(1 gravity-assist engine for space propulsion (

8/18/2019 (1 Gravity-Assist Engine for Space Propulsion (

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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: [email protected]

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:[email protected]://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:[email protected]://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.)

A. Bergstrom / Acta Astronautica 99 (2014) 99–110100


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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.)

A. Bergstrom / Acta Astronautica 99 (2014) 99–110 101


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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.)



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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