that maintains a velocity V is t = kR/ρV3, and the total distance covered is S = Vt = kRV2. Here, k is a constant that depends on the type of energy storage and on geometric factors relating to how streamlined the vehicle is. For vehicles that derive their energy from chemical storage and which are streamlined with the usual 12:1 length-to-diameter aspect ratio appropriate for conventional high-strength materials, the value of k can range up to about 1011 in metric units. This factor k can vary by a modest amount, depending on the specific chemicals used for energy storage, the fraction of total volume devoted to fuel, and so on. The Greek letter “rho” (ρ) is the density of the medium that the vehicle moves through (1,025 kg/m3 for seawater and 1.29 kg/m3 for air at sea level).

Thus, both range and endurance are expected to be roughly proportional to the scale of the vehicle. Smaller vehicles can be used if one is willing to make a proportionate sacrifice in range and endurance. The mass of the vehicle, which generally determines the cost and the difficulty of handling logistics, is proportional to R3. So range and endurance are found to be proportional to M1/3, where M is the gross (fully loaded) mass of the vehicle. Furthermore, range varies inversely with V2, and endurance varies inversely with V3. Thus, it is very important that the vehicle move only as fast as necessary to accomplish its mission, but no faster.

For endurance, a lower bound on velocity is primarily determined by the fluid disturbances that the vehicle must fight. In both atmospheric and underwater vehicles, the natural disturbances with which the vehicle must contend are driven by solar power. The fluid disturbances in the atmosphere and under water rarely exceed the power densities (½ ρV3) of the incident solar power flux (1,350 W/m2). Under water, this means that characteristic currents are 1.4 m/s (2.7 knots). In the atmosphere, because the density of air is so much lower, the velocity of typical disturbances is higher, about 14 m/s (27 knots) at sea level and 36 m/s (70 knots) at 18 km altitude (59,000 ft). Most autonomous vehicles designed for long endurance cruise at a speed that ranges from 1.3 to 5 times faster than these disturbances, so that they have sufficient command authority to loiter over a stationary point and to reach new targets of interest despite the occasional flow that exceeds the solar power flux. One side effect of the solar origin of these disturbances is that an infinite-endurance solar-powered aircraft is barely possible, since the wing area can collect power at about the same rate at which a minimal vehicle expends it.1

The simple analysis here shows that the endurance of a vehicle varies inversely with ρV3, which itself is just a fixed multiple of the solar constant, and so the endurance of a vehicle that has a fixed velocity margin over the natural


See, for example, the Web site <>. Last accessed on April 1, 2004.

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