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Stochastic Methods in the Dynamics of Satellites: Course by Peter Sagirow

By Peter Sagirow

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Extra resources for Stochastic Methods in the Dynamics of Satellites: Course Held at the Department for General Mechanics, October 1970

Sample text

4) in eq. Q(A + S - Ch; = M)' ( 3. 5) cB = M2 Here, Mx, M~, M~ • are the components of the external torques about the x, y, z-axes. 3. 2 Gravity Gradient Torques [11], [12]. [13] The gravity gradient torques acting on a satellite are Mgx ( 3. 6) Mn = = :50 2 t ¥' 2 11 M,. = 3Q

65) is stable in probability. 67) Chap. 2. Stochastic Dynamical Systems 42 2. 7. 7 The Operator L The operator L can be obtained as follows. It is = dx v fdt + crd'W' • v-(t,x) and .. 'U'tdt + r. ~ i 1 '~~~. t + ~,i. x~ dx;_ + ... - ~~ ,. + ,. t ·1. e d. '~At e)(ftdt + ... ~d- = Thus, we have finally - L a At + v a Lf~ -a ~ .. x~ n t + 2E r 1 n n ~-~ ~=t (c:ro-*)~t ()2 ax~ a . (2. 67) x~ 2. 7. 'br(t) . 68) and has to satisfy the conditions: too a') '\t( 0) C and w are in- t dependent of of f exist and are continuous at least 0• Theorem 2.

At e)(ftdt + ... ~d- = Thus, we have finally - L a At + v a Lf~ -a ~ .. x~ n t + 2E r 1 n n ~-~ ~=t (c:ro-*)~t ()2 ax~ a . (2. 67) x~ 2. 7. 'br(t) . 68) and has to satisfy the conditions: too a') '\t( 0) C and w are in- t dependent of of f exist and are continuous at least 0• Theorem 2. 4 : Suppose there exists a function ~ (x') with properties a') to d') and Chap. 2. Stochastic Dynamical Systems 44 L'U' ( 2. 69) ~ 0 holds. Then the equilibrium position x =0 of eq. 68) is stable in probability.

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