Figure 5-30 — Integrated trajectories between the Moon and L2 (after Edelbaum and D'Amario, AIAA Journal, April 1974).
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APPENDIX H: TRAJECTORIES FROM THE MOON TO L2
D'Amario and Edelbaum (ref. 27) have studied trajectories to L2, originating tangentially to the lunar surface in low lunar orbit. Such trajectories are those of concern for transport of lunar material. Their work was performed in the circular restricted three-body problem. The results are shown in figure 5-30.
The trajectories fall into two classes: "fast" transfers and "slow" transfers. The latter involve looped trajectories and transfer times over 200 hr. While they lead to arrival at L2 with velocities of only 100 m/s, approximately, they appear quite sensitive to small errors at launch and so are not of interest. The "fast" transfers involve transfer times under 100 hr and arrival velocities of approximately 200 m/s. Sensitivities of such orbits may be studied, at least when orders of magnitude only are of concern, by considering the transfer trajectories as conic sections in the two-body problem. A dynamic equilibrium may be sought between acceleration due to gravity and acceleration due to momentum flux. The equilibrium is unstable; nevertheless, the stationkeeping requirements associated with stabilizing the equilibrium may be much less than those associated with nulling out a momentum flux by continuous thrusting.
APPENDIX I: ROTARY PELLET LAUNCHER
The rotary pellet launcher (RPL) is a heavy tube rapidly rotating so as to accelerate small pellets of rock. The tube consists of a straight, nontapered section near the end and an exponentially-tapering section inward toward the hub from the nontapered section (see fig. 5-31). The notation is:
- σ = allowable stress
- ρ = material density
- A₀ = tube cross-section area at end
- r₀ = radial distance to end of tapered section
- r_c = length of nontapered section
- ω = rotation rate, rad/s
- V₀ = tip velocity = (r₀ + r_c)ω
The following equations define a design: For given V₀ and ω,
where a = V₀²/2σ - 1 and erf is error function. Now let L = r₀ + r_c. There exist the following ratios, given as functions of the dimensionless parameter a:
The mass ratio is the ratio of RPL tube mass to the mass of a tube of the same density and equal length, with constant cross-sectional area A₀. Curves of the three ratios are plotted in figure 5-32.
Typical materials of interest for use in an RPL are given in table 5-23. Here S_y is the yield strength in MPa; V_c is the critical velocity, V_c² = 2S_y/ρ. To build an RPL to eject pellets with velocity less than V_c, the launcher can be a straight tube of uniform cross section. But above V_c, the RPL must be tapered for part of its length, increasing in thickness toward the axis.
Note that a = (V₀/V_c)² - 1. Now consider a reference design. Let V₀ = 3965 m/s. The material is Kevlar, at 60 percent of yield strength, or σ = 2.2 GPa (315,000 lb/in.²). Take the density at 1.55 g/cm³, or 7 percent higher than the tabulated value. Also let A₀ = 83.6 cm², or a diameter of approximately 10.3 cm at the tip; L = 15.2 m. Then:
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Figure 5-32 — Estimating factors for RPL design.
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Figure 5-31 — RPL injector schematic.
a = 4.59 m = 8669 kg
Now consider the bending stresses due to acceleration of the pellets. Suppose pellets of 10 g mass are accelerated. Near the tip the imposed acceleration is some 100,000 g. The pellet presses on the side of the tube with a force of approximately 8.9 kN, which is denoted F. The associated stress at any distance r from the tip has maximum value at the outside of the tube. Let the tube diameter there be d; the stress σ then is:
σ = 32Fr/πd³
and d² is proportional to A. A plot of σ as a function of radial distance from the axis is given in figure 5-33. Note that for the reference design, the stress is maximum some 2.44 m inward from the point where the tube ceases to taper. But even at that maximum, the stress is only some 7 percent of the yield stress for Kevlar. Stress relief may be provided by making the tube ellipsoidal in cross section.
The reference design is a rotating tube without counterweight. The lightest counterweight is not a solid block but is an exponentially tapering shape, like the tube. A double turret has a mass 17,340 kg. Two such turrets are needed, counterrotating, for the mass-catcher to maintain zero net angular momentum.
The catchers receive 40 kg/s at, say, 200 m/s for a force of some 8 kN. To null out, some 2 kg/s are ejected at 4000 m/s. For the dual turrets described approximately 200 pellets are ejected per second; each pellet is 10 g. The theoretical power required is (1/2)mV₀² = 16 MW. If provided by a space nuclear power system at 45 kg/kW this requires 7200 t.
To achieve thrust, the RPL must be made to release its pellets with approximate uniformity in direction. This may be accomplished with the pellet injector of figure 5-31. The feed tube rotates at the same rate as the RPL tube, pressing a pellet against the restraint. There is a hole in the restraint which lets a pellet through when the tubes are pointing in the right direction. A gate in the feed tube, controlled by a cam, ensures that only one pellet goes through at a time. Use of the gate means that the hole need not be small.
The RPL is subject to considerable wear due to friction and abrasion from the pellets, and must be designed for easy maintenance. This is accomplished by providing the tube with a removable liner, and by designing other high-wear parts for easy removal and replacement.
TABLE 5-23 — MATERIALS OF INTEREST FOR A ROTARY PELLET LAUNCHER
| Material | Density (g/cm³) | Yield Strength (MPa) | Critical Velocity (m/s) | | :--- | :--- | :--- | :--- | | Aluminum | 2.7 | 450 | 577 | | Steel | 7.8 | 1500 | 620 | | Titanium | 4.5 | 1000 | 667 | | Kevlar | 1.45 | 3600 | 2227 |
Figure 5-33 — Bending stresses in reference RPL design.
<!-- image -->APPENDIX J: IMPACT UPON LUNAR ATMOSPHERE
The present lunar atmosphere, arising from natural sources with a total rate less than 0.010 kg/s, has a mass of less than 10⁴ kg and surface number densities less than 10⁷/cm³. The primary mass loss mechanism is due to the interplanetary electric field resulting from the motion of the solar wind. This causes rapid loss of gases to the lunar exosphere within 10⁶ to 10⁷ s. This loss has been confirmed by observations of lunar module exhaust gases (refs. 28, 29). If the atmosphere is dense, however, the cleansing effect of the solar wind decreases and thermal escape becomes the dominant loss mechanism due to the relatively higher collision rate among particles. The comparative effectiveness of these two loss mechanisms is illustrated in figure 5-34 for an oxygen atmosphere.
The use of the present lunar "vacuum" for industrial purposes as well as for scientific purposes (e.g., astronomical observations) will most likely necessitate the maintenance of a sufficiently "lunar-like" exosphere rather than allowing a substantial atmospheric mass to build up. Figure 5-35 presents growth curves of the lunar atmosphere for various constant gas addition rates. A release rate of about 10-100 kg/s would cause a transition to a long-lived atmosphere which occurs at a total mass of 10⁸ kg (ref. 30). Release rates at about 1000 kg/s will produce an atmosphere which will exert aerodynamic drag on orbiting or departing vehicles (ref. 31). At gas release levels at or below 0.1 kg/s the lunar atmosphere would increase at most to a mass of 10⁶ kg. Furthermore, if the artificial source of gas is shut off, the time scale for the Moon's atmosphere to return to its natural state is on the order of weeks (10⁶ to 10⁷ s). Due to the modeling techniques in determining these effects, order-of-magnitude accuracy should be attributed to these estimates.
Three sources can be identified within the framework of large-scale lunar operations as potentially releasing substantial quantities of gases into the lunar exosphere: mining and processing of lunar materials, leakages from the Moon base environment, and fuel expenditures during transportation of personnel and materials to and from the lunar surface. To build the Stanford Torus, raw material is propelled from the lunar surface by a mass launcher and not by the use of chemical rockets. Without materials processing on the lunar surface a potential source of gases is eliminated, and by using nonchemical methods to lift the required lunar materials off the lunar surface, the mass released during transport through the Moon's atmosphere is minimized.
It is believed that mining operations will not release a significant amount of gases into the atmosphere. If it is assumed that 10¹⁰ kg of lunar materials are mined during a 10 yr period and that 10 percent of the trapped gases in the lunar soil (10⁻⁴ to 10⁻⁵ of its mass) are released during normal mining operations, the average release rate is 3 X 10⁻⁴ kg/s, which is substantially less than the natural source rate (personal communication from Richard R. Vondrak, Stanford Research Institute, July/August 1975).
Losses due to leakage from a Moon base have been estimated by NASA experts (ref. 18) based upon a projected loss rate per unit surface area. The yearly leakage loss is approximated to be 18,000 kg which would result in a release rate of 6 X 10⁻⁴ kg/s. Again this is insignificant in comparison to the natural source rate. It should also be noted that the Moon base considered by Nishioka et al. (ref. 18) includes a processing plant and would most likely be larger than the lunar facility considered here. The actual release rate would then be even smaller than that given above.
By far the most significant source for release of gases into the lunar environment is the exhaust products released by chemical rockets in the initial establishment of the lunar base and its continual resupply. It has been estimated that 1 kg of propellant will be expended by the lunar landing vehicle for each kilogram of payload landed (ref. 18). The mass of lunar base, estimated at 17 X 10⁶ kg, is assumed to be delivered to the lunar surface over a 2.5 yr construction phase. An annual resupply rate of 0.4 - 0.5 X 10⁶ kg has been calculated. These figures give release rates of 0.2 kg/s during the establishment of the base and 0.02 kg/s thereafter by averaging the expenditures of propellant over an entire year. This is believed to be valid due to the rapid diffusion of gases released on the lunar surface (Vondrak, personal communication).
Although establishing a lunar base as required for the construction of the Stanford Torus will most likely result in gas release rates at times greater than that occurring naturally, a sparse lunar exosphere will still be preserved given the magnitudes of the calculated release rates. Furthermore, a long-lived atmosphere will not result and if the critical sources of gas are halted, the lunar atmosphere will return to its natural state within weeks.
Figure 5-34 — Loss rates for an oxygen atmosphere (mass = 16 amu). At a total mass of 10⁸ kg, the lunar atmosphere approaches a constant rate of mass loss (after Vondrak, 1974).