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Fig. 7-7. Coverage Figures of Merit See text for explanation.

Fig. 7-7. Coverage Figures of Merit See text for explanation.

* One advantage of response time as a Figure of Merit is that delays in processing or communications (for both data requests and responses) can be directly added to the coverage response time. This results in a total response time, which measures the total time from when users request data until they receive it We can also evaluate minimum, mean, and maximum total response times which have much more operational meaning than simple gap statistics but are still easy to compute.

The table below Fig. 7-7 shows the numerical values of the Figures of Merit defined above. The percent coverage correctly ranks constellation A better than B, but because it does not take gap statistics into account it cannot distinguish between A and C. Similarly, the maximum gap cannot distinguish between A and B, even though B is clearly worse by having an additional gap. In this case the maximum gap tells us which constellation is worst but cannot distinguish between two constellations which are clearly different

The mean gap statistic is even more misleading. By adding a short gap to constellation B, the average length of the gaps has been decreased, and consequently, this Figure of Merit ranks constellation B above constellation A. (This can happen in real constellation statistics. By adding satellites we may eliminate some of the very small gaps, thus increasing the average gap length, even though more satellites provide more and better coverage.)

Finally, the time average gap and mean response time in the fourth and fifth columns correctly rank the three constellations in order of preference by taking into account both the percent coverage and gap statistics. Consequently, both of these are better Figures of Merit than the other three. I believe the mean response time is the stronger Figure of Merit because it provides a more useful measure of the end performance of the system and because it can be easily extended to include delays due to processing, communications, decision making, or the initiation of action. However, because each of the Figures of Merit represent different characteristics we should evaluate more than one. Specifically, I recommend evaluating mean response time, percent coverage, and maximum gap, and qualitatively (not quantitatively) weighting the results in that order, keeping strongly in mind the caveat at the end of Sec. 7.2.2.

### 73 The AV Budget

To an orbit designer, a space mission is a series of different oibits. For example, a satellite may be released in a low-Earth parking orbit, transferred to some mission orbit, go through a series of rephasings or alternative mission orbits, and then move to some final orbit at the end of its useful life. Each of these oibit changes requires energy. The AV budget is traditionally used to account for this energy. It is the sum of the velocity changes required throughout the space mission life. In abroad sense the A V budget represents the cost for each mission (»bit scenario. In designing oibits and constellations, we must balance this cost against the utility achieved.

Chapter 10 shows how to develop & propulsion budget based on a given AVbudget For preliminary design, we can estimate the "cost" of the space mission by using the rocket equation to determine the total required spacecraft plus propellant mass, wys mo + mp, in terms of the dry mass of the spacecraft, mg, the total required A V, and the propellant exhaust velocity, Vg:

This is equivalent to Eqs. (17-6) and (17-7) in Sec. 17.1, with V0 replaced by Ispg, where the specific impulse, = V0 fS< and g is the acceleration of gravity at the Earth's surface. Typical exhaust velocities are in the range of 2 to 4 km/s and up to 30 km/s for electric propulsion. We can see from Eq. (7-14) that AVrequirements much smaller than the exhaust velocity (a few hundred meters per second), will require a propellant mass which is a small fraction of the total mass. If the total A V required is equal to the exhaust velocity, then we will need a total propel!ant mass equal to e - 1 » 1.7 times the mass of the spacecraft Propulsion systems require additional structure such as tanks, so a AV much greater than the exhaust velocity is difficult to achieve. It may scuttle the mission or require some alternative, such as staging or refueling.

Table 7-3 summarizes how to construct a A V budget. We begin by writing down the basic data required to compute AVs: the launch vehicle's initial conditions, the mission orbit or orbits, the mission duration, required orbit maneuvers or maintenance, and the mechanism for spacecraft disposal. We then transform each item into an equivalent A V requirement using the formulas listed in the table. The right-hand column shows how these formulas apply to the FireSat mission. Figure 7-8 shows the AV required for altitude maintenance for typical spacecraft and atmosphere parameters.

100 200 300 400 500 600 700 800 900 1,000 Altitude (km)

Fig. 7-8. Altitude Maintenance AVs for a Ballistic Coefficient of 100 kg/m2. See Sec. 8.1.3 for ballistic coefficient and atmosphere parameters. The AV for altitude maintenance is inversely proportional to the ballistic coefficient The F10.7 index is in units of 10"22 W/(m2-Hz). Ap is an index of geomagnetic activity ranging from 0 (very quiet) to 400 (extremely disturbed).

The A V budget relates strongly to the propulsion requirements and to the final cost of a space mission. Yet other conditions may vary the propellant requirements relative to the nominal AV budget For example, although rocket propulsion usually provides the AV, we can obtain very large AVs from a flyby of the Moon, other planets, or even the Earth itself [Kaufman, Newman, and Chromey, 1966; Meissinger, 1970]. In a fly-by, a spacecraft leaves the vicinity of some celestial body with the same velocity

100 200 300 400 500 600 700 800 900 1,000 Altitude (km)

Fig. 7-8. Altitude Maintenance AVs for a Ballistic Coefficient of 100 kg/m2. See Sec. 8.1.3 for ballistic coefficient and atmosphere parameters. The AV for altitude maintenance is inversely proportional to the ballistic coefficient The F10.7 index is in units of 10"22 W/(m2-Hz). Ap is an index of geomagnetic activity ranging from 0 (very quiet) to 400 (extremely disturbed).

TABLE 7-3. Creating a AP Budget See also summary tables on inside back cover.

Item

Where Discussed

Equation Source

Fire Sat Example

Basic Data

0 0