Relativistic rocket: Dream and reality

Abstract

The dream of interstellar flights persists since the first pioneers in astronautics and has never died. Many concepts of thruster capable to propel a rocket to the stars have been proposed and the most suitable among them are thought to be photon propulsion and propulsion by the products of proton–antiproton annihilation in magnetic nozzle. This article addresses both concepts allowing for cross-section of annihilation among other issues in order to show their vulnerability and to indicate the problems. The concept of relativistic matter propulsion is substantiated and discussed. The latter is argued to be the most straightforward way to build-up a relativistic rocket firstly because it is based on the existing technology of ion generators and accelerators and secondly because it can be stepped up in efflux power starting from interplanetary spacecrafts powered by nuclear reactors to interstellar starships powered by annihilation reactors. The problems imposed by thermodynamics and heat disposal are accentuated.

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Relativistic rocket: Dream and reality
Oleg G. Semyonov
State University of New York at Stony Brook, Stony Brook, NY 11794, USA
article info
Article history:
Received 1 November 2013
Received in revised form
24 January 2014
Accepted 29 January 2014
Available online 20 February 2014
Keywords:
Relativistic rocket
Interstellar flight
Relativistic propulsion
Annihilation reactor
abstract
The dream of interstellar flights persists since the first pioneers in astronautics and has
never died. Many concepts of thruster capable to propel a rocket to the stars have been
proposed and the most suitable among them are thought to be photon propulsion and
propulsion by the products of proton–antiproton annihilation in magnetic nozzle. This
article addresses both concepts allowing for cross-section of annihilation among other
issues in order to show their vulnerability and to indicate the problems. The concept of
relativistic matter propulsion is substantiated and discussed. The latter is argued to be the
most straightforward way to build-up a relativistic rocket firstly because it is based on the
existing technology of ion generators and accelerators and secondly because it can be
stepped up in efflux power starting from interplanetary spacecrafts powered by nuclear
reactors to interstellar starships powered by annihilation reactors. The problems imposed
by thermodynamics and heat disposal are accentuated.
&2014 IAA. Published by Elsevier Ltd. All rights reserved.
1. Introduction
To accelerate a starship with a reasonable payload to
a relativistic speed, the highest conceivable energy-density
fuel is needed and the prime candidate would be anti-
matter, actually, a large amount of it stored in a compact
form and safely delivered to a rocket engine with a proper
flow rate [1]. So far, there are no unambiguous conception
for annihilation thruster and no practical solution for
antimatter storage onboard. Other challenges are to be
solved before attempting extra-solar relativistic flights.
Among them, the problems of protection from hard ionizing
radiation of oncoming relativistic flow of interstellar gas
and shielding against relativistic dust bombardment, when
a spacecraft moves with a relativistic speed through inter-
stellar space, are the most arduous [2].
Putting aside such sci-fi projects as "warp drive" [3,4],
the prospects of jet propulsion for reaching the stars will
be discussed below, where the term ‘jet’is used in its
general sense as a flow of material entities either massive
or massless but carrying a mechanical momentum. As a
matter of fact, there is no other cause for locomotion of
massive object relative to other objects in the material
world but mechanical reaction-propulsion governed by
Newton's first law and the law of momentum conserva-
tion. To move forward with respect to physical environ-
ment and to acquire a relative momentum, one has to push
back something and to impart to it a backward momen-
tum: when we walk or run, we push back the Earth by our
legs; when we swim, we push back water; when we fly by
a plane, a propeller pushes back air or a jet engine jettisons
gas backward. The phenomenon of reaction–propulsion
lies in the very foundation of matter: interaction (action
and counteraction) of material objects by means of physi-
cal fields is the only driving force that makes their relative
movement.
Among the concepts of propulsion using the products
of matter–antimatter annihilation, the traditional one is
photon rocket propelled by a jet of gamma photons
resulted from annihilation of electrons e

and positrons
Contents lists available at ScienceDirect
journal homepage: www.elsevier.com/locate/actaastro
Acta Astronautica
http://dx.doi.org/10.1016/j.actaastro.2014.01.027
0094-5765 &2014 IAA. Published by Elsevier Ltd. All rights reserved.
E-mail address: oleg.semyonov.1@stonybrook.edu
Acta Astronautica 99 (2014) 52–70

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e
þ
[5,6]. Two or several e
þ
and e

beams are supposed to
cross in the focal spot of a parabolic dish. Each act of
annihilation releases two γ-photons in opposite directions,
and one or both photons impact the parabolic dish mirror
depending upon the axial extent of the dish. The attrac-
tiveness of the concept is assumed to be the complete
transformation of matter to energy with the highest
possible velocity of the efflux beam formed by the dish.
The hurdles of the method such as formation of high-
current e
þ
and e

beams, their transportation, and focus-
ing were usually omitted from consideration and the
effectiveness of annihilation in a focal spot of the crossed
beams, determined by the cross-section of annihilation
reaction, has not been taken into account. A huge problem
is the mirror itself. There is no material capable of total
reflection of γ-radiation thus parabolic mirror –a critical
part of annihilation-driven photon rocket –is virtually
improbable. The radiation-absorbing dish [5], in its turn,
means handling a huge amount of heat deposited in the
material by the flux of high-energy photons. Among other
ideas, the concept of Winterberg [7] can be mentioned,
according to which the radial collapse of a cylindrical
plasma column could cause dramatic compression of mat-
ter–antimatter ambiplasma (a mixture of two) and forma-
tion of a thin thread with near nuclear density consisting of
positronium atoms, which generates a coherent and direc-
tional beam of gamma-rays to produce thrust. However, the
assumed uniform compression is hardly realizable in prac-
tice. Self-compression of a plasma column may occur,
if pressure of magnetic field around it overwhelms thermal
pressure of compressing plasma, and this is possible when
radiative losses from plasma are sufficiently high to make
compression non-adiabatic. Magnetic compression of
current-carrying plasma (and ambiplasma is not an excep-
tion) is usually accompanied by magnetohydrodynamic
instabilities effectively disrupting the uniform radial com-
pression, so only some small point-like areas can pinch
down to relatively small radii [8,9].
Another option for annihilation propulsion is baryonic
matter–antimatter annihilation. The commonly discussed
is the thruster based on annihilation of protons p
þ
and
antiprotons p

. Two main approaches have been sug-
gested: (a) products of proton–antiproton annihilation
create the thrust directly and (2) products of annihilation
energize and heat an intermediate substance (propellant)
in the nozzle through interparticle collisions [1,10–13].
Since p
þ
p

annihilation does not produce gamma-quanta
immediately, charged and neutral mesons such as π
þ
,π

,
and π
0
-pions are generated first. Neutral π
0
-pions virtually
instantly decay into two γ-photons with their energy of
200 MeV each. Charged pions decay into μ
þ,
-mesons
(muons) which, in their turn, decay into electron or
positron and corresponding neutrinos. The mechanical
momentum of charged particles having relatively long
decay time can be used for direct thrust production in a
magnetic nozzle [1,5,11]. The inevitable energy loss to
γ-photons, which is a waste from the point of view of
engine effectiveness, inspired a number of concepts, the
main feature of which was utilization of energy of baryonic
annihilation for heating up a working substance such as
compressed gas (e.g., [11], pp.118–123), magnetically
contained plasma (ibid., p. 124), or solid core (specifically
designed multi-layered target for fission-fusion micro-
explosion) [1,14]. The fusion–fission approach outgrows
from the projects on magnetic confinement fusion [15]
and inertial confinement fusion (ICF) [16]. Despite the
enthusiasm of the proponents, the staged conversion of
annihilation energy to a heated propellant is a tricky
business especially in the case of ICF microexplosions of
pretty complicated targets, and the achievable impulse
10
4
s[1] is not impressive. As for the direct thrust
production by annihilation products, the details of forma-
tion and transportation of beams of protons and antipro-
tons are ususlly omitted from consideration and neither
cross-section of p
þ
p

annihilation, which determines the
annihilation rate and effectiveness, has been accounted for.
Putting aside antimatter confinement and storage (the
review of possible methods can be found in [11], see also
Section 5 below), the concept of direct propulsion by the
products of annihilation is revised in this article. Forma-
tion of beams of charged particles, their transportation,
and limiting factors that determine the rate of annihilation
are discussed. Photon rocket and proton–antiproton anni-
hilation rocket are analyzed accounting for electrostatic
interaction of beams, which results in high relative velo-
city of particles and antiparticles thus reducing effective-
ness of annihilation in a limited-size ambiplasma as a
consequence of the reduced annihilation cross-section.
An alternative concept of relativistic rocket propulsion by
a jet of high-energy matter accelerated in a sort of linear
accelerator and powered by a separate energy source is
outlined. The method is based on the existing technology
of accelerators of charged particles, in particular, ion
thrusters scaled to higher power with high kinetic ener-
gies of massive particles in the efflux. The limiting factors
imposed by thermodynamics on direct annihilation rock-
ets and relativistic matter propelled rockets are discussed.
2. Annihilation cross-section and transportation of
beams of charged particles
2.1. Cross-sections of electron–positron and
proton–antiproton annihilation
Positron annihilation in collisions with electrons in
substances and antiprotons annihilation in collisions with
protons or heavier nuclei have been studied both theore-
tically and experimentally. Direct annihilation of a posi-
tron–electron pair at rest results in two oppositely directed
γ-photons, each of energy E
γ
¼m
0
c
2
¼511 keV in the rest
frame, where m
0
is the mass of rest of electron. Annihila-
tion ‘in-flight’, i.e. when the relative momentum between
colliding electron and positron is not zero, is more com-
plicated. The expression for the cross-section for e
þ
e

-
annihilation as a function of their relative velocity νwas
first given by Dirac [17] and for nonrelativistic vit can be
written as
s
ee
¼πr
2
e
=β
r
;ð1Þ
where r
e
¼2.818 10
15
m is the classical radius of elec-
tron, β
r
¼ν/c, and c¼310
8
m/s is the speed of light in
vacuum. The 1/νdependence of annihilation cross-section
O.G. Semyonov / Acta Astronautica 99 (2014) 52–70 53

Author's personal copy
is usually associated with the so-called Sommerfeld
enhancement factor for the attractive electric potential in
space V(r) around a particle as felt by an incident particle
[18]. The representation of the Dirac theory in a general
case is given by Heitler [19]
s
ee
¼πr
2
e
1
γþ1
γ
2
þ4γþ1
γ
2
1lnðγþffiffiffiffiffiffiffiffiffiffiffiffi
γ
2
1
pÞ γþ3
ffiffiffiffiffiffiffiffiffiffiffiffi
γ
2
1
p
"#
¼πr
2
e
B
rel
;
ð2Þ
where γ¼E/m
0
c
2
¼1/(1ν
2
/c
2
)
0.5
is the relativistic factor of
an incident antiparticle relative to a particle (and vice
versa), E¼E
k
þm
0
c
2
is the total energy of an incident
particle, and E
k
is its kinetic energy. The annihilation
cross-section tends to infinity, when ν-0(γ-1), resulting
in catastrophic global annihilation. The cross-section s
ee
-
πr
2
e
, when ν-c(γ-1).
As for p
þ
p

-annihilation cross-section, it also tends to
grow with ν-0 as a consequence of s
pp
enhancement by
Sommerfeld factor for attractive potentials. The empirical
expression for total cross-section s
pp
, which includes
elastic scattering in addition to annihilation, is given in
[20]
s
pp
ðmbÞ¼46:6=p–30:1þ60:3p;ð3Þ
where pis the momentum of an incident particle
expressed in GeV/c(1 GeV/c¼5.344 10
19
kg m/s) and
the cross-section is expressed in millibarns: 1 mb¼
10
3
barn¼10
31
m
2
. The first term in (3) prevails at
low relative kinetic energies and tends to infinity, when
p-0. As a matter of fact, the p
þ
p

cross-section s
pp
estimated from expression (1), where the radius of proton
r
p
¼0.878 10
15
is substituted instead of electron radius,
differs from the empirical one (3) by a factor of two or less
for β
r
o0.6.
On average, a p
þ
p

-pair having low relative kinetic
energy annihilates into five pions with three charged and
two neutral pions although it can in principle decay into
3π
0
however the probability of the decay into purely
neutral mesons is relatively small and their contribution
to the energy balance is less than 4%. [21]. In addition to
pions, η-mesons can be produced with a rate of less than
7% and kaons (K
þ
,K

) with a rate of about 6% of all
annihilation events.
As seen from the dependence of annihilation cross-
section on the relative momentum of annihilating particles
expressed by Eqs. (1)–(3), the most effective annihilation
occurs when particles are nearly at rest relative to each
other. Therefore, the concepts of annihilation drivers based
on beams of charged particles/antiparticles crossing at
high angles as proposed in [22] seem to be impractical
unless both crossing beams are of very low kinetic energy.
However, the mutual attraction of oppositely charged
beams will inevitably result in gaining a high relative
velocity of particles and antiparticles (see Section 2.2)
resulting in much longer annihilation length.
The length of annihilation can be estimated in terms of
mean free path l
ann
of antiparticles in a medium containing
corresponding particles with the density n
l
ann
¼1=ns
a
;ð4Þ
where s
a
is the annihilation cross-section, either s
ee
or s
pp
.
The graphs of annihilation lengths for varying density of
target particles are shown in Fig. 1 as functions of the
relative velocity of colliding particles and antiparticles:
(a) positrons irradiating a target electron beam or a sub-
stance (Fig. 1(a) and (b) antiprotons irradiating a proton
beam or hydrogen target (Fig. 1(b)). In order to obtain the
effective annihilation rate in the reaction zone (ambiplasma)
as small as possible (small annihilation length), either the
relative velocity of colliding particle–antiparticle pairs must
be small or the density of a target material must be high. It
means the crossing beams and anti-beams must be nearly
parallel with matched velocities of particles and the inherent
velocity spread of particles in each beam must be small
while the density of matter and antimatter particles in the
crossing beams ought to be as high as possible.
The annihilation frequency per antiparticle in ambi-
plasma of mixed beams ν
a
ns
a
ν¼ns
a
β
r
c¼β
r
c/l
ann
,where
nis the density of electrons in electron–positron ambiplasma
or protons in antiproton–proton ambiplasma and ν¼β
r
cis
the average relative velocity of the colliding particles.
Fig. 1. (a) Annihilation length of positrons in an electron cloud or a substance and (b) annihilation length of antiprotons in a proton (hydrogen) cloud or
a hydrogen substance vs. their incident relative velocity fordifferent densities of particles in a target. The cross-section for positron–electron annihilation is
calculated from the expressions (1) and (2): both values coincide at the relative velocities β
r
o0.6 and slightly diverge for ν-c.p
þ
p

annihilation cross-
section s(p
þ
p

) calculated from Eq. (3) includes Coulomb scattering of antiprotons seen as the downward slope of the curves, when β
r
-1. Annihilation
prevails and can be calculated from the Eq. (1),ifβ
r
o0.6. Coulomb scattering overwhelms annihilation for β
r
40.6 and antiprotons mainly flyby protons
along hyperbolic trajectories instead of annihilating with them except for the relatively rare face-to-face collisions.
O.G. Semyonov / Acta Astronautica 99 (2014) 52–7054

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Substituting the expression for s
a
from Eqs. (1)and(2)for
nonrelativistic and relativistic approximations (E
k
⪡m
0
c
2
or
E
k
Zm
0
c
2
, correspondingly), where E
k
istherelativekinetic
energy of annihilating particles and m
0
is their mass of rest,
the annihilation frequency can be estimated as
ν
a
¼β
r
c=l
ann
¼πr
2
0
сn¼r
2
0
сN=R
2
;ðβ
r
o0:6Þ
ν
a
¼β
r
c=l
ann
¼r
2
0
сNβ
r
B
rel
=R
2
;ðβ
r
40:6Þð5Þ
where Nisthenumberofparticlesperunitoflengthofa
matter beam, r
2
0
is the radius of electron or proton, corre-
spondingly, and Ris the radius of the ambiplasma. The
number of annihilation events in the mixed particle-
antiparticle ambiplasma per time unit and per unit of length
is
R
a
¼ν
a
nπR
2
¼β
r
cN=l
ann
¼r
2
0
cN
2
=R
2
;ðβ
r
o0:6Þ
R
a
¼β
r
cN=l
ann
¼r
2
0
β
r
cB
rel
N
2
=R
2
;ðβ
r
40:6Þð6Þ
assuming equal densities of particles and antiparticles in the
matter and antimatter beams. In the case of e
þ
e

annihila-
tion, the rate of emission of γ-photons per second and per
unit of length N
ph
¼2R
a
.Inthecaseofp
þ
p

annihilation, the
rate of emission of charged pions per second and per unit of
length is 3R
a
.Itisremarkablethatiftherelativevelocityνis
non-relativistic, the total annihilation rate does not depend
on νand only concentration matters. However, the size of
ambiplasma needed for complete annihilation (annihilation
length) increases with νbecause the mean free pass of
annihilating antiparticles in the mixture is proportional to
the average relative velocity according to (4).
2.2. Formation and transportation of electron (positron) and
proton (antiproton) beams
In view of prospective utilization of beams of charged
particles and antiparticles for reactants delivery to the
annihilation zone, it is appropriate to review shortly the
methods of formation and acceleration of such beams.
Charged particles can be extracted from a heated cathode
(electrons) or a plasma (electrons and ions) by electric
field applied to a sort of cathode–anode assembly. The
accelerated particles pass through the hole in the extrac-
tion electrode and form a beam in the drift volume. There
are two limitations on the beam current in vacuum: (1)
space charge limits the beam current according to the
Child–Langmuir law [23] and (2) magnetic self-field of
a beam (every beam of charged particles is electric
current) prevents further beam propagation when the
Larmor radius of particles in this magnetic field becomes
equal to the beam radius. In the latter case, the Alfven–
Lawson limiting current [24] is estimated as
I
A
¼17β
b
γ
b
kA ð7Þ
where β
b
and γ
b
¼(1β
b
2
)
0.5
are the axial velocity of
particles in the beam with their charges equal to the
absolute value of electron charge and their relativistic
factor, correspondingly. Actually, a beam with I4I
A
will
propagate in space but its transverse kinetic energy
becomes higher then kinetic energy of axial motion of
particles near the axis of the beam [25]. To overcome the
charge limit, the beam must be charge-neutralized, i.e.
either containing the opposite charges with the same
density per unit length the beam or injected in sufficiently
dense plasma. In the latter case, the plasma electrons tend
to move radially in the electric field of the beam adjusting
the charge of plasma inside the beam volume thus
neutralizing the beam's charge. To overcome the Alfven
limit, the beam must be current-neutralized by an oppo-
sitely flowing current to zero the self-magnetic field of the
beam. Upon injecting a beam into plasma, the conditions
of charge and current neutralization can be achieved
automatically due to induction. In the space vacuum,
injecting plasma in the nozzle for neutralization of beams
is apparently not an option because it is a waste of matter
thus an extra load of the rocket unless the plasma itself
consists of particles annihilating with the particles of the
injected beam. A better solution could be mixing the
beams of conventional matter particles with the beams
of their antiparticles in the annihilation zone because the
oppositely charged particles carry the opposite currents
when moving in the same direction thus the conditions
of charge and current neutralization can be satisfied
automatically.
High-energy beams of electrons and ions have been
studied in many applications including generation of high-
power microwaves and inertial fusion [16,26–32]. The
comprehensive review of electron beam (EB) sources and
accelerators is given by Mesyats [31]. Conventionally, the
diode configuration is used to generate EB: high-voltage
electric field is applied between a cathode and an anode
and the specifically profiled cathode serves a source of
electrons that accelerate toward the meshed or annular
anode. To achieve a higher current, plasma is created on
cathode (on anode if ion beam is extracted) so the
electrons move to the anode and the positive ions move
in opposite direction to the cathode. Plasma can be also
created by electrons emitted from the cathode into
injected gas or by an external source of microwave radia-
tion or laser light to ionize the injected gas. Continuously
operating EB diodes generate relatively low-energy elec-
tron beams (up to 50–100 keV) with the current from
microamperes to several amperes. Pulsed EB diodes can
produce mega-ampere beams of much higher electron
energy up to 50 MeV in 20–100 ns pulses (the energies
of particles are commonly measured in millions (MeV)
or billions (GeV) of electron-volts (1 eV ¼1.6 10
19
J))
[29,32].
The first stage of electron or ion accelerator (EA or IA) is
usually a high-voltage diode that generates an electron
beam or a beam of dominant ions in a single-charge state.
One-stage accelerators are based on diode configurations
(mostly coaxial [31]). The ion diodes have already found
their implementation in astronautics and the ion thrusters
are currently used for stationkeeping of communication
satellites and as main propulsion engines on the deep
space probes [33] with a specific impulse up to 6400 s
[34]. One-stage and multi-stage accelerators of high-
current and high-energy beams are developed mostly for
various pulsed power applications. One-stage high-power
mega-ampere machines are relatively slow due to necessity
of changing the damaged anode or cathode because of the
destructive force of concentrated power. [32] In multi-stage
O.G. Semyonov / Acta Astronautica 99 (2014) 52–70 55

Author's personal copy
accelerators, the moderate-power electron or ion diodes are
used as a first stage of acceleration, injector, with the
additional electrodes for further acceleration of charged
particles to higher energies. The potential obtained from
a high-voltage generator is used repeatedly to produce
a running wave of potential synchronized with the velocity
of accelerated particles. If a bunch of particles arrive to
a particular gap between the electrodes in proper phase
with the radio-frequency (RF) wave, the particles are
accelerated across the gap. In a conventional RF accelerator,
the high power microwave sources feed the resonant RF
cavities with metallic or superconducting walls. The typical
frequencies are between 0.5 and 30 GHz and the cavities are
of the order of the microwaves wavelength (60–1cmlinear
dimensions). The important characteristic of an accelerator
is its accelerating gradient which is the measure of how
much the accelerator can increase the energy of a particle
over a certain stretch, typically given in volts per meter
(V/m). The higher is this gradient, the shorter, lighter,
and hence cheaper an accelerator can be made. The
machine generates a beam, consisting of bunches of charged
particles following one after another with high repetition
rate. The output energy of particles can be very high from
tens of megaelectronvolts (MeV) to several gigaelectron-
volts (GeV). The beams of electrons and ions of virtually all
elements can be accelerated. RF accelerator technology [30]
offers acceleration gradients as high as 10–100 MeV/m
[35–38]. Two aspects of RF accelerators make this approach
challenging for high-current applications. The arcing phe-
nomena limit the electric field due to microscopic imperfec-
tions (peaks and pits) in the cavity walls and therefore the
rate at which particles can be accelerated [37]. Conversion
of RF energy to beam energy can potentially reach 80% and,
providing the use of high-efficiency klystrons or magne-
trons (475%), the wallplug efficiency can exceed 50% [39].
Induction accelerators –another approach –have been
actively developed in the United States. These accelerators
can be thought of as a series of single-turn transformers,
with the beam receiving the accelerating axial E
z
from each
as if it were the secondary winding [35,36].TheE
z
pulse in
each cell is timed to coincide with the bunch arrival. Beams
with their currents of more than 100 kA can be accelerated.
Power losses, due to eddy currents, are quite small with the
use of modern ferromagnetic materials, so the induction
accelerators can be designed with wallplug-to-beam effi-
ciencies of more than 40%. The repetition rate of more than
1 kHz can be reached with the ability to accelerate several
beams in a single core [35]. The main drawback is concen-
tration of the acceleration field to the gaps between the cells,
so the gradient is constrained by the 10 MeV/m due to
surface breakdown of insulators.
Positrons and antiprotons can be accelerated in RF and
induction accelerators subject to proper phase of the applied
RF wave or high voltage current in the inductive cells. In
principle, particles and antiparticles can be accelerated
simultaneously, if the corresponding bunches are injected
in the opposite phases of the running electromagnetic wave.
Yet another type of accelerators is emerging. Plasma
wakefield accelerators utilize the wake electromagnetic
waves in plasma for acceleration of charged particles [40].
The main feature of these accelerators is a huge
acceleration gradient 410 GeV/m. The accelerating struc-
ture (or “cavity”) is sustained by plasma electrons. The
acceleration of electrons [41] and positrons [42] has been
demonstrated. The method is in its infancy and its prac-
tical application for propulsion is not clear.
Charged particle beams generated from high-power
accelerators always have velocity components perpendi-
cular to the direction of propagation. This motion originates
from imperfections in the shaping fields, electromagnetic
forces of self-fields, scattering in foils, and random thermal
motions. Transverse motions act to increase the beam
radius, which may be detrimental in applications that
require the beam transport over large distances with no
radial expansion. Since particles in the beams carry their
electrical charges, the measures must be undertaken to
compensate the beam expansion caused by repulsive Cou-
lomb forces both in the accelerators and in the drift volume.
To compensate the beam divergence inside accelerators,
electromagnetic lenses are commonly used. In the drift
volume of an annihilation chamber, the beam radius has to
be kept constant along the beam transportation length
(equilibrium beam) or focusing of the beam must be
implemented in order to increase the density of particles
in the annihilation zone thus introducing their additional
transverse motion. Self-consistent equilibrium can be
achieved in the charge-neutralized beams satisfying Ben-
nett's pinch relation using partial current neutralization or
geometrical profiling of the beam (hollowing-out) with or
without a return current [43–45].
Conventional methods of transportation of high-
current beam can be divided into three types: (1) partial
or complete neutralization of charge and current by
plasma in the drift volume, (2) plasma with the aid of
guiding magnetic field, and (3) transportation along guid-
ing magnetic field in vacuum [45]. Neither seems to be a
satisfactory solution in the magnetic nozzle for direct
rocket propulsion by the annihilation products. Wasting
matter with plasma that flows out of nozzle and loosing
beam energy to induction current and Joule heating in
collisions with plasma electrons/ions make implementa-
tion of guiding plasma undesirable. Guidance by magnetic
field leads to azimuthal motion of beam particles (beam
rotation), which is not welcomed keeping in mind the
anticipated focusing of beams and the requirement of
minimization of relative velocity of annihilating particles
(particles of opposite charge rotate in opposite direction in
a magnetic field). The means must be found to transport
an equilibrium beam in vacuum without guiding magnetic
field or plasma and the most appropriate analog could be a
beam with its electrostatic and magnetic self-fields par-
tially or completely canceled by an axial conductor carry-
ing the opposite current and having an independently
controlled potential [45].
3. Direct annihilation rocket
In order to overcome charge and current limitations
imposed on transportation of beams of charged particles
in vacuum and to minimize relative velocities of particles
and antiparticles, two ‘charge-neutralized’and ‘current-
neutralized’geometrical arrangements of beams of
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Author's personal copy
particles and antiparticles are possible: (a) ‘coaxial’geo-
metry of matter-antimatter beams and (b) simultaneous
generation of matter and antimatter beams in the form of
a series of bunches. An example of coaxial geometry of
matter and antimatter beams is shown in Fig. 2(a). A
cylindrical hollowed-out electron or proton beam with
its radius R(alternatively, a set of thin single beams
distributed over
a circle of radius R) envelopes a single positron (antiproton)
beam having a radius r
i
oR. Both beams propagate in the
same direction with the same velocity thus their axial
relative velocities are zeroed ensuring effective annihilation
of particles and antiparticles in ambiplasma after their
mixing. Remarkably, the electrical currents of the beams
are opposite and cancel each other as a consequence of
opposite charges of the carriers. In the second geometry, a
series of matter/antimatter bunches moving with the same
axial velocity can be generated by an RF accelerator, in
which the positively and negatively charged particles are
accelerated simultaneously in the opposite phases of a
running electromagnetic wave (Fig. 2(b)). Average charge
and current of the whole beam are also zeroed.
In the coordinate frame comoving with the beams
(bunches), interaction between two coaxial beams or
between two neighboring bunches in the chain-of bunches
geometry can be considered electrostatic. The particles of
a matter beam (bunch) accelerate toward an antimatter
beam (bunch) and the antiparticles of an antimatter beam
(bunch) accelerate toward a matter beam (bunch).
Dynamics of their motion in the frame axially comoving
with the beams can be described by the vector equation of
motion of charged particles in the electric field
dp
dt
!
¼7qE
!;ð8Þ
which gives a one-dimensional differential equation for
the relative velocity β
r
¼ν/c(radial for coaxial geometry
and axial for comb-of-bunches geometry)
β
0
r
¼qE=m
0
cγ
3
r
¼qE
mc ð1β
2
r
Þ
3=2
;ð9Þ
where γ
r
¼(1β
r
2
)
0.5
is the relativistic factor of the
relative movement of particles due to their accelera-
tion in the electrostatic field E,β
r
¼ν/cis the acquired
relative velocity factor in the electrostatic field, p¼γ
r
m
0
ν,
dp/dt¼m
0
cd(γ
r
β
r
)/dt¼m
0
cγ
r
3
,β
0
r
¼dβ
r
/dt,m
0
is the mass of
rest of particles, and qis their charge.
In the coaxial configuration, both beams form a cylind-
rical capacitor. The radial electric field E¼Q
l
/2πε
0
r, where
Q
1
¼eN
l
¼e(N
þ
þN

) is the total charge per unit length,
eE1.6 10
19
coulombs is the elementary charge (for
electrons, positrons, protons and antiprotons), N
þ
and
N

are the numbers of matter or antimatter particles per
unit length, ε
0
is the dielectric constant of vacuum, and ris
the radius of a particular point between the beams
(r
i
oroR). Voltage on this capacitor can be estimated as
V¼Q
l
ln(R/r
i
)/2πε
0
on the condition that the thickness ΔR
of a tubular beam is much smaller than its radius R.
Coulomb attraction of oppositely charged beams forces
the outer electron beam to contract and the inner positron
beam to diverge until the particles of both beams mix
together resulting in their relative kinetic energy E
k
numeri-
cally equal (if expressed in electronvolts) to voltage V.The
gained relative velocity νis non-relativistic, if E
k
⪡m
0
c
2
,
where m
0
is the mass of rest of annihilating particles, thus
V(volts)⪡m
0
c
2
(electronvolts)¼9.38 10
8
eV for proton and
antiproton beams and 5 10
5
eV for electron and positron
beams. Setting the initial condition in the inner beam ν
0
(t¼0)¼ν
0
(r
i
)¼0, Eq. (9) can be integrated over r(the
variable dr/dt¼ν¼β
r
c,soβ
0
r
¼(dβ
r
/dr)(dr/dt)¼(dβ
r
/dr)ν¼
(dβ
r
/dr)β
r
cto obtain an expression for dr/dt
dr=dt ¼V¼cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
11
ð1þðeA=m
0
c
2
Þlnðr=r
i
ÞÞ
2
s;ð10Þ
where A¼eN
l
/2πε
0
. Comparing the relative velocity ν¼c
[11/(E
k
/m
e
c
2
þ1)
2
]
0.5
from the equation for kinetic energy
E
k
¼m
0
c
2
(γ
r
1) and Eq. (10) one can see that E
k
(joules)¼
(e
2
N
l
/2πε
0
)ln(r/r
i
) or, alternatively, E
k
(electron-volts)¼(eN
l
/
2πε
0
)ln(r/r
i
)¼V(volts) between the radii rand r
i
in complete
accordance with the energy conservation law. If the beams
are not focused by electromagnetic lenses, the assumption E
k
(eV)¼V(volts) for r¼Rgives the estimation of relative
velocity of particles and antiparticles in ambiplasma.
In geometry of a series of matter-antimatter bunches of
charged particles, the bunches form a multi-layered capacitor
with the voltage between the bunches VQ
s
[(R
2
þD
2
)
0.5

RD]/ε
0
,whereQ
s
is the average charge per unit area of
a bunch determined by the average density of particles N
s
per unit area, Ris the radius of each bunch, and Dis the
Fig. 2. Zero-current geometrical configurations of e

e
þ
and p
þ
p

beams in vacuum: (a) coaxial configuration and (b) chain of matter and antimatter
bunches of charged particles emerged from RF accelerator.
O.G. Semyonov / Acta Astronautica 99 (2014) 52–70 57

Author's personal copy
distance between the bunches. Coulomb forces cause axial
expansion of bunches resulting in their mixing so particles
and antiparticles acquire a relative velocity ν, which can be
determined from the condition E
k
(eV)EV(volts)¼eN
s
D/2ε
0
,
if D⪡Rand the electrostatic field between bunches can be
assumed approximately constant and homogeneous. Eq. (9)
transforms to
dz=dt ¼V¼cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
11
ð1þðe
2
N
s
z=m
0
c
2
Þð1=2ε
0
ÞÞ
2
s;ð10aÞ
where zis an axial coordinate between the neighboring
matter-antimatter bunches. Higher relative velocity of anni-
hilating particles due to acceleration by the attractive forces
between the beams of charged matter-antimatter particles in
any imaginable geometrical configuration will result in lesser
annihilation cross-section in ambiplasma of the mixed beams
and longer annihilation length. Therefore, it is imperative to
maintain a given radius Rof the formed ambiplasma column
along the axial distance lZl
ann
¼1/ns
a
¼β
r
R
2
/r
2
0
Notherwise
annihilation will be ineffective.
3.1. e

e
þ
Annihilation (photon) propulsion
Positrons annihilate with electrons either directly or
through formation of positronium atoms. If positronium is
formed, annihilation occurs mostly from the ground state
with the outcome depending upon the total spin. There are
four possible spin configurations in the ground state, one
corresponding to the total spin 0 (singlet) and three
corresponding to the total spin 1 (triplet). The singlet state
has a lifetime τ
1
¼1.25 10
10
s and its annihilation
results in two 511 keV photons in the frame comoving
with the beams. The triplet has a longer lifetime τ
3
¼
1.42 10
7
s and its annihilation yields three photons
with energies o511 keV producing a continuum spec-
trum. The probability of the latter process is by the factor
of 1/372 smaller [46] and it is plausible to consider that
virtually all annihilation events emit the 0.511-MeV
spectral line in the frame axially comoving with the
e
þ
e

beams.
In comb-of-bunches geometry, V1.4 10
9
N
s
on the
condition R⪢D¼0.15 m for electromagnetic wave fre-
quency of 1 GHz and electrons and positrons attain a
relativistic relative velocity in the comoving frame, if
N
s
4410
14
m
2
. In coaxial geometry with R/r
i
5, the
voltage V510
5
Nand the relative velocity νof mixed
electrons and positrons becomes relativistic for N
l
4
10
14
m
1
. In the rocket proper frame, each beam carries
a current I¼7eN'β
b
c(total current is zero), where β
b
cis
the axial speed of electrons (positrons) in the beams
(bunches), N'
¼
γ
b
N¼0.5γ
b
N
l
due to relativistic length con-
traction, where Nis the number of electrons (positrons)
per unit length (linear density) of a beam in the frame
comoving with the beams and γ
b
¼(1β
b
2
)
0.5
.From
Eq. (6), the total number of annihilation events per unit
time N
a
in ambiplasma of a length lin the beam-comoving
frame
N
a
¼ðR
a
lÞ¼β
r
cN l
l
ann
¼cN
2
lr
2
0
=R
2
;ð11Þ
and in the rocket frame
N
0
a
¼N
a
=γ
b
¼cN
2
lr
2
0
=R
2
γ
b
if β
r
o0:6;
N
0
a
¼cN
2
lr
2
0
β
r
B
rel
=R
2
γ
b
if β
r
-1;ð12Þ
where β
r
¼[11/(1þeV/m
e
c
2
)
2
]
0.5
according to Eq. (10).
Two conditions must be fulfilled for better performance of
photon rocket: (a) to obtain the annihilation length of the
order of the focal spot of a photon mirror (otherwise,
optical performance of a mirror will be poor), the radius of
ambiplasma must be small and (b) complete annihilation
in ambiplasma means matching the rate of particles
delivery β
b
cN/γ
b
to the annihilation zone and the rate of
annihilation N
0
a
, which can be achieved with lZl
ann
. The
first condition requires sharp focusing of e

and e
þ
beams
to the point of their crossing and the second one compels
to maintain the radius of ambiplasma along the distance
not shorter than the annihilation length otherwise anni-
hilation will be incomplete. It should be noted that
particles of the focused beams attain an additional trans-
verse velocity β
┴
β
b
sin θ, where θis the focusing angle,
which is summed with β
r
thus increasing annihilation
length l
ann
.
As an example, consider a photon rocket with its launch-
ing mass, say, 1000 ton moving with a constant acceleration
a¼0.1 g¼0.98 m/s
2
. The flux of photons with E
γ
¼0.5 MeV
needed to produce this acceleration is 10
27
/s, which
corresponds to the efflux power of 10
14
Wandtherateof
annihilation events N
0
a
510
26
s
1
[47]. This annihilation
rate in ambiplasma l–l
ann
corresponds to the value of current
10
8
A and linear density N210
18
m
1
thus any hope
for non-relativistic relative velocity of electrons and positrons
in ambiplasma is groundless. Thus, β
r
1, and the annihila-
tion length is proportional to R
2
am
/(r
2
0e
N), where R
am
is the
radius of ambiplasma and r
0e
is the radius of electron. In
practice, high-intensity and high-current electron beams can
be focused down (with many efforts) to a diameter of the
order of 1 mm [48] thus ambiplasma with its radius R¼5
10
4
m and density n¼N/πR
2
10
24
m
3
must be main-
tained along the distance l
ann
1000 m or more (Fig. 1(a)),
which is absolutely unrealistic and undermines the whole
concept of photon rocket based on crossing electron and
positronbeamsinthefocusofamirror.
3.2. p
þ
p

Annihilation propulsion
The linear density of protons and antiprotons in the
beams N
p
, for which relative velocity νof mixing particles
is not yet relativistic, is by the factor of m
p
/m
e
E1836
larger than in the case of electron–positron beams. Thrust
is produced by charged π-mesons (pions) and, possibly, by
the products of π-mesons decay such as μ-mesons (muons)
on the condition of formation of a directed jet in
a magnetic nozzle. The detailed balance of annihilation
energy conversion (1877.6 MeV per proton–antiproton
pair) is given in [13]: on average, 709.1 MeV (37%) goes
to two neutral π
0
-mesons (rest mass plus kinetic energy)
virtually immediately decaying into γ-quanta, which is
complete loss, and 1167.4 MeV goes to three charged
π-mesons (total rest-mass 3m
π
¼418.8 MeV plus their total
kinetic energy 3E
π
¼748.6 MeV). As a rule, the total
O.G. Semyonov / Acta Astronautica 99 (2014) 52–7058

Author's personal copy
momentum of pions is used to calculate the thrust [13,49–52].
The average momentum P¼N
π
m
π
γ
π
β
π
cin the rocket proper
frame, where N
π
is the flux of pions per unit time in the
exhaust, γ
π
¼2.786 is their relativistic factor, β
π¼
ν
π
/c¼0.93,
and ν
π
is the average speed of pions in the efflux. Assuming
again the launching mass M
0
¼1000 ton for a rocket propelled
by an ideally aligned efflux of pions, the effective production
of pions per second N
π
M
0
a/m
π
γ
π
β
π
c,whereais accelera-
tion. If a
0
¼0.1g¼0.98 m/s
2
, the rate of pions production is
N
π
510
24
s
1
,therateofp
þ
p

-annihilation events
N
0
a
¼N
π
/3E1.4 10
24
s
1
, and the proper antimatter-matter
mass consumption rate dM/dτE4.5 10
3
kg/s, where τis
thepropertimeintherocketframe.Itcorrespondstototal
annihilation power of 2.3 10
14
W and efflux kinetic power of
510
13
W.
In the rocket frame, the linear density of protons (anti-
protons) in the beams N
0
¼N
0
a
/β
b
c210
15
/β
b
m
1
and the
beam current is quite moderate I¼eN
0
β
b
c10
5
A. Relative
kinetic energy of mixing protons and antiprotons in coaxial
geometry of p
þ
p

beams E
k
(eV)EV(volts)E610
6
β
b
1
V,
if R/r
i
3, and their relative velocity in the ambiplasma
β
r
0.1. Despite the non-relativistic relative velocity of anni-
hilationg protons and antiprotons, the annihilation length is
not as small as desired: even if we manage to focus the
beams down to the radius of 20 μm¼210
5
m(n2
10
25
m
3
), which is hardly achievable, the annihilation
length will be tens of meters (see Fig. 1(b)). It would be
not an easy task to obtain and to maintain such a thin thread
of ambiplasma along the total annihilation length. The
attracting electrostatic field between the beams accelerates
protons and antiprotons sufficiently for all reasonable values
of R/r
i
to result in multi-meter annihilation lengths despite
their higher masses with respect to electrons.
Assuming the solution for effective p
þ
p

or H–H
annihilation in a magnetic muzzle is found, the relativistic
rocket equation for the ratio of instant proper mass Mof
a rocket to its initial launching mass M
0
is [50]
M
M
0
¼1β
1þβ

1=2β
ex
;ð13Þ
where β
ex
¼u
ex
/c,u
ex
is the effective exhaust speed in the
rocket proper coordinate frame for proton–antiproton anni-
hilation propulsion, and βis the speed of a rocket in the
reference (map) frame. The effective exhaust speed u
ex
differs from the average speed of charged pions ν
π
¼0.93c
because only a fraction μγ
π
¼0.622 of the initial mass
converts into their total mass-energy, where μ¼0.223 is
the portion of the mass-energy of annihilating protons and
antiprotons converted to the rest-mass of charged pions.
The rest converts to the mass-energy of neutral pions
η¼1μγ
π
¼0.378 and it is the loss. Assuming 100% align-
ment of charged pions along the axis of the exhaust jet
formed by a magnetic nozzle, the momentum conservation
equation is
Fdτ¼dP ¼m
π
γ
π
β
π
cN
π
dτ¼μdm γ
π
β
π
c¼Mds¼Mγ
2
dV;
ð14aÞ
where sis the proper velocity (s/cis the proper Einstein
number) defined as the speed experienced by astronauts
who see the objects of outer world going by. The momen-
tum Eq. (14a) yields Eq. (13) with the effective exhaust
speed u
ex
¼μγ
π
β
π
c¼0.58c.Thevalueofu
ex
is higher than
given in [49] however it is consistent with the calculations
of Westmoreland [51], obtained from the same logical
argumentation as in [49] but with the accurate account
for the relativistic factor of charged pions (see the corre-
sponding discussions in Sections 4 and 5below). The
effective exhaust velocity of 0.58cis also given in [52].
In reality, the effectiveness of magnetic nozzle regard-
ing the alignment of pions along the efflux axis must be
taken into account, which is equivalent to variation in the
effective exhaust velocity. The average projection of pions
velocity vectors on the axis diminishes when the angular
divergence of the efflux jet increases because pions spiral
in the magnetic field and drift across the magnetic lines of
the solenoid magnet [49] thus acquiring a momentum
perpendicular to the efflux axis which does not contribute
to the thrust. Besides, there is an escape cone for pions
emitted in the direction of a magnet thus to the rocket
hull: θ¼sin
1
[(H
0
/H
m
)
1/2
], where H
0
is the magnetic field
at the position of annihilation zone and H
m
is the max-
imum magnetic field of the solenoid magnet [53]. The loss
of pions through the escape cone results in loss of thrust
and, additionally, in elevated radiation danger for a crew.
How critical is the magnetic nozzle effectiveness one can
see from the following example: 20%-reduction of the
average projection of pions velocities on the efflux jet axis
(80% nozzle efficiency in alignment of pions along the
thrust axis) results in effective β
π
¼0.744, effective γ
π
¼1.5,
and u
ex
¼0.248c. Accordingly, in order to reach the speed,
say, 0.7c, the ratio of the rocket dry mass M
d
to its
launching mass M
0
must be not more then 1/30 while in
the case of perfect alignment M
d
/M
0
E1/4.
4. Relativistic matter propulsion
The common belief in better performance of photon
rocket because of massless (energy) exhaust in comparison
with propulsion by massive matter seems to be over-
estimated. Actually, the ratio of photon momentum hν/c
which produces the thrust to photon energy E
ph
¼hνis the
lowest of all other propulsion options and equal to the
absolute minimum c
1
E310
9
s/m (Fig. 3). The only
possibility for the compact storage of positrons is, appar-
ently, antiatomic substance such as antihydrogen droplets
levitating in magnetic field. [10,13] Even if we manage to
extract positrons from antimatter for their delivery to the
nozzle, the question is what to do with antinuclei left
behind. Of course, their annihilation with matter can be
utilized for energy production however their mass-energy
is m
n
/m
e
¼180 0 Atimes higher than mass-energy of posi-
tron–electron beams, where m
n
is the mass of a nucleus,
Ais its atomic number, and m
e
is electron (positron) mass.
It would be absolutely weird to use only 1/1800 of the total
atomic mass–energy of antihydrogen fuel for propulsion
and wasting the rest.
As for direct p
þ
p

annihilation, its implementation in
relativistic rockets looks more plausible because the major
part of antihydrogen–hydrogen mass–energy may be used
for propulsion and only a relatively smaller portion is a
loss. The approach, however, has its own problems such as
transportation and focusing of high-current beams of
O.G. Semyonov / Acta Astronautica 99 (2014) 52–70 59

Author's personal copy
protons and antiprotons in the magnetic field of a mag-
netic nozzle as well as necessity of neat alignment of pions
velocities along the exhaust axis. Ineffectiveness of anni-
hilation in ambiplasma of a practically achievable size also
adds to imperfection of p
þ
p

annihilation propulsion with
no easy solution.
An alternative is to separate efflux from energy source.
A reactor (it can be nuclear, thermonuclear, or based on
matter–antimatter annihilation) supplies electrical power
to accelerator of propellant particles (ions of conventional
matter) which forms an efflux jet (beam) consisting of
high-energy ions. If we manage to accelerate conventional
matter up to relativistic velocities and form a relativistic
efflux of nuclei or ionized atoms with kinetic energies
E
k
Zm
0
c
2
, where m
0
is the mass of rest of the particles, the
exhaust beam will consist mostly of energy thus the
performance of the engine will tend to that of a photon
rocket. Graphically, it can be shown plotting the ratio of
momentum of relativistic efflux-beam particles m
0
γ
b
β
b
cto
their kinetic energy m
0
c
2
(γ
b
1), where γ
b
¼(1β
b
2
)
0.5
is
the relativistic factor, β
b
¼ν
b
/c, and v
b
is the axial velocity
of the efflux particles in the rocket proper coordinate
frame (Fig. 3).
Neglecting the loss of particles in an accelerator, the
momentum conservation equation can be written as
dM
dτγ
b
β
b
c¼Ma-dM γ
b
β
b
¼Mdβ=ð1–β
2
Þ;ð14bÞ
where Mis the proper mass of a rocket, ais its proper
acceleration (a¼γ
3
a
map
¼γ
3
cdβ/dt¼γ
2
cdβ/dτ), γ
b
and β
b
are
the relativistic factors of the efflux particles in the proper
coordinate frame, βis the relativistic velocity factor of the
rocket in the reference map-frame, and τis the proper
time. The momentum Eq. (14b) yields the relativistic
rocket Eq. (13) with the effective exhaust velocity factor
β
ex
¼γ
b
β
b
linearly increasing with γ
b
thus reducing the mass
difference ΔM¼M
0
Mand the proper mass flow in the
efflux needed to reach a given β(Fig. 4). The effective
efflux velocity factor β
ex
is actually the Einstein number
thus u
ex
can exceed the speed of light as a consequence of
tendency of the exhaust momentum to infinity when the
speed of efflux massive particles approaches the speed of
light (alternatively, it can be interpreted as relativistic
mass increase of the efflux particles).
To be specific, we will consider hydrogen for propul-
sion, namely protons extracted from ionized hydrogen and
accelerated in an accelerator to form a high-energy efflux
jet. Shown in Fig. 4 is an ideal situation, when the energy
for acceleration of propellant is taken from a hypothetical
external source (e.g. from physical vacuum [58,59]). With-
out diving too deep into sci-fi projects of vacuum energy
extraction, the option is an onboard energy source capable
of generating electric power in multi-megawatts and giga-
watts range to propel a multi-ton spacecraft. Two possibi-
lities can be considered: (A) energy production without
noticeable change of mass of fuel such as nuclear or
thermonuclear reactor, so the loss of mass is mainly due
to propellant exhaust and (B) energy production with a
significant conversion of fuel mass into energy such as
annihilation reactor, so the rocket mass reduction includes
both the propellant exhaust and mass-to-energy conver-
sion of matter-antimatter fuel.
(A) Unlike nuclear rocket engines (tested mostly in the
period between 1950 and 1970), where a nuclear
reactor was used to directly heat the gaseous propel-
lant subsequently exhausted through the nozzle [60],
the concept of thrust production by accelerated ions
presupposes utilization of electrical power produced
by a conventional nuclear reactor. This option may be
viable for interplanetary missions and, possibly, for
one-way missions to nearest stars [61] rather then for
deep interstellar flights due to limited usable energy
per unit mass. Relatively small nuclear reactors with
several years of operation without refueling such as
small reactors for ships and submarines with their
sizes of several meters are normally below 1 gigawatt
(GW) of electric power. [61] If a higher power is
needed, a combination of several modules can be
rendered. Terrapower module with a planned electri-
cal power up to 1000 MW¼1GW¼10
9
W[62] capable
of operating for decades without refueling is under
Fig. 4. Ratio of mass difference ΔMto the launching mass M
0
at the
moment when the rocket reaches the map-speed β(numbers beside the
curves) as a function of proper velocity of massive efflux particles β
b
.
Fig. 3. Ratio of relativistic momentum to kinetic energy multiplied by the
factor of c¼310
8
m/s for massive particles and photons as a function of
efflux velocity factor β
b
¼ν
b
/c.
O.G. Semyonov / Acta Astronautica 99 (2014) 52–7060

Author's personal copy
development, though it is not clear how compact it
can be made to accommodate a spacecraft. With the
advanced cooling materials and gas turbines, the
conversion efficiency of the reactors to the output
electric power can be 50% and the conversion effi-
ciency of modern high pressure gas turbines can reach
80%. Russian fast neutron reactors for submarines with
output electric power of 300 MWe or more operating
at 540 1C and supplying supercritical steam for turbine
generators can be taken as examples [62].
Assuming a sufficiently compact 600-MW nuclear
reactor can be manufactured with the total loss both
on the stage of conversion to electricity and in accel-
erator not exceeding 50%, the available efflux kinetic
power will be 300 MW. Within the solar system
there is no need for relativistic velocities of spacecrafts
(radius of the solar system is too small for a multi-ton
spacecraft to reach a cruising speed exceeding 0.1c
with a reasonable acceleration arg¼9.8 m/s). The
estimations below are performed in the range of
velocities 0.00001 (3 km/s)oβo0.01(3000 km/s)
where the non-relativistic rocket equation is also valid
[49,50]
M
M
0
¼exp β
γ
b
β
b

:ð15Þ
For the initial mass of a spacecraft M
0
¼100 ton, the
parameters of interest are listed in Table 1. As seen
from Table 1, a lower efflux velocity at the upper part
of the table results in a higher proper acceleration of
the rocket but smaller map-speed for a particular ratio
of M/M
0
. On the other hand, a very high velocity β
b
of
efflux protons near the bottom of the table produces
much smaller acceleration of the rocket thus longer
acceleration phase. The rocket can reach a significantly
higher velocity at the point where a particular M/M
0
is
reached (up to 2000 km/s at the point where
M/M
0
¼0.5) however the time for picking-up speed
becomes unacceptably long. The optimum seems to be
in the range 0.0008 oβ
b
o0.002 (kinetic energy of
protons between 300 eV and 2 keV). The map-speed
of 200–300 km/s at the point where M/M
0
¼0.5 can be
reached in several months. Within the optimum range
of β
b
, a sufficient mass of propellant can be saved for
breaking to cancel the rocket speed upon approaching
a target destination. To be specific, the map-velocity of
200 km/s can be reached in 4 months by a rocket of
M
0
¼100 ton with the efflux kinetic power of 300 MW
and β
b
¼0.001, when 50% of the launching mass has
been exhausted. If a higher electrical power is available,
the optimum exhaust velocity shifts to the bottom of
Table 1 for the same mass flow in the exhaust. Alter-
natively, the mass flow rate can be increased or a bigger
launching mass of the spacecraft can be chosen. The
tendency is clear: to reach the objects at longer distances
during a reasonably short flight-time, a rocket has to be
accelerated to higher map-velocities thus higher exhaust
velocities of propellant must be opted and higher power
must be delivered to the propellant accelerator.
The ion thruster jettisoning protons with their kinetic
energies near 1 keV is an acceptable option for inter-
planetary flights. Acceleration of a rocket is relatively
small (compared to g) due to lower mass-flow in the
exhaust but the achievable velocities are much higher in
comparison with chemical rockets thus the objects at
much longer distances can be accessible for a reasonable
time of flight while keeping a sufficient amount of
propellant for braking. It is illustrated in Fig. 5,where
the velocity of a rocket and the distance covered during
its acceleration phase are shown as functions of time
(for these nonrelativistic interplanetary flights, the
proper time and the reference map-time are virtually
equal).
Table 1
Relativistic factor γ
b
¼(1β
b
)
0.5
, kinetic energy of efflux protons E
p
(keV), the rate of mass diminishing to the efflux in the proper frame, dM/dτ(kg/s), the
flow of protons per second in the efflux beam N
p
(s
1
), map-velocity of the rocket β
0.5
at the point when M/M
0
¼0.5, its proper acceleration a
0
(m/s) at the
starting point (β¼0), map-velocity v(km/s) at the point where M/M
0
¼0.5, and the proper time of flight τ
0.5
(years) to the point where M/M
0
¼0.5 as
functions of proper efflux velocity β
b
.
β
b
γ
b
E
p
dM/dτN
p
β
0.5
a
0
V
0.5
τ
0.5
0.0001 1.000000 01 0.00469 0.66667 3.99Eþ26 6.93E 05 0.2 20.79 0.0024
0.0002 1.00000002 0.01876 0.16667 9.98Eþ25 1.39E 04 0.1 41.58 0.0095
0.0003 1.00000005 0.04221 0.07407 4.44E þ25 2.08E04 0.0666 62.38 0.0214
0.0004 1.00000008 0.07504 0.04167 2.50E þ25 2.77E04 0.05 83.17 0.038
0.0005 1.00000013 0.11725 0.02667 1.60Eþ25 3.47E 04 0.04 103.9 0.0595
0.0006 1.00000018 0.16884 0.01852 1.11Eþ25 4.16E04 0.0333 124.7 0.0856
0.0007 1.00000025 0.22981 0.01361 8.15E þ24 4.85E04 0.0285 145.5 0.1165
0.0008 1.00000032 0.30016 0.01042 6.24Eþ24 5.55E 04 0.025 166.3 0.1522
0.0009 1.00000041 0.37989 0.00823 4.93Eþ24 6.24E 04 0.0222 187.1 0.1926
0.001 1.0000 005 0.469 0.00667 3.99Eþ24 6.93E04 0.02 207.9 0.2378
0.002 1.000002 1.87600 0.00167 9.98Eþ23 0.00139 0.01 415.8 0.9513
0.003 1.0000045 4.22103 7.41E04 4.44E þ23 0.00208 0.0067 623.8 2.1404
0.004 1.000008 7.50409 4.17E04 2.49Eþ23 0.00277 0.005 831.7 3.8052
0.005 1.000 0125 11.7252 2.67E04 1.60E þ23 0.00347 0.004 1039 5.9457
0.006 1.000018 16.8844 1.85E04 1.11Eþ23 0.00416 0.0033 1247 8.5619
0.007 1.000 0245 22.9818 1.36E 04 8.15Eþ22 0.00485 0.0028 1455 11.654
0.008 1.000 032 30.0174 1.04E04 6.24E þ22 0.00555 0.0025 1663 15.221
0.009 1.000 0405 37.9913 8.23E 05 4.93E þ22 0.00624 0.0022 1871 19.265
0.01 1.00 005 46.9035 6.67E05 3.99Eþ22 0.00693 0.002 2079 23.784
O.G. Semyonov / Acta Astronautica 99 (2014) 52–70 61

Author's personal copy
It should be kept in mind that an interplanetary space-
craft propelled by the accelerated ions thruster is sup-
posedtobelaunchedeitherfromanear-earthorbitor
from a circumsolar orbit thus its orbital map-velocity
should be vector-added to its gained velocity. If propel-
lant can be found at the destination point (on a satellite
of a planet or on an asteroid), it will greatly facilitate the
mission and expand the achievable destinations. In
future, the chains of ‘gas’stations with stored propellant
could be distributed over the solar system.
(B) Implementation of annihilation reactor for powering
an accelerator of efflux matter (e.g. protons) necessi-
tates modifying the relativistic rocket Eq. (13) because
the rocket mass reduction includes now the loss of
mass of annihilating fuel in addition to the propellant
exhaust. The momentum equation is to be rewritten as
dm
b
γ
b
β
b
¼Mdβ=ð1–β
2
Þ;ð16Þ
where dm
b
is the change of propellant mass per unit
time in the proper coordinate frame. The change of the
total mass Mis dM¼dm
b
þdm
fuel
, where dm
fuel
is the
change of mass of annihilating matter and antimatter
in the reactor. A portion εof fuel mass-energy goes to
the efflux kinetic energy of the propellant and the rest
is a loss including energy escaping the reactor with
neutrinos, energy loss in the process of energy con-
version to electricity, and energy loss in accelerator.
The equation for kinetic energy of the efflux is
dm
b
c
2
ðγ
b
1Þ¼εdm
f uel
c
2
;
which gives
dm
f uel
¼γ
b
1
εdm
b
and
dM ¼dm
b
þdm
f uel
¼dm
b
1þγ
b
1
ε

:ð17Þ
Combining Eqs. (16) and (17), the relativistic rocket
equation becomes
M
M
0
¼1β
1þβ

ðð1þðχ
b
1Þ=εÞ=2γ
b
β
b
Þ
:ð18aÞ
Remarkably, if εis equal to unity, the rocket equation
returns to its familiar form
M
M
0
¼1β
1þβ

ð1=2β
b
Þ
:ð18bÞ
The last equation was also given by Walter [49] for the
direct annihilation of protons in a magnetic nozzle how-
ever mistakenly as shown in [51]. Eqs. (18a) and (18b)are
valid, when the useful portion of mass-energy of annihi-
lating matter converts exclusively into kinetic energy of
the efflux. Annihilation in the nozzle and direct thrust
production by charged pions considered in Walter's paper
presupposes conversion of mass-energy of annihilating
matter to total mass-energy of efflux pions, i.e. to their
mass of rest and kinetic energy.
The expression for the map-velocity of a rocket βas a
function of its proper mass can be readily found from (18a)
β¼1ðM=M
0
Þ
η
1þðM=M
0
Þ
η
;where η¼2γ
b
β
b
1þðγ
b
1Þ=ε:ð19Þ
The achievable map-velocity βas a function of efflux
proper velocity β
b
at the point where M¼0.5M
0
is shown
in Fig. 6. The proper propellant exhaust rate, the rocket
mass loss, the ratio of the annihilating fuel mass rate to the
propellant mass rate dm
fuel
/dm
b
¼(γ
b
1)/εand the time of
flight to the point where M¼0.5M
0
are shown in Table 2 as
functions of β
b
in the range 0.003 oβ
b
o0.5 for a rocket
with its launching mass of 1000 ton propelled by a proton
jet of 1 TW kinetic power. The graphs of map-velocity βof
a 1000-ton rocket with the efflux kinetic energy of 1 TW
(10
12
W) for ε¼0.3, 0.5, and 0.7 are plotted in Fig. 7.
The rocket map-velocity picks up faster for the same
launching mass M
0
, if efflux kinetic power increases,
however it requires the proton efflux of even higher
kinetic energy otherwise the propellant will be consumed
too fast and the achievable map-speed at M
0
¼0.5 will be
too small. For the efflux kinetic power of 100 TW (10
14
W),
the rocket launching mass will inevitably increase due to
heavier reactor and more massive power-handling equip-
ment requiring even more propellant and annihilation fuel
thus resulting in slower acceleration and longer proper
time for picking up the maximum speed at the point
Fig. 5. (a) Rocket velocity (M
0
¼100 ton) propelled by a jet of accelerated protons powered by a 60 0-MW nuclear reactor with conversion efficiency ε¼50%
and (b) distance of flight during the acceleration phase as a function of proper time for the efflux proton velocities β
b
¼0.0001, 0.0003, and 0.001. The dots
on the curves correspond to M¼0.5M
0
. The maximum length of the curves is determined by the condition ΔM¼M
0
.
O.G. Semyonov / Acta Astronautica 99 (2014) 52–7062

Author's personal copy
where a predetermined M/M
0
is reached as seen from
Table 3 and Fig. 8
For illustration, the graphs of map-velocity of a rocket
with the launching mass of 1000, 5000, and 10,000 ton
and map-distance covered by the rocket during its accel-
eration stage for the proton efflux velocity β
b
¼0.5 and
kinetic power of 100 TW (10
14
W) are shown in Fig. 8.
5. Discussion
For effective annihilation, the radial size Rof ambi-
plasma must be maintained as small as possible along the
annihilation length l
ann
⪢R, which is not an easy task. There
seems to be no practical way to focus a mega-amps
relativistic electron (positron) beam in vacuum to a dia-
meter well below 1 mm. If 1-mm focal spot is achieved,
the 1-mm ambiplasma of electrons and positrons must be
confined along the annihilation length of thousands of
meters to ensure complete annihilation. This is hardly
achievable in practice and will have no practical use
regarding formation of a photonic exhaust beam by a
mirror because the source of photons is not concentrated
in the focus of the mirror. This is true for any geometry of
crossing e

e
þ
beams: their electrostatic interaction inevi-
tably accelerates electrons and positrons of each beam to
Fig. 6. Map-velocity βof a rocket at the point where M¼0.5M
0
as
a function of efflux protons velocity β
b
for ε¼0.3, 0.5, 0.7, and 0.9
(ε¼0.9 can be thinkable, if neutrino losses are eliminated). The curves
are valid for any efflux power and launching mass. Note the existence of a
maximum of reachable β, the value and the position of which depend
on ε.
Table 2
Proper mass flow rate of propellant dm
b
/dτ, rocket mass change dM/dτ¼d(m
b
þm
fuel
)/dτ, proper time of flight τto the point where M/M
0
¼0.5, and the ratio
of annihilating mass in the reactor dm
fuel
to propellant mass dm
b
of a rocket having a launching mass M
0
¼1000 ton and propelled by 1 TW (10
12
W) proton
efflux with ε¼0.3 as functions of efflux proton velocity β
b
.
β
b
γ
b
dm
b
/dτ(kg/s) dM/dτ, (kg/s) τ(M/M
0
¼0.5), (years) dm
fuel
/dm
b
ε¼0.3 ε¼0.5 ε¼0.7 ε¼0.3 ε¼0.5 ε¼0.7
0.003 1.000005 2.469 2.469 2.469 2.469 0.006 0.006 0.006 1.62E5
0.004 1.000008 1.389 1.389 1.389 1.389 0.01 0.01 0.01 2.88E5
0.005 1.000013 0.889 0.889 0.889 0.889 0.018 0.018 0.018 4.51E 5
0.006 1.000018 0.617 0.6173 0.6173 0.6173 0.026 0.026 0.026 4.861E5
0.007 1.000025 0.453 0.4535 0.4535 0.4535 0.035 0.035 0.035 8.82E5
0.008 1.000032 0.347 0.3472 0.3472 0.3472 0.046 0.046 0.046 8.64E5
0.009 1.000041 0.274 0.2743 0.2743 0.2743 0.058 0.058 0.058 1.46E 4
0.01 1.00005 0.222 0.2222 0.2222 0.2222 0.071 0.071 0.071 1.35E4
0.02 1.0002 0.055 0.0556 0.0556 0.0555 0.286 0.286 0.286 7.2E4
0.03 1.00045 0.0247 0.0247 0.0247 0.0247 0.642 0.643 0.643 0.00162
0.04 1.000801 0.0139 0.0139 0.0139 0.0139 1.141 1.142 1.143 0.00288
0.05 1.001252 0.0089 0.0089 0.0089 0.0089 1.782 1.785 1.786 0.00451
0.06 1.001805 0.0062 0.0062 0.0062 0.0062 2.563 2.569 2.572 0.00487
0.07 1.002459 0.0045 0.0045 0.0045 0.0045 3.484 3.496 3.5 0.00885
0.08 1.003215 0.0035 0.0036 0.0035 0.0035 4.545 4.564 4.572 0.00867
0.09 1.004075 0.0027 0.0028 0.0 027 0.0 027 5.743 5.774 5.787 0.01099
0.1 1.005038 0.0022 0.0023 0.0022 0.0 022 7.078 7.125 7.145 0.01357
0.2 1.020621 5.4E04 5.8E 04 5.6E 04 5.5E 04 27.56 28.29 28.61 0.06874
0.3 1.048285 2.3E04 2.7E 04 2.5E 04 2.46E 04 59.41 62.9 64.53 0.16095
0.4 1.091089 1.2E 04 1.6E 04 1.4E 04 1.38E04 99.82 110.1 115.1 0.30363
0.5 1.154701 7.2E05 1.1E04 9.4E 05 8.8E 05 145.8 168.8 181.0 0.51566
Fig. 7. Map-velocity βof 1000-ton rocket powered by annihilation
reactor delivering power of 1 terawatt to the efflux (proton beam) as
functions of proper flight-time for β
b
¼0.01 and β
b
¼0.1. The o-marks
indicate the positions when M¼0.5M
0
.
O.G. Semyonov / Acta Astronautica 99 (2014) 52–70 63

Author's personal copy
relativistic relative velocities and, consequently, a very
long ambiplasma of mixed beams is needed to accommo-
date the annihilation length. The mismatched currents of
the beams of oppositely charged particles (nonzero total
current) can help to produce long ambiplasma as a result
of magnetic confinement however every plasma column
confined by magnetic field suffers from magnetic instabil-
ities effectively disrupting the column.
Other problems with no clear solution such as practical
impossibility to manufacture a mirror for gamma-quanta
reflection and to store positrons and electrons in a compact
form make the dream of photon rocket virtually hopeless.
The compact storage of positronic fuel onboard for many
months and years is thinkable only in the form of atomic
antimatter. Extracting positrons from antimatter for
annihilation and thrust production while leaving behind
much more powerful antinuclear fuel would be a huge
blunder regarding effectiveness of antimatter energy
utilization.
As for the direct annihilation of protons and antipro-
tons in a magnetic nozzle, the method seems to be also not
as effective as anticipated. With the radius of ambiplasma
Ras small as 0.01 mm, the length of complete annihilation
of protons with antiprotons (annihilation length) is tens of
meters thus requires either an extremely strong guiding
magnetic field or a preformed long thin plasma pinch [54]
and both will be a huge complication for the magnetic
nozzle. Besides, such a long source of pions would be of no
practical use because the travel of charged pions in the
rocket coordinate frame before the decay is 21 m only.
Table 3
Propellant exhaust rate dm
b
/dτ, total mass loss rate dM/dτ¼d(m
b
þm
fuel
)/dτ, achievable map-velocity at M/M
0
¼0.5, and proper time of flight τof a rocket
with the launching mass of 1000, 5000, and 10,000 ton to the point where M/M
0
¼0.5 as functions of proper proton velocity in the efflux for 100 TW
(10
14
W) efflux kinetic power with ε¼0.5.
β
b
γ
b
dm
b
/dτ(kg/s) dM/dτ(kg/s) dm
fuel
/dm
b
β(M/M
0
¼0.5) τ(M/M
0
¼0.5) (years)
10
3
ton 5

10
3
ton 10
4
ton
0.1 1.005038 0.221 0.2241 1.0168 0.0689 0.276 0.354 0.707
0.2 1.020621 0.0539 0.0576 1.0687 0.135 0.594 1.378 2.756
0.3 1.048285 0.023 0.0267 1.161 0.196 0.998 2.971 5.942
0.4 1.091089 0.0122 0.0159 1.304 0.25 1.458 4.991 9.982
0.5 1.154701 0.0072 0.0109 1.516 0.296 1.948 7.291 14.58
0.6 1.25 0.0044 0.0081 1.833 0.333 2.449 9.74 19.48
0.7 1.40028 0.0028 0.0065 2.334 0.36 2.955 12.25 24.49
0.8 1.666667 0.0 017 0.0054 3.222 0.377 3.479 14.78 29.56
0.9 2.294157 8.6E4 0.0046 5.314 0.379 3.535 17.39 34.79
0.91 2.411915 7.9E4 0.0045 5.706 0.378 3.591 17.67 35.34
0.92 2.551552 7.2E4 0.00442 6.172 0.377 3.649 17.95 35.91
0.93 2.720648 6.5E 4 0.00435 6.735 0.376 3.709 18.25 36.49
0.94 2.931052 5.7E 4 0.00428 7.437 0.374 3.772 19.19 37.09
0.95 3.202563 5.0E 4 0.00421 8.341 0.371 3.838 19.55 37.71
0.96 3.571429 4.3E4 0.0 0414 9.571 0.369 3.909 19.94 38.37
0.97 4.11345 3.6E 4 0.00406 11.38 0.365 3.988 20.42 39.09
0.98 5.025189 2.8E4 0.00398 14.42 0.36 4.084 20.48 39.88
0.99 7.088812 1.7E 4 0.00389 21.30 0.353 4.096 20.53 40.84
Fig. 8. (a) Map-velocity βof a rocket with launching masses of 1000, 5000, and 10,000 ton and (b) corresponding distances covered during the acceleration
stage as functions of proper time τfor 100-TW kinetic power of the proton efflux with β
b
¼0.5. The x-marks on the curves indicate positions of M/M
0
¼0.25,
and the o-marks indicate the positions of M/M
0
¼0.5 on the corresponding curves.
O.G. Semyonov / Acta Astronautica 99 (2014) 52–7064

Author's personal copy
A possible solution to maintain a relatively small radius of
ambiplasma along the long distance is its self-pinching on
the condition of mismatched currents of matter and
antimatter beams resulting in appearance of non-
compensated magnetic field around ambiplasma in ana-
logy with electron [55] and ion [56] beams injected in
plasma. This magnetic field of mismatched current can
focus the proton–antiproton beams and confine ambi-
plasma in the form of a Bennet pinch within its radius
determined by the transverse velocities (temperature) of
particles in the beams. So far, theoretical studies of the
phenomenon are performed for beams propagating
through gas or plasma, where the mismatching back
current is generated due to induction [55,56]. The theory
can be extended to ambiplasma of mixed beams of
particles and antiparticles with the mismatched charge
per unit length (it can be done, for example, by control-
lable injection of electrons in a proton beam). A plasma
column with a current, however, undergoes hydromag-
netic instabilities disrupting the uniform density distribu-
tion thus producing unstable flux of annihilation products.
Another problem could be the magnetic field of the nozzle
itself. Firstly, it would impede beams focusing and mixing
and secondly, even if we manage to focus and mix
the beams nonetheless, it could interfere with the self-
pinching magnetic field of the mismatched currents.
Anyway, the perspectives of propulsion by the direct
annihilation of beams of charged particles and antiparti-
cles in a nozzle are not very optimistic.
One possible solution to overcome the problem of small
annihilation cross-section of protons with antiprotons as
well as to subdue the influence of the nozzle magnetic
field on transportation of beams of charged particles is
utilizing neutral atomic and antiatomic beams for annihi-
lation. The cross-section of atom–antiatom annihilation is
several orders of amplitude higher than the cross-section
of p
þ
p

annihilation because the impact parameter is
now the radius of an atom (Bohr's radius for hydrogen)
instead of the radius of a proton. Beside the benefits of
much higher cross-section thus smaller annihilation
length, utilization of neutral atomic beams eliminates their
electrostatic interaction as well as their interference with
the magnetic field of magnetic nozzle. In collision, the
neutral antihydrogen atoms and molecules annihilate with
hydrogen atoms and molecules faster than protons with
antiprotons through the rearrangement reaction HþH-
PnþPs or Pnþe
þ
þe

, where Pn stays for protonium
(quasiatom consisting of proton and antiproton) and Ps stays
for positronium (quasiatom consisting of positron and elec-
tron). Annihilation cross-section for hydrogen and antihydro-
gen atoms is given in [71]:s
HH
¼1.0 6 10
7
(c/ν)
0.64
πr
0
2
,where
r
0
istheBohr'sradius.Thecross-sectionisdefinableinthe
range of relative kinetic energies of annihilating atoms and
antiatoms from 0 to 10 eV because atoms ionize each other in
collision at higher relative energies and form ambiplasma of
mixed protons, antiprotons, electrons, and positrons thus the
rate of annihilation drops down returning us to the situations
of proton–antiproton and electron–positron annihilation. It
means the beams of atoms must have nearly the same axial
velocity and their internal temperature (velocity spread of
particles) should be significantly below 10 eV (10
5
K). In
practice, atomic beam can be obtained by means of charge
neutralization of a proton beam by electrons with a matched
velocity so electrons recombine with protons and form a
beam of hydrogen atoms; this method can be also applicable
to form an antihydrogen beam. Among the problems are
difficulties of manipulating and focusing atomic beams as
well as ionization of atoms by the flux of pions from the
annihilation zone even before they arrive to the point of
crossing thus reducing their annihilation rate back to the
annihilation rate of charged particles.
The huge advantage of relativistic matter propulsion
powered by a separate power supply is much wider range
of freedom and flexibility in choosing the energy source
and the propellant as well as possibility to control inde-
pendently efflux kinetic energy and mass flow. In compar-
ison with conventional chemical rockets, propulsion by
particles of high kinetic energy results in significant
reduction of exhaust mass needed to reach a predeter-
mined cruising speed of a rocket. Another advantage is
that virtually all mass-energy of annihilating atoms and
antiatoms can be utilized for conversion to electricity with
the exception of neutrinos and antineutrinos escaping the
reactor (14.56% of the total mass-energy of annihilating
protons and antiprotons). Other annihilation options could
be chosen such as annihilation of antiprotons with heavier
nuclei without production of neutral pions and gamma
photons, which would facilitate energy consumption from
the reactor. Unexplored so far annihilation of heavy nuclei
and antinuclei can also be an alternative. Higher efflux
velocities reduce the mass exhaust rate and allow for
higher achievable map-velocities of a rocket but not with-
out a price and the price is smaller mechanical momen-
tums thus thrust for the same efflux kinetic power
resulting in smaller rocket acceleration and longer time
for picking-up the cruising speed. Nonetheless, higher
cruising velocities can be reached which will eventually
shorten the total time-of-flight to remote destinations.
Yet another plus is that the concept of relativistic
matter (ions) propulsion is based on the existing technol-
ogy of ion accelerators redesigned for operation in space
vacuum. No heavy vacuum equipment and seals are
needed and no bulky insulators; vacuum is the best
electrical insulator by itself. Thus, no sparkling on the
insulator surface and, correspondingly, higher acceleration
gradients can be tolerable, which allows designing the
low-mass accelerators. The jet of accelerated ions is
naturally aligned with the thrust axis therefore the main
problem of the magnetic nozzles can be eliminated. The
ion accelerator is friendly to any type of propellant and can
be adjusted to accelerate ions of any element from hydro-
gen to xenon. Of course, the efflux beam of ions must be
neutralized to prevent charge accumulation and the loss of
thrust in analogy with the existing ion thrusters, which are
already in use for maneuvering the interplanetary space-
crafts and under development by NASA for interplanetary
missions. [64]
When an energy source powers an accelerator of
propellant particles, the rocket design can be much more
flexible. No matter what type of electric power generator is
used: nuclear reactor, thermonuclear reactor, annihilation
reactor, or hypothetical vacuum energy source. Because
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the propellant is a natural substance, it can be found at the
destination point thus there is no need to store the
propellant reserve for the return journey thus saving
the launching mass of a rocket. As for the annihilation
reactor, various designs can be suggested. Many engineering
solutions developed for thermonuclear reactors [57] can be
applicable, such as magnetic confinement of ambiplasma in
toroidal or conventional magnetic traps for a sufficiently
long period of time to achieve effective annihilation, liquid
‘first wall’absorbing the flux of gamma radiation and high-
energy particles while maintaining integrity and function-
ality of the whole system, liquid metallic coolant to transfer
thermal energy to a heat exchanger with the secondary
coolant and then to gas turbines, fuel delivery to the
reaction zone (both matter and antimatter) in the form of
jets of neutral atoms or small pellets subsequently ionized
in the reaction zone, etc. Even a much simpler reactor
design can be imagined with a stream of antiprotons or
antihydrogen atoms incident on a sufficiently big chunk of
high-melting material to heat the chunk up to operation
temperature by annihilation products (mesons and gamma
radiation). The chunk working surface can be made porous
with a primary liquid cooler simultaneously serving as an
annihilating substance delivered to the surface through the
pores to compensate the mass loss in annihilation.
The engines based on chemical fuel and even the
conventional nuclear rockets [60] produce the efflux
velocities that correspond to kinetic energies of exhaust
atoms (molecules) significantly below 1 eV, i.e. well
beyond the upper part of Table 1. Their application is
mainly launching the rockets from planets to overcome
the gravitational pull and to reach the escape velocity in
a short time while consuming the most of its launching
mass to the efflux. The ion-thrust technique works in
vacuum therefore must be used beyond the atmosphere.
The upper portion of Table 1 (proton kinetic energies from
several electronvolts to several hundreds electronvolts) is
the domain of MHD thrusters which produce a jet of high-
temperature plasma accelerated by magnetic (Lorenz)
forces. A survey of the existing and proposed plasma
thrusters is given, in particular, in [63]. Spacecrafts
equipped with MHD thrusters are thought to be well-
suited for the missions to nearby planets (Mars and
Venus), when a return flight is planned; as for the outer
planets, the flyby missions are only possible because of
relatively high mass flow as seen from Table 1.
For higher relativistic efflux kinetic power W
k
0.3–
1 GW, a jet of accelerated ions with their energy in the
range from 0.3 to 2 keV is the best choice providing
utilization of electrical power from a small nuclear reactor.
It is applicable for interplanetary missions and may be for
one-way robotic missions to nearby stars, if dry mass is
relatively small and duration of flight is not a critical issue.
Hundreds-ton rockets can reach velocities of 100–200 km/s
after consuming 50% of their launching mass during a
reasonable time interval of several months.
Transition to the level of efflux kinetic power of 1 TW
and more requires higher velocities of the efflux protons.
Annihilation reactor will apparently be the option for
powering the ion thrusters. As seen from Fig. 7 and
Table 2, one terawatt supplied to the efflux beam of
accelerated protons for powering a 1000-ton rocket is
not yet sufficient to perform a manned interstellar flight
with a crew in a reasonably short time however it would
be very suitable for building up the space service around
the solar system (610
9
km) including Kuiper belt
(10
10
km) and even celestial objects in the Oort cloud
(10
12
10
13
km). In the optimum range 0.01oβ
b
o0.3,
the loss of annihilating fuel is much smaller than the loss
of propellant thus the value of εis virtually irrelevant to
the rocket dynamics (it is of critical issue for a cooling
system, see below).
For the efflux kinetic power of the order of 100 TW, the
proper efflux velocity β
b
r0.1 does not allow reaching
β40.1 due to fast loss of mass of the propellant (see
Table 2). The extreme relativistic efflux velocities β
b
Z0.9
in their turn result in significantly lower rocket accelera-
tion for a given launching mass and significantly longer
time for picking-up the cruising speed although the
exhaust rate is much smaller and the rocket mass reduc-
tion is determined mostly by fuel annihilation. The optimal
values of β
b
for a rocket launching mass between 1000 and
10,000 ton can be in the range from 0.5 to 0.8, which
allows reaching the map-speed β0.30.4 (at the
moment, when M/M
0
¼0.5) within a proper time from
2 to 20 years while storing a fuel reserve for braking on the
condition of rocket dry mass r0.25M
0
. A lower launching
mass would allow picking up a needed velocity for
a shorter proper time however it is hardly imaginable
a 100-TW reactor together with the corresponding pro-
pellant accelerator squeezed into the allowable dry mass
not mentioning the crew quarters, shielding, and cooling
system.
The concept of relativistic matter efflux powered by
annihilation reactor cannot be realized without solving the
problem of antimatter storage. Among the possible solu-
tions are antimatter storage in magnetic traps in the form
of free charged particles or plasma [69], cooled hydrogen
gas in the state of Bose–Einstein condensate [70], and
liquid or solid diamagnetic parahydrogen which tends to
stay in a region of minimum of magnetic field ([11], p. 96).
In principle, if antiatoms (antimolecules) could be kept
apart from atoms (molecules) of ordinary matter at the
distances Lwell above Bohr's radius r
0
10
10
m(L410r
0
would be sufficient), antiatomic and atomic substances
(liquid or solid) could coexist without annihilation. It is
worth studying the possibility of antimatter storage in
tanks made of conventional matter, if a sufficiently strong
energy (electrical or magnetic) barrier prevents immediate
contact of antimatter substance with the walls made of
matter. When an antimatter substance is formed, the
collective forces on its boundary such as surface tension
and crystalline lattice ties can play the role of energy
barrier on the boundary between antimatter and matter to
facilitate their magnetic separation. There are no experi-
mental studies of physical properties of antimatter sub-
stances however we can guess that they are similar to the
properties of matter analogs. If the boundary between
atomic (molecular) substance and antiatomic (antimolecular)
substance could prevent the close rendezvous of composing
atoms, matter and antimatter would coexist and antimatter
could be stored safely on the condition of suppressed mutual
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diffusion and quantum tunneling through the energy bar-
riers. Since antimatter substance such as antihydrogen must
be kept at a cryogenic temperature to suppress thermal
motion of molecules, the magnetic barrier can be created by
an array of superconductive coils mounted in the walls of a
storage tank. The array of coils that form mosaic of alternate
pole magnets can generate magnetic field distributed over
the surface of the wall so that the strength of magnetic field
decreases with the distance from the wall thus preventing a
diamagnetic substance from touching the wall on the con-
dition of sufficient field intensity to support the weight of the
stored antimatter parahydrogen caused by rocket accelera-
tion a⪡g. No permanent energy supply is needed to power
superconductive magnets. Electric power needed for sup-
porting cryogenic temperature in tanks as well for electro-
nics, equipment, lighting, radiation protection, etc. can be
supplied by annihilation reactor.
Summarizing, the concept of relativistic rocket pro-
pelled by an accelerated matter propellant and powered
by an annihilation reactor seems to be a hopeful bid for
interstellar journeys provided the problem of heat disposal
is solved.
5.1. The problem of heat disposal
The potential killer of the whole concept of relativistic
rocket for interstellar travels is thermodynamics. As seen
from the estimations above, interstellar missions require
multi-terawatt efflux kinetic power to peak-up relativistic
speed in a reasonably short period of time. The dry mass of
rocket grows with engine power because increasingly
bulkier and heavier equipment (reactor, turbines, transfor-
mers, accelerators, radiation shields, high-current conduc-
tors, fuel tanks, etc.) is needed to handle higher power.
Heavier load requires more propellant and annihilating
fuel thus the total launching mass grows, too. Here, the
laws of thermodynamic come into play: there is no engine
with 100% efficiency. A fraction of generated energy always
goes to thermal waste and the higher is power, the bulkier
cooling system is needed to prevent the destructive over-
heating of the power-handling parts such as magnetic
nozzle with radiation shield of a direct annihilation rocket
or reactor with ion thruster of relativistic matter propul-
sion rocket. Thermal balance of relativistic rockets has
never been considered in detail. Apparently, the first
estimations of a radiator to dispose the thermal energy
deposited into the gamma-protecting shield were per-
formed in [13]. The result is dismaying: the radiator is
huge (hundreds kilometers in length) and makes a lion's
share of rocket mass. The problem of heat disposal arises
from the fact that there is no material cooler in space
vacuum. The only available cooling mechanism is thermal
radiation.
Every object emits thermal radiation with spectral dis-
tribution close to black-body spectrum on the condition
that the object is opaque virtually over the whole spectrum.
Black body radiation power P
b
¼SsT
4
,whereSis its surface
area, s¼5.67 10
8
W/m
2
K
4
is the Stefan–Boltzmann con-
stant, and the temperature Tis measured in degrees of
Kelvin (K). Far from radiation-emitting stars, an object in
space will be irradiated by isotropic electromagnetic
radiation from the universe corresponding to radiation of
a black body with its temperature T
c
¼2.7 K [65].Ifthe
temperature Tof the object exceeds T
c
, it radiates the excess
of its thermal energy outward and cools down eventually to
T
c
provided no additional internal or external sources of
energy are present. The object can keep its temperature
T4T
c
stable, if internal energy source is present and the
heat released in the object is balanced by thermal radiation
to space. Since thermal radiation P
r
of real objects is always
less then black body radiation at the same temperature,
each object is characterized by its emissivity factor ε
T
¼P
r
/P
b
.
In the case of direct propulsion by the products of
p
þ
p

annihilation, the radiation shield between the
nozzle and the rocket body transforms the energy of
absorbed gamma radiation into heat. Total power of
absorbed gamma radiation is ηζP
a
, where η¼0.378 is the
portion of total annihilation power P
a
converted into
neutral pions thus into gamma photons and ζ¼Ω/4π¼0.5
(1cos Θ)¼0.5[1d/(d
2
þr
2
)
0.5
] is the geometric factor
defined as the ratio of a solid angle with its azimuth angle
Θof the radiation shield (as seen from the annihilation
zone) to 4π, where dis the distance between the annihila-
tion zone and the shield and ris the radius of the shield.
The absorbed energy must be radiated to space thus the
surface area Sof the radiator with its emissivity factor ε
T
S¼ηζP
a
=ε
T
sðT
4
T
4
amb
Þ;ð20Þ
where T
amb
is the ambient temperature determined mostly
by the temperature of radiation incident on radiator from
outside (in interstellar space T
amb
¼2.7 K).
As for the rocket propelled by accelerated relativistic
matter, the area Sof a radiator for cooling the reactor must
be not less than
S¼η
T
P
a
=ε
T
sðT
4
T
4
amb
Þ;ð21Þ
where η
T
¼(1εδ) is the fraction of annihilation power
P
a
converted to heat which includes the leftover thermal
energy after reactor power conversion to electricity plus
thermal energy released in the conductors, εis the
efficiency of annihilation power conversion to efflux
kinetic power, and δis non-thermal losses such as neu-
trinos and electromagnetic energy escaping accelerator.
Every reactor with turbines is a heat engine and, as any
heat engine, it possess its power conversion factor to
electric power proportional to the ratio of a difference
ΔTbetween the gas temperatures before (T
0
) and after (T)
the turbines to T
0
. The lower is Tin comparison with T
0
,
the higher is the conversion factor and higher εcan be
achieved. Here is the dilemma: to increase εwe have to
boost turbines effectiveness and to maintain either Tas
low as possible or T
0
as high as possible or both but the
problem is that the area of the radiator grows dramatically
when Tdiminishes, which is undesirable, whereas T
0
is
limited by the melting points of reactor's walls, tubing,
turbine blades, etc.
The emissivity factor ε
T
of the radiator must be as close
as possible to the black body emissivity ε
T
¼1 to ensure
maximum emission power per unit surface at a given
temperature. Emissivity varies with material: bare
polished metals have very low emissivity factor o0.1,
emissivity of heavily anodized aluminum ε
T
E0.85, and
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emissivity factor of graphite ε
T
E0.98. Since the radiator
are supposed to handle a huge amount of thermal power,
it must withhold a relatively high temperature thus the
practical choice will be a sort of ceramics or ceramic
coating on metal having high-temperature melting point.
As a rule, emissivity of ceramics is high, for example,
aluminum-nitride ceramics has emissivity factor ε
T
E0.9.
As seen from Eq. (21), the area Sof a radiator is
proportional to efflux kinetic power P
b
and grows rapidly
with diminishing temperature as T
4
. For example, if the
temperature of radiator is chosen T¼1500 K, the emitting
area should be not less than 1000 m
2
for P
b
¼1 GW, not
less than 1 km
2
for P
b
¼1 TW, and 100 km
2
for
P
b
¼100 TW, assuming ε¼0.5 and δ¼0.2. Lower tempera-
ture would require even larger radiator area to maintain
the outer temperature of the engine section stable for a
given thermal power of the reactor.
The problem of overheating is also notoriously impor-
tant for conventional chemical rockets. The solution has
been found in regenerative cooling of the engine. The only
possible option for regenerative cooling of relativistic
rocket thrusters operating for months and years of proper
time, is propellant and fuel flow through the engine (e.g.,
reactor plus turbines). If H
2
propellant and fuel are stored
in a liquid form at a temperature T
0
r20 K, the amount of
heat consumed per unit time Q
r
by regenerative cooling
under ideal conditions can be estimated from
Q
r
¼dM
dτH
vap
þC
p
ðT–T
0
Þ

;ð22Þ
where H
vap
¼4.61 10
5
J/kg is the hydrogen evaporation
heat and C
p
is the specific heat of hydrogen, the mean
value of which can be taken about 1.5 10
4
J/kg K [66] at
atmospheric pressure through the temperature range from
100 K to 2000 K. The regenerative cooling efficiency is
defined as Q
r
/η
T
P
a
, where P
a
¼P
p
/εis the power of an
annihilation reactor and η
T
is its fraction converted to
waste heat. The estimations show that regenerative cool-
ing of relativistic rocket engine is insignificant virtually
over whole range of efflux kinetic power and efflux
velocities. Thus, a heat-emitting radiator seems to be the
only workable cooler in deep space.
To increase radiator emitting area, it can be made in the
form of a system of spiraling tubes with heated gas
pumped through them. The total mass of a radiator is
proportional to its surface for a given thickness of the walls
of the tubes. The mass of tubular titanium radiator with
20-μm walls having their temperatures T¼500, 1000, and
1500 K is shown in Fig. 9 (η
T
¼0.3) as a function of kinetic
efflux power. For comparison, the optimal rocket launch-
ing mass estimated for different values of reactor power is
also plotted as a function of efflux kinetic power P
b
associated with the corresponding optimal proper efflux
velocities β
b
.
Radiator emitting area and mass grow with the efflux
kinetic energy faster than the optimal launching mass of a
rocket. The radiator takes virtually the whole rocket dry
mass for P
b
410
12
W leaving only marginal space for
equipment, radiation protection, fuel and propellant tanks
not mentioning the crew quarters with life supporting
materials and equipment. Besides, such a thin-wall
radiator requires rigid support and additional shielding
system also contributing to the rocket dry mass. The
radiator can be made up of several fins around the engine
body each filled with a lace of tubular spirals stretching
aside and backward from the engine section. The fins do
not play any hydrodynamic role. Their task is to maintain
sufficiently high emission power of thermal radiation for
keeping the body temperature of the engine section on a
predetermined level. If εr0.5, the efflux kinetic power
P
b
⪢1 TW seems to be not realistic due to fast growing
rocket mass with the increasing mass of radiator. Extreme
efflux kinetic power P
b
100 TW would require signifi-
cantly better engine performance in terms of energy
conversion efficiency ε. However, we cannot jump above
the limits imposed by thermodynamics and η
T
cannot be
decreased to zero no matter what kind of energy source
we use and what type of conductors and electrodes we
employ.
In this regard, the flights to long-distance stars require
a significant reduction of rocket dry mass, and the only
option is slashing the crew together with life supporting
materials, supplies, and equipment especially for the
missions of tens or hundreds of years. Robotic rockets
with artificial intelligence (AI) can do the job without
risking health and lives of people. The main purpose of
reconnaissance missions is gathering information on site,
and our experience with the automatic interplanetary
modules has already proved their indispensability in
performing the tasks. If the mass of robotic control,
sensors, manipulators, and instruments is minimized
down to a miniscule portion of the rocket dry mass, heat
radiator will the main part of the rocket design that
determines the engine power thus achievable cruising
speed. The time of flight is not as critical for the robotic
rockets as for the manned flights. Lower engine power
could be chosen together with higher efflux velocities at
the prize of longer time for picking-up the speed for
a given launching mass. Smaller dry mass means higher
achievable cruising speed with lower M/M
0
reserving
enough fuel for breaking. Knowledge accumulates with
Fig. 9. Mass of titanium radiator with 20-μm walls at the temperature
T¼500, 1000, and 1500 1K as a function of efflux power for η
T
¼0.3 (solid
lines). Also, the optimum launching masses of a rocket propelled by the
efflux of accelerated protons are shown (dots) as a function of efflux
kinetic power with corresponding optimum proton velocities β
b
(P
b
).
O.G. Semyonov / Acta Astronautica 99 (2014) 52–7068

Author's personal copy
practice and it is reasonable to start the road to the stars with
relatively smaller power and lower mass of spacecrafts.
6. Conclusion
Despite strict limitations imposed by thermodynamics
on thermal balance of long-distance rockets, the concept
of relativistic matter propulsion looks promising. The
technology of ion thrusters is under development and
the nearest goal is to boost it to a significantly higher
energy level. The first step will be, apparently, utilization
of small nuclear reactors to prove the concept of high-
power ion thrusters with high efflux velocity of acceler-
ated particles and to find the best design of thermal
radiator. The next step would be development of annihila-
tion reactor equipped with thermal radiators adapted for
operation in deep space. Of course, annihilation reactor
cannot be made without solving the problem of antimatter
production and storage. Currently, high-energy proton
accelerator facilities are used for antiproton production: a
small amount of antiprotons is routinely produced in
Fermi National Accelerator Laboratory (FNAL) and The
European Laboratory for Particle Physics (CERN). After
proper optimization, the ratio of output antiprotons to
incident protons can reach 0.085 [67]. The high-energy
antiprotons are supposed to be cooled down and com-
bined with the positrons to form antihydrogen atoms and
molecules for further storage. Magnetic storage in the
Penning traps can be a solution, but not an optimal one.
It is worth studying the possibility of storing atomic or
molecular antimatter substances in containers made of
conventional matter with magnetic or electrostatic bar-
riers on the surface between matter and antimatter sub-
stances to prevent their mutual diffusion and quantum
tunneling. If storage of antimatter in matter containers is
realizable, it can find many applications beyond astronau-
tics. In particular, a new type of weapon can be developed
[67]. A microgram of antihydrogen after annihilation with
equal amount of hydrogen releases 10
8
J of energy,
which is equal to explosive energy of 50 kg of TNT. To
ensure effective annihilation and conversion of energy of
charged pions to temperature in order to produce a shock
wave in air, the antimatter pellet must be rapidly com-
pressed by a material sheath in analogy with the inertial
fusion pellets [68] and the cores of thermonuclear bombs.
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