Laminar symmetry-plane boundary layer on a sharp spinning body at incidence.

Abstract

Let the true values of the orbital elements be a = 1500 miles, e = 0.1, i = 1.4, Q = - 1.0, co = 0.6, tt = 0 sec. The y-axis is in the direction of the moon's motion with the x-axis completing a right-handed system. The moon is assumed to be in a circular orbit of radius 239,000 miles with an angular speed of 0.000002646659 rad/sec. The observing station is placed at the center of the Earth. \JL = 1175.525 miles3/sec2. These values give $! = -0.95247, s2 = 0.80792, s3 = 237611.9, s3 = 0.00058475. Substituting in the formulas we obtain approximate values for the orbital elements: a = 1500.86, e = 0.1027, co = 0.6448, sin/ = 0.98930 (true value is 0.98545). From the true values we find t = 4.133 when s is first equal to w. Thus tt = 4.133 sec, the true value being 0. Completing and Improving the Solution Having found approximate values for the five elements, we can choose a sign for cosz and guess a value for the sixth element Q. With six elements we can integrate the equations of motion for the lunar satellite with major perturbations included, using rt as the starting time. As the integration proceeds residuals are produced by comparing integrated values with observed values of range and range rate. Using standard statistical theory we form a function of the residuals and the orbital elements which is to be minimized by varying Q while holding the other five elements fixed at the values found by the method previously described. This minimization problem is not difficult since it is one-dimensional. We assume data available from a sufficient number of revolutions of the satellite to determine Q. The procedure described in this paragraph must be repeated with the sign of cos i changed. With approximations for six orbital elements we can compute approximate values of uz, uz, and uz at the times associated with a set of range and range rate measurements taken during a pass of the satellite in front of the moon. The procedure for finding five elements can now be repeated, using the following exact equations. z = suz — wz, z = suz + suz— vvz z = suz + 2sii_ + su. — w7