Energy densities of magnetic field and relativistic electrons at the innermost region of the M 87 jet

We explore energy densities of magnetic fields and relativistic electrons in M87 jet. Since the radio core at the base of the M87 jet is the optically thick surface against synchrotron self absorption (SSA), observations directly give the size and turnover frequency for SSA. Using the observed angular diameter 0.11 mas, which corresponds to 16 Schwarzschild radii of the central black hole with 6 × 109 solar mass, and the flux density of the radio core at 43 GHz, we estimate the energy densities of magnetic field (UB) and relativistic electrons (Ue) by comparing the standard SSA formula to the observed radio core. Together with the allowed total kinetic power of the M87 jet, we find that (i) the allowed B is limited in the range 2 G ≤ B ≤ 13 G, and that (ii) 0.18 ≤ Ue/UB ≤ 66 holds. Our results significantly constrain formation mechanism of relativistic jets in active galactic nuclei.


Introduction
Formation mechanism of relativistic jets in active galactic nuclei (AGNs) remains as a longstanding unresolved problem in astrophysics.Although an importance of estimations of magnetic field energy density (U B ) and electron one (U e ) for resolving the formation mechanism had been emphasized (e.g., [13]; [5]), it is not observationally clear whether U B or U e is dominant at a jet base.Deviation from equi-partition (i.e., U e /U B ≈ 1) is essential for investigation of relativistic jet formation.However, no one has succeeded in firmly obtaining a robust estimation of an actual value of U e /U B at a jet base.
M87, a nearby giant radio galaxy located at a distance of D = 16.7 Mpc, hosts one of the most massive super massive black hole M • = (3 − 6) × 10 9 M and thus M87 is the best source for investigating a jet base.Furthermore, M87 has been well studied at wavelength from radio to Very High Energy (VHE) γ-ray ( [1] and reference therein) and causality arguments based on VHE γray outburst in February 2008 indicate that the VHE emission region is less than ∼ 5δ R s where δ is the relativistic Doppler factor ( [2]).VLBA beam resolution at 43GHz typically attains about 0.21 × 0.43 mas which is equivalent to 5.3 × 10 16 × 1.1 × 10 17 cm.If we take M • = 6 × 10 9 M , then VLBA beam resolution corresponds to 30 × 60 R s .Recent progress of VLBI observations reveals the innermost structure of the M87 jet, i.e., frequency and core-size relation, distance and core-size relation down to close to 16 Schwarzschild radii (R s ) ( [12], hereafter H11).Thus, a e-mail: kino@vsop.isas.jaxa.jp the jet base of M87 is the best laboratory for investigation of U e /U B in a real vicinity of a central engine.
Two significant forward steps are recently obtained in M87 observations and the present work is motivated by them.(1) H11 succeed in directly measuring core-shift phenomena at the jet base of M87 at 2, 5, 8, 15, 22 and 43 GHz.The radio core position at each frequency has been obtained by the astrometric observation of the core shift along the M87 jet.Since the radio core surface corresponds to the optically-thick surface at each frequency, a synchrotron-self-absorption (SSA) turnover frequency ν ssa is identical to the observed frequency itself.(2) We recently measure core sizes in [10] (hereafter H13).Hereafter we focus on the radio core at 43GHz.In H13, we select VLBA data observed after 2009 with sufficiently good qualities (all 10 stations participated and good uvcoverages).To measure the width of the core, a single, full-width-half-maximum (FWHM) Gaussian is fitted for observed core in the perpendicular direction to the jet axis and we derive the width of the core (θ FWHM ).We stress that the core width is free from the uncertainty of viewing angle.Therefore, using θ FWHM at 43GHz, we can estimate model-independent value of U e /U B in the 43GHz core of M87 for the first time.
EPJ Web of Conferences physical quantities, B and K e are determined by the comparison of physical quantities measured by VLBA observations (i.e., θ obs , ν ssa,obs , and S ν ssa ,obs ) and standard SSA process.In this work, we define the radio spectral index α as S ν ∝ ν −α .
Following assumptions are adopted in this work: • We assume uniform and isotropic distribution of relativistic electrons in the radio core at 43 GHz (the hatched circle in Fig. 1).For M87, polarized flux seems not very large.Therefore, we assume isotropic tangled magnetic field in this work.Hereafter, we denote B as the magnetic field strength perpendicular to the direction of electron motion.Then, the total field strength is given by B tot = √ 3B.• We assume the emission region is spherical with its radius R measured at the comoving frame.The radius is defined as 2R = θ obs D where D is the distance to a source.There might be a slight difference between θ FWHM and θ obs .VLBI measured θ FWHM is conventionally treated as θ obs = θ FWHM , while [17] (hereafter M83) pointed out a deviation expressed as θ obs ≈ 1.8θ FWHM which is caused by a forcible fitting of Gaussians to a non-Gaussian component.In this work, we introduce a factor A defined as θ obs ≡ Aθ FWHM and 1 ≤ A ≤ 1.8 is assumed.
At the radio core, (1) τ ssa = 1 at ν = ν ssa holds: where τ ν ssa , and α ν ssa are the optical depth for SSA, and the absorption coefficient for SSA, respectively.The optically thin emission equals the optically thick one at ν ssa : where ν ssa , and S ν ssa are the emissivity and flux per unit frequency, respectively.Via these two relations, we can solve B and K e .The term K e , the normalization factor of electron number density distribution n e (γ), is defined as n e (γ e )dγ e = K e γ −p e dγ e (γ e,min ≤ γ e ≤ γ e,max ), where p = 2α + 1, γ e,min , and γ e,max are a spectral index, a minimum Lorentz factor, and a maximum Lorentz factor of relativistic 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) where B is the component of magnetic field perpendicular to the direction of the electron motion.

Results
Obtained expressions for B and K e are as follows; ) 5 ( θ obs 1 mas and GHz Jy where the numerical coefficients b(p) and k(p) are shown in M83 and [15].From the above expressions, we can obtain the ratio U e /U B explicitly as GHz Jy From this, we can estimate U e /U B without minimum energy (equipartition B field) assumption.It is clear that the measurement of θ obs is crucial for determining U e /U B .We argue details on it in the next subsection.It is also evident that a careful treatment of γ e,min is crucial for determining U e /U B ( [14]).
To evaluate realistic U e /U B , we should take realistic parameter ranges.To this end, we take into uncertainty into account.
Regarding α, 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 [8] with multi-frequency ALMA observation (cycle 0) and it is robust that α > 0.5 at > 200 GHz where synchrotron emission becomes optically-thin against SSA.
As for allowed θ obs , we set 0.11 mas ≤ θ obs ≤ 0.20 mas, where we use the average value θ FWHM = 0.11 mas from H13 and maximum of θ obs is 0.11 mas × 1.8 = 0.198 mas.
The maximal value of γ e,min can be estimated with θ obs = 0.20 mas.Allowed ranges of γ e,min is given by 1 ≤ γ e,min ≤ 2 × 10 2 , here we set maximum value of γ e,min by requiring synchrotron emission is produced at least above 43 GHz.We further impose a condition that timeaveraged bulk kinetic power as inferred from its largescale radio and X-ray morphology L jet should be smaller than the power at the 43GHz core where L jet at large-scale is estimated to be a few ×10 44 ergs −1 (e.g., [20]).Hereafter, we assume Γβ = 1 for simplicity and a slight deviation from this does not influence the main results in this work.Regarding L jet in the M87 jet, we set 5 × 10 44 ergs −1 ≤ L jet ≤ 3 × 10 45 ergs −1 where upper limit of L jet includes an uncertainty due to the deviation from time-averaged L jet at while flaring phenomena at the jet base may temporally increase L jet .Xray light curve at the M87 core over 10 years showed a flux variation by a factor of ∼ 6 except for exceptionally high X-ray flux during giant VHE flares which happened in 2008 and 2010 (Fig. 1 in [1]).Based on it, we allow the largest jet kinetic power case as L jet = 6×5×10 44 ergs −1 = 3 × 10 45 ergs −1 .In this work, we demonstrate the case of L jet = 3 × 10 45 ergs −1 .In Fig. 2, we show the allowed log(U e /U B ) as a function of γ e,min and B with L jet = 3×10 45 erg s −1 and p = 3.5.Note that the field strength B has one-to-one correspondence to θ obs .From Fig. 2, one can see that the absolute value of B (equivalent to θ obs ) is more dominant factor than γ e,min in this allowed ranges.We find that energy density of relativistic electrons dominates over the one of magnetic field (log(U e /U B ) > 1) when B < 5.7 G and vice versa.

Summary and discussions
Based on VLBA observation data at 43 GHz, we explore U e /U B at the base of the M87 jet.We use standard basic theory of synchrotron radiation with the simplest geometry of one-zone sphere model for the radio core at 43 GHz.We impose the upper limit of total jet power L jet based on various previous works.We then find the following things; • We obtain the allowed range of B as 2 G < B < 13 G in the observed radio core at 43GHz with its diameter 0.11 mas ((16 R s )).Our estimate of B is basically close to the previous estimate in the literature (e.g., [19]), although fewer assumptions have been made in this work.
Our result excludes a strong magnetic field such as B ∼ 10 3−4 G which is frequently assumed in previous works in order to activate Blandford-Znajek process ( [3]).Although M87 has been a prime target for testing relativistic MHD jet simulation studies powered by black-hole spin energy, our result provides a very stringent limit on the maximum B,one of the critical parameters in relativistic MHD jets model.
• We obtain the allowed region of U e /U B in the allowed θ obs and γ e,min plane.The resultant U e /U B contains both the region of U e /U B > 1 and U e /U B < 1.What we should emphasize here is that the deviation from U e /U B ≈ 1 is not very large.It is found that the allowed range is 0.18 ≤ U e /U B ≤ 66.
This result gives a tight constraint against relativistic MHD models since they seem to postulate much larger U B /U e at a jet-base than the maximum value obtained in this work (e.g., [23]).For example, the jet flow is heavily Poynting dominated at the jet base with U B /U e ∼ 10 3 in [18].Therefore, the obtained U e /U B in this work gives a tight constraint on the initial conditions in relativistic MHD models.
Lastly, we briefly comment on key future works.Observationally, it is crucial to obtain resolved images the radio cores at 43GHz with space/sub-mm VLBI which would clarify whether there is a sub-structure or not inside ∼ 16 Rs scale at 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., [16]; [7]).Indeed, we intend to conduct a new observation of M87 at 86 GHz and we will explore this issue using the new data.Space-VLBI program also could play key role since lower frequency observation can attain higher dynamic range images with a high resolution (e.g., [6]; [22]).

γ e,min γ e m e c 2 = 8πm e c 2 3 (p − 2 )
n e (γ e )dγ e The larger γ e,min leads to smaller log(U e /U B ) because U e becomes smaller for larger γ e,min .The allowed γ e,min is limited in 30 ≤ γ e,min ≤ 2 × 10 2 .The smallest U e /U B realizes at B ≈ 8.1 G and it is U e /U B = 0.18.The allowed U e /U B is governed by the limit of L j ≥ max[L poy , L e ].The lower part of log(U e /U B ) is determined by the condition of L e = L j , while the upper part is settled by L poy = L j .The largest U e /U B is realized when B = 2.5 G which shows U e /U B = 66.

Figure 1 .
Figure 1.VLBA image of M87 at 43GHz.The the core width (i.e., length perpendicular to the jet axis) derived by the gaussian fitting by AIPS task "JMFIT" is θ FWHM = 0.11 mas (see details in H13).The hatched circle marks the 0.11 mas diameter region explored in this work.Contours start from 3σ image rms level and increasing by factors of 1.4.

Figure 2 . 4 EPJ
Figure 2. The energy density ratio of log(U e /U B ) in the allowed γ e,min − B plane with L j = 3 × 10 45 erg s −1 and p = 3.5.