Radioactive decay to propel relativistic interstellar probes along a rectilinear hyperbolic motion (Rindler spacetime)

Abstract

In this academy transactions note, we study a type of relativistic interstellar flight that would enable an unmanned space probe to reach the nearest stars in a few decades’ time. “Time” here means “proper time”, that is the time aboard the probe, a little shorter than the time elapsed on Earth for the same flight. We consider what in special relativity is called “hyperbolic motion”, that is a rectilinear, accelerated motion with a constant acceleration a in the probe's reference frame. The overall mission to any nearby star is thus subdivided into two equal halves: (1) from the Sun to half way the distance to the target star, the probe undergoes a uniform proper acceleration a; (2) at mid-point the probe is turned by an angle of 180 ∘ , so that the propulsion system becomes a braking system; (3) the probe then undergoes a uniform proper deceleration a, until it reaches the target star with zero speed. The probe's top speed is thus achieved at mid-point between the Sun and the target star, and it may equal a significant fraction of the speed of light (20% or higher). An equation is then derived expressing the hyperbolic motion's uniform acceleration a as a function of the overall mission time and of the distance of the target star from the Sun. This is the first key equation for a realistic mission plan. We next prove, by resorting to the Ackeret equation for a relativistic rocket, that, for the hyperbolic motion, the mass of the “propellant” decreases exponentially. The important point then comes: this exponential decrease of the propellant mass formally has just the same equation as the radioactive decay equation. The way thus is paved to exploit the radioactive decay as a propulsion system to achieve the uniformly accelerated mission profile of the hyperbolic motion. Finally, from this “perfect match” between hyperbolic motion and radioactive decay, one more basic equation is derived. This equation expresses the radioactive decay constant λ as the ratio between the hyperbolic uniform proper acceleration a and the proper speed w of the ejected radioactive fragments “propelling” the probe in the opposite direction. This equation is vital inasmuch as it enables one to select the appropriate radioactive material as the “propellant” capable of achieving the requested constant proper acceleration a for a given target star. In conclusion, the logical sequence of steps to design an interstellar probe based on the radioactive decay of a certain material has now been made clear: (1) First determine the uniform proper acceleration a in terms of the assumed overall flight proper time, T overall , and of the distance of the target star, D star . (2) Then select the appropriate radioactive material by selecting its radioactive decay constant λ , expressed as the ratio of the previously selected constant proper acceleration a to the (constant) proper speed of the ejected radioactive fragments w. (3) For different target stars, select different radioactive materials according to the equation λ = a / w .