Relativistic electrons and magnetic field of the M87 jet on ten Schwarzschild radii scale
We explore energy densities of magnetic field and relativistic electrons in the M87 jet. Since the radio core at the jet base is identical to the optically thick surface against synchrotron self absorption (SSA), the observing frequency is identical to the SSA turnover frequency. As a first step, we assume the radio core as a simple uniform sphere geometry. Using the observed angular size of the radio core measured by the Very Long Baseline Array at 43 GHz, we estimate the energy densities of magnetic field () and relativistic electrons () based on the standard SSA formula. Imposing the condition that the Poynting power and relativistic electron one should be smaller than the total power of the jet, we find that (i) the allowed range of the magnetic field strength () is , and that (ii) holds. The uncertainty of comes from the strong dependence on the angular size of the radio core and the minimum Lorentz factor of non-thermal electrons () in the core. It is still open that the resultant energetics is consistent with either the magnetohydrodynamic jet or with kinetic power dominated jet even on Schwarzschild radii scale.
Subject headings:galaxies: active — galaxies: jets — radio continuum: galaxies
Formation mechanism of relativistic jets in active galactic nuclei (AGNs) remains as a longstanding unresolved problem in astrophysics. Although the importance of magnetic field energy density () and relativistic electron one () for resolving the formation mechanism has been emphasized (e.g., Blandford and Rees 1978), it is not observationally clear whether either or is dominant at the jet base. Relativistic magnetohydrodynamics models for relativistic jets generally assume highly magnetized plasma at the jet base (e.g., Koide et al. 2002; Vlahakis and Konigl 2003; McKinney and Gammie 2004; Krolik et al. 2005; McKinney 2006; Komissarov et al. 2007; Tchekhovskoy et al. 2011; Toma and Takahara 2013; Nakamura and Asada 2013), while an alternative model assumes a pair plasma dominated “fireball”-like state at the jet base (e.g., Iwamoto and Takahara 2002; Asano and Takahara 2009 and reference therein). Although deviation from equi-partition (i.e., ) is essential for investigation of relativistic jet formation, none has succeeded in obtaining a robust estimation of at the jet base.
M87, a nearby giant radio galaxy located at a distance of (Jordan et al. 2005), hosts one of the most massive super massive black hole (e.g., Macchetto et al. 1997; Gebhardt and Thomas 2009; Walsh et al. 2013). Because of the largeness of the angular size of its central black hole, M87 is well known as the best source for imaging the deepest part of the jet base (e.g., Junor et al. 1999). Furthermore, M87 has been well studied at wavelengths from radio to Very High Energy (VHE) -ray (Abramowski et al. 2012; Hada et al. 2012 and reference therein) and causality arguments based on VHE -ray outburst in February 2008 indicate that the VHE emission region is less than where is the relativistic Doppler factor (Acciari et al. 2009). The Very-Long-Baseline-Array (VLBA) beam resolution at 43 GHz typically attains about which is equivalent to . When holds (Gebhardt et al. 2009), then VLBA beam resolution approximately corresponds to . Recent progresses of Very-Long-Baseline-Interferometry (VLBI) observations have revealed the inner jet structure, i.e., frequency and core-size relation, and distance and core-size relation down to close to Schwarzschild radii () scale (Hada et al. 2011, hereafter H11). Thus, the jet base of M87 is the best laboratory for investigations of in the real vicinity of the central engine.
Two significant forward steps are recently obtained in M87 observations which motivate the present work. First, Hada et al. (2011) succeeded in directly measuring core-shift phenomenon at the jet base of M87 at 2, 5, 8, 15, 24 and 43 GHz. The radio core position at each frequency has been obtained by the astrometric observation (H11). Since the radio core surface corresponds to the optically-thick surface at each frequency, the synchrotron-self-absorption (SSA) turnover frequency is identical to the observing frequency itself. 111 Difficulties for applying the basic SSA model to real sources has been already recognized by several authors ( Kellermann and Pauliny-Toth 1969; Burbidge et al. 1974; Jones et al. 1974a, 1974b; Blandford and Rees 1978; Marscher 1987) due to insufficiently accurate determination of and . Second, we recently measure core sizes in Hada et al. (2013a) (hereafter H13). Hereafter we focus on the radio core at 43 GHz. In H13, we select VLBA data observed after 2009 with sufficiently good qualities (all 10 stations participated and good uv-coverages). To measure the width of the core, a single, full-width-half-maximum (FWHM) Gaussian is fitted for the observed radio core at 43 GHz in the perpendicular direction to the jet axis and we derive the width of the core (). We stress that the core width is free from the uncertainty of viewing angle. Therefore, using at 43 GHz, we can estimate values of in the 43 GHz core of M87 for the first time.
In section 2, we derive an explicit form of by using the standard formulae of synchrotron absorption processes. As a first step, we simplify a geometry of the radio core as a single uniform sphere although the real geometry is probably more complicated. In section 3, we estimate in the M87 jet base by using the VLBA data at 43 GHz obtained in H13. In section 4, we summarize the result and discuss relevant implications. In this work, we define the radio spectral index as and we assume .
Here, we derive explicit expressions of the strength of total magnetic field and . Several papers have extensively discussed the determination of magnetic field strength . Fundamental formulae of SSA processes are shown in the following references and we follow them (Ginzburg and Syrovatskii 1965, hereafter GS65; Blumenthal and Gould 1970, hereafter BG70; Pacholczyk 1970, Rybicki and Lightman 1979, hereafter RL79). Here, we will show a simple derivation of the explicit expressions of and with sufficient accuracy.
For clarity, we briefly summarize the method for determining and in advance. The theoretical unknowns related to the magnetic field and relativistic electrons in the observed radio core with its angular diameter are following four; , (the normalization factor of non-thermal electron number density), (the minimum Lorentz factor of non-thermal electrons) 222The maximum Lorentz factor of non-thermal electrons () is not used since the case of is considered in this work based on the ALMA observation (Doi et al. 2013)., and (the spectral index of non-thermal electrons). Among them, , and are directly constrained by radio observations at mm/sub-mm wavebands. The remaining and can be solved by using the two general relations which hold at shown in Eqs. (3) and (4). The solved and are written as functions of , , , and the observed flux at .
Lastly, we further impose total jet power constraint not to overproduce Poynting- or kinetic-power shown in Eq. (3.3). This constraint can partially exclude larger value of . Then, we can determine and of M87 consistently.
Following assumptions are adopted in this work:
We assume uniform and isotropic distribution of relativistic electrons and magnetic fields in the emission region. For M87, polarized flux does not seem very large. Therefore, we assume isotropic tangled magnetic field in this work. Hereafter, we denote as the magnetic field strength perpendicular to the direction of electron motion. Then, the total field strength is
Hereafter, we define .
We assume the emission region is spherical with its radius measured in the comoving frame. The radius is defined as
where is the angular diameter distance to a source (e.g., Weinberg 1972). Because M87 is the very low redshift source, we only use throughout this paper. There might be a slight difference between and . VLBI measured is conventionally treated as , while Marscher (1983) pointed out a deviation expressed as which is caused by a forcible fitting of Gaussians to a non-Gaussian component. In this work, we introduce a factor defined as and is assumed.
We stress that the uniform and isotropic sphere model is a first step simplification and the realistic jet base probably contains more complicated geometry and nonuniform distributions in magnetic field and electron density. We will investigate such complicated cases in the future.
2.3. Synchrotron emissions and absorptions
In order to obtain explicit expression of and in terms of , , and , here we briefly review synchrotron emissions and absorptions. At the radio core, becomes an order of unity at ;
where and are the optical depth for SSA and the absorption coefficient for SSA, respectively. We impose that optically thin emission formula is still applicable at :
where and are he emissivity and flux per unit frequency, respectively. Combining Eq. 4 and the approximation of , we can solve and . This derivation is much simpler than previous studies of Marscher (1983) and Hirotani (2005) (hereafter H05). We will compare the derived in this work, Marscher (1983) and H05, and they will coincide with each other within a small difference in the range of .
Next, let us break down relevant physical quantities. The term , the normalization factor of electron number density distribution , is defined as (e.g., Eq.3.26 in GS65)
where , , and are the electron energy, the spectral index, minimum energy, and maximum energy of relativistic (non-thermal) electrons, respectively. Let us further review optically thin synchrotron emissions. The maximum in the spectrum of synchrotron radiation from an electron occurs at the frequency: (Eq. 2.23 in GS65)
Synchrotron self-absorption coefficient measured in the comoving frame is given by (Eqs. 4.18 and 4.19 in GS65; Eq. 6.53 in RL79)
where the numerical coefficient is expressed by using the gamma-functions as follows; . For convenience, we define .
Optically thin synchrotron emissivity per unit frequency from uniform emitting region is given by (Eqs. 4.59 and 4.60 in BG70; see also Eqs. 3.28, 3.31 and 3.32 in GS65)
where the numerical coefficient is . For convenience, we define .
2.4. Relations between quantities measured in source and observer frames
Let us summarize the Lorentz transformations and cosmological effect using the Doppler factor () where is the angle between the jet and our line-of-sight) and the redshift (). Hereafter, we put subscript (obs) for quantities measured at observer frame.
The observed flux from an optically thin source at a large distance is given by (Eq. 1.13 in RL79; Eqs. (7) in H05; see also Eq. C4 in Begelman et al. 1984):
2.5. Obtained and
Combining the above shown relations, we finally obtain
where the numerical value of are shown in Table 1. In the Table, we also note the values the obtained with the ones in Marscher (1983) and H05. From this, we see that the derived in this work coincide with each other within the small difference.
where . The cgs units of and depend on : ergcm. The numerical values of are summarized in Table 1 and they are similar to the ones in Marscher (1983) which are and . Using the obtained , we can evaluate as
Then, we can obtain the ratio explicitly as
From this, we find that and have the same dependence on . Using this relation, we can estimate without minimum energy (equipartition field) assumption. It is clear that the measurement of is crucial for determining . We argue details on it in the next subsection. It is also evident that a careful treatment of is crucial for determining (Kino et al. 2002; Kino and Takahara 2004).
3. Application to M87
Based on recent VLBA observations of M87 at 43 GHz, we derive at the base of M87 jet. Here, we set typical index of electrons as (Doi et al. 2013).
3.1. Electrons emitting 43 GHz radio waves
3.2. Normalized physical quantities
For convenience, we rewrite above quantities to normalized quantitates associated with the observed 43 GHz core. field inside the 43 GHz core is estimated as
Here we use as a normalization of . As mentioned in the Introduction, holds at the 43 GHz core surface, because of the clear detection of the core shift phenomena in H11. As for , we obtain
Then, we finally obtain as
This typical apparently shows the order of unity but it has strong dependences on and . Regarding , an uncertainty only comes from the bandwidth. In our VLBA observation, the bandwidth is with its central frequency 43.212 GHz. Therefore, it causes only a very small uncertainty . The accuracy of flux calibration of VLBA can be conservatively estimated as . An intrinsic flux of the radio core at 43 GHz also fluctuates with an order of during a quiescent phase (e.g., Acciari et al. 2009; Hada et al. 2012). Therefore, the flux term also causes an uncertainty of . The angular size and have much larger ambiguities than those evaluated above and we derive by taking into account of these ambiguities in the next sub-section. At the same time, we again emphasize that obtained in H13 can be bearable for the estimate of in spite of such strong dependence on .
3.3. On , , and
The most important quantity for the estimate of is . Based on VLBA observation data with sufficiently good qualities, here we set
where we use the average value from H13 and maximum of is . We note that the measured core’s FWHM overlaps with the measured width of the jet (length between the jet limb-structure) in H13. Therefore, we consider more likely for the M87 jet base. From Eq. (15), the maximal value of is given by when .
Regarding the value of , a simultaneous observation of the spectrum measurement at sub-mm wavelength range is crucial, since most of the observed fluxes at sub-mm range come from the innermost part of the jet. It has been indeed measured by Doi et al. (2013) by conducting a quasi-simultaneous multi-frequency observation with the Atacama Large Millimeter/submillimeter Array (ALMA) observation (in cycle 0 slot) and it is robust that at where synchrotron emission becomes optically-thin against SSA. Maximally taking uncertainties into account, we set the allowed range of as
in this work.
We further impose the condition that time-averaged total jet power () inferred from its large-scale jet properties should not be exceeded by the kinetic power of relativistic electrons () and Poynting power () at the 43 GHz core
where at large-scale is estimated maximally a few (e.g., Reynolds et al. 1996; Bicknell and Begelman 1996; Owen et al. 2000; Stawarz et al. 2006; Rieger and Aharonian 2012). Hereafter, we conservatively assume and a slight deviation from this does not influence the main results in this work. Regarding in the M87 jet, we set
Here we include an uncertainty due to the deviation from time-averaged at large-scale which may attribute to flaring phenomena at the jet base. X-ray light curve at the M87 core over 10 years showed a flux variation by a factor of several except for exceptionally high X-ray flux during giant VHE flares happened in 2008 and 2010 (Fig. 1 in Abramowski et al. 2012). Based on it, we set the largest jet kinetic power case as .
3.3.1 On jet speed
Jet speed in the vicinity of M87’s central black hole is quite an issue. Ly et al. (2007) and Kovalev et al. (2007) show sub-luminal speed proper motions of the M87 jet base. The recent study by Asada et al. (2014) also support it. Hada (2013b) also explores the proper motion near the jet base with the VERA (VLBI Exploration of Radio Astrometry) array. The VERA observation has been partly performed in the GENJI programme (Gamma-ray Emitting Notable AGN Monitoring with Japanese VLBI) aiming for densely-sampled monitoring of bright AGN jets (see Nagai et al. 2013 for details) and the observational data obtained by VERA also shows a sub-luminal motion at the jet base. Furthermore, Acciari et al. (2009) report that the 43 GHz core is stationary within based on their phase-reference observation at 43 GHz. Therefore, currently there is no clear observational support of super-luminal motion within the 43 GHz radio core. The brightness temperature (e.g., Lähteenmäki et al. 1999; Doi et al. 2006) of the 43 GHz radio core is evaluated as
which is below the critical temperature limited by inverse-Compton catastrophe process (Kellermann and Pauliny-Toth 1969). Because of these two reasons, we assume throughout this paper.
Here, we examine the three cases of electron indices as 2.5, 3.0, and 3.5 against the two cases of the jet power as and .
4.1. Allowed B strength
First of all, it should be noted that is primarily determined by a value of since is exactly identical to the observing frequency. By combining Eqs. (16) (3.3), and (22), we obtain the allowed range of magnetic field strength in the 43 GHz core. We summarize the obtained maximum and minimum values of in Table 2. An upper limit of is governed by the constraint of . From Eqs. (11) and (3.3), it is clear that behaves as
apart from a weak dependence on originating in . We thus obtain the allowed range in the 43 GHz core. This is a robust constraint on the M87 core’s B strength.
4.2. Allowed with
In Fig. 2, we show the allowed region in and plane (the red-colored boxed region) and the corresponding values with and which is based on the power law index measured by ALMA (Doi et al. 2013). The larger leads to smaller because becomes smaller for larger . Similar to the aforementioned constraints, the lower limit of is bounded by , while the upper limit is governed by the lowest value of . From this, we conclude that both and can be possible. We again stress that the field strength has one-to-one correspondence to . In other words, acculate determination of is definitely important for the estimate of . By the energetic constraint shown in Eq. (3.3), the maximum becomes smaller than . In this case, we obtain
The maximum is derived from . Then, the factor makes the allowed range broaden. Independent of this factor, has uncertainty about the factor of . These factors govern the overall allowed range of the order of a few times which is presented in Table 3. Additionally, we note that the right-top part is dropped out according to Eq. (15). This changes minimum values of by a factor of a few.
Additionally, we show the allowed and with and in Tables 2 and 3. Compared with the case in Fig. 2, the upper limit of becomes smaller. Then the allowed becomes
The maximum is also derived from . The decrease of the maximum value leads to the increase of correspondingly.
4.3. Allowed with
In Fig. 3, we show the allowed region in and plane (the red-colored boxed region) and the corresponding values with and . Compared with the case with , the region increases in the allowed parameter range according to the relation of . The allowed in this case is
which remains the same as Eq. (19) because both and do not exceed in this case. The factor leads to uncertainty while factor has uncertainty. Therefore, the allowed in this case has a few times of uncertainty which is shown in Table 3. The left-bottom part is slightly deficit because of .
In Tables 2 and 3, we show the allowed and with and . In this case, the allowed is
The relation of leads to the value of maximum . This upper and lower are governed in the same way as in in Fig. 3.
4.3.1 Allowed with
In Fig. 4, we show the allowed region in and plane (the red-colored boxed region) and the corresponding values with and . In this case, the allowed is
The relation of determines the maximum . The allowed is in the narrow range of . It should be stressed that this case shows the magnetic field energy dominande in all of the allowed ranges.
In Tables 2 and 3, we show the allowed and with and . Basic behavior is similar to the case shown in Fig. 4. In this case, the allowed is
The relation of determines the maximum . Corresponding to the narrow allowed range of , the allowed field strength resides in the narrow range of .
5. Summary and discussions
Based on VLBA observation data at 43 GHz, we explore at the base of the M87 jet. We apply the standard theory of synchrotron radiation to the 43 GHz radio core together with the assumption of a simple uniform sphere geometry. We impose the condition that the Poynting and relativistic electron kinetic power should be smaller than the total power of the jet. Obtained values of and are summarized in Tables 2 and 3 and we find the followings;
We obtain the allowed range of magnetic field strength in the 43 GHz core as in the observed radio core at 43 GHz with its diameter . Our estimate of is basically close to the previous estimate in the literature (e.g., Neronov and Aharonian 2007), although fewer assumptions have been made in this work. We add to note that even if of the 43 GHz core becomes larger than unity, the field strength only changes according to .
It is worth to compare these values with independently estimated in previous works more carefully. Abdo et al. (2009) has estimated Poynting power and kinetic power of the jet by the model fitting of the observed broad band spectrum and derive with , although they do not properly include SSA effect. Acciari et al. (2009) predict field strength based on the synchrotron cooling argument. Since smaller values of lead to smaller , if we assume by a factor of than the true , the predicted lies between 0.05 and 0.5 gauss which seems to be in a good agreement with previous work. However, for such a small core, electron kinetic power much exceeds the observed jet power.
Our result excludes a strong magnetic field such as which is frequently assumed in previous works in order to activate Blandford-Znajek process (Blandford and Znajek 1977; Thorne et al. 1986; Boldt & Loewenstein 2000). Although M87 has been a prime target for testing relativistic MHD jet simulation studies powered by black-hole spin energy, our result throw out the caveat that the maximum , one of the critical parameters in relativistic MHD jets model, should be smaller than G for M87.
We obtain the allowed region of in the allowed and plane. The resultant contains both the region of and . It is found that the allowed range is . The uncertainty of is caused by the strong dependence on and . Our result gives an important constraint against relativistic MHD models in which they postulate very large at a jet-base (e.g., Vlahakis and Konigel 2003; Komissarow et al. 2007, 2009; Tchekhovskoy et al. 2011). To realize sufficiently magnetic dominated jet such as , relatively large of the order of and a relatively large are required. Thus, the obtained in this work gives a new constraint on the initial conditions in relativistic MHD models.
Last, we shortly note key future works.
Observationally, it is crucial to obtain resolved images of the radio cores at 43 GHz with space/sub-mm VLBI which would clarify whether there is a sub-structure or not inside Rs scale at the M87 jet base. Towards this observational final goal, as a first step, it is important to explore physical relations between the results of the present work and observational data at higher frequencies such as 86 GHz and 230 GHz (e.g., Krichbaum et al. 2005; Krichbaum et al. 2006; Doeleman et al. 2012). Indeed, we conduct a new observation of M87 with VLBA and the Green Bank Telescope at 86 GHz and we will explore this issue using the new data. Space-VLBI program also could play a key role since lower frequency observation can attain higher dynamic range images with a high resolution (e.g., Dodson et al. 2006; Asada et al. 2009; Takahashi and Mineshige 2010; Dodson et al. 2013). If more compact regions inside the 0.11mas region are found by space-VLBI in the future, then in the compact regions are larger than the ones shown in the present work.
Theoretically, we leave following issues as our future work. (1) Constraining plasma composition (i.e, electron/proton ratio) is one of the most important issue in AGN jet physics (Reynolds et al. 1996; Kino et al. 2012) and we will study it in the future. Roughly saying, inclusion of proton powers () will simply reduce the upper limit of because would hold. (2) On scale, general relativistic (GR) effects can be important and they will induce non-spherical geometry. If there is a Kerr black hole at its jet base, for example following GR-related phenomena may happen; (i) magneto-spin effect which aligns a jet-base along black hole spin, and it leads to asymmetric geometry (McKinney et al. 2013). (ii) the accretion disk might be warped by Bardeen and Peterson effect caused by the frame dragging effect (Bardeen and Peterson 1975; Hatchett et al. 1981). Although a recent research by Dexter et al. (2012) suggests that the core emission is not dominated by the disk but the jet component, the disk emission should be taken into account if accretion flow emission is largely blended in the core emission in reality (see also Broderick and Loeb 2009). We should take these GR effects into account when they are indeed effective. (3) Apart form GR effect, pure geometrical effect between jet opening angle and viewing angle which may cause a partial blending of SSA thin part of the jet. It might also cause non-spherical geometry and inclusion of them is also important.
We acknowledge the anonymous referee for his/her careful review and suggestions for improving the paper. MK thank A. Tchekhovskoy for useful discussions. This work is partially supported by Grant-in-Aid for Scientific Research, KAKENHI 24540240 (MK) and 24340042 (AD) from Japan Society for the Promotion of Science (JSPS).
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