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2.8.2 Atmospheric drag
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For near-earth satellites, below about 1000 km, the effects of atmospheric drag are significant. Because the drag is greatest at the perigee, the drag acts to reduce the velocity at this point, with the result that the satellite does not reach the same apogee height on successive revolutions. The result is that the semimajor axis and the eccentricity are both reduced. Drag does not noticeably change the other orbital parameters, including perigee height. In the program used for generating the orbital elements given in the NASA bulletins, a pseudo-drag term is generated, which is equal to one-half the rate of change of mean motion (ADC USAF, 1980). An approximate expression for the change of major axis is a > a0 c n0 n0 nr0(t t0) d
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where the 0 subscripts denote values at the reference time t0, and n 0 is the first derivative of the mean motion. The mean anomaly is also changed, an approximate value for the change being: M nr0 2 (t t0)2 (2.17)
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From Table 2.1 it is seen that the first time derivative of the mean motion is listed in columns 34 43 of line 1 of the NASA bulletin. For the
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example shown in Fig. 2.6, this is 0.00000307 rev/day2. Thus the changes resulting from the drag term will be significant only for long time intervals, and for present purposes will be ignored. For a more accurate analysis, suitable for long-term predictions, the reader is referred to ADC USAF (1980).
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2.9 Inclined Orbits A study of the general situation of a satellite in an inclined elliptical orbit is complicated by the fact that different parameters relate to different reference frames. The orbital elements are known with reference to the plane of the orbit, the position of which is fixed (or slowly varying) in space, while the location of the earth station is usually given in terms of the local geographic coordinates which rotate with the earth. Rectangular coordinate systems are generally used in calculations of satellite position and velocity in space, while the earth station quantities of interest may be the azimuth and elevation angles and range. Transformations between coordinate systems are therefore required. Here, in order to illustrate the method of calculation for elliptical inclined orbits, the problem of finding the earth station look angles and range will be considered. It should be kept in mind that with inclined orbits the satellites are not geostationary, and therefore, the required look angles and range will change with time. Detailed and very readable treatments of orbital properties in general will be found, for example, in Bate et al. (1971) and Wertz (1984). Much of the explanation and the notation in this section is based on these two references. Determination of the look angles and range involves the following quantities and concepts: 1. The orbital elements, as published in the NASA bulletins and described in Sec. 2.6 2. Various measures of time 3. The perifocal coordinate system, which is based on the orbital plane 4. The geocentric-equatorial coordinate system, which is based on the earth s equatorial plane 5. The topocentric-horizon coordinate system, which is based on the observer s horizon plane. The two major coordinate transformations needed are:
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The satellite position measured in the perifocal system is transformed to the geocentric-horizon system in which the earth s rotation
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Orbits and Launching Methods
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is measured, thus enabling the satellite position and the earth station location to be coordinated.
The satellite-to-earth station position vector is transformed to the topocentric-horizon system, which enables the look angles and range to be calculated.
2.9.1 Calendars
A calendar is a time-keeping device in which the year is divided into months, weeks, and days. Calendar days are units of time based on the earth s motion relative to the sun. Of course, it is more convenient to think of the sun moving relative to the earth. This motion is not uniform, and so a fictitious sun, termed the mean sun, is introduced. The mean sun does move at a uniform speed but otherwise requires the same time as the real sun to complete one orbit of the earth, this time being the tropical year. A day measured relative to this mean sun is termed a mean solar day. Calendar days are mean solar days, and generally they are just referred to as days. A tropical year contains 365.2422 days. In order to make the calendar year, also referred to as the civil year, more easily usable, it is normally divided into 365 days. The extra 0.2422 of a day is significant, and for example, after 100 years, there would be a discrepancy of 24 days between the calendar year and the tropical year. Julius Caesar made the first attempt to correct the discrepancy by introducing the leap year, in which an extra day is added to February whenever the year number is divisible by 4. This gave the Julian calendar, in which the civil year was 365.25 days on average, a reasonable approximation to the tropical year. By the year 1582, an appreciable discrepancy once again existed between the civil and tropical years. Pope Gregory XIII took matters in hand by abolishing the days October 5 through October 14, 1582, to bring the civil and tropical years into line and by placing an additional constraint on the leap year in that years ending in two zeros must be divisible by 400 without remainder to be reckoned as leap years. This dodge was used to miss out 3 days every 400 years. To see this, let the year be written as X00 where X stands for the hundreds. For example, for 1900, X 19. For X00 to be divisible by 400, X must be divisible by 4. Now a succession of 400 years can be written as X (n 1), X n, X (n 1), and X (n 2), where n is any integer from 0 to 9. If X n is evenly divisible by 4, then the adjoining three numbers are not, since some fraction from 1/4 to 2/4 remains, so these three years would have to be omitted. The resulting calendar is the Gregorian calendar, which is the one in use today.
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