Observational Techniques

Space vehicles are tracked either by optical or electronic means. Typical optical instruments include:

(i) Recording optical tracking instruments which have a small field of view, and which are mounted in the horizontal system, altitude and azimuth being read automatically off graduated circles. These instruments must be calibrated frequently.

(ii) A kinetheodolite, also with a small field of view and set in the alt-azimuth system, being used to track the object and take photographs of it on 35 mm film.

(iii) A ballistic camera of very wide field, taking photographs of the object against the stellar back-ground.

(iv) A Baker-Nunn camera of very wide field, capable of registering objects and stars as faint as mag-nitude + 17·2. Hewitt cameras are also used.

(v) Orthodox astronomical telescopes for deep-space objects whose angular velocity is low and whose brightness is less than the limiting magnitude of the Baker-Nunn camera.

In astronomy, the brightness of an object is measured on the magnitude scale. This scale was first introduced in the second century^BCin an imprecise way by Hipparchus, who graded the naked-eye stars according to their brightness into six magnitudes: the first consisting of the twenty brightest, the sec-ond of the next fifty in order of brightness, until the sixth, which included the faintest stars visible to the naked eye.

Roughly speaking, a star of one magnitude is two and a half times as bright as a star of the next mag-nitude; the magnitude scale is thus basically logarithmic in character. The system has been rendered pre-cise by the following definition:

If B₁and B₂are the brightnesses of two stars and m₁and m₁are their magnitudes, then

so that

Hence a difference in magnitude of five gives a brightness ratio of exactly 100. It is to be noted that the greater the magnitude is algebraically, the fainter the object is in brightness. Thus the limiting mag-nitude (faintest possible object registered) of a Baker-Nunn camera is + 17·2^mwhile the limiting mag-nitude for the 200 inch Hale telescope at Mount Palomar is + 23·2^m.

It should also be noted that various magnitude systems exist, depending upon whether the radiation from the object enters the eye, or is allowed to fall on photographic emulsion, or on a photoelectric de-vice.

The concept of an absolute magnitude system is introduced to enable meaningful comparisons of objects’intrinsic luminosities to be made. To get rid of the effect of distance it is customary to state what the magnitude of the object would be at a standard distance. This distance is taken to be 10 pc (see sec-tion 1.3). If d is the object’s true distance in pc, and M and m are its apparent magnitudes at distances of 10 and d pc respectively, it is easy to see, taking into account that brightness falls off as the square of the distance, that

The quantity M is called the absolute magnitude of the object.

Typical electronic instruments include:

(i) Radio telescopes, used either to receive radio signals sent from the spacecraft or (if it is near) as radar instruments picking up radar echoes from the craft.

(ii) An interferometer. Two or more antennas in an array of precisely known geometry which in some instrumental designs can be varied. The principle of such a direction-finding system is that a

radiosignal arriving simultaneously at two points will show a phase difference, depending on the path difference from the signal source to the points. There are well known techniques for finding the direction of the source relative to the receiving points.

(iii) Apparatus capable of detecting Doppler shift. If a source emitting radiation has a velocity ν rela-tive to the observer, then the received radiation that normally has a wavelengthλ when the veloc-ity relative to the observer is zero will have a measured wavelengthλ , where

c being the velocity of light. The convention is made that ν is negative if the source is approach-ing and positive if it is recedapproach-ing. Wavelengthλ and frequency ν are connected by the well-known relation

ν λ = c and so we can rewrite equation (3.1) as

This change in wavelength and frequency due to relative velocity is called the Doppler effect.

It is seen that electronic apparatus capable of measuring the frequency difference will give the line-of-sight velocity of the object emitting the radio waves. It should be remarked that the above is a gross simplification of a complicated phenomenon.

There are many types of systems based on the Doppler principle. With some, the distance (range) of the object is obtained as well as the line-of-sight velocity (range rate). Accuracies attained with range and range-rate equipment are extremely high.

For natural celestial objects such as planets, stars and galaxies, optical and radio telescopes are used. Most of the work with optical telescopes is now carried out by photography.

Both optical and radio telescopes will obtain the direction coordinates of the object at the time of observation. Unless the radio telescope is used in an interferometric mode with other radio telescopes, the precision with which it pinpoints a celestial object emitting radio waves falls far short of an opti-cal telescope’s ability. As part of an interferometer with a long baseline (in some cases thousands of kilo-metres) however, its accuracy in determining position is as high as the best optical system.

A large radio telescope operating as a radar instrument is capable of measuring accurately the dis-tances of the nearer bodies in the Solar System such as the Moon, Venus, Mars, Mercury, Jupiter and Saturn.

Summarizing all these optical and electronic methods: it is seen that in general the altitude and az-imuth of the object (or its position on a photographic plate with respect to a stellar background) is ob-tained. Its distance from the observer is not usually measured unless Doppler or radar equipment is used. In addition a time is noted at which the observation was made. This time is reduced to Universal Time and then usually to local sidereal time, if not already in that system.

The main corrections to the data to obtain a geocentric equatorial position for the object are now outlined in principle. If the altitude and azimuth of the object are measured, the first corrections applied are known instrumental errors. This entails a frequent calibration of the instrument since such errors are not in general static.

3.3 Refraction

A ray of light entering the Earth’s atmosphere is refracted or bent so that the observed altitude of the source of light is increased. Thus in figure 3.1 the ray of light appears to the observer at O to come from the direction C so that the measured zenith distance ζ is ZÔC while the true zenith distance is ZÔB, where OB is parallel to the original direction in which the ray entered the atmosphere.

Then, assuming the atmosphere to consist of plane parallel layers of different densities, it is easily shown that Snell’s law of refraction leads to the relation

where r = z − ζ, and k is about 58·2 . Since the observed altitude a is too large, the angle r is subtracted from it (Roy and Clarke 2003).

Equation (3.3) is valid for zenith distances less than 45° and is a fairly good approximation up to 70°. Beyond that, a more accurate formula taking into account the curvature of the Earth’s surface is required, while for zenith distances near 90° special tables are required.

There are a number of versions of equation (3.3). Among them is Comstock’s,

where r is expressed in seconds of arc, p is the barometric pressure in inches of mercury and T is the temperature in degrees Fahrenheit.

For radio measurements refraction depends strongly upon the frequency employed. The lower at-mosphere produces refraction effects approximately twice the optical effect, decreasing rapidly with in-creasing angle of elevation. The ionosphere also refracts radio waves due to induced motion of charged particles in the ionosphere, in amounts dependent on the ion-density gradient. If N is the electron den-sity per cubic centimetre andν is the frequency in kilohertz, then the local effective dielectric constant n (which varies throughout the ionosphere) may be expressed by

Figure 3.1

As height increases above the Earth’s surface, the electron density increases then falls off again. N may become so large that n is zero or imaginary. In these cases a radio signal cannot penetrate the ion-osphere from the inside or from the outside. In other cases when the frequency is high enough, pene-tration takes place with bending of the signal. If we assume that the ionosphere consists of concentric shells about the Earth, Snell’s law enables the path of the radio signal to be calculated from the rela-tion nρ sin i = constant, where ρ is the radius of curvature of the shell of dielectric constant n, and i is the angle of incidence of the signal. Study of ionospheric refraction by comparison of optical and radio tracking of artificial satellites has yielded valuable data.

Having applied the correction for refraction, the topocentric altitude and azimuth may be converted into the topocentric equatorial coordinates hour angle and declination as insection 2.9.2, example 1.

The application of the local sidereal time using equation (2.18) enables the topocentric right ascension to be found.

The above procedure is modified if the observations give the position of the object with respect to a stellar background. The directions of the stars whose images appear on the film will be differentially affected by refraction so that suitable corrections must be applied in obtaining the right ascension and declination of the object from the position of its image among the stellar images. Various procedures have been developed in astronomy to correct for this. When such procedures are applied, the equato-rial coordinates of the object relative to the observer are obtained. In the section on precession and nu-tation (section 3.4) the outline of the method is given. An additional allowance for differential refraction must be made when the object is a rocket observed just after take-off. The stellar background will be displaced by refraction due to its light passing through the total thickness of the atmosphere, whereas the rocket’s light may have less than 50 km of atmosphere to penetrate.

The observational data can now be said to be expressed in equatorial coordinates with respect to the observer’s station on the Earth’s surface. It is necessary now to consider more closely the definition of such coordinates.