254.0 days

38.883 years

Figure 7.2. Features and average distances of objects from the Sun (1 AU = 1.496 x 108 km is the average distance of Earth from the Sun).

manned mission, both because of vehicle mass (to ensure crew sustenance and survival) and cost. It does not take long to conclude that traveling at constant speed in the Solar System becomes feasible only if the speed is higher by at least a factor 10, or if traveling at constant acceleration, rather than constant speed. In both cases the spacecraft must be accelerated far more than allowed by chemical propulsion.

It is instructive to see the consequences of shifting to a trajectory strategy based on constant acceleration. For constant acceleration, a, kept until mid-course, followed by an equal deceleration to Neptune, classical mechanics predicts a oneway time as given in equation (7.1) where S and S1=2 are distance and mid-way distance to Neptune, respectively. For a = 1 "g" (the Earth gravitational acceleration, or 9.81 m/s2), the round trip to Neptune would take 151 days, not 23 years. Such acceleration would be very convenient, freeing a crew from all undesirable effects of micro-gravity ("weightlessness"). Lowering a to 1/10 "g", the round trip would last a factor vT0 longer, or about 46 days.

These sound like awfully short travel times, but actually depend on whether or not the space ship can keep accelerating at the acceleration a chosen for the trip. Fast travel depends on "affordable" acceleration, that is, on how long the propulsion system can supply the thrust capable of maintaining it, since acceleration a = thrust F/vehicle mass M. The higher the acceleration, the shorter the trip time, but also the higher the propellant rate of consumption and the vehicle initial mass M, thereby lowering a: in fact, M must include all propellants needed by the propulsion system. The quantitative analysis of this problem is determined by the rocket equation and Newtonian mechanics. The governing equations follow:

V g Isp

Wgross = OWE(WR1/2)fr0m earth fty-by

^gross (OWE(WR1/2)toplanet)(WR1/2)fromearth rendezvous (7.2)

The WR1/2 is the weight ratio either from the Earth to the halfway point or from the halfway point to the orbit of the target planet. Table 7.1 gives the parameters for two constant accelerations and for a boost and coast mission for an Isp of 459 seconds (4,500 m/s). If the mission is a fly-by then only the weight ratio for departing Earth applies. If the mission is a rendezvous mission then the product or the two weight ratios apply. Remember for the rendezvous mission the orbital

Table 7.1. Neptune mission as a function of acceleration, a.







4.05E + 09

4.05E + 09

4.05E + 09


1/2 distance

2.02E + 09

2.02E + 09

2.02E + 09



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