© CAMBRIDGE UNIVERSITY PRESS 2000 |
Problems
with s = (r0) , e = (r0) / (r0), e = / (r0), and
Here = a sins0 and = a coss0 are dimensionless radial and azimuthal epicyclic velocities associated with the orbit at = e [cf. Eq.(13.20) and Eq.(13.22)]. From the identity r02 (r0) = r*2 * at = e, check also that, to first order, r0 = r(1 - ). (Hint: replace the r and variables in Eqs.(13.22)-(13.25) by r*() and *(), integrate the equations, and then impose the desired "boundary conditions". This is one key step towards the integration along the unperturbed characteristics of the stellar dynamical equations leading to the dispersion relation for tightly wound density waves; see Chapter 15 and the article by Shu, F.H. (1970) cited there.)