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X- AND GAMMA RAY EMISSION FROM INTRA-DAY VARIABLES
Olivia Tsang and J. G. Kirk Max-Planck-Institut fur Kernphysik, 69117 Heidelberg, Germany Ё

ABSTRACT We re-examined the brightness temperature (TB ) problem posed by rapidly varying blazars, known as intra-day variables (IDV's), such as S 5 0716 + 714, which often display TB > 1012 K, far exceed the limit imposed by inverse Compton cooling. Several of such IDV's have also shown a high degree of circular polarisation (CP) of the order of 1%, much higher that predicted by the conventional synchrotron theory. We adopted a synchrotron model that replaces the conventional power-law electron spectrum with a mono-energetic electron distribution, and found that a brightness temperature as high as TB 1014 K can be reached with a moderate Doppler factor of D 10. The predicted intrinsic degree of CP is of the order of 1%, in agreement with observations. With a predicted spectrum of I 1/3 between the radio and the infra-red frequencies, this model provides a good spectral fit to the IDV S 5 0716 + 714 in the radio band. Key words: galaxies: active ­ galaxies: high redshift ­ galaxies: jets.

power-law distribution. Kellermann and Pauliny-Toth [5] showed that the maximum brightness temperature that a source can sustain is limited by the cooling due to inverse Compton scattering in the Thomson regime, which rises rapidly when the source exceeds TB > 3 в 1011 K. As pointed out by Crusius-Waetzel [6], brightness temperatures higher than 1012 K can be produced by monoenergetic electron, since the GHz photons can emerge from the source without being absorbed by low energy electrons. The parameters of such a synchrotron model are discussed in Section 2. In Section 3, we show that brightness temperature as high as TB = 1014 can be reached. We also address another observational trend shown by several of these high brightness temperature sources - a high degree of circular polarisation (CP) at 1% level or above [e.g. 7]. In standard synchrotron theory, the degree of CP is estimated to be rc mc2 /(kB TB ), where kB is the Boltzmann's constant. For a source of TB = 1013 K, rc = 0.06%. Using S 5 0716 + 714 as an example, we computed the synchrotron and the corresponding inverse Compton spectra, and compare it with the simultaneous multifrequency study by Ostoreroe et. al. [8] in Section 4. The results will be discussed in Section 5.

1. INTRODUCTION 2. Rapid variability in radio flux have been observed in several extra-galactic sources, implying a very high brightness temperature [1]. Variability at higher frequencies have also been observed for some of these so called intraday variables (IDV's), providing the mean for studying processes extrinsic to the source as the cause of the variability. For example, cases such as P K S 1519 - 273 and P K S 0405 - 385, the variability is identified as interstellar scintillation [2, 3]. Diffractive scintillation was discovered in J 1819 + 3845 [4]. These external effects mean that the brightness temperature at the source is in fact considerably lower, however, for these sources, TB > 1013 K for the former two and TB > 2 в 1014 K for the latter are still required to account for the observed flux. Such high brightness temperatures cannot be explained within the scope of the standard synchrotron models, in which the radiating electrons are assumed to have a MODEL PARAMETERS

We consider an idealised model with a mono-energetic electron distribution with Lorentz factor and number density ne , embedded in a uniform magnetic field B . The source is characterised by a single linear length scale R, and Doppler boosting factor D = 1 - 2 /(1 - cos ), where c is the source speed with respect to the rest frame of the host galaxy and is the angle between the velocity and the line of sight. The assumed energy distribution captures the relevant properties of a power-law electron distribution with a lower energy cut-off, if the distribution rises towards the cut-off faster than d ln n/d ln > -1/3. Distribution of this type can account for the lack of Faraday depolarisation in parsec-scale emission regions [9, 10], and which has recently been discussed in connection with statistical trends in the observed distribution of superluminal velocities as a function of observing frequency and


redshift [11]. Similar models have been considered in connection with high brightness temperature sources by Crusius-Waetzel [6] and Protheroe [12]. Slysh [13] has also considered a model with mono-energetic electrons, but restricted his treatment only to optically thick sources and did not allow for multiple scattering. The source model has five parameters: D, R, B , ne and . D and R can be constrained either by variability, which puts an upper limit to the quantity R/D2 , or separately from independent observations. Surveys of superluminal motion [14] suggest that D 10 for most sources, and for a few, up to 30 or 40. An upper limit of R can be obtained if scintillation is present. The rest of the parameters, B , ne and can be constrained by eliminating them with three new parameters. Denoting the Thomson optical depth corresponding to the monenergetic electrons by T = ne RT , the first of the three new parameters is the optical depth to synchrotron absorption s is 3T me c3 K5/3 (x) s = (1) 8 e2 s 3 [e.g., 15]. Here, K
5 /3

As a consequence, the energy radiated formally diverges, a phenomenon that has acquired the name Compton Catastrophe. Therefore, to avoid large Compton losses, we require 1, which in turns ensures that T 1. Eliminating the parameters T , s and from Eq. (3) using Eqs. 1, 2 and 4, the brightness temperature can be written as 1 /5 3/2 3 1 /5 -s 6 1-e kB TB 3 me c D F (x) (5) =52 1/5 4 /5 me c2 4 e (1 + z )6 s x9/5K (x)
5/3

(x) is the modified Bessel function, (2)
2

The first term in parentheses on the right-hand side of Eq. 5 is independent of the source parameters. The third term in parentheses reaches a maximum of the order of unity at s 1. The fourth, however, diverges for small x as x-2/15 . Thus, even with < 1 and D < 10, it is possible to find source parameters for which this formula gives an arbitrarily high brightness temperature at any specified observing frequency. Ultimately, when x < -3 , the Airy integral approximation to the synchrotron emissivity, on which Eq. (5) is based, loses validity, because the harmonics of the gyro frequency L / cease to merge into a smooth continuum. However, there is a tighter constraint from observations. This model predicts an optically thin synchrotron spectrum of I 1/3 between the observing frequency and the frequency /x. Optical observations of, for example, PKS 1519 -273, PKS 0405 -385 and S5 0716+714 ([18, 8] and Wagner, priv. comm.) suggest that the optically thin synchrotron radiation cuts off above 1014 Hz. This can be quantified by introducing the third new parameter, max = /x, such that the optically thin synchrotron emission stretches from up to max < 1014 Hz.

x = (1 + z )/(Ds )

is the observing frequency, s = 3eB sin /(4 me c) is the synchrotron characteristic frequency, for an angle between the line of sight and the magnetic field B in the rest frame of the source, and z is the redshift of the host galaxy. The observed brightness temperature TB measured at frequency is related to the specific intensity of radiation I by TB = c2 I /(2 2 kB ), and is therefore kB TB me c2 = D 1+z


2x2

F (x) K5/3 (x)

1 - e-s

(3)

3.

BRIGHTNESS TEMPERATURE AND CIRCULAR POLARISATION

where F (x) = x x dtK5/3 (t) is the standard synchrotron function in the Airy integral approximation [see e.g. 16, Chapter 6]. The second new parameter, , determines the inverse Compton luminosity, and is defined as the ratio of the energy density in synchrotron photons to the magnetic energy density (and thus the ratio of the energy densities in consecutive generation of scattered photons), assuming the scattering occurs in the Thomson regime. This quantity is somewhat sensitive to the geometry and homogeneity of the source [17]. For a uniform source that can be characterised by a single length scale R we have, in the optically thin case of relevance here, = 4 2 T /3 (4)

Rewriting Eq. (5) in terms of the three new parameter s , and max,14 = max /1014 Hz, the brightness temperature is independent of the source size, and only weakly dependent on the other parameters: TB=1.2 в 10
14 6 D10 (1 + z )6 1/5

1-e

-s

1/5 s



-1/3 2/15 max,14 GHz

K (6)

where D10 = D/10 and GHz = /109 , and the approximations F (x) 2.15x1/3 and K5/3 (x) 1.43x-5/3 for x 1 are used. According to Eq. 6, brightness temperatures of up to 1014 K at GHz frequencies can be achieved with moderate Doppler boosting factors of D 10. A particularly interesting source property is the degree of intrinsic circular polarisation rc . Assuming a pure

When > 1, the energy contained in successive generations of inverse Compton scattered photons increases, provided the scattering remain in the Thomson regime.


electron-proton plasma, r
c

-8
1/3

-9 erg cm Log10 F


=

1 3

2 x

3

cot (1/3) s D10
1/5

(7) cot % (8)

2

s -10 -11 -12 -13 -14 7.5 10 12.5 15 17.5 Log10 Hz 20 22.5 25

= 1.9 в

max,

1/5

14

[19]. In the case of a power-law electron distribution, rc changes sign when entering the optically thick regime, but we have found no such calculation for the monoenergetic case so far. To order of magnitude, the peak value of rc can be estimated using this expression, which is remarkably insensitive to the source parameters except the magnetic field direction. Several extra-galactic high brightness temperature sources such as those mentioned above have been observed with circular polarisation at the percent level [7]. With a conventional power-law electron distribution, PKS 0405-385, for example, would have rc 0.06%, far too small to explain the observation. However, according to Eq. 8, the typical value of rc for a mono-energetic electron distribution is of the order of 1%.

Figure 1. Spectral energy distribution of S5 0716 +714 from observations (black dots, variation ranges are indicated by vertical bars and upper limits by arrows) and as predicted by this model. Doppler factors D are 8 (dashed) and 10 (solid), Lorentz factors are 319 (dashed) and 349 (solid), magnetic field strength B are 20G (dashed) and 0.1G (solid). We adopted an angular size of 0.16mas [21], corresponding to 0.7pc at z = 0.3. 5. DISCUSSION

4. SYNCHROTRON AND INVERSE COMPTON SPECTRA Following the approach of Georganopoulos et. al. [20], in the observer's frame, photons with energy hi-1 are scattered by a uniform distribution of electrons with number density ne at a rate dNp = dtdi
4 R/2 0

The most important feature of this model is the lack of low energy electrons, which would otherwise absorb the GHz emission. This can be in the form of a monoenergetic distribution, where all electrons have Lorentz factor = 0 , or in the form of a quasi-mono-energetic distribution, such that d ln n/d ln > -1/3 for < 0 . For example, a relativistic thermal distribution rises at low energy as 2 . The addition of a power-law tail to higher energy would not affect this assumption. (9) Continuous re-acceleration prevents the accumulation of low energy electrons, but once evacuated from the acceleration zone, the particles are subsequently cooled by synchrotron and inverse Compton scattering. Therefore, the quasi-mono-energetic assumption is only selfconsistent if these particles escape the source in a time shorter than the radiative cooling time. For the scenario discussed here, where the Thomson optical depth T 1, the dominant cooling process is synchrotron radiation. As it was shown in [22], the ratio of the synchrotron cooling time tcool to the light crossing time R/c is ctco R
ol

d
0 0

ne d3 r

di-1 dnp c dtd

i

Ii-1 hi-1

where is a constant arises from the geometry of the source, typically of the order of unity, and for the model discussed here, = 2/3. (dnp /dtdi ) is the rate of scattering per photon frequency interval from a single electron for the ith generation scattered photons, defined in [20] Eq. (4) as dnp dtdi = 3T c f (y ) 4i-1 2 (10)

and the function f (y ) is defined in Eq. (7) of [20], which takes into account of the Klein-Nishina effect. Assuming the synchrotron emission is isotropic,
SC Ii

= 2.9

13 D10 3 (1 + z )13

1/5 3 s

sin2 R

-1 -2

-1 max,14 GHz (12)

-8/5

= =

dE hi dNp = 2 d dtdi dr 4 dtdi 1 i di-1 T I f (y ) 2 4 2 0 i-1 i-1

1 4 (R/2)2 (11)

where the source size is written as R = R

-2

в 0.01 pc.

We evaluate Eq. 11 numerically over the frequency range 107 1025 Hz, and the results are shown in Fig. 1.

Fig. 1 shows the simultaneous data (black dots) obtained by Ostoreroe et. al. [8], and the theoretical spectra of S5 0716+714. Dashed lines represent the synchrotron and inverse Compton spectra produced by mono-energetic electrons of Lorentz factor 319, with optically thin synchrotron emission between GHz = 4 GHz


and max = 1013.7 Hz. The synchrotron spectrum gives a reasonable fit at radio frequencies, going through the optical points at 1014.5 Hz, although the break in the power law spectrum at 1011.5 Hz cannot be fitted. The first order inverse Compton spectrum gives emission in x-ray frequencies, and the second order inverse Compton spectrum falls outside the Thomson regime and therefore no gamma ray emission is produced. This parameter set is ruled out, however, since the ratio of the synchrotron cooling time to the light crossing time is 1, implying that the relativistic electrons are cooled by scattering before they leave the acceleration region. The shape of the electron spectrum will therefore change from being mono-energetic with (quasi-mono-energetic with a low energy cut-off at) = 0 to one which is -2 , extending to energy lower than 0 me c2 . The solid lines show the synchrotron and inverse Compton spectra produced by electrons of Lorentz factor 349. Here the optically thin spectrum is only between GHz = 4 GHz and max = 400 GHz. The synchrotron spectrum shows good agreement with the data points at radio frequencies. However, the optical points cannot be fitted with parameters that are consistent with the monoenergetic assumption. The first order inverse Compton spectrum gives emission in the x-ray frequencies, and the second order spectrum gives gamma-ray emission up to 0.1 GeV energies, higher orders are then cut off by the Klein-Nishina effect. To summarise, synchrotron radiation from a monoenergetic electron distribution reproduces the very high brightness temperatures observed in extra-galactic variable radio sources, as well as the high degree of circular polarisation. Testable predictions of the theory are a hard radio to infra-red spectrum, x-ray emission and possibly gamma-ray emission in the MeV to GeV range, depending on the specific values of the parameters.

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