3. Evolution of the old radio lobes
The complex substructure of the X-ray emission in the Perseus cluster is seen at various spatial scales. At large scales (larger than - ), excess emission to the east of NGC 1275 was observed in the HEAO-2 IPC and ROSAT images (Branduardi-Raymont et al. 1981, Fabian et al. 1981, Schwarz et al. 1992, Ettori, Fabian, White 1999). Schwarz et al. (1992), using ROSAT PSPC data, found that the temperature is lower in this region and suggested that there is a subcluster projected on the A426 cluster and merging with the main cluster. At much smaller scales (), there are two X-ray minima (symmetrically located to the north and south of NGC 1275) which Böhringer et al. (1993) explained as due to the displacement of the X-ray emitting gas by the high pressure of the radio emitting plasma associated with the radio lobes around NGC 1275. As is clear from Figs. 1,2 at intermediate scales (arcminutes), substructure is also present. We concentrate below on the possibility that at these spatial scales, the disturbed X-ray surface brightness distribution is affected by the bubbles of radio emitting plasma, created by the jets in the past and moving away from the center due to buoyancy.
Recently Heinz, Reynolds & Begelman (1998) argued that the time-averaged power of the jets in NGC 1275 exceeds . This conclusion is based on the observed properties (in particular - sharp boundaries) of the X-ray cavities in the central , presumably inflated by the relativistic particles of the jet. Such a high power input is comparable to the total X-ray luminosity of the central region (i.e. 200 kpc) around NGC 1275. If the same power is sustained for a long time (e.g. cooling time of the gas years at a radius of 200 kpc) then the entire cooling flow region could be affected. Following Gull and Northover (1973) we assume that buoyancy (i.e. Rayleigh-Taylor instability) limits the growth of the cavities inflated by the jets. After the velocity of rise due to buoyancy exceeds the expansion velocity, the bubble detaches from the jet and begins rising. As we estimate below, for the jet power of the bubble at the time of separation from the jet should have a size 10-20 kpc (). The subsequent evolution of the bubble may resemble the evolution of a powerful atmospheric explosion or a large gas bubble rising in a liquid (e.g. Walters and Davison 1963, Onufriev 1967, Zhidov et al. 1977). If the magnetic field does not provide effective surface tension to preserve the quasi-spherical shape of the bubble, then it quickly transforms into a torus and mixes with the ambient cooling flow gas. The torus keeps rising until it reaches the distance from the center where its density (accounting for adiabatic expansion) is equal to the density of the ambient gas. Since the entropy of the ICM rises with distance from the center in the cooling flow region, the torus is unlikely to travel a very large distance from the center. Then the torus extends in the lateral direction in order to occupy the layer having a similar mass density. Below we give order of magnitude estimates characterizing the formation and evolution of the bubble.
For simplicity we assume a uniform ICM in the cluster center, characterised by the density and pressure . The bubble is assumed to be spherical. During the initial phase (Scheuer 1974, Heinz et al., 1998) jets with a power L inflate the cocoon with relativistic plasma and surrounded by a shell of the compressed ICM. The expansion is supersonic and from dimensional arguments it follows that the radius of the bubble r as a function of time t is given by the expression
where is a numerical constant (see e.g. Heinz et al., 1998 for a more detailed treatment). At a later stage, expansion slows and becomes subsonic. The evolution of the bubble radius is then given by the expression
where is the adiabatic index of the relativistic gas in the bubble (i.e. ). The above equation follows from the energy conservation law, if we equate the power of the jet with the change of internal energy plus the work done by the expanding gas at constant pressure: . The expansion velocity is then simply the time derivative of Eqs. (1) or (2).
where is a numerical constant of order unity, is the mass density of the relativistic gas in the bubble, g is the gravitational acceleration, R is the distance of the bubble from the cluster center, M is the gravitating mass within this radius and, is the Keplerian velocity at this radius. In Eq. (3) we assumed that and therefore replaced the factor (Atwood number) with unity. The presently observed configuration of the bubbles on either side of NGC 1275 suggests that . Assuming that such a similar relation is approximately satisfied during the subsequent expansion phase of the bubble we can further drop the factor in Eq. (3). Thus as a crude estimate we can assume that ( is a commonly accepted value for incompressible fluids). Following Ettori, Fabian, and White (1999) we estimate the Keplerian velocity taking the gravitating mass profile as a sum of the Navarro, Frenk and White (1995) profile for the cluster and a de Vaucouleurs (1948) profile for the galaxy. For the range of parameters considered in Ettori, Fabian, White (1999), the Keplerian velocity between a few kpc and 100 kpc falls in the range 600-900 km/s. We can now equate the expansion velocity (using Eq. (2) for subsonic expansion) and the velocity due to buoyancy in order to estimate the parameters of the bubble when it starts rising:
Here and are the duration of the expansion phase and the radius of the bubble respectively. In the above equation we neglected the contribution to the radius (and time) of the initial supersonic expansion phase. Thus for and for (Böhringer et al. 1993) we expect kpc, which approximately corresponds to the size of the X-ray cavities reported by Böhringer et al. (1993). If, as suggested by Heinz et al. (1998), the jet power is larger than then the bubble size will exceed 50 kpc () before the buoyancy velocity exceeds the expansion velocity. Of course these estimates of the expanding bubble are based on many simplifying assumptions (e.g. constant pressure assumption in Eq. (2)). In a subsequent publication we consider the expansion of the bubble in more realistic density and temperature profiles expected in cluster cooling flows.
According to e.g. Walters and Davison (1963), Onufriev (1967), Zhidov et al. (1977), a large bubble of light gas rising through much heavier gas under a buoyancy force will quickly transform into a rotating torus, which consists of a mixture of smaller bubbles of heavier and lighter gases. This transformation occurs on times scales of the Rayleigh-Taylor instability (i.e. ) and during this transformation the whole bubble changes its distance from the center by an amount . The torus then rises until its average mass density is equal to the mass density of the ambient gas. The rise is accompanied by adiabatic expansion and further mixing with the ambient gas. Accounting for adiabatic expansion the mass density of the torus will change during the rise according to
where is the ICM pressure at a given distance from the center, is volume fraction of the ambient ICM gas mixed with the relativistic plasma at the stage of torus formation, and are the adiabatic indices of the relativistic plasma and the ICM. Note that in Eq. (6) we (i) neglected further mixing with the ICM during the rise of the torus and (ii) mixing was assumed to be macroscopic (i.e. separate bubbles of the relativistic plasma and ICM occupy the volume of the torus). The equilibrium position of the torus can be found if we equate the torus density and the ICM density and solve this equation for R. We consider two possibilities here. One possibility is to assume that in the inhomogeneous cooling flow, the hot phase is almost isothermal and gives the dominant contribution to the density of the gas. Adopting the temperature of keV for the hot phase and using the same gravitational potential as above, one can conclude that if a roughly equal amount (by volume) of the relativistic plasma and the ambient gas are mixed (i.e. ), during the formation of the torus, then it could rise 100-200 kpc before reaching an equilibrium position. Accounting for additional mixing will lower this estimate. Alternatively we can adopt the model of a uniform ICM with the temperature declining towards the center (e.g. temperature is decreasing from 6 keV at 200 kpc to 2 keV at 10 kpc). Then for the same value of mixing () the equilibrium position will be at the distance of 60 kpc from NGC 1275. Once at this distance the torus as a whole will be in equilibrium and it will further expand laterally in order to occupy the equipotential surface at which the density of the ambient gas is equal to the torus density. If the cosmic rays and thermal gas within the torus are uniformly mixed (or a magnetic field binds the blobs of thermal plasma and cosmic rays), the torus will not move radially. If, on the contrary, separate (and unbound) blobs of relativistic plasma exist then they will still be buoyant, but since their size is now much smaller than the distance from the cluster center the velocity of their rise will be much smaller than the Keplerian velocity. Analogously overdense blobs (with uplifted gas) may then (slowly) fall back to the center.
We now consider how radio and X-ray emission from the torus evolve with time. Duration of the rise phase of the torus will be at least several times longer than the time of the bubble formation (see Eq. (4)), since the velocity of rise is a fraction of the Keplerian velocity (see e.g. Zhidov et al., 1977), i.e., years. Adiabatic expansion and change of the transverse size of the torus in the spherical potential tend to further increase this estimate. Even if we neglect energy losses of the relativistic electrons due to adiabatic expansion we can estimate an upper limit on the electron lifetime due to synchrotron and inverse Compton (IC) losses.
where is the wavelength of the observed radio emission, B is the strength of the magnetic field, is the value characterizing the total energy density of the magnetic field and cosmic microwave background. This life time (of the electrons emitting at a given frequency) will be longest if the energy density of the magnetic field approximately matches the energy density of the microwave background, i.e., . Then the maximum life time of the electrons producing synchrotron radiation at 20 cm is years. This time is comparable to the time needed for the torus to reach its final position. Therefore, the torus could be either radio bright or radio dim during its evolution. If no reacceleration takes place, then the torus will end up as a radio dim region. We note here that, although the electrons may lose their energy via synchrotron and IC emission, the magnetic field and especially relativistic ions have a much longer lifetime (e.g. Soker & Sarazin 1990, Tribble 1993) and will provide pressure support at all stages of the torus evolution.
As we assumed above, the bubble detaches from the jet when the expansion velocity of the bubble is already subsonic. This means that there will be no strongly compressed shell surrounding the bubble and the emission measure along the line of sight going through the center of the bubble will be smaller than that for the undisturbed ICM, i.e., at the moment of detachment the bubble appears as an X-ray dim region. The X-ray brightness of the torus during final stages of evolution (when the torus has the same mass density as the ambient ICM) depends on how the relativistic plasma is mixed with the ambient gas (Böhringer et al. 1995). If mixing is microscopic (i.e. relativistic and thermal particles are uniformly mixed over the torus volume on spatial scales comparable with the mean free path) then the emission measure of the torus is the same as that for a similar region of the undisturbed ICM. Since part of the pressure support in the torus is provided by magnetic field and cosmic rays then the temperature of the torus gas must be lower than the temperature of the ambient gas (Böhringer et al. 1995). Thus emission from the torus will be softer than the emission from the ambient gas.
If, on the contrary, mixing is macroscopic (i.e. separate bubbles of relativistic and thermal plasma occupy the volume of the torus), then the torus will appear as an X-ray bright region (the average density is the same as of ICM, but only a fraction of the torus volume is occupied by the thermal plasma). For example, if half of the torus volume is occupied by the bubbles of the relativistic plasma then the emissivity of the torus will be a factor of 2 larger than that of the ambient gas. The X-ray emission of the torus is again expected to be softer than the emission of the ambient gas for two reasons (i) gas uplifted from the central region has lower entropy than the ambient gas and therefore will have lower temperature when maintaining pressure equilibrium with the ambient gas (ii) gas, uplifted from the central region, can be multiphase with stronger density contrasts between phases than the ambient gas and as a result a dense, cooler phase would give a strong contribution to the soft emission. Cosmic rays may heat the gas, but at least for the relativistic ions, the time scale for energy transfer is very long (comparable to the Hubble time). Trailing the torus could be the filaments of cooling flow gas dragged by the rising torus in a similar fashion as the rising (and rotating) torus after an atmospheric explosion drags the air in the form of a skirt.
We note here that the morphology predicted by such a picture is very similar to the morphology of the "ear-like" feature in the radio map of M87, reported by Böhringer et al. (1995). The "ear" could be a torus viewed from the side. The excess X-ray emission trailing the radio feature could then be due to the cooling flow gas uplifted by the torus from the central region. For Perseus the X-ray underluminous region to the North-West of NGC 1275 could have the same origin (i.e. rising torus). In fact, the whole "spiral" structure seen in Fig. 2 could be the remains of one very large bubble (e.g. with the initial size of the order of arminutes - corresponding to a total jet power of ) inflated by the nucleus over a period of years. Alternatively multiple smaller bubbles, produced at different periods may contribute to the formation of the X-ray feature. If the jets maintain their direction over a long time then a quasi-continuous flow of bubbles will tend to mix the ICM in these directions uplifting the gas from the central region to larger distances. If the jet direction varies (e.g. precession of the jet on a timescale of years) then a complex pattern of disturbed X-ray and radio features may develop.
© European Southern Observatory (ESO) 2000
Online publication: April 17, 2000