Section 30–4 The parabolic antenna

(Radio sources / Radio antenna / Resolving power)

 

The three interesting concepts discussed in this section are radio sources in the sky, radio antenna, and resolving power of a telescope.  

 

1. Radio sources:

“Now let us consider another problem in resolving power. This has to do with the antenna of a radio telescope, used for determining the position of radio sources in the sky, i.e., how large they are in angle (Feynman et al., 1963, p. 30–6).”

 

According to Feynman, a problem in resolving power is related to the antenna of a radio telescope that is used for determining the position of radio sources in the sky. In a sense, the phrase “radio sources in the sky” may be misleading because it takes about 23 hours and 56 minutes (a sidereal day due to the Earth’s rotation relative to the stars) for extraterrestrial radio sources to be periodically detected by the radio telescope. One should realize that radio sources can be any “warm” objects that emit radio waves. In addition, we need not criticize Feynman for not mentioning radio sources such as pulsars, quasars, active galactic nucleus, black holes, radio galaxies, or Jupiter. The term quasar was coined in May 1964 for quasi-stellar radio sources (Chiu, 1964), whereas black hole was used by Ann Ewing (a journalist) in January 1964 in an article titled Black Holes in Space.

 

“We are very interested to know whether the source is in one place or another. One way we can find out is to lay out a whole series of equally spaced dipole wires on the Australian landscape (Feynman et al., 1963, p. 30–6).”

 

In his Nobel lecture titled Radio Telescopes of Large Resolving Power, Ryle (1975) explains: “…the forerunners for this type of instrument were realized in the early days when observations in both Australia and England with aerial elements having a range of separations were used to determine the distribution of radio brightness across the solar disc.” Interestingly, Feynman says that one way to locate radio sources is to lay out equally spaced dipole wires on the Australian landscape (instead of England landscape). One may clarify that radio telescopes cover a region of the sky ±45^o from zenith, that is, mostly the southern sky if they are in Australia (northern sky if they are in England). Historically, Hanbury Brown’s research proposal was not accepted by the referees, thus he left England. Subsequently, Hanbury managed to find support and build an observatory in Australia.

 

2. Radio antenna:

“Some radio antennas are made in a different way… we may arrange them not in a line but in a curve, and put the receiver at a certain point where it can detect the scattered waves … This is an example of what is called a reciprocity principle.  (Feynman et al., 1963, p. 30–7).”

 

Feynman explains a reciprocity principle of radio antenna as “the receiving pattern of an antenna is exactly the same as the intensity distribution we would get if we turned the receiver around and made it into a transmitter.” He adds that this principle is generally true for any arrangement of antennas, angles, and so on. However, one may elaborate that the essence of reciprocity principle is similar to action and reaction are equivalent, but Newton’s third law of motion does not always hold. Better still, this principle should include a condition of validity because it relates two possible solutions in a linear system (or linear medium) where the radio sources and radio receivers are interchanged. It is worth mentioning that Rayleigh formulates the principle of reciprocity in acoustic and electromagnetism.

 

“The arranging of the antennas on a parabolic curve is not an essential point. It is only a convenient way to get all the signals to the same point with no relative delay and without feed wires (Feynman et al., 1963, p. 30–7).”

 

Feynman clarifies that we may arrange radio antennas on a parabolic curve, but this is not an essential point. However, extraterrestrial radio signals are extremely weak because the wavelengths could be 100 kilometers and longer (or billions of times weaker than the signals used by communication systems). We can apply the principles of Hanbury-Brown-Twiss effect by connecting two radio antennas to analyze the correlation between the fluctuations of radio signal intensities. In his book titled QED: The Strange Theory of Light and Matter, Feynman (1985) writes: “[t]his phenomenon, called the Hanbury-Brown-Twiss effect, has been used to distinguish between a single source and a double source of radio waves in deep space, even when the two sources are extremely close together (p. 75).” This effect has helped to develop quantum optics and it is related to Dirac’s incorrect dictum on interference: “Interference between two different photons can never occur.”

 

Many physicists including Feynman had difficulty in accepting the Hanbury-Brown-Twiss effect. In Radhakrishnan’s (2002) words: “I was present at a Caltech colloquium at which Hanbury talked about it, and Richard Feynman jumped up and said, ‘It can’t work!’ In his inimitable style, Hanbury responded, ‘Yes, I know. We were told so. But we built it anyway, and it did work.’ Late that night, Feynman phoned and woke Hanbury up to say ‘you are right.’ He also wrote a letter in which he magnanimously admitted his mistake and acknowledged the importance of this phenomenon that, at first sight, appears counterintuitive, even to quantum theorists (2002, p. 76).”

 

3. Resolving power:

“Now we are describing a telescope mirror, of course. We have found the resolving power of a telescope! Sometimes the resolving power is written θ = 1.22λ/L, where L is the diameter of the telescope (Feynman et al., 1963, p. 30–7).”

 

It may seem strange that the discussion of radio antenna is changed to the resolving power of a circular telescope within a paragraph. However, a side-view of the circular telescope is parabolic in shape, but there could be a new paragraph to explain the resolving power formula θ = 1.22λ/L (what if you substitute the wavelength of radio signals l = 100 km into the formula?). To be specific, the magnified image of a star seen through the telescope is not the star’s physical body, but it is a diffraction pattern (or “moving” diffraction pattern due to the Earth’s rotation). The resolution of image seen depends on the sky conditions as well as the diameter of the eyepiece and the size of the pupil. One may explain that the resolving power or resolution is based on at least three mathematical concepts: “Abbe’s diffraction limit,” “Airy disk diameter,” and “Rayleigh’s criterion.”

 

“… thus we can appreciate that the effective diameter is a little shorter than the true diameter, and that is what the 1.22 factor tells us. In any case, it seems a little pedantic to put such precision into the resolving power formula (p. 30–7).”

 

Feynman feels that it is pedantic to put such precision into the resolving power formula. However, one may argue that it is not strictly pedantic because we can theoretically compare Rayleigh’s criterion with Houston’s criterion, Abbe’s criterion, and Sparrow’s criterion. From a practical perspective, we can compare the resolving power of different telescopes such as a refractor telescope and reflector telescope, or other optical systems. On the other hand, the diffraction limit of the eye can be calculated using Rayleigh’s criterion where D is the diameter of the eye’s pupil. If you are wondering about the factor “1.22,” it is based on the Bessel function (of the first kind) of order one, J₁(x).

 

Review Questions:

1. How would you describe the radio sources in the sky?  

2. How would you explain a reciprocity principle of radio antenna?

3. Does the factor 1.22 seem pedantic to be included in the resolving power formula?

 

The moral of the lesson: The diffraction patterns of radio sources are so weak that we cannot simply rely on the Rayleigh’s criterion, but it is important to apply the principles of Hanbury-Brown-Twiss effect.

 

References:

1. Chiu, H. Y. (1964). Gravitational collapse. Physics Today, 17, 21–34.

2. Feynman, R. P. (1985). QED: The strange theory of light and matter. Princeton: Princeton University Press.

3. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

4. Radhakrishnan, V. (2002). Obituary: Robert Hanbury Brown. Physics Today, 55(7), 75–76.

5. Ryle, M. (1975). Radio Telescopes of Large Resolving Power. Reviews of Modern Physics, 47, 557–566.