Orbits and Launching Methods in Software

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Orbits and Launching Methods
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hence no rotation takes place. Use is made of this fact in the orbit chosen for the Russian Molniya satellites (see Probs. 2.23 and 2.24). Denoting the epoch time by t0, the right ascension of the ascending node by 0, and the argument of perigee by w0 at epoch gives the new values for and w at time t as
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d (t dt d (t dt
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t0) t0)
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(2.14) (2.15)
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Keep in mind that the orbit is not a physical entity, and it is the forces resulting from an oblate earth, which act on the satellite to produce the changes in the orbital parameters. Thus, rather than follow a closed elliptical path in a fixed plane, the satellite drifts as a result of the regression of the nodes, and the latitude of the point of closest approach (the perigee) changes as a result of the rotation of the line of apsides. With this in mind, it is permissible to visualize the satellite as following a closed elliptical orbit but with the orbit itself moving relative to the earth as a result of the changes in and w. Thus, as stated earlier, the period PA is the time required to go around the orbital path from perigee to perigee, even though the perigee has moved relative to the earth. Suppose, for example, that the inclination is 90 so that the regression of the nodes is zero (from Eq. 2.12), and the rate of rotation of the line of apsides is K/2 (from Eq. 2.13), and further, imagine the situation where the perigee at the start of observations is exactly over the ascending node. One period later the perigee would be at an angle KPA /2 relative to the ascending node or, in other words, would be south of the equator. The time between crossings at the ascending node would be PA (1 K/2n), which would be the period observed from the earth. Recall that K will have the same units as n, for example, rad/s.
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Example 2.5
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Determine the rate of regression of the nodes and the rate of rotation of the line of apsides for the satellite parameters specified in Table 2.1. The value for a obtained in Example 2.2 may be used. From Table 2.1 and Example 2.2 i
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14.23304826 day 1; a 98.6328 ; e 0.0011501; NN 7192.335 km, and the known constant: K1 66063.1704 km2
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Solution
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Converting n to rad/s: n 2 NN 0.00104 rad/s
Two
From Eq. (2.11): K nK1 a (1
6.544 deg/day From Eq. (2.12): d dt 0.981 deg/day K cos i
From Eq. (2.13): d dt 2.904 deg/day
Example 2.6 Calculate, for the satellite in Example 2.5, the new values for w and
2.5 sin 2 i)
one period after epoch. From Table 2.1: NN
Solution
14.23304826 day 1; w0
113.5534 ; 0
251.5324
The anomalistic period is PA 1 NN 0.070259 day This is also the time difference (t t0) since the satellite has completed one revolution from perigee to perigee. Hence: d
t0) 0.981(0.070259)
251.5324 251.601
t0) ( 2.903)(0.070259)
113.5534 113.349
Orbits and Launching Methods
In addition to the equatorial bulge, the earth is not perfectly circular in the equatorial plane; it has a small eccentricity of the order of 10 5. This is referred to as the equatorial ellipticity. The effect of the equatorial ellipticity is to set up a gravity gradient, which has a pronounced effect on satellites in geostationary orbit (Sec. 7.4). Very briefly, a satellite in geostationary orbit ideally should remain fixed relative to the earth. The gravity gradient resulting from the equatorial ellipticity causes the satellites in geostationary orbit to drift to one of two stable points, which coincide with the minor axis of the equatorial ellipse. These two points are separated by 180 on the equator and are at approximately 75 E longitude and 105 W longitude. Satellites in service are prevented from drifting to these points through station-keeping maneuvers, described in Sec. 7.4. Because old, out-of-service satellites eventually do drift to these points, they are referred to as satellite graveyards. It may be noted that the effect of equatorial ellipticity is negligible on most other satellite orbits.
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