For reasons that will be clear later this solution is written

where e is a constant of integration.

Combining the last two equations (13-15), we obtain d<}> _ tan 4> dijj tan <(>

The integration is immediate:

where I is a new constant of integration.

The equation for 0 can be written with i|j as the independent variable.

d0 di|j

Using the solution (13-22) for

The quadrature gives sin = sin I cos (0 - £2) (13-25)

where Si is another constant of integration.

Finally, we define a new variable a by the relation cos a = cos <j> cos (6 - ) (13-26)

Fig. 13-2. The osculating plane and the orbital elements.

Figure 13-2 displays the geometric relationship among the angles 0 , <j> , iC and I , Q , a . The angle I is the inclination, and the angle is the longitude of the ascending node. They are constants of the motion for a Keplerian orbit. The new variable a , introduced to replace the angle 9 , is simply the polar angle, measured in the plane of motion from the line of the ascending node. From relations in spherical trigonometry we have also sin cb = sin I sin a sin ip sin( 9 - )

The derivative of Eq. (13-26) with respect to r , with Eqs. (13-15), (13-22), (13-25) and (13-27) used for simplification, results in da dr

r tan y

If u is taken as the independent variable, and if the solution (13-20) is used to evaluate tan y in terms of u , then .

da du

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