MISUNDERSTANDING

OF THE

POYNTING-ROBERTSON

EFFECT

Letter to the Editor J. K L A C K A

Department of Astronomy and Astrophysics, Faculty for Mathematics and Physics, Comenius University, Bratislava, The Slovak Republic

(Received 25 August 1993) Abstract. The most frequent incorrect statements concerning derivations of the action of the solar electromagnetic radiation on the motion of interplanetary dust particles are presented. All of them are discussed and it is also explained why are they physically incorrect. It is stressed that astronomers must discuss the physics of this effect for the purpose of familiarity with it, and, may be, for better understanding of the (in-)stability of the zodiacal cloud.

1. Introduction

The action of the solar electromagnetic radiation on the motion of interplanetary dust particles - the Poynting-Robertson effect ( P - R effect) - is important in study of stability of the zodiacal cloud, orbital evolution of cometary meteor streams, asteroidal dust particles and dust rings around planets. One would expect that scientists working in the field of interplanetary matter should be familiar with the physical nature of the P - R effect. However, this is not true. Some incorrect physical statements were discussed in Kla~ka (1992a). This letter has been inspired by the fact that the article Kla~ka (1993b) was originally submitted to The Astronomy and Astrophysics and not accepted for publication. During a period of 9 months several referees have succeeded in rejection of the article, all of them using only incorrect physical arguments. Tnus, I have decided to present most frequent incorrect physical statements. I hope, this will be helpful for the better understanding of physical nature of the P - R effect. 2. Correct Statements? We can say that the current situation in physical understanding of the P - R effect is characterized by the invited review paper to Icarus - Burns et al. (1979). Already Equation (1) in that article s' = s(1 - v. ~/c),

(1)

where S' is flux of the incident energy measured by the moving observer (speed v), is incorrect. Unfortunately, people still believe that Equation (1) holds. However, correct result is (to the first order of v/c) Earth, Moon, and Planets 63: 255-258, 1993. © 1993 Kluwer Academic Publishers. Printed in the Netherlands.

256

J. KLACKA s' = s(1 - 2v. g/c).

(2)

This result, was discussed in detail by Klaeka (1992a) and correct result is also presented in Mignard (1992) and used in Klaeka (1992b, 1993a, 1993b). As for other incorrect statements used in Burns et al., I refer the reader to my papers. Equations (2) and (3) in Klaeka (1993b) yield for the flux densities S = nhvc,

(3)

S' = n'hu'c,

(4)

for unprimed and primed quantities. Equation (7) of that article reads n' = n(1 - v - g / c ) ,

(5)

for the concentration of photons; v' and v are tied through the Doppler effect. (Using these considerations, one can easily obtain Equation (2).) However, a new argument has appeared. According to this, Equation (4) is incorrect. Equation (4) seems to be relativistically correct, since the speed of light used in this equation is c, the same speed as in Equation (3). It means that the speed of light, used in Equations (3)-(4) is the same - for primed and unprimed reference frames. But this is incorrect. Correctly, Equation (4) should read S ' = n ' h v ' c'

(6)

and this equation is thus classical - nonrelativistic. It means that c' = (1 - v. S/c)

(7)

must be used in Equation (6), and, moreover, n' = n ,

(8)

which is in contradiction to Equation (5). Thus, Equations (4)-(5) are incorrect and correct are Equations (6)-(8). Moreover, Equations (6)-(8) are nonrelativistic equations which prove that the P - R effect can be derived on the basis of classical - nonrelativistic - physics (to the first order of v/c, of course). This is not the whole story. We know that according to all experiments trying to find the absolute system - ether, made in the second half of the 19-th century, measured quantities completely agree with classical-nonrelativistic calculations to the first order of v/c. And there is a difference between Equations (5) and (8) to this order. Since Equations (6)-(8) are correct from the point of view of classical physics - they are consistent with Equation (2), all again prove that Equation (5) is incorrect. This can be proved also in another way: n = N / V , where N is the number of particles (photons) in volume V. According to relativity theory, volume transforms in accordance with Lorentz-Fitzgerald contraction and this is not of the first order v/c. So, V' = V according to this argument and one obtains again Equation (8), moreover, if he uses relativity theory. In any case, one can conclude: Equations (6)-(8) are correct from the point of view of classsical physics and Equation (8)

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is also in accordance with the relativity theory to the first order of v / c . This proves that the P - R effect is classical-nonrelativistic effect, and, moreover, that Equations (4)-(5) are physical nonsenses (although it may seem, for the first glance, that these equations are relativistic equations written to the first order of v / c ) . Equations (6)-(8) lead to Equation (2), which proves this equation. Preceding text seems to be very convincing. However, if we accept the definition of the energy flux - Equation (3) - in one inertial reference frame, we have only one possibility for this flux in another inertial reference frame, if relativity theory holds. This is given by Equation (4), which takes into account that the limiting speed in nature has the same value in all inertial reference frames. The relation between frequencies is given by the Doppler effect. Now, the concentration of photons must be obtained from some general law. Continuity equation reads Oj~/Ox ~

= 0

(9)

for j " = (c n, c n g)

(10)

and generalized special Lorentz transformations yield Equation (5), to the first order of v / c ; see Equations (33)-(36) in Kla~ka (1992a). But this result is in contradiction with the result n = n' (Equation (8)), which as one has proved should be correct also in relativity theory on the basis of Lorentz-Fitzgerald contraction. Where's the problem? The central point is that the argument with the contraction of lenghts holds only for particles which are at rest in one system. But photons move with the speed c. (See also Equations (ll)-(12) in Klaeka and Saniga (1993). If we define j ~ = n u ~ - n is proper concentration of particles, then we can write no = n o y v (1 - u- v / c 2 ) , no = y u n . ) And classical physics is not acquainted with any limiting speed and even with particles which move with this speed. Current derivation of the P - R effect is presented also in Mignard (1992). Equation (18) of Mignard's article supposes that the speed of light is c in the rest frame of the source. On the other hand, Equations (19)-(20) suppose that the speed of light is c in the moving reference frame. This is not consistent with the author's opinion that his derivation is based on classical physics. Another incompleteness is presented in Equations (21)-(21') of the discussed article. In the process of transition from Equation (20) to these two equations, there is tacitly omitted index '2' on the left-hand-side. No comment is done. However, in general, the procedure is not correct. Correct transformation is given by Equation (11) in Kla6ka (1993b). Fortunately, the result for the electromagnetic radiation is correct (see Equation (141) in Kla~ka (1992a)); but the result would be incorrect for the case of solar wind - see Kla~ka and Saniga (1993).

258

J. KLACKA

3. Conclusion I have discussed several incorrect (or incomplete) considerations made in the process of derivation of the P - R effect. One cannot consider as a derivation some rearrangement of mathematical formulae for the purpose of obtaining known result. We must try to understand physics. Only in this way we can better understand problems closely connected with the orbital evolution of interplanetary dust particles, e.g., the stability of the zodiacal cloud. References Burns, J. A., Lamy, P. L., and Soter, S.: 1979, 'Radiation Forces on Small Particles in the Solar System', Icarus 40, 1-43. Klaeka, J.: 1992a, 'Poynting-Robertson Effect I. Equation of Motion', Earth, Moon and Planets 59, 41-59. Kla~ka, J.: 1992b, 'Poynting-Robertson Effect', Poster presented at the 30th Liege International Astrophysical Colloguium: Observations and Physical Properties of Small Solar System Bodies (in press). Kla~ka, J.: 1993a, 'Poynting-Robertson Effect. General Case', Earth, Moon and Planets 61, 119-124. KlaEka, J.: 1993b, 'Aberration of Light and the Poynting-Robertson Effect', Earth, Moon and Planets 62, 239-244. Klaeka, J. and Saniga, M.: 1993, 'Interplanetary Dust Particles and Solar Wind', Earth, Moon and Planets 60, 23-29. Mignard, F.: 1992, 'On the Radiation Forces, in Interrelations Between Physics and Dynamics for Minor Bodies in the Solar System', in D. Benest and C. Froeschle, Fronti~res.