Occultations of stars by the moon.

As the moon's sidereal period of orbital revolution around the earth is about 27⅓ days, it moves eastwards with reference to the stars at an average rate of rather more than half a degree per hour. In its passage over the stellar background it is continually interposing its disc between us and the stars, and the sudden disappearance of a star in this way is called the occultation of the star by the moon. After an interval, which depends on a variety of factors, the star reappears. The disappearance and reappearance of the star are generally referred to as immersion and emersion respectively. The disappearance of the star and its reappearance are instantaneous phenomena and, if the time of one or the other is noted accurately, there is obtained at that instant a definite relation between the moon's position in the sky and the position of the observer, it being assumed that the star's position is known accurately. Formerly, occultations were utilised for the determination of longitude, but the introduction of radio time-signals has rendered the occultation method obsolete.

If the moon's position is known accurately, the particulars of the occultation of a star at any place can be predicted and, under these circumstances, it is to be expected that prediction and observation would agree. Now the moon's position is predicted in the almanacs for any instant of Ephemeris Time, while the recorded time of the observation of an occultation will be in Universal Time. The study of suchoccultations, therefore, provides a ready means of determining the relationship between Universal and Ephemeris Time, and, in particular, of deriving the correction ΔT. The occultations of radio sources are also important, as precise radio positions are difficult to measure. The first positive optical identification of a quasar was made by timing the cessation of its radio signals in the course of a lunar occultation.

The geometrical conditions for an occultation.

Consider Fig. 134, in which the earth (regarded as a spheroid) and the moon (regarded as a sphere) are shown with their centres at E and M respectively.

Abstract

Introduction

Dieter Brill and one of us (JI) used to talk a lot about Mach's Principle. We both were of the Wheeler school, so our Machian discussions often focussed on issues involving the initial value formulation of Einstein's theory. One such issue was the following question: If a spacetime (M, g) is known to be globally hyperbolic, how can one tell (from intrinsic information) if a given embedded spacelike hypersurface Σ is a Cauchy surface for (M, g)? The answer to this question is important if one wants to know what minimal information about the universe “now” is needed to determine the spacetime metric (and hence its inertial frames) for all time in (M, g).

It turns out [1] that if a spacelike hypersurface Σ embedded in a globally hyperbolic spacetime is compact (without boundary), then it must be a Cauchy surface.

Abstract

In dimension eight there are three basic representations for the spin group. These representations lead to a concept of triality and hence lead to the construction of two exceptional commutative algebras: The Chevalley algebra A of dimension twenty-four and the Albert algebra I of dimension twenty-seven. All of the exceptional Lie groups can be described using triality, the octonians, and these two algebras.

On a complex four-manifold, with triality a parallel field, the Dirac and associated twistor operators can be constructed on either bundle of exceptional algebras. The geometry of triality leads to refinement of duality common to four dimensions.

In nine dimensions there is a weaker notion of triality which is related to several additional multiplicative structures on J.

Introduction

Cartan's classification of simple Lie groups yields all Lie groups with some exceptions. In his thesis Cartan described these exceptional groups but, save the one of smallest rank G2, he was unable to describe the geometry of the groups. In 1950 Chevalley and Schafer successfully identified F4 and E6 as structure groups for the exceptional Jordan algebra and the Freudenthal cross product on J, respectively. Later Freudenthal successfully identified the geometry associated with the remaining exceptional groups, E7 and E8.