The two most important advantages of photoelectric timings of occultations over visual timings are accuracy and freedom from systematic error. The observational error is so small compared with uncertainties in the lunar ephemeris, star positions, limb corrections, and such, that the value of the observations will continue to increase as time goes on and the other error sources are eliminated.

A recent result made possible largely by the availability of photoelectric timings over the last 20 yr is a new value for the secular acceleration of the Moon, with corresponding consequences for the Ephemeris Time scale.

An important current application for simultaneous photoelectric and visual observations is the determination of ‘personal equation’ for the visual observers. Experiments so far indicate the visual observers require a minimum of 0.1 sec to detect that an event has occurred, plus additional time to react to it; hence, typical personal equations are around 0.4 sec.

The timing of occultations of stars by the Moon provide data for the following studies:

• (a) Lunar theory - Constants and secular changes of its orbit.

• (b) Fundamental star systems - Systematic corrections to equinox and obliquity; individual star corrections.

• (c) Rotation of Earth - Comparison of atomic and ephemeris time scales.

With these applications in mind we consider the analysis of the [O-C]’s from occultations.

Up to about a century ago, occultations were often used to measure differences in geographical longitude. Extensive geodetic surveys, accurate chronometers, telegraphic communications, and later short-wave radio time services obviated the geodetic need for occultation observations, which are affected by geodetically severe uncertainties of stellar and lunar positions, lunar limb irregularities, and observers’ personal equations. More sophisticated methods of observation would be needed before the Moon could again be useful to geodesy.

During this century, cinematography of Bailey’s beads and the flash spectrum during total solar eclipses have been used to obtain the relative apparent position of the Sun and Moon to an accuracy which could be useful to geodesy. But observational opportunities were rare and few results of geodetic significance have been obtained.

It has long been recognized that the analysis of occultation traces from point source stars might provide a means of investigating the structure of the lunar limb on a remarkably small scale, certainly of tens of meters, possibly on a scale of meters.

The routine process of analysis of such an occultation trace produces a curve fitted to the standard model for a point source, in which the observed rate of fringe passage is matched to that computed from the rate and position angle of the relative motion of the moon with respect to the star background and the position angle of the point at which the occultation occurs. If θυ is the position angle towards which the relative motion of the lunar center takes place, θ that at which the occultation occurs, and ψ = θν — θ, then the predicted rate of the lunar limb perpendicular to itself at this point is

Rp = V cos ψ

where V is the velocity of the lunar center.

The study of occultations of radio sources by the Moon has proved a powerful method of studying the structure of radio sources with a resolution limit, in some cases, as small as 0″.1 and at the same time of obtaining positions of the radio components with an accuracy of the order of 0″.1 to 1″. Recent optical observations of occultations suggest that the method is likely to play an important role in the measurement of stellar diameters down to about 0″.001 and in the detection and measurement of binary star systems. Over the past several years considerable experience has been gained in the analysis of the occultation curves of radio sources and, since the problems encountered are common to both the optical and radio analysis, our conclusions on how best to analyse occultation curves may be of some interest to the optical workers and also to radio observers who have recently entered the field. Before discussing the methods of analysis and also before discussing some essential differences between the optical and radio work it is useful to consider in some detail the nature of the occultation curve of a simple source of small angular size. It is not proposed here to give a detailed account of the methods of analysis but to indicate the general principles along which the analysis should proceed so as to enable the choice of the most appropriate method in a particular case. A simple treatment of Scheuer’s convolution procedure is given and a simple derivation of the resolution limit imposed by the receiver bandwidth.

‘Seeing’ affects the light-curve of a stellar occultation by the Moon in two ways: the diffraction pattern on the ground is smeared out by atmospheric turbulence, and the pattern also suffers random displacements. These effects are analogous to the familiar image blur and image motion, respectively. However, there is a major difference between ordinary astronomical seeing and the effect on the lunar diffraction pattern: the former is the seeing looking up at the sky from the bottom of the atmosphere, but the latter corresponds to the seeing looking down through the atmosphere at the surface of the Earth.

This downward-looking seeing is of concern to people engaged in aerial photography and satellite reconnaissance, and has been studied theoretically from this point of view. It also enters into the theory of stellar scintillation, because the seeing blurs out the scintillation shadow pattern just as it blurs out the occultation diffraction pattern.

Nather and Evans (1970) emphasize that merely recording the light curve of an occultation is not enough. The data must be recorded in a form which is convenient for the analysis to be carried out later on. Analog methods of recording merely delay the time at which the record must be measured, usually after it has been digitized in some way. Nather and Evans have used a modified digital multiscaler with 400 storage locations for the observation. The data acquisition to be described here is in principle rather similar to a multiscaler and deserves special attention because it is in practice even more flexible and convenient due to its on-line computer. Our system has been used to measure star occultations (De Vegt and Pansch, 1970) and the Crab pulsar (Høg and Lohsen, 1970) and to sample data from a scanning photometer (Høg, 1969).

The formulae for the monochromatic radiation intensity at the Earth during the lunar occultation of an incoherent source are examined. It is shown that the oscillatory portion of the radiation intensity may be assigned an envelope amplitude and phase a+ each instant of time. Each complex number thus defined gives the amplitude and phase of one spatial Fourier component of the source brightness distribution. Resolution is limited by the maximum available spatial frequency, and is shown to depend on integration time, seeing, antenna aperture, bandwidth, and the signal-to-noise ratio of the observation. When observations are polychromatic, the monochromatic intensity is modified by a certain function of bandwidth and wavelength. This function is specified and theoretical occultation curves of a point source are given for various bandwidth-wavelength ratios. A new method of estimating a brightness distribution is illustrated by taking the inverse Fourier transform of the spatial Fourier components present in an occultation record of 3C 49. Restored distributions for the quasars 3C 273 and 3C 245 are also given. Source positions and/or lunar limb slopes may be determined from such distributions. When the width of single sources or the angular separation of double sources is all that is required, only the inner part of the envelope of the occultation record need be determined. Theoretical curves and envelopes are presented for use in comparisons with observed data.

In Japan, photoelectric observations of occultations were made for the first time in 1951.

For about ten years after that, observations were made partly for geodetic purposes in order to determine the position of Japan with reference to islands in the Pacific, Phillippine and Taiwan.

The geodetic datum of Japan, which is currently adopted, suffers seriously from the vertical deflection. The amount of this deflection is supposed to be about 10″ in latitude and 20″ in longitude.

This project of occultation geodesy was, however, not so successful, because of the following two reasons: (1) Poor efficiency of data acquisition due to weather. We need good weather at two stations; the distance between them is usually over 1000 km. (2) Difficulty of correction for the Moon’s limb. For geodetic purposes, the atlas of the Moon’s limb by Dr Watts is still insufficient.

Therefore, the project of occultation geodesy was replaced by satellite geodesy.

Photoelectric occultation traces of 5 stars are discussed which have been obtained with the 60 cm refractor of the Hamburg-Observatory during 1969/70. All traces show remarkable deviations from the usual Fresnel point-source pattern. Most details of these traces can be interpreted assuming the stars to be unresolved doubles. A short description of the equipment and the observational procedure is given.

The theory of the effect of bandwidth of lunar occultations is reviewed. It is recalled that effective beamshapes can be calculated for symmetrical bandpasses and that their widths are related to the absolute width in wavelength of the bandpasses. Restoration with the second differential of the theoretical Fresnel diffraction curves at the central wavelength, at the correct rates, yield source distributions as viewed by these beamshapes. It is shown that for asymmetric bandpasses, the real and odd parts taken about the centroids lead to equivalent even and odd beams. Assuming an approximate color temperature for the stars, the total system response can be evaluated and hence the even and odd parts. Restoration of the data should then be performed using the second differential of the Fresnel curve at the centroid wavelength to minimize the odd part, adjusting zeroes, rates, and centroids by inspection. The even part should then represent the even theoretical response convolved with the one-dimensional stellar distribution, provided the latter is circularly symmetrical.

The technique is applied to the occultation observation of λ-Aquarii by Nather et al. (1970) leading to closely similar results.

If the limb of the Moon can be regarded as a straight edge, then the diffraction pattern of a point source which it produces at the distance of the Earth is the well known Fresnel diffraction pattern. Observations of stellar occultations reveal the variation of intensity with time as the diffraction pattern passes across the detector due to the orbital motion of the Moon and the rotation of the Earth. The linear scale of the diffraction pattern in monochromatic light depends on both the wavelength of observation, λ, and on the distance of the Moon, D, so that the scale is proportional to (λD)1/2.

The technique of model fitting has been applied to the analysis of data obtained from photoelectric measurements of lunar occultations. A model occultation curve is generated and fitted, by least squares, to the observed light curve, and by this method the values of the model parameters are determined, together with their formal errors. This technique is contrasted with Scheuer’s deconvolution procedure by applying both methods to the same observed data.

The idea of finding the radius of a star from the distortion of the Fresnel pattern caused by diffraction at the edge of the Moon during an occultation, was first proposed by Williams in 1938. In subsequent years this idea was again taken up by a number of investigators: Whitford (1938, 1946,) Diercks and Hunger (1952), Evans et al. (1953), Cousins and Guelke (1953), Rakos (1964, 1967), Nather and Evans (1970). The greatest difficulty in carrying out such measures was the lack of a light detector of sufficiently high quantum efficiency and recording equipment of sufficiently short time constant. Modern advances in electronics have overcome these difficulties to such an extent that it now appears feasible to determine the apparent diameters of many stars with a minimum expenditure of observing time and equipment.

A program for the measurement of diffraction patterns resulting from the lunar occultations of stars was begun at the Lowell Observatory by Rakos (1964), resumed by him in 1967 and continued by Pettauer through the summer of 1969. The expected results were accurate timing of occultations and hence accurate position measurements, the discovery of close double stars, and the determination of stellar diameters. The author is continuing the program using new equipment designed by R. E. Nather of the University of Texas.

The equipment was built at the Lowell Observatory and put into operation on its new 42-in. reflector in March, 1970.

Angular diameters of 0″.0056 ± 0.0011 and 0″.0049 ± 0.0008 have been calculated for the red giants 46 Leo and φ Aqr respectively from occultation observations. The occultation curves of α Lib and ε Cap show that they are binary systems. The separation of the components of the visual binary ε Ari, as calculated from its occultation curve, is 1.3 times its reported value. The occultation curve of α Leo shows distortion effects which are attributed to irregularities in the lunar limb.

The dual channel photometer is mounted on the tailpiece of the 32-in.ƒ/16 reflector at McCormick Observatory’s Fan Mountain Station. A 50-50 beam splitting mirror distributes the light to two EMI 6256 S/A photomultipliers. A focal plane diaphragm admits an area of sky of 100 sec2 of arc, and UBV or narrow-band filters are used to isolate spectral regions. The photoelectrons are fed through preamplifiers and amplifiers to a specially designed integrated-circuit photon counting instrument. The preamplifiers are attached as closely as possible to the photomultiplier anodes, while the remaining circuits are housed separately. A Sony TC-200 stereo tape recorder is used to record the output signal from the counter, and an oscilloscope is used for real-time monitoring of the counter operation during occultations.

During the last year studies of lunar occupations have begun at the Dominion Astrophysical Observatory. Since the acquisition of a Fabri-Tek (model FT 1074) signal averager and analyser, twelve occultations have been observed but only one has been reduced and a comparison made with a theoretical model. The observed diffraction pattern for this occultation was found to correspond with an almost point source model.

The exposure meter photomultiplier (EMI type 6094) on the 48 in. telescope has been used for all observations to date. A preamplifier mounted close to the photomultiplier housing directs the amplified signal to a circuit which controls the D.C. level of the signal before being applied to the signal averager.

Ideally the determination of the angular diameter of a star would include the measurement of the distribution of intensity across the stellar disc. However, direct methods of measuring angular diameters have so far lacked adequate ‘signal to noise’ ratio to measure the intensity distribution and it has been the custom, in the first instance, to express the measured angular diameter in terms of the angular diameter of the equivalent uniform disc (θUD). Subsequent use of the angular diameter involves the assumption of a limb-darkening law and the application of an appropriate correction to θUD to find the ‘true’ angular diameter (θLD) of the star (e.g. Hanbury Brown et al., 1967). In this article we will discuss the determination of θUD for single stars and we will not refer further to the more difficult problems of determining intensity distributions involving limb-darkening and rotational effects and of measuring the angular parameters of binary systems.

By itself the angular diameter of a star has no intrinsic value but when it is combined with other observational data it enables basic physical properties of the star to be determined. It is then possible to make a direct comparison of the observed properties of the star with the predictions of theoretical models of stellar atmospheres and interiors. For example, the combination of an angular diameter with the absolute monochromatic flux received from the star (ƒν), corrected for interstellar extinction, yields the absolute emergent flux at the stellar surface (). If the spectral energy distribution for the star is known it can be calibrated absolutely by and hence the effective temperature (Te) of the star can be found (this is equivalent to knowing the bolometric correction for the star and using it with the angular diameter to find Te). In addition to leading to the determination of Te, the absolute surface flux distribution may be compared directly with the predicted flux distributions for theoretical model stellar atmospheres (e.g. Davis and Webb, 1970). For O and early B. type stars a large fraction of the emergent flux is in the far ultra-violet and the effective temperatures cannot be determined from the, at present, incomplete empirical flux curves. In these cases it is possible to obtain an estimate of the effective temperatures by using the values of to calibrate a grid of model atmospheres which have Te as a parameter. In this way, by measuring the angular diameters of stars of different spectral types, it is possible to establish an effective temperature scale.

Several things are clear from this Joint Discussion. Interest in this type of work is wide and could be wider. The subject touches on an immense range of topics from geodesy and celestial mechanics to astrophysics. Observations at different observatories do not conflict with each other, but rather reinforce one another. This is true of timings, but still more true of observations of binary stars and of stars with perceptible angular diameters. There is no real difficulty in identifying as double a wide pair with a separation of the order of 0˝.01. Duplicate observations from different places are necessary if we are to infer the conventional parameters of position angle and separation for a given pair. Duplicate observations are still more essential in cases explainable as close pairs with separations less than 0˝.01.