Universal scaling relation for magnetic sails: momentum braking in the limit of dilute interstellar media

Abstract: The recent progress in laser propulsion research has advanced substantially the prospects to realize interstellar spaceflight within a few decades. Here we examine passive deceleration via momentum braking from ionized interstellar media. The very large area to mass relations needed as a consequence of the low interstellar densities, of the order of 0.1 particles per $\mathrm{cm}^{3}$, or lower, are potentially realizable with magnetic sails generated by superconducting coils. Integrating the equations of motion for interstellar protons hitting a Biot Savart loop we evaluate the effective reflection area $A(v)$ in terms of the velocity $v$ of the craft. We find that the numerical data is fitted over two orders of magnitude by the scaling relation $A(v)\ =\ 0.081A_R\log^3(I/(\beta I_c))$, where $A_R=\pi R^2$ is the bare sail area, $I$ the current and $\beta=v/c$. The critical current $I_c$ is $1.55\cdot10^6$ Ampere. The resulting universal deceleration profile can be evaluated analytically and mission parameters optimized for a minimal craft mass. For the case of a sample high speed transit to Proxima Centauri we find that magnetic momentum braking would involve daunting mass requirements of the order of $10^3$ tons. A low speed mission to the Trappist-1 system could be realized on the other side already with a 1.5 ton spacecraft, which would be furthermore compatible with the specifications of currently envisioned directed energy launch systems. The extended cruising times of the order of $10^4$ years imply however that a mission to the Trappist-1 system would be viable only for mission concepts for which time constrains are not relevant.