Guiseppe Colombo, regarding the rings of Uranus, from which came the initial concept for the ring model shown here. For much of the final preparation of this thesis, I am indebted to Kay Campbell, Hallee Kelso, and Delmine Da Costa. This work will be published, as presented here, in the October 1978 issue of the Astronomical Journal.
PART 1
INTRODUCTION
The amplitude of the phase delay depends in turn on the rate of mechanical energy dissipation in the system attributable to the tidal distortion. This case deals with the question of what fraction of the energy transported by the dynamical tide is dissipated in the stellar atmosphere. In section II, the general theory of the dynamical tide in a rotating, early-type main-sequence star is developed.
THE DYNAMICAL TIDE IN A ROTATING STAR
The mth Fourier component of the total potential perturbation in equation (2.8) is therefore given by. Meanwhile, the Hough functions of the second class become less concentrated after μ = ±1 as f decreases. Neglecting the small variation in ω, equation (1.1 ) for the rotational evolution of the star can be.
The complete solution for the nt^ mode of the dynamical wave in the core is. Analytical solutions are obtained for the differential equations and (2.23) governing the radial part of the dynamic wave.
This boundary condition can be stated as follows, in a form suitable for use with the numerical solutions: the oscillating component of the solution in the envelope must take the form Let us assume that, in the envelope of the star, the oscillating parts of the particular-integral and regular homogeneous numerical solutions are given by the appropriate WKB expressions:. are adjusted to fit these solutions. It will presently be shown that these WKB expressions do indeed accurately represent the oscillatory component of the numerical solutions.).
The outer boundary condition is now satisfied and the mechanical energy flux carried by the nth component of the dynamic tide is. Since only the dynamic component of the tide is of interest in this study, the right-hand side of (2.17) and (2.18) is replaced by (2.23). The parameters b and d serve to describe locally the non-oscillating part of the particular integral solution.
Some examples of the numerical solutions for h (r) and δp (r). are presented in Figures 7 to 10. Figures 7 and 8 show the homogeneous and particular-integral solutions for σ respectively. The agreement between the WKB envelopes and the envelopes of the numerical solutions, even quite close to the core, is both impressive and typical of all the numerical solutions investigated. This aspect of the numerical solutions is also evidently in close agreement with the analytical theory.
But for simplicity, in the next section it will be assumed that the form of the function <£(K ½) -ii, approx.
SYNCHRONIZATION TIMESCALES
For simplicity, we neglect the secondary rotational energy and angular momentum and concentrate attention on the primary one. From equation (5.6>, the evolution of the tidal frequency σ2 (hereafter denoted by σ) is given by To compare these results with observational material, we ask the following question: What is the critical value of the orbital period, P .
Since σ also depends on P, it is clear that P is indeed a fundamental parameter of the tidal synchronization process. This recommendation is unwise, as the radius R of the primary does not directly play a role in the calculation of the dynamic tide. First, only the radial component of the rotational velocity is measured, i.e. v sin i, where i is the obliquity of the star.
Alternatively, the absolute radii can be estimated from the stars' apparent sizes if the distance to the binary is known. These periods are of the same order of magnitude as, but perhaps somewhat shorter than, the critical period of 4 days suggested by the observational data. The star's rotation rate has not been measured directly, but it exhibits spectral variations that are perfect.
A comparison of the critical periods for "unevolved" stars with the theoretical predictions of Table V for f = ⅛ shows excellent.
CONCLUSIONS
A rapid decrease of these coefficients as the rotation of the star approaches the synchronous state causes a decrease in the rate of energy dissipation and, therefore, in the rate of synchronization. implies that, in most cases, dynamical tides are unable to reduce the rotation rate Ω to much less than twice the average orbital motion ω over the lifetime of the star. A consequence of this enhancement is the result that gravity waves corresponding to some of the tidal oscillation modes are not reflected at the surface of the star, but can propagate outwards until they become nonlinear and damp out. The analytical expression for is found to be accurate up to a factor of 2, except for small values of the eigenvalue Kjnn∙.
Observations of the rotation velocities of early-type components of eclipsing binaries generally agree with the predicted synchronization times and critical periods for the criterion Ω∕ω ≤ 2, but in many cases the observed synchrony between rotation and orbital periods is significantly better than expected is. . Further reduction in the rotational angular velocity due to the dynamic tide proceeds very slowly due to the rapidly decreasing values of the projection coefficients (see [3j above). With the help of the analytical solutions in section 3, and of the WKB solutions of section 2, the validity of the approach is developed.
This approximation involved the neglect of the non-radial (or "horizontal") component of the rotational angular velocity when calculating the Coriolis force. We return to the linearized equations of motion and (.2.10), and rewrite them in a schematic form so that the relative magnitudes of the ternis can be ascertained. By introducing the characteristic radial and horizontal scales of the perturbed quantities, L and L , we make following schematic.
Substitution of this expression for δp in the equation for δp gives an estimate for the first term in the continuity equation (2.9.
APPENDIX 2
APPENDIX 3
PART 2
OBSERVATIONS
The observations were made using an infrared photometer mounted on the 2.5 m DuPont telescope at the Las Campanas Observatory in Chile. Sky subtraction was performed by cutting at 15 Hz a secondary beam located 30" north. Star #5 is significantly brighter than the Uranus system (planet + rings) at 2.2 pm: bright enough to give a signal-to-noise ratio of ~ 10 with a time resolution of ~ 0.1 second.
The noise was almost entirely due to background radiation from the telescope and the sky, and so it was necessary to use a small focal plane aperture centered on the star. The star was centered by finding the half-power points of the 2.2 o'clock signal one hour before the close of the first bell, and the center was then maintained using an offset/Quantex guidance system. The chopped signal from the detector was demodulated in a conventional lock-in amplifier and recorded on a strip chart, with an overall system time constant of 0.1 second.
Star No. 4 has a brightness comparable to that of the Uranus system with a size of 2.2 μm, and the signal-to-noise ratio at a time resolution of 0.1 sec was ~1.
RESULTS
The straight lines show the apparent paths of stars hidden behind the ring system, as observed by various ground stations. The top track labeled 10 March ’77 corresponds to Perth, Western Australia and the bottom track to the Kuiper Air Observatory. The timeline of the emersion records has been reversed so that they can be more easily compared to the dive records above them.
Note that the ring occultations were not, in reality, symmetrically spaced in time with respect to the other occultations. On April 4, 1978, a planetary occultation with a penumbral duration of 44 min 48 ± 10 sec occurred in Las Campanas. Only a convincing ring cover, at 9^lr31mπ'-n 00 SGC UT, is identifiable in the very noisy recording.
Guidance became impossible at dusk before the predicted time of the second ring occultation. The poor signal-to-noise ratio of the record precludes any useful analysis of the planetary occultation profiles, except perhaps to determine the mean atmospheric scale heights.
ANALYSIS a) Overall Ring Geometry
Finally, the mean close approach distance of 30.985 km defined by the β, γ and δ rings is used together with the timing data to calculate the positions in the ring plane of all the occult ring segments. As indicated in section III, a comparison of the radii in Table IV shows that the nine rings observed on April 10, 1978 can indeed be identified with the nine rings reported by Elliot. The observed radial widths of the ring segments projected onto the satellite orbital plane are given in Table IV.
There is no evidence in Figure 2 for such a broad feature in the location of the η ring. The radius of the single ring identified in the April 4, 1978 occultation data can be estimated from observations of the time and an assumed radius of Uranus. A radial width of the ring segment of 20 ± 10 km follows from the ~1 second occultation duration.
These six observations of radial width and ring radius e are shown in Fig. 4 and show a linear relationship up to a width of ~20 km. Further information on the nature of the e ring is provided by comparing occultation profiles obtained at different times. Such a correction would increase the optical depth for this occultation by ~25%, thus supporting the conclusion of Elliot et al. 1978) that the integrated optical depth of the ring e is independent of its width.
The similarity of these two profiles, observed 13 months apart, is truly remarkable when considering the orbital and precessional motion of the ring particles.
Ο TIME (secs)
AZIMUTH
