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Gravity-assist engine for space propulsion
Arne Bergstrom n
B&E Scientific Ltd, Seaford, BN25 4PA, United Kingdom
a r t i c l e i n f o
Article history:
Received 5 December 2013Received in revised form
10 February 2014
Accepted 15 February 2014Available online 28 February 2014
Keywords:
Spacecraft propulsion
Angular momentum conservation
Tidal locking
Three-body interactions
Numerical simulation
a b s t r a c t
As a possible alternative to rockets, the present article describes a new type of engine for
space travel, based on the gravity-assist concept for space propulsion. The new engine isto a great extent inspired by the conversion of rotational angular momentum to orbital
angular momentum occurring in tidal locking between astronomical bodies. It is also
greatly influenced by Minovitch's gravity-assist concept, which has revolutionized
modern space technology, and without which the deep-space probes to the outer planets
and beyond would not have been possible. Two of the three gravitating bodies in
Minovitch's concept are in the gravity-assist engine discussed in this article replaced by
an extremely massive ‘springbell' (in principle a spinning dumbbell with a powerful
spring) incorporated into the spacecraft itself, and creating a three-body interaction when
orbiting around a gravitating body. This makes gravity-assist propulsion possible without
having to find suitably aligned astronomical bodies. Detailed numerical simulations
are presented, showing how an actual spacecraft can use a ca 10-m diameter springbell
engine in order to leave the earth's gravitational field and enter an escape trajectory
towards interplanetary destinations.
& 2014 IAA. Published by Elsevier Ltd. All rights reserved.
1. Introduction
Rocket propulsion of spacecraft is technically extremely
advanced from the engineering point of view. However,
rocket propulsion is actually at the same time a very crude
and primitive method for space propagation, requiring as
it does huge amounts of propellant to transport the huge
amounts of propellant necessary to produce the massive
amounts of exhaust gases required to propel the rocket in
the opposite direction.
First after several years of space flight in this way, a
method for gravitational propulsion, now called gravity-
assist, was proposed by Minovitch [1,2] at Jet Propulsion
Laboratory (JPL) in USA. This method uses (minute) parts of
the orbital energy and momentum of a planet or moon for
the further propulsion of a space probe. The three-body
problem involved, which Minovitch thus managed to treat in
a special case, made interplanetary travel a realistic prospect.
Without this method, the exploration of the outer planets
(and now interstellar space) by the space probes Voyager I
and II (and subsequent missions like the Cassini mission)
would not have been feasible with present technology.
Inspired by the gravity-assist method for space propul-
sion described above, the present study considers an alter-
native method for space propagation without rockets. The
proposed new method also finds its inspiration in the tidal
damping of orbital motion. This indicates that there are ways
in which rotational motion of a planetary body may be
converted into orbital motion (and conversely orbital motion
be converted into rotational motion), which can be exempli-
fied as follows (cf Fig. 1, from http://en.wikipedia.org/wiki/
Tidal_locking, reprinted here in accordance with Creative
Commons Universal Public Domain Dedication).
Tidal bulges may occur on a body B that rotates around
and close to a more massive body A. If these tidal bulges
happen to be misaligned with the major axis, the tidal
Contents lists available at ScienceDirect
journal homepage: www.elsevier.com/locate/actaastro
Acta Astronautica
http://dx.doi.org/10.1016/j.actaastro.2014.02.017
0094-5765 & 2014 IAA. Published by Elsevier Ltd. All rights reserved.
n Tel./fax: þ44 1323 491310.
E-mail address: arne.bergstrom@physics.org
Acta Astronautica 99 (2014) 99–110
http://en.wikipedia.org/wiki/Tidal_lockinghttp://en.wikipedia.org/wiki/Tidal_lockinghttp://www.sciencedirect.com/science/journal/00945765http://www.elsevier.com/locate/actaastrohttp://dx.doi.org/10.1016/j.actaastro.2014.02.017mailto:arne.bergstrom@physics.orghttp://dx.doi.org/10.1016/j.actaastro.2014.02.017http://dx.doi.org/10.1016/j.actaastro.2014.02.017http://dx.doi.org/10.1016/j.actaastro.2014.02.017http://dx.doi.org/10.1016/j.actaastro.2014.02.017mailto:arne.bergstrom@physics.orghttp://crossmark.crossref.org/dialog/?doi=10.1016/j.actaastro.2014.02.017&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1016/j.actaastro.2014.02.017&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1016/j.actaastro.2014.02.017&domain=pdfhttp://dx.doi.org/10.1016/j.actaastro.2014.02.017http://dx.doi.org/10.1016/j.actaastro.2014.02.017http://dx.doi.org/10.1016/j.actaastro.2014.02.017http://www.elsevier.com/locate/actaastrohttp://www.sciencedirect.com/science/journal/00945765http://en.wikipedia.org/wiki/Tidal_lockinghttp://en.wikipedia.org/wiki/Tidal_locking
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forces will exert a net torque on body B that twists the
body towards the direction of realignment. The angular
momentum of the whole A–B system must be conserved,
so when B slows down and loses rotational angular
momentum in this way, its orbital angular momentum is
boosted by a similar amount (there are consequently also
some smaller effects on A's rotation). As a result, B's orbit
around A is raised in tandem with its rotational slowdown.
For the other case when B starts off rotating too slowly,
tidal locking both speeds up its rotation and lowers
its orbit.
In order to exploit this effect for space propagation,I will here consider a disk-shaped design containing a
spinning dumbbell. Etymologically, the word ‘dumbbell'
originates in Stuart-era England from training for ringing
church-bells by practising with dummies. To describe a
spinning dumbbell consisting of two masses connected
to each other by a spring as discussed in the following
(see Fig. 2), I will here use the term ‘springbell’.
Instead of the spacecraft interacting gravitationally
with two external celestial bodies as in the three-body
interaction in conventional gravity-assist, the onboard
gravity-assist proposed here uses weights and spring
forces between two of the three bodies involved in the
three-body interaction, and where these two bodies andthe spring replacing the gravitational interaction between
them are situated onboard the vehicle itself. The onboard
gravity-assist presented in the following thus uses only the
gravitational field of just one celestial body. Since the
method presented here thus can be described as involving
artificially manipulated tidal forces, it could perhaps also
be called tidal drive or tidal warp.
By skilful use of special trajectories, conventional
gravity-assist can have many uses in interplanetary travel,
but onboard gravity-assist always permits much faster and
more direct trajectories. The advantage with conven-
tional gravity-assist is that it uses only existing gravita-
tional fields, whereas onboard gravity-assist requires anenergy source for its trajectories. In this way, onboard
gravity-assist as presented here could be said to compare
to conventional gravity-assist in the same way as engine-
powered ships compare to sailing ships.
2. The spinning springbell
What is interesting with a spinning springbell in a
gravitational field, e.g. from the earth, is that it is actually a
three-body problem, just as the gravity-assist case dis-
cussed above. Although simple, it retains the basic char-
acteristics of a three-body problem in that it cannot be
simplified by replacing the two masses in the springbell bya mass at its centre of gravity. Instead it can for example, as
will be discussed further below, be arranged to display an
analogue to the tidal damping discussed above.
The advantage of replacing two of the three gravita-
tionally interacting bodies in the classical three-body,
gravity-assist case by a springbell is that the problem
suddenly in this way may become much more practically
useful. Now just a springbell is involved instead of having
to find planetary bodies in suitable positions. The chance
alignments of the outer planets, making possible the
gravity-assisted ‘grand tour’ mentioned above of Voyagers
I and II, will for instance not happen again for more than a
hundred years.The springbell concept is also easier to implement since
a (normal) spring force between the masses in the spring-
bell causes their mutual attraction to increase with separa-
tion, not as in the gravitational case to decrease with
separation.
I will in the following thus consider a system consisting
of a rotating springbell combined with a counter-rotating
circular flywheel. Without changing the total angular
momentum of the whole system, we can then in an
orchestrated manner adjust the angular velocity of the
springbell (and correspondingly of the flywheel), and thus
change the angular momentum of the springbell by feed-
ing energy into the system, or conversely removing energyfrom it.
Fig. 1. A spinning, deformable body exposed to the gravitational field
from a parent body (to the right), around which it rotates in a close orbit.
If the tidal bulges in the body are misaligned with the major axis (red),
then the tidal forces exert a net torque that twists the body towards the
direction of realignment and acts to change its orbital angular momen-
tum. The spinning springbell shown in Fig. 2 attempts to artificially
recreate this effect in a controllable manner. (For interpretation of the
references to color in this figure legend, the reader is referred to the web
version of this article.)
Fig. 2. Schematic springbell engine consisting of two massive weights
coupled by a strong spring and in orbit around the earth (to the right).
The spring expands or contracts in response to the gravitational force
from the earth and to the centrifugal forces from the weights when the
springbell spins around its axis (red), creating a three-body problem
analogous to Minovich’s gravity assist [1]. The springbell can be used for
converting rotational energy into orbital angular momentum as illu-
strated in Figs. 5 through 9. (For interpretation of the references to color
in this figure legend, the reader is referred to the web version of this
article.)
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The trick is thus to manipulate the spin of a springbell
in orbit around a parent gravitational body in such a way
that its gravitational interactions with the parent body
changes its orbital angular momentum around it. Just as in
the gravity-assist and tidal-locking cases discussed above,
a spacecraft containing a springbell can thus use this
technique to change its orbit around the earth or the sun.
However, a problem here is that, as shown by Poincaréalready in 1890 a general three-body problem as in the
present case has no analytical solutions given by algebraic
expressions and integrals. We are thus forced to resort to
numerical methods to solve the motion of the bodies in
this case. On the other hand, the basic equations are very
simple and permit a very straight-forward algorithm as
will be shown below. However, the price for this is that the
numerical solutions will require very small time steps and
correspondingly long computation times, as will be appar-
ent in the numerical simulations presented in the
following.
3. Computational approach
Consider a springbell with (for simplicity) two equal
masses m, located at r 1 and r 2, respectively, in a Cartesian
coordinate system with respect to a distant central, grav-
itationally dominating mass M at the origin. The two
smaller masses m are connected to each other by a spring
with rest length L and spring constant Km (in suitable
units). The accelerations of the two masses m are then the
contributions from the central gravitational field from M
supplemented with the spring forces. The spring forces are
equal but in opposite directions.
From Newton’s second law, the accelerations
€r 1 and
€r 2
of the two masses in the plane of rotation thus become as
follows (after division by m),
€r 1 ¼ GM ̂r 1=r 21þK ðLr 12Þ r̂ 12; ð1aÞ
€r 2 ¼ G M r̂ 2=r 22K ðLr 12Þ r̂ 12 : ð1bÞ
Here the first term on the right-hand sides is the
acceleration component due to the gravitational force
from the central mass at (0, 0), with G being the gravita-
tional constant, and where r̂ 1 and r̂ 2 are unit vectors in
the r 1 and r 2 directions. The second term is the accelera-
tion due to Hooke’
s law in the spring connecting the twomasses in the springbell, and where r 12 and r̂ 12 are,
respectively, a vector and a unit vector directed to r 1 from
r 2, i.e. r 12 ¼ r 1r 2.
The corresponding velocities then become in the next
time step dt
_r 1-_r 1þ €r 1dt ; ð2aÞ
_r 2-_r 2þ _r 2dt ; ð2bÞ
and r 1 and r 2 defining the trajectories of the two weights
can then be calculated from the following expressions
r 1-r 1þ _r 1 dt þ 12
€r 1 dt 2; ð3aÞ
r 2-r 2þ _r 2dt þ1
2 €r 2 dt
2: ð3bÞ
4. Numerical simulations
The above equations will now be studied in numericalsimulations for the case when we have a vertically
oriented springbell moving with velocity v clockwise in a
circular satellite orbit (red circle in Fig. 3 and following
figures). We start the springbell spinning anti-clockwise
around its centre of gravity at time t ¼0, when its bottom
weight is at position (0, 1), by giving the bottom weight a
velocity increment dv in the forward direction of the
orbital motion and its top weight the same velocity change
dv in the backward direction. The bottom of the springbell
thus starts at time t ¼0 at position (0, 1) with velocity
vþdv, and its top simultaneously at (0, 1þL) with velocity
vdv. The spring initially expands a little, and then drags
the weights along in a rotation with oscillating radius.No change has occurred in the total angular momen-
tum when we start the springbell spinning in this way,
since we assume we spin the flywheel (or a tandem
system) in the opposite direction—only energy has been
added to the system.
Fig. 3 shows a typical result from a numerical integra-
tion (using Maple, 20 digits accuracy) of Eqs. (1a) and (1b)
above for this case, and then calculating the positions of
the ends (blue and green, respectively) of the springbell
from Eqs. (3a) to (3b). The spring positions at different
times t on the two intertwined trajectories are shown (in
Fig. 3. Trajectory for a springbell in orbit around a parent gravitating
body, for comparison in this case without any velocity increment/
decrement pairs in any specific spatial direction as in the following
figures, and thus only following a normal elliptic trajectory. The spring-
bell size in this and following examples in Figs. 4 through 9 are for
illustration chosen very large compared to the size of the orbit (the
parameters K and L in Eqs. (1a) and (1b) are set to K ¼100 and L¼0.08 in
the calculations in Figs. 3 through 9). (For interpretation of the references
to color in this figure legend, the reader is referred to the web version of this article.)
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red, one in every 100 time steps calculated) as the rotating
springbell proceeds along its orbit. The points where the
two intertwined trajectories intersect thus represent the
situation when the springbell is momentarily aligned with
the direction of its orbital motion (i.e. when the red line
representing the spring lies in the direction of the orbit).
The initial conditions chosen in the case depicted in
Fig. 3 correspond to a slightly higher mean velocity at themean radius than the velocity of a circular orbit. Hence
the springbell now has a slightly excentric orbit as seen in
the figure, but it still (of course) returns to its initial
position. Even if the velocity increment/decrement dv is
chosen much larger, this slightly excentric orbit remains
the same.
However, it should be pointed out that the dynamics of
the springbell system presented can be designed to
actively manipulate the oscillations in order to achieve a
certain objective. As long as angular momentum is con-
served by the use of the flywheel, then nothing in theory
prevents supplied or extracted energy to be converted by
the springbell into changing its orbit around the parentgravitating body, just as in the case of the tidal bulge
discussed in the Introduction.
Nevertheless, it is not entirely trivial how to actually
go about to achieve conditions in springbell motion of this
kind that are equivalent to the conditions in the tidal-bulge
case. One successful way to emulate the tidal-bulge case is
described in the following and illustrated in Figs. 4–9.
5. Escape from orbit
Figs. 4 and 5 show examples of springbell simulations
as in Fig. 3, but in which we have now introduced an
additional acceleration dv in every time step. The trick is
then to introduce this additional acceleration dv in one
specific spatial direction only (otherwise we do not get any
orbit of any different type; this is how we emulate the
tidal-locking effect discussed earlier): in the x-direction, or
in the y-direction, or in any intermediate direction. Say
that we choose the x-direction. In every time step we thus
add dv to the velocity in the positive x-direction for one of the weights and, to keep the linear momentum constant,
also insert dv in the negative x-direction for the other
weight (just as what happens in the case of a tidal bulge
discussed earlier). The total angular momentum is kept
constant by the flywheel.
We then keep this specific spatial direction constant as
the springbell proceeds along its trajectory. In practical
terms, this can be arranged by suitably tailoring an accel-
eration in the spring direction (by converting energy into
increasing or decreasing the spring force) in combination
with an acceleration in the spin direction (by converting
energy into spinning-up or spinning-down the rotation of
the springbell by using the flywheel). Fig. 7 shows aschematic summary of possible combinations of orbital
rotation and spin.
In particular, as seen in Fig. 5, this scheme can be used
to achieve an outward-spiralling orbit for a spacecraft.
Suppose the springbell is at point (0, 1), moving clockwise
in its orbit and spinning anti-clockwise. When the spring
is in the radial direction, this scheme then corresponds to a
tangential force pair proportional to the velocity in the x-
direction and trying to speed up its anti-clockwise spin
(Case 2a in Fig. 7), and when the spring is in the tangential
direction at (0,1), it corresponds to a tangential expansion/
compression force pair proportional to the velocity in the
x-direction. At the point (1, 0), this extra force pair in the x-direction and proportional to the velocity in the
x-direction vanishes since the velocity in the x-direction
is then zero.
It is fortunate that a similar outward-spiralling orbit
can be obtained by an analogous scheme (Fig. 8) in the
y-direction, but then with the opposite adjustment of the
spin instead. In that case, at the point (1, 0), and when the
spring is in the tangential direction, the scheme corre-
sponds to a tangential expansion/compression force pair
proportional to the velocity in the y-direction, and when
the spring is in the radial direction (Case 1b in Fig. 7),
it corresponds to a tangential force pair proportional to the
velocity in the y-direction and trying to slow down itsanti-clockwise spin. At the point (0, 1), this extra force pair
in the y-direction and proportional to the velocity in the
y-direction vanishes since the velocity in the y-direction is
then zero.
These two schemes may thus be used in combination to
avoid a continuously increasing spin.
A tandem design may be used to remove the need for a
flywheel.
Specifically, in a simulation (Fig. 9) similar to the ones
shown in Figs. 5 and 8, we have for every time step in a
sequence of 1000 steps introduced for one of the weights
an anti-clockwise velocity increment dv in the positive
x-direction proportional to the velocity v, and simulta-neously for the other weight an identical decrement dv in
Fig. 4. Springbell dynamics can be tailored by velocity decrements/
increment pairs in some specific spatial direction to make an orbiting
springbell/flywheel system lose part of its orbital angular momentum
and (moving faster¼larger distances between the red markers) start
spiralling in towards the central gravitating body. (For interpretation of
the references to color in this figure legend, the reader is referred to theweb version of this article.)
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designed in such a way so as to make it possible to
implement technically with a springbell for actual propa-
gation of a spacecraft, as will be further discussed below.
Summarising, the most important result of the simula-
tions described above is thus that they show that it is
possible – like in tidal locking – to use energy fed into a
springbell in a spacecraft by converting it into a substantial
change of its orbit, and this without having to resort to the
disadvantages inherent in rocket propulsion.
6. Springbell-driven escape trajectories from the earth
The computer program discussed above can be usedas well to simulate actual, realistic springbell-driven
escape trajectories from the earth for a spacecraft.
However, there is then a marked difference in physical
scale compared to the simulations performed above for
illustration purposes. In a realistic case, we want to
simulate a springbell with a diameter of the order of
10 m and spinning with a realistic rotation velocity of
the order of one revolution per second. In the calcula-
tions presented below we have used the earth radius
6400 km as length unit and used a time unit such that
the corresponding velocity 8 km/s for a circular satellite
orbit can be set to v¼1, which means that a time unit
corresponds to 800 s.
This realistic case results in quite a different time scalecompared to the simulations described above. The resulting
SPIN – ORBIT ROTATION ALTERNATIVES
Alternative 1 Spinbell spins in same direction as orbital rotation (clockwise – clockwise rotation)
Case 1a Case 1b
clockwise – clockwise with x-direction increments clockwise – clockwise with y-direction increments
Relative velocity increments introduced on Relative velocity increments introduced on
spinbell specifically in x-direction only spinbell specifically in y-direction only
Alternative 2 Spinbell spins in opposite direction to orbital rotation (clockwise – anticlockwise rotation)
Case 2a Case 2b
clockwise – anticlockwise with x-direction increments clockwise – anticlockwise with y-direction increments
Relative velocity increments introduced on Relative velocity increments introduced on
spinbell specifically in x-direction only spinbell specifically in y-direction only
Fig. 7. Further to Fig. 6, the above figure shows different possible alternative combinations of spin and orbital rotation for a springbell. Thus can, e.g., a
combination of Case 2a and Case 1b be arranged to give a sustained outward motion of the springbell, and in which the spin variation is kept within a
certain range. Note that the increments are relative, i.e. there is no increment when the velocity in a certain direction is zero.
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trajectories are orders of magnitude more detailed as func-
tions of time compared to those described earlier, even
though their general characteristics remain the same (each
of the three simulations shown in Figs. 10–12 and described
below required up to two days of computing time on a
modern PC, mostly due to the storage capacity needed forthe detailed trajectories).
Fig. 10 shows for comparison the trajectory of a spinning
springbell with parameters as given in the figure caption, but
with no additional acceleration dv in any specific direction as
discussed above. The springbell in this case is thus expected
to exactly follow a stable (and in this case circular) orbit.
From the insert at the top of the figure, we see that thetrajectory in this simulation manages to retrace the trajectory
Fig. 8. As Fig. 5 but in this case instead with clockwise spin increment/decrement pairs for every time step in the y-direction.
Fig. 9. As Figs. 5 and 8 with velocity increment/decrement pairs for every time step, but here cyclically for 1000 time steps at a time with, respectively,
counter-clockwise spin change pairs in the x-direction (as in Fig. 5), followed by 1000 time steps with clockwise spin change pairs in the y-direction (as in
Fig. 8).
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In addition to the trajectory, each figure shows typical
portions of the trajectory, enlarged so that the individualtrajectories of the two weights can be seen (the time scales
shown in the enlargements are chosen so that the oscilla-
tions in the simulations can be easily seen, and should not
be taken literally).
The springbell in the simulations shown in Figs. 10
through 12 has a spring of rest length of 2107 of
the earth radius, i.e. approximately 1 m. When the spring-
bell spins, the spring oscillates between a maximum
length of ca 106 of the earth radius (E5 m) and a
minimum length of approximately its rest length. As seen
from Fig. 12, this makes it possible for the spacecraft to
reach escape velocity (E11 km/s) from orbital velocity
(E8 km/s) in less than two hours, at which time itsgravity-assist engine gives it a sustained acceleration of
approximately 0.2 g (1 gE9.8 m/s2) on top of its orbital
velocity.According to this simulation, the springbell has oscil-
lated about 3200 times to reach escape velocity from its
original orbital velocity. During this time, the weights
in the springbell have oscillated ca five meters back
and forth. This corresponds to a typical tailored average
accelerating/braking of the weights to 20 km/h and back
every two seconds, which should be within easy reach of
modern technology. In this discussion we have assumed
the springbell – with reactor and radiation shield inte-
grated into the weights – to represent the dominating part
of the mass of the spacecraft.
In addition to the figures described above, Fig. 13
shows a case when the springbell trajectory is startedfrom the earth’s surface instead from an earth orbit.
Fig. 11. Simulation for springbell in earth orbit and a modest change of orbit. Parameters as in Fig. 10, but here with a relative velocity increment/
decrement per time step dv¼70.2108E71.6105 m/s, cyclically increased in specifically the x-direction for 1000 steps and then counteracted by a
decrease specifically in the y-direction for 1000 steps. Enlargements of two small regions of the trajectories are shown to the left and right.
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However, in this case a substantially more energetic
tailored rotation of the type described above is required
in order to break free from earth’s gravitational field.
The advantage of springbell propulsion is again mani-
fested in that only conversion of rotational angular
momentum into orbital angular momentum is required,
and thus requiring nothing of the massive exhaust
gases necessary in traditional rocket propulsion. How-
ever, in contrast to starting from earth orbit, the
initially required additional rotational angular momen-
tum in this case may possibly be prohibitive for eco-
nomic use of take-offs from the earth’s surface even at
the equator.
7. Hypotrochoid motion
The specific motion described above with an additional
acceleration/deceleration pair in one specific spatial direc-
tion only, might advantageously be performed by using
hypotrochoid motion. A hypotrochoid is a curve traced by
a point attached to a circle of radius r rolling around the
inside of a fixed circle of radius R, where the point is at a
distance d from the center of the interior circle. The
parametric equations for hypotrochoid motion as function
of time t are given as follows
xðt Þ ¼ ðRr Þ cos ðt Þþd cos ððRr Þt =r Þ; ð4aÞ
Fig.12. Simulation for springbell in earth orbit and an escape trajectory. Parameters as in Fig. 11, but now with a relative velocity increment/decrement per
each time step chosen ten times larger as dv¼70.2107E71.6104 m/s, corresponding to an acceleration of about 0.2 g. This larger acceleration dv
thus leads to an escape trajectory in less than half an orbit. Enlargements of two small regions of the trajectories are shown to the left and right.
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yðt Þ ¼ ðRr Þ sin ðt Þ–d sin ððRr Þt =r Þ: ð4bÞ
In particular, the parameters R, r , and d can be chosen
so that the curve becomes an ellipse, and which in the
special case r ¼R/2, d¼r can be made to degenerate to the
straight double line x(t )¼(R-r ) cos(t ), y(t )¼0 as shown in
Fig. 14. The important point here is thus that motion along
such a hypotrochoid, including its degenerated double
line, can be arranged by two circularly rotating elements
only. This thus permits the additional acceleration/decel-
eration in one specific spatial direction as discussed above
to be achieved in a more practical fashion than for the
more straight-forward acceleration/braking motion envi-
saged in the description earlier.
8. Conclusion
Conservation of mass-energy, linear momentum, and
angular momentum, are understood to be fundamental
laws of nature. According to Noether’s theorem [3], these
laws are connected to basic symmetry properties of time
and space as expressed in the form of, respectively, the
homogeneity of time, the homogeneity of space, and theisotropy of space [4]. Any propagation in space would
seem to need to be governed by these laws. So does,
for instance, rocket propulsion use the conservation of
linear momentum for the propagation of a spacecraft,
which is why the ejection of the massive exhaust gases
in rocket propulsion is an unavoidable consequence
of such a mode of propagation based on conservation of
linear momentum.
But conservation of linear momentum is not the only
principle by which propagation in space can take place.We can also conceive of modes of propagation where the
Fig. 13. Simulation for springbell trajectory starting from earth’s surface
(red horizontal line) at the equator instead of from earth orbit as in the
previous figures. The figure shows the first ca 80 km of the trajectory,
after which the trajectory may be as in Fig. 12. The enlargement of a
portion of the trajectory shows the two weights rotating and oscillating
relative to each other in the gravitational field from the earth and thereby
converting rotational angular momentum into orbital angular momen-
tum. Parameters as in Fig. 12, but with relative velocity increment/
decrement dv per time step in this simulation a factor a thousand times
larger during the initial acceleration, necessitated by the much smaller
angular momentum available to convert in this case. (For interpretation
of the references to color in this figure legend, the reader is referred to
the web version of this article.)
Fig. 14. Examples of hypotrochoids (grey ellipses) as given by Eqs. (4a)
and (4b) and formed by the end points (black) of a rotating and
oscillating springbell (red). When a smaller interior circle (black) of
radius r rolls around the inside of a fixed exterior circle (blue) of radius R,
a hypotrochoid is formed by the motion of a point on a rolling radius of
this interior circle at a distance d (green) from its center. In the special
cases illustrated here, the hypotrochoids are ellipses. In particular, the
hypotrochoid degenerates in the special case r ¼R/2, d¼r to a double line
(bottom), thus showing a way to implement the required additional
acceleration of the springbell in only one specific spatial direction by
using two superimposed circular rotational motions. This case thus
illustrates how such springbell motion in one specific spatial direction
can be arranged and contained in this way within a disc-shaped engine
room. (For interpretation of the references to color in this figure legend,
the reader is referred to the web version of this article.)
A. Bergstrom / Acta Astronautica 99 (2014) 99–110 109
8/18/2019 (1 Gravity-Assist Engine for Space Propulsion (
12/12
conservation of linear momentum is given a subordi-
nate role, and where instead the conservation of angular
momentum plays the central part. This could then in some
respects turn out to be a much more advantageous form of
propagation from the engineering point of view. The
massive exhaust gases required in the linear-momentum/
rocket case, can in the angular-momentum/springbell case
then instead be replaced by the spinning-up/spinning-down of a counter-rotating flywheel as discussed in
this paper.
As shown by the discussion of tidal bulges in the
Introduction, conservation of angular momentum can be
instrumental in converting rotational angular momen-
tum into orbital angular momentum and raise the orbit
of a moon. Similarly, in Minovitch’s gravity-assist con-
cept [1] also discussed in the Introduction, some minute
part of the orbital angular momentum of a planet or
moon can be converted into orbital angular momentum
of a spacecraft, thus making it possible to send it even on
interstellar missions, as exemplified by the Voyager I
and II missions.Inspired by these facts, the gravity-assist engine
described here thus exploits the possibility of creating
what might perhaps best be defined as an artificial tidal-
bulge system. If a spacecraft in orbit around the earth or
the sun employs this concept, then internal energy used
for spinning-up/spinning-down a springbell as described
above can be converted into boosting the orbital energy
and orbital angular momentum of the spacecraft and thus
raising its orbit.
It should be emphasised again that the method of
propagation discussed here requires only energy and no
emission of exhaust gases. Thus, with nothing but a
sufficiently powerful internal energy source (e.g. a nuclear
reactor), a spacecraft could be sent on an escape trajectory
from the earth or the sun as shown in Figs. 11 and 12. In
this connection it should be pointed out that the elements
of such a nuclear reactor technology already exist; themodern generation of large strategic nuclear-powered
missile-carrying submarines do not require to have their
nuclear reactors refuelled during their entire planned
30-year life span (e.g. in Ohio-class submarines).
Acknowledgments
The author is grateful to Dr Hans-Olov Zetterström for
many clarifying discussions.
References
[1] M. Minovitch, An Alternative Method for Determination of Ellipticand Hyperbolic Trajectories, Jet Propulsion Laboratory TechnicalMemos TM-312-118 (July 11, 1961).
[2] M. Minovitch, A Method for Determining Interplanetary Free-fallReconnaissance Trajectories, Jet Propulsion Laboratory TechnicalMemos TM-312-130 (August 23, 1961).
[3] E. Noether, Nachr König, Gesellsch. Wiss. zu Göttingen, Math.-Phys.Klasse 235 (1918);
R. Courant, D. Hilbert, Methods of Mathematical Physics, vol. 1,Wiley, New York, 1989, 262.
[4] L.D. Landau, E.M. Lifshitz, Mechanics, third Ed. Pergamon Press, 1988.
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