- Email: [email protected]
Receding or circulating quasars?
Volume
27A.
number
PHYSICS
9
RECEDING
OR
LETTERS
23 September
CIRCULATING
1968
QUASARS?
P. BURCEV Department
of Theoretical Bmo. KotlhrskcZ Received
Physics. Bmo 2. Czechoslovakia 21 August
Unittersity.
1968
A red-shift should be seen not only in the expanding metagalaxy but also in a rotating one. where it would be due to the transverse Doppler effect. Calculation sho\vs that the metagalactic rotation could account for the red-shift of quasars.
Most workers have generally agreed that the red-shifts of quasars are Doppler shifts due to their radial recession. In our consideration submitted to discussion we shall restrict ourselves merely to the special theory of relatively. The formula for the relativistic,Doppler effect reads A = +,(l - ve/c)(l -v2/c2)-z. Choosing the direction - e away from the observer to be positive and resolving the velocity v into the radial and transversal componenets, we have p, = - ve/c and thus A - x0 l+& 2 == JI _pz _aP - 1 .
(1)
A0
A red-shift
appears
if
I
1 +p r > dl-p2-p2 (2) r t’ For a suitable value of fit the condition (2) can be fulfilled for any I&. 1 < 1. Up to now the biggest observed red-shift z = 2.223. Taking for simplicity z = 2, (1) will be fulfilled by pr = 0.80, pt = 0 or by pr = 0, pt = 0.94 or possibly by p, = = -0.80, Pt = 0.59. Consequently, the shift in question can be due to the radial recession or to the transverse motion or possibly to a non-radial approaching of the object. We see that the observed red-shift does not necessarily imply that objects recede away from us and that the universe expands! On the other hand, a blue-shift appears if j7 I +& < l-P,-& and thus only if & < 0. Consequently, a blueshift would necessarily indicate an approaching object. Since no blue-shift of quasars have been observed, there is a very small probability for quasars to be ejected from galaxies. Since,
further, assuming the red-shift due to the cosmological expansion, we can not succesfully account for the enormously great energy emission needed for quasars, we arrive at the natural conclusion. that the red-shift can be the transversal Doppler shift due to the metagalactic rotation. Let us assume, in a rough approximation, that the metagalaxy moves like a rigid body. In this case the velocity of an arbitrary metagalactic object relative to any arbitrary metagalactic object can only be a transverse one. Thus in the rotating metagalaxy any arbitrary observer can see only red-shifts, similar as in an expanding one, but due to the transverse Doppler effect. On the other hand, in the rotating metagalaxy a anisotropic distribution of the red-shifts should appear It is easy to see that the least red-shifts should be seen in the direction parallel to the rotation axis and the sources of the biggest red-shifts should lie approximately along a plane perpendicular to the rotation axis. It seems that the results recently obtained by Strittmatter et al. [l] could confirm this kind of anisotropy: In this context the study of the d.istribution of red-shifts on the celestial sphere is of great importance. In the rotating metagalaxy quasars can be at normal distances in agreement with the limits to the distances of quasi-stellar objects recently found by Burbidge et al. [2]. Denoting by d the distance of a metagalactic object from the metagalactic rotation axis, w the angular velocity and v the velocity of the object in question, we have in our approximation v = wd. Assuming v << c for our galaxy, we obtain for quasars at relativistic speeds “t = wd . Taking,
for example,
z = 2, d = 10 Mpc,
(3) vr = 0, 623
Volume
27A.
number
9
PHYSICS
LETTERS
we obtain from eqs. (1) and (3) a rotation period of about 2 x lo* years. Of course these values are illustrative only, since the metagalaxy would not move like a rigid body. A satisfactory cosmological model of the rotating metagalaxy should be worked out within the framework of the general theory of relativity. The question concerning the metagalactic rotation should be answered. At all events, the possible influence of the transverse Doppler effect at relativistic
23 September
speeds of quasars tion.
1968
must be taken into considera-
References 1. P. Strittmatter.
J. Faulkner and M. Walmsley. Nature 212 (1966) 1441. 2. G. R. Burbidge and E, M. Burbidge. Appl. J. 148 (1967) L107.
*****
A STATISTICAL
MODEL
OF
LASER
LIGHT
R. M. SILLITTO Department
of Physics. Received
A simple statistical model for laser moment of the photocount distribution model.
exp[-
(IV -IV)2/2a2]
.
that the intensity is a Gaussian variate. This is equivalent for a single-mode field to assuming the density operator to be
i.e.
o = sJP(o);o)
((Y !d20
with 1 P(cu)d20 = 1 x o(27r)i
UK
1968
(1) does, would also go over in the limit of zero mean into the form appropriate for thermal radia tion, i.e., P(a) ct exp[-
where a corresponds to an intensity. Intuitively it might seem desirable to choose a density operator which, as well as going over in the limit of zero variance into the 6-function form representing the “ideal laser” [4], as eq.
ia 12/2P2]
)
where p2 corresponds to an intensity. Eq. (1) does not behave in this way, and it is therefore of interest to consider alternative models in which the mode amplitude is a Gaussian variate, thus ensuring the suggested behaviour in both the limiting situations referred to above. Glauber’s “&i-function plus Gaussian noise” model with which Arecchi et al. compared their data is of this type [5] (see in particular ref. 3, eq. (1); and see also Laths [S], eq. (4.2a)), but it does not give as close agreement with the data as does B&lard’s model. We propose now to take as density operator 1
(1)
~exp[-(~a~~-Io,/~)~/2o~]\o~d/(~(d(argcr)
624
15 August
of Edinburgh.
light is proposed. and is sho\vn to agree as far as the fourth factorial \vith the experimental results of Arecchi et al. and \vith Bedard’s
Bedard [1,2] has recently presented a statistical model for the properties of laser light, and has obtained good agreement between calculations based on this model and experimental results published by Arecchi et al. [3]. The model is built on the assumption that the statistical properties of the radiation intensity are described by the distribution P(W) = [l/aJ2n]
University
p =
X
2 /a0 ((2np2)+xJexp[-(/a(
(2) - /ao/12/2p2]
(a)bldja/2,
which is normalized for /a0 / >> p, so that our results may be comparable with those in ref. 1. We denote by H&(X) the real polynomials
27A.
number
PHYSICS
9
RECEDING
OR
LETTERS
23 September
CIRCULATING
1968
QUASARS?
P. BURCEV Department
of Theoretical Bmo. KotlhrskcZ Received
Physics. Bmo 2. Czechoslovakia 21 August
Unittersity.
1968
A red-shift should be seen not only in the expanding metagalaxy but also in a rotating one. where it would be due to the transverse Doppler effect. Calculation sho\vs that the metagalactic rotation could account for the red-shift of quasars.
Most workers have generally agreed that the red-shifts of quasars are Doppler shifts due to their radial recession. In our consideration submitted to discussion we shall restrict ourselves merely to the special theory of relatively. The formula for the relativistic,Doppler effect reads A = +,(l - ve/c)(l -v2/c2)-z. Choosing the direction - e away from the observer to be positive and resolving the velocity v into the radial and transversal componenets, we have p, = - ve/c and thus A - x0 l+& 2 == JI _pz _aP - 1 .
(1)
A0
A red-shift
appears
if
I
1 +p r > dl-p2-p2 (2) r t’ For a suitable value of fit the condition (2) can be fulfilled for any I&. 1 < 1. Up to now the biggest observed red-shift z = 2.223. Taking for simplicity z = 2, (1) will be fulfilled by pr = 0.80, pt = 0 or by pr = 0, pt = 0.94 or possibly by p, = = -0.80, Pt = 0.59. Consequently, the shift in question can be due to the radial recession or to the transverse motion or possibly to a non-radial approaching of the object. We see that the observed red-shift does not necessarily imply that objects recede away from us and that the universe expands! On the other hand, a blue-shift appears if j7 I +& < l-P,-& and thus only if & < 0. Consequently, a blueshift would necessarily indicate an approaching object. Since no blue-shift of quasars have been observed, there is a very small probability for quasars to be ejected from galaxies. Since,
further, assuming the red-shift due to the cosmological expansion, we can not succesfully account for the enormously great energy emission needed for quasars, we arrive at the natural conclusion. that the red-shift can be the transversal Doppler shift due to the metagalactic rotation. Let us assume, in a rough approximation, that the metagalaxy moves like a rigid body. In this case the velocity of an arbitrary metagalactic object relative to any arbitrary metagalactic object can only be a transverse one. Thus in the rotating metagalaxy any arbitrary observer can see only red-shifts, similar as in an expanding one, but due to the transverse Doppler effect. On the other hand, in the rotating metagalaxy a anisotropic distribution of the red-shifts should appear It is easy to see that the least red-shifts should be seen in the direction parallel to the rotation axis and the sources of the biggest red-shifts should lie approximately along a plane perpendicular to the rotation axis. It seems that the results recently obtained by Strittmatter et al. [l] could confirm this kind of anisotropy: In this context the study of the d.istribution of red-shifts on the celestial sphere is of great importance. In the rotating metagalaxy quasars can be at normal distances in agreement with the limits to the distances of quasi-stellar objects recently found by Burbidge et al. [2]. Denoting by d the distance of a metagalactic object from the metagalactic rotation axis, w the angular velocity and v the velocity of the object in question, we have in our approximation v = wd. Assuming v << c for our galaxy, we obtain for quasars at relativistic speeds “t = wd . Taking,
for example,
z = 2, d = 10 Mpc,
(3) vr = 0, 623
Volume
27A.
number
9
PHYSICS
LETTERS
we obtain from eqs. (1) and (3) a rotation period of about 2 x lo* years. Of course these values are illustrative only, since the metagalaxy would not move like a rigid body. A satisfactory cosmological model of the rotating metagalaxy should be worked out within the framework of the general theory of relativity. The question concerning the metagalactic rotation should be answered. At all events, the possible influence of the transverse Doppler effect at relativistic
23 September
speeds of quasars tion.
1968
must be taken into considera-
References 1. P. Strittmatter.
J. Faulkner and M. Walmsley. Nature 212 (1966) 1441. 2. G. R. Burbidge and E, M. Burbidge. Appl. J. 148 (1967) L107.
*****
A STATISTICAL
MODEL
OF
LASER
LIGHT
R. M. SILLITTO Department
of Physics. Received
A simple statistical model for laser moment of the photocount distribution model.
exp[-
(IV -IV)2/2a2]
.
that the intensity is a Gaussian variate. This is equivalent for a single-mode field to assuming the density operator to be
i.e.
o = sJP(o);o)
((Y !d20
with 1 P(cu)d20 = 1 x o(27r)i
UK
1968
(1) does, would also go over in the limit of zero mean into the form appropriate for thermal radia tion, i.e., P(a) ct exp[-
where a corresponds to an intensity. Intuitively it might seem desirable to choose a density operator which, as well as going over in the limit of zero variance into the 6-function form representing the “ideal laser” [4], as eq.
ia 12/2P2]
)
where p2 corresponds to an intensity. Eq. (1) does not behave in this way, and it is therefore of interest to consider alternative models in which the mode amplitude is a Gaussian variate, thus ensuring the suggested behaviour in both the limiting situations referred to above. Glauber’s “&i-function plus Gaussian noise” model with which Arecchi et al. compared their data is of this type [5] (see in particular ref. 3, eq. (1); and see also Laths [S], eq. (4.2a)), but it does not give as close agreement with the data as does B&lard’s model. We propose now to take as density operator 1
(1)
~exp[-(~a~~-Io,/~)~/2o~]\o~d/(~(d(argcr)
624
15 August
of Edinburgh.
light is proposed. and is sho\vn to agree as far as the fourth factorial \vith the experimental results of Arecchi et al. and \vith Bedard’s
Bedard [1,2] has recently presented a statistical model for the properties of laser light, and has obtained good agreement between calculations based on this model and experimental results published by Arecchi et al. [3]. The model is built on the assumption that the statistical properties of the radiation intensity are described by the distribution P(W) = [l/aJ2n]
University
p =
X
2 /a0 ((2np2)+xJexp[-(/a(
(2) - /ao/12/2p2]
(a)bldja/2,
which is normalized for /a0 / >> p, so that our results may be comparable with those in ref. 1. We denote by H&(X) the real polynomials