The cost C of a round trip is equal to an opportunity multiple (OPm) times the investment required to make the trip. The opportunity multiple arises from the fact that investment could be made at home instead, and is equal to:
where T is the time required for one leg of the journey (out or back) and I% is the rate of return possible if the money were invested at home instead of on the interstellar voyage.
Ignoring the cost of building and maintaining a ship (as well as the costs of overhead, employees, etc.), the investment required to send a sublight trading mission using the Boardman anti-matter drive is calculated from:
where I is the investment, Rm is the number of kilograms of anti-matter required per kilogram of payload, the factor "2" entering because anti-matter must be purchased at the destination before the return trip (doubling the cost), and 5704MWy/kg is the amount of energy (in megawatt-years, MWy) required to produce a kilogram of anti-matter. For a one-way trip, the value of Rm is:
assuming the ship is capable of refueling at its destination, where c is lightspeed (3 x 105 km/sec) and u is the maximum velocity of the ship.
Cost C is then:
T, however, is a function of u, the maximum velocity. If we plug an equation for T into the equation for C and assume values for u and I%, we can calculate the cost per kilogram of trade goods. T is calculated from:
where d is the distance to be travelled and g is the rate of acceleration.
One of the interesting things about the equation for C is that the opportunity multiple decreases as u increases (because the journey takes less time) while the investment increases as u increases (because more anti-matter is required). This implies that there is, for a given set of conditions, some maximum velocity u at which minimum cost is achieved.
Table 1 (in the parent article) shows minimum costs for a number of journeys of different lengths.
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