On the interaction of jet plasma with dark matter subhaloes in active galaxies

New Astronomy 17 (2012) 362–367

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On the interaction of jet plasma with dark matter subhaloes in active galaxies A.L. Poplavsky Observatory of the Belarusian State University, 4 Nezavisimosti Avenue, Minsk 220050, Belarus

a r t i c l e

i n f o

Article history: Received 13 November 2010 Received in revised form 7 September 2011 Accepted 27 September 2011 Available online 7 October 2011 Communicated by F.D. Macchetto Keywords: Galaxies: jets Galaxies: haloes Dark matter Methods: numerical

a b s t r a c t The interaction of fragmented plasma of active galactic nuclei jets with galactic haloes via gravitational scattering and lensing by dark matter subhaloes is studied using analytical calculations and numerical Monte-Carlo method. The lensing of jet radiation by halo masses is found to be negligible and unobservable. Moving through a galactic halo jet plasma fragments are sequentially deflected on hyperbolic orbits by gravitational field of subhaloes and deviates at some angles when leaving halo, causing widening of the jet. Based on this model jet opening angles are calculated numerically for various values of jet and halo characteristics. Though these angles are very small, gravitational scattering by halo masses results in specific radial profile of jet radiation intensity, that does not depend on halo mass distribution and jet properties. The intensity of jet radiation, obeying the derived profile, decreases by reasonable observable factors giving possibility to probe the presence of dark matter subhaloes. Ó 2011 Elsevier B.V. All rights reserved.

1. Introduction The presence of dark matter was first proposed by Fritz Zwicky in 1934 as an invisible mass affecting the orbital velocities of galaxies in clusters. Later, other observations indicated the presence of dark matter in the Universe, including the rotational curves of galaxies, gravitational lensing of distant sources, and temperature distribution of hot gas in clusters of galaxies. According to the KCDM cosmological model (Bertone et al., 2005; Rich, 2010) and recent observational data (Hinshaw et al., 2009), 0.23 of the mass-energy density of the Universe is in the form of non-baryonic non-relativistic Cold Dark Matter (CDM), that cannot be observed by its electromagnetic radiation, consists of slowly moving particles, almost only takes part in the gravitational interaction, and fills galactic haloes. The most commonly proposed candidates are axions, sterile neutrinos, and Weakly Interacting Massive Particles (WIMPs) (Roos, 2010). Regardless of dark matter composition, some properties of galactic haloes can be established via observations and cosmological simulations. Dubinski and Carlberg (1991) using N-body simulations of gravitational collapse of density peaks pointed out that dark matter haloes had cuspy structure and their overall density profiles could be fitted with power laws. Later investigations showed that dark matter filled haloes of galaxies following Navarro–Frenk– White (NFW) or Einasto density profiles (Navarro et al., 1996; Merrit et al., 2006; Graham et al., 2006; Roos, 2010; Sandick, 2010). Recent results of N-body cosmological simulations (Springel

E-mail address: [email protected] 1384-1076/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.newast.2011.09.007

et al., 2008) together with main concepts of KCDM model suggest that galactic haloes could consist of the great number of subhaloes of different mass as a result of hierarchical clumping scenario (Bertone et al., 2005). However there are some important mismatches between the simulations, theoretical models and the observations: (1) the rotation curves of dark matter haloes should have, in their central regions, sharper rising slopes than it is observed in galaxies (the cuspy halo problem); (2) theory predicts large numbers of small dwarf baryonic satellites of galaxies, which are not observed (the missing satellites problem). Some more investigations are needed to cope with the discrepancies mentioned above. Probably two kinds of observation phenomena could be promising: gravitational lensing by CDM and transient events that occurs in the galactic haloes. Various aspects of gravitational lensing and evidences for CDM in the galactic haloes, observed and modelled by lensing, were considered by Hoekstra et al. (2004), Yoo et al. (2006), Viola et al. (2010) and Jösson et al. (2010). In these papers and in many others the authors investigated lensing of distant sources by haloes of galaxies and clusters. Obviously transient phenomena of galaxies could also be the tools for CDM probing. The important role among them could be played by activity of active galactic nuclei (AGN). Metcalf (2002) studied lensing of AGN relativistic jets and found some fine patterns of lensed images, that could be lensing by subhaloes. In this paper we study the scattering of AGN relativistic jet particles by gravitational field of dark matter subhaloes using Monte-Carlo numerical simulations. We also discuss the influence of scattering and effects of gravitational lensing by subhaloes on the observable structures of jets.

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particle is sequentially deflected on hyperbolic orbits by the gravitational field of subhaloes and deviates every time by a small angle. This is repeated for all of jet particles and causes widening of the jet. Every time a particle is deflected only by a single subhalo. We denote the particle multiple scattering angle, i.e. the total deflection angle after leaving halo, by h. The opening angle of a jet can be characterised by the mean square of multiple scattering angle for ensemble of jet particles. Following formalism of general scattering theory by newtonian gravity centres (Newton, 2002), we have:

hh2 i ¼

Fig. 1. CDM density curves for 1012 M halo: NFW profile (concentration parameter C = 15 Klypin, 2002; Battaglia et al., 2005); Einasto profile (slope a = 0.172, characteristic radius r2 = 20 kpc Merrit et al., 2006; Graham et al., 2006).

Z

h2 dN ¼ 2p

2. Gravitational lensing In this section we analyse lensing of the AGN jets by a dark matter subhaloes. Let I(b) be the intensity distribution in the source plane, so the observed intensity distribution in the lens plane is then

Ið/Þ ¼ I½bð/Þ;

ð1Þ

where b is vector of true angular position of source, / – vector of lensed image position. As / and b are small, the true position of the source and its observed position are related by a so called lens equation (Bartelmann and Schneider, 2001):

b¼/

DS  DL aðDL /Þ; DS

nh2 ðbÞbdb dz;

ð4Þ

where n is number density of subhaloes, b – impact parameter, z – distance along jet propagation. We use the impact parameter – scattering angle dependence derived based on the general classical theory of scattering by gravitational centres:

hðbÞ ¼ p  2cot1 In Section 2 we analyse the possibility of lensing of jet radiation by dark matter in haloes. In Section 3 we calculate opening angles of jets, widened because of gravitational scattering by dark matter subhaloes. We derive spatial and projected distributions of scattered jet plasma in Section 4 and discuss the applicability of them for the observations. When propagating through the ambient medium, jet plasma often becomes highly turbulent and is suffered from different types of instabilities. We use these properties to consider turbulent jet as a stream of independent plasma fragments when developing model of scattering by halo masses. These plasma fragments are hereinafter referred to as jet particles. We use Navarro–Frenk–White (NFW) and Einasto CDM density profiles (Fig. 1) to estimate jet scattering angles and to construct precise numerical model.

ZZ

rg ; b

ð5Þ

where rg = GM/c2, M is the mass of a scatter, G – gravitational constant, c – speed of light. This formula is derived according to (Newton, 2002) for relativistic particles deflecting by Newtonian gravitational field of point or spherically symmetrical mass. As the scattering angles h are small for reasonable impact parameters b  rg, we can expand the last expression and use the first term:

hðbÞ ’ 2

r  g : b

ð6Þ

To find the root mean square of scattering angles we use Einasto’s dark matter density profile. We also need to define limits of impact parameter integration in (4). Let bmin and bmax be the minimal and maximal values of b. Subhaloes are treated as point masses. Very small impact parameters have vanishing statistical probability as bmax  bmin. This lets us disregard the spatial structure of subhaloes. Using expression (4) we have:

hh2 i ¼ 8p

ZZ  2 rg bmax nðzÞbdb dz ¼ 8pr 2g ln Iðzmin ; zmax Þ; b bmin

ð7Þ

where I(zmin, zmax) is integrated subhalo number density n along z coordinate. As I(zmin, zmax) ’ (zmin, 1) when zmax  zmin for the selected dark matter density profile, we need to regard only initial coordinate of jet propagation zmin. To calculate an estimation of hh2i, we assume here that subhaloes have the same average mass

ð2Þ

where a is deflection angle, DL and DS – distances to the lens and the source, respectively. Emphasizing that the following condition is satisfied for the AGN jets: DS  DL  DS,

b ’ /;

ð3Þ

with high order of accuracy. Therefore, the lensing of jet radiation by dark matter subhaloes could not be observed. 3. Gravitational scattering of jet particles by subhaloes 3.1. Analytical estimation Consider multiple scattering of jet plasma fragments by numerous dark matter subhaloes. Moving through a galactic halo a jet

Fig. 2. Plot of analytical dependence of scattering angle rms from initial coordinate of jet propagation. Average masses of scatters (M/M) are free parameters.

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A.L. Poplavsky / New Astronomy 17 (2012) 362–367

M. In addition to this we let ln (bmax/bmin) ’ 10. Such a rough estimate admits sufficiently wide range of impact parameters owing to logarithm. More accurate treatment of impact parameter variation is given in the next sections. Calculated values log (hh2i1/2) are plotted in Fig. 2 as functions of initial distance zmin for different average mass of subhaloes M. 3.2. Numerical model

Table 1 Parameters of Monte-Carlo simulations. Symbol

Parameter

Value or range

Mtot Ns Np mres mlim

Total halo mass Number of subhaloes per one jet particle Number of jet particles Mass spectrum higher cutoff Mass spectrum lower cutoff Lorentz-factor of jet particles Initial distance from the jet to the galactic centre Hubble parameter

1012 M 12  6500 103–106 107 M 1  0.9  107 M 1.0001  3 1  10 kpc

C z0

Consider the same framework of multiple scattering defined in the beginning of the previous subsection. Let r0 be the vector of initial position of jet particles, X1 – solid angle, showing direction of initial motion of the jet. Position vector of a jet particle after the first scattering is:

Note: Value of Hubble parameter is taken from Hinshaw et al. (2009).

r1 ¼ r0 þ l1 X1 ;

The formula for subhalo mass random number based on the mass spectrum (13) is:

ð8Þ

where l1 is mean free path before the first scattering. We introduce Cartesian coordinate system in which z axis is oriented towards initial direction of jet propagation, x and y axes are arbitrary. Therefore

r0 ¼ ð0; 0; z0 Þ and

X1 ¼ ð0; 0; cos #1 Þ:

rk ¼ rk1 þ lk Xk ;

ð9Þ

where lk and Xk – mean free path and direction after the kth scattering in the remote reference frame. As #i are very small angles, transformation from the comoving to the remote reference frame can be found in the following way:

8 2 #k ¼ h2k þ #2k1  2hk #k1 cos vk1 ; > > > > > < sin Duk ¼ #hk sin vk1 ; k

ð10Þ

#2 þ#2k h2k > > cos Duk ¼ k1 ; > 2#k1 #k > > : uk ¼ uk1 þ Duk1 :

Cartesian coordinates of jet particles are calculated in the small angle approximation as well:

8 > < xk ¼ xk1 þ lk hk cos uk ; yk ¼ yk1 þ lk hk sin uk ; >   : zk ¼ zk1 þ lk 1  h2k =2 :

l ¼ hli ln n;

ð12Þ 1/3

where n – uniform random number at [0, 1], hli = n . Number density of subhaloes can be calculated using CDM mass distribution, taken from the results of N-body halo simulations, performed by Springel et al. (2008):

m ;

  mþ1  1 mþ1 m1 M ¼ n mlim  mres ;

ð14Þ

where n is defined above, mlim and mres – minimal and maximal values of mass of CDM subhaloes. mlim – thermal dark matter limit, mres – resolution limit in the notation of indices introduced by Springel et al. (2008), Merrit et al. (2006) and Graham et al. (2006).

Parameters used in the simulations are listed in Table 1. Both NFW and Einasto profiles are used, as NFW dependence is simpler for numerical calculations, but Einasto’s one fits a central cusp better. We use r200 as a virial radius of the halo, i.e. radius at which CDM density exceeds the critical density of the Universe by a factor of 200. Integrating density curves for the described above halo parameters and its mass 1012 M we get: r200 = 110 kpc for NFW profile and r200 = 125 kpc for Einasto’s one. Number of subhaloes per one jet particle in the simulations is kept to match the virial radius of the halo. Simulated jet shapes due to scattering by subhaloes are shown in Figs. 3–5. We can see that massive subhaloes of uniform mass distribution cause larger scattering (Figs. 4 and 5). We also consider scattering of a slow jet with (C  1) = 104. The results are shown in the Fig. 5. In this case scattering angles are the largest. In addition to the root mean square of multiple scattering angle defined earlier we use jet opening angle w and its root mean square wrms, defined in small angle approximation:

ð13Þ

where 5 m ¼ 1:9; a0 ¼ 8:21  107 =M50 ¼ 3:26  105 M1 M 50  ; m0 ¼ 10 7 ¼ 2:52  10 M .

xji ; zji  z0 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uP x 2 jN s u t zjNs z0 ; ¼2 Np

wji ¼

ð15Þ

wrms

ð16Þ

ð11Þ

In the described numerical model lk, vk, and hk are random numbers. Angles vk are uniform values from 0 to 2p. Angles hk are functions of impact parameter b, given by formula (5), b are uniform numbers from rg to n1/3, where n is number density of subhaloes. Mean free path can be found from the following expression:

dN M ¼ a0 dM m0

70.5 km s1 Mpc1

3.3. Results of Monte-Carlo simulations

In the comoving reference frame, particles are deflected after each scattering at two angles of spherical coordinate system: v – the azimuthal angle, and h – the polar one. Both of them are random numbers in the proposed numerical model. The rule for the kth step of iterative procedure is:



H0

where i = 1, . . . , Ns, j = 1, . . . , Np. Simulated data reveal some dependence of jet scattering angles both from Lorentz-factor of particles and subhalo mass lower limit. We find the correlation between these parameters (Fig. 6). We fit both Lorentz-factor (C) – logarithmic rms of jet opening angle (wrms) relationship and logarithmic lower mass (mlim) – logarithmic rms of jet opening angle one with power laws:

log wrms ¼ C m þ Bm 10Am log mlim ¼ 2:63 þ 2:36  100:050 log mlim

ð17Þ

and

log wrms ¼ BC ðC  1ÞAC ¼ 0:939ðC  1Þ0:3137 :

ð18Þ

Estimated variance for both fits are defined as:

P

EV ¼

D2i ; nd  p

ð19Þ

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A.L. Poplavsky / New Astronomy 17 (2012) 362–367

Fig. 3. Plots of simulated jet shape for Ns = 6500, Np = 1000, mlim = 1 M, C = 1.15. Top: xz coordinate plane, where z – the axis along jet propagation (z0 – initial position), x – a transversal coordinate axis. Bottom: xy transversal coordinate plane.

where Di are residuals, nd  p – number of degrees of freedom (nd – the length of dataset, p – number of fit parameters). For logmlim– logwrms relation EV(wrms) = 0.0526, for C –logwrms one EV(wrms) = 0.0719. We see that opening angles are very small for reasonable values of parameters such as mlim and C. This fact seems to prevent from direct observing of jet opening angles due to scattering by probable CDM subhaloes. But we will show in the next subsection that observing radial profiles of jet radiation could be challenging for probing this phenomenon. 3.4. Scattering profiles We can calculate jet intensity outside the wrms cones to check the possibility of detection of scattering effect. The intensity of jet radiation is proportional to the surface density of scattered jet particles. First of all we consider the distribution of surface density at the virial radius r200 of the halo. To do that we introduce normalized intensity of jet radiation (reduced intensity Jr):

J r ¼ 1 þ 9S=Smax ;

ð20Þ

where S is surface density of scattered particles. The relation is presented in Fig. 7. The simulated data are fitted with analytical curve using Nelder–Mead v2 minimization technique. The analytical fit is of the form:

log J r ¼ 

tan w sw

c

a ; þ1

ð21Þ

Fig. 4. Simulated jet shape for Ns = 15, Np = 5000, mlim = 0.9  107 M, C = 1.15. Top: xz coordinate plane (z0 – initial position). Bottom: xy coordinate plane.

where sw = 0.0007995, a = 0.942, c = 0.787. Using these data we can estimate the intensity ratio (I/Imax) at e.g. tanw = 1:

Iðp=4Þ Sðp=4Þ J r ðp=4Þ  1 ¼ 2:44  104 : ¼ ¼ Imax Smax 9

ð22Þ

This value is not negligible and could be observed. Therefore we analyse the form of projected radial jet radiation intensity profiles. First of all we find spatial radial distribution of scattered jet particles for varied sections along propagation axis (so called jet spread function F(q), where q is length of polar radius-vector). Multiple Monte-Carlo simulations show that the best possible fit for the data is asymmetric gaussian in log scale (k = logq):

8 ðklÞ2 > < e 2s2L ; k < l; A FðkÞ ¼  > ðkl2Þ2 sL þ sR : e 2sR ; k P l;

ð23Þ

From the simulations we ascertain that the function (23) is of the same form for different Lorentz factors of jet particles, values of lower mass limit of subhaloes, and sections of lengthwise coordinate (Table 2 and Fig. 8). Changing variables from k to q, taking only right-hand part of the initial asymmetric Gaussian fit, and normalizing it, we get the jet spread function: 

ðlog qÞ2

dk e 2s2 : FðqÞ ¼ FðkÞ ¼ pffiffiffiffiffiffiffi dq qs 2p ln 10

ð24Þ

If the transversal shape of a jet is described with function D(q), where q is polar radius-vector, the distribution of scattered jet particles is expressed with convolution:

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A.L. Poplavsky / New Astronomy 17 (2012) 362–367

Fig. 7. Tangent of opening angle – logarithmic reduced intensity relation (mlim = 0.9  107 M, Ns = 12, Np = 106, (C  1) = 104, initial distance of jet scattering z0 = 104 pc).

Table 2 Parameters of radial jet particle distribution.

l

sL

sR

rres/sR

mlim = 0.9  107 M 30 0.0552 50 0.0490 100 0.0452

1.054 0.474 0.0201

0.391 0.369 0.356

0.648 0.600 0.602

0.00171 0.00313 0.00321

mlim = 1 M 30 50 100

3.331 2.702 2.327

0.456 0.468 0.389

0.488 0.503 0.510

0.00436 0.01543 0.01182

z [kpc]

Fig. 5. Results of simulated jet shape for Ns = 12, Np = 104, mlim = 0.9  107 M, (C  1) = 104. Top: xz coordinate plane. Bottom: xy coordinate plane.

A

0.0354 0.00204 0.0246

Note: A, l, sR, and sL are parameters of asymmetric gaussian fit (23). rres is root mean square of residuals.

PðqÞ ¼

Z

Dðq0 ÞFðq  q0 Þdq0 :

ð25Þ

Considering the case of cylindrical jet, we derive the spatial jet spread function in axial symmetry:

PðqÞ ¼

Z

1

Fðq0 Þq0 dq0 ¼ q

1 1 þ erfða log½q=q0 Þ; 2 2

ð26Þ

where radial coordinate q is counted out from the outer border of jet, q0 is scaling radius, a — slope parameter. Using the spatial profile (25) we can write out projected jet profile as an integral:

P ðyÞ ¼

   1 1 a x2 þ y2 dx; þ erf  log 2 2 q0 1 2

Z

1

ð27Þ

where x coordinate is directed along the line of sight. Therefore, comparing intensity profiles outside the jet edge with derived function P⁄ one can probe the scattering phenomena in galactic haloes. 4. Conclusions

Fig. 6. Top: Lorentz-factor – rms of jet opening angle relation (mlim = 1 M, Ns = 6400, Np = 1000). Bottom: Minimal subhalo mass – rms of jet opening angle relation (C = 1.15, Ns = [15, 65, 150, 300, 660, 1400, 3000, 6400], Np = 5000).

Results of analytical and numerical calculations give one of a very few methods available for probing the CDM in galactic haloes. It is scattering of the AGN jet plasma by gravitational field of subhaloes. Gravitational lensing of jet radiation by dark masses in haloes is not effective as distance to the observer is much larger than that from the deflectors to the sources. Therefore lensing of jets could

A.L. Poplavsky / New Astronomy 17 (2012) 362–367

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loes. Using the same subhalo mass spectrum as derived by Springel et al. (2008), we have demonstrated that wide range of subhalo masses with mass distribution function sharply peaked at low mass has resulted in very narrow opening angle of scattered jet for mlim = 1 M, making such a widening unobservable. While using the model of uniform mass distribution of heavy subhaloes (M 107 M) we have got wider opening angles of jet up to 103 radian for nonrelativistic slow jets (C  1 = 104). In spite of seemingly measurable value of opening angle, in reality the velocities of jets are much higher. Therefore we have demonstrated that even for relativistic velocity of a jet, and for very narrow opening angle, many of jet particles have been scattered outside the cone of root mean square of opening angle. This effect leads to jet radiation intensity drop by a factor of order of thousand at the edge of the cone w = p/4. This decrease is not negligible and together with intensity distribution is quite measurable by means of radio telescopes and space observatories. Derived fits for the simulated radial spatial profiles of scattered jet plasma have the form: [1 + erf (alog (q/q0))]. This function is the same for various jet characteristics and dark mass distribution in haloes. Thus, observing the intensity of radiation outside the edges of turbulent jets and comparing them with calculated profiles one can probe the presence (or the absence) of dark matter mass concentrations. When propagating through the interstellar and intercluster media jets undergo ram pressure and Kelvin–Helmholtz instabilities, so that the scattering by dark matter subhaloes, would be detectable only as a differential effect and only if a proper assessment of the hydrodynamical effects can be carried out. References

Fig. 8. Radial distribution of jet scattered particles. Simulated data are marked with crosses, analytical fit — with solid line. Top: C  1 = 104, z = 30 kpc, mlim = 1 M. Bottom: C  1 = 104, z = 100 kpc, mlim = 0.9  107 M.

not be observed in contrast to lensing of distant sources (such as cosmic microwave background emission). Monte-Carlo simulations of jet particle multiple scattering have been performed for the dark matter halo of 1012 M. The scattering cannot be observed for fast jets with bulk Lorentz-factor C J 3. Root mean square of slow jet (C 1.15 and lower) opening angles strongly depends on the mass distribution of dark matter subha-

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