A&A 384, L27-L30 (2002)
T. A. Enßlin - S. Heinz
Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, Postfach 1317, 85741 Garching, Germany
Received 18 January 2002 / Accepted 5 February 2002
The Chandra X-ray Observatory is finding a surprisingly large number of cavities in the X-ray emitting intracluster medium (ICM), produced by the release of radio plasma from active galactic nuclei. In this letter, we present simple analytic formulae for the evolution of the X-ray deficit and for the radio spectrum of a buoyantly rising bubble. The aim of this work is to provide a theoretical framework for the planning and the analysis of X-ray and radio observations of galaxy clusters. We show that the cluster volume tested for the presence of cavities by X-ray observations is a strongly rising function of the sensitivity.
Key words: radiation mechanisms: thermal - radiation mechanism: non-thermal - galaxies: active - intergalactic medium - galaxies: cluster: general - radio continuum: general
While the early phase of radio galaxy evolution is characterized by supersonic expansion into the surrounding medium, the radio lobes quickly settle into pressure equilibrium with the ICM after the AGN has shut off. Our description sets in at this moment t1, where the bubble is located at a cluster radius r1 with volume and pressure . The bubble will then quickly approach a terminal velocity , governed by the balance of buoyancy and drag forces. During its rise, the bubble volume changes according to the adiabatic law , where the adiabatic index is close to 4/3, which we will take as our fiducial value for numerical examples. The magnetic field strength should evolve according to , if the expansion of the bubble is isotropic. For simplicity, we only consider spherical bubbles with radius , which gives sufficiently accurate estimates for most applications. If the bubble becomes highly deformed or even disintegrates, more sophisticated models than ours will have to be used. We further assume, that entrainment of environmental gas into the bubble is dynamically insignificant on the considered time-scales, implying that the bubble is X-ray dark. Numerical simulations (e.g., Reynolds et al. 2001) support the latter.
It is often convenient to express physical quantities like the bubble volume and the magnetic field strength in terms of the values they would have if the bubble were adiabatically moved to the cluster center. We denote these by the subscript 0 (e.g., ). As our working example, we will investigate a cluster described by an isothermal -profile with a density profile of , pressure , and constant sound speed .
We define the origin of the cluster coordinate system at the cluster center, with the x and y axis defining the image plane and the z-axis the line of sight to the observer, and the coordinate system oriented so that the bubble center is located in the x-z plane at . Its projected distance from the cluster center is , where . The angle should be roughly conserved along the bubble's trajectory in a spherical cluster atmosphere.
The buoyancy speed of the bubble can be estimated from the balance of
buoyancy and drag forces. The buoyancy force
If the initial bubble is large compared to the cluster core Eq. (3) can give supersonic rise velocities. In such a case Eq. (2) is no longer valid. Instead, strongly increased dissipation will limit the rise velocity to the subsonic regime. In such a case one will adopt , with .
The X-ray emissivity at cluster radius r and for a given density and pressure profile is , where n is the electron number density and is the plasma cooling function.
In the following we will assume that the gas surrounding the bubble has essentially settled back into hydrostatic equilibrium in the dark matter potential, which is unperturbed by the radio galaxy activity.
In general, the surface brightness integral must be solved
numerically, but in the case of a -model atmosphere, it can be
expressed in closed form. For an undisturbed line of sight (i.e., not
intersecting a bubble) the observed surface brightness (neglecting
cosmological corrections) is given by
For a line of sight intersecting the surface of a bubble, the intersection
points are located at
Another important diagnostic is the number of missing counts
from the cavities, i.e., the photons emitted from a
spherical region of radius
at distance r from the center
in an unperturbed cluster atmosphere during the exposure interval
For a detector with effective area
source distance of D we define
|Figure 1: Contours of the X-ray deficit significance of a bubble in a galaxy cluster. The contours mark locations at which a bubble has a significance which is lower by powers of 2 than its significance if located at the cluster center (in the lower left corner of each figure, vertical axis is parallel to the line of sight). Left: the bubble volume expands adiabatically (with ) with the pressure of the isothermal cluster ( ). Right: the bubble volume is assumed to be independent of location ( ). In both figures, an X-ray background with 1/100 of the central cluster surface brightness is assumed ( ).|
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In order to design searches for cavities in promising cluster candidates, it is useful to estimate the expected signal to noise ratio ( ) of a potential observation. The signal is the expected number of missing photons from the volume occupied by the bubble, . The background, on top of which the missing photons have to be detected, is the expected number of photons from the area on the sky subtended by the cavity, assuming there were no cavity. This assumes that the instrument point spread function is not more extended than the cavity. The noise is then given by the fluctuation in this photon number, which is .
The signal-to-noise ratio of a small bubble (so that the cluster density
does not vary significantly across its diameter,
), in an isothermal -model is well
If the question of interest is the detectability of cavities of a given fixed size at all cluster radii one can use the incompressible limit of Eqs. (11) and (13). This case is also displayed in Figs. 1 and 2.
|Figure 2: Volume of the cluster [in units of ] within which the signal to noise ratio of the X-ray deficiency of a radio bubble exceeds a minimum, for the detection required value . The latter is given in units of , the signal to noise of a comparable bubble moved to the cluster center. The solid lines are for an adiabatically expanding bubble ( ), and the dashed lines for an incompressible bubble ( ). The background to central cluster X-ray surface brightness ratio takes the values , 0.001, 0.01, and 0.1 from top to bottom. The (top) curves with are given by Eq. (13). We adopted .|
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Synchrotron, inverse Compton, and adiabatic losses cool an electron with
initial dimensionless momentum
If a bubble is rising with a constant velocity (e.g. because it approaches the sound velocity) the following parameters give the correct q value: , , , and .
An initially relativistic power-law electron spectrum
|Figure 3: Bubble's central X-ray contrast (compared to the undisturbed cluster) for various angles between plane of sky and Bubble's trajectory, its radio flux, and its rising time as a function of the (unprojected) radial position. Adopted parameters: kpc, kms-1, , r1 = 10 kpc, kpc, G, , s = 2.4, Jy.|
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As an aid and stimulus for future observations we provided simple analytic formulae of X-ray and radio properties of rising buoyant bubbles of radio plasma in galaxy clusters. The detectability of a bubble decreases with its cluster radius. The X-ray contrast of a bubble moving in the plane of the sky decreases slowly until the X-ray background dominates, then it drops rapidly. The contrast of a bubble moving along the line of sight declines quickly outside the cluster core. Similarly, the radio luminosity of the bubble at a given frequency declines rapidly with increasing cluster radius and suddenly vanishes when cooling has removed the emitting electrons. After that point, only a weak flux of synchrotron-self Comptonized emission remains (Enßlin & Sunyaev 2002). Figure 3 illustrates these dependencies for an example with parameters similar to Perseus A.
We acknowledge useful comments by Eugene Churazov, Federica Govoni, Daniel E. Harris, Gopal-Krishna, Brian R. McNamara, and the referee Paul E. J. Nulsen.