KINEMATICS
OF RELATIVISTIC
EJECTION
WITH
HUBBLE
FLOW I. Applicators to SuperluminaI Motion
S. M. G O N G
(KUNG)
Purple Mountain Observatory, Academia Sinica, Nanjing, China
(Received 1 March, 1990) Abstract. What we observe as an apparent superluminal motion is the resultant velocity of Hubble flow and
a local ejection. Taking Hubble flow into account, one can explain, problems posed for the relativistic beaming model, such as the one-sidedness problem and the untolerable extension of the deprojected structure of some superluminal sources; the misalignment of small- and large-scale jets can be analyzed more appropriately. A significant by-product of the present investigation is the initiation to find a new way to determine the value of the Hubble constant Ho, provided that one can observe the superluminal motion at much longer wavelengths, say, meter wavelengths.
1. Introduction
About two decades ago, superluminal motion of components in 3C 273 and 3C 279 were observed under the assumption that they were at cosmological distances, Since then considerably more data have been accumulated (Porcas, 1987; Witzel, 1987), and various explanations of superluminal motion have been proposed. The most generally accepted explanation (Kellermann etal., 1981; Scheuer, 1987) appears to be the relativistic beaming model first suggested by Rees (1966, 1967) and elaborated for a relativistic jet by Behr et aI. (1976) and Blandford and Konigl (1979). However, several problems have been posed for the relativistic model, such as the one-sidedness problem, the misalignment of the small-scale jet and the large-scale jet, and the uncommonly large intrinsic size of some sources (see Blandford, 1987). Some phenomena of the superluminal motion can be explained if the velocity of Hubble flow is taken into account. What is observed as apparent superluminal motion is actually the resultant velocity of Hubble flow and a local ejection. In the standard Friedmann universe model, the redshift z has, apart from the determination of cosmic distance, two effects: (1) (1 + z) = R(to)/R(t ) (Gong, 1989) expressed as the ratio of cosmic-scale factors where t and to are the cosmic and present time, respectively; (2) the velocity of Nubble flow vH = {(1 + z)2 - 1}/{(1 + z) 2 + 1}. In solving the problem of the superluminal motion, one has to take both effects into account. However, in current studies the second effect, the velocity of Hubble flow, is not considered. It is either disregarded or it is incorrectly assumed that by taking the ratio of cosmic scale factor into account the Hubble flow is also included. This is evidently not true, since the velocity of Hubble flow does not appear in any expression. With respect to the one-sidedness problem, current investigators concur that the Astrophysics and Space Science 175: 23-33, 1991. 9 1991 Kluwer Academic Publishers. Printed in Belgium.
24
s. M, GONG
visibility of the counterjets is indeed restricted by their much lower brightness due to Doppler boosting, but they do not notice that it is also conditioned by the much lower frequency as a result of considerable Doppler redshift when the counterjets, taken as an emission line, recede with large velocity. The Doppler-factor of the jet and the counterjet is as yet not well constrained by the shock model or the inverse-Compton calculation and could be significant (Motel and Philips, 1987). Introducing the velocity of Hubble flow of the quasar exact values of the Doppler factor can be determined which correspond to a set of ejection velocities. One can therefore predict the appropriate frequencies for the observation of counterjets. The following considerations explain the untolerable extension of the deprojected structure of the superluminal source. For a given quasar, the velocity of the Hubble flow is constant, the angle between the resultant velocity and the direction of the Hubble flow will decrease while the direction of a local ejection is kept constant and its speed decreases. This will in turn increase the angle between the resultant velocity and the line of sight and will thus decrease the cosecant of the angle, i.e., decrease the deprojected size of the superluminal source. In what follows, formulae will be derived describing the relationship between the various quantities based on the method of composition of velocities given by special relativity. Next, in order to investigate the change of the angle e, i.e., the angle of the resultant velocity V with the direction of Hubble flow, we keep the ejection angle 4) constant and assume for the velocity u values from 0.99 to 0.30 of the velocity of light. The method is applied to the two triple-source quasars 1928 + 738 (redshift zq = 0.302, yapp = 7.0c) and 3C 179 (zq = 0.846, Yapp = 4 . 8 C ) , The numerical results of the computations are given in three tables. The dependence of (~99 - e) on the ejection speed u is shown in Figure 1 where (e99 - c0 is the difference of the angles of the resultant velocities between the case with an ejection speed 0,99c and those with the ejection speed in question. 2. The Solution of the Superluminal Motion with Hubble Flow The superluminal motion observed in a quasar is not the ejection velocity u, but the resultant velocity V composed of u and the velocity vh of the Hubble flow. In this case, the observed transverse velocity Yap p of a separation o f a blob from the core of the quasar is related to the resultant velocity V and the angle 0~it forms with the direction of Hubble flow, through the expression
/)app --
V sin
(1)
1 + gcos~ where the velocity of light is taken as unit. Although Equation (1) has the same form as the one used, e.g., by Behr et al. (1976) and Pearson and Zensus (1987), a significant difference is that they all use directly the ejection velocity u and its angle q5instead of the resultant velocity V and its angle e with vh. All angles are defined as 0 ~ along vh, and rc along the line-of-sight. The relation between V and u can be derived as follows.
RELATIVISTICEJECTIONWITH HUBBLEFLOW
1.00
I
1
' "l'"
[
I
""
I
25 i
I
0.90
0.80
z = 0,846
0.70 = 165 ~
/i 0.60 = 175 ~ 0.50
z = 0.302
0.40 165 ~
= 165 ~
~=
0.30 I
1
20 ~
,,,L
40 ~
60 ~
I
80 ~
.l
1
100 ~
120 ~
1
140 ~
t
160 ~
180 ~
((Z99 -- (g)
Fig. 1. The angle difference (%9 - ~) vs the ejection velocity u at a given ejection angle 9. ~9 is the angle of the resultant velocity Vwith the direction of the Hubble flow in the case of an ejection velocity u = 0.99c; is the angle for the ejection velocity u in question.
L e t the e j e c t i o n v e l o c i t y u m a k e an angle q5 with the velocity vh of the H u b b l e flow, t h e n a c c o r d i n g to the f o r m u l a e o f the c o m p o s i t i o n o f velocities given b y special relativity (Einstein, 1905; MOller, 1952), w e h a v e the V h c o m p o n e n t o f the r e s u l t a n t v e l o c i t y V in the vh direction, Vh__ vh + u c o s q ~
1+
,
(2)
v~u cos4~
in w h i c h vh is u n i q u e l y d e t e r m i n e d by the redshift Zq o f the q u a s a r in q u e s t i o n , 1 + Zq - ~
l+v h
(3)
,
a n d the r e s u l t a n t v e l o c i t y V is given by V = {(vh2 + u2 + 2 v h u c ~
- ( v h u sin q})2} 1/2
1 + v,, u c o s q~ In solving E q u a t i o n
(4)
(1), there is a m i n i m u m v a l u e o f V, V o, c o r r e s p o n d i n g
c o t % = - Yapp a n d a m i n i m u m v a l u e o f c~, ~ m
to
= 2 % - n ( B e h r et al., I 9 7 6 ; P e a r s o n
26
s.M. GONG
and Zensus, 1987). Thus, we first solve for Vin Equation (1) with these two values of and some suitable values of c~. Then values of Vh = Vcos c~ are obtained. Next, we solve for u cos q~ using Equations (2) and (4) for
VhUCOS~p)2 _ {V2(1 + U2 COs2~p) + 2VhUCOSq)}]I/2
[V2(1 + u =
(1 -
v~) la
(5)
The corresponding values of q~ are derived. The Doppler factor is given by 6 _ x / i - - V2 1 + Vh
(6)
and the shift of the wavelength of the ejecta will be zE = 6 - ' -
1.
(7)
In order to derive the corresponding quantities for the counterjet, we assume the ejection velocity and angle, respectively, to be u' = u and q~' = q~ + 180 ~ and solve for V;,, V', a', a', and z~ using Equations (2), (4), (6), and (7) in which primed quantities are employed. Finally, the ratio b/b' of Doppler factors for jet to counterjet is obtained. The numerical results for the two quasars are tabulated in Table I. To investigate the change of a values with the ejection velocity u, we assume two fixed ejection angles q~ = 175 ~ and 165 ~ and vary the ejection speed as u = 0.99, 0.90, 0.80, 0.65, 0.50, and 0.30 for the two quasars. Since due to its larger redshift, the change of :r for quasar 3C 179 is too fast around u = 0.50, we add the two values o f u = 0.55 and 0.40. Thus, we compute Vh, V, ~, 6, and z E and their corresponding quantities for the counterjet using Equations (2), (4), (6), and (7). The results of these calculations for the quasars 1928 + 738 and 3C 179 are given in Tables II and III, respectively. The changes of ~ expressed as the difference (~99 - ~) with various values of u, are shown in Figure 1.
3. Numerical Results and Discussion
3.1.
T H E S O L U T I O N OF S U P E R L U M I N A L M O T I O N
In Table I, we list - c O S e o = +0.98995 = Vo for 1928+738 and - c o s % = + 0.97989 = V0 for 3C 179, corresponding to the minimum values of Vin Equation (1). The minimum angles ~min (corresponding to m a x i m u m angles r c - emin, with the line-of-sight) are 163774 and 156747 for the solution of superluminal motion for 1928 + 738 and 3C 179, respectively, when the ejection velocity u and the resultant velocity V both reach their m a x i m u m value 1. It is interesting to note in Table I that at minimum angle ~.i~, values of V, u for the jet and V', u' for the counterjet approach 1.6 and 6' decrease almost to 0 and in turn zE and z) approach + oo irrespective of whether the jet is approaching or receding, due to the time dilation factor x/1 - V 2 or x/1 - V' 2. In order to observe emissions in these extreme cases, one should use very long wavelengths!
RELATIVISTIC
EJECTION
WITH
HUBBLE
27
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RELATIVISTIC
EJECTION
WITH
HUBBLE
29
FLOW
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s.M. GONG
3.2. T H E P R O B L E M OF T H E O N E - S I D E D N E S S The brightness ratio of jet to counterjet is usually expressed as (8/5')2- ~, or (8/8')3- ~, depending on the model (Blandford and K6nigl, 1979), ~ being the spectral index of the emission (S ~ v~). The values of the ratio 5/b' for 1928 + 738 and 3C 179 given in Tables II and I I I are rather moderate both for q5 = 175 ~ and 165 ~ when the ejection velocity u = 90. With ~ = 0.5, the brightness ratio, (b/~5')25 or (5/8')15, of the jet to the counterjet is less than 1500, i.e., well below the dynamical range of 24000 : 1 (Witzel, 1987) for 1928 + 738. The invisibility of the large-scale counterjet in 1928 + 738 is, therefore, not due to its faintness but rather due to the shift of the observable frequency which is equal to ~'/b times that of the jet as the result of Doppler effect. It is straightforward to find out the appropriate wavelength to observe a counterjet when the Doppler factors b and ~', respectively, of the jet and the counterjet are determined. Let )-obs be the observed wavelength of the lobe of a given quasar, then its rest wavelength should be 1 2 r l --
1 +Zq
}~obs ,
(8)
where z a is the redshift of the quasar. The observed wavelength of its approaching jet is also hobs, and its rest wavelength should then be 1 2rj-
1 +z E
(9)
"~obs = b/]'obs 9
In order to observe the counterjets for their rest wavelengths of appropriate observed wavelengths should, respectively, be
'~rt
and Zrj, the
2c! = ,~rl(1 + Z~E) _ ~ o b s
(10)
2cj = L s ( 1 + z ~ ) = ~5-- /~obs "
(11)
1 , (1 + Zq) 6'
We can apply Equations (9)-(11) to derive the appropriate wavelength of counterjets for quasar 4C 73 = 1928 + 738. As Zq = 0.302, 2ob s = 20 cm, then 2~l = 15.4 cm. For tp = 165 ~ the following table gives the numerical results: u
6
2rj (cm)
6'
2cr (em)
2cj (cm)
0.90 0.80 0.65
2.56 2.03 1.57
51.2 40.6 31.4
0.179 0.280 0.359
85.8 59.1 43.0
286.0 156.8 87.5
If the emission of a given quasar is continuous in the range 2~t - 2rs., then 2ct - 2cj are
31
RELATIVISTIC EJECTION WITH HUBBLE FLOW
the appropriate wavelength ranges for their corresponding ejection velocities, u to observe their counterjets. The results of the present investigation m a y be used to initiate a new way to determine the value of the Hubble constant H o which is different from the traditional distanceladder method. The value of the observable wavelength range 2 d - 2cj of a counterjet for a given ejection velocity u is derived from two observed quantities, the redshift Zq of a given quasar and its apparent transverse velocity Yapp of its ejects. Since Yapp is inversely proportional to the Hubble constant Ho, as seen in Table I, the wavelength range 2ct - 2cj is mainly fixed by the value of the H o provided that an adequate ejection velocity u can be assigned. The value o f H o is at present assumed to be in the range of 50 to 1 0 0 k m s -1 M p c l We may take different values, say, 50, 75, and 100 km s 1 M p c 1 for H o and compute the values of 2 d - 2cj for a set of well-defined superluminals. A comparison of the observed and calculated values of 2c~ - 2cj might allow the determination of the most acceptable value of H 0.
3.3. THE PROBLEM OF THE UNTOLERABLE EXTENSION OF THE DEPROJECTED SIZE OF SOME SUPERLUMINALS Figure 1 shows for the two quasars the relation between the values of u with those of (~99 -- ~) both for q~ = 175 ~ and 165 ~ The figure demonstrates the effect of the ejection
speed u on the change of the angle between the directiion of the Hubble flow and that of the resultant velocity Vafter taking the Hubble flow into account when z is moderately large. In view of this effect, the unexpected extension of the deprojected structure of some superluminal sources might be alleviated, for the angle e between the resultant velocity V and the Hubble flow decreases with decreasing ejection speed, so that the angle ( r e - c0 between V and the line-of-sight increases. The deprojection factor cosec (~ - ~) is now much smaller than the case in which the Hubble flow is not taken into account. For quasar 4C 73 = 1928 + 738, we have from Table I of the paper by Simon et al. (1987) the following data: Angle size
0m~•
Apparent size h- i kpc
Deprojected size h - t kpc
80"
1275
235
1090
F r o m our Table II, we have for ~b = 165 the following result: u
c~
0
cosec 0
Apparent size h lkpc
Deprojected size h-lkpc
0.50 0.30
15072 13070
29?8 67?0
2.01 1.09
235 235
472 255
32
s.M. GONG
The untolerable deprojected size 1090 h - 1 kpc is thus reduced to 472-255 h - 1 kpc if the ejection velocity u decreases to 0.50-0.30e. 3.4. THE PROBLEM OV MISALIGNMENT OF SMALL-SCALEAND LARGE-SCALEJETS According to the principle of composition of velocities for a given ejection angle q~, the angle e of the resultant velocity Vwill turn to the direction of Hubble flow as the ejection velocity u decreases. The angle c~lies in the plane formed by u and the direction of the Hubble flow, i.e., all e angles are seen along the same position angle. It is, therefore, impossible that the misalignment of the small-scale and the large-scale jets arises from a single ejection in a fixed direction only due to the change of the speed of ejection. It should be mentioned that a few authors (Dishon and Weber, 1977; Horfik, 1978; Li, 1980) tried to solve the problem of the superluminal motion considering the Hubble flow. However, all authors treated it in a very restricted way by considering only the transverse ejection, i.e., an ejection angle q~fixed at + 90 ~ 4. Conclusion
When the treatment of the superluminal motion includes the Hubble flow, thus the angle between the resultant velocity of an ejecta and the line-of-sight increases as the ejection speed decreases. This explains the apparent uncommonly extended structure of some superluminal sources. For a quasar of known redshift Zq and an apparent transverse velocity yapp of an ejects, one can determine the values of its Doppler factor fi and wavelength shift zE and their corresponding values of 6' and z) for its counterjet for an allowable speed u and ejection angle cp. From values of b/~', one can calculate the brightness ratio of jet to counterjet and predict the adequate frequency range to search the counterjet in order to solve the problem of one-sidedness. The misalignment of the small-scale and large-scale jets, does not appear to arise from a single jet ejected at a fixed direction.
References Behr, C., Schucking, E. L., Vishveshwara, C. V., and Wallace, W.: 1976, Astron. J. 81, 147. Blandford, R. D. and KOninl, A.: Astrophys. J. 232, 34. Blandford, R. D.: 1987, in J. A. Zensus and T. J. Pearson (eds.), Superluminal Radio Sources, Cambridge University Press, Cambridge, p. 310. Dishon, G. and Weber, T. A.: 1977, Astrophys. J. 212, 31. Einstein, A.: 1905, in The Principle of Relativity, Dover Publ. Inc., New York, p. 37. Gong, S. M.: 1989~Astrophys. Space Sci. 158, 1. Hor~tk, Z.: 1978, Bull. Astron. Inst. Czech. 29, 126; 29, 368. Kellermann, K. I. and Pauling-Toth, I. I. K.: 1981, Ann. Rev. Astron. Astrophys. 19, 373. Li, Q.: 1980, Acta Astron. Sinica 21, 1. M611er, C.: 1952, The Theory of Relativity, Oxford University Press, London, p. 52. Mutel, R. L. and Philips, R. B.: 1987, in J. A. Zensus and T. J. Pearson (eds.), SuperluminalRadio Sources, Cambridge University Press, Cambridge, p. 60. Pearson, T. J. and Zensus, J. A.: 1987, in J. A. Zensus and T. J. Pearson (eds.), SuperluminalRadio Sources, Cambridge University Press, Cambridge, p. 1.
RELATIVISTIC EJECTION WITH HUBBLE FLOW
33
Porcas, R. W.: 1987, in J. A. Zensus and T.J. Pearson (eds.), Superluminal Radio Sources, Cambridge University Press, Cambridge, p. 12. Rees, M. J.: 1966, Nature 211,468. Rees, M. J.: 1967, Monthly Notices Roy. Astron. Soc. 135, 345. Scheuer, P. A. G.: 1987, in J. A. Zensus and T. J. Pearson (eds.), SuperluminalRadio Sources, Cambridge University Press, Cambridge, p. 104. Simon, R. S., Johnston, K. J., Eckart, A., Biermann, P., Schalinski, C., Witzel, A., and Strom, R. G.: 1987, in J.A. Zensus and T.J. Pearson (eds.), Superluminal Radio Sources, Cambridge University Press, Cambridge, p. 155. Witzel, A.: 1987, in J. A. Zensus and T. L Pearson (eds.), SuperluminalRadio Sources, Cambridge University Press, Cambridge, p. 83.