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From this it follows that
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This is Eq. (2.1) of the text. The equation for an ellipse in rectangular coordinates with origin at the center of the ellipse can be found as follows: Referring to Fig. B.2c, in which O is at the zero origin of the coordinate system,
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But r = ePQ = e(OZ + x) = e(a/e + x) = a + ex. Hence
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Multiplying this and simplifying gives
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Hence,
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This shows the symmetry of the ellipse, since for a fixed value of y the x2 term is the same for positive and negative values of x. Also, because of the symmetry, there exists a second directrix and focal point to the right of the ellipse. The second focal point is shown as S in Fig. B.2c. This is positioned at x = c = ae (from Eq. B.10). In the work to follow, y can be expressed in terms of x as
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(B.15)
To find the area of an ellipse, consider first the area of any segment (Fig. B.3). The area of the strip of width dx is dA = y dx, and hence the area ranging from x = 0 to x is
(B.16)
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Figure B.3 This is a standard integral which has the solution
(B.17)
where
. In particular, when x = a,
, and the area of the quadrant is
(B.18)
It follows that the total area of the ellipse is
(B.19)
In satellite orbital calculations, time is often measured from the instant of perigee passage. Denote the time of perigee passage as T and any instant of time after perigee passage as t. Then the time interval of significance is t T. Let A be the area swept out in this time interval, and let Tp be the periodic time. Then, from Kepler s second law,
(B.20)
The mean motion is n = 2 /Tp and the mean anomaly is M = n (t T). Combining these with Eq. (B.20) gives
(B.21)
The auxiliary circle is the circle of radius a circumscribing the ellipse as shown in Fig. B.4. This also shows the eccentric anomaly, which is angle E, and the true anomaly (both of which are measured from perigee). The true anomaly is found through the eccentric anomaly.
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Figure B.4 Some relationships of importance which can be seen from the figure are
(B.22)
(B.23)
The equation for the auxiliary circle is x2 + yc2 = a2. Substituting for x2 from this into Eq. (B.15) and simplifying gives
(B.24)
Combining this with Eq. (B.23) gives another important relationship:
(B.25)
The area swept out in time t T can now be found in terms of the individual areas evaluated. Referring to Fig. B.5, this is
(B.26)
Comparing this with Eq. (B.21) shows that
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Figure B.5
(B.27)
This is Kepler s equation, given as Eq. (2.27) in the text. The orbital radius r and the true anomaly v can be found from the eccentric anomaly. Referring to Fig. B.4,
But x = a cos E, and hence,
(B.28)
Also, from Fig. B.4,
and as previously shown, y = b sin E and
. Hence,
(B.29)
Squaring and adding Eqs. (B.29) and (B.28) gives
from which
(B.30)
This is Eq. (2.30) of the text. One further piece of manipulation yields a useful result. Combining Eqs. (B.28) and (B.30) gives
and hence
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Using the trigonometric identity for any angle
that
yields
(B.31)
This is Eq. (2.29) of the text. An elliptical reflector has focusing properties, which may be derived as follows. From Fig. B.6,
But AS = a(1 e), as shown by Eq. (B.8); hence,
Referring again to Fig. B.6, SP = eQP,and S P = ePQ . Hence
Figure B.6
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But SP = r1 and S P = r2, and hence,
(B.32)
This shows that the sum of the focal distances is constant, or in other words, the ray paths from one focus to the other which go via an elliptical reflector are equal in length. Thus, electromagnetic radiation emanating from a source placed at one of the foci will have the same propagation time over any reflected path, and therefore, the reflected waves from all parts of the reflector will arrive at the other foci in phase. This property is made use of in the Gregorian reflector antenna described in Sec. 6.15.
The Parabola
For the parabola, the eccentricity e = 1. Referring to Fig. B.7, since e = SA/AZ, by definition it follows that SA = AZ. Let f = SA = AZ. This is known as the focal length. Consider a line LP drawn parallel to the directrix. The path length from S to P is r + PP . But PP = AL f r cos , and hence the path length is r(1 cos ) + AL f. Substituting for r from Eq. (B.4) with e = 1 yields a path length of p + AL f. This shows that the path length of a ray originating from the focus and reflected parallel
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