile is given (in electrostatic units) by f = ( l / c ~ x H,
(10)
we obtain 1
f = - --H 4ze
x curlH.
3nkT = H2/8n,
(9)
where j is the current density vector, H the magnetic field strength and c the velocity of light. From Maxwell's equation (4zc/c)j = curl H,
encasement holding the propellant in case of the configuration shown in Fig. 3. After the vaporized and preheated propellant has left the rearside of the projectile, it is strongly heated through resistive dissipation in entering the region occupied by the strong magnetic field. Because the magnetic field is very strong the propellant becomes a plasma. If the atomic number density of the plasma is n, and if all the magnetic energy, with an energy density H 2 / 8 ~ , is converted into heat, one has (see Appendix 1)
(11)
(15)
valid for a singly ionized plasma, for example a fully ionized hydrogen plasma. Equation (15) is a special case of eqn (5). After it is heated to the temperature given by eqn (15), the plasma becomes a jet. The jet is radially confined by the wall of the gun barrel, and in case the projectile is accelereated by a travelling magnetic wave as shown in Figs. 2 and 3, also by the magnetic field. The jet will produce a recoil force acting on the projectile which is given by F~
=
- vdm/dt,
(16)
With the vector identity H x curl H = ( 1 / 2 ) V H 2 - ( H ' V)H,
(12)
where v is the jet velocity relative to the projectile and the mass ejected per unit time. I f p is the density of the jet, one has
dm/dt
and in case VIH[ is perpendicular to the direction of H, making (H • V) H = 0, we have f =
-
V(H2/87r).
(13)
In eqn (13) H2/8re is the usual expression of the magnetic pressure in the Maxwell stress tensor. For the railgun configuration, but also for the travelling magnetic wave configuration, H ± 17IHI, it follows that for a projectile of cross section A the maximum magnetic force F , acting on its rear side is given by FH = A H 2 / 8 z e .
(14)
The resistive heat produced in the projectile vaporizes the propellant which thereafter is ejected at a moderate speed through the nozzle, in case of the configuration shown in Figs. 1 and 2, and together with the metallic
-Apv,
(17)
Fr~ = A p v 2.
(18)
dm/dt
=
and hence
But since 3 n k T = pC-
= HZ/8zc,
(19)
if one neglects the electron mass against the ion mass, one also has Fr~ = A H 2 / 8 r c = FH.
(20)
The total force F acting on the rear side of the projectile is therefore F = Fu + F~c = 2Fr~.
(21)
159
Electromagnetic rocket gun With the mass flow rate given by eqn (17) one has F = -
with the asymptotic expression z
(22)
2vdm/dt.
We now make the special assumption that in each moment during the acceleration of the projectile the jet velocity v is equal to the projectile velocity. We show below that this assumption can be satisfied by a properly programmed evaporation rate of the propellant. If the jet velocity is always equal to the projectile velocity, then, as seen from a frame at rest with the magnetic gun, the jet comes to rest and no kinetic energy is lost into the jet. Under this special mode of operation the efficiency of the rocket drive is always maximized. The equation of motion of the projectile is now given by
Eliminating the time t from eqn (27) and (29) gives us a relation between v and the acceleration length z: z = ( 2 / 3 ) ( m o v ~ / F ) l ( v / v o ) 3'2 -
II.
(31)
For v > > vo this is asymptotically z
~ (2/3)(rnovJo/Z/F)v 3'2.
(32)
The time dependence of the rocket mass, rn = m ( t ) is given by eqn (25) if we insert v = v(t) given by eqn (27). The result is m/mo
mdv/dt
(30)
~ (l/12)(F2/m~,Vo)t 3.
= [1 + ( F / 2 m o V o ) t ] - ' ,
(33)
(23)
= F = - 2vdm/dt,
with the asymptotic form resulting in the differential equation m/mo
~ (2movo/F)t
~.
(34)
(24)
mdv = -2vdm.
From eqn (33) we find for the rate the rocket loses mass: By integration of eqn (24) we find
dm/dt
(25)
v/vo = ( m o / m ) 2,
where rno is the initial projectile mass at the time t = 0, and where v = vo is the initial projectile velocity. Comparing eqn (25) with the rocket equation v = c l n ( m o / m ) , where c is the constant exhaust velocity, we see that with the electromagnetic rocket gun much higher velocities can be reached than with rockets. If, for example, vo = 1 k m / s e c a n d m o / m = 10, thenv = 100km/sec is possible. Of course, the electromagnetic rocket gun needs a gun barrel and the affordable length of this gun barrel sets a practical upper limit for the attainable velocity. If we assume that the total force F = A H ~ / 4 n , acting on the rocket-like projectile during its acceleration is constant, the equation of motion of the projectile can be integrated in closed form. We have dv/dt
= F/m
(26)
= ( F / m o ) (v/vo) z,
from which we obtain the first integral
= -
(F/2v,,)[1 + (F/2moVo)t] -2,
(35)
with the asymptotic limit din~dr
~ (2m~,vo/F)t -2.
(36)
The power needed to drive the projectile is given by P = F v = A(H2/4~z)v.
(37)
Finally, the propulsion efficiency is given by (1/2)mv 2 q = (l/2)m,,v~, + fo' Pdt"
(38)
With P d t = F v d t = - 2v2dm, eqn (38) can be brought into the form q = [(moV2o/mV2) + (4/mv2) f~o v2dm]_~.
(39)
With the help of eqn (25) we then find (27)
v = vo[l + (F/2moV,,)t]%
q = 3/[4 - (m/m,,i3].
(40)
with the asymptotic expression for large t: v
(28)
~ (1/Vo) ( F / 2 m o ) 2 t "-.
If z is the distance the projectile has moved down the magnetic gun barrel, with z = 0, and v = v,, at t = 0, a second integral is obtained from eqn (27), after putting v = dz/dt: z = v,,
In the limit m / m o ~ 0 one has ~/--~ 3/4 = 75%. For larger mass ratios, r/is even larger. The electromagnetic rocket gun has a lower efficiency than a pure electromagnetic gun, but the price in a lower efficiency is well paid because it prevents the vaporization of the projectile. 4. THE MAXIMUM POSSIBLE ACCELERATION
Io
[1 + (F/2moVo)t]2dt
= (2/3)(moV~,/F)[(! + ( F / 2 m o v o ) t ) 3 -
(29) 1],
The maximum possible acceleration is determined by the tensile strength of the rails or driver coils, but also by the tensile strength of the projectile.
160
F. WINTERBERG
If the tensile strength of the rails or coils is tr. equating the magnetic shear stress H2/4~z with tr~ leads to H
=
Hma x :
(41)
4~/4~.
For ~rs -~ 10 ~° dyn/cm 2, which is typical for steel but also for other high tensile strength materials, one finds Hma~ = 3 x 105 G. The same value of H f o l l o w s for the projectile, if its tensile strength is equal to that of the rails or the coils. The maximum possible acceleration is not only determined by the tensile strength, but also by the length of the projectile. The acceleration a produces inside the projectile a pressure gradient, with the maximum pressure given by Pmax :
where lo is a characteristic length over which the propellant is heated. We thus can write for eqn (49):
pal,
ep = poV,,CpTo/lo.
Continuity requires that poV,, = p v ,
(52)
e.p = p v c p T o / l o .
(53)
and therefore also
To vaporize the propellant one must have ~ -< e. This leads to a condition for a:
(42)
a = ( c - / 2 g c p T o ) (I,,v/l-).
where p is the density of the projectile material and I its length. Equating Pmax with a~ leads to (43)
a .... = a , / p l .
The acceleration is also given by a = F/pAl
= (H~-/4n)/pl, Hma x :
(54)
Equation (54) is the condition for the evaporation rate to match the final jet velocity v. Equation (54), derived under the validity of eqn (52), therefore satisfies the assumption made following eqn (22), which was that the jet velocity is equal the projectile velocity. Typically one has c),To ~ 10~ erg/g, hence
(44) a -- 1011 l , , v / l 2.
and from which again follows
(51)
(55)
4~/~'-~G~. By order of magnitude l,, ~ I and hence
5. T H E R M O D Y N A M I C CONSIDERATIONS
The heat per unit volume generated in the projectile by resistive dissipation is (45)
e = j2/a,
where a is the conductivity of the part of the projectile which carries the current. From Maxwell's equation (10) we have j = (c/4zt)(H/l),
(46)
where l is a characteristic length and which by order of magnitude is equal to the length of the conducting projectile. We thus have C2
H 2
2~zcrl2 8~z'
(47)
With the help of eqn (19) this can be also written as follows e = (c2/2~za)(pv2/12).
a-
10 jl v / l .
(56)
For example, if v - 3 × 106 cm/sec and l - - 30 cm, one would have ~r - 10Wsec, which is in line with the conductivity of steel. After its evaporation the propellant enters the region behind the projectile where the magnetic field rapidly rises. In entering this region the propellant is shock heated as in a theta pinch and ionized. The presence of a strong magnetic field is likely to result in a collisionless shock with an anomalous resistivity. The shock speed is of the order H / ~ v / - ~ p , and which is according to eqn (19) of the same order as the final jet velocity. This suggests a rapid thermalization of the magnetic energy inside the jet. The final density and temperature of the jet are determined by eqn (19) with the result that p = H U 8 ~ v 2,
(57)
T = (a/3R)v
(58)
2.
(48)
The heating rate e; of the propellant of initial density 6. AN E X A M P L E
Po is determined by the equation e~, = poc~,OT/Ot.
(49)
If To is the vaporization temperature and v,, the evaporation velocity one has O T / a t ~- v,,T,,/l,,,
(50)
For a small, high-velocity deep-space probe we may take the following example: v = 100 km/sec, v,, = l km/sec and m = 10 kg. It then follows from eqn (25) that mr = 100 kg. For the initial projectile length we may assume that 1 - 1 m. With a tensile strength ~r, = 10 L°dyn/cm 2 p - 1 g/cm 3 (average density of hydrogen
161
Electromagnetic rocket gun propellant a n d nonpropellant c o m p o n e n t s ) , we find from eqn (43) that a = ama~ = 108 c m / s e c 2. A c c o r d i n g to eqn (44) F / A = a l p = o's. F o r A = 102 c m we thus find F = 10 ~2dyn. T h e length o f the accelerator is given by e q n (32) and we find z = 750 m. At v = 107 c m / s e c one c o m p u t e s from e q n (57) p --~ 5 × 10 -5 g / c m 3, or n = 3 × 10 '9 c m -3, and from eqn(58) T--4 × 105K. To reach for the s a m e payload a final velocity o f 300 k m / s e c w o u l d m a k e the accelerator ~ 10 k m long.
netic body force density in the conductor given by f = (l/c)j × H,
(A.2.1)
where j is the current density vector. With Maxwell's equation, (4n/c)j = curl H, one obtains from eqn (A.2.1) f = ( 4 n l / c 2 ) j 2,
(A.2.2)
where I is a length which, up to a factor of order unity, is equal to the linear dimension of the conductor. The rate of the resistive energy dissipation in the conductor is t = j2/a,
(A.2,3)
APPENDIX 1
Here we describe the formation of the arc burning behind the projectile. If a plasma of finite conductivity mixes with a magnetic field, eddy currents dissipated within the plasma heat up the plasma until the magnetic field energy becomes equal to the internal plasma energy. This fact, expressed by eqn (15), can be easily proved. The mixing and resistive heating can occur in a number of ways. The most important modes of dissipation are turbulence and shock waves, the first one for subsonic and the second one for supersonic flow of the propellant relative to the magnetic field. Furthermore, in the presence of magnetic fields, collisionless shock waves are possible which imply an anomalous high resistivity of noncollisional origin. In the second and third projectile acceleration mode, an axial magnetic field radially implodes the propellant with the heating of the propellant likely to resemble the heating of a plasma by an imploding theta pinch. In the first configuration, that is the railgun configuration, the propellant passes perpendicular through the magnetic field. There turbulent heating is probably preceded by a plane shock parallel to the magnetic field. Using electrostatic cgs units, the resistive dissipation rate per unit volume is given by e = j2/a,
(A.I.I)
where o- is the conductivity. Due to this resistive heating, the conductor melts after the time to given by pcpTm,x = tto,
where p is its density, cp its specific heat at constant pressure and T~0~ its melting temperature. The acceleration of the conductor is given by a = f/p
c2
0t - 4rco-
= 4~rljZ/pc 2,
(A.2.5)
and therefore the maximum velocity Vm,x = ato = (4rcal%T~,~)/c 2.
(A.2.6)
At temperatures near the melting point the conductivity is comparatively low, even if it was high at normal temperatures. We may therefore assume that a ~ 10 '6 sec-'. Then, for the example cpT,,,~ ~ 2 × 109 erg/g, and l ~ 10 cm, we find v.,,~ ~ 20 km/sec. In the reality the maximum velocity which can be reached is about 10 times smaller, since a space probe to be accelerated is likely to tolerate only temperatures which are ~ 10 times smaller than the melting temperature. For large projectiles, vm,~goes up in proportion to l, but only at the price of an accelerator rapidly increasing in its length. The minimum length L~m of the accelerator is given by
where j and a are the plasma current density and conductivity. The diffusion of the magnetic field into the plasma is given by 0H
(A.2.4)
L,,i. = v2.,,~/2a ....
(A.2.7)
where V-'H,
(A.I.2) a,,,x = fm,~/P = (1/pc)j.,,,H~,x ~ HZ.,J4z~pl,
which leads to the characteristic diffusion time r = 41~a22/c 2,
(A. 1.3)
where 2 is a characteristic length over which the magnetic field changes. In case of turbulent dissipation 2 is a measure for the linear dimension of a turbulent cell and in case of a shock wave of the shock thickness. The total energy which can be dissipated is e = er = 4~z(j2/c) 2.
(A.I.4)
With the help of Maxwell's equation (41t/c)j = curl H, we find that e ~ H2/8~z.
(A. 1.5)
A rigorous albeit less transparent treatment of the same problem gives e = H2/87~. For a singly ionized hydrogen plasma e = 3 n k T and we thus obtain eqn (15).
APPENDIX 2
To compute the maximum velocity attainable by the magnetic acceleration of an ordinary conductor, we start with the mag-
(A.2.8)
where we made use of Maxwell's equation (4n/c)jma, = Inserting the expressions for Vma~and a~,~ into eqn (A.2.7) we find OHmJ3Z ~ HmJl.
Lmin = 327r3(ffCpTm,x/C¥1max)2pl 3,
which shows that
L m i n o~
(A.2.9)
13.
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