*Solar physicist Tishtrya Mehta is a regular Principium contributor. Here she brings us an introduction to momentum braking, based on work by Professor Dr Claudius Gros, Johann Wolfgang Goethe-Universität, Institut für Theoretische Physik, Frankfurt/Main Germany.*

So goes the old saying ‘What goes up, must come down’. Since the dawn of human flight we’ve come to learn that this phrase no longer rings true and we have since launched probes, rockets and platforms deep into space without the intention of them ever returning to Earth. One of the most celebrated is the New Horizons probe which recently enjoyed a flyby of everyone’s favourite dwarf planet Pluto, sending back breath-taking images of the small rock in incredible detail, and has continued to hurtle through space with no intention of slowing down. This is a problem facing aerospace engineers as so far we have developed successful technologies capable of accelerating fast enough to reach interstellar distances, but now we must face the challenge of slowing the craft down sufficiently so it may fall into orbit around its target or at least attempt a flyby with a significant enough duration to collect and transmit valuable data.

Currently the most appealing and realistic target for interstellar travel is the Proxima Centauri system, headed by a low-mass red dwarf star, a mere 4.25 light-years from us. Thus, using a probe that can accelerate to around a tenth of the speed of light, a journey seems feasible within the duration of a human lifetime. A difficulty lies in decelerating a probe travelling at \(3 \times 10^4 \text{km/s}\) to a near stationary state, as our usual methods of braking involve copious friction between the moving object and a stationary medium; think of the wheels of a slowing car against the surface of a road, or an Apollo capsule crashing to a halt upon contact with the Earth. In space there is no ‘road’ as such, just a sparsely populated ‘interstellar medium’ (ISM) which is of a very low density and so offers little friction in ordinary circumstances. This is why we can accelerate probes to such high speeds in space and expect little resistance and allow the probe to continue at a largely unchanged velocity many years after the initial acceleration.

Critically, the ISM is not a complete vacuum and the material within it, even if low density, contains a significant amount of ionised molecules which can be harnessed to serve as means to slow down a probe, which is the key mechanism behind the idea of momentum braking.

So how does it work? Momentum braking relies on transferring the kinetic energy of the probe to its surroundings, in this case the ISM, which can be effectively carried out via the use of a magnetic sail. Prof Gros puts forward a candidate of a Biot-Savart magnetic sail in his 2017 paper [1] which consists of a superconducting loop tethered to a payload, the mass of which doesn’t exceed 1.5 tonnes. A Biot-Savart loop carries a current in a closed circle, which in turn creates a magnetic field. This magnetic field changes the trajectory of nearby ionized particles, (usually \(H^+\), ionised hydrogen; the simplest and most abundant element found in the universe) and causes them to be reflected, which slows the craft using conservation of momentum.

The laws of Physics dictate that in any closed system (where nothing enters or leaves the ‘frame’ during the interaction) momentum cannot be lost, only transferred between different bodies. So the momentum lost by the craft must be equal to the momentum gained by the particle.

We can look at this further by considering the equation \(p=mv\) where \(p\) is momentum, \(m\) is mass and \(v\) is velocity. So

\[mass_{craft}v_{craft}=mass_{total\ particles\ encountered}v\]

The mass of the particles encountered by the craft in a given unit of time, say \(T_{unit}\), can be given by the density of \(H^+\) ions per \(\text{m}^3\) (\(n_p\)) swept out by the area of the sail multiplied by the velocity of the craft and the mass per proton (\(m_p\)), i.e.

\[mass_{total\ particles\ encountered\ during\ T_{unit}}=Area_{sail}\times n_p \times m_p

\times v \times T_{unit}\]

The particles are assumed to perfectly reflect (which gives the greatest change in momentum, and so the most efficient braking), and velocity is a vector quantity so that we define ‘positive’ velocity in the direction of travel for the craft. Therefore the change in velocity can be found by subtracting the ‘new’ velocity from the ‘old’ velocity, i.e. \((-v)-(+v) = -2v\). Therefore the momentum change in a unit of time may be given by:

\[mass_{total} v = -2v (Area_{sail} \times n_p \times m_p \times v \times T_{unit})\]

Or by employing basic differentiation, we can see that the rate of decrease in momentum (or analogously the rate of deceleration) is given by

\[\dfrac{d}{dt}\left(mass_{total}v\right) = -2v (Area_{sail} \times n_p \times m_p \times v)\]

Therefore it can be seen that the area of the sail has a great effect on how quickly and effectively we can begin braking the craft, which intuitively makes sense.

The craft is able to reflect particles via the principle that charged particles in a magnetic field travel in circular orbits. This phenomenon can be seen in cloud chambers which trace out the path taken of incoming particles as seen in Figure 1. The spiral orbits correspond to low mass charged particles, likely electrons and positrons, which have been guided into circular trajectories via a magnetic field but due to losses of energy due to friction etc fall into smaller orbits with each turn, which presents itself as a spiral path. The sail functions by the same principle.

When a particle approaches the craft from a region of negligible magnetic field it can be treated as having a velocity perpendicular to the craft’s reference frame since the velocity of the craft is very high by comparison with that of the particle (see below).

As stronger magnetic fields (given by \(B\)) produce smaller orbits, we can examine how \(B\) affects the efficiency of reflecting particles (where \(B_1 < B_2 < B_3\)).

In \(B_1\) the magnetic field is extremely weak, and so this produces a large radius of orbit for the particle, which it cannot complete within the width of the magnetic field, and so the particle exits the region at an angle, but certainly not reflected. This is therefore inefficient.

In \(B_2\) we have a magnetic field tuned so an oncoming particle may complete exactly half an orbit before exiting the region parallel to its direction of entry. This gives perfect reflection.

For any magnetic field strength greater than \(B_3\), a particle will be drawn into a tight spiralling orbit upstream of the craft. This is less efficient than total reflection but still contributes to slowing the craft. Therefore the ideal magnetic field is in the vicinity of \(B_2\) which in the case for a 1.5 tonne craft travelling at \(0.1c\) to Proxima Centauri as Prof Gros has put forward, requires a current of around 105 Amps, the feasibility of which is discussed further in his paper. Each particle is usually a single positive Hydrogen ion which therefore has the mass of one proton and a positive charge equivalent to the magnitude of one electron, since its ‘normal paired’ electron is missing.

Hence it is plausible to consider the technology of momentum braking to be implemented on an interstellar craft in order to decelerate them as the materials, energies and mechanics underlying the principles are all within reasonable budgets and certainly within the timescales envisaged by even the most optimistic developers of interstellar craft! This technology represents an ingenious and exciting answer to a well-rehearsed problem and is one of several candidates put forward in recent years. Other issues to continue addressing and revising include the mechanisms of accelerating a craft to \(0.1c\), developing a craft with all necessary capabilities to collect and send back data which has a mass of less than 1.5 tonnes, and ensuring the craft can ‘survive’ for several decades. There are of course many more barriers to overcome but this new insight in braking technology may place us one step closer to realising the goal of sending a probe to another star system within a human lifetime.

**By Tishtrya Mehta**

## Acknowledgement

Special thanks to Prof Claudius Gros for his kind assistance in the writing of this article, errors and omissions are, of course, attributable to the author and not to Professor Gros.

Prof Gros heads a research group and lectures at the Institute for Theoretical Physics at Goethe University, Frankfurt, and recently wrote a textbook on CADS (Complex and Adaptive Dynamical Systems) [2]. He regularly gives talks on topics ranging from the philosophical “Can we personally influence the future with our present resources?” to the highly specialised “Interaction induced Fermi- surface renormalization”.

## About the author

### Tishtrya Mehta

Tishtrya Mehta is a postgraduate researcher at the University of Warwick, specialising in data analysis of quasi-periodic pulsations in solar and stellar flares. Tishtrya is Deputy Chair of the Education Committee of the Initiative for Interstellar Studies.

## References

[1] C Gros, Universal scaling relation for magnetic sails: momentum braking in the limit of dilute interstellar media, 2017, J.Phys. Commun, I, 045007, arxiv.org/abs/1707.02801

[2] Complex and Adaptive Dynamical Systems, A Primer, Claudius Gros, Springer International Publishing (2015). ISBN 978-3-319-16264-5, DOI 10.1007/978-3-319-16265-2, www.springer.com/gb/book/9783319162645